From the Science Case of LISA to the LISA Pathfinder Experiment Master Plan The LISA Pathfinder Science & Technology Operations Centre 1 1 1 2 1 M. Armano , J. Fauste , M. Freschi , P. McNamara , D. Texier on behalf of the LPF Scientific Community and ESA Project 1 ESA/ESAC, Camino bajo del Castillo s/n, Urb. Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain; 2 ESA/ESTEC, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands.
[email protected]
Writing control loops explicitly we get the fundamental science signals equations: 2 2 2 2 s + ω1 + Hdf oSC = Gn − gn1 + s + ω1 onSC , 2 2 s + ω2 + hlfs o∆ = gn2 − gn1 (13) 2 2 2 2 + ω2 − ω1 oSC − onSC + s + ω2 on∆ .
1. Introduction GWs are weak wave-like perturbations on the metric background ( gµν,BG ' ηµν ) [6, 8]: gµν = ηµν + hµν ,
hµν = 0 .
(1)
The differential sensing signal in LPF (second line of (13)) is a close representation of the LISA case but in LPF the term ω22 − ω21 oSC − onSC stems from the SC-TM1 jitter. The dynamics of the local TM-SC link for LPF is nominally identical to that of LISA.
Choosing the TT gauge: ηi j hi j = hii = 0 ,
hµ0 = 0 ,
Γi
hi j; j ' hi j, j = 0 ,
i = 1h i , 0 = Γ0 = 0 , Γ0 = − 1 h = Γ , Γ 00 0j 00 0j jk 2 jk,0 2 j ,0
(2)
bodies in free-fall act like good event markers as, from the freeµ fall equation ( x¨ µ + Γ αβ x α x β = 0, x˙ µ ≡ dx µ/dτ, τ being the proper time) coordinates x µ don’t change in time: d2 x i i k j i i k 0 v v v ' 0, (3) = −2Γ − Γ v + Γ 0j jk jk 2 dt if the speed spectrum (S stands for Power Spectral Density, PSD) of the observers is kept low: S ∆v = c
1
(4)
S∆a ' 0. 2 2 ω c
A solution of (1) is in plane waves: h(r, t) ' Ah(r) cos 2ω
r c
−t ,
(5)
From “standard” sources we get GW strain |h| ∼ 10−22 − 10−20 and GW frequency f ∼ 10−4 − 10−1 mHz. LISA will experimentally detect low-frequency GWs: each spacecraft (SC) in LISA will host 2 test masses (TM’s), and the sensing interferometric arm will be created shining a laser beam along the L = 5×106 km between TM’s residing in different SC. LISA Pathfinder[1, 5, 7] (LPF) is an ESA mission to demonstrate geodesy and technology for LISA. In LPF only one arm is left into a single SC, the distance between TM’s being shrinked to 38 cm (see figure 2). The mission aims at: 1. putting Test Masses (TM’s) in free-fall as observers; 2. shielding them from local environmental noise; 3. tracking their mutual position; 4. minimizing the experimental disturbances; 5. testing key technology for LISA.
2. Sensitivity The relative frequency shift between emitter and receiver due to GW’s is, from GR [8, 3]: S ∆ωGW ω
ω2 L 2 ' 4c 2 Sh ,
(6)
4. Experiment approach and masterplan
Figure 1: Comparison of acceleration sensitivity curves for various space science missions. LISA and LPF differ in three main effects: 1. LPF is o(1 m) size: the acceleration and displacement noise upon the two TM’s will be correlated; 2. chasing two TM’s along a collinear sensing line within the same SC is impossible: one TM is electrostatically suspended; 3. sensitivity to gravitational waves strain scales like δL/L , thus LTP will be GW-insensitive. LTP will then be a local acceleration difference detector, not the first in the timeline, but the most precise insofar! (See figure 1 and table 1). GRACE Accelerometer (< 100 g)
TM’s
Tracking
Microscope Differential accelerometer (< 0.5 kg) Capacitive TM-SC
GOCE Accelerometer (320 g)
LISA Pathfinder Gravity Reference Sensor (2 kg)
Capacitive TM-SC
∼ 10−10
Low Earth, drag-free 2 × 10−12
Low Earth, drag-free 3 × 10−12
Differential TM-TM interferometry Interplanetary (L1), drag-free 3 × 10−14
No
3 × 10−10
3 × 10−8
3 × 10−13
Radio-link + capacitive Low Earth
Orbit TM’s geodesic motion performance p (m/s2 Hz at 1 mHz) Residual SC acceleration p (m/s2 Hz at 1 mHz)
Reducing and subtracting the different noise contribution to demonstrate (10) is the key strategy of the LPF mission: 1. the experimental hardware is designed to minimize noise excess; 2. the measurement procedures will explore the noise contributions attributed to different sources. The first goal is to measure the PSD of the total ∆F/m noise on the TM’s, and to put an upper limit to any unexplained source of noise. The experiment will prove the physical model to be used to calculate and confirm LISA sensitivity. Internal noise characteristics will be at a level comparable to that of LISA but LPF’s test tolerance towards force-per-unit-mass gradient is raised to 2 × 10−6 s−2, one order of magnitude worse than LISA’s (4 × 10−7 s−2) on account of extra suspension loop stiffness and static gravity gradients. The noise level in (10) is just the starting point of a detailed set of experimental procedures [1, 4] to quantify the leading noise contributions. Noise is estimated with the method of System Identification by orthogonalizing the system (subtraction of readout cross-talk) and injecting specific stimuli (bias voltages, extra electrode potentials, heating, magnetic fields) and reading the effect on the ∆a PSD (see figure 3 for a simile with the ground-based torsion pendulum facility).
Table 1: Space experiments comparison: geodesy, tracking and acceleration performances.
3. LISA Pathfinder and LISA as dynamical systems
h ' ∆L/L being the GW strain, ∆L the interferometer arm path variation in time. Three categories of disturbances affect the ideal measurement by causing time-variations of the frequency mimicking S∆ωGW/ω in(6). 1. relative motion of emitter and receiver due to nongravitational forces: S∆a,fcs ' S ∆F ' ω2 c 2S ∆v ,
(7)
c
m
2. readout noise: S∆a,rdo ' ω4S∆x ,
(8)
3. departure of the actual LISA scheme from the ideal 5 million kilometre link, point-to-point measurement: each link measures the relative acceleration of two somewhat arbitrary points on the two linked extended bodies. Summing up (no correlations): S ∆ω ' ω
1
2
S∆a,rdo + 2 2 S∆a,fcs + S ∆ωGW 2 2 ω c ω c ω
2 ω L = ω S + S + S ∆F c 2 ∆x ω2 c 2 m 4c 2 h 2
2 2
(9)
The LISA technology Package (LTP) experiment on LPF aims at delivering an experimentally tested physical model for all three categories of disturbances above in a space environment similar to that of LISA, with an accuracy sufficient to quantitatively predict LISA performance. At test is the ∆a sensitivity threshold ( f ≥ 0.1 mHz) built from (9): Ç 1/2
S∆a,LPF
≤ 3 × 10−14
1+
4 p f 2 m/s Hz , 3 mHz
(10)
given the laser metrology sensitivity (first time at test in space): Ç 1/2
S∆x,LPF
≤ 9 × 10−12
1+
−4 p f m/ Hz . 3 mHz
8th Edoardo Amaldi Conference on Gravitational Waves, New York, USA, June 2009
(11)
Figure 2: Schematic (left) and artist’s impression (right) of the experiment. The TM’s (yellow and green cubes) are hosted by the SC. The (phase) signal oSC of an interferometer reads out the SC-TM1 displacement x SC along the sensitive axis x with noise onSC. The signal o∆ of a second interferometer reads out the TM2-TM1 displacement ∆x with noise on∆. Forces are applied to both TM’s via the GRS (green and yellow plates), and to the SC through the micro-thrusters. A drag-free control loop, of gain Hdf driven by oSC and acting on thrusters, forces the spacecraft to follow TM1. A suspension loop of gain hlfs, driven by o∆, applies an electrostatic force on TM2, forcing it to follow again TM1. Bias signals oiSC and oi∆ can be injected in both loops. Command forces (per unit mass of each TM) gi1, gi2 and Gi can directly be applied to TM’s and SC respectively. Noisy accelerations gn1, gn2 and Gn contribute to total accelerations g1, g2 and G on TM1, TM2 and SC. The linearized, Newtonian equations for the dynamics for the LTP in Laplace angular frequency s along the x axis are: 2 2 x SC s + ω1 = G − g1 , 2 2 2 2 ∆x s + ω2 − x SC ω2 − ω1 = g2 − g1 .
(12)
The forces gradients on the TM’s have been linearized as ω21 x SC 2 for TM1 and −ω2 ∆x − x SC for TM2 (the negative “stiffness” coefficients ω21 and ω22 will be dominated by the electric and gravitational field gradients). In (12) terms proportional to the mass ratio between TM’s and SC and to the direct gravitational coupling between the TM’s were neglected.
Figure 3: The noise projection paradigm from torsion pendulum reference experiments [?, ?]: (top) subtraction of readout cross-talk from residual torque acceleration PSD, subtraction of magnetic and thermal contributions, (bottom) comparison with model.
References [1] M Armano, M Benedetti, J Bogenstahl, et al. Classical and Quantum Gravity, 26(9):094001 (18pp), 2009. [2] D Bortoluzzi, L Baglivo, M Benedetti, et al. Classical and Quantum Gravity, 26(9):094011 (11pp), 2009. [3] A Cavalleri, G Ciani, R Dolesi, et al. Classical and Quantum Gravity, 26(9):094012 (10pp), 2009. [4] B. Chauvineau, T. Regimbau, J.Y. Vinet, et al. Phys. Rev. D, 72(12):122003, 2005. doi:10.1103/PhysRevD.72.122003. [5] K. Danzmann, C. Dunn, O. Jennrich, et al. ESA-SCI, 1, 2007. [6] P. McNamara, S. Vitale, and K. Danzmann. Class. Quant. Grav., 25(11):114034 (8pp), 2008. [7] C.W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. W.H. Freeman and Company, San Francisco, 1973. [8] G. Racca and P. McNamara. Space Sci. Rev., 2008 (submitted to). [9] J. L. Synge. Relativity : the General Theory. North-Holland, 1964.
