COSMOLOGICAL IMPLICATIONS OF SCALAR FIELDS AND VARYING-CONSTANTS by
Douglas J. Shaw Trinity College
This dissertation is submitted for the degree of
Doctor of Philosophy University of Cambridge Department of Applied Mathematics and Theoretical Physics
Supervisor: Prof. John D. Barrow
29th November 2006
i
Cosmological Implications of Scalar Fields and Varying-Constants Douglas J. Shaw Abstract There is wide-spread interest in the possibility that our Universe may be populated by one or more scalar fields and that these fields could cause some or all of the fundamental ‘constants’ of nature to vary. In this thesis, we study the behaviour of these new scalar fields, detail their predictions for alterations to the standard laws of physics and discuss how current experiments constrain their existence and the prospects for detecting them, and through them new physics, in the near future. Recent observations of quasar absorption systems have suggested that the fine structure constant, αem , has increased by a few parts in a million over the last 10-12 Gyrs. In this thesis we analyse what new phenomena emerge when theories that describe varying-αem are made consistent with electroweak unification. We calculate how the scalar fields in these theories interact with both baryonic and dark matter through quantum corrections, and show that experiments should be able to detect or rule out the simplest electroweak varying-αem theories in the near future. In order to properly constrain theories that predict scalar fields and varying ‘constants’, one needs know how to combine observational evidence from quasar spectra with indirect solar system data and laboratory constraint. There is no a priori reason why these very different types of measurement should be directly comparable yet they are almost always assumed to be so. In this thesis, we use matched asymptotic expansions to provide the first rigorous proof that, under very general conditions, local experiments do measure variations in the ‘constants’ that occur over cosmological scales. This result applies to almost all physicallyviable theories but not to the so-called chameleon theories. In chameleon theories the scalar fields possess highly non-linear self-interactions that allow them to evade the most restrictive of the current experimental bounds. In this thesis we conduct a detailed analysis of these chameleon theories and show that scalar fields that couple to matter much more strongly than gravity are not only viable but in fact less well constrained than those with weaker couplings. Nonetheless strongly coupled scalar fields could well be detected by a number of future experiments provided they are properly designed to do so.
Contents
Abstract
i
Contents
iii
List of Figures
ix
Acknowledgements
xi
Preface
xiii
1 Introduction 1.1
1
The Fundamental Constants of Nature . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
What are the Constants of Nature? . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
A remarkable coincidence . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.3
Varying-Constants? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.4
Constraints on Varying-Constants . . . . . . . . . . . . . . . . . . . . .
5
1.1.5
Varying constants and scalar fields . . . . . . . . . . . . . . . . . . . . .
8
Theoretical Motivations and Models . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.1
Higher Dimensional Theories . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2.2
Phenomenological models . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3
Fifth-force tests for scalar fields . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Aims and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.4.1
Aims of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.4.2
Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.2
iii
iv
CONTENTS
2 Varying-αem in the context of Electroweak theory
17
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Simplest Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.1
A Single-Dilaton Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.2
A two dilaton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.3.3
Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.3.4
Classical Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
23
The Dilaton-to-Matter Coupling . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.4.1
Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.4.2
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4.3
The Unbroken Symmetry Case: MA = 0 . . . . . . . . . . . . . . . . . .
27
2.4.4
The Broken Symmetry Case: MA 6= 0 . . . . . . . . . . . . . . . . . . .
27
2.4.5
Application to a cosmological setting . . . . . . . . . . . . . . . . . . . .
28
2.4.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Effective field equations for KM models . . . . . . . . . . . . . . . . . . . . . .
33
2.5.1
Single dilaton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.5.2
Two dilaton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Homogeneous and Isotropic Cosmology . . . . . . . . . . . . . . . . . . . . . . .
34
2.6.1
Cosmological Field Equations . . . . . . . . . . . . . . . . . . . . . . . .
34
2.6.2
Reduced Cosmological Equations . . . . . . . . . . . . . . . . . . . . . .
35
2.6.3
The Radiation Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.6.4
The Matter Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.6.5
A Reduced WIMP Dominated System . . . . . . . . . . . . . . . . . . .
45
2.6.6
The Full System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.6.7
The Dark Energy Dominated Era . . . . . . . . . . . . . . . . . . . . . .
53
2.6.8
WEP Violations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.6.9
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
New phenomena due to a varying-θw . . . . . . . . . . . . . . . . . . . . . . . .
59
2.7.1
Charge non-conservation and simple varying-αem theories . . . . . . . .
59
2.7.2
A new interaction from varying-θw . . . . . . . . . . . . . . . . . . . . .
61
2.4
2.5
2.6
2.7
v
2.8
2.7.3
Induced currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
2.7.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3 Locally observable effects of varying constants and scalar fields
69
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.2
Past Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.2.1
Review of Wetterich’s analysis . . . . . . . . . . . . . . . . . . . . . . .
74
3.2.2
Jacobson’s result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.2.3
The Spherical Infall Approximation . . . . . . . . . . . . . . . . . . . .
78
Matched Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.3.1
Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.3.2
Singular problems and ones with multiple scales . . . . . . . . . . . . .
79
3.3.3
Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
3.3.4
Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.3.5
Application to General Relativity . . . . . . . . . . . . . . . . . . . . . .
85
3.3
3.4
3.5
3.6
Geometrical Set-Up
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.4.1
The McVittie background . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.4.2
The Szekeres-Szafron background . . . . . . . . . . . . . . . . . . . . . .
90
Virialised Case: Spherically symmetric backgrounds . . . . . . . . . . . . . . .
92
3.5.1
Case I: McVittie background . . . . . . . . . . . . . . . . . . . . . . . .
93
3.5.2
Case II: Tolman-Bondi background . . . . . . . . . . . . . . . . . . . . .
95
3.5.3
Boundary Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
3.5.4
Solution of φ equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3.5.5
Matching Conditions
3.5.6
Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Virialised Case: Non-spherically symmetric backgrounds . . . . . . . . . . . . . 109 3.6.1
Szekeres-Szafron backgrounds . . . . . . . . . . . . . . . . . . . . . . . . 109
3.6.2
Exterior Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.6.3
Interior Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.6.4
Solution of φ equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.6.5
Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
vi
CONTENTS
3.6.6 3.7
3.8
Matching and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Collapsing Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.7.1
The Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.7.2
The Exterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.7.3
Matching and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Results and Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4 Chameleon Field Theories
125
4.1
Introduction
4.2
Chameleon Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.3
4.4
4.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.1
The Thin-Shell Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.2
Chameleon to Matter Coupling . . . . . . . . . . . . . . . . . . . . . . . 130
4.2.3
A Lagrangian for Chameleon Theories . . . . . . . . . . . . . . . . . . . 131
4.2.4
Intrinsic Chameleon Mass Scale . . . . . . . . . . . . . . . . . . . . . . . 132
4.2.5
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.2.6
The Importance of Non-Linearities . . . . . . . . . . . . . . . . . . . . . 134
4.2.7
The Chameleon Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.8
Chameleon Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2.9
Natural Values of M and λ . . . . . . . . . . . . . . . . . . . . . . . . . 135
One body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3.1
Linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.3.2
Pseudo-Linear Regime
4.3.3
Non-linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.3.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
. . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Effective Macroscopic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.4.1
Averaging in Linear Theories . . . . . . . . . . . . . . . . . . . . . . . . 152
4.4.2
Averaging in Chameleon Theories . . . . . . . . . . . . . . . . . . . . . 154
Force between two bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.5.1
Force between two nearby bodies
. . . . . . . . . . . . . . . . . . . . . 158
4.5.2
Force between two distant bodies
. . . . . . . . . . . . . . . . . . . . . 161
4.5.3
Force between a large body and a small body . . . . . . . . . . . . . . . 163
4.5.4
Force between bodies without thin-shells . . . . . . . . . . . . . . . . . . 164
vii
4.5.5 4.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5 Evading Experimental Constraints with Strong Coupled Scalar Fields
169
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2
Laboratory Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.3
5.4
5.2.1
E¨ot-Wash experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.2.2
Casimir force experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.3
WEP violation experiments
5.2.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
. . . . . . . . . . . . . . . . . . . . . . . . 178
Implications for Compact Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.3.1
The Mass-Radius Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.3.2
General Relativistic Stability . . . . . . . . . . . . . . . . . . . . . . . . 191
5.3.3
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Cosmological and Other Astrophysical Bounds . . . . . . . . . . . . . . . . . . 195 5.4.1
Nucleosynthesis and the Cosmic Microwave Background . . . . . . . . . 195
5.4.2
Variation of fundamental constants . . . . . . . . . . . . . . . . . . . . . 204
5.5
Combined bounds on chameleon theories
. . . . . . . . . . . . . . . . . . . . . 206
5.6
Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6 Conclusions
217
6.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.2
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A Explicit matching for the McVittie background
227
A.1 Exterior Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 A.2 Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 B Pseudo-Linear regime for single-body problem
231
B.1 Inner approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 B.2 Outer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 B.3 Matching Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 B.4 Conditions for Matching
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
viii
CONTENTS
B.5 Case: n < −4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 C Far field in n = −4 theory for a body with thin-shell
243
D Effective macroscopic theory
245
D.1 Linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 D.2 Pseudo-Linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 D.3 Case n < −4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 D.4 Case n > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 D.5 Case n = −4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 D.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 D.7 Non-linear Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 D.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 E Evaluation of αΦ for white-dwarfs
255
Bibliography
257
List of Figures
1.1
Allowed couplings and force ranges for scalar fields for linear theories . . . . . . . .
13
2.1
Exact and asymptotic baryon-dominated ∆αem /αem solutions . . . . . . . . . . . .
43
2.2
Exact and asymptotic baryon-dominated ∆ sin2 θw / sin2 θw solutions . . . . . . . .
44
2.3
Baryon-dominated evolution of αem and sin2 θw for ζem /ω2 > 0. . . . . . . . . . . .
44
2.4
Baryon-dominated evolution of αem and sin2 θw for ζem /ω2 < 0. . . . . . . . . . . .
45
2.5
Baryon-dominated evolution of αem and sin2 θw with different values of ω2 /ω1 . . .
46
2.6
Exact and asymptotic WIMP-dominated ∆αem /αem solutions. . . . . . . . . . . .
47
2.7
Exact and asymptotic WIMP-dominated ∆ sin2 θw / sin2 θw solutions. . . . . . . . .
48
2.8
WIMP-dominated evolution of αem and sin2 θw for ζW /ω2 > 0. . . . . . . . . . . .
49
2.9
WIMP-dominated evolution of αem and sin2 θw for ζW /ω2 < 0. . . . . . . . . . . .
50
2.10 WIMP-dominated evolution of αem and sin2 θw with different values of ω2 /ω1 . . . .
52
2.11 Effect of a cosmological constant on αem and sin2 θw . . . . . . . . . . . . . . . . .
54
3.1
Exact and asymptotic solutions to 2δy 00 + (1 + 2δ)y 0 + y = 0 . . . . . . . . . . . . .
82
3.2
Sketch of the construction of the 5-D manifold N . . . . . . . . . . . . . . . . . . .
87
3.3
Sketch of the geometrical set-up used sections 3.5 and 3.6. . . . . . . . . . . . . . .
93
3.4
Sketch of a quasi-spherical Szekeres-Szafron spacetime. . . . . . . . . . . . . . . . . 110
3.5
Plot of (φ,t0 − φc,t )/φc,t vs. time for Brans-Dicke theory. . . . . . . . . . . . . . . . 123
4.1
Sketch of the chameleon mechanism for a runaway potential . . . . . . . . . . . . . 128
4.2
Sketch of the chameleon mechanism for a potential with a minimum . . . . . . . . 129
4.3
Illustration of the model used for calculating the macroscopic field theory. . . . . . 151
4.4
Dependence of the critical chameleon mass on D/R. . . . . . . . . . . . . . . . . . 157 ix
x
LIST OF FIGURES
5.1
E¨ot-Wash constraints on chameleon theories. . . . . . . . . . . . . . . . . . . . . . 174
5.2
Casimir force experiment constraints on chameleon theories. . . . . . . . . . . . . . 177
5.3
WEP violation constraints on chameleon theories . . . . . . . . . . . . . . . . . . . 183
5.4
Compact object constraints on chameleon theories . . . . . . . . . . . . . . . . . . 193
5.5
Cosmological (BBN and CMB) constraints on chameleon theories . . . . . . . . . . 203
5.6
Combined constraints on chameleon theories
5.7
Allowed couplings and cosmological force ranges for chameleon theories. . . . . . . 211
5.8
Allowed couplings and solar system force ranges for chameleon theories. . . . . . . 212
. . . . . . . . . . . . . . . . . . . . . 208
Acknowledgements
It is my pleasure to express my very deep and sincere thanks to my supervisor, John D. Barrow, for his supervision and expert guidance throughout the course of my studies, and also, not least, for suggesting such an interesting field of research. It has been truly gratifying to work with and learn from him throughout my PhD. I am also very grateful to Christos Tsagas and David Mota for our fruitful collaborations and willingness to share their ideas and listen to mine. I would also like to thank Tim Clifton, Jo˜ao Magueijo, Peter D’Eath, Thomas Dent, Justin Khoury, Amol Upadhye, Jacob D. Bekenstein, Michael T. Murphy, John K. Webb, Christof Wetterich, Michael Doran, Holger Geis, Fernando Quevedo, Kerim Suruliz and Joe Conlon for helpful discussions and PPARC for funding my research. I also wish to thank all of my friends for making my seven or so years in Cambridge such an enjoyable time. To my family, I extend my endless thanks for their continual love, support and encouragement throughout the whole of my life. I am eternally indebted to them for all that they have given and know that none of what I already achieved in life would have been possible if it were not for them. I am forever grateful to my fianc´ee, Andreea, for her unceasing support, her continual belief in me and, most of all, her love that fills my life with such joy each and every day. Finally, my deepest thanks I give to God for all of the many ways in which he has blessed my life.
xi
Preface In the most part, the work presented in this thesis has been (or is to be) published as the following series of papers: • D. J. Shaw and J. D. Barrow, “Varying couplings in electroweak theory,” Physical Review D 71 (2005) 063525 [arXiv:gr-qc/0412135]: Chapter 2. • D. J. Shaw, “Charge non-conservation, dequantisation, and induced electric dipole moments in varying-alpha theories,” Physics Letters B 632 (2006) 105-108 [arXiv:hep-th/0509093]: Chapter 2. • D. J. Shaw and J. D. Barrow, “The local effects of cosmological variations in physical ’constants’ and scalar fields. I: Spherically symmetric spacetimes,” Physical Review D 73 (2006) 123505 [arXiv:gr-qc/0512022]: Chapter 3. • D. J. Shaw and J. D. Barrow, “The local effects of cosmological variations in physical ‘constants’ and scalar fields. II: Quasi-spherical spacetimes,” Physical Review D 73 (2006) 123506 [arXiv:gr-qc/0601056]: Chapter 3. • J. D. Barrow and D. J. Shaw,“Observable effects of scalar fields and varying constants,” invited contribution to Obregon’s Festschrift (to appear later this year) (2006) [arXiv:grqc/0607132]: Chapter 3. • D. F. Mota and D. J. Shaw, “Evading equivalence principle violations, astrophysical and cosmological constraints in scalar field theories with a strong coupling to matter,” [arXiv:hep-ph/0608078]: Chapters 4 and 5. The some of the results presented here were also summarised in the following letters: xiii
xiv
PREFACE
• D. J. Shaw and J. D. Barrow, “Local experiments see cosmologically varying constants,” Physics Letters B 639 (2006) 596-599 [arXiv:gr-qc/0512117]: Chapter 3. • D. F. Mota and D. J. Shaw, “Strongly coupled chameleon fields: New horizons in scalar field theory,” accepted for publication in Physical Review Letters (2006) [arXiv:hep-ph/0606204]: Chapters 4 and 5. The work presented in this thesis is my own but I have, as a matter of style, chosen to retain the use of the first person plural throughout. This dissertation is not substantially the same as any that I have submitted, or am submitting, for a degree, diploma or other qualification at any other university. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text.
Signed:............................................
Dated:............................................
Chapter 1
Introduction
Since ’tis Nature’s law to change, Constancy alone is strange. John Wilmot (1647-1680)
There is wide-spread interest in the possibility that, in addition to the matter described by the Standard Model of particle physics, our Universe may be populated by one or more scalar fields. These fields are a general prediction of modern proposals for theories of quantum gravity, e.g. String Theory, and other models of high energy physics beyond the Standard Model. In this thesis, we study the behaviour of these new scalar fields, detail their predictions for alterations to the standard laws of physics, and discuss the extent to which these fields are constrained by current experiments and could be detected in the future. In this chapter, we describe some common aspects of the theories that predict and describe such scalar fields. It is almost always the case that these scalar fields interact with matter, and when they do so they generically result in a variation of one or more of the fundamental constants of Nature. The study of scalar fields that couple to matter and the idea that some or all of these fundamental ‘constants’ can and do vary are, therefore, inextricably linked. As a motivation to what follows, we begin in section 1.1 by describing the crucial rˆole that the fundamental constants of Nature play in defining the laws of physics, and by reviewing the current experimental limits on any variation in these ‘constants’. 1
2
CHAPTER 1. INTRODUCTION
Scalar fields emerge in the low energy limit of a large number of proposals for new physics beyond General Relativity and the Standard Model. In section 1.2, we review these theoretical motivations for studying scalar fields and discuss some of the more popular phenomenological descriptions of varying-constants. Scalar fields generically result in the existence of a new, or ‘fifth’, force between bodies. The experimental constraints on ‘fifth’-forces are very strong, and finding models that naturally satisfy or evade these constraints represents a major theoretical challenge. We briefly discuss the experiment bounds on new forces in section 1.3. We conclude in section 1.4 by stating our main aims in this thesis, and by outlining the following chapters.
1.1 1.1.1
The Fundamental Constants of Nature What are the Constants of Nature?
A Fundamental Constant of Nature is any parameter of any physically viable theory whose value is: 1) independent of any other measured quantity and 2) not predicted or explained by any scientific theory. Although the definition of what makes a parameter a ‘constant of Nature’ does not change, as time goes by our knowledge and understanding of the world around us increases and parameters that were once thought to be fundamental constants are found to be predicted by some new theory, or to depend on some other quantities. The Constants of Nature define the laws of physics and their independence of time and position ensures that these laws are the same at every point in space and time. It is because of this time and position invariance that were are able to predict how the universe would have been billion of years ago. The constants of Nature specify, amongst other things, the strength of the different forces and their values determine which chemical elements and compounds are stable and which are not. If the constants of Nature were to take other values than they do then our universe would be a different place. Despite this important rˆole, we know of no fundamental reason for preferring any one set of values for these constants to any other. Of all the parameters that are currently believed to be fundamental constants, the one which we have known about for longest is Newton’s Constant of Gravitation, G. The value of G determines the strength of the gravitational force, and one of Newton’s most important observations was that the same value of G was sufficient to explain both the motion of planets and comets around Sun, and the way in which an apple falls to ground: G was a constant
1.1. THE FUNDAMENTAL CONSTANTS OF NATURE
3
of nature. The constancy of G transformed Newton’s theory from a mere phenomenological description of the motion of the planets in our solar system into a fundamental law of physics: equally applicable in the far cosmos as it was on Earth. Nowadays, the Standard Model of particle physics is known to contain some 26 fundamental constants in addition to G. Three of these constants, like G, determine the strength of a force of nature, they are: αem , αW and αs . The fine structure constant, αem , sets the strength of electromagnetism, whereas the strength of the weak and strong nuclear forces are respectively determined by αW and αs . Unlike G, αem , αW and αs are dimensionless numbers. Indeed, all truly fundamental constants should be expressible in some dimensionless form. The reason for this is that the measured value of a dimensionful constant will always inherently depend upon one’s choice of units. Consider, for instance, the speed of light in a vacuum as measured in m s−1 . In 1983, the General Conference on Weights and Measures (CPGM) changed the definition of a metre to make the speed of light in a vacuum exactly 299, 792, 458 m s−1 . When the metre was first conceived in 1795, however, it was defined, without any reference to the speed of light, to be one ten-millionth of the distance from the north pole to the equator passing along the meridian that runs through Paris and Barcelona. At the time, a small error was made in measuring the equator to pole distance and, as a result, when the first prototype metre was manufactured its length differed from the length it was defined to have by one part in four thousand. Therefore, if one were to measure the speed of light in m s−1 , using the original definition of a metre, one would find that is differs from the value chosen by the CPGM at the 0.1% level. Without there having been, to the best of our knowledge, any actual change in the speed of light over the last 211 years, its value in m s−1 has not remained the same. At the same time, even if the speed of light to change in the future, its value in m s−1 would always remain the same because the modern definition of a metre is not independent of the speed of light. The above example highlights the major problem with dimensionful constants: one can only truly talk about their value as being fundamental if they have been measured with respect to some fundamental, independent and unchanging set of units. Therefore, whilst the value of G in m3 s−2 kg−1 is essentially just an artifact of our definitions of a metre, a second and a kilogram, the dimensionless quantity αG = Gmp /~c ≈ 5.9 × 10−39 , where mp is the proton mass and ~ is Planck’s constant, is currently believed to be a true fundamental constant of
4
CHAPTER 1. INTRODUCTION
Nature. The dimensionless fine structure constant is similarly constructed from dimensionful constants: αem = e2 /~c ≈ 1/137 with e being the charge on an electron. At energies ∼ MZ , the strong force coupling constant is measured to be αs = gs2 /~c ≈ 0.12 , and αW = GF m2p c/~3 ≈ 1.03 × 10−5 .
1.1.2
A remarkable coincidence
Although, we know of no deep reason for αG , αem , αW and αs taking the values that they do, if their values were different by even as little as few percent the consequences for life, as we understand it, would be disastrous [24]. For instance: if αem were increased by more than about 4%, or αs reduced by more than 0.4%, then the amount of carbon produced in stars would drastically reduced. Similarly, a 4% or greater decrease in αem , or a 0.4% increase in αs , would cause stars to produce far less oxygen. Either of these scenarios would result in the universe being a far less hospitable place, and it is questionable as to whether human life could exist at all. An even greater disaster would occur if αs were larger by as little as 4%, or smaller by more than 10%. In the former case, Helium-2 would be stable, the process of nuclear fusion would be much faster and stars would quickly use up all of their fuel and die. In the latter scenario, deuterium could not exist and, as a result, elements such as oxygen and carbon would not be produced in the cores of stars at all. Another constant whose value has important consequences for life is the electron-proton mass ratio: µ = mp /me ≈ 1836. A much smaller value of µ would result in there being no ordered molecular structures; additionally, the cores of stars would not be hot enough for nuclear fusion to occur. We are faced, therefore, with a seemingly remarkable coincidence: although we know of no explanation for the values that the Fundamental Constant of Nature take, they appear to be have been fine-tuned for the existence of human life.
1.1.3
Varying-Constants?
If the values of the constants of Nature are the same throughout all space and time then it seems that we are either incredibly lucky to exist at all, or that the whole Universe has been specifically designed for the purpose of producing life; right or wrong, neither of these hypotheses are very testable. If, however, the constants of nature are not fixed but can, and do, take different values at different points in space and time then the fact that we live in a region of
1.1. THE FUNDAMENTAL CONSTANTS OF NATURE
5
space where they just happen to take the right values for life is no longer that remarkable. We could not, after all, live anywhere else. If the constants vary, we might be living in one of the few inhabitable pockets of an otherwise uninhabitable universe. Importantly, if the constants can vary then we should be able to detect such them doing so; variation of the constants is, at least in principle, a testable hypothesis. Over the past few years there has been a resurgence of observational and theoretical interest in the possibility that some of the fundamental ‘constants’ of Nature might be varying over cosmological timescales [1]. In respect of two such ‘constants’, αem and G, the idea of such variations is not new, and was proposed by authors such as Milne [120], Dirac [121], and Gamow [122] as a solution to some perceived cosmological problems of the day [123]. At first, theoretical attempts to model such variations in the constants were rather crude and equations derived under the assumption that constants like G and αem are truly fixed were simply altered by writing-in an explicit time variation. Most observational constraints in the literature have been derived in this fashion. A more refined approach was first introduced in the case of varying G by the creation of scalar-tensor theories of gravity [124], of which Brans-Dicke theory is perhaps the most famous example [116]. In all of these theories, G is promoted to a dynamical quantity by associating it with the expectation of scalar field, φ, which conserves energy and momentum. Like any other form of matter or energy, φ contributes to the curvature of spacetime. More recently, such self-consistent descriptions of the spacetime variation of other constants, like αem [37, 36], the electroweak couplings [44], and the electron-proton mass ratio, µ, [125] have been formulated. Much of the resurgence of interest in possible time variations of the constants has focused on αem and µ. This has been brought about by the significant progress that has recently been made in high-precision quasar spectroscopy. These improvements have allowed astronomers to measure the values that αem and µ took in the past to an unprecedented precision. These measurements have produced what might prove to be the first evidence that at least some of the traditional constants of nature are not so constant after all. We discuss these results below.
1.1.4
Constraints on Varying-Constants
In recent years, some of the most exciting experimental results pertaining to varying-constants have come from studies of the quasar spectra. In addition to this data, we also have available
6
CHAPTER 1. INTRODUCTION
a growing number of laboratory, geochemical, and astronomical observations with which to constrain any variation in the constants. It should be noted that these different sources of data are drawn from vastly different scales and environments and it is not at all clear, a priori, that they are measuring the same thing. This issue is discussed in more detail in Chapter 3. The quasar data analysed by Webb et al. in refs. [1, 2, 3] consists of three separate samples of Keck-Hires observations which combine to give a data set of 128 objects at redshifts 0.5 < z < 3. Using the many-multiplet technique, it was found that their absorption spectra are consistent with a shift in the value of αem between these redshifts and the present day of ∆αem /αem ≡ [αem (z) − αem ]/αem = −0.57 ± 0.10 × 10−5 , where αem ≡ αem (0) is the present value of the fine structure constant [1, 2, 3]. Extensive analysis has yet to find a selection effect that can explain either the sense or the magnitude of the relativistic line-shifts underpinning these deductions. Two further observational studies, refs. [8, 9], analysed a different, but smaller, data set of 23 absorption systems in front of 23 VLT-UVES quasars at 0.4 ≤ z ≤ 2.3. They obtained ∆αem /αem ≡ −0.6 ± 0.6 × 10−6 ; a figure that disagrees with the results of refs. [1, 2, 3]. Reanalysis is needed, however, in order to understand the accuracy that has been claimed. A number of other astronomical searches for varying-αem have been conducted. Whilst these have all found results consistent with no variation, they have yet to reach a precision that in comparable to that achieved by Webb et al. Data sets of 42 and 165 quasars from the Sloan Digital Sky Survey (SDSS) provided the constraints ∆αem /αem ≡ 0.51 ± 1.26 × 10−4 and ∆αem /αem ≡ 1.2±0.7×10−4 respectively for objects in the redshift ranges 0.16 ≤ z ≤ 0.8 [10]. Quast et al. [11] found ∆αem /αem ≡ −0.1 ± 1.7 × 10−6 at z = 1.15 using observations of single quasar absorption system, and Levashov et al. [12] gave ∆αem /αem ≡ 4.3 ± 7.8 × 10−6 for a single absorption system at z = 1.839. A preliminary analysis the OH microwave transition from a quasar at z = 0.2467, a method proposed by Darling [13], has given ∆αem /αem ≡ 0.51 ± 1.26 × 10−4 , [14]. Drinkwater et al. made a comparison of redshifts measured using molecular and atomic hydrogen in two cloud systems at z = 0.25 and z = 0.68 [15]. Their analysis gave a bound of ∆αem /αem < 5 × 10−6 for temporal variation, and an upper bound on spatial variations of δαem /αem < 3 × 10−6 over 3 Gpc at these redshifts. Tzanavaris et al. [16] used 21cm absorption lines from the same cloud in a sample of
1.1. THE FUNDAMENTAL CONSTANTS OF NATURE
7
2 g m /m . 8 quasars to probe the constancy of αem and µ together in the combination αem p e p
Assuming that the electron-proton mass ratio and proton g-factor, gp , are both constant, they found ∆αem /αem ≡ 0.18 ± 0.55 × 10−5 . Observation bounds on αem have also been derived from the structure of the Cosmic Microwave Background (CMB) [17] and from Big-Bang Nucleosynthesis (BBN). These studies give ∆αem /αem . 10−2 at z ∼ 103 and z ∼ 109 − 1010 respectively. Although these constraints are not competitive with the those coming from studies of QSO spectra, they do probe the evolution of αem at much higher redshifts. Recently, Reinhold et al. reported a 3.5σ indication of a variation in the electron-proton mass ratio over the last 12 billion years: ∆µ/µ = 2.0 ± 0.6 × 10−5 [20]. This result combines high-quality astronomical data with improvements in the measurement of crucial laboratory wavelengths to new levels of precision. Although they report a variation in µ at a level comparable to that claimed for αem (which is partly a reflection of the recent improvements in spectroscopic measurements), the Reinhold et al. result is theoretically much harder to understand. Variations in αem and µ lead to violations of the Weak Equivalence Principle (WEP) and these appear to be unacceptably large if ∆µ/µ varies at the 10−5 level [125]. We discuss WEP violation constraints further in section 1.3. Laboratory-based experiments probe the rate at which αem is varying today. However, if αem has varied at constant rate over the last 10-12 Gyrs, the astronomical studies mentioned above are, in fact, currently more sensitive than the laboratory probes of the constancy of the αem . This said, there is no particular reason to believe that αem should vary at a constant rate. Comparisons of atomic clock standards based on different sensitive hyperfine transition frequencies give α˙ em /αem ≡ −0.4 ± 16 × 10−16 yr−1 , [4], |α˙ em /αem | < 1.2 × 10−15 yr−1 , [5], α˙ em /αem ≡ −0.9 ± 2.9 × 10−15 yr−1 , [6]. A bound of α˙ em /αem ≡ −0.3 ± 2.0 × 10−15 yr−1 was found by comparing two standards derived from 1S-2S transitions in atomic hydrogen after an interval of 2.8 years [7]. By comparison, if αem has been changing linearly with time over the last 10-12 Gyrs then a total a variation of ∆αem /αem ∼ −0.57 ± 0.10 × 10−5 corresponds to α˙ em /αem ∼ 5.3 ± 1.5 × 10−16 yr−1 . In the near future the laboratory precision should be increased by at least a factor of 10 [51]. Such an improvement would, at least in cases where αem varies linearly with time, make laboratory tests more sensitive to local variations in αem than observations of QSO spectra are to global changes.
8
CHAPTER 1. INTRODUCTION
Other local bounds on the possible variation of the fine structure constant have been derived from geochemical studies [19, 24, 25] using data from the Oklo natural reactor in Gabon, and from studying the ratio of rhenium to osmium in meteorites [28]. The Oklo natural reactor was operational about two billion years ago (z ' 0.15). In order for the measured value of the samarium-149:samarium-147 ratio at the reactor site to be consistent with theory, one must require that the
149 Sm + n
→150 Sm + γ capture resonance was in place 2 billion years ago at
an energy level within about 90meV of its current value. As first noted by Shlyakhter [19, 24], this leads to very stringent bounds on all interaction coupling constants that contribute to the energy level. The the capture cross-section’s response to small changes in αem has a doublevalued form and, as a result, the latest analyses by Fujii et al [25] allow two solutions: one consistent with no variation, ∆αem /αem ≡ −0.8 ± 1.0 × 10−8 , the other with a small change in αem , ∆αem /αem ≡ 8.8 ± 0.7 × 10−8 . This latter possibility might be excluded by further studies. More recently, Lamoureax [26] has argued that a better (non-Maxwellian) model for the thermal neutron spectrum in the reactor leads to 6σ lower bound on the variation of ∆αem /αem > 4.5 × 10−8 at z ' 0.15. Peebles and Dicke [27] were the first to realise that studies of the rhenium to osmium ratio in meteorites could be used to constrain any variation in the fine structure constant over the lifetime of the solar system. The β-decay
187 Re 75
→187 ¯e + e− is very sensitive to αem 76 Os + ν
and the analyses of new meteoritic data, when combined with new laboratory measurements of the decay rates of long-lived beta isotopes, have led to a limit of ∆αem /αem = 8 ± 16 × 10−7 [28] for a sample that spans the age of the solar system (z ≤ 0.45). For a more detailed and excellent review of the current observational status of varyingconstants we refer the reader towards ref. [29].
1.1.5
Varying constants and scalar fields
In order to interpret the many different experimental and observations bounds mentioned above, it is essential that one has a self-consistent theory which describes the spacetime variation of one or more of the constants and from which further detectable consequences of any variation can be inferred. In general, a constant, C, is promoted to be a dynamical quantity by associating it with some scalar field or “dilaton”, φ, i.e. C → C(φ). We usually assume that the scalar field
1.2. THEORETICAL MOTIVATIONS AND MODELS
9
theory associated with φ has a canonical kinetic structure and the variations of this scalar field contribute to the spacetime curvature like all other forms of mass-energy. Variations in φ must therefore conserve energy and momentum and so their dynamics are constrained by a non-linear wave equation of the form − ¤φ =
X
fj,φ (φ)Lj (εk , pk ) + V,φ (φ),
(1.1)
j,k
where φ is associated with the variation of one or more ‘constants’, Cj , via a relation Cj = fj (φ); fj,φ (φ) = dfj (φ)/dφ. The Lj (εk , pk ) are some linear combinations of the density, εk , and pressure, pk , of the k th species of matter that couples to the field φ. The mass and selfinteractions of φ are determined by its potential, V (φ). Included in this formulation are all standard theories for varying constants, like those for the variation of the Newtonian gravitation ‘constant’ G, αem , and the electron-proton mass ratio, as described in refs. [116, 37, 36, 125]. It also includes the so-called ‘chameleon theories’ [69], which we shall discuss in more detail in Chapters 4 and 5. Throughout this thesis, we are interested in the behaviour, predictions and delectability of scalar fields whose evolution is determined by an equation with the form of eqn. (1.1).
1.2
Theoretical Motivations and Models
Scalar field theories with evolution equations of the form of eqn. (1.1) are well motivated by a number of proposals for high energy physics beyond the Standard Model. In this section, we briefly review some of these theoretical motivations. We also detail some of the more phenomenological models of varying-constant.
1.2.1
Higher Dimensional Theories
Many of the modern proposals for physics beyond the Standard Model predict that, on a fundamental level, spacetime has more than the usual four dimensions. The earliest example of such a theory, the 5-D Kaluza-Klein (KK) scenario for unifying gravity and electromagnetism [31, 32], was proposed in 1921. Although we only observe four dimensions, Klein noted in 1926 that the fifth dimension could have gone unnoticed if it was curled up and, as a result, very small. By compactifing the extra dimensions, it is possible to construct KK inspired models in
10
CHAPTER 1. INTRODUCTION
4 + D dimensions, for any D, that look four dimensional over large scales. Nowadays, String Theory, with its 10 dimensions, presents a natural home for the KK scenario. A feature common to all these models, it that, when the extra dimensions are compactified, the effective 4-D dimensional couplings αem , αW , αs and G are found to depend on the size of the compactified dimensions, which themselves are determined by the expectation of a scalar field, φ; φ is commonly called the ‘dilaton’. For instance: in the low energy limit of heteoritic string theory, αem is proportional e2φ . More generally, the dilaton-dependence of the constants is not universal but depends on the compactification scheme and the version of string theory that is being studied (see ref. [29] for a more in depth discussion). In these multi-dimensional scenarios, it is almost always the case that if one of the 4dimensional coupling ‘constants’, αem changes, then G and the other couplings do so as well. This linked variation in the ‘constants’ is also a prediction of grand unified theories (GUTs) with a dynamical coupling, whether they postulate the existence of extra dimensions or not. For instance: it the context of grand-unified supersymmetric theories, it was argued in [21] that variations in the QCD scale, ΛQCD , and αem are linked by δΛQCD /ΛQCD ≈ 38δαem /αem . Varying-constants and the existence of new scalar fields that interact with matter are, therefore, a generic prediction of higher-dimensional theories such as String Theory. However, given the large number of such theories, one cannot even find even a unique, order of magnitude estimate for the rate at which the constants should be changing today. In the field of varyingconstants, it has therefore proved popular to construct theories along more phenomenological lines.
1.2.2
Phenomenological models
Higher-dimensional theories provide an important motivation for the existence of new scalar fields and varying-constants, however, they do little to single out a low-energy preferred theory to describe those fields. Even if one restricts oneself to considering models that can come from String Theory, recent studies [22] have shown that the number of physically acceptable vacua is so vast that unique predictions for low energy physics beyond the Standard Model are virtually impossible to make. For this reason, varying-constant theories are generally constructed in the manner described in section 1.1.5. Using these theories, the ways in which varying-constants affect the laws of physics can be studied and experimental data can be used
11
1.3. FIFTH-FORCE TESTS FOR SCALAR FIELDS
to constrain the unknown parameters that are generally present in such theories. By bounding these phenomenological theories it is possible to constrain proposals for new fundamental physics. When taking this approach, the simplest models tend, until they are ruled out, to be preferred to more complicated ones. The simplest varying-G model, for instance, is Brans-Dicke theory [116]. The first, and simplest, self-consistent phenomenological model of varying-αem was proposed by Bekenstein in 1982. Bekenstein’s model reduces to the standard Maxwell theory of electromagnetism in the limit of no variation.
Sandvik, Barrow and Magueijo
(SBM)[36] later extended Bekenstein’s (B) [37] original theory to include general relativity. The resulting BSBM theory provides a framework for studying effects of varying αem . A number of detailed cosmological studies of the behaviour of this theory have been made in refs. [39, 40, 41, 42, 43, 36]. Recently a simple varying electron mass model was proposed by Barrow and Magueijo [125]. In ref. [38], a extended version of BSBM theory was used to study the simultaneous variation of αem and G. However, in phenomenological models such as these, it is more often than not the case than only one ‘constant’ is allowed to vary, and that the linked variation predicted by higher-dimensional and grand unified theories is not accounted for. In the simplest models, the scalar fields are assumed to be massless, and their cosmological behaviour would be radically altered if they had a mass larger than about H0 ~/c2 ∼ 10−33 eV /c2 . Although, classically there is nothing wrong with such a small mass, before such a simple model can be seen as a description of physics at a fundamental level, one would need to find some mechanism to prevent the mass from gaining large radiative corrections. Whilst such mechanisms are known to exist in String Theory [61, 63], it is not clear to what extent they can be applied to models constructed on purely phenomenological grounds. Issues such as this mean that, as attractive as the simplest models seem to be, we should not be too surprised the low-energy theory that describes the evolution of constants is ultimately found to be relatively complicated.
1.3
Fifth-force tests for scalar fields
In addition to causing variations in one or more of the fundamental ‘constants’, scalar fields that interact with matter also induce a new attractive ‘fifth’ force between bodies. Constraints
12
CHAPTER 1. INTRODUCTION
on the existence of such new forces are very tight and generally fall into two categories: 1) bounds on the violation of the weak equivalence principle (WEP) and 2) bounds on alterations to the 1/r2 behaviour of gravity [102, 103, 104, 106, 105, 87, 78]. In deriving these bounds, it is standard practice to assume that the self-interactions are of scalar field are negligible over the scales of the experiments and that, as a result, the dynamics of the scalar field, φ, are well approximated by an essentially linear field equation of the form: − ¤φ = m2 φ + βG1/2 ε,
(1.2)
where m = const is interpreted as the mass of the scalar field, ε is the local density of matter, and β quantifies the strength of the scalar field’s interaction with matter. We normalise φ so that it has a canonical kinetic term. We have also set c = ~ = 1. If β ∼ O(1) then the scalar field and gravity have similar strength interactions with normal matter. The attractive force, induced by φ, between two bodies with masses M1 and M2 that are separated by a distance r takes the Yukawa form F12 =
β 2 G(1 + r/λ)e−r/λ M1 M2 , 4πr2
(1.3)
where λ = ~(mc)−1 is the range of the force. In most theories, the strength, β, with which φ interacts with a body actually depends slightly on that body’s composition. 400 years ago Galileo famously pronounced that, in the absence of air resistance, any two bodies would fall to Earth at precisely the same rate. This universality of free-fall is otherwise known as the weak equivalence principle (WEP) and it has been tested and verified to a precision of few parts in 1013 [102, 103, 104, 78]. In scalar field theories, however, the composition dependence of the coupling, β, causes different bodies to fall to at different rates and, as a result, the weak equivalence principle is violated. For a scalar field theory to be physically-viable these violations must be too small to have been detected thus far. Assuming that scalar field self-interactions have only a negligible effect, WEP violation constraints impose a very stringent upper bound on β. This bound may well become even stronger in the near future as planned space-based tests, such as STEP, MICROSCOPE and SEE [97, 99, 96], promise to be able to detect WEP violations down to a precision of one part in 1018 . The first of these missions, MICROSCOPE, is scheduled to be launched as early as 2007 and will be able to probe for violations as small as one part in 1015 .
13
1.3. FIFTH-FORCE TESTS FOR SCALAR FIELDS
-0.5 Excluded region
-1
Irvine Boulder
-1.5
log10 (β)
-2 -2.5
Seattle
-3 -3.5 -4 Seattle -4.5 -5
-2
0
2
4 6 log10 (λcos / 1 m)
8
10
12
Figure 1.1: Allowed couplings and force ranges for scalar fields which couple to baryon number and have approximately linear field equations. The shaded area shows the parameter space that is allowed by the current bounds on scalar field theories with linear field equations. λ = 1/m is the range of the force, and m is the mass of the scalar field. β is the matter coupling of the scalar. β > 1 is ruled out for all but the smallest ranges (currently λ < 10−4 m). We refer the reader to [87] for a plot of the allowed parameter space for λ < 10−3 m. Irvine, Seattle and Boulder refers to [105], [102, 103, 104] and [106] respectively.
WEP violation bounds on β are strongest for long ranges forces: λ & 108 m. For forces with considerable shorter ranges, the best bounds on β come from experiments, such as that conducted at the University of Washington [87], which search for alterations to 1/r2 behaviour of gravity. Massive scalar fields (m > 0) produce such alterations, and at present these tests have been able to probe for scales fields with ranges as small as 0.1 mm. A more detailed discussion of fifth-force tests for scalar fields is given in Chapter 5.2. FIG. 1.1 shows the 95% confidence limits on m and β for a scalar field with an essentially linear field equation under the conservative assumption that it couples to baryon number. The shaded area is the allowed region of parameter space. The white region is excluded. A light scalar
14
CHAPTER 1. INTRODUCTION
field, i.e. λ & 108 m, with negligible self-interactions can therefore only couple to matter very weakly: |β| . 10−4 . The bounds on a scalar field that, as it often the case, couples differently to protons and neutrons are even stronger for λ & 1012 m, requiring |β| . 10−6 .
1.4 1.4.1
Aims and Outline Aims of thesis
Varying-constants and, more generally, the existence of new scalar fields are a generic prediction of modern theories of high energy physics beyond the Standard Model. Unlike many other features associated with these proposals for new physics, there is real hope that the effects of these new scalar fields could be detected in the near future. If such a detection were to be made, either as the result of incontrovertible evidence for variation of the constants or through fifth-force tests, it would be the first piece of experimental evidence for String Theory over General Relativity. Additionally, if the ‘constants’ of nature are shown to be, in reality, dynamical: able to take different values at different points in space and time, then the fact that we observe them to take values that are perfect for human life would no longer be quite so mysterious. The better we understand the way in which these new scalar fields behave, the better the chances of detecting them, if they exist, become. The more accurate our models for them are, and the more that is known about their effects, the more straightforward it becomes to design experiments to find them. Our aim in this thesis is, therefore, to expand upon what is currently known about the behaviour of varying-constants and scalar fields, so that experimental and observational data can be better combined to constrain and, maybe one day, verify or rule out theories of new physics. Throughout this thesis we set c = ~ = 1 and define Mpl = G−1/2 = 2.176×10−8 kg to be the Planck mass. We work in signature (+ − − −).
1.4.2
Outline of thesis
In Chapter 2, we study the new effects that occur when varying-αem theories are generalised to be consistent with the Glashow-Salam-Weinberg (GSW) theory of electroweak interactions. We study the ways in which the scalar fields in these models interact with matter and consider the cosmological predictions of the simplest electroweak varying-αem theories. We also note how
1.4. AIMS AND OUTLINE
15
the inclusion of electroweak effects leads to new, and perhaps one day detectible, phenomena associated with the spatial variations of the Weinberg mixing-angle, θw . The data that one uses to bound or constrain a scalar field theory is often a combination of laboratory, geochemical, and astronomical observations. In Chapter 3 we consider how these disparate observations, made over vastly differing scales, can be combined to give reliable constraints on the allowed cosmological variations in the constants of Nature. If local observations are directly comparable with cosmological ones then the next-generation of laboratory searches for varying-αem should be able to confirm or refute the apparent variation in αem that has been seen in QSO spectra. In Chapter 3 we derive a sufficient condition that must be satisfied for local and global tests to be directly comparable and show that this condition is satisfied by a large class of physically viable theories. One important class of theories that does not satisfy this condition is the subject of Chapters 4 and 5. In this class of theories, the selfinteractions of the scalar field play an important rˆole and the field’s mass is highly dependent on its surroundings: it is large in high density regions, but small when the density of matter is very low. This property allows the field to act like a ‘chameleon’ and ‘hide’ from the most restrictive experimental bounds on its coupling to matter. We show that, unlike other types of scalar fields, chameleon fields that interact with matter much more strongly than gravity, and yet are light on cosmological scales, are not ruled out at present. We conclude in Chapter 6 by summarising our main results and discussing their consequences for the detection of scalar fields and varying-constants in the near future.
Chapter 2
Varying-αem in the context of Electroweak theory
The value of e stands as the Rock of Gibraltar for the last 6 × 109 years! George Gamow (1904-1968)
2.1
Introduction
Much of the recent interest in varying-constants theories has been focussed on scalar theories where αem = e2 /~c can vary [37, 39, 40, 41, 42, 43, 36, 46, 47, 48, 49]. The simplest such theory was initially conceived by Bekenstein (B) [37] and later extended by Sandvik, Barrow and Magueijo (SBM)[36]. The behaviour of this BSBM theory has been the subject of a number of detailed cosmological studies (see refs. [39, 40, 41, 42, 43, 36]). In order to evaluate the constraints introduced by a programme of unification, it is important to extend simple varying-αem models, such as BSBM, to include electroweak and grand unification. Kimberly and Magueijo [44] recently proposed an extension to the Glashow-Salam-Weinberg (GSW) electroweak theory in which the weak couplings can also vary in space and time and which reduces to the standard GSW theory in the limit of no variation. This allows the consequences of electroweak unification to be investigated self consistently for the first time. Chamoun et al. 17
18
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
[45] have proposed an extension of QCD in which the strong coupling constant (and therefore ΛQCD ) vary. In this chapter we consider varying-coupling extensions to the GSW electroweak model in more detail. We begin in section 2.2 by discussing some of the general features present in electroweak varying-coupling theories. In section 2.3 we review the simplest such theories, i.e. the Kimberly-Magueijo (KM) models, and state the evolution equations for the scalar fields (dilatons) that are responsible for the variation of the coupling constants. In the form that they are presented in section 2.3, the classical dilaton evolution equations are not very usable in a cosmological setting. The reason for this is that in the KM-models, as with all varying-coupling electroweak theories, the dilaton interacts with normal matter through quantum corrections and, as a result, it is not initially clear how the terms that source the evolution of the dilaton fields are related to easily understood and measurable quantities such as the local energy density, ε, and pressure, P , of matter. In section 2.4 we calculate the quantum loop corrections that are responsible for the interaction between the dilatons and normal matter and express them in terms of ε and P . The analysis of this section is not specific to the Kimberly-Magueijo models but applies equally well to all varying coupling extensions of GSW electroweak theory and to extensions of Maxwell’s theory that allow for varying-αem e.g. BSBM. We apply our analysis to a universe populated by the particle spectrum of the standard model. We also consider the interaction between the dilaton field and some putative species of weakly-interacting dark matter. In section 2.5, we use our results to write down effective quantum-corrected dilaton evolution equations for the Kimberly Magueijo models in the presence of a background cosmological matter density that is composed of weakly interacting, and non-weakly-interacting, non-relativistic dark-matter as well as the usual baryonic matter. The source terms to these effective field equations are written in terms of the local energy density and pressure of matter. In section 2.6, we solve the cosmological field equations and use our analysis to provide a brief overview of the general features of the homogeneous and isotropic cosmology predicted by the KM models. We are specifically interested in the ways in which these electroweak varyingαem models distinguish themselves from simpler varying-αem based on Maxwell theory. We also consider the bounds that WEP violation experiments place of the parameters of the KM models.
19
2.2. GENERAL THEORY
One new feature that is predicted by most electroweak varying-αem theories is the variability of another fundamental ‘constant’ of Nature: the Weinberg-mixing angle, θw . We show that it is generally the case that cosmological observations of varying-αem alone provide us with little or no insight into the way in which θw varies. Fortunately, as we show in section 2.7, spacetime variations of θw result in several new and potentially detectible phenomena, specifically: non-conservation and dequantisation of electric charge and the creation of an effective electric dipole moment (EDM) on particle species which carry a weak neutral charge. In section 2.8, we discuss our results and summarise the new features that emerge when varying-αem theories are generalised to include electroweak effects. We also consider the prospects for detecting these theories in the near future.
2.2
General Theory
The electroweak couplings, gw and gy , are promoted to dynamical quantities by associating them with scalar fields ϕ(x) and χ(x): gw := eϕ and gy := eχ . We do not, at this stage, exclude the possibility that ϕ and χ may be functions of each other. Gauge-invariance fixes the Lagrangian density of the gauge-kinetic sector of such a theory to be: µν Lg = − 14 tr FW µν FW − 14 FY
µν µν FY
µν where the field strengths, FW and FYµν are given by: µν FW
µ ν ν µ = Fµν W + ∂ ϕW − ∂ ϕW ,
FYµν
= FYµν + ∂ µ χY ν − ∂ ν χY µ .
µν The Fµν are respectively given by standard expressions for the SU (2) and U (1) W and FY
Yang-Mills field strengths; Wµ and Y µ are the gauge fields. In all cases, apart from the one where ϕ ≡ χ + const, the ratio of gy /gw = tan θw is not constant. Moreover, as a result of renormalisation, even if θw is truly constant at one particular energy scale it will generally be dilaton dependant at all other energies. The fine structure constant is given by: 2 α = gW sin2 θw := e2φ ;
φ = φ(ϕ, χ).
To fully specify an electroweak theory with varying couplings one must specify the dynamics of the scalar fields ϕ and χ. This is done by specifying a Lagrangian density: Lϕ, χ = f (X, Y, Z, ϕ, χ),
20
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
where f is some function to be specified and X = ∇µ ϕ∇µ ϕ, Y = ∇µ χ∇µ χ and Z = ∇µ χ∇µ χ. If one demands a canonical kinetic term then f is linear in X, Y and Z and, provided that χ 6= g(ϕ) for any g, we write: Lϕ, χ =
ω1 (ϕ, χ) ω2 (ϕ, χ) ∇µ ϕ∇µ ϕ + ∇µ χ∇µ χ + ω3 ∇µ χ∇µ χ + V (ϕ, χ). 2 2
Whereas, if the two fields are related, i.e. χ = g(ϕ) for some g, we have: Lϕ, χ =
ω(ϕ) ∇µ ϕ∇µ ϕ + V (ϕ). 2
We identify V (ϕ, χ) as the dilaton potential and the ωi (ϕ, χ), which have dimensions of (mass)2 , as the dilaton couplings. The smaller the ωi are, the stronger the coupling of the dilaton to matter.
2.3
Simplest Models
The simplest electroweak varying-coupling theories arise when one requires that the theory have (classical) global shift symmetries: ϕ → ϕ + const and χ → χ + const. These simple theories do not favour any particular values for the constants over any other. The shift symmetries ensure that the ωi and V are constants. Further simplification emerges from the requirement that ω3 = 0. A constant potential has no effect on the dynamics of ϕ and χ and so it may be ignored. Up to a determination of the ωi , two potential theories now remain. In the single dilaton model (KM-I), ϕ and χ are related and the shift symmetry ensures we can set ϕ = χ. In the second model (KM-II), ϕ and χ are independent. These are precisely the two extensions of GSW electroweak theory proposed by Kimberly and Magueijo in ref. [44] and we shall discuss them further below.
2.3.1
A Single-Dilaton Theory
2 and α The first model (KM-I) contains a single dilaton and allows both αW := gw em to vary but
their ratio, and hence the mixing angle θW , are true constants. In this model the electroweak sector of the Standard Model is described by the following lagrangian density: 1 LKM −I = − e−2ϕ [wµν · wµν + yµν y µν ] 4 ´2 ω λ³ † + (Dµ Φ)† (Dµ Φ) + Φ Φ − v 2 + ϕ,µ ϕ,µ , 4 2
(2.1)
21
2.3. SIMPLEST MODELS
where wµν = 2w[ν,µ] − g W wµ ∧ wν ,
(2.2)
yµν = 2y[ν,µ] , µ ¶ i i Dµ Φ = ∂µ − g W t · wµ − g Y yµ Φ 2 2
(2.3) (2.4)
and Φ is the Higgs field. The dilaton field is ϕ and ω is the dilaton coupling. Since this theory ³ ´ 2 is perturbatively non-renormalisable, we would like ω & O Mpl so that the dilaton enters at the same level as gravity, and we are justified in ignoring any quantum fluctuations of the dilaton field when quantising with respect to the other fields. The auxiliary gauge fields, wµ and yµ , take values in the adjoint representations of su(2) and u(1) respectively. They are not the physical gauge fields, which will be denoted by capital letters, but are related to them by the transformations g¯w wµ = gw Wµ ,
(2.5)
g¯y yµ = gy Yµ ,
(2.6)
gw = g¯w eϕ ,
(2.7)
gy = g¯y eϕ .
(2.8)
The distinction between the physical and auxiliary fields will only become important at the quantum level (see section 2.4 below). When written in terms of these auxiliary gauge fields the covariant derivatives which act upon matter species are independent of ϕ. This makes it simpler to derive the classical field equations. The physical gauge couplings, gw and gy , are dynamical whereas the auxiliary couplings, g¯w and g¯y , are arbitrary constants. At tree level the Fermi constant, GF , and the fermion masses do not vary, whereas the W and Z boson masses do. We will also define the physical field strength tensors by Wµν := Yµν :=
g¯w wµν , gw
g¯y yµν . gy
(2.9) (2.10)
These field strengths reduce to the standard definitions of the weak and hypercharge field strengths with gauge couplings gw and gy respectively whenever the dilaton field ϕ is constant.
22
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
2.3.2
A two dilaton model
In the second model (KM-II), proposed in ref. [44], a second dilaton field is added. This results is the weak mixing angle, θW , becoming a dynamical quantity. The lagrangian density of the electroweak sector in this model is: 1 1 LKM −II = − e−2ϕ wµν · wµν − e−2χ yµν y µν + (Dµ Φ)† (Dµ Φ) 4 4 ´2 ω λ³ † ω2 1 2 + Φ Φ−v + ϕ,µ ϕ,µ + χ,µ χ,µ 4 2 2
(2.11)
The definitions (2.2)-(2.4) still hold, as do relationships (2.5)-(2.6) but the physical coupling constants are now related to their auxiliary values by the transformations gw = g¯w eϕ , gy = g¯y eχ , ¸ · g¯y := tan θ¯W eχ−ϕ . tan θW = g¯w The dilaton fields are ϕ and χ and their respective dimensionful couplings are ω1 and ω2 . As ¡ ¢ remarked above, we would like ωi & O MP2 l . In accord with KM-I, GF and the fermion masses are constant at tree-level whereas the boson masses are dynamical.
2.3.3
Symmetry Breaking
At low energies and temperatures, the most important feature of the GSW electroweak model is that the SU (2)L × U (1)Y symmetry of the lagrangian is spontaneously broken to U (1)em via the Higgs doublet, Φ, assuming a vacuum expectation value, Φ0 , which minimises its potential. At tree-level this value is
0 Φ0 = . v
A perturbative expansion about this vacuum can be written as Φ= v+
0 H(xµ ) √ 2
.
Expanding out the kinetic Higgs term gives: ¤ v2 1 v 2 (g W )2 £ 1 2 (Dµ Φ)† (Dµ Φ) = H,µ H ,µ + (wµ ) + (wµ2 )2 + (g W wµ3 − g Y yµ )2 . 2 4 4
(2.12)
23
2.3. SIMPLEST MODELS
When written in terms of the physical gauge fields, the broken-phase boson fields of both KM-I and KM-II are given by the usual formulae: ¢ 1 ¡ Wµ± := √ Wµ1 ± iWµ2 , 2 gw Wµ3 − gy Yµ Zµ := q , 2 + g2 gw y Aµ :=
gy Wµ3 + gw Yµ q . 2 + g2 gw y
Hence, the tree-level boson masses and their dilaton field dependence can read off from (2.12): v MW = √ gw , 2 v q 2 MZ = √ gw + gy2 , 2 MA = 0.
2.3.4
Classical Field Equations
In order to make and test the predictions of these theories we need to know the field equations. In both KM-I and KM-II, the generalised Einstein and Yang-Mills equations are: ( ) 2 g¯y2 y g¯w w φ matter dilaton Gµν = 8πG T + 2 Tµν + Tµν + Tµν + Tµν , 2 µν gw gy µ 2 ¶ ¯w δLmatter µ g D wµν = − , 2 gw δwν Ã ! g¯y2 δLmatter µ ∂ yµν = − , 2 gy δy ν w and T y are the standard Yang-Mills energy-momentum tensors written in terms of where Tµν µν
the auxiliary fields and couplings. In KM-I the dilaton conservation equation is − ¤ϕ = −
1 1 −2ϕ e (wµν · wµν + yµν y µν ) = − (Wµν · Wµν + Yµν Y µν ) . 2ω 2ω
(2.13)
and in KM-II we have conservation equations for the two fields: 1 1 −2ϕ e wµν · wµν = − Wµν · Wµν , 2ω1 2ω1 1 1 −2χ e yµν · y µν = − Yµν Y µν . −¤χ = − 2ω2 2ω2
−¤ϕ = −
(2.14) (2.15)
24
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
The conservation equations for other matter fields, like perfect fluids, are unchanged. Although this system represents the full classical field equations, their current form is not very useful. In order study the cosmological behaviour of the couplings, and make experimentally testable predictions, we need to understand how the right-hand sides of (2.13, 2.14, 2.15) depend on macroscopic quantities such as the background matter density, ε, and the expectations of the dilaton field or fields. It is this question that we address in the next two sections.
2.4
The Dilaton-to-Matter Coupling
In this section we derive the precise functional way in which the coupling of the dilaton fields to matter depends on ϕ and χ. Although we have the Kimberly-Magueijo varying-coupling models, as described in section 2.3, in mind, the derivations of this section apply to any electroweak varying-coupling theory. Most tests which we might wish to apply to these, and other, varying-constant theories require knowledge of how the dilaton fields will evolve in the presence of some background matter density, ε, and pressure, P : D E ε := T0 0(matter) ,
P :=
1 D i(matter) E Ti , 3
where h · i denotes the quantum expectation. In order to understand the dilaton evolution, we must therefore evaluate the terms on the right-hand sides of (2.13)-(2.15) under theh · i operation. Those terms all consist of terms quadratic in the Yang-Mills field strengths, i.e. Wµν · Wµν and Yµν Y µν , henceforth we refer to them collectively as Fµν F µν or F 2 terms. When we quantise this theory (leaving ϕ as a classical field) we must do so with respect to the physical gauge fields rather than the auxiliary ones, since it is only the physical fields whose kinetic terms possess the correct normalisation. This feature is important if we want the renormalisation procedure to go through as usual when ϕ = const. It is for this reason that we have labeled the capitalised fields as physical.
2.4.1
Derivation
® We need to understand how these F 2 terms depend, in a functional sense, on the dilaton fields and the background matter density, ε. For simplicity, we consider the case of a single
25
2.4. THE DILATON-TO-MATTER COUPLING
Dirac fermion, Ψ, of mass m, coupled to a U (1) gauge field Aµ with Higgs scalar Φ and dilaton ϕ. The effective lagrangian, including ’t Hooft’s gauge-fixing term, is: L = Lgauge,0 + Lf ix + LΨ,0 + Lint °³ ´ °2 ° ° MA 1 µν Lgauge,0 = − 4 F Fµν + ° ∂µ + i v Aµ Φ° −
λ 4
³ Φ† Φ −
v2 2
´2
1 µ (∂ Aµ + ξMA Im (Φ))2 2ξ ¯ (iγ µ ∂µ − m) Ψ =Ψ
Lf ix = − LΨ,0
¯ µ ΨAµ Lint = −e (ϕ) Ψγ The ghost fields have been excluded since in this gauge they decouple from the abelian sector. By taking a spacetime region R, of volume V and temporal extent T , that is large compared to the scale of quantum fluctuations of the matter and gauge fields, and yet small when compared to the scale over which the dilaton field varies, we can quantise this theory inside R by taking ϕ = const and defining the partition function in the usual manner. At energies far below the Higgs mass, mH , we can ignore quantum Higgs fluctuations in the first approximation. Doing this and integrating out the gauge fields we determine that − 14
hFµν F
µν
Z Z Z
D ¡ ν ¢ ¡ ε ¢ gauge ¯ Ψ (y)Dµ ε gauge (x − z) Ψγ ¯ Ψ (z) d4 xd4 yd4 z ∂x2 Dµν (x − y) Ψγ ¡ ν ¢ ¡ ε ¢ E gauge ¯ Ψ (y)∂ τ x Dτgauge ¯ Ψ (z) , + ∂ µ x Dµν (x − y) Ψγ (x − z) Ψγ ε
e2 (ϕ) i= 2V T
Ψ
(2.16) where the gauge-field propagator is given by: Z gauge Dµν (x)
=
· ¸ d4 k ikx (1 − ξ) k µ k ν 1 µν −g + 2 4 ie 2 2 k − ξMA k − MA2 (2π)
The quantum expectation operator, h·iΨ , used in equation (2.16), is defined thus: ´ R ³ ¡ ¢ R£ ¤ RR 4 4 ν e2 gauge (x − y) J ν (y) ei d4 xLΨ,0 ¯ ¯ d xd yJ (x) D d Ψ [dΨ] A Ψ, Ψ exp −i ¡ ¢® 2 ¯ ´ R ³ := A Ψ, Ψ R£ ¤ R R Ψ 2 4 xd4 yJ ν (x) D gauge (x − y) J ν (y) ei d4 xLΨ,0 ¯ [dΨ] exp −i e d dΨ 2 (2.17) R The [dΨ] represents a functional integral with respect to the Ψ-field, and we have defined µ Ψ(x). The term with an exponent that is quadratic in the J’s in equation ¯ J µ (x) := Ψ(x)γ
(2.17) encodes the self-energy corrections to the fermion and gauge boson propagators. By writing both the fermion and gauge boson propagators as full propagators we are justified in
26
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
dropping the aforementioned exponential term because its effect is of sub-leading order. Finally then, by writing this expression in momentum space we reduce it to the more transparent and manifestly gauge invariant form: − 14
hFµν F
µν
e2 (ϕ) i= 2V T
Z
E ¢ µν ¤¢−2 D µ d4 p 2 £¡ 2 µν ¡ 2 2 ˜ (p) J˜ν (−p) p p − M (ϕ) g + Π p, e (ϕ) J , A A Ψ∗ (2π)4 (2.18)
where J˜µ is the momentum space representation of J µ and Πµν A is the vacuum polarisation of the gauge boson. The expectation here is defined with respect to the partition function µ Z ¶ Z £ ¤ 4 4 ¯ ¯ ZΨ∗ := dΨ [dΨ] exp i d xd y Ψ(x)K(x − y)Ψ(y) with K(x − y) denoting the inverse of the full fermion propagator.
2.4.2
Interpretation
By Wick’s theorem it is apparent that the remaining quantum expectation in (2.18) contains two distinct contributions. The first contribution will always involve boson exchange between two fermionic particles and so is proportional to (ε − 3P )2 /m2 to leading order. The second contribution results from a single fermionic particle admitting and reabsorbing a gauge boson (i.e. the process that results in the fermion’s self-energy). This second term will clearly be proportional to m2 (ε − 3P ) to leading order. In almost all cases of experimental interest (ε − 3P ) /m4 ¿ 1. Only in objects whose density approaches that of nuclear matter does it fail to hold and this na¨ıve quantisation procedure is not suitable for dealing with such high-density backgrounds. For this reason we drop the contribution due to photon exchange. When the perturbation theory holds it is appropriate to expand (2.18) as a series in the gauge coupling, e2 (ϕ), giving − 14 hFµν F µν i =
³ ³ ε ´´ ¡ 2 ¢´³ e2 ζ (ϕ) (ε − 3P ) 1 + O e ln e 1 + O e¯2 m4
(2.19)
where, for this model, ζ is a defined by: Z Z ζ (ϕ) (ε − 3P ) = e¯2
µ ¶ d4 p d4 q p2 (p − q) · γ + m νq·γ +m tr γ γ ¡ ¢ ν q2 − m (p − q)2 − m (2π)4 (2π)2 p2 − MA2 (ϕ) 2 (2.20)
When inter-fermion strong interactions are introduced, confinement may occur. If this happens the trace factor in (2.20) will take on a more complicated form. At leading order,
27
2.4. THE DILATON-TO-MATTER COUPLING
however, it will remain independent of ϕ so long as ΛQCD is. At densities much lower than the nucleon mass, the right-hand side will be proportional to the hadronic energy density.
2.4.3
The Unbroken Symmetry Case: MA = 0
It is helpful to have a physical interpretation of the ζ parameter. When dynamical symmetry breaking does not occur (and so we have MA2 = 0) the right-hand side of (2.20) is proportional to the 1-loop fermion field self-energy resulting from its interaction with the Aµ gauge field. Defining this self-energy to be δm (ϕ) we have: ζ=
e¯2 δm (ϕ) δm ¯2 = = const. e2 (ϕ) m m2
(2.21)
In this case ζ is ϕ-independent at leading order. δ m ¯ is defined as the electromagnetic mass correction when the electric charge takes some value e¯ = const.
2.4.4
The Broken Symmetry Case: MA 6= 0
When MA2 6= 0 the above interpretation of ζ in terms of the self-energy mass-correction will, in general, fail. The correct physical interpretation will depend heavily on the size of
m MA (ϕ) .
We consider the three cases: •
m MA
À 1. The dominant contribution to the ζ integral will come from momenta p2 À MA2 ,
and so we can ignore MA at leading order and interpret ζ just as we did in the MA = 0 case. Hence, ζ is dilaton independent to leading order. In the simple model used above we should expect ζ ∼ O(¯ e2 ). •
m MA
¿ 1. The dominant contribution to the (2.20) will come from momenta p2 ≈ m2 .
Hence
µ ζ (ϕ) =
m MA (ϕ)
¶4 ζ0 ,
where ζ0 is of the order the value of ζ which we would have had if MA = 0. In this case ζ will vary with ϕ like •
m MA
1 4 . MA
= O (1). This is by far the most difficult case to analyse. In general ,ζ will have a
leading-order dependence on the dilaton field through MA (ϕ), but the precise form of ζ (ϕ) will depend on the nature of the trace term in equation (2.20). This in turn rests
28
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
on the precise details of the microscopic physical model for the matter fields in question. This chain of complications means that a general prescription for the dilaton dependence of ζ, in this case, is not possible. Roughly though we expect, in the context of the simple model described above, that ζ ∼ O(¯ e2 ) or so.
2.4.5
Application to a cosmological setting
We can now use the above results to evaluate the dominant contributions to the quantities hWµν · Wµν i and hYµν Y µν i in a cosmological setting. The values of hWµν · Wµν i and hYµν Y µν i must be known before the dilaton evolution equations can be solved, for example consider the eqns. (2.13-2.15) which describe the KM-models. We consider four important classes of matter: ultra-relativistic particle species e.g. light neutrinos, the top-quark, ‘light’ charged particles i.e. charged leptons and baryonic matter, and WIMP dark matter. We also consider any potential contribution from heavy neutrinos. Ultra-Relativistic Matter At the very high energies required to restore the broken gauge symmetry in these theories, most matter species will be ultra-relativistic and we will assume this to be the case for all species. We will also assume that there is a discrete spectrum of spin states. Matter will therefore behave like black-body radiation with ε = 3P . It is clear from equation (2.19) that ® these ultra-relativistic species do not contribute to the F 2 terms which source the dilaton fields’ evolution. The only uncharged fundamental field is the neutrino, which we will assume to be so light that it remains relativistic at experimental and cosmological temperatures. Such neutrinos make only a very small contribution to the dilaton source terms. The corollary of this is that amongst the known fundamental matter species, we are justified in assuming that only those with non-zero charge contribute to the right-hand sides of equations (2.13)-(2.15). The see-saw mechanism for neutrino mass-generation results in three light neutrinos, masses (i)
mν (corresponding to the currently observed particles) and three very heavy neutrinos, masses (i)
MN ≈ a few tens of GeV. The light neutrinos are formed mostly from the weakly interacting left-handed components, whilst the heavy neutrinos are primarily composed of the weak singlet right-handed particles. Such heavy neutrinos would therefore interact with the weak bosons
29
2.4. THE DILATON-TO-MATTER COUPLING
(i)
(i)
much more weakly that other massive particle species. To zeroth order in mν /MN ¿ 1, we can assume these heavy neutrinos to be non-interacting with either electromagnetic or weakly ® interacting particles. Their contribution to the F 2 terms will be negligible compared to that of the other matter species. The Top Quark With a mass of about 180 GeV, the top quark is the only fermionic species present in the Standard Model that does not fall into the
m MW
¿ 1 category. Whilst in principle its contribution
to the dilaton terms could be calculated, it would be a difficult procedure which would require accounting for gauge boson self-interactions, Higgs boson fluctuations, and non-perturbative effects - as well as having to do QCD calculations. However, the results would actually have very little bearing upon most cosmological tests of this theory, since the background energy density of top quarks is entirely negligible when compared to that of all the other forms of matter. ‘Light’ Charged Matter We have just argued that, amongst the Standard Model particle species, the only non-negligible contributions to the dilaton source terms will come from particles with mass m and charge Q 6= 0, which are light compared to MW . We must also require the particles to be relativistic, allowing us to restrict to the low-energy broken symmetric phase of the KM electroweak theories. This is appropriate for energies well below the 100GeV level. At these energies, perturbation theory will be appropriate and non-abelian effects will be of sub-leading order. In this case then, the results of Section II should be valid. Writing the hWµν · Wµν i and hYµν · Y µν i quantities in terms of the low-energy physical fields we find: E ® hD 1 1 2 ® † 2 − 4 Wµν · Wµν = − 12 FW · F W ± + sin θW − 4 Fem ± ® ®¤ + cos2 θW − 41 FZ2 + sin θW cos θW − 12 Fem · FZ
(2.22)
· [1 + O (∂µ θW ) + O (gw , gy )] and
1 ® £ ® ® 2 − 4 Yµν · Y µν = cos2 θW − 14 Fem + sin2 θW − 41 FZ2 ®¤ − sin θW cos θW − 12 Fem · FZ · [1 + O (∂µ θW )]
(2.23)
30
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
The O (∂µ θw ) terms produce only negligible corrections. The O (gw , gy ) symbol represents the additional terms that arise from gauge boson self interactions; these only contribute at sub-leading order. Thus for a non-relativistic particle species, Ψi , of mass mi ¿ MW , charge Qi 6= 0, and ® 2 weak isospin t3i we see that only − 14 Fem will contribute to (2.22, 2.23) at leading order. The 1 2 ® 1 ¯ 2 ¯® 1 ® − 4 FZ , − 4 ¯FW ± ¯ and − 2 Fem · FZ terms are suppressed by relative factors of order of ¢2 gw2 m4i 2 ¢ 2 m4i ¡ m2i ¡ gw gw 2 2 t and t − 2Q sin θ , t − Q sin θ respectively. 3i 3i i W W 2 4 2 4 2 2 2 2 3i Qi e cos θW M Q e cos θ M Q e M i
W
i
Z
W
Z
The leading-order contributions of such a ‘light’ matter species to (2.22) and (2.23) therefore reduce to 1 ® e2 δ m ¯i − 4 Wµν · Wµν i = sin2 θW 2 εi , e¯ mi 1 ® e2 δ m ¯i − 4 Yµν Y µν i = cos2 θW 2 εi , e¯ mi
(2.24)
where δ m ¯ is defined as the electromagnetic mass correction when the electric charge is some value e¯ = const. For leptons, with a mass ml , we expect that, to within an order of magnitude, δml /ml ≈ e2 δ m ¯ l /¯ eml ∼ e2 . Defining e¯ = e(today) ⇔ φ(today) = 0, we find ζleptons ∼ 10−2 . The situation is more complicated for baryonic matter due to the fact that the largest contribution to its mass comes from QCD corrections. ζbaryon can, however, be estimated from the electromagnetic contribution to the proton and neutron mass, [23]: δmp ≈ 0.63 MeV,
δmn ≈ −0.13 MeV.
Defining e¯ = e(today), we find ζp ≈ 6.7 × 10−4 and ζn ≈ −1.4 × 10−4 . Since me ¿ mp , we expect ζH ≈ ζp for Hydrogen and ζHe ∼
ζp +ζn 2
for Helium-4. Assuming that, in line
with observations, three quarters of the mass of baryonic matter in the universe to located in hydrogen and taking the other 25% to be Helium, we estimate the average cosmological value of ζ for baryonic matter to be: ζbaryon ∼
ζHe 3ζH + ∼ 6 × 10−4 . 4 4
Although the precise numerical value for ζbaryon given above should not be taken too seriously, we do expect ζbaryon ∼ O(10−4 ) and positive.
2.4. THE DILATON-TO-MATTER COUPLING
31
Dark Matter There is a great deal of evidence from cosmological and astronomical data for the existence of dark matter which contributes about 27% of the gravitating mass density of the universe. Such matter must be non-baryonic, electromagnetically non-interacting and non-relativistic (‘cold’). It appears that the Standard Model lacks any suitable dark matter candidates. Weakly Interacting Massive Particles (WIMPS), such as the lightest putative supersymmetric partners in MSSM, are one possible class of candidates for dark matter. The masses of these particles tend to be of the order of a few tens of GeVs and so fall into the m ≈ MW category. Locally, if dynamically virialised, they will have keV energies which may allow them to be detected in underground nuclear recoil experiments. They are necessarily uncharged ( Q 6= 0) but they ¯ 2 ¯® ® ¯ terms. can however interact weakly and as such contribute to the − 14 FZ2 and − 14 ¯FW ± Thus the leading-order contribution from these WIMPs to (2.22, 2.23) would be given by: " à ! µ ¶# 2 2 m 1 ® m 2 g wimp wimp − 4 Wµν · Wµν wimp = w FW + FZ εwimp , 2 2 g¯w MW MZ2 ! à (2.25) 2 2 m 1 ® g wimp − 4 Yµν Y µν wimp = w εwimp , tan2 θW FZ 2 g¯w MZ2 where we have defined FW and FZ to be WIMP-model dependent ‘structure’ functions. These encode precisely how the WIMP’s ζ parameters depend on the gauge boson masses. We expect |FW | and |FZ | to be ¿ 1. Not much more can be said about their precise structure as functions of the dilaton field without the aid of a microscopic model for dark matter. Although, provided that the dark matter particles interact only through the weak force, we 2 , g 2 ) ∼ 10−2 when estimate that |δmwimp /mwimp |, and therefore also |FW | and |FZ |, to be O(gw y
mwimp & MW , MZ . For smaller values of mwimp , we expect, for the reasons given in section 4 . 2.4.4, that the structure functions would be suppressed by a factor of m4wimp /MW
2.4.6
Summary
In this section we have derived the form of the effective dilaton field equations for an electroweak theory with varying couplings in the presence of a background matter density. It is this form that is of most use for experimental, observational and cosmological tests of electroweak varying-coupling theories. We took particular care in the identification of the physical gauge fields. In previous work on BSBM theory, the dilaton dependence of the F 2 source term was
32
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
incorrectly determined as a result of the auxiliary fields mistakenly taken to be the physically propagating modes. This error resulted in the statement: e¯2 e2
¿ À 1 e¯2 δ m ¯2 µν − fµν f = 2 2 = e−2ϕ ζε 4 e m
(2.26)
Where fµν = 2∂[µ aν] was the auxiliary field strength, and aµ the auxiliary photon field. δ m ¯ is defined as the electromagnetic mass correction when the electric charge is some value e¯ = const. This lead to the conclusion that the leading-order dilaton field dependence went like: e¯2 e2
= e−2ϕ . However, it is clear from equations (2.20)-(2.21) that in reality the leading-order
dilaton dependence should be
e2 e¯2
= e+2ϕ and equation (2.26) should read:
¿ 2 À e¯ 1 e2 δ m ¯2 µν − 2 fµν f = 2 2 ε = e2ϕ ζε e 4 e¯ m
(2.27)
Indeed equation (2.26) is problematic because, in the limit of zero electric charge, matter 2 term. This situation decouples from the photon, and so cannot possibility contribute to the Fem 2 term as given by the right-hand side corresponds to ϕ → −∞. However, in this limit, the Fem
of equation (2.26) grows infinity large. Equation (2.27) shows the correct behaviour, and vanishes, as it should, it this limit. We have also seen that if the gauge bosons become massive then, for particles which are much lighter than the gauge boson in question, the leading-order dilaton dependence of the F 2 term changes from g 2 (g is the physical gauge coupling) to
g2 4 Mgauge
∼
1 . g2
For particles that are
much heavier than the gauge boson in question, provided perturbation theory is still valid at energies of the order of the particle’s mass, the leading-order dilaton dependence remains as g 2 . We also briefly discussed the complication of mparticle ≈ Mgauge , whereby the leading-order dilaton dependence will be highly susceptible to the details of the matter model in question, and noted that this effect might be important in cosmology if dark matter is weakly interacting. In all of this analysis we assumed both that perturbation theory holds and that any nonabelian effects are negligible. Whilst this is true for electroweak theory at energies well below the Higgs boson mass, it will not be true for QCD. If we were to construct a BSBM-like 2 (or equivalently, varying ΛQCD ) theory, we would expect the leadingvarying αstrong = gstrong 2 term to come from a complicated function of gstrong . order dilaton dependence of the Fstrong
Evaluating this function would, at the very least, require us to be able to predict quark and nucleon masses accurately via a QCD calculation which is not yet possible. In the absence of
2.5. EFFECTIVE FIELD EQUATIONS FOR KM MODELS
33
such calculations, varying ΛQCD theories will be difficult to test accurately or make use of in the early universe except at the very highest energies.
2.5
Effective field equations for KM models
Using the results of section 2.4, we can now evaluate the right hand sides of eqns. (2.13-2.15) and write down the effective dilaton field equations for the KM-models.
2.5.1
Single dilaton model
In the single dilaton (KM-I) theory, the effective dilaton evolution equation is: " Ã ! Ã !# 2 m2wimp m 2 2ϕ X 2 2ϕ wimp − ¤ϕ = e ζi εi + e FW + sec2 θW FZ εwimp, ω ω MW (ϕ)2 MZ (ϕ)2
(2.28)
i
where the sum is over all charged matter species (with m ¿ MW ). The first term is identical to the properly evaluated source term in BSBM. Hence the KM-I theory differs (at low energies and densities) from BSBM only in the putative WIMP matter contribution.
2.5.2
Two dilaton model
In the KM-II theory the effective dilaton equations are: 2 αem (ϕ, χ) X sin2 θW ζi εi ω1 α ¯ em i " à ! à !# m2wimp m2wimp 2 2ϕ + e FW + FZ εwimp ω1 MW (ϕ)2 MZ (ϕ, χ)2 2 αem (ϕ, χ) X −¤χ = cos2 θW (ϕ, χ) ζi εi ω2 α ¯ em i à ! 2 m 2 2χ wimp + e tan2 θw FZ εwimp ω2 MZ (ϕ, χ)2
− ¤ϕ =
(2.29)
(2.30)
In the absence of the permitted dark matter contributions, this two-dilaton theory will only reduce to BSBM when ω2 sin2 θW = ω1 cos2 θW . In all other cases θW will vary and lead to an evolution of αem that slightly different from that of BSBM theory. However, as we shall see below, unless a variation in θw can be directly detected, or the unknowns ω1 , ω2 , FW and FZ independently evaluated, it will be almost impossible to distinguish a two dilaton model from a single dilaton one using observations of time evolution αem alone.
34
2.6 2.6.1
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
Homogeneous and Isotropic Cosmology Cosmological Field Equations
We now consider the cosmological evolution of the coupling constants in the Kimberly-Magueijo models. We focus on the way in which these models behave in a homogeneous and isotropic background. The effect of inhomogeneities on the evolution of the dilaton fields is dealt with, in a more general setting, in Chapter 3. Assuming isotropy and homogeneity requires the background spacetime to be described by the FRW line element with some scalar factor a(t) and curvature parameter k: ·
dr2 ds = dt − a (t) + r2 dθ2 + r2 sin2 θdφ2 1 − kr2 2
2
¸
2
(2.31)
In following analysis, we only explicitly consider the cosmological behaviour of the two dilaton model (KM-II). However, as we shall see, the KM-II model reduces to the single dilaton (KM-I) theory whenever certain conditions hold.
Full Cosmological Equations Using the KM-II dilaton conservation equations (2.29, 2.30) derived above, we write down the full cosmological Friedman and mass conservation equations: µ ¶2 a˙ ω2 i k 8πG h ω1 Λ H ≡ = εrad + εbaryon + εwimp + ϕ˙ 2 + χ˙ 2 − 2 + a 3 2 2 a 3 2
(2.32)
and ε˙rad = −4Hεrad
(2.33)
£ ¤ αem ζbaryon εbaryon ε˙baryon = −3Hεbaryon + 2 sin2 (θw ) ϕ˙ + 2 cos2 (θw ) χ˙ α ¯ em ! Ã m2wimp εwimp ε˙wimp = −3Hεwimp + 2χ˙ tan2 θw FZ mZ (ϕ, χ)2 ! Ã !# " Ã m2wimp m2wimp + FZ εwimp , +2ϕ˙ FW mW (ϕ)2 mZ (ϕ, χ)2 where we have defined ζbaryon εbaryon =
P
i ζi εi .
(2.34) (2.35)
We have also assumed that the dark matter in
the universe is composed mostly of weakly interacting massive particles (WIMPs). The WIMP
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
35
density is εwimp . The dilaton evolution equations are: 2 αem sin2 (θw ) ζbaryon εbaryon ω1 α ¯ em " ! !# Ã Ã m2wimp m2wimp 2 2ϕ − e FW + FZ εwimp ω1 mW (ϕ)2 mZ (ϕ, χ)2 2 αem cos2 (θw ) ζbaryon εbaryon χ ¨ + 3H χ˙ = − ω2 α ¯ em ! Ã m2wimp 2 2χ 2 − e tan θw FZ εwimp ω2 mZ (ϕ, χ)2
ϕ¨ + 3H ϕ˙ = −
2.6.2
(2.36)
(2.37)
Reduced Cosmological Equations
The standard cosmological picture of a radiation dominated era, followed by period of matter domination and then, most recently, a period of accelerated expansion fits very well with all currently available observational data. If either 4πGω1 ϕ˙ 2 or 4πGω2 χ˙ 2 were of the order of the H 2 then the presence of the dilaton fields would result in significant changes to this standard cosmological model. Such changes would certainly result in the KM-models being ruled out by the current observational data. We therefore assume that (4πGω1 )1/2 |ϕ|, ˙ (4πGω2 )1/2 |χ| ˙ ¿ H; we show that this assumption is valid below. Whenever these conditions hold, the ϕ˙ and χ˙ terms in eqns. (2.32), (2.34) and (2.35) have only a small effect and so can be safely dropped when working to leading order. When these terms are discarded, we are left with a simplified set of cosmological equations which nonetheless form a self-consistent closed system. The simplified matter evolution equations have solutions εbaryon = εwimp =
ε¯baryon , a3 ε¯wimp . a3
To solve the dilaton evolution equations exactly we need a prescription to deal with the WIMP matter structure functions: FW and FZ . Constraints coming from microwave background radiation structure [17] and BBN [18] ensure that ∆αem /αem < 10−2 during the matter era. In the context of the KM models this implies that only small changes in values of the dilaton fields can have occurred during the matter era and so, to leading order, we can approximate FW and FZ by constants.
36
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
We define ζwimp = const by: " 2ϕ
ζwimp := e
Ã
FW
!
m2wimp
à 2
mW (ϕ)2
+ sec θw FZ
m2wimp mZ (ϕ, χ)2
!# ,
and parameterise the fraction of ζwimp that comes from Z-boson loops by νZ = const: µ sec2 θw FZ µ
νZ := FW
¶
m2wimp mW (ϕ)2
¶
m2wimp mZ (ϕ,χ)2
µ
+ sec2 θw FZ
m2wimp
¶.
mZ (ϕ,χ)2
4 ) We expect that ζwimp ∼ O(10−2 ) when mwimp & MW and ζwimp ∼ O(10−2 m4wimp /MW
otherwise. Estimates of the electromagnetic contribution to the proton and neutron mass suggest that ζbaryon ∼ O(10−4 ). Coupling Evolution Equations The fields ϕ and χ are respectively responsible for the evolution of the couplings gw and gy , and so the evolution of these coupling ‘constants’ is described by eqns (2.36) and (2.37) respectively. At energies below MW , however, it is more natural to think in terms of αem and θw than gw and gy . Noting that: α˙ em = 2αem sin2 (θw ) ϕ˙ + 2αem cos2 (θw ) χ, ˙ we find the following explicit differential equations for the evolution of the observable couplings αem and sin2 (θw ): φ¨ + 3H φ˙ = −
2 (α) ωem (θw )
e2φ ζbaryon
ε¯baryon a3
2ψ ε¯dm ˙2 e − 2 ψ , (α) a3 1 − e2ψ ωW (ψ) ³ ´ ε¯baryon 2 ψ¨ + 3H ψ˙ = − (θ ) 1 − e2ψ e2φ ζbaryon a3 ωemw (θw ) ³ ´ 2 e2ψ ε¯dm + (θ ) , 1 − e2ψ ζwimp 3 − 2ψ˙ 2 a 1 − e2ψ ωWw (ψ)
−
2
(2.38)
e2ψ ζwimp
where we have defined αem ≡ e2φ α ¯ em sin2 (θw ) ≡ e2ψ .
(2.39)
37
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
We also define effective dilaton scales by: (α) ωem (θw ) = (α)
ωW (θw ) = (θw ) ωem (θw ) = (θ )
ωWw (θw ) =
ω1 ω2 2
+ ω1 (1 − e2ψ ) ω1 ω2 (1 − νZ )ω2 + νZ (ω1 + ω2 ) (1 − e2ψ ) ω1 ω2 ω1 − (ω1 + ω2 ) e2ψ ω1 ω2 νZ (ω1 + ω2 ) e2ψ − ω2 ω2
e4ψ
(2.40) (2.41) (2.42) (2.43)
Reduction from KM-II to KM-I When certain relationships hold between ω1 /ω2 , νZ and ψ, equations (2.38) and (2.39) emit a constant ψ, i.e. constant θw , solution. This appears to happen when ψ = 0, i.e. gw = 0, but if gw = 0 then the derivation of the symmetry broken version the electroweak varying constant theory performed in section 2.2 is no longer applicable. Hence, ψ = 0 solutions are to be discarded as unphysical. A physically acceptable constant θw solution occurs when: ω1 = tan2 θw ω2 , νZ
= cot2 θw .
(2.44) (2.45)
It is clear that, in the context of the KM-II models, having such a constant θw solution would require a remarkable degree of fine-tuning - particularly in the value of νZ . In this analysis, we have approximated νZ to be a constant. More generally however it will have some dependence on the dilaton fields. Finding this dependence would, however, require a specific particle-level model for dark matter. If νZ has non-trivial dilaton field dependence then it is possible that there might be an attractor mechanism, unknown to us here, that could decrease the amount fine-tuning that would be otherwise be necessary to have a constant θw solution. The constant-θw solution is most useful in that it allows us to pass from the 2 dilaton, KM-II, scenario to the single dilaton KM-I model. In the KM-I model, νZ does not appear, θw is constant and there is only one dilaton scale: ω. Equations (2.44), (2.45) link the value of νZ and the second dilaton scale, ω2 , to the first dilaton scale, ω1 , and the required constant value of θw . When the values of νZ , ω1 , ω2 and θw are connected in this way, equation (2.38) describes the evolution of the KM-I dilaton. The scale, ω, of the KM-I dilaton is given by: (α)
(α) ω = ωem (θw ) = ωW (θw ) cosec2 θw .
38
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
All cosmological solutions of the KM-I model emerge in this way. It is for this reason that it is sufficient to only explicitly consider the dynamics of the two dilaton theory.
2.6.3
The Radiation Era
From the time that temperature of the universe dropped below about 100 MeV (i.e. after the QCD phase transition) right up until the epoch of matter-radiation equality (T ∼ 1 eV), we can accurately model the matter content of the universe as a radiation bath with ε = 3P . In 2 and F 2 quantities, which drive the dilaton fields, vanish since such a radiation bath the FW Y
they are proportional to ε − 3P . Therefore, during the radiation era, the dilaton fields behave as free massless scalar fields. This is no different from the behaviour of the dilaton field in the BSBM theory of varying αem [36]. Integrating eqns. (2.36, 2.37) gives ¡ 3 ¢ a ϕ˙ = const, ¡ 3 ¢ a χ˙ = const, and so ¡ ¢ ϕ(p) = ϕ¯ + C1 1 − e−p ¡ ¢ χ(p) = χ ¯ + C2 1 − e−p ³ where p ≡ ln
a(t) a ¯
´
³ ´ = ln
T¯ T
. C1 and C2 are constants of integration. T¯ and a ¯ are respectively
the arbitrary reference temperature and cosmological scale factor. Both dilatons tend to a constant as p → ∞, with the p-time velocity of the dilatons decaying exponentially to 0. For instance, if ϕ, ˙ χ˙ ∼ O(H) when T ∼ 100 MeV, then by the epoch of matter-radiation equality: ϕ, ˙ χ˙ < 10−8 O(H). To a good approximation, we may therefore take the dilatons fields to be stationary at the beginning of matter era.
2.6.4
The Matter Era
The radiation era dynamics ensure that both dilaton fields are almost stationary at the beginning of the matter era. In the matter era, the universe is dominated by pressureless dust and
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
39
so: 8πG ε¯matter , 3 a3 3 H˙ = − H 2 . 2
H2 =
We proceed by considering two limits: In the first limit the baryon source term to the dilaton equations is much larger than the source term due to WIMP matter. In the second limit, WIMP matter dominates the evolution of the dilatons. We refer to the former limiting case as baryon dominated evolution, and the latter as WIMP dominated evolution.
Baryon Dominated Evolution In the baryon dominated limit we take the effects of WIMP matter to be negligible i.e. ζwimp = 4 ) when m 0. If ζbaryon ∼ O(10−4 ) and ζwimp ∼ O(10−2 m4wimp /MW wimp . mwimp , then baryon
dominated limit would applicable if mwimp . 10 GeV. We define ζem ≡ ζbaryon εbaryon /εmatter . ζem is the fraction of the energy density of matter which is due to electromagnetic effects. The 2 2 quantities ζbaryon /4πGω1 and ζbaryon /4πGω2 quantify the strength of the dilaton to baryon
coupling. Since the dilaton fields in the KM-models are massless, WEP violation constraints , [102, 103, 104, 78], (see section 2.6.8)imply that ¯ ¯ ¯ ¯ ¯ ζ2 ¯ ¯ ζ2 ¯ ¯ baryon ¯ ¯ baryon ¯ ¯ ¯,¯ ¯ ¿ 1. ¯ 4πGω1 ¯ ¯ 4πGω2 ¯ It it the smallness of this matter coupling that allows us to consistently ignore dilaton corrections to Friedman equation and the matter evolution equations. We define a ‘cosmological time coordinate’ p = log a(t). The coordinate p is related to the cosmological redshift, z, by p = − ln(1 + z). Baryon-dominated dilaton evolution is described by the following equations: 3ζem e2ψ 3 e2φ − 2ψp2 , φpp + φp = − (α) 2 1 − e2ψ σem (θw ) ´ e2ψ 3ζem ³ 3 2ψ = − ψpp + ψp + 2ψp2 1 − e e2φ , (θw ) 2 1 − e2ψ σem (θw )
(2.46) (2.47)
where φp = dφ/dp and we have defined σi = 4πGωi ; the σi are dimensionless representations of the dilaton scales. String theory, and other compactification scenarios, prefer O(1) values for the σi .
40
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
Contrast with BSBM We compare equation (2.46) which describes the evolution of αem ∝ e2φ in the KM-models with the BSBM αem evolution equation. The (baryon-dominated) matter era BSBM equation for φ is 3 3ζem 2φ φpp + φp = − e 2 σbsbm where again we have defined ζem ≡ ζbaryon εbaryon /εmatter ; σbsbm = 4πGωbsbm where ωbsbm is the scale of BSBM dilaton. Using the WMAP values for εbaryon and εmatter , [92], we find ζem /ζbaryon = 0.17 ± 0.02. When one reduces to KM-I theory we have ψp = 0 ⇔ θw = const (α)
and σem (θw ) → 4πGω = σ. In the baryon-dominated limit, the KM-I φ evolution, eqn. (2.46), is equivalent to the one coming from BSBM with σ = σbsbm . It is only in the dark sector that BSBM theory and the KM-I model make different predictions. The KM-II model differs from BSBM in two ways: (α)
1. The effective dilaton scale, σem (θw ), depends on θw . 2. θw is itself is evolving. The smallness of the dilaton-to-matter couplings results in the dilaton fields varying only very slowly over cosmological time scales. As a result we expect |ψp | ¿ 1 and so the −2ψp2
e2ψ 1 − e2ψ
term is negligible. When this term is neglected, the KM-II αem evolution equation has the same structure as the BSBM equation for αem . The only difference is that in the KM-II we have an (α)
effective, θw -dependent, dilaton scale: σem (θw ). Despite this difference, for small variations in θw , and without any prior knowledge of the values of the values of ω1 , ω2 and ωbsbm , it impossible to distinguish the baryon-dominated limit of the KM-II model from BSBM using cosmological observations of αem alone. The only new feature of the KM-II models, that is potentially detectible using cosmological observations, is a varying-θw . Unfortunately however, the value of θw is very hard to measure using astronomical observations and to date the only accurate evaluations of it have been in the laboratory. If baryons provide the dominant source to the evolution of the coupling constants then it seems unlikely that cosmological observations alone will be able to distinguish between the BSBM, KM-I and KM-II models. In section 2.7 we uncover some new phenomena that could be potentially detected in the laboratory and which would provide a ‘smoking gun’ for θw variation.
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
41
Late Time Solutions Exact solutions to equations (2.46) and (2.47) must be found by numerical integration. At late times, however, we can find accurate ‘asymptotic’ approximations to the exact solutions. We assume that at the beginning of the matter era the dilaton fields were stationary to a good approximation. We show below that |ζem /σi | ¿ 1 generally implies that |φp |, |ψp | ¿ 1. Provided that φp and ψp are ¿ 1 we can drop the φpp , ψpp and ψp2 terms as there are small compared to φp and ψp ; we justify this procedure below. When these terms are dropped equations (2.46, 2.47) become: φp = − ψp = −
2ζem e2φ , (α) σem (θw ) 2ζem (θ )
σemw (θw )
³
(2.48)
´ 1 − e2ψ e2φ .
(2.49)
It is now clear that φpp ∼ O(φ2p , φp ψp ), ψpp ∼ O(ψp2 , φp ψp ) and so |φp |, |ψp | ¿ 1 implies that |φpp | ¿ |φp | and |ψpp | ¿ |ψp | and we are justified in ignoring the φpp and ψpp terms in eqns. (2.46, 2.47). The ψp2 terms can clearly ignored at leading order. Immediately we can see from the above equations that (θ )
dφ σemw (θw ) = (α) . dψ σem (θw )(1 − e2ψ ) (θ )
(α)
The sign of σemw /σem determines the relative sense of the variations in αem and θw . Since we expect only very small changes in φ and ψ we can, to leading order, approximate exp(2ψ) = sin2 θw by its current value of 0.231. Under this approximate we find: 0.090 ωω21 + 1 dφ ≈ . dψ 1 − 0.301 ωω21 Thus dφ/dψ > 0 provided that ω2 /ω1 < cot θw ≈ 3.33, and φ and αem are changing faster than ψ and θw whenever 0<
ω2 cos2 θw < 2/(tan2 θw − tan4 θw ) ≈ 9.51. ω1 1 + sin2 θw
In all other cases there is a larger relative change in sin2 θw than in αem . Integrating equations (2.48) and (2.49) gives: £ 2 ¤z tan θw − cot2 θw + 2 ln tan2 θw 0 =
4ζem e2φ(z=0) ln(1 + z) (θ ) sin2 (θ¯w ) cos2 (θ¯w )σemw (θ¯w )
(2.50)
(θ )
e2φ (t) =
e2φ(z=0) sin2 θw cos2 θw σemw (θw ) (θ ) sin2 (θ¯w ) cos2 (θ¯w )σemw (θ¯w )
(2.51)
42
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
where z is the redshift, z = 0 being today and θ¯w = θw (z = 0). These solutions are only valid (θ )
so long as σemw 6= 0 i.e. ω2 /ω2 6= cot θw ≈ 3.33. We define tan2 θw := tan2 θ¯w exp(2γ) where γ = χ − ϕ. Our previous assumptions require that |γ(z) − γ(0)| ¿ 1 and so ¶ µ tan θw 2ζem e2φ(z=0) γ(z) − γ(0) = ln = ln(1 + z) (θ ) tan θ¯w σemw (θ¯w ) e2φ (z) = 1 +
4ζem e2φ(z=0) ln(1 + z). (α) ¯ σem (θ)
(2.52) (2.53)
These solutions are valid so long as ψp , φp ¿ 1 and φpp , ψpp ¿ ψp , φp . These conditions are clearly satisfied for the range of redshifts that of interest to us, z . 100, so long as |ζem | ¿ 1. The smallness of the change in the dilaton fields also justifies the previous assumption that it was acceptable to ignore the alterations to the cosmological expansion caused by the dilaton fields. The baryon-dominated system is defined by two parameters ζem /σ2 and ω2 /ω1 ; observations of ∆αem /αem can only provide us with information about one combination of α (θ ). Without knowing ω /ω we cannot infer anything about the these parameters: ζem /σem w 2 1
way in which θw varies from observations of αem . In figures 2.1-2.5 we plot ∆αem (z)/αem (0) and
∆ sin2 θw (z) sin2 θw (0)
vs. redshift, z, for various values of ζem /σ2 and ω2 /ω1 .
In FIG. 2.1 we compare the exact numerical solution for ∆αem /αem with the asymptotic approximation found above i.e. eqn. (2.53). We have taken ω2 /ω1 = 1 and ζem /σ2 = −2.0 × 10−6 which gives ∆αem (z = 2)/αem (0) = −0.57 in line with Webb observations [3, 2, 1]. The plot on the left hand side shows the two solutions on the same axes. The solid black line is the exact solution whilst the dotted line is the approximate solution. We can clearly see that the asymptotic solution is a very good approximation for z . 400. The plot of the right hand side shows the relative error in the asymptotic approximation. We see that for z < 100, eqn. (2.53) gives the correct value of ∆αem (z) to within 0.04%. Fig. 2.2 is laid out in the same way as FIG. 2.1 and shows how the exact and asymptotic solutions (see eqn. (2.52)) for ∆ sin2 θw (z)/ sin2 θw (0) compare. We have, once again, used ω2 /ω1 = 1 and ζem /σ2 = −2.0 × 10−6 . As in the ∆αem case, it is clear that the asymptotic solution is a good approximation for z . 400. For z < 100, the asymptotic approximation gives the correct value for ∆ sin2 θw / sin2 θw to within 0.04%. In FIGs. 2.3 and 2.4 we fix ω2 /ω1 = 1 and plot ∆αem /αem and ∆ sin2 θw / sin2 θw vs. redshift for different values of ζem /ω2 . In 2.3 positive values of ζem /ω2 are used in FIG. 2.3 whilst negative ones are used in FIG. 2.4. If ω2 /ω1 = 1, then the sign of ∆ sin2 θw / sin2 θw
43
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
−5
x 10
0.04 0.035
−1
0.03 approx.
−0.5
em
−1.5 −2
error (%) in ∆ α
∆ αem(z) / αem(0)
0
−2.5 −3 −3.5
0.02 0.015 0.01 0.005
−4 −4.5
0.025
0
approx. solution exact solution 0
10
1
2
10
3
10
10
redshift
Figure 2.1:
−0.005 0
20
40
60
80
100
redshift
Comparison of the baryon-dominated, exact and asymptotic solutions for
∆αem /αem in matter era. We have taken ω2 /ω1 = 1 and ζem /σ2 = −1.9 × 10−6 . The left hand plot shows the exact solution (solid line) on the same axes as the asymptotic solution (dotted line). For z . 400, the two solutions are almost identical. The plot on the right hand side shows the relative error in the asymptotic solution. For z < 100, this error is always smaller than 0.04%.
is the same as that of ∆αem /αem for baryon-dominated evolution. In general, the larger the magnitude of ζem /ω2 , the larger the change in αem and sin2 θw . To match the variation in αem seen by Webb et al. [1, 2, 3], i.e. ∆αem (z ≈ 2)/αem (0) = −0.57 × 10−5 , we require ζem /ω2 = −2.0 × 10−6 when ω2 /ω1 = 1. More generally, i.e. for other values of ω2 /ω1 , we α (θ ¯w ) ≈ −1.2×10−6 . In deriving this value of ζem /σ α (θ¯w ) we have require that today: ζem /σem em
not included the effects of the recent acceleration of the universe’s expansion on the evolution of the dilaton fields. These effects are considered in section 2.6.7 below. (α)
In FIG. 2.5, we fix ζem /σem (θw ) to be −1.3 × 10−6 today and plot the exact, baryondominated, matter era evolution of αem and θw for ω2 /ω1 . Ignoring the recent acceleration of the universe, this gives an αem evolution that is consistent with the observations of Webb et al. [1, 2, 3]. As we should expect given the asymptotic solution for ∆αem i.e. eqn. 2.53, varying (α)
ω2 /ω1 whilst keeping ζem /σem (θw ) fixed does little to alter the evolution of αem . However, the sense and magnitude of the variation in θw does change. Without any prior knowledge of the correct value of ω2 /ω1 , and with only observations of ∆αem /αem to go by, we are therefore unable to determine much about the way in which θw has evolved cosmologically. In most
44
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
−5
0
x 10
0.04 0.035 0.03
error (%) in ∆ sin θ approx.
−1
0.025
w
∆ sin θw(z) / sin θw(0)
−0.5
−1.5
−2
0.02 0.015 0.01 0.005
−2.5 0
approx. solution exact solution −3
0
1
10
2
10
10
3
10
−0.005 0
20
40
Figure 2.2:
60
80
100
redshift
redshift
Comparison of the baryon-dominated, exact and asymptotic solutions for
∆ sin2 θw / sin2 θw in matter era. We have taken ω2 /ω1 = 1 and ζem /σ2 = −1.9 × 10−6 . The left hand plot shows the exact solution (solid line) on the same axes as the asymptotic solution (dotted line). For z . 400, the two solutions are almost identical. The plot on the right hand side shows the relative error in the asymptotic solution. For z < 100, this error is always smaller than 0.04%.
−5
−5
x 10
4 ζ ζ
/ω = 2e−006
ζ
/ω = 3e−006
2
3
2
2
ζem/ω2 = 1e−006 ζ
/ω = 2e−006
ζ
/ω = 3e−006
em em
2 2
w
em
/ω = 5e−007
em
3.5
4 3 2
2 1.5 1
1 0
2.5
w
∆ αem(z) / αem(0)
2
ζem/ω2 = 1e−006 em
5
x 10
ζ
/ω = 5e−007
em
6
∆ sin2 θ (z) / sin2 θ (0)
7
0.5
0
10
1
2
10
10 redshift
3
10
0
0
10
1
2
10
10
3
10
redshift
Figure 2.3: The above figures show the exact, numerical evaluations of ∆αem (z)/αem (0) and ∆ sin2 θw (z)/ sin2 θw (0) vs. redshift (z) in the matter era for different positive values of ζem /ω2 . Baryons are assumed to dominate the evolution of the couplings. We have fixed ω2 /ω1 = 1. Larger values of ζem /ω2 produce larger changes in αem and sin2 θw . Positive ζem /ω2 always gives ∆αem /αem > 0.
45
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
−5
0
−5
x 10
0 −0.5
−1
−1
2
∆ sin θw(z) / sin θw(0)
−3 −4
2
∆ αem(z) / αem(0)
−2
−5
ζem/ω2 = −5e−007
−1.5 −2 −2.5 −3
ζem/ω2 = −1e−006 −6
ζem/ω2 = −2e−006
−3.5
ζem/ω2 = −3e−006 −7
x 10
0
10
ζem/ω2 = −5e−007 ζem/ω2 = −1e−006 ζem/ω2 = −2e−006 ζem/ω2 = −3e−006
1
2
10
10
3
−4
10
redshift
0
10
1
2
10
3
10
10
redshift
Figure 2.4: The above figures show the exact, numerical evaluations of ∆αem (z)/αem (0) and ∆ sin2 θw (z)/ sin2 θw (0) vs. redshift (z) in the matter era for different negative values of ζem /ω2 . Baryons are assumed to dominate the evolution of the couplings. We have fixed ω2 /ω1 = 1. More negative values of ζem /ω2 produce larger changes in αem and sin2 θw . Negative ζem /ω2 always gives ∆αem /αem < 0.
cases, however, the magnitude of the cosmological variation in θw is similar to the magnitude of the change in αem . (α)
We note that the Webb data seems to prefer negative values of ζem /σem . However, we expect that ζem > 0 for baryons. If we are to match the Webb data, we must therefore have at least one of ω1 and ω2 being negative. This, in turn, implies that at least one of the dilaton fields behaves like a ghost field, which is a potentially serious flaw. This said, ghost fields can result in some interesting effects such as the initial cosmological singularity being replaced by a bounce [50]. If neither of the dilaton fields are ghosts, and ∆αem /αem < 0 then either baryons do not dominate the cosmological evolution of the dilatons or we must rule out the KM-models. If ∆αem /αem ∼ O(10−6 ) over the last 10-12 Gyrs then WEP violation constraints can be used to rule out this baryon dominated scenario (see section 2.6.8 below); another possibility was discussed by Bekenstein in [48].
2.6.5
A Reduced WIMP Dominated System
In the WIMP dominated limit we set ζem = 0. The WIMP dominated limit would seem to be the natural one if most dark matter is WIMPs and mwimp & MW . WIMP particles are
46
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
−5
0
−5
x 10
5
x 10
ω /ω = 0.1
∆ sin2 θw(z) / sin2 θw(0)
∆ αem(z) / αem(0)
−1.5 −2 −2.5
−3.5
ω2/ω1 = 0.1
−4
ω /ω = 5
0 −1 −2 −3
0
1
2
10
10
3
1
1
ω2/ω1 = 5
10
2
ω2/ω1 = 10
2
ω2/ω1 = 1 ω2/ω1 = 10
1
ω2/ω1 = 1
3
−1
−3
2
4
−0.5
−4
0
10
10
redshift
1
2
10
3
10
10
redshift
Figure 2.5: The above figures show the exact, numerical evaluations of ∆αem (z)/αem (0) and ∆ sin2 θw (z)/ sin2 θw (0) vs. redshift (z) in the matter era for different values of ω2 /ω1 . Baryons (α)
are assumed to dominate the evolution of the couplings. We have fixed ζem /σem (θw ) = −1.2 × 10−6 at z = 0. The evolution of αem is then almost independent of ω2 /ω1 . The sense of magnitude of ∆ sin2 θw / sin2 θw is, however, very sensitive to different values of ω2 /ω1 . We cannot, therefore, predict the evolution of θw using only observations of αem .
not electrically charged and so does not directly source the BSBM theory dilaton. We should therefore expect the greatest differences between the predictions KM-models and BSBM to occur in this limit. In section 2.6.8 below we see that this WIMP dominated scenario is strongly preferred to the baryon dominated one by WEP violation constraints if ∆αem /αem ∼ O(10−6 ). In the WIMP dominated limit, the dilaton equations read: 3 3ζW e2ψ e2ψ φpp + φp = − (α) − 2ψp2 , 2 1 − e2ψ σW (θw ) ³ ´ 3 e2ψ 3ζW 2ψ ψpp + ψp + 2ψp2 = − 1 − e , (θ ) 2 1 − e2ψ σWw (θw )
(2.54) (2.55)
and a certain linear combination of the ϕ and χ equations emits an exact solution. Defining εdm u (p) ≡ σ1 ϕ + σ2 χ and ζW = ζwimp εmatter :
3 upp + up = −3ζW , 2 with solution: 4ζW u(p) = 3
¶ µ 3(p−peq ) − 2 − 1−e
2ζW 3
(2 + νZ ) (p − peq ) .
47
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
At the beginning of the matter era, when p = peq , we have u(p) = 0 and up (p) = 0 and so: A1 = 2ζW ,
A2 = 3ζW ,
which gives σ1 ϕ + σ2 χ = (σ1 + σ2 ) φ + ln [sinσ1 θw cosσ2 θw ] · ¸ 2 2 − 3 (p−peq ) − (p − peq ) . = 3ζW 1 − e 3
(2.56)
We extract an equation that involves only θw with the definition γ(p) ≡ ln tan (θw ): · ¸ νZ e2γ (σ1 + σ2 ) 3 3ζW −σ2 + . γpp + γp = − 2 σ1 σ2 e2γ + 1
(2.57)
For particular values of νZ , γ, σ1 and σ2 the right hand side of the above equation vanishes. This represents the constant θw solution that allows us to reduce to the single dilaton KM-I model.
−5
0
x 10
0.045
−0.5
0.04 error (%) in ∆ αem approx.
∆ αem(z) / αem(0)
−1 −1.5 −2 −2.5 −3 −3.5 approx. solution: νZ = 0.33333
−4 −4.5
0.035 0.03 0.025 0.02 0.015 0.01
exact solution: νZ = 0.33333 0
10
1
2
10
10
3
10
0.005 0
20
redshift
Figure 2.6:
40
60
80
100
redshift
Comparison of the baryon-dominated, exact and asymptotic solutions for
∆αem /αem in matter era. We have taken ω2 /ω1 = 1 and νZ = 1/3. ζW /σ2 is chosen so that ∆αem (z = 2)/∆αem = −0.57 × 10−5 . The left hand plot shows the exact solution (solid line) on the same axes as the asymptotic solution (dotted line). For z . 400, the two solutions are almost identical. The plot on the right hand side shows the relative error in the asymptotic solution. For z < 100, this error is always smaller than 0.045%.
48
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
−4
1
x 10
0.035 approx. solution: ν = 0.33333 Z
0.9
exact solution: νZ = 0.33333
0.03 error (%) in ∆ sin θ approx.
0.7 0.6
w
∆ sin θw(z) / sin θw(0)
0.8
0.5 0.4 0.3
0.025 0.02 0.015 0.01 0.005
0.2 0
0.1 0
0
1
10
2
10
−0.005 0
3
10
10
20
40
Figure 2.7:
60
80
100
redshift
redshift
Comparison of the baryon-dominated, exact and asymptotic solutions for
∆ sin2 θw / sin2 θw in matter era. We have taken ω2 /ω1 = 1 and νZ = 1/3. ζW /σ2 is chosen so that ∆αem (z = 2)/∆αem = −0.57 × 10−5 . The left hand plot shows the exact solution (solid line) on the same axes as the asymptotic solution (dotted line). For z . 400, the two solutions are almost identical. The plot on the right hand side shows the relative error in the asymptotic solution. For z < 100, this error is always smaller than 0.031%.
Late Time Solutions Late time ‘asymptotic’ solutions are found under the same assumptions that were made in the baryon-dominated case: γ(z) − γ(0) ∼ − 2φ(z)
e
∼ e
2ζW (θ )
σ ¯Ww
2φ(0)
ln(1 + z),
(1 + z)
4ζW sin2 θw (α) σ ¯ W
(2.58) Ã 2φ(0)
∼e
1+
4ζW sin2 θw (α)
σ ¯W
! ln(1 + z) ,
(2.59)
where the final approximation in the equation for φ(z) is accurate provided that: ¯ ¯ ¯ ¯ 4ζ sin2 θ ¯ ¯ W w ln(1 + z) ¯ ¿ 1. ¯ (α) ¯ ¯ σ ¯ W
We compare these late time, asymptotic solutions with the exact numerical solutions in FIGs. 2.6 and 2.7. In both of these figures we set ω2 /ω1 and chosen ζW so that we have ∆αem /αem (z = 2) = −0.57 × 10−5 . In the WIMP dominated case, we have an extra parameter, νZ , which quantifies the fraction of ζW that is due to Z-boson loops. Since there are two W bosons and only one Z boson we might naturally Z-boson loops to contribute one third of ζW i.e.
49
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
νZ ≈ 1/3. If, however, all of ζW comes from Z-boson effects, then we would have νZ = 1. In FIGs. 2.6 and 2.7, we plotted for νZ = 1/3. As in the baryon dominated case, the asymptotic solutions are very good approximations for z . 400. The approximation to ∆αem /αem is accurate to with 0.045% for z < 100. The ∆ sin2 θw / sin2 θw approximation is slightly better, being accurate to within 0.031% for z < 100.
−5
3
−5
x 10
3.5 ζ
/ω = 5e−007, ν = 0.33333
em
ζ
Z
ζ
/ω = 2e−006, ν = 0.33333
ζ
/ω = 3e−006, ν = 0.33333
em em
2
2
Z
2
/ω = 5e−007, ν = 1
em
1.5
1
Z
ζ
/ω = 2e−006, ν = 1
ζ
/ω = 3e−006, ν = 1
em
2.5
Z
2
ζem/ω2 = 1e−006, νZ = 1
3
∆ αem(z) / αem(0)
∆ αem(z) / αem(0)
2
ζem/ω2 = 1e−006, νZ = 0.33333
2.5
x 10
em
2
Z
2
Z
2 1.5 1
0.5
0.5
0
0
10
1
2
10
10
0
3
0
10
10
1
0
2
∆ sin θw(z) / sin θw(0)
−4
2
∆ sin2 θw(z) / sin2 θw(0)
−1
−3
ζem/ω2 = 5e−007, νZ = 0.33333
−1.5 −2 −2.5 −3
ζem/ω2 = 1e−006, νZ = 0.33333 ζem/ω2 = 2e−006, νZ = 0.33333
−3.5
ζem/ω2 = 3e−006, νZ = 0.33333 −7
x 10
−0.5
−2
−6
3
10
−5
x 10
−1
−5
10 redshift
−5
0
2
10
redshift
0
10
1
10 redshift
ζem/ω2 = 1e−006, νZ = 1 ζem/ω2 = 2e−006, νZ = 1 ζem/ω2 = 3e−006, νZ = 1
2
10
ζem/ω2 = 5e−007, νZ = 1
3
10
−4
0
10
1
2
10
10
3
10
redshift
Figure 2.8: The above figures show the exact, numerical evaluations of ∆αem (z)/αem (0) and ∆ sin2 θw (z)/ sin2 θw (0) vs. redshift (z) in the matter era for different positive values of ζW /ω2 . WIMPs are assumed to dominate the evolution of the couplings. We have fixed ω2 /ω1 = 1 and show plots for both νZ = 1/3 and νZ = 1. Larger values of ζem /ω2 produce larger changes in αem and sin2 θw . The positive ζW /ω2 always gives ∆αem /αem > 0 for νZ ≤ 1.
By studying the late time solutions, we note that, for z . 400, the αem evolution produced
50
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
−5
0
−5
x 10
0
x 10
−0.5
−0.5
∆ αem(z) / αem(0)
∆ αem(z) / αem(0)
−1 −1
−1.5
−2
ζem/ω2 = −5e−007, νZ = 0.33333
−1.5 −2 ζem/ω2 = −5e−007, νZ = 1
−2.5
ζem/ω2 = −1e−006, νZ = 0.33333
−2.5
ζem/ω2 = −1e−006, νZ = 1
ζem/ω2 = −2e−006, νZ = 0.33333
ζem/ω2 = −2e−006, νZ = 1
−3
ζem/ω2 = −3e−006, νZ = 0.33333 −3
0
1
10
2
10
10
ζem/ω2 = −3e−006, νZ = 1 −3.5
3
0
10
1
10
4 /ω = −5e−007, ν = 0.33333
em
ζ
/ω = −2e−006, ν = 0.33333
ζ
/ω = −3e−006, ν = 0.33333
Z
2
/ω = −5e−007, ν = 1
em
3.5 3
Z
2
Z
ζem/ω2 = −1e−006, νZ = 1 ζ
/ω = −2e−006, ν = 1
ζ
/ω = −3e−006, ν = 1
em em
2
Z
2
Z
w
em
2
x 10
ζ
Z
∆ sin2 θ (z) / sin2 θ (0)
4 3 2
2 1.5 1
1 0
2.5
w
∆ sin2 θw(z) / sin2 θw(0)
2
ζem/ω2 = −1e−006, νZ = 0.33333 em
5
3
10
−5
x 10
ζ 6
10 redshift
−5
7
2
10
redshift
0.5
0
10
1
2
10
10 redshift
3
10
0
0
10
1
2
10
10
3
10
redshift
Figure 2.9: The above figures show the exact, numerical evaluations of ∆αem (z)/αem (0) and ∆ sin2 θw (z)/ sin2 θw (0) vs. redshift (z) in the matter era for different negative values of ζW /ω2 . WIMPs are assumed to dominate the evolution of the couplings. We have fixed ω2 /ω1 = 1 and show plots for both νZ = 1/3 and νZ = 1. More negative values of ζem /ω2 produce larger changes in αem and sin2 θw . Negative ζW /ω2 always gives ∆αem /αem < 0 for νZ ≤ 1.
51
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
by
ζW sin2 θ¯w (α)
σ ¯W
= A,
ζem ¿ ζW
for some A = const, is indistinguishable from the αem evolution given by ζem (α)
σ ¯em
= A,
ζW ¿ ζem .
Late time observations of αem alone cannot distinguish a baryon dominated evolution from WIMP dominated one. Such observations are also unable to produce us with much information about the expected change in θw . The full WIMP dominated system is described by three parameters: ζW /ω2 , ω2 /ω1 and νZ , but the late time cosmological evolutions of θw and αem are described by only two parameters:
ζW (θ ) σ ¯Ww
and
ζW (α) . σ ¯W
Measurements of αem can only provide
us with accurate information about the latter of these two parameters. We plot the evolution of αem and θw for different values of ζW /ω2 , ω2 /ω1 and νZ in FIGs. 2.8 - 2.10. We note that, in contrast to the baryon-dominated case, ∆αem and ∆ sin2 θw have opposite signs for when ω2 /ω1 = 1 (with νZ = 1/3). In each of these figures we have plotted for both νZ = 1/3 and νZ = 1. In order to reproduce ∆αem (z = 2)/αem (0) = −0.57 × 10−5 we need: ζW (α) σW (θw )
= −5.6 × 10−6 .
Assuming that neither of the two dilatons are ghosts (ω1 , ω2 > 0) and that, as we expect, ζem is positive then the Webb data suggests that either ζW < 0 or νZ > 1. This implies that at least one of the structure functions, FW and FZ , is negative. Without a preferred model for WIMP dark matter it is not possible for us to say to what extent it is reasonable to have the structure functions taking negative values. If it is later concluded that both structure functions must be positive, and the Webb measurement of ∆αem is confirmed as a true detection of a varying-constant, then we must either exclude the KM-models or the dilaton evolution must be sourced, in the main part, by form of matter more exotic than that considered here (e.g. superconducting cosmic strings would naturally have a ζem < 0).
2.6.6
The Full System
Whilst, in general, neither ζem nor ζW vanish, without any prior constraints on their values, the vast majority of parameter space has of the ζs much larger than the other. In these
52
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
−5
−5
x 10
0
−0.5
−0.5
−1
−1 ∆ αem(z) / αem(0)
∆ αem(z) / αem(0)
0
−1.5 −2 −2.5
−1.5 −2 −2.5
ω2/ω1 = 0.01, νZ = 0.33333
−3
ω2/ω1 = 0.01, νZ = 1
−3
ω2/ω1 = 0.1, νZ = 0.33333 ω2/ω1 = 1, νZ = 0.33333
−3.5
x 10
ω2/ω1 = 0.1, νZ = 1 ω2/ω1 = 1, νZ = 1
−3.5
ω2/ω1 = 10, νZ = 0.33333 −4
0
1
10
ω2/ω1 = 10, νZ = 1 2
10
10
−4
3
0
10
1
10
12 2
1
2
10
ω /ω = 1, ν = 0.33333 2
1
8
1
8
Z
6 4 2
Z
2
1
Z
ω /ω = 10, ν = 1 2
1
Z
6 4 2 0
0
−2
−2 −4
1
ω2/ω1 = 0.1, νZ = 1 ω /ω = 1, ν = 1
Z
ω /ω = 10, ν = 0.33333 2
x 10
ω /ω = 0.01, ν = 1
Z
ω2/ω1 = 0.1, νZ = 0.33333
∆ sin2 θw(z) / sin2 θw(0)
∆ sin2 θw(z) / sin2 θw(0)
10
3
10
−5
x 10
ω /ω = 0.01, ν = 0.33333 12
10 redshift
−5
14
2
10
redshift
0
10
1
2
10
10 redshift
3
10
−4
0
10
1
2
10
10
3
10
redshift
Figure 2.10: The above figures show the exact, numerical evaluations of ∆αem (z)/αem (0) and ∆ sin2 θw (z)/ sin2 θw (0) vs. redshift (z) in the matter era for different values of ω2 /ω1 . (α)
WIMPs are assumed to dominate the evolution of the couplings. We have fixed ζW /σW (θw ) = −5.6 × 10−6 at z = 0 and plot for both νZ = 1/3 and νZ = 1. The evolution of αem is then almost independent of ω2 /ω1 . The sense of magnitude of ∆ sin2 θw / sin2 θw is, however, very sensitive to different values of ω2 /ω1 . We cannot, therefore, predict the evolution of θw using only observations of αem .
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
53
cases, one or other of above scenarios provides a good approximation. We may, however, find that both ζem and ζW are of the same order. If the two ζs are of the same sign, then the qualitative behaviour of αem and θw is similar to that observed in both the baryon-dominated and WIMP-dominated scenarios. If ζem and ζW have opposite signs then it is possible that some cancelation effects may occur, which could result in a more complicated evolution for the coupling constants. However, as we discuss in section 2.6.8, WEP violation constraints, when combined with the observations of QSO spectra by Webb et al. [1, 2, 3], show a strong preference for WIMP dominated scenario. In general, ∆αem (z)/αem (0) is, at late times (z < 100), given by: Ã ! ∆αem ζem ζW sin2 θw ∼ 4 ln(1 + z), + (α) (α) αem (0) σem (θw ) σ (θw ) W
when the effects of our universe’s recent period accelerated expansion on the evolution of the coupling constants are ignored.
2.6.7
The Dark Energy Dominated Era
It is now known that the universe is today undergoing a period of accelerated expansion and it is widely believed that this acceleration is due to new form of matter called dark energy. Without an uncontested model for dark energy at our disposal, it is not possible to say if, and how, the dilaton fields interact with this dark energy. Most current surveys seem to prefer dark energy with an equation of state parameter that is very close to that of a cosmological constant, Λ. For this reason, we model the dark energy as a cosmological constant and assume no interaction between it and the dilatons. The dilaton evolution is then described by eqns. (2.38) and (2.39) and the Friedman equation is: ´ 8πG ³ ε¯matter + ε H2 = de 3 a3 We use Ωm to quantify the fraction of the energy density of the universe that is today located in pressureless matter. Ωde is the fraction of the energy in the universe that comes from the dark energy. WMAP gives Ωm = 0.27 ± 0.04 and Ωde = 0.73 ± 0.04, [92]. At late times, the a−3 factors on the right hand sides of the dilaton evolution equations cause the source terms to vanish exponentially quickly and, as in BSBM theory [36, 40], we tend towards a scenario where the dilatons behave as free scalar fields in an exponentially expanding universe (H = const → a ∝ eHt ). As t → ∞, we find asymptotic solutions:
54
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
ϕ ∼ ϕ¯ + C3 e−3Ht , χ ∼ χ ¯ + C4 e−3Ht . −5
0
−5
x 10
0
x 10
−0.2 −0.2 ∆ sin θw(z) / sin θw(0)
−0.4
2
−0.6 −0.8 −1
−0.6
2
∆ αem(z) / αem(0)
−0.4
−1.2
−0.8
−1.4 −1.6 −1.8 0
−1
Ωm = 0.27 Ωm = 1 2
Ωm = 0.27 Ωm = 1
4
6
8
10
−1.2 0
2
redshift
4
6
8
10
redshift
Figure 2.11: The above plot shows the way in which a cosmological constant effects the evolution of αem and sin2 θw . We set ω2 /ω1 = 1, ζem /σ2 = −2.86 and ζW = 0. The solid and dotted lines respectively show the evolution of the couplings in realistic Ωm = 0.27 universe, and a matter dominated Ωm = 1 universe.
In a dark energy dominated universe, the frictional effect of the expansion of the universe causes the dilaton fields to asymptote towards constant values. As a result, we predict that the coupling constants are varying more slowly today than they did before dark energy began to dominate the expansion of the universe. As in BSBM [36, 40], this feature allows for the lack of geochemical and meteorite evidence for varying αem to be squared with the value of ∆αem (z = 2)/αem ∼ O(10−6 ) suggested by QSO observations. In FIG. 2.11, we compare the evolutions of ∆αem /αem and ∆ sin2 θw /∆ sin2 θw in a realistic model of the universe (Ωm = 0.27, Ωde = 0.73) with what would occur if there was no dark energy (Ωm = 1, Ωde = 0.73). If these plots we have taken ζW = 0 and ω2 /ω1 = 1, ζem /σ2 = −2.86 × 10−5 . Note that the solution with a cosmological constant (Ωm = 0.27) is shallower than the one without one and it is decelerating today (z = 0). The Ωm = 1, Λ = 0 solution, however, continues to accelerate right up to the present day. A similar qualitative behaviour is seen for all solution in which |ζem /σi , ζW /σi | ¿ 1.
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
55
In a realistic universe (Ωm = 0.27), we find that, in order for the KM-II models to reproduce the change in αem seen by Webb et al. [3, 2, 1] i.e. ∆αem /αem (0) = −0.57 ± 0.10 × 10−5 , we require:
Ã
ζem (α)
σem (θw )
+
ζW sin2 θw
!
(α)
σW (θw )
= −1.8 ± 0.3 × 10−6 .
Similarly, in KM-I we need: µ
ζem ζW + σ σ
¶ = −1.8 ± 0.3 × 10−6 .
If the WIMP contribution is negligible, ζW ¿ ζem , and ω1 = ω2 then, in the KM-II model, we must have: ζbaryon = −1.7 ± 0.5 × 10−5 , 4πGω2 where we have used ζem /ζbaryon = 0.17±0.02 as given by WMAP [92]. In the baryon-dominated KM-I model, as in BSBM [36], we require: ζbaryon = −1.1 ± 0.3 × 10−5 . 4πGω If the WIMP contribution if dominant, ζW À ζem , then the KM-II model with ω1 = ω2 and νZ = 1/3 predicts that: ζwimp = −8.1 ± 2.1 × 10−6 , 4πGω2 whereas the KM-I model predicts: ζwimp = −2.2 ± 0.6 × 10−6 . 4πGω In all cases, ∆αem (z = 2)/αem = −0.57 ± 0.10 × 10−5 implies that, cosmologically, α˙ em /αem = 1.1 ± 0.2 × 10−16 yr−1 today. In Chapter 3, we prove that, for a large class of theories of which the KM-models are particular examples, the local and cosmological values of αem ˙ αem differ by less than 1%. In the near future it should be possible to measure the local value of α˙ em /αem down to about the 10−16 yr−1 level and proposals have recently been made which would allow experiments to measure α˙ em /αem to a precision of 10−23 yr−1 , [51]. If this can be achieved then, at least in the context of the KM models, local experiments will soon be able to detect any cosmological variation of αem at the level reported by Webb et al. [1, 2, 3] i.e. ∆αem /αem ∼ O(10−6 ) over the law 10-12 Gyrs.
56
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
2.6.8
WEP Violations
Although cosmological observations of ∆αem alone cannot be used to distinguish between a WIMP dominated evolution of the coupling constants and baryon dominated one, by combining these observations with the experimental bounds on any violation of the weak equivalence principle (WEP) it is possible to rule out the latter scenario. Consider the motion of a body comprised of baryonic matter. As a result of quantum corrections, its mass, mt , depends on the value of αem ∝ e2π , [42]: mt (φ) = mt (0)(1 + ζt (e2φ − 1)). The parameter ζt generally depends on the composition of the body. If 100fp % of the mass of the test body lies in protons and 100fn % in neutrons then we estimate: ζt = ζp fp + ζn fn ,
(2.60)
where ζp ≈ 6.7 × 10−4 and ζn ≈ −1.4 × 10−4 represent the fractional electromagnetic contribution to the mass of the proton and neutron respectively. In the non-relativistic limit, the body’s position, ~x(t), satisfies: d2 ~x ~ N − 2ζt ∇φ, ~ = −∇Φ dt2 where ΦN is the Newtonian gravitational potential. The composition dependence of ζt results in WEP violations. To estimate the magnitude of these violations, we consider the form that ~ takes near a body such as the Sun. ∇φ In KM-II theory, the dilaton field equations, eqns. (2.29) and (2.30), imply that outside an isolated spherical body composed of baryonic matter, with mass MS (φ) = MS (0)(1 + ζS (e2φ − 1)), we have: 2 sin2 θw ζE GMS ζS ~ ~ ∇ϕ ≈ − = ∇ΦN , 2 σ1 r σ1 ζS ~ 2 cos2 θw ζE GMS ~ = ∇ΦN . ∇χ ≈ − 2 σ2 r σ2 From the definition of αem in terms of gw and gy we find: ~ = sin2 θw ∇ϕ ~ + cos2 θw ∇χ ~ = ∇φ
2ζS ~ ∇ΦN . (α) σem (θw )
As a result, a test mass will move towards this isolated spherical body with acceleration: Ã ! 2GMS 2ζS ζt a= 1 + (α) . r2 σem (θw )
57
2.6. HOMOGENEOUS AND ISOTROPIC COSMOLOGY
(α)
The same formula, with σem → σ = 4πGω, applies in the KM-I models. For two test bodies with ζt = ζE and ζM and accelerations a = aE and a = aS respectively, the magnitude of the WEP violations are quantified by the E otvos parameter, η: η=
2|aE − aM | |ζE − ζM | . = 4|ζS | (α) |aE + aM | σem
The best current experimental bounds on WEP violation come from measuring the differential acceleration towards the Sun of two test masses which have been manufactured to have a similar composition to the Earth’s core and the Moon’s mantle respectively. Using a modified torsion balance the E ot-Wash group, [103, 104], found that: η = +0.1 ± 2.7 ± 1.7 × 10−13 . We estimate ζM , ζE and ζS using eqn. (2.60), and find that the KM models predict: η ≈ 0.94 × 10−4
ζbaryon (α)
σem
,
for this experiment. If the baryonic matter is the dominant driving force behind the evolution of αem , and that evolution is well described by the KM-models, then value of ∆αem /αem reported by Webb et al., [1], requires: ζbaryon (α) σem (θw )
= −1.1 ± 0.3 × 10−5 .
It follows that, in both the KM-I and KM-II models, the baryon-dominated scenario predicts η ≈ 1.0 ± 0.3 × 10−9 , which is three to four orders of magnitude larger than the upper bound established by the E ot-Wash group [103, 104] and so very clearly ruled out. By comparison, if WIMP dark matter is responsible for the evolution of the fine structure constant then QSO observations of ∆αem /αem imply: ζwimp sin2 θw εwimp = −1.8 ± 0.3 × 10−6 . (α) ε m σ (θw ) W
We define fwimp to be the fraction of the energy density of dark matter that comes from weakly interacting massive particles. In section 2.4.5, we argued for WIMP particles with masses & MW we expect |ζwimp | ∼ 10−2 , and consequently |ζbaryon /ζwimp | ∼ 10−2 . We find
58
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
that, in the KM-II model, a WIMP dominated αem evolution that is consistent with the quasar observations reported in refs. [1, 2, 3] predicts: −12
η ≈ (4.8 ± 1.2 × 10
)f
¯ ¯· ¸ ¯ ζbaryon ¯ 0.083ω + 0.917ω 2 1 ¯ ¯ 10−2 ζwimp ¯ 0.848(1 − νZ )ω2 + 0.652νZ (ω1 + ω2 ) .
−1 ¯
where the quantity in [..] is unity when νZ = 1/3 and ω1 = ω2 and we expect the quantity in |..| to be O(1). The KM-I model, on the other hand, predicts −12
η ≈ (2.0 ± 0.6 × 10
¯ ¯ ¯ ζbaryon ¯ ¯ ¯, ) ¯ −2 10 ζwimp ¯
for a WIMP dominated evolution of αem . It should be noted that estimated error in the above predictions does not any inaccuracies that are introduced by the use of eqn. 2.60 to find ζE , ζM , ζS . Given this, and the uncertainties in the values of ζwimp , ω2 /ω1 and νZ , the WIMPdominated KM-models cannot be conclusively ruled out at this stage. We do note, however, the KM-I model predicts slightly smaller WEP violations, for a given ζwimp , than the KM-II model with ω2 /ω1 = 1 and νZ = 1/3. The MICROSCOPE satellite [99], which is due to be launched as early as next year, promises to be able to detect an η as small as 10−15 . If this, and other future tests of WEP, do not detect any violations then it would become very difficult to reconcile the KM models with ∆αem /αem ∼ O(10−6 ) over the late 10-12 billion years.
2.6.9
Summary
In this section, we have studied the cosmological evolution of the ‘constants’, αem and θw as predicted by the KM-I and KM-II models in a homogeneous and isotropic background. The rˆole of inhomogeneities is dealt with, in a more general setting, in section 3. The overall picture that has emerged is as follows: • In the radiation era, the velocities of dilaton fields and related couplings are damped by the expansion of the universe and the dilaton fields come to rest. By the end of the radiation era the coupling ‘constants’ are almost stationary. • During the matter era, the ‘constants’ evolve very slowly and in monotonic fashion. Recent observations of quasar absorption systems suggest that either at one of ζbaryon and ζwimp is negative, or that at least one of the dilatons is a ghost.
2.7. NEW PHENOMENA DUE TO A VARYING-θW
59
• As dark energy comes to dominate the expansion of the universe, the frictional effect of the expansion brings the dilaton fields almost to rest in about one third of the Hubble time. Although cosmological measurements of αem alone cannot distinguish between the two KMmodels, or tell us if the evolution of the couplings has been driven predominantly by the baryon content of the universe or via a WIMP component of dark matter, we saw that, by combining these observations with WEP violation bounds, it is possible to baryon-dominated scenario. We also found that, with reasonable values for the ζ, the WIMP-dominated scenario was, itself, only marginally allowed. This combination of WEP violation data and QSO spectra also showed a slight preference for the KM-I model over the KM-II model with natural values of ω2 /ω1 and νZ (i.e. 1 and a 1/3 respectively). Near future searches for WEP violations will have the precision to either detect or rule out the WIMP-dominated scenario for both KMmodels, and, in the context of the KM-models, the next generation of αem ˙ /αem measurements should detect a variation if ∆αem /αem ∼ O(10−6 ) over the last 10-12 Gyrs.
2.7
New phenomena due to a varying-θw
We saw above that, using observations of the cosmological time evolution of αem alone, it is difficult to gain much information about any variation in θw . It is interesting, therefore, to search for new predictions or phenomena that are specific to theories in which θw , as well as αem , varies. In this section we show that, with the exception of the cases where the ratio gy /gw := tan θw is a true constant, charge non-conservation, dequantisation and a charged neutrino are generic features of almost all varying-coupling electroweak theories. These features also ensure that all matter species, which couple to the Z-boson, gain an electric dipole moment (EDM). These phenomena are not present in varying-αem theories, such as BSBM, that ignore electroweak effects. In this section, we are concerned with a general electroweak varying coupling with the properties described in section 2.2.
2.7.1
Charge non-conservation and simple varying-αem theories
It follows from Noether’s principle that any gauge-invariant varying-αem theory contains a conserved current. The class of theories described in section 2.2 is symmetric under modified
60
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
U (1)em gauge transformations Aµ → Aµ + e−φ ∇µ Λ, where α = e2φ . The rest of the gauge symmetry must be broken by a Higgs sector if it is to describe our universe. This conserved matter current is however not the one conjugate to Aµ , i.e. J µ (x) := δS δAµ (x) (with Smatter being the ¡ ¢ matter action). Noether’s principle dictates that ∇µ eφ J µ = 0 and so it is j µ := eφ J µ , rather
than J µ , that is conserved. The question of the two currents, J µ and j µ , should be considered ‘physical’ rests in the form of the dilaton-to-matter coupling. In our above evaluation of the dilaton-to-matter coupling, see section 2.4, we argued that Aµ was the ‘physical’ photon field and, as result, J µ plays the rˆole of the ‘physical’ current. In ref. [48], Bekenstein presents an alterative argument for the way in which the dilaton couples to matter. When this viewpoint is taken, it is actually j µ that plays the rˆole of the physical current. If j µ should be viewed as the physical current then charge is clearly conserved. Bekenstein’s evaluation relies on the idea that self-interactions of the dilaton field become non-negligible near nucleons. These self-interactions emerge because, in the theories that 2 (φ)/ω where α Bekenstein analysed, the dilaton-to-matter coupling is αem em ∝ exp 2φ and ω,
as above, has units of mass-squared. Self-interactions are negligible whenever it is acceptable to approximate αem (φ) by αem (φb ), where φb is the background value of φ. For this approximation to be valid we need |∆φ = φ − φb | ¿ 1 so that exp(φ) ∼ exp(φb ) + O(∆φ). If we make this approximation and consider how φ behaves near and inside a spherically particle, mass M and radius R, we see that the maximum value of |∆φ| occurs as the centre of the particle and that |∆φ|max ∼ O(αem (φb )GM/σR) where σ = 4πGω. For the approximation to be valid, we must have αem (φb )GM/σR ¿ 1. In Bekenstein’s analysis, the nucleons were taken to be classical point particles i.e. R = 0 and so |∆φ|max = ∞; in this model of the nucleons, self-interactions of φ are clearly important. The reason that we disagree with results presented in ref. [48] is simply that, in this context, we not do believe that the point particle model for nucleons to be appropriate. Protons and neutrons are not classical particles but have an inherently quantum nature. Quantum uncertainty ensures that energy of the nucleons is not concentrated at a point but is instead spread out of a roughly spherical region with radius R = λ where λ = 1/M is the Compton wavelength of the nucleon. When these quantum effects are taken into account, we see that |∆φ|max ∼ O(αem (φb )GM 2 /σ). For protons and neutrons αem (φb )GM 2 ∼ 10−40 and WEP violation bounds imply that, in theories such as the KM-models and the theory considered by
2.7. NEW PHENOMENA DUE TO A VARYING-θW
61
Bekenstein, σ À 1 and so |∆φ|max ¿ 10−40 ¿ 1. It follows that non-linear effects do not play any great rˆole in this case. Despite our objections to the results presented in ref. [48] we do, below, consider how our analysis would be altered if our objections are later shown to be invalid. We refer to theories which rely on our evaluation of the matter coupling as being ‘J µ − physical0 and those that use the prescription given in ref. [48] as ‘j µ − physical0 . If J µ is naturally interpreted as the physical current then there is a form of non-conservation of charge. The total charge, Q, in a volume V is: Z
Z φ(t,x)
Q: =
j0 e
= e
V φ(t,0)
dV = V
³ ´ ~ φ(t,0) + ... j0 eφ(t,0) + x · ∇e
(2.61)
~ φ(t,0) · d + ... + ∇ ~ i∇ ~ j ...∇ ~ k eφ(t,0) d(n) + ... q + ∇e ij..k
(n)
where dij..k is the nth electric multipole moment w.r.t. to the conserved current j µ . A collection of neutral particles cannot develop an electric charge in such theories. Similarly an initially electrically neutral, perfect fluid (containing a mixture of negatively and positively charged components) cannot become charged since all multipole moments vanish for such a fluid. This implies that cosmologically, at least, charge is conserved to a very good approximation. The universe cannot develop a non-negligible overall charge in this way. Particle level interactions also conserve charge at each vertex as a result of the conservation of j µ . We now show that when θw and αem vary then a stronger form of non-conservation of charge arises in ‘J µ -physical’ theories and that, even in ‘j µ -physical’ theories, fermions develop an EDM.
2.7.2
A new interaction from varying-θw
A Higgs sector must break the SU (2)L × U (1)Y symmetry down to the U (1) of electromagnetism. The physically propagating fields, the photon, Aµ , and the Z-boson, Z µ , are given in terms of Y µ and W3µ in the usual way. Their field strengths are: FAµν FZµν
³ ´ = 2e−1 ∂ [µ eAν] ,
³ ´ = 2(gw cos θw )−1 ∂ [µ gw cos θw Z ν] ,
62
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
where e = gw sin θw := eφ is the fundamental electric charge. The kinetic terms for W 3 and Y now become: LZ,A := −
1 2 4 FW 3
− 14 FY2 = − 14 FA2 − 14 FZ2
+ 2FAµν ∂µ θw Zν − 2 tan θw FZµν ∂µ θw Zν h i − sin2 θ 2cos2 θ (∂θw )2 Z 2 − ∂µ θw ∂ν θw Z µ Z ν w
w
The first two terms are the standard kinetic terms for the photon and Z-boson. The term in square brackets and the one before it provide only minor corrections to the Z-boson propagator. The third term, 2FAµν ∂µ θw Zν , is the one that interests us. It produces a coupling between the photon and the Z boson that was not previously present. It means that all particle species with a weak neutral charge induce an electric current density.
2.7.3
Induced currents
µ µ At energies well below the Z-boson mass, MZ ∼ 91 GeV, Z µ ≈ JN /MZ2 where JN is the weak
neutral current density. The new interaction therefore produces an effective electromagnetic current density, Jˆµ , given by: µ Jˆµ ≡ eφˆj µ := 2eφ ∇ν
ν − ∇ν θ J µ ∇µ θw JN w N eφ MZ2
¶ .
The nature of the physical electric potential depends on whether J µ or j µ is the physical current density. When the magnetic field vanishes, B = 0, the physical potential is defined by the condition that the electric field E should vanish if and only if the potential is constant. When B = 0, the modified Maxwell equations are: ~ · (e−φ E) = ε := J 0 eφ ∇ ³ ´ ~ × eφ E ∇ = 0 Provided that the local charge density has only a very slight perturbing effect on the gradients ³ ´2 ~ of φ then we can define $ = ∇φ − ∇2 φ ≈ const and the electric field is given by E := ¡ ¢ ~ eφ Ψ where: e−φ ∇ 1 Ψ(x) = − 4π
Ã
Z 3 0
d x Re
√
0
e− $|x−x | |x − x0 |
! ε(x0 )
63
2.7. NEW PHENOMENA DUE TO A VARYING-θW
This case corresponds to J µ being the physical current density. Here, Φ := eφ Ψ(x) is deemed to be the physical potential. ~ ×E = 0 This is not the only possibility. If the φ equation of motion is such that ∇φ ~ with: whenever B = 0, then E = eφ ∇Υ Υ(x) = −
1 4π
Z d3 x0
e−φ ε(x0 ) . |x − x0 |
It is clear that here e−φ ε(x) = j 0 is the physical charge density; Υ(x) is identified as the ~ × E = 0 might seem quite contrived. It might, physical potential. The requirement that ∇φ however, arise as an integrability condition for the φ-equation of motion; for example as in ref. [48]. This condition defines how the mass of any charged particle should depend on αem . All charged particles must develop an αem -dependent mass through photon and dilaton loop corrections. Chiral fermions are protected against becoming massive in this way, therefore all viable ‘j µ -physical’ theories cannot contain charged chiral fermions. This statement applies equally to all charges associated with varying-gauge couplings. Weakly charged neutrinos must therefore be massive in ‘j µ -physical’ varying-αem theories. Consider a point particle with weak neutral charge QN at x = 0. In a ‘J µ -physical’ theory the new interaction term described above makes the following contribution to the physical electric potential Φ(x): Φ(x) ≈
Ã
~ w · ∇φ ~ QN ∇θ − MZ2
!
~ w xeφ(x) eφ(x) QN ∇θ + · , 2πr 2πr3 MZ2
where r = |x|. The first term in Φ represents a point electric charge qef f = ~
N ∇θw second term is the potential of an electric-dipole moment def f = − 2QM . 2
~ w ·∇φ ~ 2QN ∇θ . The 2 MZ In ‘J µ -physical’
Z
theories all weak neutrally-charged particles will become effectively electrically charged when θw varies. Such particles will also develop an effective EDM. The form of qef f means that it will not be quantised in units of e. There is effective dequantisation of electric charge in these theories. In ‘j µ -physical’ theories we do not see an induced charge effect. Weak neutrallycharged particles still develop an EDM however. The electric potential, Υ(x), is: Υ(x) =
~ w x QN ∇θ · . 2 φ(0) e MZ 2πr3
The induced EDM is d0ef f := −
~ w 2QN ∇θ 2
eφ(0)MZ
.
Order of magnitude estimates for qef f , def f and d0ef f are discussed below.
(2.62)
64
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
2.7.4
Discussion
µ The weak neutral current, JN , is not conserved. In general, massive bodies, such as our
Sun and the universe as a whole, have a large net weak-neutral charge density compared to their net electric charge density. In ‘J µ -physical’ electroweak varying-αem theories, particles develop an electric charge proportional to their weak-neutral charge. It is possible then to have non-conservation of electric charge. The universe can develops a non-negligible overall charge in this way. The charges that are induced are, in general, not quantised in units of the fundamental charge e. Spatial variations in θw also induce EDMs on the fundamental ~ w . In a region where ∇θ ~ w ≈ const, fermion species. The EDMs all point in the direction of ∇θ therefore, these EDMs will line up and produce an overall macroscopic EDM. Numerically the sizes of the effective charges and EDMs are: qef f
¡ ¢ ∼ 10−31 e k∇θw k · k∇ ln αk cm2
(2.63)
def f
≈ d0ef f ∼ 10−31 e−cm (k∇θw k cm)
(2.64)
~ ln α ≈ ζα ∇φ ~ N , ∇θ ~ w ≈ ζθ ∇φ ~ N , where φN = GM/r2 In many varying-αem theories one finds ∇ is the Newtonian gravitational potential. We expect ζα , ζθ ¿ 1. Near the surface of Earth such theories would induce qef f ∼ ζα ζθ 10−66 e, def f ≈ d0ef f ∼ ζθ 10−48 e-cm. Any physically viable, varying-αem and varying-θw theory must satisfy all relevant bounds on the neutrino and neutron charges and on the EDMs of the fundamental particles. The most restrictive upper bound on the electron-neutrino charge, qν , has been given by Caprini and Ferreira in ref. [128]. They considered the isotropy of the Cosmic Microwave Background (CMB) and found: qν < 4 × 10−35 e. In the same way they also bounded the charge difference between a proton and an electron: qe−p < 10−26 e. An upper bound on the neutron charge, qn < −0.4 ± 1.1 × 10−21 , is given by Baumann et al. in [126]. The Particle Data Group, see ref. [127], gives the upper bound on the electron EDM as de < 6.9 ± 7.4 × 10−28 e-cm. Experiments are planned that would be able to detect any electron EDM at the 10−31 e-cm level, [129]. Ref. [127] also gives upper bounds on the EDMs of the proton, dp < 0.54 × 10−23 e-cm, and the neutron, dn < 0.63 × 10−25 e-cm. It is clear that all current bounds are easily satisfied by most physically viable, varying-αem theories. It is normally the case that intrinsic EDMs on Dirac fermions are indicators of CP-violation. In electroweak varying-αem theories we have seen that it possible to induce such EDMs with-
2.8. SUMMARY
65
out adding any explicit CP-violating term to the Lagrangian and that varying-αem theories generically result in some manner of charge non-conservation and effective dequantisation of charge without breaking the U (1)em symmetry. These effects, if detectable in the context of a given theory, could provide us with a new way of probing the rate of spatial variation in θw and αem .
2.8
Summary
In this chapter, we have considered the new types of behaviour and phenomena that emerge when varying-αem theories are generalised to include electroweak effects. We have also analysed the way in which the dilaton fields in these theories couple to ordinary matter. Any physically viable theory of varying-αem must include these electroweak effects, as it must reduce to the GSW theory of electroweak interactions in the limit of no variation. The first consequence of this generalisation is that, in general, if αem varies then so does θw . Although there do exist theories, such as the KM-I model, where, because the couplings gw and gy depend on the dilaton field or fields in the same manner classically, θw is a true constant at tree level, when quantum corrections are included, even if there is only one dilaton field, gw and gy will generally depend on the dilaton field(s)in different ways and so θw will vary. Indeed, since gw and gy run in different ways even if θw is a true constant at one energy scale it will generally depend on the dilaton(s) at all other energy scales. In our study of the Kimberly-Magueijo models we have not included the effects of running couplings. We have instead assumed that the lagrangians of the KM-I and KM-II models provide an effective description of the dilatons at low energies. As much as varying-θw should be considered a generic prediction of varying-αem theories, we have seen that, using the KM models as examples, measurements of any cosmological change in αem alone provide us with very little information about the way in which θw has changed with time. That said, we should, in most cases, expect the magnitude of ∆ sin2 θw / sin2 θw to be similar to that of ∆αem /αem , however an increase in αem does not necessarily imply an increase in θw . Detecting a cosmological change in θw is very difficult goal to achieve since there are very few low-energy observables that actually depend on the mixing-angle. However, if θw can vary then, as we showed in section 2.7, a degree of non-conservation and dequantisation of electric charge exists. Additionally, varying-θw results in fundamental particle species, which
66
CHAPTER 2. VARYING-α IN THE CONTEXT OF ELECTROWEAK THEORY
couple to the Z-boson, developing an effective intrinsic electric dipole moment (EDM). Intrinsic EDMs are usually taken to be a sign of CP violation and a number of experimental searches for such EDMs already exist. Ultimately, any component of an intrinsic EDM of a given particle due to varying-θw could be distinguished from the component coming from CP violation by ~ w whereas the fact that the former would depend on the particle’s surroundings through ∇θ the latter would not. EDM searches might well prove to be our best probe of any spacetime variation in θw . The major new feature present varying-αem theories that generalise GSW theory is that the cosmological evolution of the coupling constants is driven, at least in part, by any WIMP dark matter in the universe. Since WIMP matter is not electrically charged it does not contribute to the evolution of αem in theories, such as BSBM, that ignore electroweak effects. Although late time observations of ∆αem /αem alone tell us very little about the strength of the WIMP contribution to the evolution of the couplings, WEP violation constraints imply that, at least in the context of the simplest models, it must be dominant if ∆αem /αem ∼ O(10−6 ) over the last 10-12 Gyrs. The studies of QSO absorption spectra made by Webb et al. [1, 2, 3] suggest that ∆αem /αem < 0, which implies that ζwimp < 0. Until we have a definitive and uncontestable particle level explanation for dark matter it is not possible to say whether or not a negative value for ζwimp is acceptable. Some of this ambiguity may be lifted if supersymmetry is detected at the LHC and a preferred WIMP candidate can be singled out. If, as we believe to be reasonable, ζwimp ∼ O(10−2 ) then the KM-models predict violations of the weak equivalence principle at the η = 10−13 − 10−12 level; this is only just allowed by the current experimental constraints [102, 103, 104, 78]. Future space-based tests of WEP, e.g. STEP [97], MICROSCOPE [99], SEE [96], will have the precision to unambiguously detect η ∼ 10−13 . In the near future, WEP violation searches will therefore be able to confirm or rule out at least the simplest models of varying-αem . It should also be noted, if ∆αem (z = 2)/αem ∼ O(10−6 ) then |ζwimp | ¿ 10−2 would produce unacceptably large violations of WEP. For acceptably sized WEP violations, we must, as a result, require |ζwimp | & 10−2 which implies mwimp & MW , MZ ; we must also require that the fraction of dark matter that it comprised of WIMPs is O(1). The WIMP-dominated, KM-models predict WEP violating forces between particles of dark matter that are about (ζwimp /ζbaryon )2 ∼ O(104 ) larger than those between baryonic matter. This implies an E otvos parameter for WIMP matter of ηwimp ∼ O(10−9 − 10−8 ). Dark matter
67
2.8. SUMMARY
WEP violation constraints are, however, much weaker than those for baryonic matter. Kesden and Kamionkowski, [52, 53], used Two-Micron All-Sky Survey (2MASS) and SDSS observations of the tidal tail of the Sagittarius dwarf galaxy to find what is presently the best limit on WEP violating forces in the dark-sector: ηwimp . 0.09. This bound is very easily satisfied by KM models. Dark sector WEP violation constraints are expected to improve as further observations of the tidal tails of satellite galaxies are made and analysed. To summarise: when electroweak effects are accounted for, varying-αem theories display many new features such as varying-θw , charge non-conservation and dequantisation, induced EDMs on fundamental particles and a new interaction between the dilaton fields and WIMP dark matter. However, until experimental precision is improved, an unambiguous detection of any of these new features remains beyond our grasp. This said, if ∆αem /αem ∼ O(10−6 ) over the last 10-12 Gyrs then, at least in the context of the simplest models, the prospects for an unambiguous detection of varying-αem in the near future are very good, with both the next-generation of laboratory observations of αem ˙ /αem and WEP violation searches promising to have the required precision.
Chapter 3
Locally observable effects of varying constants and scalar fields
Thus the partial differential equation entered theoretical physics as a handmaid, but has gradually become mistress. Albert Einstein (1879-1955)
3.1
Introduction
In any study of varying constants, the data used to constrain the theories which allow variations to occur comes from a number of very different environments and scales: with densities differing by a factor of 1030 or more, and spanning some 12 billion years. In order to be able to use all of the information available we need to know how the results of local laboratory experiments, terrestrial or solar-system bounds from the Oklo natural reactor, and from isotope ratios in meteorites, are related to data coming from astronomical observations on extragalactic scales. This is the ‘Local vs. Global’ problem for varying-constants. It is an important problem and yet most commentators invariably assume that the local and cosmological observations are directly comparable [29, 24]. This is strong assumption and is almost invariably made without any proof. A priori, it is far from obvious that this assumption is even true; indeed, in many other theories, not least that for gravity itself, it is not: we do not expect to be able 69
70
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
to measure the expansion of the Universe by observing an expansion of the Earth. In this chapter, we describe the first rigorous proof of why, in almost all varying-constant theories, local experiments can and do ‘see’ any variations in ‘constants’ which occur on cosmological scales. The general way in which a constant of Nature is promoted to become a dynamical quantity that is consistent with Einstein’s conception of gravity was discussed in Chapter 1.1.5. For local observations to be directly comparable with cosmological ones we need to know the conditions under which ˙ x, t) ≈ φc (t) φ(~
(3.1)
to some specified precision, with ~x taking values in the solar system. The subscript c labels the large-scale cosmological value of the field φ. The validity of this approximate equality, the accuracy to which it holds, and the accompanying preconditions needed to support its validity are the subject of the rest of this chapter. Prior to the onset of the matter era the universe is homogeneous to a very high precision inside the horizon. Any inhomogeneities that do exist, and the evolution of φ within them, can be consistently and accurately described by linear perturbation theory and φ˙ ≈ φ˙ c holds. But the study of the evolution of ‘constants’ becomes mathematically challenging when linear theory breaks down and the inhomogeneities become non-linear. This only starts to occur during the matter era; at these epochs it is an acceptable approximation to consider the Universe to be comprised of only pressureless dust (baryonic and dark matter), density ε, and some cosmological constant, Λ. We usually expect that the scalar field φ only couples to some fraction of the total dust density; for example, in varying-αem theories it couples to the fraction that feels the electromagnetic and weak forces, and in varying me theories it couples only to the electron density [125]. We assume, as is almost always the case, that the fraction of matter to which the field couples is approximately constant during the epoch of interest. Under these simplifications eqn. 1.1 reduces to: −¤φ = B,φ (φ)κε + V,φ (φ) where κ = 8πG. For simplicity we set G = 1 for the remainder of this chapter. V (φ) is the dilaton potential, and B,φ (φ) is the effective dilaton-to-matter coupling. In what follows we assume that the varying-constant evolves according to the above conservation equation, which
71
3.1. INTRODUCTION
is certainly true of Brans-Dicke theory, BSBM and BM varying-µ theory. We further assume that the cosmological value of φ, denoted by φc , is sufficiently far away from any extrema of the matter coupling, B(φ), and that V,φ (φ) is not too large. Our conditions on B(φ) and V (φ) are summarised as follows: ¯ ¯ ¯ B,φφ (φc )(φ(~x, t) − φc (t)) ¯ ¯ ¯ ¿ 1, ¯ ¯ B,φ (φc )
|V,φ (φ(~x, t))| . Λ,
for all values of φ within the range of interest (i.e. those that can be reached from the evolution of some given initial data). The condition on the matter coupling is usually equivalent to |B,φφ (φc )| ¿ 1 and B,φ (φc ) 6= 0. The condition on the potential must hold for φ = φc to prevent the varying “constant” evolving at an unphysically fast rate cosmologically; the assumption that holds everywhere is then valid provided that V,φ (φ) is suitably flat. As a result of this final assumption our results do not apply to the chameleon field theories, [45], discussed in Chapters 4 and 5. With the major exception of theories with ‘chameleon behavior’, our model includes almost all physically viable proposals for varying-constant theories. Our results are also applicable to any scalar-field theory, not just those that describe varying-constants, provided that the scalar satisfies a conservation equation of the above form. For a general matter distribution the dilaton conservation equation is a second-order, non-linear PDE, and there is no reason to suspect that it should be easily solvable, or indeed analytically solvable at all. Even numerical calculations are generally difficult to set-up and control. Cosmologically, we assume homogeneity and isotropy which leads us to a FRW background and to a solution for φ = φc (t). Under these specifications the conservation equation for φ reduces to an ODE in time, and can be solved. The other scenario in which it reduces to an ODE is near a spherically symmetric, static body which couples to the dilaton strongly enough so that any temporal gradients of φ are negligible compared to the spatial ones. In these cases we can easily find the leading-order static mode of φ, but to find the temporal evolution of φ we need to enforce the boundary condition that it match up to its cosmological value at large distances. The central technical problem is that the local, static solution was found under the assumption that the temporal derivatives were negligible, and at infinity this is no longer the case. Indeed, to get to a region of space where we know φ ≈ φc (t) we must certainly pass through some zone where the temporal and spatial gradients of φ are of comparable magnitude. As soon as we reach this zone, the assumptions under which the local solution was derived break down. In short: we cannot consistently apply the boundary condition at infinity to the approximate local
72
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
solution since spatial infinity is far outside the range of validity of that approximation. We express this idea more formally below and see that it is associated with the fact that the local asymptotic approximation for φ is not uniformly valid. To circumvent this problem, created by the presence of multiple length scales, we use the method of matched asymptotic expansions (MAEs). The method of matched asymptotic expansions is discussed further in section 3.3 below. ¯ ¯ ¯˙ ¯ In this chapter we use MAEs to derive an expression for the quantity ¯φ(~ x, t)/φ˙ c (t) − 1¯ both near the surface of a virialised over-density of matter and in a collapsing region of spacetime. ˙ x, t) ≈ φ˙ c (t) and & O(1) whenever that condition does not This quantity must be small if φ(~ hold. As far as collapsing regions of spacetime go we limit ourselves to considered spherically symmetric spacetimes. In the virialised case we take our local observer to be near the surface of roughly spherical static mass which we refer to as a ‘star’, this mass could be taken to be a planet (e.g. the Earth), a black-hole, or even a galaxy or cluster of galaxies. In the virialised case we do not demand that the spacetime external to the ‘star’ be spherically symmetric. We are mostly interested in the (realistic) case where the surface of our ‘star’ lies far-outside its own Schwarzschild-radius. By applying our results to black-hole case, however, we are able to comment on the problem of ’gravitational memory’; that is, is the cosmological background value of φ on the horizon at the time when a black hole forms, frozen-in, or ’remembered’, or does it continue to track the background cosmological evolution? This chapter is organised as follows: in section 3.2 we review some of the previous studies into the problem of local vs. global dilaton evolution and note where our work extends and improves upon these studies. We focus on the analytical studies of Wetterich, [130], Jacobson, [131] and Mota & Barrow [107, 108]. We then introduce the method of matched asymptotic expansions (MAEs). MAEs are the crucial piece of mathematical machinery that is used in our study. Since MAEs are more commonly used in the study of fluid flow than in that of cosmology, section 3.3 provides a detailed description of the method and its application. In particular, we briefly review how the method of MAEs applies to general-relativistic problems and provide two guiding examples. The second of these examples provides a very simple toy model of the full problem of the time evolution of a dilaton field around a spherical star in an expanding Universe. This toy model is worked through in full and provides a helpful tool in understanding the full system. In section 3.4, precise details of the geometrical set-up that
3.2. PAST WORK
73
we use throughout this chapter are given. When there is a virialised central mass, or ‘star’, we consider, separately, the cases where the spacetime external to the star is given by the spherically-symmetric McVittie (1933) [144] solution and by the Szekeres-Szafron solutions [146, 148], this latter class of spacetimes generically have no symmetries. When studying collapsing inhomogeneities we model the spacetime using the spherically-symmetric limit of the Szekeres-Szafron solutions i.e. the Tolman-Bondi solutions [149]. The McVittie (1933) solution is not as realistic as either the Tolman-Bondi or full Szekeres-Szafron solutions but it is analysed here because of its historic use in the study of the effect of a expansion of the universe on the solar system. Both the McVittie and Szekeres-Szafron cosmological backgrounds are introduced in section 3.4. In section 3.5, we apply the method of MAEs to spherically symmetric spacetimes with a ˙ x, t) ≈ φ(~ ˙ x, t) virialised central mass and construct a necessary and sufficient condition for φ(~ near the surface of a virialised over-density of matter or ‘star’. We also derive the conditions that must hold for our analysis to be valid. We find that for realistic choices of the initial matter density the method of MAEs and our analysis do indeed work. In section 3.6, we extend the results of section 3.5 to include the non-spherically symmetric Szekeres-Szafron backgrounds and in section 3.7 we consider spacetime models where there is a collapsing inhomogeneity but no central virialised region. Finally in section 3.8 we discuss the potential observable ˙ x, t) ≈ φ˙ c (t) to hold here on consequences of our results and show that we do indeed expect φ(~ Earth for a wide-class of physically-viable varying constant theories.
3.2
Past Work
Several authors have, in the past, claimed to have shown that condition (3.1) holds close to the surface at r = Rs of a spherically-symmetric, massive body embedded, in some particular way, into an expanding universe. The results derived are usually only valid for a particular choice of B(φ) and V (φ) and under some, usually restrictive, assumptions about the background distribution of matter. In this section we review the pioneering analyses carried out by Wetterich in [130], Jacobson in [131] and Mota & Barrow in [107, 108] and describe how our study goes further.
74
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
Table 3.1: Density distribution in Wetterich’s model
3.2.1
Region
Range
²dust
Description
a
r < Rs
² = ²E
local planet
b
Rs < r < rc
²=0
intermediate space
c
r > rc
² = ²c (t)
Hubble flow
Review of Wetterich’s analysis
In ref. [130], Wetterich claimed that condition (3.1) would hold for any potential, V (w) (φ), with the properties V,φ (φc ) À B,φ (φc )κ²c . The demonstration was confined to a given background and particular potential V (w) (φ) ∝ exp(−λφ). It was argued that similar behaviour should be found whenever the potential dominates the cosmic evolution of the dilaton field i.e. V,φ (φc ) À B,φ (φc )κ²c . In what follows we show that condition φ˙ ≈ φ˙ c also holds in many situations where the potential does not dominate the cosmic evolution of the dilaton field. It is our belief, however, that the reasoning given in ref. [130] is incomplete and does not show that condition (3.1) holds even under the restrictive conditions specified there. We briefly outline the arguments given in ref. [130] below, and show where we believe they fail. Wetterich considered a universe filled with pressureless dust and a cosmological constant. Under these assumptions we have T = ²dust , the dust density, and: (w)
¤φ = −B,φ (φ)κ²dust − V,φ (φ) , where ² is the local matter density and it was assumed that the potential is V (w) (φ) = ωe−φ , ³ ´ 2 . The function B (φ) represents the coupling of the dilaton field to where ω ∼ O Mpl ,φ the dust. Experimental bounds on the largest allowed violations of the Weak Equivalence Principle (WEP) that would be created by a dilaton field that couples to matter suggest that |B,φ (φ)| < 10−4 . Wetterich considered a particularly simple example of a spherically symmetric massive body superimposed onto the cosmological background; table 3.1 shows the local energy density budget in this model. For r À rc , φ takes it cosmological value φc (t), which satisfies: φ¨c + 3Hφc = ωe−φ − B,φ (φc )κ²c .
75
3.2. PAST WORK
Wetterich only considered the case where ωe−φc À B,φ (φc )κ²c . Assuming that the scale factor a ∝ tn , we find: φc (t) = φ0 + 2 ln(t/t0 ), ³ ´ 2 where t = t0 today, and φ(t0 ) = φ0 ≈ 140 + ln ω/Mpl is the present value of the scalar field. For r ≈ Rs , dilaton field in ref. [130] is written as: φ = φc (t) + φl (r, t) + φe (r, t), where φl (r, t) was defined as the ‘local’, quasi-static field configuration satisfying: ∇2 φl − µ2 φl = B,φ (φc ) κ²E θ (Rs − r) , (w)
with ∇2 the 3-D Laplacian and θ (Rs − r) the Heaviside function; µ2 = V,φφ (φc ) and µ2 Rs2 ¿ 1. Near r = Rs it was found that φ˙ l /φ˙ c ∼ B,φ (φc )2M/Rs ¿ 1. If φ˙ e (r, t)/φ˙ c (t) ¿ 1 for r ∼ Rs then the dilaton field satisfies condition (3.1). If |¤φe /¤φc | ¿ 1 locally then, as stated in ref. [130] we have the required result. Wetterich argues that this is indeed the case in his chapter. However, this does not appear to be the case. Assuming 2m/Rs ¿ 1, from r = Rs spacetime is approximately Minkowski and so: ¤φe = φ¨e − ∇2 φe ≈ −µ2 φe − φ¨l + 3H φ˙ c + B,φ (φc )²c
(3.2)
−B,φφ (φc )²E θ(Rs − r) (φl + φe ) , (w) ¤φc = φ¨c = −3H φ˙ c − V,φ (φc ).
(3.3)
Now, φ¨l ¿ φ¨c and µ2 Rs2 ¿ 1 and so, near the surface of our body, the µ2 φe term represents a negligible correction to the φe dynamics. Therefore, if ¤φc À ¤φe we must have: ´ ³ 3H φ˙ c + B,φ (φc )²c − B,φφ (φc )²E (φl + φe ) ¿ φ¨c . However, under Wetterich’s assumptions that B,φφ (φc ) ∼ O (B,φ (φc )) and ²E ∼ 1030 ²c , as is the case for the density of the Earth, we find: µ ¶2 µ ¶2 B,φφ (φc )²E φl 2m 22 B,φ (φc ) 13 B,φ (φc ) ≈ 10 ≈ 10 À 1, 10−4 Rs 10−4 φ¨c where we have taken
2m Rs
∼ 10−9 in the final deduction. Hence, we have shown that in general
the condition |¤φe /¤φc | ¿ 1 does not hold; indeed ¤φe À ¤φc . This result is opposite to the one found in ref. [130]. We conclude then that Wetterich’s analysis does not prove that
76
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
condition (3.1) holds. If we approximate our local solution by φ = φl + φc then we have seen that correction terms, φe , to this solution vary on scales much smaller than 1/φ˙ c . As a result we cannot conclude that φ˙ ≈ φ˙ c . In fact, at the epoch t = t0 , the asymptotic expansion of the local solution should be correctly written as: φ ∼ φ0 + φl (r, t0 ), We see later, by using more detailed methods, that, even though the above analysis fails to show it, that we should expect condition (3.1) to hold for this set-up.
3.2.2
Jacobson’s result
In ref. [131], the problem of gravitational memory [132] is considered in the context of BransDicke (varying-G) theory. If Newton’s constant, G, can and does vary in time and space then one must ask which value of G is appropriate for use on the horizon of a black hole after it forms. One motivation was to discover if the black hole possesses a type of gravitational memory, freezing-in the value of G that existed cosmologically at the moment when it formed in the early universe, or whether the value of Newton’s ‘constant’ on the horizon changes over time so as to track its changing cosmological value in the background universe. This has been investigated by several different methods and found to be a small effect [133], but Jacobson’s argument was that if the cosmological variation in G, and the related Brans-Dicke field, φ ∝ G−1 , is slow over time-scales of the order of the intrinsic length scale of the black-hole (∼ 2M ) then, at each epoch t = t0 , one can expand the cosmological value of φ as a Taylor series in (T = t − t0 ) and, to a good approximation drop all O(T 2 ) and higher-order terms, so φc ≈ φ1 = φc (t0 ) + φ˙ c (t0 )T. Jacobson noted that φ1 is a complementary solution to the Brans-Dicke field conservation equation in an (empty) Schwarzschild background: ¤s φ1 = 0, where ¤s is the d’Alembertian operator for the Schwarzschild metric. Therefore φ1 can be added to any known Schwarzschild-background solution of Brans-Dicke field equations to gain a new solution. Jacobson took T to be the ‘curvature’ (or Schwarzschild) time so that the
77
3.2. PAST WORK
metric is given by: ¶ µ ¶ µ 2m 2m −1 2 dT − 1 − dR2 − R2 {dθ2 + sin2 θdφ2 }. ds = 1 − R R 2
This time coordinate diverges as we move towards the horizon, hence φ1 (T ) also diverges in this limit. The unique static solution of the dilaton conservation equation, which vanishes as r → 0, is given by:
µ ¶ 2m φ2 = ln 1 − r
Whilst both φ1 and φ2 diverge on the horizon, Jacobson found that there is a unique linear combination of them that is well-defined as r → 2m: φjac = φ1 (t) + 2mφ˙ c (t0 )φ2 (r) = φ˙ c (v − r − 2m ln(r/2m))
(3.4)
where v = t + r + 2m ln(r/2m − 1) is the advanced time coordinate. By construction, it can then be shown that there exists a unique solution for φ in the Schwarzschild background that is non-singular on the horizon and has φjac (r = ∞, t) ≈ φc (t) for the particular case where φc (t) ∝ t ∝ G−1 . If the approximation used is valid, then this suggests that the value of G on the horizon lags slightly behind the cosmological value, but that over time-scales much larger than m there is no gravitational memory when G ∝ t−1 falls in this extreme fashion. Jacobson’s result assumes that the cosmological region can be taken to be infinity far away, so that the entire spacetime is Schwarzschild; in reality the cosmological matter becomes gravitational dominant over the black-hole at some finite-distance. Another issue with Jacobson’s method is that, since T diverges on the horizon, the expansion of φc as a Taylor series in small T is not valid near the horizon. In realistic models the space surrounding the black hole is also not be totally empty, indeed there will be an accretion disk surrounding the black-hole. The p time-scale for matter to fall into the black-hole is order r3 /2m and this is relatively short compared with the time-scale over which cosmological expansion occurs. Accretion of matter into the black-hole, therefore, might well result in a significant difference in the time-variation of G close to the horizon or cosmologically. In this chapter we consider a more realistic embedding of a Schwarzschild mass in an expanding universe; and build our asymptotic expansions in such a way that they are well-defined on the horizon. We conclude that Jacobson’s result [131] (see also refs. [133, 134] for similar conclusions arrived at by different methods) does indeed give the correction behaviour of φ whenever the energy-density in the region surrounding
78
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
the black-hole is low enough and for more general time-variations of G(t) in the background universe.
3.2.3
The Spherical Infall Approximation
˙ x, t) ≈ φc (t) to hold here on Earth, The main conclusion of this chapter: that we expect φ(~ differs from the conclusions of another analysis of this problem by made Mota & Barrow, see refs [107, 108, 109], where it was claimed that the time evolution of the local value of the dilaton field inside a collapsing region of spacetime would become disconnected from its cosmological value. It was also claimed that virialisation would result in the local value of the dilaton field becoming static. In these previous works, the spherical infall approximation was used to match together two spatially homogeneous Friedmann metrics of different spatial curvature (see section 5.10 of [145] for a introduction to the spherical infall approximation). This method implicitly assumes that the spatial derivatives of the dilaton are everywhere negligible compared to the temporal ones and can thus be consistently ignored. This assumption is, however, only valid when the scale of the inhomogeneous region of spacetime in which the observer sits, be it collapsing or virialised, is of the order of, or greater than, the Hubble radius. The results of [107, 108, 109] are therefore not applicable to the situation of a planet-based local observer in a large universe. We say more about the way in which the results derived using the spherical infall model compare to those found using our analysis in section 3.8 below.
3.3
Matched Asymptotic Expansions
The method of matched asymptotic expansions that we use in this chapter was first developed to solve systems of PDEs that involve multiple length scales. It is often used in the field of fluid dynamics to study systems where there is thin boundary layer, inside which the length scale of variation is much smaller than outside it. It also has applications in the study of slender bodies, with widths much smaller than their lengths. Other problems with multiple length scales include the interaction of greatly separated particles and the evaluation of the influence of a slowly changing background field on the dynamics of a small body. The books by Cole, [135], and Hinch, [136], provide a more detailed treatment of this subject. In this section we briefly introduce the method and give some simple examples of its applications in order to fix
79
3.3. MATCHED ASYMPTOTIC EXPANSIONS
ideas.
3.3.1
Asymptotic expansions
Critical to this method is the requirement that the approximations we work with are asymptotic expansions rather than convergent (Taylor-like) power-series. In general, an asymptotic expansion is not convergent. Thus, we define asymptotic approximations and expansions as follows: Definition 3.3.1.
PM
fn (δ) is an asymptotic approximation to f (δ) as δ → 0, if for each
M ≤ N the remainder term is much smaller than the last included term: P f (δ) − M fn (δ) → 0 as δ → 0. fM (δ) One then writes: f (δ) ∼
N X
f (δ) as δ → 0.
Definition 3.3.2. If definition 3.3.1 holds, in principle, for all N , i.e. we can take N = ∞, then the we deem the approximation to be an asymptotic expansion of f (δ): f (δ) ∼
P∞
fn (δ) as δ → 0.
(3.5)
In many cases fn (δ) ∝ (δ p )n for some constant p and we have an asymptotic power series. It is also quite common to find fi (δ) ∝ δ j ln(δ), for some i and j. The sum in eq. (3.5) is a formal one, and it is not required that it does converge. The property required by definition 3.3.1 is, however, more useful that convergence in many cases; it ensures that one only needs the first few terms of the expansion to create a good numerical approximation to f (δ).
3.3.2
Singular problems and ones with multiple scales
Consider a differential operator Lx (δ) which defines some function f (x, δ) by Lx (δ)f (x, δ) = 0. We can attempt to solve this equation by making an asymptotic expansion of f (x, δ) and solving the resultant equation order-by-order in δ: f (x, δ) ∼
∞ X
fn (x)γn (δ)as δ → 0,
with x fixed. A ’singular’ problem is one where the above asymptotic expansion is not uniformly valid, i.e. it breaks down for certain ranges of x, typically x = O (δ) or x = O (1/δ). Singular
80
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
behaviour such as this can be divided into two distinct classes. In the first, the highest derivative in Lx is multiplied by some power of δ and so can be ignored everywhere apart from in those regions where the variation in f (x, δ) is fast enough to ensure that the highest derivative make a significant contribution. In the second class, the problem has more than one intrinsic length (or time) scale, one much larger than the other. The application of this method to physical problems generally tends to fall into this latter class. In both cases one proceeds by constructing two (or more) asymptotic approximations to the solutions which are valid for different ranges of x, e.g. for x ∼ O(1) and x/δ = ξ ∼ O(1), with f (x, δ) ∼ f (x, δ) ∼
PQ
n=0 fn (x)δn
PP
n=0 gn (ξ)δn
as δ → 0, x fixed,
(3.6)
as δ → 0, ξ = x/δ fixed,
(3.7)
and solving Lx (δ)f (x, δ) order by order in δ for both expansions w.r.t. to some boundary conditions. We call expansion (3.7) the outer solution, and (3.6) the inner solution. The inner expansion is not uniformly valid in the region ξ = O(1/δ), as the outer one is not valid where x = O(δ). Because of these restrictions on the size of x, we are only able to apply a subset of the boundary conditions to each expansion; in general we are, therefore, left with unknown coefficients in our asymptotic approximations. This ambiguity can be lifted by matching the inner and outer solutions in an intermediate region where they are both valid.
3.3.3
Matching
We can match the inner and outer solutions together if there exists some intermediate range of x where both (3.6) and (3.7) are valid asymptotic approximations to f (x, δ). By the uniqueness properties of asymptotic expansions, see [136] for a proof, if they are both valid in some region then the two expansions must be equal. We take x to scale as some intermediate function, η(δ), with magnitude between 1 and δ, e.g. η(δ) = δ α where 0 < α < 1, and define a new X coordinate appropriate for the intermediate region by µ X = x/η(δ) =
δ η(δ)
¶ ξ.
We then write both the inner and outer approximations in terms of X and, keeping X fixed, make an asymptotic expansion of each of them in the limit δ → 0; this is called the intermediate limit. We must be careful to neglect any terms in the approximations, that we find in this
81
3.3. MATCHED ASYMPTOTIC EXPANSIONS
way, which would be smaller than the first excluded term; equality is then demanded between the remaining terms: PQ
n=0 fn (x(X, η))δn
PP
n=0 gn (ξ(X, η))δn (out)
hn
∼ ∼
PS
(out) (X)γn (δ) n=0 hn
as δ → 0; X fixed,
PS
(in) n=0 hn (X)γn (δ)
as δ → 0; X fixed,
(in)
(X) = hn (X)
for 0 ≤ n ≤ S.
Usually, the precise form we choose for η(δ) is unimportant and the matching can be done regardless; there are some cases, however, where we must be more careful (see Hinch [136] for more).
3.3.4
Simple Examples
Example 1: Simple matching We start with a boundary layer example where the exact solution is known. Consider y(x, δ) which satisfies: 2δy 00 + (1 + 2δ)y 0 + y = 0, in 0 ≤ x ≤ 1, with boundary conditions y = 0 at x = 0 and y = e−1 at x = 1. The exact solution is:
à y(x, δ) = e−1
e−x − e−x/2δ e−1 − e−1/2δ
! .
(3.8)
If we apply the method of matched asymptotic expansions to this, we would find that the outer approximation is given (to all orders) by: y(x, δ) ∼ xe−x 6= 0 fixed. We have been able to apply the boundary condition at x = 1; the x = 0 boundary condition cannot be reached given our assumptions about the size of x. For the inner approximation we keep ξ = x/δ fixed. In terms of ξ our differential equation reads: 2y,ξξ + (1 + 2δ)y,ξ + δy = 0. The inner solution is found to be: ³ ´ ³ ³ ´ ´ ³ ³ ´ ´ ¡ ¢ y(x, δ) ∼ a0 1 − e−ξ/2 +δ a1 1 − e−ξ/2 − a0 ξ +δ 2 a2 1 − e−ξ/2 − a1 ξ + 12 a0 ξ 2 +O δ 3 ,
82
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
1
δ = 0.00001
0.9 0.8 0.7
y(x)
0.6 0.5 0.4 0.3
interior
exterior
0.2
x=O(δ)
x=O(1)
0.1 0 1e-006
intermediate 1e-005
0.0001
0.001 x
outer approx. inner approx. 0.01 0.1
1
Figure 3.1: The leading order, inner and outer approximations to the solution of problem given in section 3.3.4, with δ = 0.00001. The exact solution is not visible on this plot since it lies underneath one or other of the approximations everywhere. with ξ = x/δ held fixed. Again we can only apply one boundary condition, this time the one at x = 0. The boundary condition at x = 1 is beyond our reach in this approximation. In this problem the choice of the intermediate scale turns out to be unimportant; we take η = δ 1/2 and define X = δ −1/2 x = δ 1/2 ξ. The intermediate limit of the inner approximation is: ³ ´ ¡ ¢ inner approx. ∼ a0 − δ 1/2 a0 X + δ a1 + 12 a0 X 2 + O δ 3/2 . The first excluded term is order δ 3/2 ; we have neglected all terms of, or below, this order including the exponentially small e−X/2δ have:
1/2
terms. Expanding the outer approximation we
³ ´ outer approx. ∼ 1 − δ 1/2 X + δ 12 X 2 + O δ 3/2 .
By our matching criteria, eq. 3.8, we must have: a0 = 1, a1 = 0. If we were to perform this process to all orders we would find that all ai = 0 for i ≥ 1. The fully specified inner approximation is therefore y(x, δ) ∼ −e−ξ/2 +
∞ X n=0
δ n (−1)n ξ n .
3.3. MATCHED ASYMPTOTIC EXPANSIONS
83
This is precisely what we would have found by performing a Taylor series expansion of the exact solution, (3.8), in the inner limit and then dropped all exponentially small terms i.e. e−1/δ . By performing the matching we have been able to lift the ambiguity in the coefficients, ai , and fully specify the inner approximation. The leading order inner and outer approximations are displayed alongside the exact solution in FIG. 3.1. Example 2: Scalar field in an expanding universe In this second example we consider a toy model of our full problem. For the purposes of this model we assume that the spacetime background is everywhere FRW, and that the appropriate scalar wave operator is that of FRW spacetime. These assumes are clearly false near a black-hole, or other dense body, where the spacetime is, to a very good approximation, Schwarzschild. We also assume that the dilaton potential V (φ) vanishes, and that dilaton-tomatter coupling B,φ (φ) = const. From the point of view of the full problem, this is a severe over-simplification but it is useful to illustrate the method of matched asymptotic expansions. Consider a spherically-symmetric, scalar field, φ(r, t), like the dilaton, in a flat FRW cosmology with metric ¡ ¡ ¢¢ ds2 = dt2 − a2 (t) dr2 + r2 dθ2 + sin2 θdφ2 . To make contact with our problem we assume that the only source term in the φ evolution equation is homogeneous and proportional to the background dust density, so ¤φ =
˙ · (a3 (t)φ) (rφ)00 λH02 − = a3 (t) a2 (t)r a3 (t)
with H0 the Hubble parameter at some arbitrary time t = t0 and λ is a constant parameter of the theory. We simplify the problem further by considering only the matter era, where a(t) ∝ t2/3 . As boundary conditions we take φ → φ0 (t) as r → ∞ and φ0 /a(t)|r=r0 /a(t) = µ2m/r02 = const; µ is a dimensionless constant, and m is a length scale such that δ = 2mH0 ¿ 1. We assume that r0 /a(t0 ) ∼ O(2m). We could visualise this problem arising from the presence of a massive body in the region a(t)r < r0 that creates a spatial gradient in φ. Near a(t)r = r0 ˙ We take our problem to be similar to one we consider later and determine we have φ0 À φ. ˙ 0 /a(t), t). the relation between φ˙ 0 and φ(r The small parameter in this problem is δ = 2mH0 . We work at the epoch when t = t0 ; for simplicity we take a(t0 ) = 1. In the inner approximation we define 2mξ = a(t)r, and
84
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
ξ0 := r0 /2m ∼ O(1). We assume that the inner solution is quasi-static, so it depends on time only through the slowly increasing cosmological t, rather than (t − t0 )/2m. We define a dimensionless time coordinate by τ = H0 t, τ0 = H0 t0 = 2/3. In this interior region φ = φI (ξ, τ ; δ) which satisfies · ¸ ¡ 2 ¢ 4λ 2 4 4 1 2 2 2 1 ∂τ τ ∂τ φI − 2 + 2 ξ∂ξ φI + 2 (ξ∂ξ ) φI + ξ∂ξ ∂τ φI . ∂ (ξφI ) = δ ξ ξ τ2 9τ 3τ 9τ 3τ ¡ 2¢ Solving this to order O δ we find: Ã ¡ 3 ¢! µ ¶ 3 2 ¡ ¢ g(τ ) ξ − 2ξ µ µ ξ 0 φI (τ, ξ) ∼ φI0 (τ ) − + δ 2 φI1 (τ ) + 2 ξ + 0 + + O δ4 ξ 9τ ξ 6ξ where φI0 (τ ) and φI1 (τ ) are ‘constants’ of integration, to be determined via the matching pro¡ ¢ 4λ cedure. g(τ ) = τ12 ∂τ τ 2 ∂τ φI0 − 9τ 2 . In the exterior we define ρ = H0 r to be our dimensionless radial coordinate, and φ = φE . Our boundary conditions determine the leading-order exterior (i)
term to be φ0 (τ ) (= φ0 (t) with some abuse of notation). The sub-leading order terms, δ i φE (i)
are then given by ¤φE = 0. So, to order δ, R∞ φE (τ, ρ) ∼ φ0 (τ ) + δ
−∞ dγTγ (τ )Xγ (ρ)
ρ
+ O(δ 2 )
where Xγ (ρ) = A(β) cos (βρ) + B(β) sin (βρ) where γ = −β 2 < 0, Xγ (ρ) = A(0) Xγ (ρ) = C(α)e−αρ
(3.9)
where γ = 0, where γ = α2 > 0,
τ 2 Tγ (τ ),τ τ + 2τ Tγ (τ ),τ = γτ 2/3 Tγ (τ ),
(3.10)
and A(β), B(β), C(α) are all to be determined by the matching procedure. As in the previous example, the precise position of the intermediate region is not important. We choose an intermediate coordinate z = δ −1/2 ρ = δ 1/2 ξ and take the intermediate limit of both the interior and exterior approximations. By equating our two approximations in the intermediate region we find B(β) = C(β) = 0 from the O (δ) terms and O (1) : ¡ ¢ O δ 1/2 : ¡ ¢ O δ 3/2 :
φI0 (τ ) = φ0 (τ ) → g(τ ) = 0, Z ∞ µ dβA(β)T−β 2 (τ ) = 2/3 τ Z0 ∞ 2µ dββ 2 A(β)T−β 2 (τ ) = 4/3 . 9τ 0
(3.11) (3.12) (3.13)
85
3.3. MATCHED ASYMPTOTIC EXPANSIONS
By differentiating twice and applying (3.9) we can check that conditions (3.12) and (3.11) can be simultaneous satisfied for some choice of A(β). The Tγ (τ ) can be made orthonormal w.r.t. to some inner product and so eq. (3.12) can, in principle, be inverted to find A(β). We omit this step, however, since we are mostly concerned with the effect of the exterior on the behaviour of φ in the interior rather than vice versa. By finding the interior solution to ¡ ¢ ¡ ¢ O δ 4 and the exterior to O δ 2 we can show that φI1 (τ ) = 0.; we have now fully specified ¡ ¢ the interior solution to O δ 2 : φI (t, R = a(t)r) ∼ φ0 (t) −
2mµ 4mµ + R 9t2
µ ¶ ¡ ¢ r2 R + 0 + O δ4 R
We have that: φ˙ I |ar=r0 16mµr0 −1≈ ˙ φ0 9t3 φ˙ 0 and φ˙ 0 ∼ 4λ/9t for large t. Therefore we have shown that for the time variation of φ at ar = r0 to track the cosmological time variation we need λÀ
¡ 2¢ 2µmr0 = O δ t2
where we mean that this taken to hold numerically and not asymptotically. In most physically reasonable scenarios, if the massive body is taken to be a black-hole then we must have µ = 0, whereas for bodies that are much larger than their Schwarzschild radius we generally have µ ∼ O(λ), in a numerical sense. We therefore expect the above condition to hold, and as such the local time variation of the dilaton in this toy model tracks the cosmological one. It is clear that, whatever the value of λ, the rate of time variation in φ tends to homogeneity as t → ∞. This example is a greatly simplified version of the problem we consider in the rest of this chapter.
3.3.5
Application to General Relativity
The use of matched asymptotic expansions in general relativity was pioneered by Burke and Thorne [137], Burke [138], and D’Eath [139, 140] in the 1970s. These authors used them to the study how the laws of motion of a test body were affected by the background spacetime. We now outline how matched asymptotic expansions are applied in general relativity. We assume that the universe
86
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
³ ´ C = M, gab , T ab , φ , with Tab is the energy-momentum tensor and φ the dilaton, can be viewed as a background cosmology, ³ ´ 0 C0 = M0 , gab , T0ab , φ0 , onto which some localised, interior configuration has been superimposed in a non-linear fashion. In what follows, for simplicity, we require the interior configuration to be static in some coordinate system. We demand that the ‘size’ of the interior region be given by a single parameter m; and that as m → 0, with {xµ } fixed, the interior region disappears and C → C0 . We formalise this statement later. Length scales The length scale of the interior region, as defined by its curvature invariants is denoted by LI (m), with LI (m) → 0 as m → 0. The length scale of the background (exterior) region is written LE . For the asymptotic expansion method to be viable we need δ = LI /LE ¿ 1. The effect of the interior on the background configuration can then be treated as a linear perturbation to C0 , with δ playing the rle of a small parameter. We can similarly treat the effect of the background on the interior as a linear perturbation. Five-dimensional manifold For each m, in some interval [0, mmax ), we write the global configuration as ³ ´ Cm = Mm , gab (m), T ab (m), φ(m) . Following Geroch [141] and D’Eath [139], we consider a five-dimensional manifold with bound(0)
ary, N , that is built up from spacetimes (Mm , gab (m)) for m ∈ [0, mmax ); gab (0) = gab . As D’Eath noted we should properly exclude from M0 a smooth time-like world line, l0 , that corresponds to the ‘position’ of the interior region; this is illustrated in figure 3.2. We require that the contravariant metrics g ab (m) define a smooth tensor field on N ; in addition, we require that the dilaton, φ(m), defines a smooth scalar field on N . These conditions are required by
87
3.3. MATCHED ASYMPTOTIC EXPANSIONS
our assumption that the interior region has only a small perturbing effect over length scales >> LI (m).
Figure 3.2: The 5-D manifold N is built up from spacetimes (Mm , gab (m)) for m ∈ [0, mmax ).
In an open subset on N , we choose a coordinate chart (t, r, θ, φ, m) such that the aforementioned world line l0 is given by (r = 0, m = 0). In the limit −1 (tL−1 E , rLE , θ, φ) → consts, m → 0, (0)
ab and φ , respecwe can give gab (m), T ab (m) and φ(m) as asymptotic expansions about gab , T(0) 0
tively. This is the exterior approximation, and is appropriate for considering the perturbing effect that the interior has on the background universe. For each epoch t = t0 , we also define an interior approximation. For this, we take the asymptotic expansions of gab (m), T ab (m) and φ(m) in the limit ¢ ¡ −1 (t − t0 )L−1 I (m), rLI (m), θ, φ → consts, m → 0. This approximation is appropriate for considering the perturbations produced in the interior region by the background cosmology. This is the problem that we are most interested in. We assume that we know the leading-order configuration in this limit and in what follows we take it to be a Schwarzschild metric with a quasi-static dilaton field.
88
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
Intermediate region and matching We match the interior and exterior approximations in some intermediate region, where ¡ ¢ −1 (t − t0 )L−1 int , rLint → consts, m → 0, and where Lint = LE δ 1−α = LI δ −α , with 0 < α < 1. In many cases the precise value of α is not important (see section 3.3.3). Following D’Eath, [139], we assume that all the functions in our approximations are sufficiently well-behaved in this region so as to admit a power-series expansion in the radial coordinate; given this we do not need to examine this region explicitly. A typical term in the interior expansion, in this matched region, looks like δ i ξ j fij (t0 ; L−1 int (t−t0 )) where ξ = rL−1 I and, i and j are rational numbers; i ≥ 0. In the exterior expansion, a typical −1 term looks like δ k ρl gkl (t0 ; L−1 int (t − t0 )), again with k and l rational and k ≥ 0 and ρ = LE r.
Matching requires that g(k+1)l = fij .
3.4
Geometrical Set-Up
The experimental bounds on the permitted level of violations of the WEP due to the presence of light scalar fields demand that the dilaton field couples to matter much less strongly than gravity (see Chapters 1.3 and 5.2), so |B,φ | < 10−4 . As a result, the dilaton field is only weakly coupled to gravity, and so its energy density and motion create metric perturbations which have a negligible effect on the expansion of the background universe. This feature allows us to consider the dilaton evolution on a fixed background spacetime. We consider below both the extent to which condition φ˙ ≈ φ˙ c is satisfied near the surface of some spherical virialised over-density of matter, e.g. a the Earth, a star, black-hole, galaxy or galaxy cluster, and also the degree to which it is valid during the collapse of an over-dense region. We treat the two cases separately. In the first case, we refer to the virialised over-density as our ‘star’ and take it to have mass m and radius Rs at some time of interest t = t0 . Although we require that the ‘star’ itself be spherical, we do not demand that the background spacetime possess any symmetries. We do require however that, at t = t0 , the metric is approximately Schwarzschild, with mass m, inside some closed region of spacetime bounded by a surface at r = Rs ; this region is called
89
3.4. GEOMETRICAL SET-UP
the interior. The metric for r < Rs is left unspecified. We allow for the possibility that r = Rs is a black-hole horizon. In the second case, we only consider the spherically-symmetric spacetimes and label the mass of the collapsing region by m. We assume that the spatial extent of the collapsing region is small compared to the Hubble scale. We also demand there are no black-hole horizons in the interior of the collapsing region. By using the results of the first case, however, we can, in some cases, allow for the formation of a horizon. In both cases we demand that: • Asymptotically, the metric must approach the FRW metric and the whole spacetime should tend to the FRW solution in the limit m → 0. • The spacetime is approximately FRW in some open region that extends to spatial infinity, this is called the exterior. We are concerned with spacetimes where the matter is a pressureless dust of density ε, with cosmological constant, Λ. We further require that the motion of the dust particles be geodesic. In the spherically-symmetric case, all such solutions to Einstein’s equations with matter fall into the Tolman-Bondi class of metrics (for a review of these and other inhomogeneous spherically symmetric metrics see ref [142]), however when condition of spherical symmetry is dropped, the general solution is not known. We can, however, simplify our analysis greatly by specifying four further requirements: 1. The flow-lines of the background matter are non-rotating. This implies that the flow-lines are orthogonal to a family of spacelike hypersurfaces, St . 2. Each of the surfaces St is conformally flat. 3. The Ricci tensor for the hypersurfaces St ,
(3) R , ab
has two equal eigenvalues.
4. The shear tensor, as defined for the pressureless dust background, has two equal eigenvalues. These conditions are automatic if spherical symmetry is required, and in general they specify the Szekeres-Szafron class of solutions, [146, 148], of which the Tolman-Bondi solutions, [149, 150], are the spherically symmetric limit.
90
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
When considering spherically symmetric spacetimes with a central ‘star’ we also perform our analysis using the McVitte (1933) background [142, 144]. In the exterior limit the McVittie metric asymptotes to a dust + Λ FRW cosmology. For radii r where r À 2m, r ¿ H0−1 , the McVittie metric looks like Schwarzschild spacetime. We introduce both the McVittie and Szekeres-Szafron solutions below.
3.4.1
The McVittie background
The earliest studies of the gravitational field produced by a spherically-symmetric, massive body in an expanding universe were based upon McVittie’s solution to Einstein’s equations [142, 144]. McVittie’s solution is a superposition of the Schwarzschild metric and a FRW background. In isotropic coordinates the metric, is · ¸ ¤ 1 − µ(t, r) 2 2 [1 + µ(t, r)]4 2 £ 2 2 2 2 2 2 ds = dt − a (t) dr + r {dθ + sin θdϕ } , 1 + µ(t, r) (1 + 14 kr2 )2
(3.14)
where µ(t, r) =
¢1/2 m ¡ 1 + 14 kr2 , 2ra(t)
(3.15)
and m and k are arbitrary constants. In the limit a = 1, m is the Schwarzschild mass. k is the curvature of the surfaces (t, r) = const in the m = 0 (i.e. FRW) limit. As a model of a black hole in an expanding universe it has the distinct disadvantage that the horizon is a naked singularity. This defect aside, the McVittie metric is believed to be a good approximation to the exterior metric of a massive spherical body with a physical radius much larger than its Schwarzschild radius. In the flat (k = 0) case, the local energy density depends only on time and is the same as the FRW energy density to which the solution matches smoothly as r → ∞. The pressure, however, is not the same as the FRW pressure. Apart from the vacuum-energy dominated case (i.e. where ² = −P ), it is not possible to prescribe a dust, P = 0, or any other, non-vacuum, barotropic, P = P (²), equation of state to hold apart from in the asymptotic FRW limit.
3.4.2
The Szekeres-Szafron background
We require the inhomogeneity to be of finite spatial extent, this limits us to considering only the quasi-spherical Szekeres solutions, which are described by the metric: ds2 = dt2 −
2 dr 2 ¡ ¢ (1 + ν,R R)2 R,r − R2 e2ν dx2 + dy 2 , 1 − k(r)
(3.16)
91
3.4. GEOMETRICAL SET-UP
where ν,R := ν,r /R,r and e−ν = A(r)(x2 + y 2 ) + 2B1 (r)x + 2B2 (r)y + C(r),
AC − B12 − B22 = 14 , and: 1 R,t2 = −k(r) + 2M (r)/R + ΛR2 . 3 In this quasi-spherically symmetric subcase of the Szekeres-Szafron spacetimes the surfaces of constant curvature, (t, r) = const, are 2-spheres [147]; however, they are not necessarily concentric. These 2-spheres have surface area 4πR2 , and so we deem R to be the physical radial coordinate. In the limit ν,r → 0, the (t, r) = const spheres becomes concentric. We can make one further coordinate transformation so that the metric on the surfaces of constant curvature, (t, r) = const, is the canonical metric on S 2 i.e. dθ2 + sin2 θ dφ2 : x → X = 2 (A(r)x + B1 (r)) , y→Y
= 2 (A(r)y + B2 (r)) ,
where X + iY = eiϕ cot θ/2. This yields −ν,r |x.y =
λz (X 2 + Y 2 − 1) + 2λx X + 2λy Y = λz (r) cos θ + λx (r) sin θ cos ϕ + λy (r) sin θ sin ϕ, X2 + Y 2 + 1
where we have defined: λz (r) :=
A0 , A
µ λx (r) :=
2B1 A
¶0
µ A,
λy (r) :=
2B2 A
¶0 A.
With this choice of coordinates, the local energy density of the dust separates uniquely into a spherical symmetric part, εs , and a non-spherical part, εns : ε = εs (t, R) + εns (t, R, θ, ϕ), where: 2M,R , R2 µ ¶ Rν,R 2M = − · . 1 + ν,R R R3 ,R
κεs = κεns
(3.17) (3.18)
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
We define M,R = M,r /R,r . We use the remaining freedom to choose r to demand that r = R at t = t0 . In the virialised case, we write M (r) := m + Z(r) where m is the gravitational mass of our ‘star’. In the spherically-symmetric case κεns = 0 and the metric is of Tolman-Bondi form: ds2 = dt2 −
3.5
2 © ª R,r − R2 dθ2 + sin2 θdϕ2 . 1 − k(r)
Virialised Case: Spherically symmetric backgrounds
In this section we consider how the local time evolution of φ near a virialised overdensity of matter, or ‘star’, is related to the cosmological time evolution of φ. We choose our background spacetime to be spherically symmetric and we further require that: • The metric is approximately Schwarzschild, with mass m, inside some closed region of spacetime outside a surface at r = Rs . The metric for r < Rs is left unspecified. • Asymptotically, the metric must approach FRW and the whole spacetime should tend to pure FRW in the limit m → 0. • As the local inhomogeneous energy density of asymptotically FRW spacetimes tends to zero, the spacetime metric exterior to r = Rs must tend to a Schwarzschild metric with mass m . This set-up is illustrated in FIG. 3.3. This list of requirements is far from exhaustive, and they are re-expressed in more rigorous fashion later. In the local, approximately Schwarzschild, region the intrinsic interior length scale, LI , of a sphere centred on the Schwarzschild mass with surface area 4πRs2 , is given by considering the curvature invariant: ³ LI =
abcd 1 12 Rabcd R
´−1/4
3/2
=
Rs
(2m)1/2
.
(3.19)
In the asymptotically FRW region, the intrinsic length scale is proportional to the inverse root √ of the local energy density: 1/ κ² + Λ. We assume that, in line with current observations [92], the FRW region is approximately flat, and so we set our exterior length scale appropriate for the epoch at t = t0 is the inverse Hubble parameter at that time: LE = 1/H0 .
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
93
Figure 3.3: We are concerned with the evolution of the dilaton field, φ, in spacetimes where a mass m, of radius Rs , has been embedded into the cosmological fluid. Our small parameter is then defined to be
δ = LI /LE . Formally we require that, as δ → 0, our choice of spacetime background should be FRW at zeroth-order in the exterior limit, i.e. LE t and LE r fixed, and Schwarzschild to lowest order in the interior limit i.e. LI (t − t0 ) and LI r fixed. These conditions are satisfied by the McVittie (1933) solution and by Tolman-Bondi models where M → m > 0 in the interior limit. If we wish to require that the matter be pressureless dust with or without a cosmological constant then we must limit ourselves to the Tolman-Bondi models. We consider the expansion of the background spacetime in the exterior and interior limits.
3.5.1
Case I: McVittie background
The McVittie metric is ¸ · ¤ 1 − µ(t, r) 2 2 [1 + µ(t, r)]4 2 £ 2 2 dt − a (t) dr + r2 {dθ2 + sin2 θdϕ2 } , ds = 1 2 2 1 + µ(t, r) (1 + 4 kr ) where µ(t, r) =
¢1/2 m ¡ . 1 + 14 kr2 2ra(t)
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
Exterior Metric We work in the (r, t) coordinates introduced above. In the exterior dr ∼ H0−1 ∼ dt. We define dimensionless exterior coordinates τ = H0 t and ρ = H0 . Then, with δ = LI /LE = 3/2
H0 Rs (2m)1/2 ¿ 1 we have: n ¡ ¢o (0) (1) ds2ext ∼ H0−2 gab + δgab + O δ 2 dxa dxb (3.20) (· ) ¸2 4 £ ¤ 1 − δ µ ˜ (τ, ρ) [1 + δ µ ˜ (τ, ρ)] = H0−2 dt2 − a ˜2 (τ ) dρ2 + ρ2 {dθ2 + sin2 θdϕ2 } , 1 + δµ ˜(τ, ρ) (1 + 14 Ω0k ρ2 )2 where µ ˜(τ, ρ) =
1 4
(2m/Rs )3/2 (ρ˜ a(τ ))−1 (1 + 14 Ωk ρ2 ), a ˜(τ ) := a(t) and Ω0k = kH0−2 .
In the McVittie metric the pressure of the matter is non-vanishing and so the dilaton generally couples not only to the local energy density but to the trace of the energy momentum tensor of the matter: Tµµ = (² − 3P ). In the exterior expansion: H0−2 κTµµ = εe0 (τ ) + δεe1 (τ, ρ) + O(δ 2 ) µ ¶ 1 + δµ ˜(τ, ρ) 2 0 = 12h (τ ) + 6h (τ ) 1 − δµ ˜(τ, ρ) ¶ µ 2 − δµ ˜(τ, ρ) (1 + δ µ ˜(τ, ρ))−5 − 4Ω0Λ . + 3Ωk (τ ) 1 − δµ ˜(τ, ρ)
(3.21)
Interior Metric In the interior it is most convenient to work with R = [1 + µ] (1 + 14 kr2 )−1/2 a(t)r as the radial coordinate. If k = 0, then the surface (t, R) = const has area 4πR2 . When t = t0 the interior geometry is approximately Schwarzschild over scales where dR ∼ O(Rs ) and d(t − t0 ) ∼ O(LI ). We define new dimensionless coordinates by
T = (t − t0 )/LI and ξ = R/Rs . In these coordinates the interior metric is: n ¡ ¢o (0) (1) ds2int ∼ Rs2 jab (ξ) + δjab (δT, ξ) + O δ 2 dxa dxb (3.22) ½ µ ¶ Rs = Rs2 A(ξ) dT 2 2m ) µ ¶−1 £ 2 ¤ δ 2 Ωk (δT ) 2m 2 −1 2 2 2 2 − 1− X (ξ) A(ξ) ψ + ξ {dθ + sin θdϕ } , 4 Rs
95
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
where A(ξ) = 1 − 2m/(Rs ξ), µ ψ = dξ − δh(δT )ξ
Rs 2m
¶1/2
A(ξ)1/2 dT,
and h(δT ) = a ˜0 (τ = δT )/˜ a(τ ), Ωk (δT ) = Ω0k a ˜−2 (τ ). We have defined X(ξ) =
ξ 4
¡ ¢2 1 + A(ξ)1/2 .
0 , j 1 etc. can, at each order, be written in quasi-static Note that the interior metric functions jab ab
form, as functions only of ξ and δT . In the interior expansion the trace of the energy momentum tensor of matter, Tµµ , is given by: Rs2 κTµµ = δ 2 εi2 (δT, ξ) (3.23) µ ¶ µ ¶ 2m 2m = 12δ 2 h2 (δT ) + 6δ 2 h0 (δT )A(ξ)−1/2 Rs Rs à !à !5 µ ¶ ¶ µ −1/2 (ξ) 1/2 (ξ) 2m 3 + A 1 + A 2m 2 2 + 6δ Ωk (δT ) Ω0Λ −δ 4 Rs 4 2 Rs When written in quasi-static form given above, the only term in the interior expansion of Tµµ is at O(δ 2 ).
3.5.2
Case II: Tolman-Bondi background
In general, to specify initial data for the Tolman-Bondi (and Szekeres-Szafron) solution we must give both the energy density on some initial hypersurface, κε, and the spatial curvature of that hypersurface (given by k(r)). In two relatively simple cases, both compatible with our geometric set-up, the solution is completely specified by giving just the energy density, κε, on some initial hypersurface: • The ‘Gautreau’ case: the hypersurfaces t = const are spatially flat, k(r) = 0. In this case the big-bang is not simultaneous along the past world lines of all geodesic observers; i.e. it does not occur everywhere for a single value of t, at t = 0 say. In these cases the flow lines of matter move out of our ‘star’ - and the mass of ‘star’ decreases of time. • The simultaneous big-bang case: the big-bang singularity occurs at t = 0 for all geodesic observers. In these cases the flow lines of matter move into the ‘star’ - and its mass increases with time.
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
The first of these cases is simpler to analyse but the second is more physically reasonable, since we expect the gravity to pull matter onto our star rather than expel it. For this reason we only explicitly consider the simultaneous big-bang case here. The results for the Gautreau case are very similar to this case. The simultaneity of the big bang is actually not a significant factor for the late-time evolutionary problem that we are considering. The Tolman-Bondi metric is: ds2 = dt2 −
2 (t, r) R,r dr2 − R2 (t, r){dθ2 + sin2 θdϕ2 }, 1 − k(r)
where R,t2 = −k(r) +
2m + 2Z (r) 1 + ΛR2 . R 3
The dust energy density is given by: κ² = 2Z,r /(R2 R,r ). We use the freedom we have in the definition of R to prescribe that at some epoch of interest, t = t0 , R is the physical radial coordinate i.e. R(t0 , r) = r and the surface (t, R) = const always has area 4πR2 ; this requirement, combined with the simultaneity of the initial big-bang curvature singularity , determines the form of k(r). The surface of our massive spherical body is at R = Rs . m + Z(r) is the active gravitational mass interior to the surface (t = t0 , r) = const and m is the active gravitational mass of the central body at t = t0 . The mass of the central object grows over time as a result of accretion. We also require the dust density to be everywhere positive and tend to spatial homogeneity for large r, hence we need Z(Rs ) = 0, Z,r > 0, 1 (0) 2 3 lim Z(r) = Ω H r . r→∞ 2 dust 0
(3.24) (3.25) (3.26)
We want the zeroth-order, exterior limit to be a FRW spacetime with curvature parameter k; this requires limr→∞ k(r) = kr2 . We must require k(r) > 0 in the interior region; however we do not require k(r) > 0 everywhere. The exact solution for R(r, t) was found by Barrow and Stein-Schabes and is given in ref. [151].
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
97
Exterior Metric We define dimensionless coordinates ρ = H0 r and τ = H0 t, and expand Z(r) order by order in the small parameter δ. We write: 1 (0) H0 Z(r) ∼ Ωdust ρ3 + δ p z1 (ρ) + o (δ p ) . 2 (0)
We require 2δ p z1 (ρ)/Ωdust ρ3 ¿ 1 for ρ ∼ O(1) to ensure that this is valid asymptotic expansion; i.e. p > 0. The unperturbed spacetime is then FRW. The value of p is model dependent. With Z(r) specified, we can use the exact solutions of Barrow and Stein-Schabes, [151, 152], for R(t, r), and the requirement that R(r, t0 ) = r to find the expansion of k(r) and from there the expansion of the metric; schematically we have: (0)
k(r) ∼ Ωk ρ2 + δ p E1 (ρ) + o (δ p ) , ³ ´ (0) (1) ds2ext ∼ H0−2 gab (τ, ρ) + δ p gab (τ, ρ) + o(δ p ) dxa dxb ,
(3.27) (3.28)
(0)
where gab is the FRW metric. Interior Metric We take T = (t − t0 )/LI , and ξ = R/Rs to be our coordinates in the interior, and: Z(r)/m ∼ δ q µ1 (η) + o (δ q ) , ¡ ¢2/3 where η = ξ 3/2 + 3T /2 ; Rs η = r + O(δ q , δ 2/3 ). From the exact solutions we find: ³ ´ k(η) = δ 2/3 k0 (1 + δ q µ1 (η) + o (δ q )) + O δ 5/3 , where k0 (δT ) =
2m Rs
µ
π H0 t0 + δT
¶2/3 .
(3.29)
The interior expansion of the metric is written: ds2int (0)
∼
Rs2
³
(0) jab (ξ)
+
(q) δ q jab (ξ, η)
+
(2/3) δ 2/3 jab
q
´
+ o(δ ) dxa dxb + o(δ q ).
(3.30)
(q)
where jab and jab are given by: (0) jab dxa dxb (q)
jab dxa dxb
³ ´2 Rs 2 −1/2 = dT − dξ + ξ dT − ξ 2 {dθ2 + sin2 θdϕ2 }, 2m µ1 (η) µ1 (η) 2 = − 1/2 dξdT − dT . ξ ξ
(3.31) (3.32)
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
and (2/3)
jab
³ ´2 dxa dxb = k0 (δT ) dξ + ξ −1/2 dT .
(3.33)
(0)
The zeroth order metric jab is the Schwarzschild metric in Painlev-Gullstrand coordinates. Unlike in the McVittie solution, the next-to-leading order term in the interior metric here is p not quasi-static, however by defining Tˆ = Rs /2mT to be the time coordinate, the metric can be seen to be quasi-static in a different sense. Instead of only depending on T through δT , it depends on Tˆ only through the combination 2mTˆ/Rs , and for most bodies, with the exception of black holes and neutron stars, 2m/Rs ¿ 1. For the above approximation to be a valid asymptotic expansion we require δ q µ1 (η) = o(1) for η ∼ O(1) i.e. q > 0. Energy - Momentum Tensor The energy density of the dust locally is given by: Rs2 κTµµ = δ q εi1 (ξ, η) + o (δ q ) ¶ µ 2mδ q µ1 (η),η = + o (δ q ) . Rs ξ 3/2 η 1/2
(3.34)
We do not give the exterior expansion of Tµµ here explicitly, although it can in principle be found with reference to the exact solutions for R(r, t) if required.
3.5.3
Boundary Conditions
Before we can solve the field equations for φ we need boundary conditions. As the physical radius tends to infinity, R → ∞, we demand that the dilaton tends to its homogeneous cosmological value: φ(R, t) → φc (t). We can apply this boundary condition in the exterior region but not in the interior. To solve the interior dilaton field equations we need to specify the dilaton-flux passing out from the surface of our ‘star’ at R = Rs . At leading order we parameterise this by: µ ¶ Z Rs ¡ ¢ 2m 2 ¯ Rs 1 − ∂ξ φ0 |ξ=Rs = 2mF φ0 = dR0 R02 B,φ (φ0 (ξˆ0 ))κ²(R0 ), Rs 0
(3.35)
where φ¯0 = φ0 (R = Rs ). The function F (φ) can be found by solving the dilaton field equations ¡ ¢ to leading order in the R < Rs region. For black-holes F φ¯0 = 0, whereas for objects much ¡ ¢ larger than their Schwarzschild radius F φ¯0 ≈ B,φ (φc ). Without considering the sub-leading order dilaton evolution inside our ‘star’, i.e. at R < Rs , we cannot rigorously specify any
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
99
boundary conditions beyond leading order. Despite this, we can guess at a general boundary condition by perturbing eq. (3.35): µ ¶ ¯ ¯ ¡√ ¢ ¡ ¢ 2m ¯ 2 ˜ ¯¯ ˜ )F φ¯0 Rs 1 − ∂R δ(φ) = δ˜ −gg RR ∂R φ0 ¯ + 2δ(M Rs ξ=Rs R=Rs ¡ ¢ + 2mF,φ (φ¯0 )δ˜ φ¯0 + smaller terms,
(3.36)
˜ where δ(X) is the first sub-leading order term in the interior expansion of X; M is the total mass contained inside ξ < Rs and is found by requiring the conservation of energy; at t = t0 ¡ ¢ we have M = m. Only δ˜ φ¯0 remains unknown, however we assume it to be the same order ˜ as δ(φ) and see that this unknown term is usually suppressed by a factor of 2m/Rs relative to the other terms in eq. (3.36). Our assumptions about the nature of B and V ensure that: B,φφ (φE )F (φ¯0 ) R2m sξ ¿ 1, B,φ (φE ) V,φφ (φE )F (φ¯0 ) R2m sξ ¿ 1, V,φ (φE )
(3.37) (3.38)
and that V (φ) is the same order of magnitude for the local value of φ as it is cosmological one. For physically acceptable theories, this assumption only fails for special choices of V (φ), such as those required when one considers chameleon scalar field theories, [45].
3.5.4
Solution of φ equations
Zeroth-order solutions In accord with our prescription, all of the models that we have considered share the property that, to lowest order in δ, the interior is pure Schwarzschild, and the exterior is pure FRW. Exterior In the exterior, the dilaton field takes its cosmological value: φ = φ(t)c to zeroth order and so satisfies the homogeneous conservation equation: ∂t2 φ(t)c + 3H∂t φ(t)c = −B,φ (φc ) κ²cdust (t) − V,φ (φc ) , The effect of the interior region on the exterior should, even for finite δ, become increasingly smaller as r → ∞. As a result there are no sub-leading order, r-independent, terms in the exterior expansion of φ. Equivalently, the homogeneous mode of φ in the exterior is given by the cosmological term, φc , alone to all orders
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
Interior In the interior the dilaton field satisfies the wave equation on a Schwarzschild background at zeroth order in δ; we take φ to be quasi-static to leading order. By applying the zeroth-order boundary condition, eq. (3.35), we find: (0) φI
µ ¶ ¡ ¢ 2m ¯ = φe (δT ) + F φ0 ln 1 − , Rs ξ
(3.39)
where φe (δT ) is a constant of integration to be determined by the matching procedure. Matching As specified above, we should rewrite the exterior and interior expansion of φ in some intermediate region, with length scale, LI < Lint < LE . We define
Lint T = (t − t0 ). The only homogeneous mode in the exterior expansion was the cosmological term, φc (t) = φhom (Lint T /LE ). This term therefore appears at leading order in the intermediate expansion. Any other homogeneous terms that result from taking the intermediate limit of the exterior expansion must be sub-leading order. Therefore, we conclude that φhom = φc (t) is the only leading order, homogeneous term in the intermediate expansion of φ. In addition, all subleading-order homogeneous terms that arise in this way are functions of t only and so will be quasi-static (Lint T /LE dependent) in the intermediate regions. When the intermediate limit of the interior approximation is taken, it is clear that homogeneous terms in the interior will map to homogeneous terms in the intermediate zone. The matching criteria therefore implies that all homogeneous terms in the interior must be quasi-static, i.e. only depending on time through δT , and that at leading order: φe (δT ) = φe (Lint T /LE ) = φhom (Lint T /LE ) = φc (t) The interior solution therefore reads: ¶ µ ¡ ¢ 2m ¯ , φ0 = φc (t) + F φ0 ln 1 − Rs ξ ´ ¡ ¢ ³ and φ¯0 = φc (t) − F φ¯0 ln 1 − 2m Rs . We see directly the effect of the cosmological evolution of φ in the local region.
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
101
Case I: The McVittie Background When the matched asymptotic expansion method is applied to the McVittie background, the analysis goes through in much the same way as it did in example 2 above. In the interior we find: (0)
φI = φI + δ 2 (1)
where φI
¡ ¢ 2m (1) φI + O δ 4 , Rs
satisfies:
´ 1 ³ 2 (1) ∂ ξ A(ξ)∂ φ = ξ ξ I ξ2
µ ¶ 00 φE (δT ) 3 1 2m + h (δT ) + φ0E (δT ) (3.40) A(ξ) A1/2 (ξ) 2 Rs RA3/2 (ξ) µ ¶ ¡ ¢ ¡ ¢ Ωk (δT )X 2 2m h0 (δT ) 2m h2 ξ 2 ¯ + F φ − ∂ − F φ¯0 0 ξ Rs A1/2 (ξ)ξ Rs ξ 2 8 A(ξ) !5 Ã !Ã ³ ´ 1/2 1/2 1 + 3A 1 + A (0) +B,φ φI 12h2 + 6h0 A(ξ)1/2 + 3Ωk (δT ) 2 A1/2 ¡ ¢ −12ΩΛ ] + Rs2 V,φ φ0I .
Given our that our assumptions about B and V ensure that B,φφ (φE )F (φ¯0 ) R2m sξ B,φ (φE ) V,φφ (φE )F (φ¯0 ) 2m
Rs ξ
V,φ (φE ) (1)
then we can solve for φI
¿ 1, ¿ 1,
as an asymptotic series in 2m/Rs (provided this quantity is small).
To O (2m/Rs ) we find: (1)
φI
∼
¡ ¢´ 1 ¡ ¢ ¡ ¢ 2mξ ³ 1 ³ (0)00 (0)0 0 ¯0 − Ωk (δT )F φ¯0 + h2 F φ¯0 φ + hφ + h F φ E 8 Rs 2 E ¡ ¢ 3 1 ¡ ¢ (0) − B,φ (φE ) h0 (δT ) + 11 Ωk (δT ) − F φ¯0 B,φφ (φE ) Rs2 κ²dust 4 2 2 µ ¶ µ ¶ ¡ ¢ 2 ¢ 2m C(δT ) 2m 2 ¯ +F φ0 Rs V,φφ (φE ) + + D(δT ) + O , Rs ξ Rs
(3.41)
where C(δT ) is determined by the boundary conditions on R = Rs and D(δT ) comes from the matching conditions. The matching procedure would set D(δT ) = 0 if were to continue our asymptotic expansions to an order smaller than we do here. We do not expect C(δT ) to be any larger than the other terms in this expression when ξ = 1. The exterior expansion, to O(δ), along with the matching conditions that determine the otherwise unknown functions in it, is given in Appendix A. In this appendix, we show that these matching conditions are selfconsistent, and hence that, to the order considered, the matching procedure works correctly.
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
As with the simple example we considered in section 3.3.4, we see that when the matching conditions are applied, the background sets the form of the homogeneous terms in the interior solution, and the interior solution gives us the behaviour of the inhomogeneous terms in the exterior expansion. This is to be expected, since in these models the interior region is the sole source of the inhomogeneities in both the spacetime and the dilaton field. This picture would be more complicated if we were to consider more than one interior region, as in the case of a nested series of shells of matter of differing densities Our original concern was the behaviour of the time derivative of φ in the interior and we have shown that in the McVittie background: ³ ´ ¡ ¢ ∂t φI ∼ 1 + F,φ φ¯0 ln(1 − 2m/R) + O δ 2 (2m/R))2 . ∂t φc Since, in most cases of interest, the dilaton to matter coupling is weak and we are far from the ¡ ¢ Schwarzschild radius of our ‘star’, both F ( φ¯0 and 2m/R are ¿ 1. Hence: ∂t φI /∂t φc ≈ 1, and the local time evolution of the dilaton tracks the cosmological one. The strength of this result arises partly from our restrictive choice of background metric. If the background is taken to be Tolman-Bondi rather than McVittie, we can find quite different behaviour. Case II: Tolman-Bondi Backgrounds We assume that we have specified some Z(r; δ) that has exterior and interior approximations as given in section 3.5.2 and 3.5.2 respectively. But we must be careful to ensure that the form of Z(r; δ) is such that these two approximations can be matched in some intermediate region. We can now proceed to solve the dilaton evolution equation in the inner and outer limits and then apply our matching procedure. We have already seen that at zeroth order the matching conditions imply that φ is approximated by interior approx ∼
(0)
φI (ξ; δT ) + O (δ q ) = φc (τ = τ0 + δT ) µ ¶ ¡ ¢ 2m ¯ +F φ0 ln 1 − + O (δ q ) , Rs ξ
exterior approx ∼ φc (τ ) + O (δ p ) . We now consider the next-to-leading order terms with a view to determining how time variation of φ in the interior is related to its cosmological rate of change.
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3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
In the interior the local energy density, to O (δ q ), depends on T only through η. The O(δ q ) metric perturbation similarly depends only on T and η. These O(δ q ) corrections induce order δ q corrections to φ. There is also a perturbation at order δ that is sourced by the ¤0 φc term, where ¤0 is the d’Alembertian of the Schwarzschild metric. The O(δ 2/3 ) metric corrections coming from k0 only induces corrections to φ at O(δ 5/3 ) and order O(δ q+2/3 ) it can that therefore be ignored when considered only next-to-leading order terms. Whether or not the O(δ) or O(δ q ) correction is dominant depends on the value of q. (1)
We define φI
(q)
and φI
by: (0)
(q)
(1)
φ ∼ φI (ξ; δT ) + δ q φI (T, η; δT ) + δφI (T, η) + ... , (q)
and we find that φI 2m − Rs
satisfies:
à ! µ ¶ ¡ 0 ¢ µ1 (η),η ξ (q) 1 ξ 5/2 (q) 2m 3/2 (q) φI ξ φI,T T + φI,T + 1/2 B φ = ,φ I,η 2η Rs η η 1/2 η 1/2 ,η µ ¶ µ1 (η),η 2m 2 ¡ ¯ ¢ 2µ1 (η) ³ ´− − F φ0 ³ ´2 . 2m Rs ξη 1/2 1 − Rs ξ ξ 5/2 1 − 2m
(3.42)
Rs ξ
In general, the above equation is difficult to solve exactly, however, in most of the cases of interest the surface of our ‘star’ is far outside its Schwarzschild radius and so 2m/Rs ¿ 1. We may then solve eq. (3.42) as an asymptotic series in 2m/Rs , valid where ξ À
2m Rs .
Using our
prescription for the R = Rs boundary condition given in section 3.5.3 above, we find that we must require: (q)
∂ξ φI |ξ=1 =
2m ¯ F (φ0 )µ1 (χ(ξ = 1, T )) . Rs
(q)
To leading order in 2m/Rs we find φI to be given by ÃZ µ ¶ η µ (η 0 ),η µ1 (η) F (φ¯0 ) µ1 (η(ξ = 1, T ) 2m (q) 0 1 B,φ (φc ) dη − + 1− φI ∼ Rs ξ(η 0 , T ) ξ B(φc ) ξ õ ¶2 ! 2m +D(T )) + O . Rs
(3.43)
(q)
In the cases where 2m/Rs ≈ 1 we were unable to find a closed analytical expression for φI ; (q)
however, from the its equation of motion, it can be easily seen that φI
is of the same order
of magnitude as the above expression. The function D(T ) is a constant of integration, and it must be found via the matching procedure.
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
(1)
The order δ term, φI , is found to satisfy: µ r ¶ µ µ ¶ ¶ ´ ¡ ¢ 1 ³ 2m 1 1 2 2m ¯0 (1) 0 ¯ ξ(ξ − 2m/Rs )φI,ξ = − ξ φe (δT ) + F,φ φ0 ln 1 − φ0 (δT ) . ξ2 Rs ξ 2 ξ Rs ξ ,ξ ,ξ Solving this we find: ¯q ¯ ¯ Rs ξ ¯ − 1 ¯ Rs ξ 1 ¯¯ 2m 2m ¯ φ0e (δT ) = 2 + ln ¯ q Rs 2m 2 ¯ Rs ξ + 1 ¯¯ 2m µ ¶3/2 µ ¶ µ ¶ ¡ ¢ 2m 2m 3/2 2m + l(ξ)F,φ φ¯0 φ¯00 (δT )) − A ln 1 − . Rs Rs Rs ξ µ
(1)
φI
¶3/2
r
(3.44)
where A is a constant of integration and µ ¶ õ ¶ µ ¶ µ ¶ 2m −1 Rs ξ 3/2 2m Rs ξ 1/2 l(ξ) = − dξ ξ 1− ln 1 − −2 Rs ξ 0 2m Rs ξ 0 2m q q ¯ ¯ ¯ 1 + 1 − 2m − 2m ¯ ¯ Rs ξ 0 Rs ξ 0 ¯ ¯ ¯ . q q −2 ln ¯ ¯ ¯ 1 + 1 − R2mξ0 + R2mξ0 ¯ s s Z
ξ
0 0−2
The term proportional to l(ξ) dies off as ξ 1/2 , and is suppressed by a factor of 2m/Rs ξ relative to the first term in eq. (3.44). Both these terms only depend on time through δT and are thus deemed quasi-static. These terms do not, therefore, change the leading-order behaviour of the time derivative of φ in the interior. Although we are concerned mostly with objects which are much larger than their Schwarzschild radii it would be nice to be able to address the problem of black-hole gravitational memory via this method. For the zeroth-order approximation to φ to be well-defined on the horizon we need F (φ¯0 ) = 0 (this is just a statement of the ‘no-hair’ theorem for Schwarzschild black (δ)
holes). We can remove any divergence in φI
near the horizon by an appropriate choice of
the constant of integration, A. We take A = φ0e (δT ). This is analogous to what was done by Jacobson in [131]. We can now absorb the first and last terms on the RHS of eq. 3.44 into the definition of φe (δT ) by a definition of the time coordinate:
where
φe (δT ) → φe (δT 0 ) r ¯ ¯! µ ¶3/2 Ãr ¯ ¯ 2m R ξ 2m s ¯ . T0 = T + 2 − ln ¯¯1 + Rs 2m Rs ξ ¯
The zeroth-order matching is not affected since δ(T 0 −T ) remains sub-leading order in any intermediate region. We note that LI T 0 = v −R−2m ln(R/2m) where v = ts +R+2m ln(R/2m−1)
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
105
and ts is the usual Schwarzschild time coordinate. Thus the leading-order homogeneous term in φI has the same behaviour as that predicted by Jacobson (see ref. [131] and section 3.2.2 above). By expanding φe w.r.t. to co-moving time, LI T , rather than the Schwarzschild time, ts , as Jacobson did, we avoid one of the problems encountered in his analysis: that the time coordinate use to make the expansion diverges on the horizon. At lower orders we can always ensure, by choice of the constants of integration, that limR→0 (1 − 2m/R)∂R φ|T = 0. This boundary condition ensures that the inner approximation does not diverge as R → 2m, and remains valid on the horizon. This justifies the statement that, if p > 1, Jacobson’s prediction is itself a valid asymptotic approximation to φ on the horizon. The condition p > 1 is equivalent to: ²local /²c ¿ 1/(2mH0 )(À 1), with ²l being the average value of the energy density, outside the black-hole, over length scales O(2m) from the horizon. We are now in a position to study the conditions under which the local time variation of φ tracks its cosmological value. In the interior we have: φI,T
µ µ ¶¶ ¡ ¢ 2m ¯ ≈ 1 + F,φ φ0 ln 1 − (3.45) Rs ξ "Z # ¶ µ η µ1 (η 0 ),η 2m F (φ¯0 ) µ1,η (ξ = 1, T ∗ ) 0 q dη 5/2 0 + ... +δ B,φ (φc ) + 1− Rs B(φc ) ξ (η , T ) ξ 3/2 δφ0e (δT 0 )
The excluded terms are then certainly smaller than the second term, however, since they might be larger than the first term, both numerically and in the limit δ → 0, the above expression is not a formal asymptotic approximation. When the first term in eq. (3.45) dominates, the condition φ˙ ≈ φ˙ c holds, and the local time evolution of the scalar field is, to leading order, the same as the cosmological one. Assuming our background choice is suitable for the application of this method, condition φ˙ ≈ φ˙ c fails to hold if, and only if, the second term in this expression dominates over the first.
3.5.5
Matching Conditions
By considering the growing modes in the interior expansion for φ, and the singular modes in the exterior expansion, we can when an intermediate matching region does and does not exist. It is imperative, if our method is to be valid, that a matching region does indeed exist.
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
Case I: McVittie background At O(δ 2 ), the interior expansion of φ in the McVittie metric has a mode that grows like ξ. This asymptotic expansion certainly ceases to be valid if, when rewritten in some intermediate scaling region, where ξ ∝ δ −α ξ say with 0 < α < 1, terms appear that scale as inverse powers of δ. In such a region δ 2 ξ ∼ δ 2−α ξ, and so such terms are always suppressed by at least a (0)
single power of δ. The zeroth-order term, φI , in the interior has itself ceased to be a valid asymptotic approximation to φ when the δ 2 ξ terms dominate over the 1/ξ term. This will occur only when ξ ∝ δ −1 , i.e. in the exterior region. At order δ 4 , the fastest growing mode goes like ξ 3 , and this will also only conflict with the first two terms in the expansion when ξ ∝ δ −1 . We conclude that eq. (3.41) is a valid asymptotic approximation to φ everywhere outside the exterior region. Similarly we note that the most singular term in the first two terms ¡ ¢ of the exterior expansion, as given in Appendix A, goes like δ/ρ. At O δ 2 we would find terms that behave as δ 2 /ρ2 . The exterior expansion therefore only ceases to be valid when ρ ∼ δ, i.e. in the interior region. Therefore, in the McVittie case, the position of the intermediate region is not important; we can choose any intermediate scaling and the matching procedure will be valid. As a final check on our method, we have explicitly performed the matching for the McVittie background in Appendix A, and shown it to be self-consistent.
Case II: Tolman-Bondi models We assume that as µ1 (η) ∼ η n as η → ∞ for some n > 0. At order δ q , the growing mode in the interior approximation grows like δ q η n /ξ for the simultaneous big-bang models. Assuming that in some intermediate region η, ξ ∼ δ −α with 0 < α < 1 we can see that this growing mode dominates over the zeroth order 1/ξ when α = q/n, and the asymptotic approximate fails altogether if n > 1 and α > q/(n − 1). We did not explicitly find the exterior expansion of φ, since we were only really concerned with its behaviour in the interior, however the first non-homogeneous mode should behave as δ p z1 (ρ)/ρ if p ≤ 1 or δ/ρ if p > 1. If z1 (ρ) ∼ ρm as ρ → 0, then the exterior expansion break downs if m<1−p α≤1− m ≥ 1 − p α = 0.
p , 1−m
(3.46)
3.5. VIRIALISED CASE: SPHERICALLY SYMMETRIC BACKGROUNDS
107
Just by considering the behaviour of the next-to-leading order terms we see that if both the interior and exterior zeroth-order approximations are to be simultaneously valid in some intermediate scaling region we must require µ max 0, 1 −
p 1−m
¶ <α<
q . n
If such an intermediate region does indeed exist then the zeroth-order matching performed in section 3.5.4 is valid. The general form of the interior approximation to O (δ p ) is then be correct; the only unknown function in this term is D(δT ). If the matching works to order δ p , as well as zeroth-order, then we have argued that D(δT ) is quasi-static. If the matching procedure does not work to this order then its quasi-static character may be lost. However, we would not expect it to vary in time any faster than the other O (δ p ) terms. So long as we can match the zeroth-order approximations in some region, we can find the circumstances under which φ˙ ≈ φ˙ c by comparing the sizes of the two terms in eqn. (3.45).
3.5.6
Interpretation of results
˙ x, t) ≈ φ˙ c always holds. In the Tolman-Bondi In the McVittie background we found that φ(~ backgrounds we found φI,T to be given by eqn. (3.45). Using the leading definition of κε and R,t we can rewrite the expression for φI,T in a more usable and understandable form. We are interested in time derivatives of the φ field. From the expression for φ in the interior, we see that t0 = LI T 0 seems to play the role of a natural time coordinate. At radii where 2m/R ¿ 1, the interior metric is close to diagonal when written in (t0 , R) coordinates; it is in this sense a natural time coordinate for an observer at fixed R. In this region, t0 coincides with the standard Schwarzschild time coordinate. As R → ∞, t0 → t. We therefore consider φ,t0 . Whenever the required conditions of the previous section hold, the matching procedure is valid, and we find (0)
(q)
φ,t0 (r, t) ≈ φI,t0 + δ q φI,t0 + o(δ q ) in the interior, where the ≈ sign means that whilst this is not a formal asymptotic series (since there may be excluded terms that are bigger than some of the included ones) this is a good numerical estimate since at least one of the included terms is bigger than all the excluded ones.
108
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
The terms in this expression are given by: (0)
∼ φ˙ c (t) + ∆t(r, t)φ¨c (t), Z r ∆(R,t )R2 − Rs2 ∆(R,t )|R=Rs ∆t(r, t) = dr0 R,r R2 (1 − k(r0 ) − R,t2 ) ∞ φI,t0
(3.47) (3.48)
where ∆R,t = R,t − HR, µZ (q) δ q φI,t0
r
dr0 R,r ∆(R,t κε)(r0 , t) ∞ ! õ ¢ µ ¶¡ 2 ¶ ! ¯ Rs ∆(R,t κε) R=Rs F (φ0 ) 2m 2 + 1− . +O B,φ (φc ) R Rs
∼ −B,φ (φc )
(3.49)
where ∆R,t ε = R,t ε − Hεc . Equation (3.49) has been derived for the case 2m/Rs ¿ 1. For 2m/Rs ≈ 1, they are accurate when R À 2m and otherwise provide order-of-magnitude (q)
0
(q 0 )
estimates for δ q φI,t0 and δ q φI,t0 respectively. ˙ x, t) ≈ φ˙ c (t) and also state the We can now evaluate these terms to find out when φ(~ precision to which this approximate equality holds. In many cases, however, a lot of the terms in the above expression are negligible or cancel, and so we can find a more succinct ˙ x, t) ≈ φ˙ c (t) to hold. When 2m/Rs À 1 we expect necessary and sufficient condition for φ(~ ¡ ¢ ¡ ¢ F φ¯0 ≈ B,φ φ0I (1 + O(2m/Rs )), and so: Z r 0 φI,t − φc,t ≈ −B,φ (φc ) dr0 R,r κ∆(R,t ε(r0 , t)) + ∆t(r, t)φ¨c (t). (3.50) ∞
We refer to this last term as the drag term, and it is responsible for the local value of φ˙ lagging slightly behind the cosmological one - this effect was first observed by Jacobson in the study of gravitational memory, [131]. Whenever the cosmological φ is not potential dominated, and our ‘star’ resides in a local overdensity of matter, the drag term is negligible compared to the other terms in this expression. If the potential term dominates the cosmological evolution then it is possible for the drag term to give the dominant effect, even if we have an local over-density of matter. But whenever this happens we always have |∆t φ¨c /φ˙ c | ¿ 1 and so we ˙ x, t) ≈ φ˙ c (t). have φ(~ Ignoring the drag term we find: φI,t0 − φc,t = −B,φ (φc ) φc,t
Z
R
∞
dR0
∆v(R0 , t)κε(R0 , t) , φ˙ c
where v(R, t) is the radial component of the velocity of the matter particles at position (R, t) relative to an observer fixed at (R, t). In the Tolman-Bondi models v = R,t , cosmologically
3.6. VIRIALISED CASE: NON-SPHERICALLY SYMMETRIC BACKGROUNDS
109
v = H. Provided that the matching conditions hold, a necessary and sufficient condition for ˙ x, t) ≈ φ˙ c near the surface of a virialised over-density of matter in spherically symmetric φ(~ dust spacetimes is, therefore, that ¯ Z ¯ ¯ I := ¯B,φ (φc )
R
dR
0 ∆v(R
∞
¯ ¯ ¿ 1. ¯
0 , t)κε(R0 , t) ¯
φ˙ c
The matching conditions can be rephrased as: lim R2 κ∆ε = o(1)
(3.51)
lim 2(m + Z)/R = o(1)
(3.52)
δ→0 δ→0
α 1−α as δ → 0, with LαI L1−α E (t − t0 ), LI LE R held fixed, for all α ∈ (0, 1). In the next section we
consider how this result generalised when non-spherically symmetric spacetime backgrounds are used.
3.6
Virialised Case: Non-spherically symmetric backgrounds
In this section we show how the results derived in the previous section can be extended to include a class of background spacetimes which have no symmetries. We limit ourselves to considering spacetimes in which the background matter density satisfies a physically realistic equation of state, specifically that of pressureless dust (P = 0). To simplify matters further we require that the conditions listed in section 3.4 hold which means that we are dealing with a Szekeres-Szafron background spacetime.
3.6.1
Szekeres-Szafron backgrounds
The metric and equations that describe Szekeres-Szafron backgrounds are given in section 3.4.2 above. We use the freedom we have in the definition of R to prescribe that at some epoch of interest, t = t0 , R is the physical radial coordinate i.e. R(t0 , r) = r and the surface (t, R) = const always has area 4πR2 ; this requirement, combined with the simultaneity of the initial big-bang curvature singularity , determines the form of k(r). The surface of our massive spherical body is at R = Rs . The active gravitational mass interior to the surface (t = t0 , r) = const is given by M (r) = m + Z(r). We identify m as the gravitational mass of the central body, or ‘star’, at t = t0 . Over time the mass of this body grows due to accretion. We must require that the density of matter be positive definite and that spatial homogeneity
110
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
emerges as r → ∞. As in the spherically symmetric Tolman-Bondi case, these requirements provide us with three condition on Z(r):
Z(Rs ) = 0, Z,r > 0, 1 (0) 2 3 lim Z(r) = Ω H r . r→∞ 2 dust 0 To zeroth-order in exterior limit we wish to have a FRW spacetime with curvature parameter k; this requires limr→∞ k(r) = kr2 . We must require k(r) > 0 in the interior region; however we do not require k(r) > 0 everywhere. The exact solution for R(r, t) was found by Barrow and SteinSchabes and is given in ref. [151]. We will shortly see that we can split the both the exterior and interior metric perturbations into spherically symmetric perturbations and non-spherically symmetric ones. The spherically symmetric perturbations are precisely the same as they where in the Tolman-Bondi case. Similarly the spherically symmetric dilaton perturbations have precisely the same form as they do in the Tolman-Bondi case and we have already calculated these. We need only, therefore, concentrate on the non-spherically-symmetric perturbations. We now consider the exterior and interior expansions on the Szekeres-Szafron backgrounds.
Figure 3.4: Sketch of a quasi-spherical Szekeres-Szafron spacetime. The surfaces (t, r) are spheres, which are concentric to leading order in δ in both the exterior and interior limits.
3.6. VIRIALISED CASE: NON-SPHERICALLY SYMMETRIC BACKGROUNDS
3.6.2
111
Exterior Expansion
As a result of the way that the inhomogeneity is introduced in these models, we want the FRW limit to be ‘natural’ , that is for the O(3) orbits to become concentric in this limit; we therefore require νr ∼ o(1) as δ → 0 in the exterior. This follows from the requirement that the whole spacetime should become homogeneous in a smooth fashion in the limit where the mass of the ‘star’ vanishes: m → 0. In other words, the introduction of our star is the only thing responsible for making the surfaces of constant curvature non-concentric. We define, as in the previous section, dimensionless ‘radial’ and time coordinates appropriate for the exterior by
τ = H0 t,
ρ = H0 r.
The exterior limit is defined by δ → 0 with τ and ρ fixed. In the exterior region we find asymptotic expansions in this limit. According to our prescription, we write
1 H0 Z(ρ) ∼ Ωm ρ3 + δ p z1 (r) + o(δ p ), 2 and introduce functions li (ρ) so
H0−1 λi ∼ δ s li (ρ) + O(δ s ). Since H0−2
¡ 2M ¢ R3
,R
∼ O(δ p , δ), we have that: H0−2 κεns ∼ O(δ p+s , δ 1+s ) whereas H0−2 κεs ∼
O(δ p , δ). Thus, the non-spherical perturbation to the energy density is always of subleading order compared to the first order in spherical perturbation. The first-order, non-spherical, metric perturbation appears at O(δ s ); however, since this is equivalent to a coordinate transform on (r, θ, ϕ) and the dilaton field, φ, is homogeneous to leading order in the exterior, this perturbation does not make any corrections to the dilaton conservation equation at O(δ s ). Thus, both at leading order, and at next-to-leading order, both the energy density and the dilaton field behave in the same way as in the spherically-symmetric Tolman-Bondi case. As in the preceding section we are not especially interested in the exterior solution for φ beyond zeroth order, just the effect of any background variation of φ on what is measured on the surface of a local ’star’.
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
3.6.3
Interior Expansion
We define dimensionless coordinates for the interior in the same way as we did for the spherically symmetric case: T = L−1 I (t − t0 ) and ξ = R/Rs . We define the interior limit to be δ → 0 with T and ξ fixed, and perform out interior asymptotic expansions in this limit. To lowest order in the interior region, we write Z ∼ δ q Rs µ1 , and λi := δq0 Rs−1 bi , where i = {x, y, z}. The condition that κε > 0 everywhere requires q 0 ≥ q and then, to next-to-leading order, the interior expansion of φ is the same as it was in the spherically-symmetric Tolman-Bondi case. We can potentially include a non-spherical vacuum component for φ; however, this is entirely determined by a boundary condition on R = Rs and the need that it should vanish for large R. To find the leading-order behaviour of the φ,T we need to know φ at next-to-leading order. The only new case we need to consider therefore is when q 0 = q, i.e. κεns ∼ O (κεs ). We define a new coordinate η in the same way as we did in ¡ ¢2/3 the Tolman-Bondi case: η = ξ 3/2 + 3T /2 ; Rs η = r + O(δ q , δ 2/3 ). The interior expansion of k(r) is unaltered by the inclusion of non-spherically symmetric modes i.e. k ∼ δ 2/3 k0 (δT ) with k0 as given by eqn. (3.29). The spherically symmetric metric perturbations are the same as they were in Tolman-Bondi case - see eqn. (3.30). There the first non-spherically symmetric 0
metric perturbation occurs as order δ q : ³ ´ 0 (q 0 ) 0 s ds2int ∼ Rs2 jab + δ q jab dxa dxb + o(δ q ), (q)
(2/3)
s ∼ j 0 + δq j 2/3 j where jab ab ab + δ ab
(0)
(q)
(2/3)
+ ... and jab , jab and jab
are given by eqns. (3.31), (q 0 )
(3.5.2) and (3.33) respectively. The non-spherically symmetric perturbation, jab is given by (q 0 )
jab dxa dy b = 2(bz cos θ + bx cos ϕ sin θ + by sin ϕ sin θ)ξdη 2
(3.53)
−2ξ 2 [bz sin θ + bx cos ϕ(1 − cos θ) + by sin ϕ(1 − cos θ)] dθdη −2ξ 2 (1 − cos θ) sin θ(bx sin ϕ − by cos ϕ)dϕdη. The interior expansion of the spherically symmetric part of the local energy density, κεs is the same as it was in the Tolman-Bondi case Rs2 κεs = δ q
2m µ1,ξ . Rs ξ 3/2 η 1/2
3.6. VIRIALISED CASE: NON-SPHERICALLY SYMMETRIC BACKGROUNDS
113
The non-spherically symmetric part is: Rs2 κεns = −δ q
3.6.4
0
6m (bz cos θ + bx sin θ cos ϕ + by sin θ sin ϕ) . Rs ξ 3/2 η 1/2
Solution of φ equations
The boundary conditions and zeroth order behaviour of φ are given by the same formulae as they were in the spherically symmetric case (see sections 3.5.3 and 3.5.4). As mentioned above, in the exterior limit the next-to-leading order behaviour of φ is precisely the same as it was in the spherically symmetric case. In the interior limit, the expansion of φ to next-to-leading order can be broken into spherically symmetric modes and non-spherically ones: 0
(q 0 )
φ ∼ φsI + δ q φI
0
+ o(δ q ),
where (0)
(1)
(q)
φs ∼ φI + δφI + δ q φI + ... , (0)
(q)
and φI , φI
(1)
and φI
are given by eqns. (3.39), (3.43) and (3.44) respectively. We split the (q 0 )
non-spherically symmetric mode φI (q 0 )
φI
(q 0 )z
:= φI
(q 0 )i
We find that the φI
up as follows: (q 0 )x
(ξ, T ) cos θ + φI
=
(ξ, T ) sin θ sin ϕ.
evolve according to:
³ ´ 0 0 3/2 φ(q )i + 3 φ(q )i + − 2m ξ I,T T Rs 2 I,T − 6m Rs B,φ
(q 0 )y
(ξ, T ) sin θ cos ϕ + φI
³
1 η 1/2
¡ 0 ¢ bi (η) ³ 2m ´ 1 ¡ ¢ φI η1/2 − Rs η1/2 F φ¯0
´
ξ 5/2 (q 0 )i φ η 1/2 I,η ,η
−
(1)i 2 φ ξ 1/2 I
(3.54)
"µ # µ 2m ¶¶ 1+ R ξ s bi (η)ξ 1− 2m − 2bi (η) . Rs ξ
,η
As in the previous section, when 2m/Rs ¿ 1, we can find analytic expressions for these modes: (q 0 )i φI
Z η Z 0 ¡ 0¢ ¡ 0¢ 1 η 2m 2m 0 bi (η ) B,φ φI ξ B,φ φI 2 ∼ − dη 0 2 + dη 0 ξ 0 bi (η 0 ) Rs Rs ξ ξ=1 ξ Z 2m ¡ ¯ ¢ 1 η Ci (T ) − F φ0 2 + Di (T )ξ + O((2m/Rs )2 ), dη 0 bi (η 0 )ξ 0 + Rs ξ ξ=1 ξ2
(3.55)
where i = x, y z. When 2m/Rs ≈ 1 the above expression can be seen as an order of magnitude (q 0 )i
estimate for the φI
. The Di (T ) and Ci (T ) are constants of integration, with Di (T ) deter-
mined by the matching procedure. The value of Ci (T ) should be set by a boundary condition on R = Rs . We cannot specify Ci (T ) exactly without further information about the interior
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CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
of our ‘star’ in R < Rs . If we assume that the prescription for the sub-leading order boundary condition given above is correct then we find: (q 0 )i ∂ξ φI |ξ=1
⇒ Ci
¯ ¡ ¢ 2m bi ¯¯ ∼ − φ¯0 + O((2m/Rs )2 ) F ¯ Rs η 1/2 ξ=1 Z ¡ 0 ¢ ξ=1 0 bi (η 0 ) 1 m = − B,φ φI dη 0 2 + 2 Di Rs ξ
From now on, we set Ci = 0, for simplicity. Even when this is not exactly satisfied, we do not (1)i
expect the magnitude of Ci or Ci,T to be larger than any of the other terms in φI
(1)i
or φI,T ,
respectively.
3.6.5
Matching Conditions
All of the conditions found above for the matching of the spherically symmetric parts of φ to be possible still apply here. However, we must now satisfy some extra conditions that come from the requirement that the non-spherical parts should also be matchable. We assume that bi (η) ∝ η di as η → ∞ for some di > 0. At order δ q , the growing mode in the non-spherically symmetric part of the interior approximation will then grow like δ q η di +1 /ξ. In the intermediate, or matching, region we have that η, ξ ∼ δ −α for some α ∈ (0, 1). We require φI to have a valid asymptotic expansion this region. This implies that there exists some α ∈ (0, 1) such that, for each i, we have α − q/di > 0. 0
In the exterior we write H0−1 λi ∼ δ pi li (ρ), where p0i > 0 comes from the requirement that the 2-spheres of constant curvature become concentric in the exterior limit. As ρ → 0 we assume that li (ρ) ∝ ρ−fi . We previously stated that Z ∼ 12 Ωm ρ3 + δ p z1 + o(δ p ) in the exterior. We assume that as ρ → 0, we have z1 ∝ ρ−m . Although we did not explicitly consider the exterior expansion of φ we can now examine the behaviour of the leading-order non-spherically symmetric mode in the intermediate limit of that exterior expansion. We noted above that 0
there is no O(δ pi ) correction resulting from the li . The leading-order mode therefore either ´ ´ ³ ³ 0 0 goes like maxi δ p+pi z1 (ρ)li (ρ) if p < 1 or maxi δ 1+pi (ρ)li (ρ) otherwise, and ρ ∼ O(δ 1−α ) in the intermediate region. For the exterior expansion to be valid in the intermediate region we therefore require max(p0i + (1 − α)(fi + m)) > −p if p ≤ 1, i
max(p0i + (1 − α)fi ) > −1 if p ≥ 1. i
3.6. VIRIALISED CASE: NON-SPHERICALLY SYMMETRIC BACKGROUNDS
115
These conditions on α are equivalent to the following: there exists α such that the interior expansion of R2 κεns is o(1) as δ → 0 for all 0 < α0 < α where ξ, T ∼ O(δ −α ), and the exterior expansion of R2 κεns is also o(1) as δ → 0 for all 0 < α00 < α where ρ, τ − τ0 ∼ O(δ 1−α ). This suggests that the condition for the matching procedure to work, as far as the spherically non-symmetric modes are concerned, is simply that R2 κεns ¿ 1 everywhere. If we combine these conditions with those found for the spherically symmetric modes, we see α 1−α that our analysis is valid provided that: for all α ∈ (0, 1), and keeping LαI L1−α E (t−t0 ), LI LE R
fixed, we have limδ→0 R2 κ∆ε = o(1) and limδ→0 2(m + Z)/R = o(1). Strictly speaking, since α ∈ (0, 1) (as opposed to [0, 1), (0, 1] or [0, 1]) we can also replace ∆ε by just ε since R2 κεF RW is small everywhere outside the exterior region. For Szekeres-Szafron backgrounds the first of these conditions implies the second everywhere outside the interior region. These conditions are equivalent to the statement that, in any intermediate region, the background spacetime is asymptotically Minkowski as δ → 0. The power of our method is that we do not require this to be true of the interior and exterior regions. So long as this condition holds in the intermediate region, we can match the interior and exterior approximations in some region and therefore find the circumstances under which condition φ˙ ≈ φ˙ c holds.
3.6.6
Matching and Results (q 0 )i
We rewrite the expression for the φI
in terms of the non-spherical part of local energy
density: 0
(q 0 )ns
δ q φI
³ 0 ´ (q )z (q 0 )x (q 0 )y φI cos θ + φI sin θ cos ϕ + φI sin θ sin ϕ Z r ¡ 0¢ 1 R ˆ ∼ B,φ φI R D(T, θ, φ) dr0 R,r κεns (r0 , t) − 3 Rs Z r ¡ ¢ 1 1 − B,φ φ0I dr0 R,r R3 κεns (r0 , t) 3 R2 R=Rs Z r 1 ¡ ¢ 1 + F φ¯0 dr0 R,r R3 κεns (r0 , t) 3 R2 R=Rs = δq
0
ˆ where D(T, θ, φ) := Dz cos θ+Dx sin θ cos ϕ+Dy sin θ sin ϕ. By examining the dilaton equations of motion in the FRW region, we can see there the leading-order (θ, ϕ)-dependent term in the
116
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
exterior expansion or φ behaves like 1 B,φ (φc )R 3
Z
r
∞
dr0 R,r κεns (r0 , t)
ˆ such that for R ¿ H0−1 and t fixed. Therefore matching requires that we choose D (q 0 )ns
δ q φI
³ 0 ´ (q )z (q 0 )x (q 0 )y = δ q φI cos θ + φI sin θ cos ϕ + φI sin θ sin ϕ Z r ¡ 0¢ 1 ∼ − B,φ φI R dr0 R,r κεns (r0 , t) 3 ∞ Z r ¡ 0¢ 1 1 dr0 R,r R3 κεns (r0 , t) + B,φ φI 3 R2 R=Rs Z r 1 ¡ ¢ 1 + F φ¯0 dr0 R,r R3 κεns (r0 , t). 3 R2 R=Rs
(3.56)
0
The interior expansion is now fully specified to order O(δ q ). As in the spherically symmetric case we are interested in time derivatives of the φ field and t0 = LI T 0 seems to play the role of a natural time coordinate for an observer at fixed R. When 2m/R ¿ 1, t0 coincides with the usual Schwarzschild time coordinate and as R → ∞, t0 → t. We consider φ,t0 in the interior. The spherically symmetric contribution to this quantity, φs,t0 , is the same as it was in the previous section - see eqn. (3.49). We therefore focus on the non-spherically-symmetric (q 0 )
modes. Rewriting φI
in terms of R,t and κεns and applying the matching we arrive at:
0 (q 0 ) δ q φI,t0
Z r ¡ 0¢ κεns (r0 , t) 2 dr0 R,r R,t ∼ − B,φ φI R 3 R ∞ Z r ¡ 0¢ 1 1 − B,φ φI dr0 R,r R,t R2 κεns (r0 , t) 3 R2 R=Rs Z r 1 ¡¯ ¢ 1 + F φ0 dr0 R,r R,t R3 κεns (r0 , t) 3 R2 R=Rs õ ¶ ! 1 ¡¯ ¢ 2m 2 − F φ0 RR,t κεns (r, t) + O . 3 Rs
(3.57)
Equation (3.57) is derived under the assumption 2m/Rs ¿ 1. If 2m/Rs ≈ 1 then the above expression is accurate when R À 2m. In other circumstances it provides an order-of-magnitude 0
(q 0 )
estimate for δ q φI,t0 . ˙ x, t) ≈ φ˙ c (t) and also state the We can now evaluate these terms to find out when φ(~ precision to which this approximate equality holds. In many cases, however, a lot of the terms in the above expression are negligible or cancel, and so we can find a more succinct ˙ x, t) ≈ φ˙ c (t) to hold. When 2m/Rs À 1 we expect necessary and sufficient condition for φ(~
117
3.6. VIRIALISED CASE: NON-SPHERICALLY SYMMETRIC BACKGROUNDS
¡ ¢ ¡ ¢ F φ¯0 ≈ B,φ φ0I (1 + O(2m/Rs )), and so: Z
r
dr0 R,r R,t κ∆εs (r0 , t) Z r 2 κεns (r0 , t) dr0 R,r R,t − B,φ (φc ) R 3 R ∞ 1 − B,φ (φc ) RR,t κεns (r, t) + ∆t(r, t)φ¨c (t). 3
φI,t0 − φc,t ≈ −B,φ (φc )
(3.58)
∞
We refer to the last term as the drag term as it results in the local value of φ˙ lagging slightly behind the cosmological one. As we noted for the spherically symmetric case, the drag term is usually unimportant for when there is a local over-densities of matter and so can generally be ignored. The non-spherically symmetric parts of energy density enter into the expression differently. The magnitude of the terms on the left-hand side of eq. (3.58) is, as in the spherically symmetric case, still hH0 R∆R,t ε/εi (R, t) where h·i (R, t) represents some ‘average’ over the region outside the surface (R, t) = const. We should note that, given the condition on κε that has been required for matching, the leading-order contribution to κεns is everywhere of dipole form and this is responsible for the special form of the average over the non-spherically symmetric terms. We can also see that, as a result of form of eqn. (3.58), peaks in κεns that occur outside of the interior region will, in the interior, produce a weaker contribution to the left-hand side of eqn. (3.58) than a peak of similar amplitude in a spherically symmetric energy density κεs . This behaviour would continue if we were also to account for higher multipole terms in κεns . The higher the multipole, the more ‘massive’ the mode, and the faster it dissipates. If we are interested in finding a sufficient condition (as opposed to a necessary and sufficient one) for φ˙ ≈ φ˙ c to hold locally, then in most circumstances we are justified in averaging over the non-spherically symmetric modes in the same way as we average over the spherically symmetric ones. In most cases, this will over-estimate rather than under-estimate the magnitude of the right-hand side of our condition, (3.58). We therefore alter the definition of the quantity I from that which was given for it in the spherically symmetric case in the following way: Z I := B,φ (φc )
R
∞
dR0
maxθ,φ (sin θ∆(vε)) ¿1 φ˙c
(3.59)
where v is the radial velocity of the matter particles (i.e. v = R,t ). When the background
118
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
spacetime is Szekeres-Szafron we have that: ¯ ¯ ¯ φ,t0 − φc,t ¯ ¯ ¯ ¯ φc,t ¯ . I with equality in the spherically symmetric case. The strong inequality I ¿ 1 is therefore a ˙ x, t) ≈ φ˙ c (t), and the value of I gives a measure of the amount by sufficient condition for φ(~ which φ,t0 and φc,t differ. In addition to having shown this for cases where the background is Szekeres-Szafron, and that matching conditions hold, we also conjecture that, even if the matching conditions formally fail, and for other classes of spacetime, that I ¿ 1 is a sufficient ˙ x, t) ≈ φ˙ c (t) on the surface of a virialised over-density of matter. This completes condition for φ(~ the extension of our main result to non-spherically symmetric backgrounds. We see in section 3.8 below that I ¿ 1 holds for physically realistic distributions of matter, and that as we ˙ x, t) ≈ φ˙ c,t to hold for an Earth-based observer. expect φ(~
3.7
Collapsing Backgrounds
When a spacetime undergoes gravitational collapse it is possible for a black-hole to form inside the collapsing region. In the Tolman-Bondi model a black-hole horizon appears when 2M (r)/R = 1. If we have κ∆εR2 = (κε − κεc )R2 ¿ 1 outside the horizon then we can apply the results of the sections 3.5 and 3.6 taking the surface of our ‘star’ to be the black-hole horizon. This is also true for any virialised region in the interior of the collapsing region, not just for black holes. In this section we extend the results of previous two sections to the case where the collapsing region has no central black-hole or virialised region. For simplicity we consider only sphericallysymmetric, dust-plus-Λ cosmologies i.e. Tolman-Bondi models. We do not require the big bang to be simultaneous for all observers. This extension requires that curvature of the interior spacetime be in some sense weak so that the metric is Minkowski to zeroth order. We require: R2 κ∆ε ¿ 1,
2∆M/R ¿ 1
everywhere; ∆ε = ² − ²c and ∆M = M − 16 κ²c . in the interior. In the exterior we assume, as before, that the spacetime is FRW to zeroth order. It is clear that in this model the following
119
3.7. COLLAPSING BACKGROUNDS
parameters, δ1 and δ2 are everywhere small: δ1 (R, t) = R2 (κε − κεc ) = δ2 (R, t) =
2M,r − 3Ωm H 2 R2 , R,r
2M − Ωm H 2 R 2 . R
In addition, the following parameter, δ3 , is small in interior region but O(1) in the exterior: δ3 (R, t) = H 2 R2 . In the interior δ3 ¿ δ1 , δ2 . For the purposes of our asymptotic expansions we treat and δ1 and δ2 as being of the same order. For the interior to be collapsing we need k(r) > 0. The condition that R,t2 = −k(r) + δ2 + (Ωm + Ωλ )δ3 > 0 implies that k(r) . O(δ2 ), and R,t2 ∼ O(δ2 ) in the interior. We perform the matching in an intermediate region where δ1 ∼ δ2 ∼ δ3 ¿ 1. It is clear that with these definitions that such an intermediate region must always exist.
3.7.1
The Interior
In the interior we write the metric as: ds2 =
(1 − δ2 + δ3 (Ωm + Ωλ ))dt2 2R,t dRdt dR2 + − − R2 {dθ2 + sin2 θdϕ2 }, 1 − k(r) 1 − k(r) (1 − k(r))
which is flat spacetime to lowest order in the δi . The dilaton, φ, obeys: −R2 ¤φ = B,φ (φ)(δ1 + 3Ωm δ3 ) + V,φ (φ)R2 , and, in line with our previous assumptions, we have V,φ (φ)R2 ∼ O(ΩΛ δ3 ). We note that 1/2
3/2
∂t δ1 ∼ O(δ1 δ2 ) = o(δ1 ) and so δ1 is quasi-static; as such we expect φ to also be quasi-static in the interior. We can solve the equations for φ order-by-order in the interior, requiring (as a boundary condition) that φ is regular at R = 0: Z R Z R ¡ ¢ dR0 0 ˙ φ ≈ φe (t) + B,φ (φe ) δ (R , t) + φ (t) dR0 R,t − HR0 (3.60) 2 e 0 C R D i ³ ´ 1h 2 + R V,φ (φe ) + 3B,φ Ωm δ3 + (φ¨e (t) + 3H φ˙ e (t))R2 + O δ12 , δ22 , δ32 , (Rφ˙ e )3 , δ3 (Rφ˙ e ) , 6 1/2
1/2
1/2
where φe (t) is O(1) but quasi-static i.e. Rφ˙ e = o(δ1 , δ2 ); the third term is O(δ2 Rφ˙ e ). Since the above expression is not a formal asymptotic expansion as such we cannot be sure
120
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
that the neglected terms are smaller than all of the included terms; indeed, as we show below, the matching ensures the vanishing of the term in [..]. This is because we do not know precisely how the sizes of δ3 and Rφ˙ e relate to those of δ1 and δ2 . What we do know is that, in the interior, the excluded terms are smaller than at least one of the included terms. The limits C and D as well as φe (t) must be found matching the interior expansion to the exterior one.
3.7.2
The Exterior
In the exterior, we define a coordinate % = R/a(t) ∼ r where a(t) is the FRW scalar factor. In (t, %) coordinates the metric reads: ds2 = dt2 (1 + O(δ22 , (∆k)2 ) +
2 (R,t − HR) ad%dt a2 d%2 − − a2 %2 {dθ2 + sin2 θdϕ2 } 1 − k(r) 1 − k(r)
where ∆k = k(r) − k0 %2 , and k0 = limr→∞ k(r)/r2 , (1 − Ωm − ΩΛ )H 2 = −k02 /a2 and 2 (R,t − HR) ∼ (−∆k + δ2 )/HR. As R, % → ∞ and the inhomogeneity is removed (i.e. ∆k, δ1 , δ2 → 0) we require that φ → φc (t). As with the virialised case our only interest in the subleading order behaviour of φ in the exterior is so as to match it to the interior approximation. It is only necessary therefore to consider how the exterior approximation to φ˙ behaves in the intermediate region where all the δi are small. In the intermediate region the exterior approximation is: Z φ ∼ φc (t) + φs (%, t) + B,φ (φc )
R
∞
dR0 δ2 (R0 , t) + φ˙ c (t) R0
Z
R
∞
dR0 (R,t − HR0 ) + O(δ12 , δ22 , δ33 ),
where φs (%, t) = o(1) is some vacuum mode i.e. ¤F RW φs = 0.
3.7.3
Matching and Results
Now we match the interior and exterior expansions and find that C = D = ∞, φe (t) = φc (t) and φs = 0. The requirement that φe (t) = φc (t), combined with the cosmological evolution equation for φc (t) ensure that the term in [..] in eqn. (3.60) vanishes. We find that the matched interior approximation is given by: Z R dR0 0 ˙ δ (R , t) + φc (t) dR0 (R,t − HR0 ) + O(δ12 , δ22 , δ33 ) φ ∼ φc (t) + B,φ (φc ) 0 2 R ∞ ∞ Z R 0 dR ∼ φc (t0 = t + ∆t) + B,φ (φc ) δ (R0 , t) + O(δ12 , δ22 , δ33 ). 0 2 R ∞ Z
R
121
3.8. RESULTS AND CONSEQUENCES
where the lag in the time coordinate, ∆t, is given by: Z R ∆t = dR0 (R,t − HR0 ). ∞
This coincides (to leading order) with the expression for the virialised case, eqn. (3.48), for Rs = 0. It seems natural, in the interior, to consider the time derivative of φ w.r.t. t0 . As noted in the previous section, t0 looks like the Schwarzschild time coordinate near the surface of a massive body (that is far outside its out Schwarzschild radius), and t0 → t as R → ∞. We find that: Z 0
φ,t0 (r, t ) − φc,t ∼ −B,φ (φc )
R
∞
dR0 ∆(R,t κε)(R0 , t) + φ¨c (t)∆t + O(δ12 , δ22 , δ33 )
where ∆(R,t κε)(R, t) = R,t κε(R, t) − HRκεc . We note that this is the same as the sphericallysymmetric limit of the result found in eqn. (3.58) for the virialised case. This completes the extension of our analysis to the case of spherically symmetric collapsing spacetimes. It is clear that the quantity I, defined in the analysis of the virialised case, is also a good measure of |(φ˙ − φ˙ c )/φ˙ c | in the collapsing case. The validity of the matching procedure is this case is assured by the condition that δ1 , δ2 ¿ 1 holds everywhere.
3.8
Results and Consequences
We now consider the astronomical consequences of our results for observations here on Earth, and answer the basic question of whether local experiments can detect cosmologically varying constants. We can evaluate the quantity I explicitly for an Earth-based experiment assuming the varying constant to be the Newtonian gravitation ‘constant’ G governed by Brans-Dicke theory (since in this case the cosmological evolution of φ is easy to solve). We expect similar values for BSBM, BM and other non-potential-dominated theories for varying α and µ [36, 125]. We consider a star (and associated planetary system) inside a galaxy that is itself embedded in a large galactic cluster. The cluster is assumed to have virialised and be of size Rclust . Close to the edge of the cluster we allow for some dust to be unvirialised and still undergoing collapse. There are three main contributions to I coming from the star, the galaxy, and the galaxy cluster, respectively, and of these the galaxy cluster contribution is by far the biggest. This can be understood by noting that the galaxy cluster is the deepest gravitational potential well, and the galaxy and star are only small perturbations to it. The contribution to I from
122
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
the galaxy cluster is found to be: Iclust .
3 2 H0 (s
− 1/2)−1
p εclust 2Mclust Rclust εc 1/2
=
2 10[3/(2s − 1)]vclust (1 + zvir )3/2 ∆vir 1/2
à −5
= 1.01 × 10
3Ωm
5/6
[3/(2s − 1)](1 + zvir )5/2 ∆vir 1/6
Ωm
!µ
hMclust 1015 M¯
¶2/3 ,
³
vclust ´2 (1 + zvir )3/2 , 103 km s−1 ≈ 1.61 × 10−3 [3/(2s − 1)]Ω−1/2 (1 + zvir )3/2 ¿ 1 m = 4.95 × 10−4 [3/(2s − 1)]Ω−1/2 m
2 3 where we have used 3Mclust /5Rclust = vclust = 3σv2 and κεclust = 6Mclust /Rclust ; σv is the
1-D velocity dispersion and ∆vir ≈ 178 is the density contrast between the cluster and the background at virialisation. In the final line of the approximation we have used the representative value σv = 1040km s−1 ⇒ vvir = 1800km s−1 appropriate for a rich cluster like Coma, [153]. Taking a cosmological density parameter equal to Ωm = 0.27, in accordance with WMAP [92], we expect that for a typical cluster which virialised at a redshift zvir ¿ 1, we would have Iclust ≈ 0.31[3/(2s − 1)] × 10−2 . The term in [..] is unity when s = 2, i.e. 2GM/R → const; such a matter distribution is characteristic of dark matter halos. Different choices of s > 1/2 only change this estimate by a factor that is O(1). We note that, since 2G∆M/R (where ∆M = M − 16 εc R3 ) is required to be small as R → ∞ by the matching conditions, the model used here is only valid for s ≤ 2 and hence the singularity in Iclust at s = 1/2 is fictitious. If we were to have Iclust & 1 then we would require a large virial velocity: vclust & 32, 400[3/(2s − 1)]−1/2 (1 + zvir )−3/4 km s−1 . It is clear that in theories such as BSBM and the KM-models of Chapter 2 which, like BransDicke, have their cosmological evolution dominated by the matter-to-dilaton coupling, B,φ , the local time variation of φ and the associated constant differs from its cosmological value by at most about 1%. In theories where the potential dominates the cosmological evolution this result becomes even stronger and we expect any deviations to occur only at the 0.4|B,φ (φc )/V,φ (φc )|% level, where |B,φ (φc )/V,φ (φc )| ¿ 1. We have seen that I is also good measure of |(φ˙ − φ˙ c )/φ˙ c | inside spherically-symmetric regions that are still undergoing gravitational collapse. Let us now evaluate I for the case of a collapsing cluster in Brans-Dicke theory in the matter era. We assume that the cluster is approximately homogeneous. We further assume that when the cluster eventually virialises,
123
3.8. RESULTS AND CONSEQUENCES
at time tvir , it has a virialisation velocity vvir . We use the spherical approximation detailed in Chapter 5 of [145] to model the collapse of the cluster. We define θ by t = tvir (θ − sin θ)/2π, and find:
¯ à !¯ √ 1/2 ¯ 5v 2 ¯ ∆vir f (θ)(θ − sin θ) 2πg(θ) ¯ vir ¯ √ I(t) = ¯ − + 1/2 ¯ ¯ 2 2 2π ∆vir (θ − sin θ) ¯ ¯ 2 µ ¶¯ ¯ ¯ 5vvir 2g(θ) ¯, ¯ −3f (θ)(θ − sin θ) + = ¯ 4 3(θ − sin θ) ¯
where ∆vir = 18π 2 ≈ 178 is the density contrast at virialisation when θ = 2π. When θ ≤ 3π/2 f (θ) = sin θ(1 − cos θ)−3 and g(θ) = sin θ; for θ ≥ 3π/2, f (θ) = g(θ) = −1. In this evaluation we have included the effect of the ‘drag term’; this is important up to turnaround but it becomes negligible soon afterwards. Turnaround occurs at θ = π, t = tturn . In deriving the −4
10
x 10
Virialisation 8
6 Turnaround 4
2
0 time
Figure 3.5: Plot of (φ,t0 −φc,t )/φc,t vs. time for Brans-Dicke theory at the centre of a collapsing cluster with vvir = 1800km s−1 . 2 , where R above expression we have used 3Mclust /5Rvir = vvir vir is the radius of the cluster
after virialisation and Mclust its mass. The conditions required for the matching procedure to 2 /(1 − cos θ) ¿ 1, and it is clear that this is not satisfied all be valid are equivalent to 10vvir
the way down to θ = 0. For vvir = 1800kms−1 , our method is valid for θ > 0.027 and for the matching conditions to hold from turnaround to virialisation we require vvir ¿ 95000km s−1 . Assuming that the cluster virialises at an epoch that is close to the present day, this bound on vvir translates to requiring Rvir ¿ 432h−1 (1 + zvir )−3/2 Mpc, where H0 = 100h km s−1 Mpc−1 . We observe that I is small up until turnaround and then grows quickly until virialisation. At
124
CHAPTER 3. LOCALLY OBSERVABLE EFFECTS OF VARYING CONSTANTS
turnaround I = 0, and at virialisation we find µ ¶ ´2 ³ 15π 5 vvir 2 I(tvir ) = ≈ 0.85 × 10−3 . − vvir = 2.61 × 10−4 2 12π 103 km s−1 For the final evaluation we have taken vvir = 1800km s−1 (as appropriate for the Coma cluster). The vanishing of I at turnaround is specific to Brans-Dicke theory, more generally: I(tturn ) = 2 [(φ ¨c −B,φ (φc )κ²c )/H φ˙ c ]/27π 2 . In theories where the matter coupling is strongly dominant 40vvir 2 (φ )κ² | À |V (φ )|, we find I(t 2 2 cosmologically, |B,φ c c turn ) ≈ 160vvir |B,φφ (φc )|/27π ¿ 1. ,φ c
Our results differ greatly from those that were found in [107, 108, 109] using the spherical infall model; where I(tvir ) ≈ 200. In that model, the spatial derivatives of φ were assumed to be negligible and are neglected. However, this is can only be a realistic approximation when the collapsing region is as large as the cosmological horizon; for a cluster virialising today that would require Rvir & 5Gpc. Since our method fails for Rvir & 432h−1 (1 + zvir )−3/2 Mpc, there must be some region of intermediate behavior, 500Mpc . Rvir . 5Gpc, that is not described by either the spherical collapse model or our present analysis. We derived these results for Brans-Dicke theory, where φ ∝ G−1 , however we should expect similar numbers for all varying-constant theories where the cosmological dilaton evolution is dominated by its matter coupling, B,φ κεc . In potential-dominated theories, the above numbers are reduced by a factor of |B,φ (φc )κεc /V,φ (φc )| ¿ 1. As in the post-virialisation case, potential domination of the cosmological evolution increases the precision to which local experiments can see cosmologically varying constants. In conclusion: we have used the method of matched asymptotic expansions to find a sufficient condition for the time-variation of a scalar field, and any varying physical ‘constants’ whose variation is driven by such a field, to track its cosmological evolution. We have also proposed a generalisation of this condition for local variations to follow global cosmological variations and conjectured that this generalised condition is applicable to scenarios more general than those we have explicitly considered here. We have seen that this sufficient condition is always satisfied for typical distributions of matter, and we have provided a proof of what was previously merely assumed: terrestrial and solar system based observations can legitimately be used to constrain the cosmological time variation of many supposed ‘constants’ of Nature.
Chapter 4
Chameleon Field Theories
The difficulty lies, not in the new ideas, but in escaping the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds. John Maynard Keynes (1883-1946)
4.1
Introduction
It is almost always the case that scalar fields interact with matter: either due to a direct Lagrangian coupling, as in the BM theory of varying-µ [125], or indirectly through a coupling to the Ricci scalar, as in Brans-Dicke theory [116], or as the result of quantum loop corrections, as was seen to occur in varying-αem the theories discussed in chapter 2, [60, 61, 62, 64, 65]. It is also generally the case that scalar fields possess a potential which describes their selfinteractions and dictates their masses. For the vast majority of physically viable and cosmologically interesting scalar field theories such self-interactions are negligible over sub-Hubble length and time scales. If the scalar field self-interactions are negligible then the experimental bounds on such a field are very strong: requiring it to either couple to matter much more weakly than gravity does, or to be very heavy [29, 66, 67, 68] (see Chapter 1.3). In the latter case, the scalar field would have had little or no detectable effect on the history of Universe, and would generally be indistinguishable from a cosmological constant. 125
126
CHAPTER 4. CHAMELEON FIELD THEORIES
Recently, a novel scenario was presented by Khoury and Weltman [69] that employed strong self-interactions of the scalar-field to avoid the most restrictive of the current experiment bounds. In the models that they proposed, a scalar field couples to matter with gravitational strength, in harmony with general expectations from string theory, whilst, at the same time, remaining very light on cosmological scales. Over the next two chapters, we will go much further and show, contrary to most expectations, that the scenario presented in [69] allows scalar fields, which are very light on cosmological scales, to couple to matter much more strongly than gravity does, and yet still satisfy all of the current experimental and observational constraints. Importantly, the cosmological value of such a field evolves over Hubble time-scales and could potentially cause the late-time acceleration of our Universe [70]. The crucial feature that these models possess is that the mass of the scalar field depends on the local background matter density. On Earth, where the density is some 1030 times higher than the cosmological background, the Compton wavelength of the field is sufficiently small as to satisfy all existing tests of gravity. In the solar system, where the density is several orders of magnitude smaller, the Compton wavelength of the field can be much larger. This means that, in those models, it is possible for the scalar field to have a mass in the solar system that is much smaller than was previously thought allowed. In the cosmos, the field is lighter still and its energy density evolves slowly over cosmological time-scales and it could function as an effective cosmological constant. While the idea of a density-dependent mass term is not new [71, 72, 73, 74, 75, 76], the work presented in [69, 70] is novel in that the scalar field can couple directly to matter with gravitational strength. If a scalar field theory contains a mechanism by which the scalar field can obtain a mass that is greater in high-density regions than in sparse ones, we deem it to possess a chameleon mechanism and be a chameleon field theory. When referring to chameleon theories, it is common to refer to the scalar field as the chameleon. In Chapters 2 and 3 we defined the dilaton field, Φ, χ and ϕ, to be dimensionless. When dealing with chameleon theories however it is standard practice normalise the chameleon field, Φ, so that it has dimensions of mass and a canonical kinetic term:
1 µ 2 ∇µ Φ∇ Φ;
we adopt
this latter convention in this chapter and the next. The two conventions are related by a simple transformation. A dimensionless dilaton field Φ, with dilaton scale ω, has kinetic term:
1 µ 2 ω∇µ Φ∇ Φ.
We then define a field, Φ, with a canonically normalised kinetic term by
Φ = ω 1/2 Φ. To minimise confusion, we use the upper-case, Φ, to refer to canonically normalised
4.2. CHAMELEON FIELD THEORIES
127
fields in what follows. In this chapter we focus on the dynamics of chameleon fields whereas the potential delectability of chameleon fields is dealt with in following chapter. We begin this chapter by reviewing the main features of scalar field theories with a chameleon mechanism. In section 4.3, we study how Φ behaves both inside and outside an isolated body and derive the conditions that must hold for such a body to have a thin-shell. We show how non-linear effects ensure that the value that the chameleon takes far away from a body with a thin-shell is independent of the matter-coupling, β. Whilst such β-independence as been noted before for Φ4 -theory in ref. [86], this is the first time that it has been shown to be a generic prediction of a large class of chameleon theories. In section 4.4 we show the internal, i.e. microscopic, structure of macroscopic bodies can unexpectedly alter the macroscopic behaviour of the chameleon. Using the results of sections 4.3 and 4.4 we are then able to calculate the Φ-force between two bodies; this is done in section 4.5. We discuss this chapter’s results in section 4.6.
4.2
Chameleon Field Theories
In the theories proposed in [69], the chameleon mechanism was realised by giving the scalar field both a potential, V (Φ), and a coupling to matter, B(βΦ/Mpl )ε; where ε is the local density of matter. We shall say more about how the functions V and B are defined, and the meaning of β, below. The potential and the coupling-to-matter combine to create an effective potential for the chameleon field: V ef f (Φ) = V (Φ) + B(βΦ/Mpl )ε. The values Φ takes at the minima of this effective potential will generally depend on the local density of matter. If at ef f a minima of V ef f we have Φ = Φc , i.e. V,Φ (Φc ) = 0, then the effective ‘mass’ (mc ) of small ef f perturbations about Φc , will be given by the second derivative of V ef f , i.e. m2c = V,ΦΦ (Φc ). It
is usually the case that |V,ΦΦ | À |B,ΦΦ ε| and so mc will be determined almost entirely by the form of V (Φ) and the value of Φc . If V (Φ) is neither constant, linear nor quadratic in Φ then V,ΦΦ (Φc ), and hence the mass mc , will depend on Φc . Since Φc depends on the background density of matter, the effective mass will also be density-dependent. Such a form for V (Φ) inevitably results in non-linear field equations for Φ. For a scalar field theory to be a chameleon theory, the effective mass of the scalar must increase as the background density increases. This implies V,ΦΦΦ (Φc )/V,Φ (Φc ) > 0. It is
128
CHAPTER 4. CHAMELEON FIELD THEORIES
important to note that it is not necessary for either V (Φ), or B(βΦ/Mpl ), to have any minima themselves for the effective potential, V ef f , to have minimum. A sketch of the chameleon mechanism, as described above, is shown in FIGS. 4.1 and 4.2. In FIG 4.1 the potential is Sketch of chameleon mechanism: Low Density Background
Sketch of chameleon mechanism: High Density Background
−4
V(φ) ~ φ B(β φ/Mpl)ρ V (φ) eff
Effective minimum φ = φ (ρ) c
Effective minimum φ = φ (ρ)
Mass of φ near φ is c large because Veff is quite steep near φ .
c
c
V(φ) ~ φ−4 B(β φ / Mpl)ρ
Mass of φ near φc is small because Veff is quite flat near φc.
φ
V (φ) eff
φ
Figure 4.1: Sketch of the chameleon mechanism for a runaway potential: V ∼ Φ−4 . The sketch on the left is for a low density background, whereas the drawing of the right shows what occurs when there is a high density of matter in the surroundings. We can clearly see that the position of the effective minimum, Φc , and the steepness of the effective potential near that minimum, depends on the density. A shallow minimum corresponds to a low chameleon mass. The mass of the chameleon can be clearly seen to grow with the background density of matter.
taken to be of runaway form and has no minimum itself. However, It is clear from the sketches that V ef f does have an minimum, and that the value Φ takes at that minimum is density dependent. In FIG 4.2 the potential is taken to behave like Φ4 and so does have a minimum at Φ = 0. However, the minimum of the effective potential, V ef f , does not coincide with that of V . Once again, the minimum of V ef f is seen to be density dependent.
4.2.1
The Thin-Shell Effect
This chameleon mechanism often results in macroscopic bodies developing what is called a “thin-shell”. A body is said to have a thin-shell if Φ is approximately constant everywhere inside the body apart from in a small region near the surface of the body. Large (O(1)) changes in the value of Φ can and do occur in this surface layer or thin-shell. Inside a body with a
129
4.2. CHAMELEON FIELD THEORIES
Sketch of chameleon mechanism: Low Density Background
Sketch of chameleon mechanism: High Density Background
4
V(φ) ∼ φ B(β φ / Mpl)ρ
Mass of φ near φc is small because V is eff quite flat near φc.
V (φ) eff
Effective minimum φ = φ (ρ) c
4
V(φ) ∼ φ B(β φ/Mpl)ρ
Effective minimum φ = φ (ρ) c
V (φ) eff
Mass of φ near φ is c large because Veff is quite steep near φc.
φ
φ
Figure 4.2: Sketch of the chameleon mechanism for a potential with a minimum at Φ = 0: V ∼ Φ4 . The sketch on the left is for a low density background, whereas the drawing of the right shows what occurs when there is a high density of matter in the surroundings. We can clearly see that the position of the effective minimum, Φc , and the steepness of the effective potential near that minimum, depends on the density. A shallow minimum corresponds to a low chameleon mass. The mass of the chameleon near Φc can be clearly seen to grow with the background density of matter.
~ vanishes everywhere apart from in a thin surface layer. Since the force mediated thin-shell ∇Φ ~ by Φ is proportional to ∇Φ, it is only that surface layer, or thin-shell, that both feels and contributes to the ‘fifth force’ mediated by Φ. It was noted in [69, 70] that the existence of such a thin-shell effect allows scalar field theories with a chameleon mechanism to evade the most stringent experimental constraints on the strength of the field’s coupling to matter. For example: in the solar system, the chameleon can be very light thus mediate a long-range force. The limits on such forces are very tight, [78, 79]. However, since the chameleon only couples to a small fraction of the matter in large bodies i.e. that fraction in the thin-shell, the chameleon force between the Sun and the planets is very weak. As a result the chameleon has no great effect on planetary orbits, and the otherwise tight limits on such a long-range force are evaded, [77]. In section 4.3, we will show that the presence of a thin-shell effect is intimately linked to non-linear nature of chameleon field theories.
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4.2.2
Chameleon to Matter Coupling
When a scalar field, Φ, couples to a species of matter, the effect of that coupling is to make the mass, m, of that species of particles Φ-dependent. This can happen either at the classical level (i.e. in the Lagrangian) or a result of quantum corrections. We parameterise the dependence of m on Φ by
µ m(Φ) = m0 C
βΦ Mpl
¶ (4.1)
where Mpl is the Planck mass and m0 is just some constant with units of mass whose definition ´ ³ βΦ . β defines the strength of the coupling will depend on one’s choice of the function C M pl of Φ to matter. We shall say more about the definition of β below. A Φ-dependent mass will cause the rest-mass density of this particle species to be Φ dependent, specifically µ ε(Φ) = ε0 C
βΦ Mpl
¶ .
(4.2)
The coupling of Φ to the local energy density of this particle species is given by: ∂ε(Φ)/∂Φ which is: ∂ε(Φ) = B0 ∂Φ
µ
βΦ Mpl
¶
βεΦ) , Mpl
(4.3)
where B(x) = ln C(x) and B 0 (x) = dB(x)/dx. Throughout this work we will, for simplicity, assume that our chameleon field, Φ, couples to all species of matter in the same way, however we will keep in mind the fact that, generically, different species of matter will interact with the chameleon in different ways. We shall see in section 5.4 that, if C, and hence B, are at least approximately the same for all particle species, then cosmological bounds on chameleon theories will require that |βB 0 (βΦ/Mpl )Φ/Mpl | < 0.1 everywhere since the epoch of nucleosynthesis. We preempt this requirement and use it to justify the linearisation of B(βΦ/Mpl ): µ B
βΦ Mpl
¶ ≈ B(0) +
βB 0 (0)Φ . Mpl
(4.4)
For this to be a valid truncation we require (B 00 (0)/B 0 (0))βΦ/Mpl ¿ 1. So long as |B 00 (0)| < 10|B 0 (0)|, the cosmological bounds on Φ will then ensure that the above truncation of the expansion of B is a valid one. The only forms of B that are excluded from this analysis are the ones where |B 00 (0)| & 10|B 0 (0)|; we generally expect B 00 (0) ∼ O(B 0 (0)). Provided B 0 (0) 6= 0, we can use the freedom in the definition of β to set B 0 (0) = 1. When this is done, β quantifies the strength of the chameleon-to-matter coupling.
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4.2. CHAMELEON FIELD THEORIES
For example: a particular choice for C that has had some favour in the literature, [69, 70], is C = ekΦ/Mpl for some k. It follows that B = kΦ/Mpl , and so we choose β = k which ensures B 0 (0) = 1, B 00 (0) = 0. We wish to construct our chameleon theories to be compatible with Einstein’s conception of gravity. By this we mean that we wish them to display diffeomorphism and Poincar´e invariance at the level of the Lagrangian. A natural consequence of Poincar´e invariance is that the chameleon couples to matter in a Lorentz invariant fashion. For a perfect fluid this implies that Φ will generally couple to some linear combination of ε and the fluid pressure (P ) i.e ε + ωP . The simplest way for the chameleon to couple to matter in a relativistically invariant fashion is for it to couple to the trace of the energy momentum tensor; in this case ω = −3. This said, apart from in the early universe and in very high density objects such as neutron stars, P/ε ¿ 1, and so the precise value of ω is not of great importance. Apart from where such an assumption would be invalid, we will take P/ε ¿ 1 and set P = 0.
4.2.3
A Lagrangian for Chameleon Theories
It is possible to couple the chameleon to matter in a number of different ways, and as such it is possible to construct many different actions for chameleon theories. A reasonably general example of how the chameleon can couple to trace of the energy momentum tensor is given by following Lagrangian density " # 2 Mpl √ 1 µ (i) L = −g − R(g) − ∂µ Φ∂ Φ + V (Φ) + Lm (ψ (i) , gµν ), 16π 2
(4.5)
where Lm is the Lagrangian density for normal matter. This Lagrangian was first proposed in ref. [77]. The index i labels the different matter fields, ψ (i) , and their chameleon coupling. (i)
(i)
The metrics gµν are conformally related to the Einstein frame metric gµν by gµν = Ω2(i) gµν where Ω(i) = C(i) (β (i) Φ/Mpl ). The C(i) (·) are model dependent functions of β (i) Φ/Mpl . The 0 (0) := (ln C )0 (0) = 1. R(g) is the Ricci-scalar associated with the β (i) are chosen so that B(i) (i)
Einstein frame metric. For simplicity we will restrict ourselves to a universal matter coupling √ (i) g )δLm /δ˜ gµν . It follows that i.e gµν = g˜µν , C(i) = C and β (i) = β. We define T µν = (2/ −˜ T = T µν g˜µν = ε − 3P , where ε is the physical energy density and P is the sum of the principal pressures. In general ε and P are Φ-dependent. With respect to this action, the chameleon
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CHAPTER 4. CHAMELEON FIELD THEORIES
field equation is − ¤Φ = V,Φ (Φ) +
βB 0 (βΦ/Mpl )(ε − 3P ) . Mpl
(4.6)
As mentioned above, |βΦ/Mpl | < 0.1 is required for the theory to be viable and so it is acceptable to approximate B 0 (βΦ/Mpl ) by B 0 (0). We then scale β so that B 0 (0) = 1. The requirement, |βΦ/Mpl | < 0.1, also ensures that ε and P are independent of Φ to leading-order. The field equation for Φ is therefore − ¤Φ = V,Φ (Φ) +
βB 0 (βΦ/Mpl )(ε − 3P ) . Mpl
(4.7)
The above Lagrangian should not be viewed as specifying the only way in which Φ can couple to matter. When one considers varying constant theories, the matter coupling often results from quantum loop effects (see Chapter 2). However, despite the fact that many different Lagrangians are possible, it is almost always the case that the field equation for Φ takes a form very similar to the one given above.
4.2.4
Intrinsic Chameleon Mass Scale
β quantifies the strength of the chameleon coupling. MΦ := Mpl /β can then be viewed as the intrinsic mass scale of the chameleon. Although precise calculations of scattering amplitudes fall outside the scope of this work, we expect that chameleon particles would be produced in large numbers in particle colliders that operate at energies of the order of MΦ or greater. It is generally seen as ‘natural’, from the point of view of string theory, to have MΦ ≈ Mpl . When this happens the chameleon has the same energy-scale as gravity. It has also been suggested that the chameleon field arises from the compactification of extra dimensions, [54], if this is the case then there is no particular reason why the true Planck scale (i.e. of the whole of space time including the extra-dimensions) should be the same as the effective 4-dimensional Planck scale defined by Mpl . Indeed having the true Planck scale being much lower than Mpl has been put forward as a means by which to solve the Hierarchy problem (e.g. the ADD scenario [117, 118, 119]). In string-theory too, there is no particularly reason why the stringscale should be the same as the effective four-dimensional Planck scale. It is also possible that the chameleon might arise as a result of new physics with an associated energy scale greater than the electroweak scale but much less than Mpl . In light of these considerations it would
4.2. CHAMELEON FIELD THEORIES
133
be pleasant if MΦ = Mpl /β ¿ Mpl , say of the GUT scale, or, if we hoped to find traces of it at the LHC, maybe even the TeV-scale. A positive detection of a chameleon field with such a sub-Planckian energy scale could provide us with the first evidence for new physics beyond the standard model, but below the Planck scale. As pleasant as it might be to have MΦ ¿ Mpl , it is generally agreed that the current experimental bounds on the existence of light scalar fields rule out this possibility [29, 66, 67, 68]. Indeed, in the absence of a chameleon mechanism similar to that proposed in [70, 77], bounds on the violation of the weak equivalence principle (WEP) coming from Lunar Laser Ranging (LLR), [78, 79], limit |β| ≤ 10−5 for a light scalar field. This implies MΦ À Mpl . If the Planck scale is supposed to be associated with some fundamental maximum energy, such a large value of MΦ seems highly unlikely. Even if a (non-chameleon) scalar field has a mass of the order of 1mm−1 , then we must have β < 10−1 , [87]. One of the major successes of the proposal of chameleon field by Khoury and Weltman, [69, 77], was that chameleon fields can, by attaining a large mass in high density environments such as the Earth, Sun and Moon, evade the experimental limits coming from LLR and other laboratory tests of gravity. In this way, it has been shown the scalar fields in theories that possess a chameleon mechanism can couple to matter with the strength of gravity, β ∼ O(1) and still coexist with the best experimental data currently available. Even though β ∼ O(1) has been shown to be possible, β À 1 is still generally assumed to be ruled out. Over the next two chapters, however, we challenge this assumption and show that it is indeed feasible for β to be very large. Moreover MΦ ≈ MGU T ∼ 1015 GeV or MΦ ∼ 1 TeV are allowed. Tantalisingly the experimental precision required to detect such a sub-Planckian chameleon theory is already within reach. Large matter couplings are allowed in normal scalar field theories but only if the scalar field has a mass greater than (0.1 mm)−1 . This is not the case for chameleon theories. We shall show that the mass of the chameleon in the cosmos, or the solar system, can be, and generally is, much less than 1 m−1 .
4.2.5
Initial Conditions
Even though the term chameleon field sounds rather exotic, in a general scalar field theory with a matter coupling and arbitrary self-interaction potential, there will generically be some values
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CHAPTER 4. CHAMELEON FIELD THEORIES
of Φ about which the field theory exhibits a chameleon mechanism. Whether or not Φ ends up in such a region will depend on its cosmological evolution and one’s choice of initial conditions. The importance of initial conditions was discussed in [70]. In that paper the potential was chosen to be of runaway form V ∝ Φ−n , n > 0. We will review what is required of the initial conditions for such potentials in section 5.4 below. We shall see that the larger β is, the less important the initial conditions become. We will also see that the stronger the coupling, the stronger the chameleon mechanism and so the more likely it becomes that a given scalar field theory will display chameleon-like behaviour. This is one of the reasons for wanting to have a large value of β.
4.2.6
The Importance of Non-Linearities
Chameleon field theories necessarily involve highly non-linear self-interaction potentials for the chameleon. These non-linearities make analytical solution of the field equations much more difficult, particularly when the background matter density is highly inhomogeneous. Most commentators therefore linearise the equations of chameleon theories when studying their behaviour in inhomogeneous backgrounds [80, 81, 54, 82]. Such an approximation may mislead theoretical investigations and result in erroneous conclusions about experiments which probe fifth force effects. In this chapter we shall show, in detail, that this linearisation procedure is indeed very often invalid. When the non-linearities are properly accounted for, we will see that the chameleon mechanism becomes much stronger. It is this strengthening of the chameleon mechanism that opens up the possibility of the existence of light cosmological scalars that couple to matter much more strongly than gravity (β À 1).
4.2.7
The Chameleon Potential
The key ingredient of a chameleon field theory, in addition to the chameleon-to-matter coupling, is a non-linear and non-quadratic self-interaction potential V (Φ). It has been noted previously that V (Φ) could play the role of an effective cosmological constant [70]. There are obviously many choices one could make for V (Φ), and whilst we wish to remain suitably general in our study, we must go some way to specifying V (Φ) if we are to make progress. One quite general form that has been widely used in the literature is the Ratra-Peebles potential, V (Φ) = M 4 (M/Φ)n [83], where M is some mass scale and n > 0; chameleon fields have also been studied
135
4.2. CHAMELEON FIELD THEORIES
in the context of V (Φ) = kΦ4 /4! [84]. In this chapter and the next we will consider both of these types and generalise a little further. We take: V (Φ) = λM 4 (M/Φ)n ,
(4.8)
where n can be positive or negative and λ > 0. If n 6= −4 then we can scale M so that without loss of generality λ = 1. When n = −4, M drops out and we have a Φ4 theory. When n > 0 this is just the Ratra-Pebbles potential.
4.2.8
Chameleon Field Equation
With these assumptions and requirements, the chameleon field, Φ, obeys the following conservation equation:
µ − ¤Φ = −nλM
3
M Φ
¶n+1 +
β(ε + ωP ) . Mpl
(4.9)
For this to be a chameleon field we need the potential gradient term, V,Φ = −nλM 3 (M/Φ)n+1 , and the matter coupling term, β(ε + ωP )/Mpl , to be of opposite signs. It is usually the case that β > 0 and P/ε ¿ 1. If n > 0 we must therefore have Φ > 0. In theories with n < 0 we must have Φ < 0 and n = −2p where p is a positive integer. We must also require that the effective mass-squared of the chameleon field, m2c = V,ΦΦ , be positive, non-zero and depend on Φ. These conditions mean that we must exclude the region −2 ≥ n ≥ 0. If n = −2, n = −1 or n = 0 then the field equations for Φ would be linear.
4.2.9
Natural Values of M and λ
When n 6= −4, one might imagine that our choice of potential has arisen out of an expansion, ˆ /Φ)n , of another potential W (Φ) = M ˆ 4 f ((M ˆ /Φ)n ) where f is some function. We for small (M could then write:
à ˆ 4 f (0) + M ˆ 4 f 0 (0) W ∼M
ˆ M Φ
!n ,
(4.10)
ˆ is some mass-scale. We define M so that the second term on the right hand side of where M the above expression reads M 4 (M/Φ)n . The first term on the right hand then plays the rˆole ˆ 4 f (0) ≈ εΛ . Assuming that both f (0) and f 0 (0) are O(1), we of a cosmological constant M ˆ ≈ (εΛ )1/4 ≈ (0.1 mm)−1 . It is for this reason that one will often find would then have M ≈ M (0.1 mm)−1 referred to as a ‘natural’ value for M , [70, 77]. When n = −4 we naturally expect λ ≈ 1/4!, [85].
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CHAPTER 4. CHAMELEON FIELD THEORIES
4.3
One body problem
In this section, we consider the perturbation to the chameleon field generated by a single body embedded in background of uniform density εb . For simplicity we shall model the body to be both spherical and of uniform density εc . This analysis will prove vital when we come to calculate the force between two bodies that is induced by the chameleon field. We assume space-time to be Minkowski (at least to leading order) and we also assume that everything is static. Under these assumptions ¤ → −∇2 where ∇2 Φ = r−2 (r2 Φ0 )0 ; Φ0 = dΦ/dr. Whilst this problem has been considered elsewhere in the literature [77, 69, 70], most commentators have chosen to linearise the chameleon field equation, eq. (4.9), before solving it. This linearisation is, however, often invalid. In this section, we begin by briefly reviewing what occurs when it is appropriate to linearise eq. (4.9), and, in doing so, note where the linear approximation breaks down. In some cases, even though it is not possible to construct a linearised theory that is valid everywhere, we shall demonstrate, using the method of matched asymptotic expansions, how to construct multiple linearisations of the field equation, each valid in a different region, and then match them together to find an asymptotic approximation to Φ that is valid everywhere. When this is possible, the chameleon field, Φ, will behave as if these were the solution to a consistent, everywhere valid, linearisation of the field equations; for this reason we deem this method of finding solutions to be the pseudo-linear approximation. If a body is large enough, however, both the linear and pseudo-linear approximations will fail. We shall see that, when this happens, Φ behaves in a truly non-linear fashion near the surface of the body. The onset of non-linear behaviour is related to the emergence of a thin-shell in the body. The linear approximation is discussed in section 4.3.1 whilst the pseudo-linear approximation is considered in section 4.3.2. We discuss the non-linear regime in section 4.3.3. We take the body that we are considering to be spherical with radius R and uniform density εc . Assuming spherical symmetry, inside the body (r < R), Φ obeys: d2 Φ 2 dΦ + = −nλM 3 dr2 r dr
µ
M Φ
¶n+1 +
βεc , Mpl
(4.11)
+
βεb . Mpl
(4.12)
and outside the body, (r > R), we have: d2 Φ 2 dΦ + = −nλM 3 dr2 r dr
µ
M Φ
¶n+1
137
4.3. ONE BODY PROBLEM
The right hand side of eq. (4.11) vanishes when Φ = Φc where µ Φc = M
βεc nλMpl M 3
¶−
1 n+1
.
This value of Φ corresponds to the minimum, of the effective potential of the chameleon field, inside the body. Similarly, the right hand side of eq. (4.12) vanishes when Φ = Φb where µ Φb = M
βεb nλMpl M 3
¶−
1 n+1
.
This value of Φ corresponds to the minimum, of the effective potential of the chameleon field, outside the body. For large r we must have Φ ≈ Φb . Associated with every value of Φ is an effective chameleon mass, mΦ (Φ), which is the mass of small perturbations about that value of Φ. This effective mass is given by: µ m2Φ (Φ)
=
ef f V,ΦΦ (Φ)
= n(n + 1)λM
2
M Φ
¶n+2 .
(4.13)
We define mc = mΦ (Φc ) and mb = mΦ (Φb ). We shall see below that the larger the quantity mc R, the more likely it is that a body will have a thin-shell. In this section we shall see both why this is so, and precisely how large mc R has to be for a thin-shell to appear. Throughout this section we will require, as boundary conditions, that ¯ ¯ dΦ ¯¯ dΦ ¯¯ = 0 and = 0. dr ¯r=0 dr ¯r=∞
4.3.1
Linear Regime
We assume that it is a valid approximation to linearise the equations of motion for Φ about the value of Φ in the far background, Φb . For this to be possible we must require that certain conditions, which we state below, hold. Writing Φ = Φb + Φ1 , the linearised field equations are: d2 Φ1 2 dΦ1 + = −nM 3 dr2 r dr
µ
M Φb
¶n+1
+ m2b Φ1 +
β(εc − εb ) βεb H(R − r) + , Mpl Mpl
(4.14)
where H(R − r) is the Heaviside function: H(x) = 1, x ≥ 0, and H(x) = 0, x < 0. For this linearisation of the potential to be valid we need: V,ΦΦ (Φb )Φ1 < 1. V,Φ (Φb )
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CHAPTER 4. CHAMELEON FIELD THEORIES
This translates to |Φ1 /Φb | < |n + 1|−1 . Also, for this linearisation to remain valid as r → ∞, we need Φ1 → 0, which implies that: µ nM
3
M Φb
¶n+1 =
βεb . Mpl
Defining ∆εc = εc − εb , and solving the field equations, we find that outside the body (r > R) we have: β∆εc emb (R−r) Φ1 = Mpl m2b mb r
µ
tanh(mb R) − mb R 1 + tanh(mb R)
¶ .
Inside the body (r < R), Φ1 is given by Φ1 = −
β∆εc (1 + mb R) e−mb R sinh(mb r) β∆εc + . mb r m2b Mpl Mpl m2b
The largest value of |Φ1 /Φb | occurs at r = 0 and so, for this linear approximation to be valid, we need: |Φ1 (r = 0)/Φb | < |n + 1|−1 . This requirement is equivalent to the statement that ¯ ¯ ¯(1 + mb R)e−mb R − 1¯ ∆εc ∼ 1 m2 R2 ∆εc = ∆εc εb 2 b εb εc
µ
εb εc
¶
1 n+1
(mc R)2 < 1. 2
where ‘∼’ means “asymptotically in the limit mb R → 0”. It is often the case that the background is much less dense than the body i.e. εb ¿ εc . If this is the case then it is clear, from the above expression, that there will be a distinct difference between theories with n > 0 and those with n ≤ −4. In theories with n > 0, the lower the density of the background, the better the linear theory approximation will hold, whereas when n ≤ −4 the opposite is −1/(n+1)
true. This can be understood by considering the relation Φb ∝ εb
. If n > 0, the smaller
εb becomes, the larger Φb will be. It is therefore possible for larger perturbations in Φ to be treated consistently in terms of the linearised theory. If, however, we have that n ≤ −4 then Φb → 0 as εb → 0 and the opposite is true. We can, however, use the method of matched asymptotic expansions to show that the region where behaviour, similar to that which would be predicted by linearised theory, occur is significantly larger than one would have guessed simply by requiring that the linear approximation hold. The results of this section, as well as those of sections 4.3.2 and 4.3.3, are summarised in section 4.3.4 below.
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4.3. ONE BODY PROBLEM
4.3.2
Pseudo-Linear Regime
The defining approximation of the pseudo-linear regime (for both positive and negative n) is that inside the body:
µ
Φc Φ(r)
¶n+1 ¿1
¯ This is equivalent to βεc /Mpl À nM 3 (M/Φ(r))n+1 . When this holds we find Φ ∼ Φ(r) inside ¯ the body, where this defines Φ(r) and: ¯ 1 d2 (rΦ) βεc = . 2 r dr Mpl It follows that 2 ¯ = Φ0 + βεc r . Φ∼Φ 6Mpl
In this case ‘∼’ means “asymptotically as (Φc /Φ(r))n+1 → 0”. Outside the body, we can find a similar asymptotic approximation: 2 3 ¯ = Φ0 + βεc R − βεc R . Φ∼Φ 2Mpl 3Mpl r
For this to be valid we must ensure that the neglected terms, in the above approximation to Φ, are small compared to the included ones; this requires that: R3 À 3
Z
Z
r
dr 0
0
r0
µ 00 00
dr r 0
Φc ¯ Φ(r00 )
¶n+1 .
For large r we expect, as we did in the previous section, that Φ → Φb , and so: Φ ∼ Φ∗ = Φ b −
Ae−mb r , r
¯ as the inner which will remain valid whenever Ae−mb r Φb r ¿ |1/(n + 1). We shall refer to Φ approximation to Φ. Similarly, Φ∗ is the outer approximation. So far both A, and the value of Φ0 , remain unknown constants of integration. In general, when Φ ∼ Φ∗ we will not also ¯ (and vice versa). If, however, there is some intermediate region where both the have Φ ∼ Φ inner and outer approximations are simultaneously valid, then we can match both expressions in that intermediate region and determine both Φ0 and A, [135, 136]. A detailed explanation of the use of matched asymptotic expansions was given above in chapter 3.3. For the moment we shall assume that such an intermediate region does exist. We check what is required for this assumption to hold in appendix B and present the results of that
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CHAPTER 4. CHAMELEON FIELD THEORIES
analysis below. Given an intermediate region, we find: A =
βεc R3 , 3Mpl
Φ0 = Φb −
βεc R2 βεc mb R3 + . 2Mpl 3Mpl
The external field produced by a single body in the pseudo-linear approximation is: Φ ∼ Φb −
βεc R3 e−mb r Φb (mc R)2 e−mb r Φ ∼ − ⇔ , 3Mpl r Φc Φc 3(n + 1)(r/R)
(4.15)
and the field inside the body is given by: Φ ≈ Φb −
βεc R2 βεc mb R3 βεc r2 + + . 2Mpl 3Mpl 6Mpl
(4.16)
In appendix B, we find that for the pseudo-linear approximation to hold, we must require ¯³ ´ 2 ¯¶ µ ³ ´ n+4 p ¯ m |n+2| ¯ 1/6 mc 3(n+2) b mc R ¿ min (18) , 2|n + 1| ¯¯ mc − 1¯¯ , n < −4, (4.17a) mb ¯³ ´ 2 ¯¶ µ ³ ´ n+4 √ m 3(n+2) p ¯ m |n+2| ¯ c c ¯ mc R ¿ min 3 mb , 2|n + 1| ¯ mb − 1¯¯ , n > 0, (4.17b) mc R ¿ 1,
n = −4.
(4.17c)
When n = −4, we actually find a slightly different asymptotic behaviour of Φ outside the body, precisely: s Φ ∼ Φb −
e−mb r 1 , 2λ(y0 + ln(min(r/R, 1/mb ))) 2r
(4.18)
where y0 is an integration constant and: (mc R)3 = 9
r
3 1 + 2y0 3
µ
3 2y0
¶3/2 .
The conditions given by eqs. (4.17a-c) ensure that the pseudo-linear approximation is everywhere valid. When these conditions fail, non-linear effects begin to become important near the centre of the body. As mc R is increased further the region where non-linear effects play a rˆole moves out from the centre of the body. Eventually, for large enough mc R, the non-linear nature of chameleon potential, V (Φ), is only important in a thin region near the surface of the body; this is the thin-shell. Since the emergence of such a thin-shell is linked to non-linear effects becoming important near the surface of the body, it must be the case that the assumption that Φ is given by equation (4.15) (or by eq. (4.18) in n = −4 theories) breaks down for
4.3. ONE BODY PROBLEM
141
some r > R. By this logic, we find, in appendix B, that a thin-shell occurs when: ¯³ ´ 2 ¯¶ µ ³ ´ n+4 p ¯ m |n+2| ¯ 1/6 mc 3(n+2) b ¯ ¯ , n < −4, (4.19a) mc R & min (18) , 3|n + 1| − 1 mb ¯ mc ¯ ¯³ ´ 2 ¯¶ µ ³ ´ n+4 √ m 3(n+2) p ¯ m |n+2| ¯ b ¯ , n > 0, mc R & min 3 mcb , 3|n + 1| ¯¯ m − 1 (4.19b) ¯ c mc R & 4,
n = −4.
(4.19c)
In both eqs. (4.17a-b) and (4.19a-b) the second term in the min( · , · ) is almost always smaller than the first when εb /εc ¿ 1 ⇔ mb /mc ¿ 1. The behaviour of Φ both near to, and far away from, a body with thin-shell is discussed below in section 4.3.3. The results of this section are summarised in section 4.3.4. Note that the thin-shell conditions , eqs. (4.19a-c), necessarily imply that mc R À 1.
4.3.3
Non-linear Regime
We have just seen, in eqs. (4.19a-c), that for non-linear effects to be important, and the pseudo-linear approximation to fail, we must have mc R À 1. In this regime the body is, necessarily, very large compared to the length scale 1/mc . We expect that all perturbations in Φ will die off exponentially quickly over a distance of about 1/mc and, as such, Φ ≈ Φc will be almost constant inside the body. Any variation in the chameleon field, that does take place, will occur in a ‘thin-shell’ of thickness ∆R ≈ 1/mc near the surface of the body. It is clear that mc R À 1 implies ∆R/R ¿ 1. In this section we will consider both the behaviour of the field close to the surface of the body, and far from the body. Close to the body We noted above that mc R À 1 implies ∆R/R ¿ 1, we shall demonstrate this is a rigorous fashion below. Given ∆R/R ¿ 1, when we consider the evolution of Φ in the thin-shell region, we can ignore the curvature of the surface of the body, to a good approximation. We therefore treat the surface of the body as being flat, with outward normal in the direction of the positive x-axis. The surface of the body defined to be at x = 0 (i.e. x = r − R). Since the shell is thin compared to the scale of the body, we are interested in physics that occurs over length-scales that are very small compared to the size of the body. We therefore make the approximation that the body extends to infinity along the y and z axes and also along the negative x axis. Given these assumptions, we have that Φ evolves according to
142
CHAPTER 4. CHAMELEON FIELD THEORIES
d2 Φ = −nλM 3 (M/Φ)n+1 + βεc /Mpl dx2 As a boundary conditions (BCs) we have Φ → Φc and dΦ/dx → 0 as x → −∞. With these BCs, the first integral of the above equation is: µ ¶ 1 dΦ 2 βεc ≈ λM 4 [(M/Φ)n − (M/Φc )n ] + (Φ − Φc ). 2 dx Mpl
(4.20)
Outside of the body, we assume that Φ → Φb as x → ∞, and that the background has density εb ¿ εc . Assuming that:
¯ 2 ¯ ¯ ¯ ¯d Φ¯ ¯ ¯ ¯ ¯ À ¯ 2 dΦ ¯ ¯ dx2 ¯ ¯ r dx ¯
then we can ignore the curvature of the surface of the body and, in x > 0, we have: µ ¶ 1 dΦ 2 βεb (Φ − Φb ), = λM 4 (M/Φ)n − λM 4 (M/Φb )n + 2 dx Mpl
(4.21)
where Φb is as we have defined above. Our assumption that |d2 Φ/dx2 | À (2/r)dΦ/dx then requires that: Ã √ r µ ¶n µ ¶n+1 !1/2 2 2 Φ n+1 Φ 1 − (n + 1) ¿ 1. +n mΦ (Φ)r n Φb Φb Provided the pseudo-linear approximation breaks down, and that the body has a thin-shell, we expect that, near the surface of the body, Φ ∼ O(Φc ). It follows that, whenever εc À εb , (Φ/Φb )n ¿ 1 and (Φ/Φb )n+1 ¿ 1. The above condition will therefore be satisfied provided p that mc R À 8(n + 1)/n; this is generally a weaker condition than the requirement that the body satisfy the thin-shell conditions, eqs (4.19a-c). On the surface at x = 0, both Φ and dΦ/dx must be continuous. By comparing the expressions for dΦ/dx inside and outside the body we have: Φ(0) − Φc 1 = . Φc n We now check that we do indeed have a thin shell i.e. ∆R ¿ R. We expect that, near the surface of the body, almost all variation in Φ will concentrated into a shell of thickness ∆R. We define msurf by: Z
0
dx −∞
dΦ dΦ = Φ(x = 0) − Φc = Φc /n = msurf (x = 0). dx dx
m−1 surf is then, approximately, the length scale over which any variation in Φ dies off. As it happens, msurf is also the mass of the chameleon field at x = 0. It follows that ∆R ≈ m−1 surf .
143
4.3. ONE BODY PROBLEM
For this shell to be thin, and for us to be justified in ignoring the curvature of the surface of the body, we need ∆R/R ¿ 1 or equivalently msurf R À 1. We find (assuming εb ¿ εc ) that: µ msurf R ≈
n n+1
¶n/2+1 mc R ∼ O(mc R),
and so msurf R À 1 follows from mc R À 1, and ∆R ∼ O(m−1 c ). msurf R À 1 will be automatically satisfied whenever the thin-shell conditions eqs. (4.19a-c) hold. Whenever εc À εb , eq. 4.21 will, near r = R, be well-approximated by: µ ¶ 1 dΦ 2 ≈ λM 4 (M/Φ)n . 2 dx Solving this under the boundary conditions Φ(x = 0) = (1 + 1/n)Φc and Φ/Φc → Φb /Φc ≈ 0 as x → ∞ we find 1 |n + 2|(r − R) ∼ p + mΦ (Φ) 2n(n + 1)
µ
n+1 n
¶n/2+1
1 mc
(4.22)
This approximation will therefore break-down when mΦ (Φ)r ∼ O(1), which occurs when r − p R ∼ O(R). We can see that, if r − R À 2n(n + 1)/(|n + 2|msurf ) then mΦ , and hence also Φ, will be independent of mc and hence also of Φc and β at leading order. Since msurf R À 1, there will be some region where eq. (4.22) is both valid and, to leading order, independent of β. Although, in this approximation, we cannot talk about what occurs for (r−R) & R, it seems likely, in light of the behaviour seen when (r − R) ¿ R, that, whenever r À 1/msurf ≈ 1/mc , the perturbation in Φ, induced by an isolated body with thin-shell, will be independent of the matter coupling β. We confirm this expectation in section 4.3.3 below. Far field of body with thin-shell We found above that the emergence of a thin-shell was related to non-linear effects being non-negligible near the surface of the body. We noted that a thin-shell will exist whenever conditions (4.19a-c) hold. However, even when these conditions hold, we do not expect nonlinear effects to be important far from the surface of the body. Indeed, for large r we should expect that Φ takes a functional form similar to that found in the pseudo-linear approximation i.e. as given by eq. (4.15) (or eq. (4.18) if n = −4). Although the functional form should be similar, in order to find the correct behaviour, one must replace (mc R) in equations (4.15)
144
CHAPTER 4. CHAMELEON FIELD THEORIES
and (4.18) by some other quantity C, say. We show that, to leading order as r → ∞, C is independent of the matter coupling β and the density of the central body ε. This confirms the expectation of section 4.3.3 above. The analysis for n = −4 is slightly more involved than it is for other values of n. We therefore consider the n = −4 case separately below and in appendix C. The analysis for theories with runaway potentials that become singular at Φ = 0 (i.e n > 0 theories) is much simpler than it is for theories where the potential has a minimum at Φ = 0 and which are non-singular for all finite Φ i.e. (n < −4 theories): we therefore consider the n < 0 and n > 0 cases separately. Runaway Potentials (n > 0) Away from the surface of the body we expect that non-linear effects will be negligible and as r → ∞ we will have: Φ ∼ Φ(0) = Φb −
De−mb r , r
for some D where Φb and mb are the values of the chameleon and its mass in background. It is clear from the field equations however that ∇2 Φ < ∇2 Φ(0) and so, given the boundary conditions at infinity, Φ < Φ(0) outside the body. In n > 0 theories there is a singularity of the potential, and hence also of the field equations, at Φ = 0. It is clear that this singularity cannot be reached in any physically acceptable evolution and so we must always have Φ > 0, which in turn implies Φ(0) > 0 outside the body. The minimum value of Φ(0) outside the body occurs at r = R and so we must have: D < Φb emb R R. In most cases of interest mb R ¿ 1 and so we have: D < Φb R. This upper bound on D defines a critical form for the field outside the body: Ã ! emb (R−r) R Φcrit = Φb 1 − . r No matter what occurs inside the body (r < R) we must have Φ > Φcrit outside the body as r → ∞. This implies that: ¯ ¯ ¯ dΦ ¯ (1 + mb r)Φb emb (R−r)R Φb R ¯ ¯< < 2 , ¯ dr ¯ 2 r r
145
4.3. ONE BODY PROBLEM
as r → ∞. Ignoring non-linear effects, Φ > Φcrit is satisfied by all bodies that satisfy the conditions for the pseudo-linear approximation (eqs. (4.17b)) but would be violated, in the absence of non-linear effects, by those that satisfy the thin-shell conditions (eqs. (4.19)). We must therefore conclude that non-linear effects near the surface of body with thin-shells ensure Φ > Φcrit is always satisfied as r → ∞. Furthermore, if Φ À Φcrit then ¯ ¯ ¯ dΦ ¯ ¯ ¯ ¿ Φb R , ¯ dr ¯ r2 and it follows from section 4.3.2 that the pseudo-linear approximation is valid for all r, which further implies that the body cannot have a thin-shell. Thus thin-shelled bodies must actually have Φ being only greater less than Φcrit as r → ∞. We are therefore justified in using Φcrit to approximate the far field of a body with a thin-shell. In summary: in n > 0 theories, the far field of a body with a thin-shell has the following form: Φ ∼ Φb − Φb emb (R−r)R r. We note that this form, and the arguments with which we have derived it, do not depend, in any way, on the physics inside r < R. The critical form of Φ is determined entirely by the form of the potential and the background value of Φ. Potentials with minimum (n < 0) For n > 0 theories the singularity of the potential at Φ = 0 allowed us to determine asymptotic form of Φ outside a body with a thin-shell. In n < 0 theories, however, the potential is welldefined for all finite Φ and so we cannot play the same trick as we did above. The n = −4 case is special and treated in great detail in appendix C. When n = −4, we find that the far field of a body with thin-shell is given by: e−mb r e−mb r Φ ≈ Φb − p ∼ Φb − p . r 2λ(1 + 4 ln(min(r/R, 1/mb R))) 2r 2λ ln(min(r/R, 1/mb R)) A thin-shell is certainly present whenever mc R & 4. For other negative values of n we use a semi-analytical method. We saw when deriving the thin-shell conditions for n < 0 theories that the background value of Φ plays only a negligible rˆole since Φ/Φb À 1 near the body and, in most cases, mb R ¿ 1. Assuming mb R ¿ 1, we simplify our analysis by setting Φb = 0. Far from the body non-linear effects are sub-leading order and we expect: Φ ∼ Φ(0) = −
D + o(1/r). r
146
CHAPTER 4. CHAMELEON FIELD THEORIES
We now define a new coordinate s =
p |n|A−(n+2) M r and u = −Φ/AM for some constant A.
With these definitions the full field equation for Φ outside the body (with Φb = 0) becomes: µ ¶ 1 d 2 du s = u−n−1 , s2 ds ds and as s → ∞:
n+4
DA− 2 u∼ p + o(1/s). |n|s
We set An+4 = D2 /|n| so that u ∼ 1/s and define t = 1/s so that the field equations become: d2 u u−n−1 = . dt2 t4
(4.23)
The asymptotic form of u as r → ∞, t → 0, requires that u(t = 0) = 0 and du/dt(t = 0) = 1 exactly. With these boundary conditions we numerically evolve eqn. (4.23) towards larger t (smaller r). As one might expect from such an elliptic equation, with these boundary conditions, a singularity occurs at some finite t which we label tmax . We use our numerical evolutions to determine tmax for each n. For the evolution of Φ to remain non-singular up to the surface of the body, we need r = R to correspond to a value of t < tmax . The limiting case is given by t(R) = tmax . This limiting case determines a critical form for the Φ field which occurs when A = Amin where: 1
2
Amin = |nt2max | n+2 (M R) n+2 . This corresponds to the following critical asymptotic form for Φ: Ã Φcrit ∼ Φb −
|n+4|
tmax |n|
!
1 |n+2|
n+4
(M R) n+2
e−mb r , r
where we have reinserted the (almost always negligible) Φb and mb dependence. We use nu|n+4|
merical integration to calculate the value of γ(n) := tmax for different values of n. Our results are displayed in table 4.1. Physically acceptable non-singular evolution implies that asymptotically Φ/Φcrit < 1. If Φ/Φcrit ¿ 1 then the conditions of the pseudo-linear approximation are satisfied and so the body cannot have a thin-shell. Thin-shelled bodies must therefore almost saturate this bound on Φ and so Φcrit provides a good approximation to the asymptotic behaviour Φ outside thin-shelled bodies. We note that, as in the n > 0 case, the existence of a critical form for Φ depends in no way on the what occurs inside the body and, as such, is independent of both β and the density of the body.
147
4.3. ONE BODY PROBLEM
|n+4|
Table 4.1: Values of γ(n) = tmax n
γ(n)
-12
14.687
-10
10.726
-8
6.803
-6
3.000
Critical Behaviour The existent of a critical form for Φ when r À R implies that, no matter how massive our central body, and no matter how strongly it couples to the chameleon, the perturbation it produces in Φ for r À R takes a universal value whenever the thin-shell conditions, eqns (4.19a-c), hold. When n 6= −4, the critical form of the far field, depends only on M , n, R and on the chameleon mass in the background, mb . When n = −4 the critical form for the far field depends only on λ, R and mb . For all n, the far field is, crucially, found to be independent of the coupling, β, of the chameleon to the isolated body. This is one of the main reasons why β À 1 is not ruled out by current experiments. The larger β becomes, the stronger the chameleon mechanism and so the easier it is for a given body to have a thin-shell. However, the far field of a body with a thin-shell is independent of β, and so, in stark contrast to what occurs for linear theories, larger values of β do not result in larger forces between distant bodies. Defining the mass of our central body to be M = 4πεc R3 /3 we can express this critical behaviour of the far field in terms of an effective coupling, βef f , defined by: Φ ∼ Φb −
βef f Me−mb r , 4πMpl r
when r À R. Assuming εb /εc ¿ 1 we find that: βef f
=
βef f
=
βef f (r) =
¶ 1 n+4 γ(n) |n+2| (M R) n+2 , n < −4, |n| ¶ 1 µ 4πMpl n(n + 1)M 2 n+2 MR , n>0 M m2b 2πMpl (2λ ln(min(r/R, 1/mb R)))−1/2 , n = −4. M 4πMpl M
µ
(4.24a) (4.24b) (4.24c)
148
CHAPTER 4. CHAMELEON FIELD THEORIES
The β independence of βef f was first noted, in the context of Φ4 theory, in [86]. However, the authors were mostly concerned with region of parameter space β < 1, λ ¿ 1; in our analysis we go further: considering a wider range of theories and also the possibility that β À 1. βindependence was also present in the original work of Khoury and Weltman [45, 69] for n > 0 theories. However, in those works, the β independence together with its important implications for experiments, was not commented on. Especially those that search for WEP violations were not considered. As we shall see in section 5.2 below, this β independence means that if one uses test-bodies with the same mass and outer dimensions then in chameleon theories, no matter how much the weak equivalence principle is violated at a particle level, there will be no violations of WEP far from the body. Simply because the far field is totally independence of both the body’s chameleon coupling and its density. In this work, we have also shown that this β independence is a generic feature of all V ∝ Φ−n chameleon theories and it is not simply as artifact of the runaway (n > 0) potentials considered in [45, 69]. Indeed there are good reasons to believe that similar behaviour will be seen in chameleon theories with other potentials. As we mentioned in the introduction, the field equations for chameleon theories are necessarily non-linear. It is well-known that, that in non-linear theories, the evolution of arbitrary initial conditions will generically be singular. If one wishes to avoid singularities then tight constraints on the initial conditions must be satisfied. When considering the field outside an isolated body, these conditions will, generally, require that |dΦ/dr| is smaller than some critical, r-dependent, value. As a result, there will a critical, or maximal, form that the field produced by a body can take. This precisely what we have found for Φ−n theories. The form of this critical far field will depend on the nature of the non-linear potential, and possibly the coupling of Φ to any background matter, but, since we are outside the body, it cannot depend on the coupling of the chameleon to the body itself. Again, this is precisely what we have seen for Φ−n chameleon theories. We can understand the β-independence, in a slightly different way, as follows: just outside a thin-shelled body, the potential term in eq. (4.9) is large and negative (∼ O(−βε/Mpl )), and it causes Φ to decay very quickly. At some point, Φ will reach a critical value, Φcrit , that is small enough so that non-linearities are no longer important. Since this all occurs outside the body, Φcrit can only depend on the size of the body, the choice of potential (M, λ, n) and the mass of Φ in the background, mb . This is precisely what we have found above.
149
4.3. ONE BODY PROBLEM
We have seen above that the far field of a body with thin-shell is independent of the microscopic chameleon-to-matter coupling, β. This is one of the vital features that allows theories with β À 1 to coexist with the current experimental bounds. It is also of great importance when testing for WEP violations, since any microscopic composition dependence in β will be invisible in the far field of the body. We discuss these issues further in section 5.2, where we consider the experimental constraints on β, M and λ in more detail. The results of this section are summarised in section 4.3.4 below.
4.3.4
Summary
We have seen in this section that there are three important classes of behaviour for Φ outside an isolated body: the linear, pseudo-linear and non-linear regimes. In fact, although the mathematical analysis differs, Φ behaves in same way in both the pseudo-linear and linear regimes. We have shown that linear, or pseudo-linear, behaviour will occur whenever conditions (4.17a-c) on mc R hold. As mc R is increased, conditions (4.17a-c) will eventually fail. As mc R is increased still more, a thin-shell forms and we move into the non-linear regime. A thinshell will exist whenever the thin-shell conditions, eqs. (4.19a-c), hold; these are equivalent to mc R > (mc R)ef f . We have seen that, in the non-linear regime, the far field is independent of the coupling of the chameleon to the isolated body. The main results of this section are summarised below. We have been concerned with a spherical body of uniform density εc and radius R. The background has density εb ¿ εc . The chameleon in background (r À R) takes the value Φb and its mass there is mb = mΦ (Φb ). We also have µ Φc = M
βεc nλMpl M 3
¶−
1 n+1
,
mc = mΦ (Φc ).
Linear and Pseudo-Linear Behaviour Non-linear effects are negligible when: ¯¶ ¯³ ´ 2 µ ³ ´ n+4 p ¯ ¯ m |n+2| 1/6 mc 3(n+2) b ¯ , 2|n + 1| ¯ mc − 1¯¯ , n < −4, mc R ¿ min (18) mb ¯¶ ¯³ ´ 2 µ ³ ´ n+4 √ m 3(n+2) p ¯ ¯ m |n+2| c c ¯ mc R ¿ min , 2|n + 1| ¯ mb − 1¯¯ , n > 0, 3 mb mc R ¿ 1,
n = −4,
150
CHAPTER 4. CHAMELEON FIELD THEORIES
and outside the body, r > R, Φ behaves like: (mc R)2 Φc Re−mb r , n 6= −4, 3(n + 1)r s e−mb r 1 , Φ ≈ Φb − 2(y0 + ln(min(r/R, 1/mb ))) 2r Φ ≈ Φb −
where y0 is given by: (mc R)3 = 9
r
3 1 + 2y0 3
µ
3 2y0
n = −4,
¶3/2 .
When non-linear effects are negligible a body will certainly not have a thin-shell. Bodies with thin-shells A body with have a thin-shell when: ¯³ ´ 2 ¯¶ µ ³ ´ n+4 p ¯ m |n+2| ¯ 1/6 mc 3(n+2) b ¯ ¯ , n < −4, , 3|n + 1| − 1 mc R & min (18) mb ¯ mc ¯ ¯³ ´ 2 ¯¶ µ ³ ´ n+4 √ m 3(n+2) p ¯ m |n+2| ¯ b ¯ , n > 0, 3 mcb , 3|n + 1| ¯¯ m − 1 mc R & min ¯ c mc R & 4,
n = −4.
Outside the body there are two regimes of behaviour. If (r − R)/R ¿ 1 and mΦ (Φ)/mb À 1 then Φ is given by |n + 2|(r − R) 1 ∼ p + mΦ (Φ) 2n(n + 1) and:
µ Φ = sgn(n)M
µ
n+1 n
¶n/2+1
n(n + 1)λM 2 m2Φ (Φ)
1 ¶ n+2
1 , mc
.
If, however, (r − R)/R & 1 then Φ ∼ Φb −
βef f Me−mb r , 4πMpl r
where M is the mass of the body, and βef f is the effective coupling βef f
=
βef f
=
βef f (r) ≈
¶ 1 n+4 γ(n) |n+2| (M R) n+2 , n < −4, |n| ¶ 1 µ 4πMpl n(n + 1)M 2 n+2 MR , n>0 M m2b 2πMpl (2λ ln(min(r/R, 1/mb R)))−1/2 , n = −4. M
4πMpl M
µ
4.4. EFFECTIVE MACROSCOPIC THEORY
151
We note that βef f is independent of coupling of the chameleon to the body, and that Φ is independent of the body’s mass, M. When (r − R)/R & 1, Φ only depends on r, R, M , λ, n and mb .
4.4
Effective Macroscopic Theory
Figure 4.3: Illustration of the model for the microscopic structure of the body considered in this section. The constituent particles are assumed to be spherical of radius R, mass mp and of uniform density. They are separated by an average distance 2D. The average density of the body is defined to be εc . Eq. (4.9) defines the microscopic, or particle-level, field theory for Φ, whereas in most cases, which we wish to study, we are interested in the large scale or coarse grained behaviour of Φ. In macroscopic bodies the density is actually strongly peaked near the nuclei of the individual atoms from which it is formed and these atoms are separated from each other by distances much greater than their radii. Rather than explicitly considering the microscopic structure of a body, it is standard practice to define an ‘averaged’ field theory that is valid over scales comparable to the body’s size. If our field theory were linear, then the averaged equations would be the same as the microscopic ones e.g. as in Newtonian gravity. It is important to note, though, that this is very much a property of linear theories and is not in general true of non-linear ones.
152
CHAPTER 4. CHAMELEON FIELD THEORIES
Non-linear effects must therefore be taken into account. Using similar methods to those that were used in section 4.3 above, we derive an effective theory that describes the behaviour of the course-grained or macroscopic value of Φ in a body with thin-shell. We will identify the conditions that are required for linear theory averaging to give accurate results and consider what happens when non-linear effects are non-negligible. In this section, we derive an effective macroscopic theory appropriate for use within bodies that possess a thin-shell and which are made up of small particles, radius R and mass mp . These particles are separated by an average distance 2D À R. The average density of the body is εc . We illustrate this set-up in FIG. 4.3. A thin-shell, in this sense, means that the average value of Φ inside a sphere of radius & D, will be approximately constant, Φ ≈ Φc , everywhere inside the body apart from in a thin-shell close to the surface of the body. Generally the emergence of a thin-shell is related to a breakdown of linear theory on some level. The conditions for a body to have a thin-shell are given by eqs. (4.19a-c). The outcome of this section will be to slightly modify these conditions. Precisely, we will find that there is maximal, or critical, value for the average chameleon mass, mc . Oddly, this critical, macroscopic chameleon mass depends only on the microscopic properties of the body.
4.4.1
Averaging in Linear Theories
We are concerned with finding an effective theory that will give correct value of Φc . We have defined mc = mΦ (Φc ). The microscopic field equations for Φ, as given by eq. (4.9), are: µ −¤Φ = −nλM
3
M Φ
¶n+1 +
βε(~x) , Mpl
where the microscopic matter density, ε(~x), is strongly peaked about the constituent particles of the macroscopic body but negligible in the large spaces between them. Before considering what occurs in a non-linear theory, such as the chameleon theories being studied here, we will review what would occur in the linear case. For the field equation to be linear, the potential must be, at worst, a quadratic in the scalar field Φ. With the potentials considered in this chapter, a linear theory emerges if n = 0, −1 or −2. To examine why averaging, or coarse graining, is not an issue if field equations are linear, and make reference to what actually occurs in chameleon theories, we shall linearise eq. (4.9) about Φ = Φ0 for some Φ0 . It is important to note that we are performing this linearisation only for the purpose of showing
153
4.4. EFFECTIVE MACROSCOPIC THEORY
what occurs in linear theories; we are not claiming that a linearisation, such as this, is actually valid. Defining Φ = Φ0 + Φ1 , and neglecting non-linear terms, we obtain: µ − ¤Φ1 = −nλM
3
M Φ0
¶n+1
µ + n(n + 1)λM
2
M Φ0
¶n+2 Φ1 +
βε~x) . Mpl
(4.25)
We will write the averaged, or coarse-grained, value of a quantity Q(~x) as < Q > (~x) and define it by: R < Q > (~x) =
d3 y Q(~y )Θ(~x − ~y ) R , d3 yΘ(~x − ~y )
(4.26)
where the function Θ(~x − ~y ) defines the coarse graining and the integral is over all space. Different choices of Θ will result in different coarse-grainings. If we are interested in averaging over a radius of about D around the point ~x, then a sensible choice for Θ would something like: Θ1 (~x − ~y ) = e−
|~ x−~ y |2 D2
,
or Θ2 (~x − ~y ) = H (D − |~x − ~y |) , where H(x) is the Heaviside function. For the coarse graining process to be well-defined we must require that, whatever choice one makes for Θ, it vanishes sufficiently quickly as |~x −~y | → ∞ that the integrals of eq. (4.26) converge. This will usually require Θ ∼ O(|~x −~y |−3 ) as |~x − ~y | → ∞. Consider the application of the averaging procedure to the linear field equation given by eq. (4.25). It follows from the assumed properties of Θ that < ¤Φ1 >= ¤ < Φ1 > and so µ ¤ < Φ1 >= −nλM
3
M Φ0
¶n+1
µ + n(n + 1)λM
2
M Φ0
¶n+2 < Φ1 > +
β < ε > (~x) . Mpl
This is the averaged field equation for Φ1 . It is clear from the above expression, that, although the precise definition of the averaging operator depends on a choice of the function Θ, the averaged field equations are independent of this choice. This independence is a property of linear theories but it is not, in general, seen in non-linear ones. The averaged field equations for a non-linear theory will, generally, depend on ones choice of averaging. In this section, we take our averaging function to be Θ2 as defined above; this is equivalent to averaging by volume in a spherical region of radius D. It is also clear that, for a linear theory, the averaged
154
CHAPTER 4. CHAMELEON FIELD THEORIES
field equation for Φ1 is functionally the same as the microscopic equation. This, again, would not be true if non-linear terms where present in the equations; in general < Φn >6=< Φ >n unless n = 0 or 1 or Φ is a constant.
4.4.2
Averaging in Chameleon Theories
Our aim, in this section, is to calculate the correct value of < Φ > and < mΦ (Φ) > inside a body with a thin-shell. We have defined Φc =< Φ > and εc =< ε >. Although these calculations will implicitly depend on our choice of averaging function, our results should also be approximately equal, at least to an order of magnitude, to those that would be found using any other sensible choice of coarse-graining defined over length scales of about D or greater. (lin)
If our chameleon theories were linear, we have seen that we would expect Φ = Φc à −¤Φc(lin) (lin)
and so, for Φc
= −nλM
3
!n+1
M
+
(lin) Φc
where
βεc , Mpl
≈ const, we have: µ Φc(lin)
=M
βεc nλMpl M 3
¶−1/(n+1) .
In appendix D, we show that, for some values of R, D and mp , linearised theory will give (lin)
the correct value of Φc to a high accuracy i.e. Φc ≈ Φc
. This happens when there either
exists a consistent, everywhere valid, linearisation of the field equations or we can construct a pseudo-linear approximation along the same lines as was done in section 4.3.2 . However, for some values of R, D and mp , we find that non-linear effects are unavoidable. When R, D and mp take such values, we will say that we are in the non-linear regime. We find that, just as it did in section 4.3.3 above, the non-linear regime features β-independent critical behaviour. The details of these calculations can be found in appendix D. We define: Dc D∗
¶ n+2 n+4 3βmp = (n(n + 1)) M , 4πMpl |n| µ µ ¶ n+1 ¶ n+2 3 3βmp n(n + 1) 3 −1 M , = MR 4πMpl |n| n+1 n+4
µ
−1
(4.27)
155
4.4. EFFECTIVE MACROSCOPIC THEORY
and note that D∗ /Dc = (Dc /R)(n+1)/3 . For the linear approximation to be valid we need both mc D ¿ 1 and m2c D3 /2R ¿ 1. This is equivalent to: Dc ¿ D ¿ D∗ , max(Dc , D∗ ) ¿ D, When n = −4 we require D ¿ D∗ and:
µ
3/2 1/2
(12)
λ
n < −4
n > 0.
3βmp 4πnMpl
¶ ¿ 1.
We can see that, for given mp and R, it is always possible to find a D such that the linear approximation is valid when n > 0. However, when n ≤ −4, it is possible that there will exist no value of D for which the above conditions hold. Whenever the linear approximation holds we have:
³ ´ mc ≈ mΦ Φc(lin) .
We can construct a pseudo-linear approximation whenever: ¶ µ 3 ¡ Dc ¢ n+4 ¡ R ¢ |n+1| n+1 , n < −4 < 2|n + 1| 1 − D R ·
D Dc
> 1 and
D∗ D
µ ¶¸ n+1 3 3 ¡ R ¢ n+1 < 2(n + 1) 1 − D ,
(4.28a)
³ ´ ³ ´1/6 (lin) 243 mΦ Φc D ¿ 2 ln(D/R) ,
n>0
(4.28b) (4.28c)
n = −4.
When the pseudo-linear approximation holds we again find: ³ ´ mc ≈ mΦ Φc(lin) . As the inter-particle separation, D, is decreased we will eventually reach a point where eqs. (4.28a-c) fail to hold. When this occurs it is because non-linear effects have become important inside the individual particles that make up the body. As D decreases still further these particles will eventually develop thin-shells of their own. Non-linear effects become important when: µ
Dc R
Ã
¶ n+4 n+1
D∗ D
µ
> 3|n + 1| 1 − " >
Ã
R D µ
3(n + 1) 1 −
µ ³ ´ (lin) mΦ Φc D &
243 2 ln(D/R)
¶
R D
3 |n+1|
¶
!
3 n+1
,
n < −4
(4.29a)
,
(4.29b)
!# n+1 3 n>0
¶1/6 ,
n = −4.
(4.29c)
156
CHAPTER 4. CHAMELEON FIELD THEORIES
These conditions define the non-linear regime. Between the pseudo-linear, and fully non-linear regimes, there is, of course, some intermediate region, however this has proven too difficult to analyse analytically. We therefore leave the detailed analysis of this intermediate behaviour to a later work. This intermediate region is, however, in some sense small and so we do not believe it to have any great importance with respect to experimental tests of chameleon theories. When the individual particles develop thin-shells, the Φ-field external to the particles will be, by the results of section 4.3, independent of β. This ensures that the chameleon mass far from the particles is also independent of β. Therefore, whenever a body falls into the non-linear regime, the average chameleon mass will take a critical value, mc = mcrit c . This is defined in a similar way to which (mc R)crit was in section 4.3.3, i.e. mcrit is the maximal mass c that the chameleon may have when r ∼ O(D) such that, when the microscopic field equations are integrated, (Φc /Φ)n is finite for all r > R. This definition implies a relationship between crit is also found to depend mcrit c , R and D, however it does not depend on either M or λ. mc
on n; this is because n defines precisely how quickly (Φc /Φ)n blows up. We derive expressions mcrit in appendix D finding: c p
mcrit c mcrit c
µ ¶ 3|n + 1| R q(n)/2 ≈ S(n), D D = X/D, n = −4,
n 6= −4
(4.30)
where q(n) = min((n + 4)/(n + 1), 1), S(n) = 1 if n > 0 and S(n) = (γ(n)/3)1/2|n+1| if n < 0. X is given by:
√ 3 3
p ≈ X cosh X − sinh X. 2 ln(D/R) We plot mcrit c D vs. ln(D/R) in figure 4.4. For an everyday body with density similar to water, we approximate R and mp respectively by the radius and mass of carbon nucleus (say) and ≈ 1.4/D when n = −4. When n 6= −4, mcrit find ln(D/R) ≈ 11, and so mcrit c D ¿ 1 follows c from R/D ¿ 1. is a macroscopic quantity, it depends It is interesting to note that, even though mcrit c entirely on the details of the microscopic structure of the body i.e. R and D. By combining the results of this section, we find that the average mass of the chameleon inside a body with thin-shell that is itself made out of particles is given by: ! Ã µ ¶ n+2 |n|λMpl M 3 2(n+1) crit , mc (n, R, D) . mc = min M βεc
157
4.5. FORCE BETWEEN TWO BODIES
Plot of m
crit
2.6
Plot of ln(mcritD) vs. ln(D/R) for n ≠ −4
D vs. ln(D/R) for n = −4 0
n = −10 n = −8 n = −6 n>0
−2
2.4
−4
2.2
−6 ln(mcritD)
mcritD
2 1.8 1.6
−8 −10 −12 −14
1.4 −16 1.2 1 0
−18 10
20
30
40
50
ln(D/R)
−20 0
10
20
30
40
50
ln(D/R)
Figure 4.4: Dependence of the critical chameleon mass on D/R. The above plots show how mcrit D depends on ln(D/R) for different values of n. The cases n = −4 and n 6= −4 are qualitatively different and are therefore shown on separate plots. mcrit is the maximal mass the chameleon can take inside a thin-shelled body. 2D is the average separation of the particles that comprise that body and R is the average radius of the constituent particles. Typically we find that ln(D/R) ≈ 11 for bodies with density ε ∼ 1 − 10 g cm−3 ; ln(D/R) = 11 is indicated on the plots. Note that when n > 0, mcrit D is independent of n. Also note that in Φ4 theory mcrit D ∼ O(1) whereas for other value of n it is generically much smaller.
When evaluating the thin-shell conditions, eqs. (4.19a-c), it is therefore this value of mc that should be used.
4.5
Force between two bodies
In the previous two sections, we have considered how the chameleon field, Φ, behaves both inside and outside an isolated body. In this section we study the form that the chameleon field takes when two bodies are present, and use our results to calculate the resultant Φ-mediated force between those bodies. The results of this sections will prove to be especially useful when we come to consider the constraints on chameleon field theories coming from experimental tests of gravity in section 5.2 below. Chameleon field theories, by their very nature, have highly non-linear field equations. This non-linear nature is especially important when bodies develop thin-shells. As a result of their
158
CHAPTER 4. CHAMELEON FIELD THEORIES
non-linear structure, one cannot solve the two (or many) body problem by simply superimposing the fields generated by two (or many) isolated bodies, as one would do for a linear theory. When the two bodies in question have thin-shells, we shall see that the formula for the Φ-force is highly dependent on the magnitude of their separation relative to their respective sizes. We shall firstly consider the case where the separation between the two bodies is small compared the radius of curvature of their surfaces, and secondly look at the force between two distant bodies. Finally we will consider the force between a very small body and a very large body. We will also look at what occurs when one or both of the bodies does not have a thin-shell.
4.5.1
Force between two nearby bodies
We consider the force between two bodies (hereafter body one and body two) whose surfaces are separated by a distance d. Both bodies are assumed to satisfy the thin-shell conditions. The two bodies are taken to be nearby in the sense that: d ¿ R1 , R2 where R1 and R2 are respectively the radii of curvature of the surface of body one and body two. Since d ¿ R1 , R2 we can ignore the curvature of the surfaces of bodies to a first approximation. With this simplification we treat the bodies as being infinite, flat slabs and take body one to occupy the region x < 0, and body two the region x > d. We use a subscript 1 to refer to quantities that are defined for body one: e.g. the density of body one is ε1 and the chameleon mass deep inside body one is m1 , and a subscript 2 for quantities relating to body two. Additionally a subscript or superscript s is uses to refer to quantities that are defined on the surfaces of the two bodies e.g. ms1 is the chameleon mass of the surface of body one. Subscript 0 is used two label quantities defined at that point between body one and body two where dΦ/dx = 0. We assume also that the background chameleon mass, mb , obeys mb d ¿ 1, we discuss later what occurs if this is not the case. We now consider the Φ-mediated force on body one due to body two. With the above definitions, and Φ obeys: d2 Φ = V,Φ (Φ) dx2 in 0 < x < d and βε1 d2 Φ = V,Φ (Φ) + 2 dx Mpl
159
4.5. FORCE BETWEEN TWO BODIES
in x < 0. Integrating these equations we find: µ ¶2 dΦ = 2(V (Φ) − V0 ), dx
(4.31)
in 0 < x < d, and in x < 0 we have: µ ¶ 1 dΦ 2 βε1 (Φ − Φ1 ) = V (Φ) − V1 + . 2 dx Mpl Matching these expressions at x = 0 we have: Φs1 − Φ1 =
Mpl (V1 − V0 ) , βε1
(4.32)
¯ s where: If the second body where not present then V0 = 0 and Φs1 = Φ 1 ¯ s − Φ1 = Mpl V1 . Φ 1 βε1 The attractive force per unit area of body one due to body two is therefore: ¯ βε1 ¯¯ ¯ s FΦ = Φ1 − Φs1 ¯ = V0 . A Mpl This holds for all V (Φ) not just the Φ−n potentials considered in this work. To find V0 we integrate eqn. (4.31) in the region 0 < x < d and find: √ µZ y1 ¶ Z y2 √ V0 dx dx √ √ + , 2d = |V0,Φ | W (x) x − 1 W (x) x − 1 1 1
(4.33)
where y1 = V1s /V0 and y2 = V2s /V0 and W (x = V /V0 ) = VΦ /V0,Φ . We evaluate the integrals in the above expression in two important limits. Limit 1: y1 = V1s /V0 = 1 + δ In this limit we assume V1 ≈ V1s ≈ V0 , this would occur if V1 < V2 and d is suitably small. In this limit:
Z
Z δ1/2 dz dx √ =2 ≈ 2δ 1/2 + O(δ). W (1 + z 2 ) W (x) x − 1 0 1 We also define Φs1 − Φ1 = Φs1 ² > 0. With this definition equation (4.32) becomes: µ ¶ Mpl (V1 − V1s ) Mpl V0 δ Mpl 1 s2 2 2 s s 3 Φ1 ² = + ≈ Φ1 ² + − Φ1 m1 ² + V0 δ + O(² ) , βε1 Mpl βε 2 y1
thus
s Φs1 ²
≈
and
2V0 δ , m21 s
βε V0 = V1 − Mpl
2V1 δ + O(δ). m21
160
CHAPTER 4. CHAMELEON FIELD THEORIES
Limit 2: y1 = V1s /V0 À 1 This limit occurs when either d is suitably large or if V1 À V2 . We take 1/y1 = δ ¿ 1. We consider V ∝ Φ−n potentials and so W = x1+1/n . In this limit we have: µ ¶ µ ¶ Z y1 Z 1 1 1 1 1 1 dx 2n 1 1 1 − − √ dz z n 2 (1 − z) 2 ≈ B + , − δ n + 2 + ... = n 2 2 n+2 W (x) x − 1 1 δ where B( · , · ) is the Beta function. To leading order in δ we have: Φs1 = Φ1 +
Mpl V1 . βε1
We are now in a position to evaluate V0 (d) and hence FΦ /A. Without loss of generality we take V1 ≤ V2 and consider three limits: Large separations If
√ µ ¶− n+1 µ ¶ 2 n 1 1 1 2 2n B + , . m1 d, m2 d À n+2 n+1 n 2 2
then we have V2 ≥ V1 À V0 and so: ¯ ¯ n +1 ¶ µ ³ ´ √ ¯M ¯2 1 1 1 1 1 1 1 2λ|n|M ¯¯ ¯¯ + , + O (V0 /V1 ) n + 2 , (V0 /V2 ) n + 2 . d = 2B Φ0 n 2 2 In this limit: 2 FΦ = V0 = λ n+2 M 4 A
where we have defined
2n Ã√ ¡ ¢ ! n+2 2 2n 2B n1 + 12 , 12 = λ n+2 Kn M 4 (M d) n+2 , |n|M d
2n Ã√ ¡ ¢ ! n+2 2B n1 + 12 , 12 Kn = . |n|
Small separations: m2 ≈ m1 If
µ m1 d >
n+1 n
then V2 ≈ V1 ≈ V0 and: " µ V0 ≈
1/2 1/2 V1 V2
1−
¶ õ
n n+1
¶
m2 m1
¶n/(n+2) −
1/2
V0
1/2
V1
µ
m0 d 1/2
+ V2
m1 m2
¶n/(n+2) !
#
, "
≈
1/2 1/2 V1 V2
1−
µ
m2 d ¿ 1
n n+1
¶
1/2
V1
1/2
V1
m1 d 1/2
+ V2
# .
The force per area is therefore: " # " # µ ¶ µ ¶ 1/2 1/2 n V m d n V m d FΦ 1/2 1/2 1/2 1/2 0 1 0 1 = V1 V2 1− ≈ V1 V2 1− . A n + 1 V 1/2 + V 1/2 n + 1 V 1/2 + V 1/2 1
2
1
2
161
4.5. FORCE BETWEEN TWO BODIES
Small separations: m2 À m1 When m2 À m1 it is generally the case that V2 > V1s À V0 > V1 . In this case we have: 2 Ã ! n+2 µ ¶ n+2 2 FΦ m1 n (n + 2)m1 d . p = V0 = (n + 1)V1 1 − + A m2 n+1 2n(n + 1) This limit is valid provided that V0 > 0 and V1s À V0 , which requires: m1 + m2
µ
n n+1
¶ n+2
(n + 2)m1 d p ¿ 1. 2n(n + 1)
2
If mb d is not ¿ 1 then the FΦ /A given above are further suppressed by a factor of exp(−mb d).
4.5.2
Force between two distant bodies
We shall now consider the force between two bodies, with thin-shells, that are separated by a ‘great distance’, d. By ‘great distance’ we mean d À R1 , R2 , where R1 and R2 are respectively the length scales of body one and body two. Given that d À R1 , R2 , then, to a good approximation, we can consider just the monopole moment of the field emanating from the two bodies, and model each body as a sphere with respective radii R1 and R2 . We expect that, outside some thin region close to the surface of either body, the pseudolinear approximation (with the field taking its critical value i.e.
(mc R) → (mc R)crit ≈
(mc R)ef f ) is appropriate to describe the field of either body. In the region where pseudolinear behaviour is seen, we can safely super-impose the two one body solutions to find the full two body solution. Close to the surface of body one the mass of the chameleon induced by body one will act to attenuate the perturbation to the chameleon field created by body two. This effect can be quite difficult to model correctly. We can predict the magnitude of the field, however, by noting that the perturbation to Φ, induced by body two, near to body one will be (2)
δΦ2 = −
βef f M2 e−mb d 4πMpl d
.
(2)
From the results of section 4.3.3, we know that βef f M2 /Mpl only depends on the radius of body two and the theory dependent parameters, M , λ and n; βef f is as given in eqs. (4.24a-c); mb is the mass of the chameleon field in the background i.e. far away from either of the two bodies. M2 is the mass of body two.
162
CHAPTER 4. CHAMELEON FIELD THEORIES
We define the perturbation to Φ induced by body one near to body one in a similar manner (1)
δΦ1 = −
βef f M1 e−mb d 4πMpl d
.
From eqs. (4.24a-c) we know that δΦ1 is independent of β and the mass of body one, M1 . The force on body one due to body two will be proportional to ∇δΦ2 , however, since this must also be the force on body two due to body one, it must also be proportional to ∇δΦ1 evaluated near body two. ¿From this we can see that the force on one body due to the other must, up to a possible O(1) factor, be given by (1)
FΦ =
(2)
βef f M1 βef f M2 (1 + mb d)e−mb d (4πMpl )2 d2
.
The functional dependence of this force on M , n, R1 , R2 and λ depends on whether n < −4, n = −4 or n > 0. We consider these three cases separately below. In all cases the force is found to be independent of β, M1 and M2 . Case n < −4 When n < −4, eq. (4.24a) gives µ
(1)
βef f M1 4πMpl
=
γ(n) |n|
¶
1 |n+2|
n+4
(M R1 ) n+2 .
(2)
The expression for βef f is similar but with 1 → 2. The force between two spherical bodies, with respective radii R1 and R2 , separated by a distance d À R1 , R2 , is therefore given by µ FΦ =
γ(n) |n|
¶
2 |n+2|
n+4
n+4
(M R1 ) n+2 (M R2 ) n+2 (1 + mb d)e−mb d . d2
(4.34)
Case n > 0 (1)
If n > 0 then βef f is given by eq. (4.24b) to be µ
(1)
βef f M1 4πMpl
= M R1
n(n + 1)M 2 m2b
1 ¶ n+2
.
(2)
Again the expression for βef f is similar. The force between two distant bodies is therefore found to be µ FΦ =
n(n + 1)M 2 m2b
2 ¶ n+2
M 2 R1 R2 (1 + mb d)e−mb d . d2
(4.35)
163
4.5. FORCE BETWEEN TWO BODIES
Case n = −4 (1,2)
When n = −4, βef f are given by eq. (4.24c) and they are actually weakly dependent on r. Using eq. (4.24c) we find that FΦ =
4.5.3
(1 + mb d)e−mb d p . 8λ ln(d/R1 ) ln(d/R2 )d2
(4.36)
Force between a large body and a small body
One subcase that is not included in the above results is the force between a very large body with radius of curvature R1 , and a very small body with radius of curvature R2 , that are separated by an intermediate distance d i.e. R2 À d À R1 . We assume that both bodies have thin-shells. In this case we find a behaviour that is half-way between the two cases described above in sections 4.5.1 and 4.5.2. The magnitude of field produced by the large body will be much greater than that of the small body. If we ignore the small body and assume that the average mass of the chameleon in the background obeys mb d ¿ 1, then the field produced by body one is given by eq. (4.22). Using this equation, we find that 1 dΦ (x = d) ≈ λ n+2 dx
µ
2 (n + 2)2
¶
n 2(n+2)
n+4
(M d) n+2 d−2 .
The effective coupling of body two to this Φ-gradient will be βef f as it is given by eqs. (4.24ac). If mb d & 1 then this gradient will be, up to an order O(1) coefficient, attenuated by a factor of (1 + mb d) exp(−mb d). The force between the two bodies is therefore given by ¶ 1 µ µ ¶ n 2(n+2) n+4 γ(n) |n+2| 2 FΦ = (M R2 ) n+2 2 |n| (n + 2) −m d b n+4 (1 + mb d)e (M d) n+2 , n < −4 (4.37a) 2 d µ ¶ 1 µ ¶ n 2(n+2) n(n + 1)M 2 n+2 2 FΦ = M R2 2 2 (n + 2) mb (1 + mb d)e−mb d , n>0 d2 1 (1 + mb d)e−mb d p , n = −4. d2 2λ 2 ln(d/R2 ) n+4
(M d) n+2 FΦ =
(4.37b) (4.37c)
As before, d is the distance of separation. These formulae will prove useful when we consider the Φ-force between the Earth and a test-mass in laboratory tests for WEP violation in section 5.2.3.
164
4.5.4
CHAPTER 4. CHAMELEON FIELD THEORIES
Force between bodies without thin-shells
If neither of the two bodies have thin-shells, Φ behaves just like a standard, linear, scalar field with mass mb . The force between the two bodies, with masses M1 and M2 , is given by: FΦ =
β 2 M1 M2 (1 + mb d)e−mb d . 2 d2 4πMpl
As above, mb is the mass of the chameleon in the background. If one of the bodies has a thin-shell, body one say, but the other body does not, then the force is given by (1)
FΦ =
βef f βM1 M2 (1 + mb d)e−mb d 4πd2
, (1)
whenever d À R1 where R1 is the radius of curvature of body one and βef f is as given by eqs. (4.24). If d ¿ R1 then FΦ = λ
1 n+2
µ
2 (n + 2)2
¶
n 2(n+2)
n+4
(M d) n+2
βM2 (1 + mb d)e−mb d . Mpl d2
As above, d is the distance of separation.
4.5.5
Summary
In this section we have considered the force that the chameleon field, Φ, induces between two bodies, with masses M1 and M2 and radii R1 and R2 , separated by a distance d. The chameleon mass in the far background is taken to be mb . When both bodies have thin-shells, we found that, to leading order, the Φ-force between them is independent of the matter coupling, β, provided that m1 d, m2 d À 1; m1 is the mass that the chameleon has inside body one, and m2 is similarly defined with respect to body two. The force between two such bodies is also independent M1 and M2 but does in general depend on R1 and R2 , as well as on M , n and λ. The main results of this section are summarised below.
Neither body has a thin-shell
FΦ =
β 2 M1 M2 (1 + mb d)e−mb d . 2 d2 4πMpl
165
4.5. FORCE BETWEEN TWO BODIES
Body one has a thin-shell, body two does not If the two bodies are close together (m−1 1 ¿ d ¿ R1 ) then
FΦ = λ
1 n+2
µ
2 (n + 2)2
¶
n 2(n+2)
n+4
(M d) n+2
βM2 (1 + mb d)e−mb d . Mpl d2
If the bodies are far apart d À R1 then µ
FΦ FΦ
¶ 1 n+4 βM2 (1 + mb d)e−mb d γ(n) |n+2| = , (M R1 ) n+2 |n| Mpl d2 µ ¶ 1 n(n + 1)M 2 n+2 βM2 (1 + mb d)e−mb d = M R1 , Mpl d2 m2b
FΦ = (2λ min(d/R, 1/mb R))−1/2
βM2 (1 + mb d)e−mb d , 2Mpl d2
n < −4, n > 0, n = −4.
This force is independent of the mass of body one, and the chameleon’s coupling to it.
Both bodies have thin-shells −1 If the two bodies of close by and m−1 1 , m2 ¿ d ¿ R1 , R2 then the force per unit area is
2 FΦ ≈ λ 2+n M 4 A
"√ ¡ ¢ # 2n 2B 12 , 12 + n1 n+2 . |n|M d
Different formulae for FΦ /A apply if either m1 d < 1 or m2 d < 1, these are given at the end of section 4.5.1 above. µ
FΦ FΦ
¶
2 |n+2|
n+4
n+4
(M R1 ) n+2 (M R2 ) n+2 (1 + mb d)e−mb d = , n < −4 d2 µ ¶ 2 n(n + 1)M 2 n+2 M 2 R1 R2 (1 + mb d)e−mb d = , n>0 d2 m2b
FΦ =
γ(n) |n|
(1 + mb d)e−mb d p , 8λ ln(d/R1 ) ln(d/R2 )d2
n = −4,
166
CHAPTER 4. CHAMELEON FIELD THEORIES
where γ(n) is given in table 4.1. If body one is much larger than body two and they are at an intermediate separation: R2 , m−1 1 À d À R1 then µ FΦ =
γ(n) |n|
¶
1 |n+2|
(M R2 )
n+4 n+2
µ
2 (n + 2)2
¶
n 2(n+2)
(1 + mb d)e−mb d , n < −4 d2 µ ¶ n ¶ 1 µ 2(n+2) n(n + 1)M 2 n+2 2 = M R2 2 (n + 2)2 mb n+4
(M d) n+2
FΦ
(1 + mb d)e−mb d , n>0 d2 1 (1 + mb d)e−mb d p , n = −4. d2 2λ 2 ln(d/R2 ) n+4
(M d) n+2 FΦ =
4.6
Discussion
In this chapter we have studied the dynamics of a class of scalar field theories with potentials V ∝ Φ−n . These theories were seen to exhibit a chameleon mechanism which causes their mass to be strongly dependent on the background density of matter. The larger the local density of matter, the more massive the fields become. As we shall see in the next chapter, this unique property allows such scalar fields to hide from experimental tests of gravity and it is for this reason that they deemed to be chameleon fields. Chameleon field theories generically involve highly non-linear field equations. It is therefore often very difficult to solve the equations exactly. Although it is often appealing to linearise the chameleon field equation, we have seen above that it is also often not acceptable to do so. In this chapter, we have avoided the temptation to linearise the field equation for the chameleon, eq. (4.9), when it is not valid to do so. We have instead combined matched asymptotic expansions with approximate analytical, and exact numerical, solution of the full non-linear equations to study the behaviour of chameleon field theories in more detail. The main results of our analysis are: • We found the conditions under which a body would have a thin-shell and saw how the development of a thin-shell is related to the onset of non-linear behaviour. • We showed that the far field of a body with a thin-shell is independent of the chameleonto-matter coupling β. We also found that this was a generic property of all the chameleon
167
4.6. DISCUSSION
field theories with a power-law potential.
As we shall see in Chapter 5.2, this β-
independence has important consequences for the design of experiments that search for WEP violations, and it allows chameleon fields with strong matter couplings, β À 1, to be compatible with all current experimental bounds. • The β independence of the far-field of thin-shelled bodies was also shown to result in the Φ-force between two such bodies being independent of the strength with which the chameleon interacts with those bodies. The precise form of the Φ-force was seen to depend, in a significant manner, upon the separation of the two bodies. • Non-linear effects were shown to limit the magnitude of the average chameleon mass in a thin-shelled body to be smaller than some critical value, mcrit c . Intriguingly, we found mcrit to be independent of β, M and λ, and depend only on n and the microscopic c properties of the thin-shelled body. In the next chapter we analysis the experimental, cosmological and astrophysical bounds on chameleon field theories with power-law potentials. We shall see that the β-independence of the chameleon force between two thin-shelled bodies, exponentially weakens the bounds of the coupling, β, that can be derived from table-top and solar system tests of gravity.
Chapter 5
Evading Experimental Constraints with Strong Coupled Scalar Fields
... having observed this I came to the conclusion that, if one could totally remove the resistance of the medium, all substances would fall at equal speeds. Galileo Galilei (1564-1642)
5.1
Introduction
In Chapter 4 we considered the dynamics of a scalar field theory with power-law potential V = λM 4 (M/Φ)n . We denoted the strength of the this scalar field’s coupling to matter coupling by β. If the scalar field interacts with matter with gravitational strength then β ∼ O(1). These scalar field theories were seen to possess a chameleon mechanism: they are chameleon field theories. The rˆole of scalar field self-interactions is usually ignored when one considers the experimental bounds on their parameters. Indeed, if these self-interactions are truly negligible, then the experimental bounds on the properties of scalar fields are, as we noted in the introduction, very strong (see Chapter 1.3). Chameleon field theories however generically exhibit strong self-interactions which, as we saw in Chapter 4, drastically alter the way they behave near macroscopic bodies. We saw that if a body is large enough it develops 169
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CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
a thin-shell and the chameleon then only effectively couples to a thin layer near the surface of the body. In Chapter 4.5 the force between two bodies with thin-shells was shown to be independent of the coupling of the chameleon to either of those bodies. In this chapter we show that the results of Chapter 4 have important consequences for the experimental constraints on the chameleon to matter coupling (β) as well as the design of future experiments. In this chapter we consider the experimental, cosmologically and astrophysical constraints on chameleon theories with a power-law potential. Solar system bounds on WEP violation require β < 10−4 for non-chameleon theories [66] whereas chameleon theories have previously been shown to be compatible with β ∼ O(1) [77]. As we show in this chapter, however, it is actually possible for a chameleon field to couple to matter much more strongly than gravity does (i.e β À 1) and yet for it to have remained thus far undetected. Indeed the bounds on the other parameters (M and λ) are weaker for large β than they are for β . O(1). Laboratory bounds on chameleon field theories are analysed in section 5.2. We focus on the short-range E¨ot-Wash experiment reported in [87], which tests for corrections to the 1/r2 behaviour of gravity, and on the variety of laboratory and solar-system based tests for violations of the weak equivalence principle (WEP). The extent to which proposed satellite-based searches for WEP violation will aide in the search for scalar fields with a chameleon-like behaviour is also considered in section 5.2. In sections 5.3 and 5.4 we show how the stability of white-dwarfs and neutron stars, as well as requirements coming from BBN and the CMB, can be used to bound the parameters of chameleon field theories. We see that such considerations result in the best upper-bound on β. Finally, in sections 5.5 and 5.6 we collate all of the different experimental and astrophysical restrictions on chameleon theories, use them to plot the allowed values of β, M and λ. We discuss our results and their potentially important implications for our understanding of the rˆole played by scalar fields in cosmology and for the design of the experiments which aim to detect them.
5.2
Laboratory Constraints
The best bounds on corrections to General Relativity come from laboratory experiments such as the E¨ot-Wash experiment, [87] and Lunar Laser Ranging tests for WEP violations, [78, 79].
5.2. LABORATORY CONSTRAINTS
171
At very small distances d . 10µm, the best bounds on the strength of any fifth force come from measurements of the Casimir force. In this section we will consider, to what extent, the results of these tests constrain the class of chameleon field theories considered here. We will find the rather startling result that β À 1 is not ruled out for chameleon theories. One of original reasons for studying chameleon theories with n > 0 potentials, [77], was that the condition for an object to have a thin-shell, eq. (4.19b), was found to depend on the background density of matter. It is clear from eq. (4.19b), that the smaller εb is, the larger mc R must be for a body to have a thin-shell. This property lead the authors of ref. [77] to conclude that the thin-shell suppression of the fifth-force associated with Φ would be weaker for tests performed in the low density vacuum of space εb ∼ 10−25 g cm−3 , than it is in the relatively high density laboratory vacuum εb ∼ 10−17 g cm−3 . As a result, it is possible that, if the same experimental searches for WEP violation, which were performed in the laboratory in [102, 103, 104, 93, 94], were to be repeated in space, they would find equivalence principle violation at a level greater than that already ruled out by the laboratory-based tests. It is important to note that this is very much a property of n > 0 theories. It is clear from eqs. (4.19a) and (4.19c) that, when n ≤ −4, the thin-shell condition, for a body of density εc , is only very weakly dependent on the background density of matter when εb ¿ εc . As a result, space-based searches for WEP violation will not detect any violation at a level that is already ruled by lab-based tests for n ≤ −4 theories. Planned space-based tests such as STEP [97], SEE [96], GG [98] and MICROSCOPE [99] promise a much greater precisions than their labbased counterparts. MICROSCOPE is due to be launched in 2007. This improved precision will, in all cases, provide us with better bounds on chameleon theories.
5.2.1
E¨ ot-Wash experiment
The University of Washington’s E¨ot-Wash experiment, [87, 88], is designed to search for deviations from the 1/r2 drop-off gravity predicted by General Relativity. The experiment uses a rotating torsion balance to measure the torque on a pendulum. The torque on the pendulum is induced by an attractor which rotates with a frequency ω. The attractor has 42 equally spaced holes, or ‘missing masses’, bored into it. As a result, any torque on the pendulum, which is produced by the attractor, will have a characteristic frequency which is some integer
172
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
multiple of 21ω. This characteristic frequency allows any torque due to background forces to be identified in a straightforward manner. The torsion balance is configured so as to factor out any background forces. The attractor is manufactured so that, if gravity drops off as 1/r2 , the torque on the pendulum vanishes. The experiment has been run with different separations between the pendulum and attractor. The Eot-Wash group recently announced some new results which go a long way towards better constraining the parameter space of chameleon theory, [88]. The experiment has been run for separations, 55µm ≤ d ≤ 9.53mm. Both the attractor and the pendulum are made out of molybdenum with a density of about εM b ∼ 10 g cm−3 and are 0.997 mm thick. Electrostatic forces are shielded by placing a 10 µm thick, uniform BeCu sheet between the attractor and pendulum. The density of this sheet is εBeCu ∼ 8.4 g cm−3 . The rˆole played by this sheet is crucial when testing for chameleon fields. If β is large enough, the sheet will itself develop a thin-shell. When this occurs the effect of the sheet is not only to shield electrostatic forces, but also to block any chameleon force originating from the attractor. The force per unit area between the attractor and pendulum plates due to a scalar field with matter coupling β and constant mass m, where 1/m ¿ 0.997mm is: |
αe−md 2πGε2M b F |= A m2
where α = β 2 /4π and d is the separation of the two plates. The strongest bound on α coming from the Eot-Wash experiment is α < 2.5 × 10−3 for 1/m = 0.4 − − 0.8mm. Depending on the values of β, M and λ there are three possible situations: • The pendulum and the attractor have thin-shells, but the BeCu sheet does not • The pendulum, the attractor and the BeCu sheet all have thin-shells. • Neither the test masses nor the BeCu sheet have thin-shells. In the first case the Φ-mediated force per unit area in a perfect vacuum is given by one of the equations derived in chapter 4.5.1 (depending on the separation d). In reality the vacuum used in these experiments is not perfect actually has a pressure of 10−6 Torr which means that the chameleon mass in the background, mb , is non-zero and so FΦ /A is suppressed by a factor of exp(−mb d). Fortunately however mb d ¿ 1 for all but the largest β. A further, and far more important, suppression occurs when the BeCu sheet has a thin-shell. If mBeCu is the chameleon
5.2. LABORATORY CONSTRAINTS
173
mass inside the BeCu sheet and dBeCu the sheet’s thickness, the existence of a thin-shell in the electromagnetic shield causes the chameleon mediated force between the pendulum and attractor to be suppressed by a further factor of exp(−mBeCu dBeCu ). The thin-shell condition for the BeCu sheet implies that mBeCu dBeCu À 1, and so this suppression all but removes any detectible chameleon induced torque on the pendulum due to the attractor. The BeCu sheet will itself produce a force on pendulum but, since the sheet is uniform, this force will result in no detectible torque. If M and λ take natural values the electromagnetic sheet develops a thin-shell for β & 104 : as a result of this the Eot-Wash experiment can only very weakly constrain large β theories. If neither the pendulum or the attractor have thin-shells then we must have mb d ¿ 1 and the chameleon force is just β 2 /4π times the gravitational one. Since this force drops of as 1/r2 it will be undetectable from the point of view this experiment. In this case, however, β is constrained by other experiments such as those that search for WEP violation [78, 79, 102, 103, 104] or those that look for Yukawa forces with larger ranges [105]. Given all of the considerations mentioned above, we used the formulae given in chapter 4.5.1 to evaluate the latest E¨ot-Wash constraints on the parameter space of chameleon theories. Our results are shown in FIG 5.1. In these plots the shaded region is allowed by the current bounds. When β is small, the chameleon mechanism present in these theories becomes very weak, and from the point of view of the E¨ot-Wash experiment, Φ behaves like a normal (nonchameleon) scalar field. When β À 1, the Φ-force is independent of the coupling of the chameleon to attractor or the pendulum, but does depend on the mass of the chameleon in the BeCu sheet, mBeCu . The larger mBeCu is, the weaker the E¨ot-Wash constraint becomes. Larger β implies a larger mBeCu , and this is why the allowed region of parameter space increases as β grows to be very large. When n = −4, we can see that a natural value of λ is ruled out for 10−1 . β . 104 , but is permissible for β & 104 . This is entirely due to the that BeCu sheet has a thin-shell, in n = −4 theories with λ = 1/4!, whenever β & 104 . It is important to stress that, despite the fact that the E¨ot-Wash experiment is currently unable to detect β À 1, this is not due to a lack of precision. One pleasant feature, of the β-independence of the Φ-force, is that if you can detect, or rule out, such a force for one value
174
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
Eot−Wash Bounds on n = −8 theories
Eot−Wash Bounds on n = −4 theories
30
log10 k, λ = k/4!
,M
nat
M = M /β pl
5
10
0
15 10
log
5
20
M/M
nat
= (0.1 mm)−1
25
−5 −10 −15
0 −20
−5
−25 −5
0
5 log10 β
10
15
20
−5
0
5 log10 β
10
15
20
Eot−Wash Bounds on n = 4 theories 30
20 M = M /β
10 5
pl
log
10
nat
,M
15
M/M
nat
= (0.1 mm)
−1
25
0 −5 −5
0
5 log10 β
10
15
20
Figure 5.1: Constraints on chameleon theories coming from the E¨ot-Wash bounds on deviations from Newton’s law. The shaded area shows the regions of parameter space that are allowed by the current data. The solid black lines indicate the cases where M and λ take ‘natural values’. The dotted-black line indicates when M = MΦ := Mpl /β i.e. when the mass scale of the potential is the same as that of the matter coupling. Other n < −4 theories are similar to the n = −8 case, whilst the n = 4 plot is typical of what is allowed for n > 0 theories. The amount of allowed parameter space increases with |n|.
175
5.2. LABORATORY CONSTRAINTS
of β À 1, then you will be able to detect it, or rule out, all such β À 1 theories. If design of the experiment can be altered so that electrostatic forces are compensated without using a thin-sheet then the experimental precision already exists to detect, or rule out, almost all β À 1, Φ4 theories with λ ≈ 1/4!. In conclusion, an experiment, along the same lines of the E¨ot-Wash test, could detect, or rule out, the existence of sub-Planckian, chameleon fields with natural values of M and λ in the near future, provided it is designed to do so.
5.2.2
Casimir force experiments
Short distance tests of gravity fail to constrain strongly coupled chameleon theories as a result of their use of a thin metallic sheet to shield electrostatic forces. However, experiments designed to detect the Casimir force between two objects, control electrostatic effects by inducing an electrostatic potential difference between the two test bodies. By varying this potential difference and measuring the force between the test masses, it is possible to factor out electrostatic effects. As a result, Casimir force experiments provide an excellent way in which to bound chameleon fields where the scalar field is strongly coupled to matter. Casimir force experiments measure the force per unit area between two test masses separated by a distance d. It is generally the case that d is small compared to the curvature of the surface of the two bodies and so the test masses can be modeled, to a good approximation, as flat plates. In chapter 4.5.1 we evaluated the force per unit area between two flat, thin-shelled slabs with densities ε1 and ε2 . The Casimir force is between two such plates is: FCas π2 = . A 240d4 Whilst a number of experimental measurement of the Casimir force between two plates have been made, the most accurate measurements of the Casimir force have been made using one sphere and one slab as the test bodies. The sphere is manufactured so that its radius of curvature, R, is much larger than the minimal distance of separation d. In this case the total Casimir force between the test masses is: µ FCas = 2πR
1 π2 1 3 240 d3
¶
µ = 3.35
R (µm)3 d3 cm
¶ µdyn.
176
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
In all cases, apart from when n = −4 and m1 d, m2 d À 1, the chameleon force per area grows more slowly than d−4 as d → 0. When n = −4 and m1 d, m2 d À 1 we have FΦ /A ∝ d−4 . It follows that the larger the separation, d, used, the better Casimir force searches constrain chameleon theories. Additionally, these tests provide the best bounds when the test masses do have thin-shells as this results in a strongly d dependent chameleon force. Large test masses are therefore preferable to small ones. Note that if the background chameleon mass is large enough that mb d & 1 then FΦ is suppressed by a factor of exp(−mb d). The smaller the background density, εb , is, the smaller mb become. Since small mb is clearly preferably, the best bounds come from experiments that use the lowest pressure laboratory vacuum. In [89], Lamoreaux reported the measurement of the Casimir force using a torsion balance between a sphere, with radius of curvature 12.5cm ± 0.3cm and diameter of 4cm, and a flat plate. The plate was 0.5cm thick and 2.54cm in diameter. The apparatus of placed in a vacuum with a pressure of 10−4 mbar. Distances of separation of 6 − 60µm we used and in the region d ≈ 7 − 10µm it was found that: theory |F measured − FCas | . 1µ dyn.
Another measurement, this time using a microelectromechanical torsional oscillator, was performed by Decca et al. and reported in [90]. In this experiment, the sphere was much smaller than that used by Lamoreaux, being on 296 ± 2µm in radius; the plate was made of 3.5µm thick, 500 × 500µm2 polysilicon. The smallness of these test masses means that they will only have thin-shells when β is very large. In the region d ≈ 400nm − 1200nm, theory |F measured − FCas | . 7.5 × 10−2 µdyn. We show how these experiments constrain the pa-
rameter space of Φ−n chameleon theories in FIG 5.2. Other Casimir force tests (e.g. [91] ) are less suited to constraining chameleon theories such as those considered here. As in the previous plots, the shaded area is allowed, the solid black line is M ∼ (εΛ )1/4 or λ = 1/4!, and the dotted black line is M = Mpl /β. We note that Casimir force experiments provide very tight bounds on λ and M when β À 1. A natural value of λ when n = −4 is ruled out for all β ≥ 104 . When this is combined with the latest E¨ot-Wash data we can rule out λ = 1/4! for all β > 1. For other n, we see that we cannot have M much larger than its ‘natural’ value (εΛ )1/4 for large β. If the bounds on extra forces at d ∼ 1 − 10µm can be tighten by roughly an order of magnitude then a natural value of M can be ruled out for all large β. Casimir
177
5.2. LABORATORY CONSTRAINTS
Casimir force bounds on n = −8 theories
Casimir force bounds on n = −4 theories
5
20
0
10 5
M = M /β pl
log
10
nat
,M
15
log10 k, λ = k/4!
25
M/M
nat
= (0.1 mm)−1
30
−5 −10 −15
0 −20 −5 −25 −5
0
5 log10 β
10
15
20
−5
0
5 log10 β
10
15
20
Casimir force bounds on n = 4 theories 30
20
10 5
M = Mpl/β
log
10
nat
,M
15
M/M
nat
= (0.1 mm)
−1
25
0 −5 −5
0
5 log10 β
10
15
20
Figure 5.2: Constraints on chameleon theories coming from experimental searches for the Casimir force. The shaded area shows the regions of parameter space that are allowed by the current data. The solid black lines indicate the cases where M and λ take ‘natural values’. The dotted-black line indicates when M = MΦ := Mpl /β i.e. when the mass scale of the potential is the same as that of the matter coupling. Other n < −4 theories are similar to the n = −8 case, whilst the n = 4 plot is typical of what is allowed for n > 0 theories. The amount of allowed parameter space increases with |n| i.e. as the potential becomes steeper.
178
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
force tests provided by far the best bounds on M and λ for large β. We note that n = −4 theories are the most tightly constrained by Casimir force experiments, this is not surprising since FΦ /FCas ∼ const and is > 1 in the region when m1 d, m2 d À 1 in this model, whereas in all the other theories FΦ /FCas decreases as d is made smaller. More generally, the steeper the potential in a given theory is, the more slowly FΦ /A increases as d → 0 and, as a result, the weaker Casimir force bounds on the theories parameter space are.
5.2.3
WEP violation experiments
The weak equivalence principle (WEP) is the statement that the (effective) gravitational and inertial mass of a body are equal. If it is violated then the either the strength of gravitational force on a body depends on its composition, or there is a composition dependent ‘fifth-force’. Since we believe that gravity is geometric in nature, most commentators, ourselves included, would tend to interpret any detection of a violation of WEP in terms of the latter option. The existence of light scalar fields that couple to matter usually results in WEP violations. As we mentioned in the introduction to this thesis, the experimental bounds on WEP violation are exceeding strong, [78, 79, 93, 94, 95, 102, 103, 104], and, at present, they represent the strongest bounds on the parameters of, non-chameleon, scalar-tensor theories such as Brans-Dicke theory, [29, 116]. A number of planned satellite missions promise to increase the precision to which we can detect violations of WEP by between 2 to 5 orders of magnitude [99, 96, 97, 98]. The precision that is achievable in laboratory based tests also continues to increase at a steady rate. Experiments that search for violations of the weak equivalence principle generally fall into two categories: laboratory based experiments, which often employ a modified torsion balance, [102, 103, 104], and solar system tests such as lunar laser ranging (LLR) [78, 79]. The laboratory based searches use a modified version of the E¨ot-Wash experiment mentioned above. In these experiments the test masses are manufactured to have different compositions. The aim is then to detect, and measure, any difference in the acceleration of test-masses towards an attractor, which is usually the Earth, the Sun or the Moon. In some versions of the experiment a laboratory body is used as the attractor. If the test masses have thin-shells then the Φ-force pulling them towards the Earth is given
5.2. LABORATORY CONSTRAINTS
179
by equations (4.37a-c). The chameleon force towards the Moon or Sun is given by eqs. (4.34)(4.36). If the attractor is a laboratory body then, depending on the separations used, the force is given by either eqs. (4.34)-(4.36) or by eq. (4.5.1). We label the mass and radius of the attractor by M3 and R3 respectively, and take the mass and radius (or size) of the two test-masses to be given by {M1 , R1 } and {M2 , R2 }. We define α13 to be the relative strength, compared to gravity, of Φ-force between the attractor (body three) and the first of the test masses (body one). α23 is defined similarly as a measure of the Φ-force between body two and body three. The difference between the acceleration of the two test masses towards the attractor is quantified by the E¨otvos parameter, η, where η=
2|α13 − α23 | ≈ |α13 − α23 |. |2 + α13 + α23 |
When the test-bodies have thin-shells, we found, in chapter 4.5, that the Φ-force is independent of the masses of the test-bodies, the mass of the attractor and the coupling of the test-masses and attractor to the chameleon. The only property of the attractor and test bodies, which the Φ-force does depend on, is their respective radii. Since the gravitational force between the test-masses and the attractor does depend on the masses of the bodies, it follows that α13 only depends on M1 , M3 , R1 , R3 , M (or λ), mb and n, where mb is the chameleon mass in the background. It does not depend on the chameleon’s coupling to the test-mass, β. The situation with α23 is very similar. Since the Φ-force is independent of the coupling, β, any microscopic composition dependence in β will be hidden on macroscopic length scales. The only ‘composition’ dependence in α13 is through the masses of the bodies and their dimensions (R1 and R3 ). Taking the third body to be the Earth, the Sun or the Moon, experimental searches for WEP violations have, to date, found that η . 10−13 [102, 103, 104]. Future satellite tests promise to be able to detect violations of WEP at between the 10−15 , [99], and the 10−18 level, [97]. It also is claimed that future laboratory tests will be able to see η ∼ 10−15 , however, whilst the precision to detect at such a level is achievable, there are a number of systematic effects that need to be compensated for, before an accurate measurement can be made. In most of these searches, although the composition of the test-masses is different, they are manufactured to have the same mass (M1 = M2 ) and the same size (R1 = R2 ). Therefore, if the test-masses have thin-shells, we will have α13 = α23 and so η = 0 identically. As a result, a chameleon field will produce no detectable WEP violation in these experiments. The only
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CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
implicit dependence of this result on β is that, the larger the coupling is, the more likely it is that the test-masses will satisfy the thin-shells conditions. If one wishes to detect the chameleon using WEP violations searches, then one must either ensure that test-masses do not satisfy the thin-shell conditions, or that they have different masses and/or dimensions. We have just argued that all of the chameleon theories considered here will automatically satisfy all laboratory bounds on WEP violation, provided the test-masses have thin-shells. This occurs entirely as a result of the design of those experiments. Let us consider then a putative experiment, which could be conducted, that would, in principle, be able to detect the chameleon through a violation of WEP. In this experiment the test masses are of different densities (ε1 and ε2 ) but of the same mass, Mtest . Crucially the radii (size) of the two bodies are taken to be different: R1 and R2 . We now calculate the E¨otvos parameter, η, taking the attractor to be either the Earth, the Sun or the Moon.
Attractor is the Earth If the attractor is the Earth then we obtain à !µ ¶ 1 2 (1 + m d)e−mb d ¯ Mpl n+4 n+4 ¯ 3 |n+2| ¯¯ b η = ¯(M R1 ) n+2 − (M R2 ) n+2 ¯ ME Mtest |n| µ ¶ n 2(n+2) n+4 2 (M d) n+2 , n < −4, 2 (n + 2) à !µ ¶ 1 2 (1 + m d)e−mb d Mpl n(n + 1)M 2 n+2 b η = |M (R1 − R2 )| ME Mtest m2b µ ¶ n 2(n+2) n+4 2 (M d) n+2 , n > 0, 2 (n + 2) ¯ ¯ à ! 2 (1 + m d)e−mb d ¯ ¯ Mpl 1 1 b ¯ ¯ √ n = −4, η = − ¯p ¯, ¯ ln(d/R1 ) ln(d/R2 ) ¯ 4 2λME Mtest where mb is the mass of the chameleon in the background region between the test masses and the surface of the Earth; d is the distance between the test masses and the surface of the Earth. MEarth is the mass of the Earth, and REarth its radius. Current experimental precision bounds η . 10−13 . We shall assume that our putative experiment, if conducted, would find η . 10−13 . However, even if this is the case, we are still only able to recover very weak bounds on {β, M, λ}. The bounds on β are especially weak
181
5.2. LABORATORY CONSTRAINTS
due to the β-independence of the Φ-force whenever the test-bodies have thin-shells. The only real bound on β comes out of requirement that it be large enough for the test-masses to have thin-shells. For definiteness, we take the test-masses to be spherical, with a mass of Mtest = 10 g. We assume that one of them is made entirely of copper and the other from aluminum. If the test-masses have thin-shells then, even if mb d ¿ 1, finding η < 10−13 would, when n = −4, only limit
λ & 10−30 .
For theories with n < −4, η < 10−13 is easily satisfied provided that: M < 1010 mm−1 . When n > 0, the resultant bound is on a combination of M and m2b . The WEP bounds on the parameter space of n > 0 theories are generally stronger than those for other n. We plot the effect of these bounds on the parameter space of our chameleon theories in FIG 5.3.
Attractor is the Sun or the Moon
Constraints on chameleon theories can also be found by considering the differential acceleration of the test masses towards the Moon or the Sun, rather than towards the Earth. The analysis for both of these scenarios proceeds along the much same lines. Since, for the reasons we explain below, the lunar bound will be by far the stronger, we will only explicitly consider the case where the third body is the Moon. In this case, the force between the test mass and the Moon is given by equations (4.34), (4.35) and (4.36) for n < −4, n > 0 and n = −4 respectively. We define mb to be the average chameleon mass in the region between the Earth and the Moon, and matm to be the mass of the chameleon in the Earth’s atmosphere; mlab is the background mass of the chameleon in the laboratory. Ra is the thickness of the Earth’s atmosphere. d is the distance of separation between the laboratory apparatus and the Moon.
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Evaluating η we find: η =
η =
Ã
!µ
3 |n|
¶
2 |n+2|
MM oon Mtest ¯ ¯ n+4 ¯ n+4 ¯ n+4 (M RM oon ) n+2 ¯(M R1 ) n+2 − (M R2 ) n+2 ¯ , ! Ã 2 (1 + m d)e−mb d−ma Ra Mpl b MM oon Mtest µ ¶ 2 n(n + 1)M 2 n+2 M RM oon |M (R1 − R2 )|, mb mlab Ã
η =
2 (1 + m d)e−mb d−ma Ra Mpl b
2 (1 + m d)e−mb d−ma Ra Mpl b
8λMM oon Mtest
(5.1) n < −4 (5.2) n > 0,
!
¯ ¯ ¯ ¯ 1 1 ¯ ¯ p p − ¯ ¯, ¯ ln(d/R1 ) ln(d/RM oon ) ln(d/R2 ) ln(d/RM oon ) ¯
(5.3) (5.4)
where MM oon is the mass of the Moon and RM oon its radius. If the Sun is used as the attractor then η is given by a similar set of equations to those shown above only with MM oon → MSun and RM oon → RSun . MSun and RSun are respectively the mass and the radius of the Sun. If the attractor is taken to have fixed density, then we can see that η dies off at least as quickly as 1/R2 , where R is the radius of the attractor. It follows that the predicted value η, induced by chameleon, is always smaller when the Sun is used as the attractor than when the Moon is. This is because RSun À RM oon . The predicted values of η have a similar dependence on the radii of the test-masses i.e. dying off at least as quickly as 1/R2 . The corollary of this result is that if we are unable to detect Φ in lab-based, micro-gravity experiments where the radii of the test-masses and the attractor are both of the order of 10 cm, or smaller, then the Φ-force between larger (say humansized) objects would also be undetectably small. For this reason, in the context of chameleon theories, measurements of the differential acceleration of the Earth and Moon towards the Sun, e.g. lunar laser ranging [78, 79], are not competitive with the bounds on WEP violation found in laboratory-based searches. Summary We show how bounding the E¨otvos parameter by η < 10−13 , with the attracting body being either the Moon or the Earth, constrains chameleon theories in FIG. 5.3. As with the E¨otWash plots: the whole of the shaded area is currently allowed, whilst the more lightly shaded
183
5.2. LABORATORY CONSTRAINTS
WEP violation bounds on n = −8 theories
WEP violation bounds on n = −4 theories
50 0
pl
−10 30
log10 k, λ = k/4!
log10 M/Mnat, Mnat = (0.1 mm)
−1
M = M /β 40
20
10
−20 −30 −40 −50
0 −60 −5
0
5 log10 β
10
15
20
−5
0
5 log10 β
10
15
20
WEP violation bounds on n = 4 theories 30
20
,M
5
10
nat
10
log
M = Mpl/β
15
M/M
nat
= (0.1 mm)
−1
25
0 −5 −5
0
5 log10 β
10
15
20
Figure 5.3: Constraints on chameleon theories coming from WEP violation searches. The whole of shaded area shows the regions of parameter space that are allowed by the current data. Future space-based tests could detect the more lightly shaded region. The solid black lines indicate the cases where M and λ take ‘natural values’. The dotted-black line indicates when M = MΦ := Mpl /β i.e. when the mass scale of the potential is the same as that of the matter coupling. Other n < −4 theories are similar to the n = −8 case, whilst the n = 4 plot is typical of what is allowed for n > 0 theories. The amount of allowed parameter space increases with |n|.
184
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
area is that which could be detected by proposed space-based tests of gravity such as SEE, STEP, GG and MICROSCOPE. It is claimed, [97, 96, 99, 98], that these experiments will be able to detect η down to 10−18 . This improved precision is responsible for most of the increased ability of spaced based tests to detect a chameleon field. When n > 0, the thinsell condition is stronger in the low density background of space than it is in the relatively higher density background of the laboratory; this effect accounts for some of extra ability that future space-based experiments have to detect the chameleon. We note that WEP violation searches only have any real hope of detecting the chameleon field, if M take a ‘natural’ value i.e. M ∼ (0.1mm)−1 , in n > 0 theories.
5.2.4
Discussion
In FIG. 5.3, we plot how all of the bounds on WEP violation mentioned in refs. [102, 103, 104, 78, 79], as well as the putative bound resulting from the modified WEP violation test we considered above, constrain the parameter space of our chameleon theories. The E¨ot-Wash bounds are shown in FIG. 5.1, and the Casimir bounds in FIG. 5.2 These E¨ot-Wash, Casimir and WEP violation bounds are also included in the plots of section 5.5. It is important to note that, the larger β is, the stronger the chameleon mechanism becomes. A strong chameleon mechanism results in larger chameleon masses, and larger chameleon masses in turn result in weaker chameleon-mediated forces. A stronger chameleon mechanism also increases the likelihood of the test masses, used in these experiments, having thin-shells. Large values of β cannot therefore be detected at present by the E¨ot-Wash and WEP tests, if λ and M take natural values. If the matter coupling is very small, β ¿ 1, then the chameleon mechanism is very weak and Φ behaves as a normal (non-chameleon) scalar field. The current precision of laboratory tests of gravity prevent them from seeing even non-chameleon theories with β < 10−5 . Experiments that search for the Casimir force are better able to detect large β theories primarily due to the way in which they cancel electrostatic forces. The β ∼ O(1) region is, however, currently inaccessible to Casimir force experiments. Casimir force experiments provide upper bounds on 1/λ and M but do not directly rule out large β. Upper bounds on β do arise, however, from astrophysical considerations. It so happens that for β . 1020 , these astrophysical constraints are weaker than those coming from Casimir force searches, however they are important since they constrain how the chameleon
5.3. IMPLICATIONS FOR COMPACT BODIES
185
field can behave in much lower (and higher) density backgrounds than those that are easily accessible in the laboratory, and thus effectively probe a different region of the chameleon potential. These bounds are discussed in sections 5.3 and 5.4 below. In conclusion: contrary to most expectations laboratory tests of gravity do not rule out scalar field theories with a large matter couplings, β À 1, provided that they have a strong enough chameleon mechanism. When n = −4, a natural value of λ is ruled out for all α = β 2 /4π < 10−2 , and even λ ∼ O(1) requires β ∼ 107 , i.e α ∼ 1013 . When n 6= −4, large β theories with natural values of M are allowed for all n 6= −6. Chameleon theories, with natural values of M , could well be detected or ruled out by a number of future experiments provided they are properly designed to do so.
5.3
Implications for Compact Bodies
In this section we will consider the effect that the fifth force associated with the chameleon field has upon the physics of compact bodies such as white dwarfs and neutron stars. In the preceding analysis, we have shown that the Φ-force is only comparable in strength to gravity over very small scales. In neutron stars and white dwarfs, however, the average inter-particle separations are very small, about 10−13 cm and 10−10 cm respectively, and the physics of such compact objects can therefore be very sensitive to additional forces that only become important on small scales. The stability of white dwarf and neutron stars involves a delicate balancing act between the degeneracy of, respectively, electrons or neutrons and the effect of gravity. If the presence of the chameleon field were to alter this balance significantly, one might find oneself predicting that such compact objects are always unstable. If chameleon theories were to make such a prediction, for some values of the parameters {β, M, λ}, then we could, obviously, rule out those parameter choices. As well as the issue of stability, we must also consider potential astrophysical observables such as the Chandrasekar mass limit, and the mass-radius relationship. In 1930, Chandrasekar made the important discovery that white dwarfs had a maximum mass ∼ 1.4M¯ , [113, 114]. The precise value depends on the composition of the star. A similar maximum mass was found for neutron stars by Landau [115]. It was additionally noted that the mass, Mstar , of a white dwarf or neutron star would depend on its radius, R, in a very special manner. This is the mass-radius relationship. It is possible to extrapolate both Mstar
186
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
and R from astronomical data, for example see [110, 111, 112]. In all of those works, and others like them, the mass-radius relationship, as predicted by General Relativity , has been found to be in good agreement with the data. It is, therefore, important that the addition of a chameleon field should not greatly alter this relationship. The quoted 1σ error bars on most of the determinations of Mstar and R are between about 3 and 10% of the central value. It would certainly be fair to say then that any new theory, which predicts deviations in the value of Mstar (R) from the GR value of less than about 10%, is consistent with all current data. Much greater deviations, are however ruled out. We shall use this criterion to bound the parameters of our chameleon theories. We firstly consider how the presence of a chameleon alters the Chandrasekar mass limit and the mass-radius relation for both white dwarfs and neutron stars. Our analysis proceeds along the same lines as that presented in [100]. We will start by considering a white dwarf and then note how our results carry over to neutron stars. We suppose that the white dwarf contains N electrons. Charge neutrality then implies that there are N protons. There will also be neutrons present (approximately N of them) but for this calculation we will merely group the protons and neutrons together into N nucleons where each nucleon contains one proton. We denote the mass of a nucleon by mu and take it to be the atomic mass unit, mu = 1.661 × 10−24 g. White dwarfs are kept from collapsing by the pressure of degenerate electrons, whilst their gravitational potential comes almost entirely from the nucleons (as mu À me ). We model the white dwarf as being at zero temperature. If should be noted that we are not interested, so much, in accurately determining the mass-radius relationship or Chandrasekar mass limit, as we are in seeing to what extent they deviate from the general relativistic form. In the limiting cases of non-relativistic, Γ = 5/3, and relativistic, Γ = 4/3, behaviour, the equation of state for the electrons can be written in polytropic form: P = KεΓ with K a constant. For relativistic electrons: K=
31/3 π 2/3 1 , 4/3 5/3 4 mu µe
where µe ≈ 2 is the chemical potential for the electrons in the white-dwarf, [100]. We require that the white dwarf be in hydrostatic equilibrium. Ignoring general relativistic corrections,
187
5.3. IMPLICATIONS FOR COMPACT BODIES
this implies
~ ~ = −ε∇Φ ~ N − βε∇Φ , ∇P Mpl
where the last term is the additional element that comes from the chameleon field. ΦN is the 2 . In most realistic scenarios the density Newtonian gravitational potential: ∇2 ΦN = 4πε/Mpl
inside the white dwarf will only change very slowly over length scales comparable to the inverse chameleon mass. We can therefore take µ Φ(x) ≈ M
Mpl M 3 nλ βε(~x)
1 ¶ n+1
,
to hold inside the white dwarf. It is standard practice, [100, 101], to solve for hydrostatic equilibrium by minimizing an appropriately chosen energy functional. In the absence of a chameleon field this is: ¯ = U + W, E where W is the gravitational potential energy: Z W =−
1 d3 x ΦN ε, 2
and U is the internal energy of the white dwarf. It is shown in [100] that when one perturbs ~ for some vector field ~ · (εξ) the density in such a way that the total mass is conserved (δε = −∇ ~ we have: ξ) ¯ = δU + δW = δE
Z ³
´ ~ 3 x. ~ + ε∇Φ ~ N · ξd ∇P
¯ vanishes for a star in hydrostatic equilibrium. To solve for hydrostatic It follows that δ E equilibrium in the presence of a chameleon field, we need to minimise the following energy functional: ¯ + WΦ = U + W + WΦ , E=E where: WΦ =
n+1 n
Z d3 x
βΦ ε. Mpl
To see that this is the correct expression we consider δWΦ : δWΦ
Z Z n+1 βΦ 3 βδ(Φε) = d x = d3 x δε n Mpl Mpl à ! Z Z ~ βε ∇Φ 3 βΦ ~ 3 ~ = d x = − d x ∇ · (εξ) · ξ. Mpl Mpl
188
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
With this definition δE = 0 is seen to be equivalent to requiring hydrostatic equilibrium: ! Z Ã ~ βε ∇Φ ~ + ε∇Φ ~ N+ δE = · ξ~ d3 x = 0. ∇P Mpl
5.3.1
The Mass-Radius Relation
Schematically we have WΦ ∝
n+1 n (β<
Φ >/Mpl )Mstar where < .. > indicates an average and
Mstar = mu N is the mass of the star. We note that WΦ ∼ ε−1/(n+1) , and that it is negative for n ≤ −4 and positive for n > 0. To study the effect of the chameleon upon the Chandrasekar mass limit, and the mass-radius relationship, we assume that the density of the white dwarf is uniform. Whilst this is not at all accurate, it is sufficient to see when the chameleon does, or does not, have a significant effect. Also, whilst not being accurate, this approximation still gives the mass-limit and mass-radius relationship up to an O(1) factor. This said, we shall consider a more accurate model later when we look at general relativistic corrections. The total internal energy of the white dwarf is given by: U =N
³¡ ´ ¢1/2 p2F + m2e − N me > 0,
where pF = N 1/3 /R is the Fermi momentum of degenerate electrons, [100, 101]. For W we find: W =−
3m2u N 2 2R . 5Mpl
Evaluating WΦ in this approximation yields: WΦ = αΦ
n + 1 βΦ(ε) mu N. n Mpl
We have included a numerical factor αΦ in the definition of WΦ given above. Although in the uniform density approximation we have αΦ = 1, we chosen to leave αΦ in the above equation so that our results can be more easily reassessed in light of the more accurate evaluation of WΦ performed in appendix E. The total energy is then: µ³ ¶ ´1/2 n + 1 βΦ(ε)mu N 3m2u N 2 2/3 2 2 E(R) = N N /R + me + αΦ , − N me − 2 n Mpl 5Mpl R and ε ∝ 1/R3 . We will have equilibrium when dE(R)/dR = 0 which implies: ¯ ¯ N 1/3 3m2u N 2/3 3αΦ ¯¯ βΦ(ε) ¯¯ −1/3 . + ¡ ¢1/2 = ¯ Mpl ¯ mu RN 2 |n| 5M 2/3 2 2 N + me R pl
189
5.3. IMPLICATIONS FOR COMPACT BODIES
We note that the term on the left-hand side of the above expression is always less than 1, and that both terms on the right hand side are positive definite. This implies that there is a maximal value of N . This maximal value is found by setting the left hand side to 1 and solving for N . Because both terms on the right hand side are positive definite, the maximum value of N , with a chameleon field present, will be less than or equal to the value it takes in pure General Relativity . We can see that, both with an without a chameleon, we have: Ã N < Nmax =
2 5Mpl
!3/2
3m2u
.
Following what was done for the braneworld corrections to gravity in [101] we define: R0 = N 1/3 /me , and x = R/R0 . We also define ε0 = 3mu N/4πR03 = 3mu m3e /4π, and Y = (N/Nmax )2/3 . Hydrostatic equilibrium, dE/dR = 0, is therefore equivalent to: 1 3αΦ √ =Y + 2 |n| 1+x
¯ ¯ ¯ βΦ(ε0 ) ¯ mu n+4 ¯ ¯ n+1 . ¯ Mpl ¯ me x
This is the mass-radius relationship for a white-dwarf star. The Chandrasekar mass-limit follows from setting x = 0. We can see that it is unchanged by the presence of a chameleon field. The chameleon field will, however, alter the mass-radius relationship. We shall assume, and later require, that the second term on the right hand side, i.e. the chameleon correction, is small compared to the first. Solving perturbatively under this assumption we find: r x≈
¡ ¢ n+4 ¯ ¯ 1 3αΦ ¯¯ βΦ(ε0 ) ¯¯ mu Y12 − 1 2(n+1) √ −1− . Y2 |n| ¯ Mpl ¯ me Y 2 1 − Y 2
The maximum value of x for relativistic white-dwarfs (pF ≥ me ) is x = 1. In order for our assumption that the effect of the chameleon field could be treated perturbatively to be valid, we need:
¯ ¯ n+4 3αΦ ¯¯ βΦ(ε0 ) ¯¯ mu p 1 + x2 x 2(n+1) ¿ 1. ¯ ¯ |n| Mpl me
The right hand side is clearly increasing with x, and so we evaluate it at x = 1. For corrections to the mass-radius relationship to be smaller than 10% we must therefore require: √ ¯ ¯ 3 2αΦ ¯¯ βΦ(ε0 ) ¯¯ mu < 0.1. |n| ¯ Mpl ¯ me
190
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
Evaluating this expression, for a white dwarf, we find ¡ ¢ 5.065 × 10−41 αΦ
µ
3.19β × 108 |n|
n µ ¶ n+1
M 1 mm−1
¶ n+4 n+1
1
λ n+1 < 0.1,
The above expression provides us with an upper-bound on β. For natural values of M and λ, this upper-bound is strongest for n = −4 theories. For the corrections to the mass-radius relationship to be smaller than 10% when n = −4 we need β < 6.70λ−1/4 × 1018 . Alternatively, for the corrections to be smaller than 1% we require β < 1.19λ−1/4 × 1018 . As the data improves it might, in future, to be able to limit any such corrections to being smaller than 0.1%. In this case we would need β < 2.18λ−1/4 ×1017 . In evaluating these limits, we have used the accurate value for αΦ (n = −4) found in appendix E: αΦ (n = −4) = 0.58. Despite the fact that these represent some of the best upper bounds on β for chameleon fields, it is clear that β À 1 is still allowed. Even if the astronomical data improves to the point where we can rule out corrections at the 0.1% level, we would still be unable to rule out Mpl /β & 1 TeV. The calculation for a neutron star proceeds along similar lines. In a neutron star, the neutrons provide both the degeneracy pressure and the gravitational potential. We must therefore replace both mu and me by mn . For the correction for the mass-radius relationship to be less than 10% we must require: ¡ ¢ 8.15 × 10−51 αΦ
µ
1.98β × 1018 |n|
n µ ¶ n+1
M 1 mm−1
¶ n+4 n+1
1
λ n+1 < 0.1.
n+4
The left hand side of the above expression is a factor of (me /mn ) n+1 smaller than the equivalent expression for a white dwarf. It follows that, for all n = −4, a weaker bound on the parameters results. When n = −4 we find the same bound. In FIG. 5.4 we have plotted the white-dwarf bounds on β, M and λ. We have, conservatively, assumed that corrections to the mass radius relationship are smaller than 5%. The plots are similar to those done for the E¨ot-Wash and WEP bounds. The whole of the shaded region is allowed and the plots for other theories with n < −4 are similar to those with n = −8. Similarly, the plots for other n > 0 theories looks much the same as the n = 4 plot does. These white dwarf bounds are included in the plots of section 5.5, where all the bounds on these chameleon theories are collated.
191
5.3. IMPLICATIONS FOR COMPACT BODIES
5.3.2
General Relativistic Stability
We have already derived conditions for the effect of the chameleon to be small compared to that of the Newtonian potential and thus produce only a negligible change to the mass-radius relation. It also is necessary to consider how the inclusion of a chameleon affects the stability of white dwarfs and neutron stars. This requires the inclusion of general relativistic effects. A compact body in hydrostatic equilibrium (dE/dR = 0) will be stable against small perturbations whenever d2 E/dR2 > 0, i.e. we are at a minima of the energy. For a proof of this result see [100, 101]. The onset of instability occurs when d2 E/dR2 = 0. For Newtonian gravity this occurs when the star becomes relativistic i.e. Γ = 4/3. Ignoring general relativistic effects but including chameleon effects we find that d2 E/dR2 is given by: à ! ¯ ¯ ¶ µ d2 E N 1/3 m2e R2 2 dE N 4/3 n + 4 3αΦ ¯¯ βΦ(ε) ¯¯ −1/3 + =− + mu N R . dR2 R dR R2 n + 1 |n| ¯ Mpl ¯ (N 2/3 + m2e R2 )3/2 When dE/dR = 0, the contribution from the chameleon field to the right hand side of this equation is positive. It follows that the effect of the chameleon field is to increase the stability of white dwarfs and neutron stars i.e. it makes d2 E/dR2 more positive. It is well known, [100], that General Relativity alters the stability of white dwarfs and neutrons stars. When GR effects are included gravity is generally stronger. As a result, it tends to have a destabilising effect on configurations that are stable when studied in Newtonian physics. In the absence of chameleon corrections, but including GR effects (assuming GMstar /R ¿ 1) the criterion for stability is roughly: Γ − 4/3 > κ
Mstar 2 , RMpl
where κ ∼ O(1), [100]. Whilst GR destabilises white-dwarf stars, we have just noted that the chameleon force acts to stabilise them. In this section we will see how the leading order general relativistic effects balance out against the chameleon force, and study their cumulative effect on the stability of compact objects. The assumption that general relativistic effects are small means that these results will be more accurate for white dwarfs than they will for neutron stars. A full derivation of the potential energies associated with the leading order general relativistic effects can be found in [100]. For the sake of brevity we shall not repeat that analysis here but merely quote the results.
192
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
We shall assume that the electrons in the white dwarfs are approximately relativistic and so satisfy P = Kε4/3 . We shall also assume spherical symmetry. Just as we do in appendix E, we evaluate the different contributions to the energy of the white-dwarf under the assumption that the fluid satisfies the Newtonian equation of hydrostatic equilibrium at leading order. We then solve the resulting Lane-Emden equation for P numerically. This procedure is valid if one only wishes, as we do, to calculate the GR and chameleon field corrections to leading order. The internal energy of a n = 3 polytrope is: U = k1 Kεc1/3 mu N, and the gravitational potential of the star is given by: −2 5/3 W = −k2 ε1/3 , c Mpl (mu N )
where k1 and k2 are found in ref. [100] to be (for Γ = 4/3): k1 = 1.75579,
k2 = 0.639001.
In addition to the correction coming from General Relativity (which we will consider shortly) we also need to account for the fact that the electrons are not entirely relativistic. At leading order, this gives the following correction to the internal energy: ∆U = k3 m2e /(µe mu )2/3 mu N ε−1/3 , c where k3 = 0.519723, [100]. Finally, the leading order general relativistic contribution to the energy is found to be of the form: −4 ∆WGR = −k4 Mpl (mu N )7/3 ε2/3 c
where k4 = 0.918294, [100]. Including the effect of the chameleon, the energy of the white dwarf is given by: E = U + W + ∆U + ∆WGR + WΦ . 2/3
−2 We define B = k1 K, C = k2 Mpl , D = k3 m2e /mu 1/(n+1)
−1 H = αΦ /|n||βΦ(εc )Mpl |εc
−4 and F = k4 Mpl . We also define
. As defined, H is actually independent of εc . With these
definitions, the energy is extremised when: − n+2 dE 5/3 7/3 = 13 (BMstar − CMstar )ε−2/3 − 13 DMstar ε−4/3 − 32 F Mstar ε−1/3 − HMstar εc n+1 = 0. c c c dεc
193
5.3. IMPLICATIONS FOR COMPACT BODIES
White Dwarf bounds on n = −8 theories
White Dwarf bounds on n = −4 theories
45 M = Mpl/β
log
10
0
35 −10
30 log10 k, λ = k/4!
M/M
nat
,M
nat
= (0.1 mm)−1
40
25 20 15 10
−20 −30 −40
5 0
−50
−5 −60 0
5
10 log10 β
15
20
0
5
10 log10 β
15
20
White Dwarf bounds on n = 4 theories 30 M = Mpl/β
20
10 5
log
10
nat
,M
15
M/M
nat
= (0.1 mm)
−1
25
0 −5 0
5
10 log10 β
15
20
Figure 5.4: Constraints on chameleon theories coming from white dwarfs and neutron stars. The shaded area shows the regions of parameter space that are allowed assuming that any alterations to the white-dwarf mass-radius relationship are at the 5% level or smaller and that the chameleon only induces changes that are smaller than 10% in the maximum white-dwarf density, εcrit . Neutron star bounds are not competitive with the white-dwarf constraints. The solid black lines indicate the cases where M and λ take ‘natural values’. The dotted-black line indicates when M = MΦ := Mpl /β i.e. when the mass scale of the potential is the same as that of the matter coupling. Other n < −4 theories are similar to the n = −8 case, whilst the n = 4 plot is typical of what is allowed for n > 0 theories. The amount of allowed parameter space increases with |n|.
194
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
At leading order, we drop the terms proportional to D, F and H and recover the standard Chandrasekar limit for the mass of a white dwarf: µ ¶3/2 ³ µ ´−2 A e Mstar = = 1.457 M¯ , B 2 where M¯ is the mass of the Sun. Instability begins to occur when d2 E/d2 εc = 0 which, given that dE/dεc = 0, is equivalent to: 2 −4/3 2 n + 4 − n+2 4/3 Dεc − F Mstar ε−1/3 + Hεc n+1 = 0. c 3 3 n+1
(5.5)
Solving this equation for εc gives a critical density, εcrit , at which instability occurs. We find that, for all n 6= −4, there will be a chameleon induced correction to εcrit . When n 6= −4, we must either have n ≤ −6 or n > 0 and so then (n + 2)/(n + 1) > 4/5 > 1/3. This observation means that eq. (5.5) still has solutions. The effect of the chameleon is to raise the value of εcrit . Even in pure General Relativity it turns out that this critical density is so high that it will only be important for white dwarfs in which the core is 4 He, [100]. In all other cases, except for that where the core is 12 C, εcrit is greater than the neutronisation threshold, and so such high-density white dwarfs will not occur. In the case of carbon white dwarfs, the central density is in fact limited by pyconuclear reactions, [100]. Since the addition of a chameleon field raises εcrit , this change can only be potentially important for Helium white dwarfs. If the effect of the chameleon is small then, at leading order, εcrit = CB 2 /DA2 = 2.65 × 1010 g cm−3 for 4 He stars. The leading order chameleon correction to εcrit is à !¯ ¯ 1/3 δεcrit 3(n + 4) αΦ (2mu )2/3 εcrit ¯¯ βΦ(εcrit ) ¯¯ = ¯ Mpl ¯ εcrit 2(n + 1) |n|m2e k3 µ ¶ n µ ¶ n+4 n+1 1.22β × 1012 n+1 M −44 (n + 4) = 1.33 × 10 αΦ . (n + 2) |n| 1 mm−1
(5.6)
If we wish to require that the presence of a chameleon do little to alter the stability properties of white dwarfs in general relativity, we will need δεcrit /εcrit ¿ 1. This gives us another upper bound on β. In general, however, it is not competitive with the white-dwarf mass-radius relation bound on β. The requirement δεcrit /εcrit < 0.1 is included in the plots of FIG. 5.4.
5.3.3
Discussion
It should be noted that, in this section, the bounds that have been derived on β have been found under the assumption that the chameleon field couples to relativistic matter in the same
195
5.4. COSMOLOGICAL AND OTHER ASTROPHYSICAL BOUNDS
way as it does to normal matter i.e. it just couples to the rest mass energy density of matter. As we noted in the introduction, however, it is usually the case that the chameleon in fact couples to some linear combination of the energy density and the pressure e.g. ε + ωP . In the simplest models ω = −3. The total energy density, εtot , of the star with equation of state P = KεΓ0 is given by: εtot = ε0 +
P = ε0 + pP Γ−1
where ε0 is the rest mass energy density and Γ = 1 + 1/p. In calculations presented above, we have implicitly assumed that εtot + ωP = ε0 + (p + ω)P ≈ ε0 . The bounds that we have derived come from the sector where the matter in the star is relativistic i.e. p = 3. If the chameleon couples to matter through the trace of the energy momentum tensor i.e. ω = −3 then we do, in fact, have εtot = ε0 , just as we have assumed. In the non-relativistic case, P ¿ ε0 , and so εtot + ωP ≈ ε0 is always true. Even in the 1/3
relativistic case, since P/ε = Kε0
1/3
¿ 1 for white dwarfs, and P/ε = Kε0
∼ O(1) for
neutron stars, we always have P . O(ε), and so different values of ω will only alter our bounds by at most an O(1) quantity. There is one caveat: if ω is chosen so that εtot + ωP can become negative, then the n > 0 chameleon field theories will cease to display chameleon behaviour. This would immediately rule them out for all β & 1.
5.4 5.4.1
Cosmological and Other Astrophysical Bounds Nucleosynthesis and the Cosmic Microwave Background
The compact object bounds present above constrain a chameleon field behaves in very high density backgrounds whereas cosmological bounds on chameleon theories constrain how the behaviour of the chameleon field in low-density backgrounds. We have assumed that the chameleon couples to the energy density and pressure of matter in the combination: ε + ωP. In the radiation era P ≈ 3ε. Provided then that ε1 + ω/3) > 0, i.e. ω > −3, and β is large enough, the chameleon will simply stay at the minima of its effective potential, which is itself slowly evolving over time. For this to be the case it is required that: ¯ ¯ ¯ βε(1 + ω/3) ¯ ¯, ¯ |¤Φc | ¿ ¯ ¯ Mpl
196
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
where Φc is the value of Φ at its effective minima: µ Φc = M
M 3 Mpl λn βε(1 + ω/3)
1 ¶ n+1
,
and so: ¨ c − 3H Φ˙ c = − 4(n + 5) H 2 Φc ¤Φc = −Φ (n + 1)2 µ ¶µ ¶µ ¶ 4(n + 5) 8π βΦc βε(1 + ω/3) = − , (n + 1)2 3β 2 (1 + ω/3) Mpl Mpl
(5.7)
2 and ε˙ = −4Hε as appropriate for the radiation era. where we have used H 2 = 8πε/3Mpl
We shall show below that we must require that |βΦc /Mpl | < 0.1 since the epoch of nucleosynthesis. When β & 1, it follows from eq. (5.8) that this requirement is enough to ensure that |¤Φc | ¿ βε(1 + ω/3)/Mpl provided that ω > −3. The simplest, and perhaps the most natural way, for the chameleon to interact with matter in a relativistically invariant fashion, however, is for it to couple to the trace of the energy momentum tensor i.e. ω = −3. When ω = −3 the above analysis does not apply. The strongest bounds on the parameters of chameleon theories arise in ω = −3 case. When ω = −3 we must evaluate ε − 3P . Although ε ≈ 3P in the radiation era, ε − 3P is not identically zero. Following [70], we find, for each particle species i: εi − 3Pi = where g∗ =
P
2 2 45 H Mpl gi τ (mi /T ), π 4 8π g∗ (T )
boson (T /T )4 + (7/8) i bosons gi
P
f ermion (Ti /T )4 f ermions gi
is the standard expression
for the total number of relativistic degrees of freedom; gi and Ti are respectively the degrees of freedom and temperature of the ith particle species. The function τ is defined by: Z ∞ √ 2 u − x2 2 τ (x) = x du u e ±1 x with the + sign for fermions and the − for bosons. This function goes like x2 when x ¿ 1 and e−x when x À 1, but it is O(1) when x ∼ O(1). The case β ∼ O(1) and n > 0 was analysed very thoroughly in [70]. We will therefore restrict ourselves to looking at the n ≤ −4 cases. We will also consider, for all n, what new features emerge when we take β À 1. In both cases we will see that theories with n ≤ −4 and/or β À 1 are much less susceptible to different initial conditions than those with n > 0 and β ∼ O(1).
5.4. COSMOLOGICAL AND OTHER ASTROPHYSICAL BOUNDS
197
We consider the cases n ≤ −4 and n > 0 separately below. Before we do so, we note some similarities between the two cases. Whatever the sign of n, the effective potential will have a minima at: µ Φmin (T ) = M
Mpl M 3 nλ β εˆ(T )
1 ¶ n+1
,
where we define: εˆ(T ) =
X
(εi − 3Pi ) .
i
From the form of εi − 3Pi , it is clear that εˆ will be dominated by the heaviest particle species that satisfies mi . T . When mi ¿ T , τ ∝ (mi /T )2 and so the largest value of mi dominates, whereas the contribution from species with mi À T are exponentially suppressed. The function τ (mi /T ) is peaked near mi /T = 1. If the chameleon is not at Φmin , and (Φmin /Φ)n+1 ¿ 1, then this peak will result in the value of Φ being ‘kicked’ towards Φmin (T ). We label the distance that Φ moves due to this ‘kick’ by (∆Φ)i where 7 βgi Mpl 8 (∆Φ)i ≈ − . 8πg∗ (mi ) 1 The 7/8 is for fermions and the 1 for bosons. This formula is valid so long as |Φmin (T ) − Φ| > |∆Φi | i.e. so long as the kick is not large enough to move Φ to its minimum. The largest jump of this sort will occur for the smallest value of gi /g∗ . It will therefore occur when electrons decouple from equilibrium at T ≈ 0.5 MeV. If, however, |Φ − Φmin (T )| is smaller in magnitude than this above quantity, then (∆Φ)i = Φmin − Φ. Whether or not Φ will stay at Φmin (T ), as it evolves with time, will depend on the mass of the chameleon at Φmin (T ). If m2Φ À H 2 then it will stay at the minimum. Otherwise it will tend to slowly evolve towards values of Φ for which (Φmin /Φ)n+1 < 1. If (Φmin /Φ)n+1 > 1 then Φ will very quickly (in under one Hubble time) roll down the potential. It will either overshoot Φmin , or if the mass of the chameleon at Φmin is large enough, stick at Φmin . We can therefore assume that our initial conditions are such that, in the far past, Φ is either at Φmin or (Φmin /Φ)n+1 < 1.
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CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
Case n ≤ −4 We note that when n ≤ −4, before any jump, we have Φmin /Φ > 1 and so |Φmin − Φ| ≤ |Φmin |. It follows that: Φmin (T ≈ mi ) 45(n + 1) H 2 (T ≈ mi ) ≈ . (∆Φ)i π4 m2Φ (Φmin ) This seems to suggest that, if (∆Φ)i is large enough to move Φ to Φmin (T ), then we will necessarily have m2Φ (Φmin ) > H 2 , and so Φ will stay at Φmin (T ) in the subsequent evolution. However, this is not quite the case. As T drops below mi , the ith species decouples and its energy density decreases exponentially. This causes εˆ, and consequently mΦ (Φmin (T )), to decrease quickly. Roughly, εˆ shortly after decoupling will be a factor of (mj /mi )2 smaller than it was before, where the jth species is the most massive species of particle obeying mj < mi . If Φ reaches Φmin (T ≈ mi ) with the ith kick, then as T decreases the chameleon will roll quickly down to the potential towards Φmin (T < mi ); |Φmin (T < mi )/Φmin (T ≈ mi )| < 1. For Φ to stick at Φmin , for mj < T < mi , and not overshoot it, we need: H 2 (T ) ≈ 2 mΦ (Φmin (T ))
µ
T mi
¶
2n n+1
µ
mi mj
¶ 2(n+2) n+1
H 2 (mi ) 2 mΦ (Φmin (mi ))
¿ 1.
Since T < mi in this region, and n/(n + 1) > 0, it is sufficient to require: µ
mi mj
¶ 2(n+2) n+1
H 2 (mi ) 2 mΦ (Φmin (mi ))
¿ 1.
We know that mΦ ∝ β (n+2)/(n+1) and so the above condition is more likely to be satisfied for larger values of β than for smaller ones. We note that, even when Φ 6= Φmin , we cannot have Φ/Φmin À 1. If this were the case initially, when Φ = Φi say, then the gradient in the Φ potential would be very steep and in one Hubble time Φ would move a distance ∆Φ where: ¶(n+2) ³ m ´2 µ M ¶ µ Φ ∆Φ pl i min ∼ −β . Φi T Φmin Φi It is clear from this expression that for β & O(1), Φ/Φmin will decrease very quickly and pretty soon Φ/Φmin . 1. When β & O(1) it is therefore valid to assume that, for almost of all of the radiation era evolution, Φ(T )/Φmin (T ) . 1. The larger β becomes, the greater the extent to which this assumption holds true.
5.4. COSMOLOGICAL AND OTHER ASTROPHYSICAL BOUNDS
199
The main purpose of the above discussion is to illustrate that, for theories with n ≤ −4, the late time behaviour of Φ will be virtually, independent of one’s choice of initial conditions. The same is not true, or at least not true to the same extent, of theories in which n > 0. The corollary of this result is that |βΦ/Mpl | could be very large at the beginning of the radiation epoch and still be less than 0.1 from the epoch of nucleosynthesis onwards. This would correspond to the masses of particles during the early radiation era being very different from the values they have taken since the epoch of nucleosynthesis until the present day. The larger β is, the easier it is to support large changes in the particle masses. This is one reason why β À 1 is theoretically very interesting. In chameleon theories with β À 1, the constants of nature that describe the physics of the very early universe (i.e. pre-nucleosynthesis) could take very different values from the ones that they do today. This could have some interesting implications for baryogenesis at the electroweak scale. The best radiation-era bounds on β, M and λ come from big-bang nucleosynthesis (BBN) and the isotropy of the cosmic microwave background (CMB). As noted in [70], a variation in the value of Φ between now and the epochs of BBN and recombination would result in a variation of the particle masses (relative to the Planck mass) of about: ¯ ¯ ¯ ∆m ¯ β|∆Φ| ¯ ¯ ¯ m ¯ ∼ Mpl . BBN constrains the particle masses at nucleosynthesis to be within about 10% of their current values, [70]. This requires: |ΦBBN | . 0.1β −1 Mpl . We have argued above that ΦBBN /Φmin (TBBN ) . 1 and so the above condition will be satisfied whenever: |Φmin (TBBN )| . 0.1β −1 Mpl . Nucleosynthesis occurs at temperatures between 0.1 MeV and 1.3 MeV. Since Φmin (T ) decreases with temperature, we conservatively evaluate the above condition with TBBN = 2 MeV. At such temperatures the largest contribution to εˆ will come from the electrons (with me (today) = 0.511 MeV) and: εˆ ≈
ge (MeV)4 ge T 2 m2e ≈ , 24 24
200
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
where ge = 4 (2 from the electrons and 2 from the positrons). The BBN constraint on β, λ and M for n ≤ −4 is therefore: µ λ|n|
1.8β × 106 λ|n|
n µ ¶ n+1
M 1 mm−1
¶ n+4 n+2
. 1.1 × 1037 .
This is, however, a weaker bound on {β, M, λ} than the white dwarf mass-radius relation constraint discussed in section 5.3 above. Another important restriction on these chameleon theories comes out of considering the isotropy of the CMB [92]. As is mentioned in [70] a difference between the value of Φ today and the value it had during the epoch of recombination would mean that the electron mass at that epoch differed from its present value ∆me /me ≈ β∆Φ/Mpl . Such a change in me would, in turn, alter the redshift at which recombination occurred, zrec : ¯ ¯ ¯ ¯ ¯ ∆zrec ¯ ¯ β∆Φ ¯ ¯ ¯≈¯ ¯ ¯ zrec ¯ ¯ Mpl ¯ . WMAP bounds zrec to be within 10% of the value that has been calculated using the present day value of me , [70]. We define εˆrec and εˆBBN to be, respectively, the value that εˆ takes at the recombination and BBN epochs. Now εˆrec À εtoday , where εtoday is the cosmological energy density, and Φ ∝ ε−1/(n+1) , it follows that |∆Φ| ≈ |Φrec | for n ≤ −4. Φrec is the value of Φ had during the epoch of recombination. However, since εˆrec < εˆBBN , we also have Φrec /ΦBBN < 1, and so this CMB bound is always weaker than the one coming from BBN. For this reason we do not evaluate the CMB constraint here. Case n > 0 When we considered theories with n ≤ −4, the analysis was made easier by the fact that V (Φ) had a minimum. This allowed us to bound the magnitude of Φ to be approximately less than that of Φmin (T ). However, when n > 0, the potential is of runaway form and we cannot bound the magnitude of Φ in such a way. Many of the issues involved with n > 0 potentials were discussed very thoroughly in ref. [70]. In that work it was assumed that β ∼ 1 and so, generically mΦ (Φmin ) ¿ H in the radiation era, due the fact that εˆ ¿ ε. However, if β ¿ 1, it is possible for mΦ to be large compared to H. As with the n ≤ −4 case, it is not necessary to require that Φ is at its effective minima during the whole of the radiation era. All that is really needed is for Φ be sufficiently close
5.4. COSMOLOGICAL AND OTHER ASTROPHYSICAL BOUNDS
201
to the minima at the epochs of BBN and recombination that we are able to satisfy their constraints. It was shown in [70] that the total sum of all of the kicks that occur before BBN will move the chameleon a distance of approximately: βMpl . BBN requires that: |∆ΦBBN | = Φtoday − ΦBBN . 0.1β −1 Mpl . Provided that, at the beginning of the radiation era, Φ ¿ βMpl /8π, then Φ at BBN will easily satisfy the above bound provided that βΦtoday /Mpl < 0.1. The first of these requirements is a statement about initial conditions. It is clear that the larger β becomes, the less restrictive this condition is. Indeed for β À 1, it is quite possible to have had βΦ/Mpl ∼ 0.1O(β 2 ) À 1 at the beginning of the radiation era and still satisfy this bound. The larger the matter coupling is, the less important the initial conditions become, and the greater the scope for large changes in the values of the particle masses and other constants to have occurred between the pre-BBN universe and today. Recombination enforces a similar bound to BBN: β(Φtoday − Φrec ) ∆zrec ≈ . 0.1. zrec Mpl If the requirements on the initial conditions are satisfied, we will have Φtoday À Φrec , ΦBBN and so, in both cases, ∆Φ ≈ Φtoday . We must therefore require that βΦtoday /Mpl . 0.1. −1/(n+1)
Φtoday ∝ εc
where εc is that part of the cosmological energy density of matter that
the chameleon couples to. For the most part, we have, in this chapter, assumed that the chameleon couples to all forms of matter with the same strength. However, up to now, we have only been concerned with baryonic matter. Even if the chameleon to baryon coupling is virtually universal, there is not necessarily any reason to expect the chameleon to couple to dark matter with the same strength. It is possible that the chameleon does not interact with dark matter at all. Since it clear that Φtoday is a very important quantity when it comes to bounding n > 0 chameleon theories, it is crucial to know what fraction of the cosmological energy density the chameleon actually couples to. The smaller εc is, the larger Φtoday will be. The larger the value of Φtoday , the more restrictive the resultant bound on {β, M, λ}. This implies that cosmological bounds on a chameleon theory that couples only to baryonic matter will be stronger than those on a chameleon that also couples to dark matter. Not
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CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
knowing how the chameleon couples to dark matter, we chose to be cautious, and err on the side of specifying a bound that is perhaps slightly too tight, rather than too loose. We therefore work on the assumption that the chameleon only couples to baryonic matter, and so εc ≈ 0.42 × 10−30 g cm−3 , [92]. Under this assumption, we find: µ −3
8.34 × 10
|n|
1.93 × 10−29 β n
n µ ¶ n+1
M 1 mm−1
¶ n+4
n+1
< 1.
If the chameleon to dark matter coupling is similar in magnitude to the baryon coupling then εc ≈ 2.54 × 10−30 g cm−3 and we have the less restrictive bound: µ −3
1.39 × 10
|n|
1.17 × 10−28 β n
n µ ¶ n+1
M 1 mm−1
¶ n+4
n+1
< 1.
The bound that we have just found has come about from the requirement that the particle masses at BBN and recombination are within 10% of the values they take in regions of cosmological density i.e. εtoday ∼ 10−30 g cm−3 . However, all cosmological determinations of the particle masses, and indeed of the other constants of nature, have come from analysing measurements made in regions with densities much greater than the cosmological one. For instance, recent cosmological determinations of mp /me have employed the absorption and emission spectra of dust clouds around QSOs [20, 16]. These dust clouds have typical densities of the order of ε ∼ 10−25 − 10−24 g cm−3 . If we take εc to be 10−25 g cm−3 , then the BBN and CMB bounds would only require: µ −8
3.52 × 10
|n|
4.59 × 10−24 β n
n µ ¶ n+1
M 1 mm−1
¶ n+4
n+1
< 1.
Summary We have plotted the BBN and CMB constraints on {β, M λ} in FIG. 5.5. We have plotted what occurs in the most restrictive case i.e. when the chameleon couples only to baryons. The whole of the shaded region is currently allowed. Another class of potentially important cosmological bounds can be derived by employing astronomical bounds on the allowed variation of the fundamental constants of nature during the matter era. We discuss this further below.
203
5.4. COSMOLOGICAL AND OTHER ASTROPHYSICAL BOUNDS
BBN and CMB bounds on n = −8 theories
BBN and CMB bounds on n = −4 theories
60 0
40
−10 log10 k, λ = k/4!
log10 M/Mnat, Mnat = (0.1 mm)
−1
M = Mpl/β 50
30 20
−20 −30
10
−40
0
−50 −60 0
5
10 log10 β
15
20
0
5
10 log10 β
15
20
BBN and CMB bounds on n = 4 theories 30 M = Mpl/β
20
10 5
log
10
nat
,M
15
M/M
nat
= (0.1 mm)
−1
25
0 −5 0
5
10 log10 β
15
20
Figure 5.5: Constraints on chameleon theories coming from the particle masses at big-bang nucleosynthesis and the constraints the redshift of recombination. The shaded area shows the regions of parameter space that are allowed by the current data. The solid black lines indicate the cases where M and λ take ‘natural values’. The dotted-black line indicates when M = MΦ := Mpl /β i.e. when the mass scale of the potential is the same as that of the matter coupling. Other n < −4 theories are similar to the n = −8 case, whilst the n = 4 plot is typical of what is allowed for n > 0 theories. The amount of allowed parameter space increases with |n|. In these plots we have assumed that the chameleon couples only to baryons. Slightly weaker constraints result if the chameleon additionally couples to dark matter.
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CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
5.4.2
Variation of fundamental constants
Any variation in a chameleon field will lead to a variation in the masses of the particle species to which the chameleon couples. This variation is relative to the fixed Planck mass, Mpl = G−1/2 . If the chameleon couples to all matter particles in the same way, then all the fundamental particle masses will vary in the same fashion and so their ratios will remain constant. It is also feasible to construct theories where a variation in the chameleon leads to a variation in some other fundamental ‘constants’ of nature. For instance one might propose a theory where the fine-structure constant, αem , is given by αem = f (βΦ/Mpl ) for some function f . If this is the case then, in addition to bounds on any allowed variation in the particle masses, we would also have to apply the whole range of very stringent bounds on the possible variation of αem mentioned above. For the purposes of this section, however, we assume that αem is a true constant. It should be noted that even if αem does vary with Φ at the same level as the particle masses do, the resulting bounds on the parameters of the theory are only slightly more stringent than those already found. The best matter era bounds on the variation of the particle masses come from measurements of the ratio µ = mp /me , [20, 16, 29]. We shall assume that the chameleon couples to protons with strength βp and to electrons with strength βe . The relative change in the proton and electron masses is then given by: ∆mp mp ∆me me
≈ ≈
βp ∆Φ , Mpl βe ∆Φ . Mpl
Without any a priori knowledge about the magnitude, or sign, of βe − βp it is difficult to derive any bounds on chameleon theories simply by considering ∆µ/µ, where µ = mp /me . The simplest assumption one could make about the matter coupling, β, is that it is universal i.e. βp = βe = β. If this is the case then ∆µ = 0 identically. An alternative, but still very reasonable, assumption about the chameleon coupling, which would produce ∆µ 6= 0, is that the chameleon couples to all fundamental particles with the same universal coupling, βU , but that the QCD scale, ΛQCD , is independent of Φ. When the quark masses vanish the proton mass is proportional to ΛQCD . The masses of the three lightest quarks, mu , md and ms , are considerably smaller than ΛQCD and as a result contribute only a small correction to
205
5.4. COSMOLOGICAL AND OTHER ASTROPHYSICAL BOUNDS
the proton mass (at approximately the 10% level). If ΛQCD is Φ-independent, we expect the proton mass to depend only weakly on Φ, through the quark masses, and so βp ∼ O(βU /10) whereas βe = βU . Since the mass of the electron is so much smaller than the proton mass, the coupling of the chameleon to baryons is approximately given by βp , and so it is βp that is constrained by the experiments mentioned in sections 5.2 and 5.3. However, the BBN and CMB requirements constrain βe = βU . A change in Φ of ∆Φ would therefore induce a change in µ of: ∆µ β∆Φ ≈ −9 . µ Mpl
(5.8)
As we reported above, the Reinhold et al., [20], result is consistent with a difference between the laboratory value of µ and the one measured in such a dust cloud at the 3.5σ level: ∆µ/µ = 2.0 ± 0.6 × 10−5 where ∆µ = µdust − µlab . It should be noted that in the context of the chameleon theories considered here ∆Φ = Φdust − Φlab is always positive and so ∆µ/µ < 0 if βp = βe /10 as we have assumed. Under this assumption, it is not possible to reproduce the result of Reinhold et al.. If we had alternatively assumed that the fundamental particle masses are true constants but that ΛQCD is Φ-dependent, then we would be able to have ∆µ/µ > 0. We interpret the Reinhold result conservatively i.e. as limiting any variation in µ to be at or below the 2 × 10−5 level. We shall also assume that the laboratory experiments that measure µ are performed close enough to the Earth’s surface that the background value of Φ for these experiments is approximately the value the chameleon takes inside the Earth. This is last assumption is also a conservative one i.e. it will result in a tighter bound on the chameleon theory parameters. Taking the density of Earth to be εEarth ≈ 5.5 g cm−3 and the density of the dust clouds from which the absorption spectra come to be εdust ∼ 10−25 g cm−3 we find that we must require: µ −29
2.88 × 10
|n|
253β |n|
¶
n n+1
µ
M 1 mm−1
¶ n+4 n+1
1
λ n+1 < 1,
β < 1.42λ−1/4 × 1019 , n = −4, µ ¶ n µ ¶ n+4 n+1 M 4.59 × 10−24 β n+1 −3 1.58 × 10 |n| < 1, n 1 mm−1
n ≤ −4
n > 0.
For fixed M and λ this places an upper-bound on β. When n > 0, the bounds coming from varying-µ are competitive with the other cosmological constraints and they provide a weak upper bound on β. When n ≤ −4, however, the white dwarf bounds of section 5.3 still provide
206
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
the best upper bound on β. It should be noted that the bounds on {β, M, λ} deduced from measurements of ∆µ/µ are highly model-dependent. For this reason we do not plot them here.
5.5
Combined bounds on chameleon theories
In this chapter, we have combined the results of chapter 4 with a variety of experimental and astrophysical bounds to constrain the parameter space of our chameleon theories. All of the chameleon theories considered in this work have a two-dimensional parameter space, spanned either by M and β (n 6= −4), or by λ and β (n = −4). We combine the constraints found in sections 5.2-5.4 to bound the values of β and M (or λ) for different n. We plotted the constraints for n = −8, −6, −4, 1 4 and n = 6 in FIG. 5.6. In these figures we have included all the bounds coming from the E¨ot-Wash experiment, [87, 88], as well as those coming from Casimir force searches, [89, 90]. We also include the bounds (labeled Irvine) coming from another search for Yukawa forces, [105]. We also show how current WEP violation experiments constrain these theories - i.e. experiments that have actually been done as opposed to the putative WEP violation experiment described in section 5.2.3. The white-dwarf and BBN constraints are also included in the plots, however, for these theories, they are always weaker than those laboratory tests (when β . 1020 ). The plots for other n < −4 theories are very similar to the n = −10 and n = −8 plots, whilst the n = 4 and n = 6 cases are representative of n > 0 theories. In general, the larger |n| is, the larger the region of allowed parameter space. This is because, in a fixed density background, the chameleon mass, mc , scales as |n + 1|1/2 /|n|1/2(n+1) , and so mc increases with |n|. The larger mc is, in a given background, the stronger the chameleon mechanism, and a stronger chameleon mechanism tends to lead to looser constraints. We have indicated on each plot how the variety of different bounds, considered above, combine to constrain the theory. In each plot, the whole of the shaded area indicates the allowed values of M and β (or λ and β). Three satellite experiments (SEE [96], STEP [97] and GG [98]) are currently in the proposal stage, whilst a fourth one (MICROSCOPE [99]) will be launched in 2007. These experiments will be able to detect WEP violations down to η = 10−18 . The more lightly shaded region on the plots indicates those regions of parameter space that could be detected by these experiments - we have assumed that their WEP violation experiments to be along the lines of the putative experiment described in section 5.2.3. In n > 0 theories, the increased precision, promised by these satellite tests, is not the only
207
5.5. COMBINED BOUNDS ON CHAMELEON THEORIES
advantage that they offer over their lab-based counterparts. In space, the background density is much lower than in the laboratory. As a result, the thin-shell condition, eq,. (4.19b), is generally more restrictive for bodies in the vacuum of space than it is for the same bodies here on Earth. It is therefore possible for test-bodies, that had thin-shells in the laboratory, to lose them when they are taken into space [77, 69, 70]. If such a thing occurs for the test-masses used in the aforementioned satellite experiments, then it is possible that they might see WEP violations in space at a level that had previously been thought ruled out by Earth-based tests i.e. η > 10−13 . This interesting feature of n > 0 chameleon theories was first noted in [77]. It is important to stress that this effect will not occur if β is so large that the satellites themselves develop thin-shells [77, 69]. In chameleon theories where the potential has a minimum, i.e. n ≤ −4, the thin-shell condition, eqs. (4.19a & c), is only weakly dependent on the background density of matter. As a result, n ≤ −4 theories will generally be oblivious to the background in which the experimental tests of it are conducted. It is for this reason that future space-based tests will be better able to constrain n > 0 theories than they will n ≤ −4 ones. In all of the plots, we show β running from 10−10 to 1020 . β < 10−10 will remain invisible to even the best of the currently proposed space-based tests, and β > 1020 corresponds to Mpl /β < 500 MeV. The region in which Mpl /β . 200 GeV is, in fact, probably already ruled out. If β were so large that Mpl /β < 200 GeV, then we would probably have already seen some trace of the chameleon in particle colliders. This said, without a quantum theory for the chameleon it is hard to say how chameleon theories behave at high energies. A result a detailed calculation of the chameleon’s effect on scattering amplitudes is not possible at this stage. A full quantum mechanical treatment of the chameleon is very much beyond the scope of this work, but remains one possible area of future study. n
The chameleon mass (mc ), in a background of fixed density, scales as λ n+1 M
n+4 − 2(n+1)
. As
we mentioned above, the larger mc , is the easier it is to satisfy the thin-shell conditions, eqs. (4.19a-c), and the stronger the chameleon mechanism becomes. Since (n + 4)/(2(n + 1)) ≥ 0 and n/(n+1) > 0, for all theories considered here, the chameleon mechanism becomes stronger as M → 0, or λ → ∞, and all of the constraints are more easily satisfied in these limits. It is for this reason that we truncate our plots for some small M and, when n = −4, for a large value of λ. Values of M that are smaller than those shown, or values of λ that are larger, are
208
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
Combined Bounds on n = −8 theories
Combined Bounds on n = −6 theories
45
45 WEP
40
= (0.1 mm)−1
30 25
Casimir
∼ (ρ )1/4 Λ
0
Irvine
20
,M
nat
nat
25
10
nat
M=M
5
30
15
log
10
log
Eot−Wash
10
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10
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nat
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M = Mpl/β
35
M/M
= (0.1 mm)−1
35
15
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40 M = Mpl/β
Eot−Wash
M=M
5
nat
Casimir
∼ (ρ )1/4 Λ
0 −5
−5
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5 log10 β
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Combined Bounds on n = −4 theories
0
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Eot−Wash
−20
Irvine
= (0.1 mm)−1
−10
λ = 1/4!
,M
−40
M = Mpl/β
10 1/4
M=M
Irvine
nat
∼ (ρ ) Λ
Casimir
5
Eot−Wash
nat
−30
M/M
nat
Casimir
WEP
15
0
log
10
log10 k, λ = k/4!
0
−50
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−60 −5
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5 log10 β
10
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20
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Combined Bounds on n = 4 theories
15
20
= (0.1 mm)−1
M = Mpl/β
10 M=M
Irvine
1/4
∼ (ρ ) Λ
nat
nat
5
,M
Casimir Eot−Wash
WEP
15
M = Mpl/β
10 M=M
Irvine
1/4
nat
∼ (ρ ) Λ
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5
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M/M
nat
= (0.1 mm)−1 nat
WEP
15
nat
,M
10
20
0
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5 log10 β
Combined Bounds on n = 6 theories
20
log
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−5
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20
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−5
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10
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20
Figure 5.6: Combined constraints on chameleon theories. The whole of shaded area shows the regions of parameter space that are allowed by the current data. Future space-based tests could detect the more lightly shaded region. The solid black lines indicate the cases where M and λ take ‘natural values’. For n 6= −4, a natural value for M is required if the chameleon is to be dark energy. The dotted-black line indicates when M = MΦ := Mpl /β i.e. when the mass scale of the potential is the same as that of the matter coupling. Other n < −4 theories are similar to the n = −6 and n = −8 cases, whilst the n = 4 and n = 6 plots are typical of what is allowed for n > 0 theories. The amount of allowed parameter space increases with |n|.
5.5. COMBINED BOUNDS ON CHAMELEON THEORIES
209
still allowed. The upper limit on M (and lower limit on λ) has been chosen so as to show as much of the allowed parameter space as possible. When β is very small, the chameleon mechanism is so weak that, in all cases, Φ behaves like a standard (non-chameleon) scalar field. When this happens, the values of M and λ become unimportant, and the bounds one finds are on β alone. This transition to non-chameleon behaviour can be seen to occur towards the far left of each of the plots. It is clear from FIG. 5.6 that β À 1 is, rather unexpectedly, very much allowed for a large class of chameleon theories. We can also see that, rather disappointingly, future space-based searches for WEP violation, or corrections to 1/r2 behaviour of Newton’s law, will only have a small effect in limiting the magnitude of β. If Mpl /β ∼ 1 TeV is feasible, pending a detailed calculation of chameleon scattering amplitudes, that chameleon particles might be produced at the LHC. The solid black line on each of the plots indicates the ‘natural’ values of M and λ i.e. M ∼ (0.1 mm)−1 and λ = 1/4!. For all n, these values of M and λ are allowed for all 104 . β . 1018 . In particular, Mpl /β ∼ 1015 GeV ≈ MGU T , i.e. the GUT scale, and Mpl /β ∼ 1 TeV are allowed. The dotted black line indicates the cases where M = Mpl /β i.e. when there is only one mass scale associated with the chameleon theory. It is clear, however, that no such theories are allowed if β . 1020 . In all cases we must require M ¿ Mpl /β. As noted in [77, 69], this requirement introduces a hierarchy problem in the chameleon theory itself. This problem is not, however, present in Φ4 (n = −4) theory. The larger β is the stronger the chameleon mechanism becomes. A strong chameleon mechanism results in larger chameleon masses, and larger chameleon masses in turn result in weaker chameleon-mediated forces. A stronger chameleon mechanism also increases the likelihood of the test masses, used in the experiments consider in section 5.2, having thinshells. Large values of β cannot, therefore be detected, at present by the E¨ot-Wash and WEP tests, if λ and M take natural values. It is for this reason that, as can be seen in FIG. 5.6, the E¨ot-Wash experiment and the WEP violation searches place the greatest constraints on the parameter space in a region about β ∼ O(1). Casimir force tests are much better able to detect large β, but they can ultimately only place an upper-bound on M or 1/λ.
210
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
For large β, Casimir force test succeed where the E¨ot-Wash experiment fails because in the former does not use a thin metal sheet to cancel electrostatic effects. If it were not for the presence of the BeCu sheet, the E¨ot-Wash test would be able to detect, or rule out, almost all β À 1 theories with M ∼ (εΛ )1/4 . The fact that we can have β À 1, in scalar theories with a chameleon mechanism, is entirely due to the non-linear nature of these theories. Almost all the quoted bounds on the coupling to matter are for scalar field theories with linear field equations. As we noted in chapter 1.3, in such theories Φ evolves according to:
−¤Φ = m2c Φ +
βε , Mpl
where the field’s mass, m, is constant (i.e. not density dependent). The Φ-force between two bodies with masses M1 and M2 , which are separated by a distance d, takes the Yukawa form
F12 =
β 2 (1 + d/λ)e−d/λ M1 M2 . 2 d2 Mpl
where λ = 1/mc is the range of the force. The best limits of λ and β come from WEP violation searches, [102, 103, 104, 78, 79], and searches for corrections to the 1/r2 behaviour of gravity, [87]. The 95% confidence limits on m and β for such a linear theory, where the field couples to baryon number, were plotted in FIG. 1.1 with the allowed regions shaded and the excluded regions left white. It is clear that β > 1 is ruled out for all but the smallest ranges (currently λ . 10−4 m = 0.1 mm). To make the comparison with the linear case more straightforward, we have replotted the allowed parameter space for chameleon theories with n = −10, −8, −4, 4 and n = 6 in terms of its coupling to matter, β, and the range of the chameleon force cosmologically, λcos , and in the solar system, λsol , in FIGS. 5.7 and 5.8. FIG. 5.7 shows the cosmological range, whilst λsol is shown in FIG. 5.8. The solid black line, in each of these plots, indicates the case where M and λ take their ‘natural’ values. We can clearly see that, in stark contrast to the linear case, chameleon theories are easily able to accommodate both β À 1 and λ À 1 m. This underlines the extent to which non-linear scalar field theories are different from linear ones, and the important rˆole that is played by the chameleon mechanism.
211
5.5. COMBINED BOUNDS ON CHAMELEON THEORIES
Cosmological force ranges for n = −8 theories
15
15
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10 10
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15
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20
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Figure 5.7: Allowed couplings and cosmological force ranges for chameleon theories. The shaded area shows the allowed parameter space with all current bounds. λcos is the range of the chameleon force in the cosmological background with density ε ∼ 10−29 g cm−3 . It is cos related to the cosmological mass of the chameleon, mcos c , by λcos = 1/mc . The solid black
lines indicate the cases where M and λ take ‘natural’ values. Plots for theories with n < −4 or n > 0 are similar to the cases n = −8, −10 and n = 4, 6 respectively.
212
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
Solar system force ranges for n = −8 theories
15
15
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10 10
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β 5
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15
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20
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15
20
Figure 5.8: Allowed couplings and cosmological force ranges for chameleon theories. The shaded area shows the allowed parameter space with all current bounds. λsol is the range of the chameleon force in the solar system, where the average density of matter is ε ∼ 10−24 g cm−3 . sol It is related to the cosmological mass of the chameleon, msol c , by λsol = 1/mc . The solid black
lines indicate the cases where M and λ take ‘natural’ values. Plots for theories with n < −4 or n > 0 are similar to the cases n = −8, −10 and n = 4, 6 respectively.
5.6. CONCLUSIONS AND DISCUSSION
5.6
213
Conclusions and Discussion
In this chapter, we have investigated the experimental constraints on scalar field theories which possess a chameleon mechanism. In such theories self-interactions of the scalar-field are employed to avoid the most restrictive of the current bounds on such fields and its coupling. The chameleon theories were first considered by Khoury and Weltman in ref. [69]. In the models that they proposed, the scalar fields coupled to matter with gravitational strength, in harmony with general expectations from string theory, whilst, at the same time, remaining very light on cosmological scales. Over the last couple of chapters we have gone much further and shown that, contrary to most expectations, the scenario presented in [69] allows scalar fields, which are very light on cosmological scales, to couple to matter much more strongly than gravity does, and yet still satisfy all of the current experimental and observational constraints. Previous investigations into such scenarios, [77, 70, 86, 85], noted that an important feature of chameleon field theories is that they make unambiguous and testable predictions for nearfuture tests of gravity in space. This is timely because, as we reported in Chapter 1, there are currently four satellite experiments either in the proposal stage or due to be launched shortly (SEE [96], STEP [97], GG [98] and MICROSCOPE [99]). A reasonably sized region of the parameter space of the chameleon theories, with runaway potential potentials (n > 0) and β ∼ O(1) or smaller, will be visible to these missions. The prospects for detecting chameleon theories in which the potential has a minimum (n ≤ −4) are not so good and, in all cases, theories with very large matter couplings β À 1 will remain undetectable. The ability of these planned missions to detect large β theories could, however, be exponentially increased if the experiments they carry were to be redesigned slightly in the light of our findings. Past studies claimed that typical test masses in the above satellite experiments do not have a thin shell. Therefore, the extra force is comparable to their gravitational interaction. If this is the case, then chameleon model predicts that MICROSCOPE, STEP and GG could measure violations of the weak equivalence principle in space that are stronger than those currently allowed by laboratory experiments. Furthermore, the SEE project could measure an effective Newton’s constant different, by order of unity, from that measured on Earth. We shown in this chapter that both of these features are very much properties of chameleon theories with runaway (n > 0) potentials. In general, they do not occur if the chameleon potential has a minimum (e.g. n < −4 theories). These features are also very much associated with a
214
CHAPTER 5. EVADING EXPERIMENTAL CONSTRAINTS ...
gravitational strength chameleon coupling i.e. β ∼ O(1), and will not occur if β À 1, or β ¿ 1. The major result presented in this work is that current experiments do not limit the coupling of the chameleon to matter, β, to be order O(1) or smaller. Indeed, if we wish to have a ‘natural’ value of M in a theory where V ∼ 1/Φ we must require β & 104 . If β À 1 the testmasses in the planned satellite experiments will still have thin-shells. As such SEE, STEP, GG and MICROSCOPE, as they are currently proposed, will be unable to detect strongly coupled chameleon field and they cannot therefore place an upper-bound on β. We have shown that the best upper-bounds on the matter coupling, β, come from astrophysical and cosmological considerations. If β is very large, of the order of 1017 or greater, then it might even be possible to detect the effect on the chameleon on scattering amplitudes in particle colliders. This possibility is one avenue that requires further in depth study. We noted in the introduction that β À 1 could be seen as being pleasant in light of the hierarchy problem [117, 118, 119]. If a chameleon with a large β where detected it would imply that new physics emerges at a sub-Planckian energy-scale: MΦ = Mpl /β. We saw in section 5.4 that a large value of the matter coupling is also preferable to an order unity value in that it leads to the late time behaviour of the chameleon being much more weakly dependent on the initial conditions than it would be if β . O(1). The magnitude of β(∆Φ = Φ1 − Φ2 )/Mpl quantifies the relative difference between the particle masses in a region where Φ = Φ1 and one where Φ = Φ2 . The larger the coupling is, the easier it produce large changes in the particle masses. In particular, the values that the particle masses, and the other ‘constants’, had in the very early universe (i.e. pre-BBN) could be very different, in these models, from the ones that they take today. If the particle masses were very different from their present values at the epoch of the electroweak phase transition then the predictions of electroweak baryogenesis could be significantly altered. In this chapter we took the chameleon potential to have a power-law form i.e V ∝ Φ−n . This is certainly not the only class of chameleon potentials that it is possible to have. In general, any potential that satisfies βV,Φ < 0, V,ΦΦ > 0 and V,ΦΦΦ /V,Φ > 0 in a region near some Φ = Φ0 will produce a chameleon theory. In fact, a generic potential might contain many different regions in which chameleon behaviour is displayed. In some of these regions the potential may appear to have a runaway form and so behave qualitatively as an n > 0
5.6. CONCLUSIONS AND DISCUSSION
215
theory. In other regions the potential might have a local minimum leading to n ≤ −4 type behaviour. The existence of the matter coupling provides one with a mechanism, along the lines of that considered by Damour and Polyakov, [72], by which the scalar field Φ can, during the radiation era, be moved into a region where it behaves like a chameleon field. The larger β is, the more effective this mechanism becomes. Given this mechanism, one important avenue for further study is to see precisely how general late-time chameleon behaviour is of a generic scalar field theory with a strong coupling to matter. In summary: • The experimental constraints on the coupling of chameleon fields to matter are much weaker than those on non-chameleon fields. In fact, the larger the matter coupling, β, the weaker the constraints from table-top and solar system tests of gravity are. Almost paradoxically, strongly-coupled scalar fields are actually harder to detect than weakly coupled ones. • Perhaps the most important result, though, is that the ability of table-top gravity tests to see strongly coupled, chameleon fields could be exponentially increased if certain features of their design could be adjusted in the appropriate manner. The detection, or exclusion, of chameleon fields with β À 1 represents a significant but ultimately, we believe, achievable challenge to experimentalists. Searches for large β chameleon fields represent one way in which table-top tests of gravity could be used to probe for new physics beyond the standard model. Whether or not chameleon fields actually exist, it is important to note those areas into which our current experiments cannot see and, if possible, design experiments to probe them. We have shown, in this chapter, that scalar field theories that couple to matter much more strongly than gravity are not only viable but could well be detected by a number of future experiments provided they are properly designed to do so. This result opens up an altogether new window which might lead to a completely different view of the rˆole played by light scalar fields in particle physics and cosmology.
Chapter 6
Conclusions
Yet all experience is an arch wherethro’ Gleams that untravell’d world, whose margin fades For ever and for ever when I move. Alfred, Lord Tennyson (1809-1892)
6.1
Summary
In this thesis we have studied several important aspects of theories which predict the existence of new scalar fields. Our conclusions have considerable implications for the experimental programmes that aim to detect these fields and, through such a detection, open a new window onto high energy physics beyond the Standard Model. The main results of our analyses are summarised below: 1. In Chapter 2 we considered the new effects that emerge when theories that describe a variation in the fine structure constant, αem , are generalised so that they include the GSW theory of electroweak interactions. We found that if αem varies then, in general, another constant of Nature, θw - the Weinberg mixing angle, does so as well. Although models in which θw is a true constant at tree level can be constructed, quantum loop effects will generally result it becoming variable. 217
218
CHAPTER 6. CONCLUSIONS
2. We analysed the way in which the scalar field (or fields) predicted by electroweak varyingαem interact with matter as a result of quantum corrections. We derived the effective quantum-corrected dilaton evolution equations in the presence of a background cosmological matter density that is composed of the particles present in the Standard Model as well as weakly interacting and non-weakly-interacting non-relativistic dark-matter components 3. We found that, in contrast to earlier varying-αem theories, such as BSBM, which ignore electroweak effects, the cosmological evolution of the fine structure constant could be driven, in the main part, by weakly interacting dark matter particles (WIMPs) with masses & MW , MZ . This result has important consequences for future observations. In order to produce a cosmological change in αem of the order of that reported by recent observations of QSO spectra, i.e |∆αem /αem | ∼ O(10−6 ) over the last 10-12 Gyrs, we saw that WEP violation constraints required that WIMP matter provide the dominant source for the evolution of the coupling ‘constants’. Given these same constraints, the KM-models predict that dark matter must, in the main part, be comprised of WIMP particles and that mwimp & MW , MZ . 4. If ∆αem /αem ∼ O(10−6 ) in the last 10-12 Gyrs, we concluded that, at least in the context of the simplest models, the prospects for local experiments making an unambiguous detection of αem -variation in the near future were very good. Planned and proposed satellite tests of WEP will have the precision required to detect the presence of the scalar field that is responsible for the variation in αem , and the next generation of laboratory observations of α˙ em /αem will have the precision required to directly detect a cosmological variation in the fine structure constant. If no such detection is made then either αem is varying much more slowly than suggested by recent observations of quasar absorption systems or we must rule out the simplest models. 5. We found that spatial variations in the Weinberg mixing-angle, θw , result in a form of non-conservation and dequantisation of electric charge, and that they induce an electric dipole moment (EDM) on all particle species that carry a weak neutral charge. 6. In Chapter 3 we showed, using the method of matched asymptotic expansions, that a sufficient condition can be derived which determines when a local experiment will
6.1. SUMMARY
219
detect the cosmological variation of a scalar field which is driving the spacetime variation of a supposed constant of Nature. We studied both spherically symmetric, and nonspherically symmetric, spacetimes in which there was some central virialised structure (e.g a galaxy, black hole, star or planet) and spherically symmetric spacetime backgrounds that had yet to virialise. 7. We showed that, in the context of a very large class of physically-viable scalar field theories, ground and solar system based experiments can detect any cosmological variation of the constants. In the specific example of Brans-Dicke theory, we found that the lo˙ cal value G/G differs from the cosmological value by less than 1%. When applied to BSBM and the KM-models our analysis implies that the local and cosmological values of α˙ em /αem also differ by less than 1%. If |∆αem /αem | ∼ 10−6 over the last 10-12 Gyrs then, in the context of the KM-models, the results of Chapters 2 and 3 imply that laboratory tests should detect |α˙ em /αem | ∼ 10−16 . 8. We found that, contrary to what is suggested when spatial derivatives of the scalar field and ignored and the spherical infall approximation is used, the temporal rate of change of a scalar field inside an inhomogeneous collapsing region of spacetime would not deviate significantly from its cosmological value so long as the virialisation velocity, vvir , of the cluster is not too large: vvir ¿ 60, 000 km s−1 . This condition is easily satisfied for a collapsing region that with properties similar to a rich cluster like Coma (vvir ≈ 1, 800 km s−1 . Contrary to what had been previously stated elsewhere in the literature, we showed that virialisation of a collapsing cluster does not stabilise the local values of cosmologically varying-constants. 9. In Chapters 4 and 5, we studied a class of physically viable, and potentially highly testable, scalar field theories that possess a chameleon mechanism. In such theories the presence of a self-interaction potential causes the mass of the scalar field to be highly dependent on the background density of matter. These chameleon fields were seen to be very light in sparse backgrounds, e.g. space, but very heavy in a high density background such as that here on Earth. In Chapter 4, we considered the dynamics of chameleon theories with power law potentials and calculated the ‘fifth’ force that these chameleon fields generate between bodies composed of normal matter. Crucially, we found that the
220
CHAPTER 6. CONCLUSIONS
chameleon force between two suitably large bodies was independent of: a) the mass of those bodies and b) the strength with which scalar field couples to the matter in the bodies. 10. In Chapter 5, we analysed the experimental, cosmological and astrophysical constraints on chameleon theories. As a result of the coupling (β) independence of the chameleon force, we saw that experiments which search for fifth-forces could no longer place an upper bound on the size of β. By comparison, in non-chameleon theories these experiments, which include WEP violation searches, require that β ¿ 1 for all but the heaviest scalar fields. β ¿ 1 implies that the scalar interacts with matter much more weakly than gravity does. 11. In stark contrast to that which is usually thought to be the case, we showed that, as a result of self-interactions, scalar fields which are light over cosmological scales and yet couple to matter much more strongly than gravity are not ruled out my experiments. We also showed that such strongly-coupled scalars could well be detected by a number of future experiments provided they are properly designed to do so.
6.2
Outlook
Our aim in this thesis was to expand upon what was previously known about the behaviour of theories that predict the existence of new scalar fields, so that the vast array of applicable experimental and observation data can be better combined to detect or rule out these scalar field theories. By constraining the properties of any new scalar fields, one also places limits on those theories of new physics, e.g. String Theory, that predict their existence. Searches for varying-constants and the other scalar field effects provide an excellent low energy test for new high energy physics, and any unequivocal detection of a new scalar field would therefore have important consequences for our understanding of physics. Accordingly, we conclude this thesis by briefly discussing what the prospects are, in light of our results, for making such a detection in the near future. Recent observations of the spectra of quasar absorption systems suggest that the fine structure constant has increased by a few parts in a million over the last 10-12 billion years [1, 2, 3]. This reported detection of varying-αem remains highly controversial, although in seven years
221
6.2. OUTLOOK
since it was first announced, no-one has yet to identify any selection effect or other source of error which can explain either the sense or the magnitude of the observed change. The status of the recent 3.5σ indication of a variation in the electron-proton mass ratio, µ, over the last 12 billions, [20], is even less clear. Later this year, a major effort to produce a very large new data set of observations of QSO spectra should be reported. This will hopefully clarify the status of these earlier investigations. However, even if this new data set confirms the earlier reports of a variation in αem , it is unlikely, given the controversial nature of the result and potential unknown sources of systematic errors, that it will be taken as definitive evidence for varying-constants. If a variation in αem , or µ, could be independently verified in a controlled laboratory experiment however, then the combination of local and global evidence would make a highly compelling case for variation of the ‘constants’. In the context of the simplest varying-αem models, at least, the prospects for making such a laboratory-based detection in the near future are very good, if the QSO value for ∆αem /αem is indeed accurate. Currently laboratory experiments are able to bound α˙ em /αem . 10−15 yr−1 locally, whereas the simplest varying-αem theories predict that today α˙ em /αem ∼ O(10−16 ) yr−1 on cosmological scales if ∆αem /αem ∼ O(10−6 ) over the last 10-12 Gyrs. The next generation of tests promise the precision to see α˙ em /αem ∼ O(10−16 ) and, as Flambaum recently reported in ref. [51], by making use of enhancement effects associated with the very light UV transition between the ground and first excited state of
229 T h,
it should be possible detect relative variations in αem
and the quark masses at the 10−23 yr−1 level in the near future. In Chapter 3 we proved that, in a wide-class of theories, these local tests directly measure the rate at which the constants are changing cosmologically (with a relative error of less 1%). At least in the context of the class of theories considered in Chapter 3, there is therefore a real prospect that, within the next decade, laboratory tests will be able to directly detect a cosmological change in αem and µ. If these tests produce a null result then, in most cases, it would imply that the values αem and µ have, to a precision of one part in 1013 or so, remained unchanged over the last 12 billion years. When interpreted in the context of String Theory, this would imply that the dilaton fields which set the values of αem and µ are either very massive on cosmological scales or have strong self-interactions and behave like the chameleon fields described in Chapter 4 and 5. There is also a real hope that scalar fields could be detected in the near future through their effective alternations to the 1/r2 behaviour of gravity over short distances and through
222
CHAPTER 6. CONCLUSIONS
the WEP violations that they induce. Four planned space-based tests of gravity, STEP [97], SEE [96], GG [98] and MICROSCOPE [99], promise to be able to such alterations to General Relativity with unprecedented precision. MICROSCOPE is due to be launched in 2007/08 and it should be able to detect WEP violations at the one part in 1015 level, whilst GG promises the precision to detect down to one part in 1017 and STEP should offer an order of magnitude improvement over that. The proposed SEE satellite will test not only for WEP violations but also for alterations to the inverse-square law of gravity. It will also be able to measure G to ˙ within 1ppm and directly measure G/G down to a precision of 10−13 yr−1 . These experiments will improve the bounds on all scalar fields theories. In particular, as we showed in Chapter 2, MICROSCOPE alone will be able to rule out, or verify, the simplest varying-αem theories if ∆αem (z = 2)/αem ∼ O(10−6 ), and in Chapter 5 we noted that the fact that these tests will be conducted in space, rather than in a ground-based laboratory, allows them to better detect chameleon theories with runaway potentials. In Chapters 4 and 5, we showed that constraints provided by table-top and solar system gravity tests are exponentially weakened if the scalar fields have strong self-interactions i.e. if they are chameleon fields. Whilst the next-generation of space-based tests will just about be able to detect chameleon fields with runaway Φ−n potentials (n > 0) and a gravitational strength interaction with matter, the prospects of detecting theories in which the potential as a minimum (e.g. n < 0 theories) in future tests is not so good. This said, it has recently been shown elsewhere [154] that next-generation E¨ot-Wash experiment, [87], will be able to detect or rule out the existence of a scalar field with gravitational strength interactions with matter and a Φ4 potential at a 3σ confidence level. Unfortunately, planned future experiments will still be unable to detect chameleon fields which couple to matter much more strongly than gravity does. The ability of table-top gravity tests to see strongly coupled, chameleon fields could, however, be exponentially increased if certain features of their design can be adjusted in light of the analyses presented in this thesis. In conclusion: if our universe is populated by one or more scalar fields that interact with matter and whose self-interactions are negligible over sub-Hubble scales then, due to expected increases in experimental precision, the prospects for detecting such fields are very good. As we illustrated in Chapters 4 and 5, however, the presence of non-negligible self interactions can radically alter this picture and make any scalar fields much harder to detect and, in many
6.2. OUTLOOK
223
cases, experiments must be redesigned if they are to be able detect scalar fields with strong self-interactions.
Appendices
225
Appendix A
Explicit matching for the McVittie background
A.1
Exterior Solution (0)
(1)
In the exterior region φE ∼ φE + δφE + O(δ 2 ) where: ! ¢3 Ã ¡ 1 + 14 Ω0k ρ2 ρ2 (1) ∂ρ ∂ρ φE a ˜2 ρ2 1 + 14 Ω0k ρ2 µ ¶ ¡ 2m 3/2 (1 + 14 Ωk ρ2 )1/2 n (0)00 (0)0 = φE + hφE − 3B,φ (φE ) h0 + Rs a ˜(τ )ρ
³ ´ 1 (1) 3 ∂ a ˜ ∂ φ − τ τ E a ˜3
(0)
(1)
(0)
(A.1) ¢o
11 4 Ωk (τ )
(1)
−B,φφ (φE )κ²0dust φE − V,φφ (φE )φE . Despite the complexity of this formula we can solve it admits separable solutions in τ and ρ: µ (1) φE
=
2m Rs
¢1/2 ¶3/2 ¡ 1 + 14 Ω0k ρ2 Υ(τ ) + Φ(τ ρ). ρ
where Υ(τ ) satisfies the ODE: ¡ 3 ¢ 1 ∂ a ˜ ∂ Υ − τ τ a ˜3
3Ωk (τ ) (0) (0) Υ + B,φφ (φE )H0−2 κ²0dust Υ + H0−2 V,φφ (φE )Υ 4 ¡ ¢o 1 n (0)0 (0)0 = − φE + hφE − 3B,φ (φE ) h0 + 11 Ω (τ ) . k 4 a ˜ 227
228
APPENDIX A. EXPLICIT MATCHING FOR THE MCVITTIE BACKGROUND
When k > 0, Φ =
P∞
q
γ=−∞
2 + Ω0k cγ · T+ γ (τ )Xγ (ρ), cγ ∈ C :
eiγα(ρ) sin α q Ω0k ρ , α(ρ) = sin−1 1 + 41 Ω0k ρ2 Xγ
=
³ ´t +,1 +,2 and T+ , where Tγ+,1 and Tγ+,2 are linearly independent solutions of the folγ = Tγ , Tγ lowing ODE: ¡ 3 ¢ 1 ∂τ a ˜ ∂τ Tγ+,i (τ ) + (γ 2 − 1)Ωk (τ )Tγ+,i (τ ) 3 a ˜ (0) (0) = −B,φφ (φE )H0−2 κ²0dust Tγ+,i (τ ) − H0−2 V,φφ (φE )Tγ+,i (τ ). R∞ If k = 0. then Φ = −∞ dγ c(γ) · T0γ (τ )Xγ0 (ρ), c(γ) ∈ C2 , and Xγ0 (ρ) = eiγρ /γρ. T0γ (τ ) = ³ ´t Tγ0,1 , Tγ0,2 satisfies: ¡ 3 ¢ 1 (0) (0) 0,i a ˜ ∂ T (τ ) + γ 2 Tγ0,i (τ ) = −B,φφ (φE )H0−2 κ²0dust Tγ0,i (τ ) − H0−2 V,φφ (φE )Tγ0,i (τ ), ∂ τ τ γ a ˜3 q R∞ − 2 where i = 1, 2. Finally if k < 0 then Φ = −∞ dγ −Ω0k c(γ) · T− γ (τ )Xγ (ρ), c(γ) ∈ C : Xγ− (ρ) =
eiγα sinh α
q −Ω0k ρ , α = sinh−1 1 + 14 Ω0k ρ2
³ ´t −,1 −,2 and T− (τ ) = T , T : γ γ γ ¡ 3 ¢ ¡ ¢ 1 ∂τ a ˜ ∂τ Tγ−,i (τ ) − γ 2 + 1 Ωk (τ )Tγ−,i (τ ) 3 a ˜ (0) (0) = −B,φφ (φE )H0−2 κ²0dust Tγ−,i (τ ) − H0−2 V,φφ (φE )Tγ−,i (τ ), where i = 1, 2. In all cases we shall fix our definition of the Tγ by the normalisation: Tγ (τ0 ) = 1.
A.2
Matching Conditions
¯−γ for k > 0, and By making our interior and exterior solutions for φ we see that: cγ = c P n ¯(−γ); z¯ is the complex conjugate of z. Defining An (τ ) = γ γ cγ · T+ c(γ) = c γ or An (τ ) =
229
A.2. MATCHING CONDITIONS
R
0/−
dγ γ n c(γ) · Tγ
for k > 0 and k ≤ 0 respectively we see that:
F (φ¯0 ) − Υ(τ ) a A1 (τ ) = 0 ⇔ c(γ) = c¯(−γ) A0 (τ ) = −
(A.2)
(A.3) ´ ³ ¯ 0¯ (0) ¯Ω ¯ A2 (τ ) = −(h0 + 2h2 )˜ ˜F (φ¯0 ) aF (φ¯0 ) + B,φφ (φE )κ²0dust + V,φφ a (A.4) k ³ ¢´ 1 0 ¡ (0)00 (0)0 − Ωk Υ. Ω (τ ) −Ωk (τ )˜ aF (φ¯0 ) − a ˜ φE + hφE − 3B,φ h0 + 11 k 4 4 In principle we can invert the equation for A0 (τ ) to find c(γ). Given the definition of the An , the expressions for A0 and A2 must satisfy a consistency relation (if there were not to satisfy this, the matching procedure would be invalid). We have checked that this relation does indeed hold here. The relation is: |Ωk (τ )| A2 − Ωk (τ )A0 = −
´ ¢ ³ ¡ 3 1 (0) 0 − B (φ )κ² + V A0 . a ˜ ∂ A ∂ τ 0 τ ,φφ ,φφ dust E a ˜3
We can see, explicitly, that the matched asymptotic expansion method works for the McVittie background.
Appendix B
Pseudo-Linear regime for single-body problem
In the pseudo-linear approximation, we assume that non-linear effects are, locally, everywhere sub-leading order. The cumulative, or integrated, effect of the non-linearities is not, necessarily small. This means that, whilst we assume that there always exists at least one self-consistent linearisation of the field equations about every point, we do not require there to be a linearisation that is everywhere valid. Instead we aim to contrast two linearisations of the field equations: the inner and the outer approximations to Φ. The inner approximation is intended to be an asymptotic approximation to the chameleon that is valid both inside an isolated body, and close to the surface of that body. We take the isolated body to be spherically symmetric, with uniform density εc and radius R. Far from the body, r À R, the inner approximation will, in general, break down. The outer approximation is an asymptotic approximation to Φ that is valid for large values of r. We require that it remains valid as r → ∞. In general the outer approximation will not be valid for r ∼ O(R). The boundary conditions on the evolution of Φ are: ¯ ¯ dΦ ¯¯ dΦ ¯¯ = 0, = 0. dr ¯r=0 dr ¯r=∞ The first of these conditions is defined at r = 0. We will generally find that only the inner approximation is valid at r = 0. As a result we cannot apply the r = 0 boundary condition 231
232
APPENDIX B. PSEUDO-LINEAR REGIME FOR SINGLE-BODY PROBLEM
to the outer-approximation. Similarly the r = ∞ boundary condition will be applicable to the outer-approximation but not to the inner one. Since we cannot directly apply all the boundary conditions to both approximations, there will generally be undefined constants of integration in both the inner and outer expansions for Φ. This ambiguity in both expansions can, however, be lifted if there exists some intermediate range of values of r (rout < r < rin say) where both the inner and outer approximations are valid. Asymptotic expansions are locally unique [135, 136]. Thus, if both the outer and inner approximation are simultaneously valid in some intermediate region, then they must equal to each other in that region. By appealing to this fact, we can match the inner and outer approximations in the intermediate region. In this way, we effectively apply all of the boundary conditions to both expansions. This method of matched asymptotic expansions is described in more detail in Chapter 3.3.
B.1
Inner approximation
Inside the body, 0 ≤ r ≤ R, the chameleon obeys: µ 3 ¶n+1 d2 Φ 2 dΦ βεc 3 M = −nλM + . + 2 dr r dr Φ Mpl
(B.1)
The inner approximation is defined by the assumption µ 3 ¶n+1 βεc 3 M nλM ¿ . Φ Mpl Defining
µ Φc = M
βεc nλMpl M 3
¶−
1 n+1
,
we see that the above assumption is equivalent to: µ ¶ Φc n+1 δ(r) := ¿1 Φ(r) We define the inner approximation by solving eq. (B.1) for Φ as an asymptotic expansion in the small parameter δ(0); we shall see below that δ(r) < δ(0) := δ. Whenever the inner approximation is valid we have: Φ ∼ Φ0 +
βεc r2 + O(δ), 6Mpl
233
B.2. OUTER APPROXIMATION
where the order δ term is: δβεc Φδ (r) = rMpl
Z
Z
r
dr
0
0
r0
µ 00 00
dr r 0
Φ0 ¯ Φ(r00 )
¶n+1 .
¯ We have defined Φ(r) := Φ0 +βεc r2 /6Mpl for r < R. Φ0 is an undefined constant of integration. It will be found by the matching of the inner approximation to the outer one. For the inner approximation to remain valid inside the body we need: Φδ (r) ¿ 1. ¯ Φ(r) Outside the body, r > R, Φ obeys: d2 Φ 2 dΦ + = −nλM 3 dr2 r dr
µ
M3 Φ
¶n+1 .
Whenever δ(r) < 1, we can solve the above equation in the inner approximation finding: ¯ Φ ∼ Φ(r) + Φδ (r). ¯ Outside the body, r > R, we define Φ(r) to be βεc r2 βεc R3 ¯ − . Φ(r) = Φ0 + 2Mpl 3Mpl r
(B.2)
The order δ term, Φδ (r), is given by the same expression as it was for r < R. The inner approximation will therefore be valid, both inside and outside the body, provided that: Φδ (r) ¿ 1. ¯ Φ(r) In general, this requirement will only hold for r less than some finite value of r, r = rin say. To fix the value of Φ0 , and properly evaluate the above condition, we must now consider the outer approximation.
B.2
Outer Approximation
When r is very large, we expect that the presence of the body should only induce a small perturbation in the value of Φ. Assuming that, as r → ∞, Φ → Φb , where Φb is the value of Φ in the background, then the outer approximation is defined by the assumption |(Φ − Φb )/Φb | < 1/|n + 1|. We may therefore write: µ 3 ¶n+1 µ 3 ¶n+1 ¡ ¢ 3 M 3 M ∼ −nλM + m2b (Φ − Φb ) + O (Φ/Φb − 1)2 −nλM Φ Φb
234
APPENDIX B. PSEUDO-LINEAR REGIME FOR SINGLE-BODY PROBLEM
where
µ m2b
= λn(n + 1)M
2
M Φ
¶n+2 ,
is the mass of the chameleon in the background. The assumption that |(Φ−Φb )/Φb | < 1/|n+1| is essentially the same assumption as was made in the linear approximation in chapter 4.3.1. In the linear approximation, however, this assumption was required to hold all the way up to r = 0. All that is required for the pseudo-linear approximation to work, is that the outer approximation be valid for all r > rout , where rout is any value of r less than rin . This is to say that, we need there to be some intermediate region where both the inner and outer approximations are simultaneously valid. Outside of the body Φ obeys: d2 Φ 2 dΦ = −nλM 3 + dr2 r dr
µ
M3 Φ
¶n+1 +
βεb . Mpl
For the outer approximation to remain valid as {r → ∞, Φ → Φb } we need µ nλM
3
M3 Φb
¶n+1 =
βεb . Mpl
Solving for Φ in the outer approximation, we find Φ ∼ Φ∗ where: Φ∗ = Φb −
Ae−mb r . r
(B.3)
A is an unknown constant of integration. It will be determined through the matching procedure.
B.3
Matching Procedure
We assume that there exists an intermediate region, rout < r < rin , where both the inner and outer approximations are valid. This region does not need to be very large. All that is truly needed is for there to exist an open set, about some point r = d, where both approximations are valid. We shall consider what is required for such an open set to exist in section B.4 below. For the moment we shall assume that it does exist, and evaluate Φ0 and A. In the intermediate region we must have ¯ ∼ Φ∗ , Φ∼Φ
235
B.4. CONDITIONS FOR MATCHING
by the uniqueness of asymptotic expansions. The requirement that δ(r) ¿ 1 ensures that mb r ¿ 1 in any intermediate region. Expanding Φ∗ to leading order in mb r and equating it ¯ we find that to Φ Φ0 = Φb − A =
βεc , 2Mpl
βεc . 3Mpl
Now that the previously unknown constants of integration, A and Φ0 , have been found, we can evaluate the conditions under which an intermediate region actually exists.
B.4
Conditions for Matching
¯ = 0)/Φc )−(n+1) ¿ For the inner approximation to be valid we must certainly require that (Φ(r 1. This is equivalent to: ³ ´ (mc R)2 ¿ 2|n + 1| (εc /εb )1/(n+1) − 1 .
(B.4)
It is also interesting to note what is required for the pseudo-linear approximation to be valid outside the body i.e. without requiring it to be valid for r < R. This gives the weaker condition: ¯ ¯ ¯ ¯ (mc R)2 ¿ 3|n + 1| ¯(εc /εb )1/(n+1) − 1¯ . If this weaker condition also fails then, irrespective of what occurs inside the body, we cannot ¯ outside of the body. As we show below, the condition that a even have a solution where Φ ∼ Φ body have a thin-shell is equivalent to the requirement that this weaker condition fail to hold. For an intermediate matching region to exist, there must, at the very least, exist some d such that, in an open set about r = d, both the inner and outer approximations are valid. We must also require that the inner approximation be valid for all r < d, and that the outer approximation hold for all r > d. For the outer approximation to hold for all r > d we need: ¯ ¯ ¯ (n + 1)(Φ∗ − Φb ) ¯ ¯ ¿ 1. ¯ ¯ ¯ Φb Using A = βεc /3Mpl and eq. (B.3) we can see that this is equivalent to: R (mc R) ¿ 3d 2
µ
εc εb
¶1/(n+1) .
(B.5)
236
APPENDIX B. PSEUDO-LINEAR REGIME FOR SINGLE-BODY PROBLEM
Using this condition, we define dmin to be the value of d for which the left hand side and right hand side of the above expression are equal: (mc R)2 R/3dmin = (εb /εc )−1/(n+1) . For the outer approximation to be valid at d we require d > dmin . ¯ to hold in the intermediate region we require that, For the inner approximation, Φ ∼ Φ, for all R < r < d:
Z
R3 À 3
Z
r
dr
µ
r0
0
00 00
dr r
0
0
Φc ¯ Φ(r00 )
¶n+1 .
¯ = 0))n+1 i.e. that condition (B.4) holds. Provided that this We must also have that (Φc /Φ(r is the case then, for all r in (R, d), we need R3 À 3
Z
Z
d
dr
0
µ
d
00 00
dr r r0
R
Φc ¯ Φ(r00 )
¶n+1 .
(B.6)
If both eqs. (B.4) and (B.6) hold then the inner approximation will be valid for all r < d. We ¯ − Φb = Φb . evaluate eq. (B.6) approximately as follows: For n ≤ −4 we define r = d0 by Φ ¯ in the above integral by Φb , and for all r in (R, d0 ) we For all r in (d0 , d) we approximate Φ ¯ by (Φ ¯ − Φb ). approximate Φ ¯ − Φb ) À Φc and so such no d0 exists. When When n > 0, eq. (B.4) implies that Φb > (Φ n > 0 and condition (B.4) holds, we can therefore find a good estimate for the validity of the pseudo-linear regime by setting Φ = Φb in the above integral. We consider the cases n ≤ −4, n = −4 and n > 0 separately below.
B.5
Case: n < −4
We shall deal with the subcases d0 ≤ R and d0 > R separately. Subcase: d0 ≤ R The subcase where d0 < R includes those circumstances where the linear regime is valid (see chapter 4.3.1). Since we already found, in chapter 4.3.1, how Φ behaves when linear approximation holds, we shall only consider what occurs when the linear approximation fails. If the linear approximation fails, then the outer approximation must break down outside the body i.e. dmin /R > 1. Evaluating eq. (B.6), we find that we must, at the very least, require that: µ
dmin R
¶3
εb < 1, εc
237
B.5. CASE: N < −4
which, for all n, is equivalent to: √ 3 mc R < mc R <
µ
εc εb
¶
n+4 6(n+1)
µ
p
3|n + 1|
εc εb
, ¶1/2|n+1| .
The second criteria is just the statement that d0 < R. When εb ¿ εc , this latter condition is more restrictive than the former. Subcase: d0 > R ¿From the definition of d0 we find d0 /R = (mc R)2 /(3|n+1|) (εc /εb )1/|n+1| and d0 = dmin /|n+1|. It follows that d ≥ dmin > d0 . When d0 > R, an intermediate region will exist so long as: 3d3 εb 3 + 3 2R εc (n + 4)(n + 3)
µ
(mc R)2 3|n + 1|
¶|n+1| ¿ 1.
The smaller d is, the more likely it is that this condition will be satisfied; we therefore evaluate the condition at d = dmin . Both of the terms on the left hand side are positive, and so for an intermediate region to exist we must have: µ
¶ µ ¶(n+4)/6(n+1) εc (n+4)/6(n+1) εc mc R < (18) ≈ 1.6 , εb εb µ ¶ p (n + 4)(n + 3) 1/2|n+1| 3|n + 1| mc R < . 3 1/6
For n < −4, the second of these conditions is usually the more restrictive. However, since n < −4 implies n ≤ −6, this second condition is itself, in general, less restrictive than requiring ¯ = 0))n+1 ¿ 1 or (Φc /Φ(r ¯ = R))n+1 ¿ 1. either (Φc /Φ(r Conditions Putting together all of the conditions found above, we see that, for the pseudo-linear approximation to be valid all the way from r = ∞ to r = 0 for n < −4, we must, at the very least, have: Ã mc R < min (18)1/6
µ
mc mb
¶(n+4)/3(n+2) ,
¯ ¯1/2 ¯ ¯ 2|n + 1| ¯(mb /mc )2/|n+2| − 1¯
p
where we have used ρc /ρb = (Φb /Φc )n+1 = (mc /mb )2(n+1)/(n+2) .
! ,
238
APPENDIX B. PSEUDO-LINEAR REGIME FOR SINGLE-BODY PROBLEM
Thin-Shell Condition If the above condition fails to hold, then it is a sign that non-linear effects have become important. However, even when non-linear effects are important, we only expect them to be so in a region very close to the surface of the body itself. In chapter 4.3.3 we assumed that our body had a thin-shell and considered the behaviour of the field close to the surface of the body. We found there to be pronounced non-linear behaviour near the surface of such bodies. This implies that the pseudo-linear approximation must break down for r > R for bodies with thin-shells. We further found that the assumption that the body had a thin-shell required: µ
n n+1
¶n/2+1 mc R À 1.
For a body to have a thin-shell we must therefore require that this condition hold and that the pseudo-linear approximation would break down for r > R: these are the thin-shell conditions. It is almost always the case that the latter of these two conditions is the more restrictive. In chapter 4.3.2 we found this condition to be:
µ
(mc R) > min (18)1/6
mc mb
¶(n+4)/3(n+2)
¯µ ¶ ¯1/2 ¯ m 1/|n+2| ¯ p ¯ ¯ b , 3|n + 1| ¯ − 1¯ . ¯ mc ¯
Since this condition implies mc R À 1, it is the thin-shell condition for n < 0 theories.
Case n > 0 The case of n > 0 is actually slightly simpler than the n ≤ −4 one because we cannot have d0 > R. Other than that, the analysis proceeds in much that same way as it does for the n < −4 case. For this reason, we will not repeat the details of the calculations here.
Conditions For the pseudo-linear approximation to be valid all the way up to r = 0 we must, at the very least, have: Ã
µ 1/6
mc R < min (18)
ρc ρb
¶
n+4 6(n+1)
¯ ¯1/2 p 1 ¯ ¯ , 2(n + 1) ¯(ρc /ρb ) n+1 − 1¯
! .
When ρb ¿ ρc , the most restrictive bound comes from the second term on the right hand side.
239
B.5. CASE: N < −4
Thin-Shell Condition As in n < 0 theories, a body with have a thin-shell provided that mc R À 1 and non-linear effects are important near the surface of the body which implies that the pseudo-linear approximation breaks down outside the body. This latter condition in fact implies the former and is therefore the thin-shell condition for n > 0 theories. This condition reads: Ã µ ¶ ! ¯ ¯1/2 √ mc (n+4)/6(n+2) p ¯ ¯ 1/(n+2) , 3(n + 1) ¯(mc /mb ) mc R > min 3 − 1¯ , mb where the second term on the right hand side is usually the more restrictive when ρb ¿ ρc . Case n = −4 The n = −4 case, i.e. Φ4 theory, requires a more involved analysis. The reason for this is that, unlike the n < −4 theories, there does not exist a solution to this theory where Φ ∼ −A/r as r → ∞. If we propose such a leading order behavior for the inner approximation, then it can be easily checked that the next-to-leading order term dies off as ln(r)/r, i.e. more slowly than the leading order one. This means that, for some finite r, the next-to-leading order term will dominate over the leading order one. When this happens the inner approximation will break down. It can also be checked that higher order terms will always die off more slowly than the terms of lower order. This complication will only manifest itself, however, when the conditions for the pseudo-linear approximation fail, or almost fail, to hold i.e when (mc R) is large. Inner approximation and matching To avoid these difficulties, we shall use a different form for the leading order behaviour for Φ when n = −4. We write: Φb α(r)e−mb r Φ ∼ + . Φc Φc mc r To leading order in the inner approximation we neglect terms of O(mb r) and smaller. The field equation for Φ is then found to be equivalent to: d2 α dα α3 − ∼ dy 2 dy 3 where y = y0 + ln(r/R). The outer approximation is still given by Φ ∼ Φ∗ . To perform the matching we need to know the large r behaviour of the inner approximation, and the small r behaviour of the outer one. Solving for α we find that: r 3 α∼ . 2y
240
APPENDIX B. PSEUDO-LINEAR REGIME FOR SINGLE-BODY PROBLEM
The next-to-leading correction to α is: ln(y) 2
µr
3 2y
¶3 .
It is clear that, as we would wish, the next-to-leading order term dies off faster than the leading order one. We shall see below that, for this approximation to be valid near the body, we need y0 À 0.634. This approximation also breaks down when mb r À 1. Matching the inner and outer approximations in the same manner as we did before, we find for r > R Φ Φb α(min(r, 1/mb ))e−mb r ≈ + . Φc Φc mc r
(B.7)
Inside the body, r < R, we assume that, at leading order, Φ/Φc behaves in the same way as if did for all the other values of n: Φ Φ0 (mc R)2 r2 ∼ + . Φc Φc 6(n + 1)R2 By requiring Φ to be C1 continuous at r = R, we find y0 to be (mc R)3 = 9
r
3 1 + 2y0 3
µr
3 2y0
and so Φ0 Φb (mc R)2 1 = + + Φc Φc 18 mc R
¶3
r
(B.8)
3 . 2y0
Conditions For this approximation to be a valid approximation, we must firstly require that the next-toleading order correction to α is always small compared to the leading order one. This implies y0 À 0.508,
mc R ¿ 3.13.
We must also require Φ(r = 0) = Φ0 ¿ Φc which gives (assuming ρb /ρc ¿ 1): y0 À 0.634,
mc R ¿ 2.915.
This is the stronger of the two bounds. When the field equations are solved numerically we find that the form for Φ given above is an accurate approximation whenever mc R < 1.
241
B.5. CASE: N < −4
Critical Far Field The form of the far field for bodies with thin-shells in n = −4 theories is examined in appendix C below. However, it is interesting to note that, even in the pseudo-linear approximation for n = −4, there is already the first hint of β-independent behaviour in the far field. When mb r ¿ 1, we found that Φ ≈ Φb −
(y0 + ln(r/R))−1/2 e−mb r √ . 2 2λr
Thus, when ln(r/R) À y0 (provided mb r ¿ 1), we have, to leading order: e−mb r Φ ∼ Φb − p , 2 2λ ln(r/R)r which is manifestly independent of Φc and hence also of β. In n 6= −4 theories, β-independent critical behaviour was reserved for bodies with thin-shells. When n = −4, we can see that leading order, β-independent behaviour in the far field (r À R) can occur for all y0 i.e. all values of (mc R). However if (mc R) ¿ 1, this critical behaviour will only be seen for exponentially large values of r/R. If, however, mc R & 2 then Φ will be β-independent at leading order for all r/R & 10.
Appendix C
Far field in n = −4 theory for a body with thin-shell
Using eq. (4.22) and the other results of chapter 4.3.3, we find that, if a body has a thin-shell, then for 0 < (r − R)/R ¿ 1 we have Φ≈
1 √ − 2λ(r − R) +
4 3Φc
,
when n = −4. In appendix B, we saw that, far from the body and when mb r ¿ 1, we have: Φ ≈ Φb −
(y0 + ln(r/R))−1/2 e−mb r √ . 2 2λr
We can find the value that y0 takes for a thin-shelled body, and hence determine the behaviour of the far field, by matching the leading order large r behaviour of the first expression to the leading order behaviour of the second expression as r → R. This gives: 4y0 = 1 ⇒ y0 = 1/4. Numerical simulations show that this tends to be a slight over-estimate of the true far-field. We find therefore that far from a body with thin-shell, the chameleon field is approximately given by: Φ ≈ Φb −
(1 + 4 ln(r/R))−1/2 e−mb r √ . 2λr 243
244
APPENDIX C. FAR FIELD IN N = −4 THEORY FOR A BODY WITH THIN-SHELL
Indeed, since we are far from the body, it is almost always that case that r/R À 1.3 and so 4 ln(r/R) À 1. As a result, it is usually a very good approximation to take e−mb r Φ ∼ Φb − p . 2 2λ ln(r/R)r We note that, whenever the pseudo-linear approximation holds, y0 = 1/4 is equivalent, by eq. (B.8), to an effective value of mc R of (mc R)ef f = 4.04. Numerical simulations confirm that there is pronounced, thin-shell behaviour whenever mc R & 4. When n = −4, the condition for a body to have a thin-shell is therefore mc R & 4. This condition is shown to be a sufficient condition for thin-shell behaviour in chapter 4.3.3 above. It is clear that, far from the body, Φ is independent of both β and the mass of the body, M. There is a very weak, log-type, dependence on the radius of the body, R. The strongest parameter dependence is on λ. We can use the form of Φ given above to define an effective coupling, βef f . βef f is defined so that Φ ≈ Φb −
βef f (r/R)Me−mb r . 4πMpl r
It follows that, for r & 2R: βef f ≈
2πMpl p . M 2λ ln(min(r/R, 1/mb R))
Appendix D
Effective macroscopic theory
The analysis of the effective field theory for a macroscopic body proceeds along the same lines as the analysis that was performed to find the field about an isolated body in chapter 4.3, and appendices B and C. A detailed discussion of precisely what is meant by ‘effective’, or ‘averaged’, macroscopic theory is given chapter 4.4 above. For the purposes of this appendix, our aim is to find the average value of the chameleon mass inside a body in a thin-shell. We firstly consider what is required for linearisation of the field equations to be a good approximation. We then introduce a pseudo-linear approximation. Finally, we consider what we expect to see when non-linear effects are strong. We assume that the macroscopic body has a thin-shell, and is composed of spherical particles of mass mp and radius R. The average inter-particle separation is taken to be 2D. The average density of the macroscopic body is: εc = 3mp /4πD3 . The density of the particles is εp = 3mp /4πR3 . We label the average (i.e. volume averaged over a scale & D) value of Φ deep inside the body by < Φ >. We shall assume that the effect of the other particles, on a particle at r = 0, is sub-leading when r ¿ D. In general, the surfaces on which dΦ/dr = 0 will not be spherical, but their shape will depend on how the different particles are packed together. It will, however, make the calculation much simpler, and easier to follow, if we assume that the field about every particle is approximately spherically symmetric for 0 < r < D, and that at r = D, dΦ/dr = 0. We take everything to be approximately symmetric about r = D. We define Φ(r = D) = Φc ; mc = mΦ (Φc ). We argue below that < Φ >≈ Φc , and that the average chameleon mass is approximately equal to mc = mΦ (Φc ). Our aim therefore is to find Φc and mc . 245
246
APPENDIX D. EFFECTIVE MACROSCOPIC THEORY
We shall see below that it is possible, for all values of the parameters {D, R}, to construct an outer approximation that is valid near r = D. The outer approximation will be valid so long as (Φ − Φc )/Φc ¿ 1/|n + 1|. In the linear regime, this outer approximation will be valid everywhere. In the pseudo-linear and non-linear regimes, however, the outer approximation will only be valid for r > dmin where dmin =
m2c D3 . 3
When n 6= −4, we shall find that R ¿ D implies that we always have mc D ¿ 1. When √ n = −4, R ¿ D implies that mc R . 3 is always true. It follows that (dmin /D)3 ¿ 1. The volume in which the outer approximation holds is: Vout = 4π(D3 − d3min )/3. Since (dmin /D)3 ¿ 1, Vout ≈ 4πD3 /3 i.e. the entire volume of the region 0 < r < D. This means that the volume averaged value of Φ, and mΦ (Φ), will be dominated by the value Φ takes in the outer expansion. Since Φ ≈ Φc in the outer expansion, and mΦ ≈ mc , it follows that < Φ >≈ Φc and < mΦ >≈ mc . Throughout this appendix, we shall therefore refer to Φc as the average value of Φ, and mc as the average chameleon mass. The averaged behaviour of Φ, in a body with a thin-shell, is entirely determined by Φc and mc . Our aim of finding an effective macroscopic theory is therefore equivalent to calculating Φc and mc . (lin)
If linear theory holds inside the body then Φc = Φc µ Φc(lin)
=M
We also define:
µ Φp = M
D.1
βεc nλMpl M 3
βεp nλMpl M 3
where:
¶−1/(n+1) ,
¶−1/(n+1) .
Linear Regime
We write Φ = Φc + Φ1 , where |Φ1 /Φc | ¿ 1, and Φ(r = D) = Φc . Linearising the equations about Φ0 , one obtains: d2 (Φ1 /Φc ) 2 d(Φ1 /Φc ) m2c 3βMb + = − + m2c (Φ1 /Φc ) + H(R − r), dr2 r dr n+1 4πR3 Φc
(D.1)
247
D.1. LINEAR REGIME
and 3βMb m2c D3 = 4πR3 Φc (n + 1)R3
Ã
Φc (lin)
Φc
!n+1 .
For this linearisation of the potential to be valid we need, just as we did in chapter 4.3.1, |Φ1 /Φc | ¿ 1/|n + 1|. Solving these equations is straightforward and we find that for r > R: (n + 1)
cosh(mc (r − D))D sinh(mc (r − D)) Φ1 =1− − . Φc r mc r
Inside the particle, r < R, we have: Ã !n+1 µ ¶ D3 sinh(mc r)R Φ1 Φc = 1− 3 1− (n + 1) − (lin) Φc R sinh(mc R)r Φc (cosh(mc (R − D))mc D + sinh(mc (R − D))) sinh(mc r) . sinh(mc R)mc r
−
For dΦ/dr to be continuous at R = d, we need: Ã !n+1 Φc R3 mc D cosh(mc D) − sinh(mc D) . = (lin) D3 mc R cosh(mc R) − sinh(mc R) Φc
(D.2)
The largest value of |Φ1 /Φc | occurs when r = 0. For this linearisation to be valid everywhere we must therefore require: 1 − −
µ ¶ mc R mc D cosh(mc D) − sinh(mc D) 1− mc R cosh(mc R) − sinh(mc R) sinh(mc R) (cosh(mc (R − D))mc D + sinh(mc (R − D))) ¿ 1. sinh(mc R)
(D.3)
We shall see below that mc R ¿ 1. Given that mc R is small, the left hand side of the above condition becomes: · ¸ m3c D3 3(sinh(mc D) − mc D cosh(mc D)) + 1 + sinh(mc D)mc D − cosh(mc D) + O(mc R) 2mc R m3c D3 For this quantity to be small compared with 1, we need both mc D ¿ 1, which implies mc R ¿ 1, and m2c D3 /2R ¿ 1. For all n 6= −4 we define: Dc D∗
¶ n+2 n+4 3βmp , = (n(n + 1)) 4πMpl |n| µ ¶ n+1 µ ¶ n+2 3 3βmp n(n + 1) 3 = , MR 4πMpl |n| n+1 n+4
µ
(D.4) (D.5)
we note that D∗ /Dc = (Dc /R)(n+1)/3 . With these definitions the requirement that (mc D)2 ¿ 1 is equivalent to D À Dc . When n < −4 we need D ¿ D∗ for m2c D3 /R ¿ 1, whereas when
248
APPENDIX D. EFFECTIVE MACROSCOPIC THEORY
n > 0 we need D À D∗ for the same condition to hold. Therefore, when n > 0 we need D À Dc , D∗ , whereas for n < −4 we require Dc ¿ D ¿ D∗ . When n > 0, no matter what value R takes, there will always be some range of D for which the linear approximation holds. If n < −4, however, we must require that D∗ À Dc for there to exist any value of D for which the linear approximation is valid. It follows that, for the linear approximation to be valid for any D (when n < −4) we need R À Dc , i.e. mΦ (Φp )R ¿ 1. When n = −4 we need both D ¿ D∗ and: µ ¶ 3βmp 3/2 1/2 (12) λ ¿ 1. 4πnMpl This second condition implies both that mc D ¿ 1 and mΦ (Φp )R ¿ 1. We conclude that, for large enough particle separations, it is always possible to find some region where the linear approximation holds when n > 0. However, when n ≤ −4 we must also require that mΦ (Φp )R ¿ 1 for there to be any value of D for which the linear approximation is appropriate. Whenever the linear approximation holds, it follows from mc D ¿ 1 and eq. (D.2) that we must have: Φc ≈ Φc(lin)
D.2
⇔
mc ≈ mΦ (Φ(lin) ). c
Pseudo-Linear Regime
The pseudo-linear approximation proceeds in much the same way as it did in chapter 4.3.2, and appendix B, for an isolated body. Near each of the particles we can use the same inner approximation as was used for a single body. This is because, near any one particle, the other particles are sufficiently far away that their effect is very much sub-leading order. Inner and Outer Approximations The inner approximation (for n 6= −4) is therefore: 2 3 ¯ c = A − mc D Φ/Φc ∼ Φ/Φ 3(n + 1)r
Ã
Φc (lin)
Φc
!n+1 ,
where A is to be determined by the matching procedure. We will deal with the n = −4 case separately later.
249
D.2. PSEUDO-LINEAR REGIME
Previously, the outer approximation was defined so that it remained valid as r → ∞. In this model, however, we are assuming that everything is symmetric about r = D, and so we need only to require that the outer approximation remain valid up to r = D. Near r = D, we assume that Φ ≈ Φc and linearise about Φc . Requiring that dΦ/dr = 0 when r = D, we find: µ ¶ 1 cosh(mc (r − D))D sinh(mc (r − D)) Φ/Φc ∼ 1 + 1− − . n+1 r mc r The outer approximation, as defined above, is also good for n 6= −4. Matching For the pseudo-linear approximation to work we need, just as we did in appendix B, there to exist an intermediate region where both the inner and outer approximations are simultaneously valid. We will discuss what conditions this requirement imposes shortly, but, before we do, we shall assume that such a region does exist, and match the inner and outer approximations. Matching the inner and outer approximations we find: A = 1+ Ã
Φc
!n+1
(lin) Φc
=
1 (1 − cosh(mc D) + mc D sinh(mc D)) , 1+n
3(cosh(mc D)mc D − sinh(mc D)) , (mc D)3
(D.6) (D.7)
Conditions for matching For the outer approximation to remain valid in the intermediate region, where r ≈ d say, we require that, for all r ∈ (d, D): ¯ ¯ 2 3 ¯ ¯ ¯1 − cosh(mc D) + mc D sinh(mc D) − mc D 3(cosh(mc d)mc D − sinh(mc D)) ¯ ¿ 1. ¯ ¯ 3d (mc D)3 (lin)
This is equivalent to mc D ¿ 1 ⇒ Φc ≈ Φc
. We must also require
m2c D3 ¿ 1. 3d We define dmin to be the smallest value of d for which the above condition holds. ¯ to hold in the intermediate region, we require that For the inner approximation, Φ ∼ Φ, conditions, similar to those that were found in the isolated body case, hold (see chapter 4.3.2 and appendix B). Specifically, we require that for all r in (R, d) µ ¶n+1 Z d Z d Φp R3 0 00 00 À dr dr r . ¯ 00 ) 3 Φ(r R r0
(D.8)
250
APPENDIX D. EFFECTIVE MACROSCOPIC THEORY
(lin)
We also require that (Φc /Φ(r = 0))n+1 ¿ 1. We note that (Φc
/Φp )n+1 = εp /εc = D3 /R3 .
We consider the subcases n < −4, n > 0 and n = −4 separately.
D.3
Case n < −4
The analysis proceeds in the same way as it did for an isolated body in appendix B. We find that we must require µ m2c R2
=
Dc R
Ã
¶ n+4 n+1
µ
< 2|n + 1| 1 −
R D
¶3/|n+1| ! .
If we only wish for the pseudo-linear approximation to remain valid up to r = R, then we must ¯ = R)/Φ < 1. This is equivalent to require the weaker condition Φ(r à µ ¶ n+4 µ ¶3/|n+1| ! Dc (n+1) R 2 2 . mc R = < 3|n + 1| 1 − R D Provided that either of these conditions hold, the conditions for the outer approximation to be valid in the intermediate region are automatically satisfied. Whenever the first of the above requirements holds, the pseudo-linear approximation will give accurate results. When the latter (and weaker) of the two conditions fails, we expect pronounced non-linear behaviour near the particles. When this happens the far field induced by each particle becomes β-independent. We will discuss how this affects the values of Φc and mc in section D.7 below. (lin)
mc D ¿ 1 implies Φc ≈ Φc
via eq. D.7. It follows that mc ≈ mΦ (Φ(lin) ). The resulting
macroscopic theory, therefore, looks precisely like it did in the linear regime.
D.4
Case n > 0
The analysis for the n > 0 case proceeds in the much same way as it did for a single body (see appendix B). For the outer approximation to hold, we require that: D/Dc > 1, which implies mc D ¿ 1. We also need "
Ã
D∗ < 2(n + 1) 1 − D
µ
R D
¶
3 n+1
!# n+1 3 .
251
D.5. CASE N = −4
If we only wish to require that the pseudo-linear approximation hold up to r = R, then we can relax this second condition to: " Ã µ ¶ 3 !# n+1 3 D∗ R n+1 < 3(n + 1) 1 − . D D When the first of the two conditions holds, the pseudo-linear approximation gives accurate results. Whenever the second condition fails, we expect pronounced non-linear behaviour near the particles. When this happens, we expect that the far field (in r À R) will attain a critical form. We discuss the consequences of this in section D.7 below. As in the n < −4 case, when the pseudo-linear approximation holds we have mc D ¿ 1 and (lin)
so Φc ≈ Φc
D.5
. This implies mc ≈ mΦ (Φ(lin) ).
Case n = −4
The case of n = −4 is, as it was in the one particle case, the most complicated to study. However, the analysis proceeds almost entirely along the same lines as it did in the one particle case. We find, using the results of appendix B, that the field near the particles will behave like:
q
3 2y R Φ ∼A+ , Φc mc r
(D.9)
where y = y0 + ln(r/R), and, provided non-linear effects are small i.e. y0 & 0.6, we find y0 to be given by: (mΦ (Φp )R)3 = 9
r
3 1 + 2y0 3
µr
3 2y0
¶3 .
When n = −4, we have that mΦ (Φlin )D = mΦ (Φp )R. Using this, and matching the inner approximation to the outer one, we find: A = 1+ s 3 2(y0 + ln(D/R))
1 (1 − cosh(mc D) + mc D sinh(mc D)) , 1+n
= (cosh(mc D)mc D − sinh(mc D)) /3.
where y(D) = y0 + ln(D/R). Whenever non-linear effects are small we have y0 & 0.6, which implies r ¶3 µr 3 3 /3 ¿ 2y0 2y0
(D.10) (D.11)
252
APPENDIX D. EFFECTIVE MACROSCOPIC THEORY
and so
r q
Since y0 > 0.6 implies
3 2y0
(mΦ (Φp )R)2 (mΦ (Φ(lin) D))3 3 ≈ = . 2y0 9 9 < 1.6 and so it follows eq. (D.11) that: s
(mc D)3 ≈ (mΦ (Φ(lin) D))3
1 . 1 + ln(D/R)/y0
The above approximation is very accurate when it predicts mc D < 1, and even when mc D & 1 it gives a good estimate for mc D. For macroscopic, everyday, bodies with densities of the order of 1 − 10 g cm−3 , we tend to find ln(D/R) ≈ 11. If linear theory is to give a good estimate of mc , we need: ln(D/R)/y0 ¿ 1 ⇒ y0 À 11 ⇒ mc (Φ(lin) ) ¿ 1.5. More generally, linear theory gives a good approximation to mc whenever: 2(mΦ (Φ(lin) )D)6 ln(D/R)/243 ¿ 1. (lin)
Therefore, unless D/R is improbably large (D & Re243/2 ), we will have we will have mc ≈ mc (lin)
whenever mc
D . 1.
When mc (Φ(lin) ) & 1.5, we actually move into the a regime of β-independent, critical behaviour, more about which shall be said below.
D.6
Summary
When the pseudo-linear approximation holds (and mΦ (Φ(lin) )D . 1), we have found that mc ≈ mΦ (Φlin ). As a result, despite the fact that there does not exist an everywhere valid linearisation of the field equations, linear theory actually gives the correct value of mc , at least to a good approximation.
D.7
Non-linear Regime
When the pseudo-linear approximation fails it is because non-linear effects have become important near the surface of the particles, and they have developed thin-shells of their own. Far from the particles we expect that the field will take its critical form as given by eqs. (4.24a-c).
253
D.7. NON-LINEAR REGIME
We can find the value of Φc , in this case, by matching our the outer approximation for Φ to the critical form of the far field around the particles. When n < −4 we will have critical behaviour whenever: Ã µ ¶ n+4 µ ¶3/|n+1| ! (n+1) D R c m2Φ (Φp )R2 = > 3|n + 1| 1 − . R D When this happens the far-field (r À R, r ¿ D) behaviour of the chameleon takes its critical form. We have found this to be well approximated by: µ ¯ = AΦc − Φ∼Φ
γ(n) |n|
¶1/|n+2|
n+4
(M R) n+1
1 . r
Performing the matching to the outer approximation we find that: 1 (1 − cosh(mc D) + mc D sinh(mc D)) , 3 Φc (cosh(mc D)mc D − sinh(mc D)) (n + 1)mc µ ¶ n(M D)3 M n+1 3 (cosh(mc D)mc D − sinh(mc D)) 3 Φc (mc D)3 µ ¶ n+1 n(M D)3 M . 3 Φc
A = 1− µ
γ(n) |n|
¶1/|n+2|
n+4
(M R) n+2
= = ≈
We therefore have that: m(crit) c
p µ ¶ n+4 µ ¶ 1 3|n + 1| R 2(n+1) γ(n) 2|n+1| . ≈ D D 3
When n > 0, a similar analysis finds that " Ã !# n+1 3 3 R n+1 D∗ > 3(n + 1) 1 − , D D and that n(M D)3 3
µ
M Φc
¶n+1
We therefore find m(crit) c
µ ≈ MR
n(n + 1)M 2 m2c
¶1/(n+2) .
p µ ¶ 3|n + 1| R 1/2 ≈ . D D
In the n = −4 case, critical behaviour will actually emerge whenever ln(D/R)/y0 & 1. Non-linear effects are still responsible for this critical behaviour but it is not necessarily the case that the particle have developed thin-shells. Indeed, the thin-shell condition requires that
254
APPENDIX D. EFFECTIVE MACROSCOPIC THEORY
mΦ (Φp )R & 4, whereas critical behaviour (for ln(D/R) ≈ 11 as is typical), emerges whenever mΦ (Φp )R & 1.4. When the particles do have thin-shells, y0 = 1/4 (see appendix C) and so critical behaviour is seen whenever D & 1.3R. We find the β-independent critical mass for n = −4 theories using eq. (D.11) and taking y0 ¿ ln(D/R). It follows that, the critical value of mc for n = −4 theories is given by: m(crit) = X/D, c where X satisfies
√ 3 3
X cosh X − sinh X ≈ p . 2 ln(r/R) When ln(D/R) = 11, we find mcrit c D ≈ 1.4.
D.8
Summary
In this appendix, we have performed a very detailed analysis of the way in which the chameleon behaves inside a body that has a thin-shell and which is made up of many small particles. In this way, we have been able to calculate the average chameleon mass inside such a body. Despite the in depth nature of the analysis, our results can be summarised in a very succinct fashion. The average mass, mc , of the chameleon field inside a thin-shelled body of average density εc is mc = min
à p
µ n(n + 1)λM
βεc λ|n|Mpl M 3
¶
n+2 2(n+1)
! , m(crit) (R, D, n) , c
where R is the radius of the particles that make up the body, and 2D is the average inter(crit)
particle separation. The critical mass, mc , is given by: p µ ¶ q(n) 3|n + 1| R 2 (crit) mc ≈ S(n), D D m(crit) ≈ X/D, c
n 6= −4,
n = −4,
where q(n) = min((n + 4)/(n + 1), 1), S(n) = 1 if n > 0, S(n) = (γ(n)/3)1/2|n+1| if n < 0, and √ 3 3 X cosh X − sinh X ≈ p . 2 ln(D/R) (crit)
The dependence of mc implies
(crit) mc D
D vs. ln(D/R) is shown in FIG. 4.4. We can clearly see that D À R √ (crit) ¿ 1 when n 6= −4. When n = −4, mc can be seen to be less than 3
whenever ln(D/R) & 2.
Appendix E
Evaluation of αΦ for white-dwarfs
In this appendix, we evaluate αΦ for white-dwarfs under a more accurate approximation than that used in chapter 5.3. This accurate evaluation allows for the effect of a chameleon on the general relativistic stability of white dwarfs to be studied in more detail. The effect of the chameleon on compact objects such as these is considered in chapter 5.3. The chameleon contribution to the energy of the white-dwarf was found, in chapter 5.3, to be: n+1 WΦ = n
Z d3 x
βΦ ε. Mpl
For the chameleon theory to be valid, we must require that the chameleon force is weak compared to gravity inside the white dwarf. We shall therefore ignore the chameleon corrections in the behaviour of ε when evaluating WΦ , because they must be sub-leading order. In this way, we are able to accurately find the leading order behaviour of WΦ . By leading order we mean: ignoring general relativistic and chameleon corrections. It is important to stress that we are not ignoring special relativistic corrections, which are very important; we are merely assuming that the only volume force inside the white dwarf is, to leading order, Newtonian gravity. This approximation is similar to the one used in [100] to evaluate the general relativistic corrections to the energy of the white dwarf. We shall assume that the polytropic equation of state, P = KεΓ , holds everywhere. We will also approximate the white dwarf to be spherically symmetric. Defining ε = εc θp and r = aξ, hydrostatic equilibrium implies that, to leading order, θ satisfies the Lane-Emden equation 255
256
APPENDIX E. EVALUATION OF αΦ FOR WHITE-DWARFS
Table E.1: Values of αΦ for different n n
αΦ
n
αΦ
-12
0.8493
4
1.523
-10
0.8207
5
1.410
-8
0.7786
6
1.337
-6
0.7110
7
1.286
-4
0.5853
8
1.249
1
3.613
9
1.220
2
2.144
10
1.197
3
1.720
11
1.178
[100]: 1 d 2 dθ ξ = −θp , ξ dξ ξ where
" a=
(1/p−1) #1/2
2ε (p + 1)KMpl c
.
4π
We define εc to be the density of matter in the centre of the star. The boundary conditions θ(0) = 1, θ0 (0) = 1 follow. The index p is related to Γ by Γ ≡ 1 + 1/p. This equation must be solved numerically. The surface of the star is at r = R = aξ1 , which is defined to be the point where ε = 0. We will mostly be interested in the case of relativistic matter, Γ = 4/3 → p = 3. For p = 3 we have ξ1 = 6.89685. The chameleon potential is then given by: n + 1 βΦ(εc ) 3 WΦ = a εc n Mpl
Z
ξ1
pn
dξξ 2 θ n+1 .
0
We evaluate this integral numerically assuming a relativistic equation of state (i.e p = 3) and find: WΦ =
n + 1 βΦ(εc ) αΦ Mstar n Mpl
where Mstar = mu N is the mass of the star. The values of αΦ are given in table E.1. As n → ±∞ we find αΦ → 1.
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