PHYSICAL REVIEW D 77, 063523 (2008)

Noncommutative geometry modified non-Gaussianities of cosmological perturbation Kejie Fang,1 Bin Chen,1,2 and Wei Xue1 1

Department of Physics, Peking University, Beijing 100871, People’s Republic of China The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy (Received 6 September 2007; published 20 March 2008)

2

We investigate the noncommutative effect on the non-Gaussianities of primordial cosmological perturbation. In the lowest order of string length and slow-roll parameter, we find that in the models with small speed of sound the noncommutative modifications could be observable if assuming a relatively low string scale. In particular, the dominant modification of the non-Gaussianity estimator fNL could reach O
1 in Dirac-Born-Infeld (DBI) inflation and K-inflation. The corrections are sensitive to the speed of sound and the choice of string length scale. Moreover the shapes of the corrected non-Gaussianities are distinct from that of ordinary ones. DOI: 10.1103/PhysRevD.77.063523

PACS numbers: 98.80.Cq, 98.70.Vc

I. INTRODUCTION Inflation [1] is a very successful paradigm of the very early universe. It can naturally solve several very tough cosmological problems without fine-tuning. Furthermore, it predicts a nearly scale invariant Gaussian cosmic microwave background (CMB) spectrum, which has been confirmed in the experiments [2]. However, one of the problems with inflation is that there are too many inflationary models. It is necessary to find the signatures which could distinct various models. With the development of precise cosmology, we expect new experiments to constrain a large amount of inflation models and to make the paradigm more clear. The scalar spectral index and its running, the gravitational wave, and the non-Gaussian component of the primordial fluctuations are among the important probes to detect different inflation models. In this paper, we mainly discuss non-Gaussianity. In recent years, the non-Gaussianities of primordial perturbation open another window other than power spectrum to study different scenarios of inflation models. After a systematic analysis [3–6], it is found that the nonGaussianity estimator fNL is of order O

 in the ordinary ‘‘slow-roll’’ inflation and thus is unobservable. In some particular models with nontrivial dynamics, such as ghost inflation [7] and those with sufficiently small speed of sound, including the Dirac-Born-Infeld (DBI) model [8,9] and K-inflation [10,11], the unsuppressed nonGaussianity is potentially detectable in future experiments [9,12]. Furthermore, since the shape of non-Gaussianity is more multiple than that of power spectrum, it could be used to distinguish different inflation models through the classification of those configuration of k modes that determine the maximum of the three-point function [13]. Non-Gaussianity could also be used to study various trans-Planckian physics proposals. Deviation from the standard Bunch-Davies vacuum was considered in [9,12] where the correction is found to be O
H
, where
is the energy scale on which the modes are generated. In [14], higher dimensional operators were introduced in the infla-

1550-7998= 2008=77(6)=063523(12)

ton Lagrangian. This modification could enhance the nonGaussian effect if the energy cutoff is not too high; but it is difficult to exceed fNL  1. In this paper, we consider another trans-Planckian scenario which is based on noncommutative geometry. In string theory, a promising candidate of quantum gravity, one may use a perturbative string or nonperturbative object D-brane to probe the space-time geometry. Because of the extensive nature of the string, or stringy effect, or strong string interaction, the picture of spacetime geometry in string theory could be very different from the usual one. Especially, very near the cosmological singularity, the usual concept of commutative geometry may break down completely. A better description could be noncommutative geometry [15], in terms of the algebra generated by noncommutative coordinates x ; x   i :

(1.1)

One natural way to get noncommutative geometry is to consider the D-brane in the presence of the constant NSNS magnetic B field [16]. In this case, the spatial coordinates are noncommutative xi ; xj   iij ;

(1.2)

where ij depends on the background flux, while spacetime is commutative. Similarly one can obtain space-time noncommutativity by placing the D-brane in a constant electric field, but the theory is no more unitary in this case [17–21]. Another way to realize the noncommutative relation among coordinates, especially between space and time, is to start from the stringy uncertainty relation xp t  l2s :

(1.3)

Equivalently one may assume t; xp   il2s . This kind of noncommutative relation has been applied to the study of the inflationary universe [22 –24]. In [22], Brandenberger and Ho started from (1.3) and discussed the cosmological implications of such a relation. To keep the background

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© 2008 The American Physical Society

KEJIE FANG, BIN CHEN, AND WEI XUE

PHYSICAL REVIEW D 77, 063523 (2008)

isometry intact, they actually considered the noncommutative relation between radial coordinate r and time t, however, as we mentioned before, the space-time noncommutative relation may violate the unitarity of the theory. In this paper, in order to keep unitarity, we choose to set the space-time components 0i zero. The cosmological imprint based on this kind of noncommutativity has been studied by many authors [25–29]. Meanwhile, without losing generality, we choose a particular frame in which the only nonvanishing space-space component of  in a comoving coordinate is x1 ; x2   i12 :

(1.4)

We find the calculation is much more involved if one chooses to keep all possible spatial noncommutativity but this will not change the main result we obtained from this particular frame. We also require that noncommutativity following from (1.4) only dominates at the small length scale. A simple form of 12 realizes this condition is [29] 12

l2  s2 ; a

(1.5)

where ls is the string length. In this case, the noncommutativity in the physical coordinate remains to be a constant, x1p ; x2p   il2s , and the noncommutativity of the comoving coordinate is diluted by the expansion of the universe. However, with this choice, the isotropy of the space-time is now broken. Nevertheless, if the noncommutativity is very small, the breaking of isotropy could be ignorable but the physical implication could be observable. In this paper, we will focus on this case. In most of the inflation models, the non-Gaussianity is quite small and hard to be detected. The noncommutative corrections of non-Gaussianity is even smaller. This drives us to work on the models with big non-Gaussianity. We will discuss the noncommutative effect in the stringinspired DBI inflation model and k-inflation model. We find that in both models the noncommutative modification of the non-Gaussianity estimator can reach O
1 or even larger with a small speed of sound and a relatively low string scale. We also determine the shape of corrected nonGaussianity, which is different from the usual ones. Though the non-Gaussianity estimator could be overshadowed, the shape is a distinctive signature to be probed by the future experiments. On the contrary, future experiments could set a bound of the noncommutativity and be used for cross-checking along with other experiments, like atomic experiments. The paper is organized as follows. In Sec. II, we give a brief review of the Arnowitt-Deser-Misner (ADM) formalism to discuss the perturbations in a general inflation Lagrangian. In Sec. III, we study the noncommutative modifications to general inflation models. In Sec. IV, we focus on two inflation models with observable non-

Gaussianity. In Sec. V, we end with a conclusion and discussions. II. PERTURBATIONS IN GENERAL INFLATION MODELS Let us start with a general Lagrangian P
X;  which can be used to describe a broad class of inflation models. The action is of the form as follows: S

1 Z 4 p






d x gR 2P
X; ; 2

(2.1)

where  is the inflaton field and X   12 g @ @ . We set the Planck mass Mpl 
8G
1=2  1 and adopt the metric signature
1; 1; 1; 1. In order to proceed, it is convenient to work in the ADM metric formalism, d2 s  N 2 d2 t hij
dxi N i dt
dxj N j dt;

(2.2)

where hij  a2
1 2ij , with a the scale factor which grows quasiexponentially during inflation and  the scalar perturbation of the metric. In this paper, we do not consider the tensor perturbation. Since in the ADM formalism N and N i are Lagrangian multipliers, the metric in terms of  can be determined by solving the constraint equations of N and N i in a given gauge, rather by solving the Einstein equation, which makes the calculation relatively simpler. There are two main gauges in which we can calculate the perturbation action. One gauge is the comoving gauge, that is   0. In this gauge,  is the curvature scalar on a comoving hypersurface and it directly seeds on the later generation of large scale structure and anisotropy of microwave background radiation. It is straightforward to find out that  remains constant after the horizon exit. Although the physical meaning is manifest within this gauge, it is rather complicated to analyze the order of the perturbation action with respect to the slow-roll parameters, which is important in determining the magnitude of correlation functions. In this gauge, the third order action of perturbation is apparently of order O

0 , where
represents the slow-roll parameter; however after doing a lot of integration by part the action is actually of order O

2  [3,12]. The other gauge with   0 is called uniform density gauge. The gauge transformation linking the two gauges is   H , where H is the Hubble parameter H  aa_ . One could _ easily recover the exact order O

2  of the action by doing a gauge transformation of the perturbation action of  [3]. In order to calculate the correlation function of metric perturbation in this gauge, we first calculate the correlation function of  and then transform it to  just after the horizon exit, which remains constant ever since. Below we adopt the uniform density gauge   0 to carry out the calculation. Substituting the ADM metric into the action (2.1), we get

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NONCOMMUTATIVE GEOMETRY MODIFIED NON- . . .

p


1Z dtd3 x hN
R
3 2P S 2 p


1Z dtd3 x hN 1
Eij Eij  E2 ; 2

PHYSICAL REVIEW D 77, 063523 (2008)

c2s  (2.3)

where h  det
hij , Eij  12
h_ ij  ri Nj  rj Ni , E  Eii . R
3 is the Ricci scalar calculated in the three-dimensional hypersurface with metric hij , and ri is the covariant differential coefficient defined on the hypersurface. Since N and N i are Lagrangian multipliers, we can obtain two constraint equations from them, which are _ i  N i N i @i @j  R
3 2P  2N 2 P;X
_ 2  2N i @  N 2
Eij Eij  E2   0; (2.4)   2N j 2 _ NP;X 2N @  2 @i @j  2
rj
N 1 Eji  N  ri
N 1 E  0: (2.5) We divide  into the isotropic background 0
t and the fluctuation ’,   0 ’. In order to evaluate the third order action of perturbation, we only need to solve the equations of N and N i to the first order of ’, since the second and third order of solutions, when substituted into the action, will be multiplied with the first and zeroth order Hamiltonian constraint of the action, respectively, and thus vanish. In fact, to calculate the nth order action of perturbation, one only needs the solutions of N and N i to
n  2th order [12]. Following [3], we decompose N i into two ~ i @i , where @i N~ i  0 and Ni is lowered parts Ni  N by hij through N i . Then we expand them to the first order of ’, N~ i  Ni
1 ;

N  1 1 ;



1;

(2.6)

where 1 , Ni
1 , and 1 are of order O
’. Substituting (2.6) into (2.4) and (2.5), and solving them to O
’, we obtain the solution 1 

@2

1



P;X0 _ 0 ’; 2H

Ni
1  0;

(2.7)

1

2P;0 _ 30 H 1 P2;X0  2P;X0 0 _ 20 4H _ 50 H 1 P;X0 P;X0 X0 ’  6HP;X0 _ 0 ’ _ 
2P;X0 _ 0 2P;X0 X0 _ 0 ’;

(2.8)

where the subindex 0 represents the background value. It will be more succinct to express the solutions using the slow-roll parameter
and the ‘‘speed of sound’’ cs ,


X0 P;X0 H_  ; 2 H 2 H 2 Mpl

(2.9)

P;X0 dP  ; dE P;X0 2X0 P;X0 X0

(2.10)

where E  2XP;X  P is the energy of the inflaton field. Keeping with the lowest order of slow-roll parameter, Eqs. (2.7) and (2.8) can be written as 1 

@2

1



H
’; _ 0

  P;0 3H2 H

’ ’: _ 2H _ 0 _ 0 c2s

(2.11)

(2.12)

We omit the subindex 1 in these expressions in the following section for simplicity. Substituting (2.11) and (2.12) into (2.3) and expanding it to the second order of perturbation, we attain the free field action of fluctuation   1 Z 4 3 P;X0 2 2 d xa S2  ’ _  P
@’ : (2.13) ;X0 2 c2s The equation of motion is   a00 a’  0;
a’00 c2s k2  a

(2.14)

where the prime denotes derivative with respect to conformal time and we have assumed P;X0 and cs to be time independent. The classical solution is _ 0
1 ics k eics k ; ’  p













3 4
cs k

(2.15)

where we choose the standard Bunch-Davies vacuum to fix the coefficient. With the same procedure one could obtain the third order action of perturbation which has been studied thoroughly in [12] in the comoving gauge. III. NONCOMMUTATIVE MODIFICATION In the noncommutative space-time, the functions are better described by the operators in Hilbert space. This is very similar to the case in quantum mechanics, where one has a noncommutative phase space. The product of functions may be taken as the multiplication of the operators. But an efficient way to define the product is by the socalled Moyal product, whose expansion in curved spacetime gives [29]   1 X 1 i k 1 1 f?g

    k k
D1    Dk f k0 k! 2
D1    Dk g;

(3.1)

where D is the covariant derivative coefficient in the curved space-time. The star product could be extended to a multiple function situation, at quadratic order in  ,

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KEJIE FANG, BIN CHEN, AND WEI XUE

 X i 1 f1 ?    ? fn  1  Da Db    2 8 a
PHYSICAL REVIEW D 77, 063523 (2008)

(3.2)

a
To incorporate the noncommutative effect, we replace the ordinary product in the inflaton Lagrangian P
X;  with the star product. Suppose that P
X;  relies on  through function V
,1 which could be the inflaton potential or the warping factor in the DBI model. Without losing generality, we simply consider V
 of the form V
  n , n  1, since one could always do Taylor’s expansion of a generic function. When we apply the star product in the generic Lagrangian, noncommutativity changes the forms of the dynamic term X   12 @ @  and scalar function V
 as follows:

One should keep in mind that all the covariant derivatives are calculated within the ADM formalism, which makes the evaluation much more involved. In some other papers in calculating large unsuppressed non-Gaussianity [7,8,14], because of the particularity of the actions (they could themselves, by doing serial expansion, generate cubic terms of perturbation without involving metric corrections, i.e. , ), the authors ignore the correlation between the metric correction and inflaton perturbation in the uniform density gauge, which is subleading in the slowroll parameter, and do the calculation with the isotropic Friedmann-Robertson-Walker (FRW) metric. Our calculation turns out to be the same situation when considering the noncommutativity correction. The change of the inflation action is

X  12
@  ?
@ 

 S 

Z

p






d4 x g
P;X  X P;V  V:

(3.7)

 12
@ 
@  18
12 2 g
D1 D1 D D2 D2 D   D1 D2 D D2 D1 D  O
3 ;

(3.3)

and V   ?  ?  n
n  1 12 2 n2
  
D1 D1 D2 D2  8 n
n  1
n  2  D1 D2 D2 D1   24
12 2 n3
D1 D1 D2 D2  D2 D2 D1 D1 

 n 

 D1 D2 D1 D2   D2 D1 D2 D1  O
3 :

To obtain the third order action of perturbation, one needs to serially expand P;X and P;V around the background value and multiply with the corresponding terms in  X and  V which generate overall cubic terms of perturbation. To simplify the calculation, we need to pick out the terms of leading order of the slow-roll parameter. We find in this case, the leading order is O

 for  after gauge transformation. We decompose  X into terms of a different order of perturbation,

(3.4) We find that the lowest order noncommutative modification term is of order O
2 . We only consider the correction of the lowest order 2 in this paper, and denote them as  X and  V, respectively,  X  18
12 2 g
D1 D1 D D2 D2 D   D1 D2 D D2 D1 D ;

(3.5)

n
n  1 12 2 n2
  
D1 D1 D2 D2  8 n
n  1
n  2  D1 D2 D2 D1   24 12 2 n3
  
D1 D1 D2 D2 

 V  

D2 D2 D1 D1   D1 D2 D1 D2   D2 D1 D2 D1 : 1

(3.6)

If there are more than one such functions, one just needs to include all the corrections from each function in (3.14).

 X 
 X3
 X2
 X1
 X0 ;

(3.8)

where
 X3 represents the cubic terms, etc., and we do not consider the higher order of perturbation since we are only going to calculate the three-point function. The terms of lowest order of the slow-roll parameter in
 X3 are those composed of the product of two inflaton perturbation ’ and one metric correction, i.e. and . However, we do not need to take them into account to obtain the final third order action of perturbation by the reason that they will generate terms of second order in the slow-roll parameter after doing the gauge transformation to , which turns out to be the subleading terms in our calculation. As for
 X0 which is composed of terms of _ 0 , it also results in subleading order terms when multiplied with the serially expanding terms of P;X . In short, we only need
 X2 and
 X1 . The terms of lowest order of the slow-roll parameter in
 X2 are the products of two inflaton perturbation, which are summarized as follows:

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PHYSICAL REVIEW D 77, 063523 (2008)



1 1 1
 X2 
12 2 2 @21 @i ’@22 @i ’  2 @1 @2 @i ’@1 @2 @i ’  H@21 @i ’@i ’_  H@22 @i ’@i ’_ H 2 @21 @i ’@i ’ H 2 @22 @i ’@i ’ 8 a a 2 2 _ 2 @1 ’@1 ’ a_ @1 ’@ _ 1 ’_ a_ @2 ’@ _ 2 ’_ a_ 2 @i ’@ _ i ’_  2Ha_ 2 @1 ’@ _ 1 ’  2Ha_ 2 @2 ’@ _ 2 ’  2H a_ 2 @i ’@ _ i ’
H a _ 2 @2 ’@2 ’
Ha _ 2 @i ’@i ’  @21 ’@ _ 2 ’_ 2  Haa_ ’_ @21 ’_ Haa_ ’_ @22 ’_
H a _ 22 ’_ @1 @2 ’@ _ 1 @2 ’_ 
Haa 2H@21 ’@ _ 22 ’ 2H@21 ’@22 ’_ 2H 2 aa_ ’_ @21 ’ 2H 2 aa_ ’_ @22 ’  4H@1 @2 ’@ _ 1 @2 ’  4H 2 @21 ’@22 ’  2 2 _ 21 ’_ ’ aa@ _ 22 ’_ ’ a2 a_ 2 ’ ’ a_ 2 ’
2@ _ 4H 2 @1 @2 ’@1 @2 ’ aa@  ’ 2@ ’  a a ’ _ ; 1 1

where i should be summed from 1 to 3. And
 X1 is

 X0 X0 X0 P;X0 X0 X0 :


 X1  18
12 2 Haa_ _ 0
@21 ’_ @22 ’_  2H@21 ’ 

2H@22 ’

2Haa_ ’: _

(3.10)

Following the same procedure, we decompose V as  V 
 V3
 V2
 V1
 V0 :

(3.11)

According to the same reason as that of the case of  X, we can just consider the leading terms in
 V2 and
 V1 , which are n
n  1 12 2 n2 2 2
  0
@1 ’@2 ’  @1 @2 ’@1 @2 ’ 8 _ 2 ’_ 2  aa_ ’_ @21 ’  aa_ ’_ @22 ’; (3.12)
aa


 V2  

n
n  1 12 2 n2 _ 2 _ 0 ’_
  0
2
aa 8  aa_ _ 0 @21 ’  aa_ _ 0 @22 ’:


 V1  

(3.13)

The whole change of the third order action of perturbation due to noncommutative geometry in leading order of the slow-roll parameter can be written as  S 3 

Z

P;X0 X0
g X2
 X1 P;X0 0 ’
 X2 1 P;X0 X0 X0
g X21
 X1 P;V0 X0
g X1
 V2 2 P;V0 X0
g X2
 V1 ; (3.14) where g X  X  X0 , and
g X1  _ 0 ’, _
g X2  12
’_ 2 
@’2 , where we have picked out the terms with least _ 0 to reduce the order of the slow-roll parameter. Although we have picked out the leading order terms, the result is still lengthy. We write the result of (3.14) in Appendix A. To write the result, we define two new parameters, c2V 

P;X0

P;X0 ; 2V0 P;X0 V0

(3.15)

(3.16)

As we pointed out above, all the terms except those with the coefficient are of order O

 in  after a gauge _ . These two parameters vanish in transformation ’  H the general ‘‘slow-roll’’ inflation models, but could be nontrivial in some particular models. We will see in some case, they determine the dominance of the correction of non-Gaussianity. There are some subtle differences between calculating the modification of the three-point function and two-point function due to noncommutativity. In calculating the twopoint function, one has to solve the equation of motion of perturbation which is in general hard to solve with the presence of noncommutative coordinates (for a solvable example, see [29]). In Ref. [22], the author developed another way to encode the noncommutative effect into the power spectrum without solving the equation of motion. However, even with a solvable equation, we do not need the modified classical solution to calculate the noncommutative correction of the three-point function. The threepoint function is calculated through h 3
ti  i

Zt t0

p


d4 x h
P;X0 X0
g X1
 X2

(3.9)

dt0 h 3
t; Hint
t0 i

(3.17)

in tree level. So, to evaluate the modification of the threepoint function, which is denoted by h 3
ti below, in the lowest order of , we divide the Poisson bracket into two groups: h 3
ti  i

Zt

i

t0

Zt t0

dt0 hc3
t;  Hint
t0 i dt0 h3c2
t 
t; Hint
t0 i;

(3.18)

where c is the commutative solution. Since the primordial Hamiltonian without noncommutative modification is of order
2 in the slow-roll parameter, thus the second group contributes terms subleading in the slow-roll parameter, as we emphasize the leading order of the modification part of the Hamiltonian is O

. While the order of  are 2 in both cases, so the leading modification is obtained from the first

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KEJIE FANG, BIN CHEN, AND WEI XUE

PHYSICAL REVIEW D 77, 063523 (2008)

group of the Poisson bracket which is summarized in Appendix B. Another point deserving mention is that the constraint equations of N and Ni , and thus the solutions (2.11) and (2.12) obtain corrections of order 2 in the noncommutative coordinate. However, the same as the above analysis, they contribute terms of subleading order of  or
either in  S or S and thus we do not need to consider them. IV. MODEL TESTING In this section, we evaluate the effect of noncommutativity on the non-Gaussianities of perturbation in some particular models. Note all the background field and Hubble scale are estimated at the horizon crossing, namely, at the time about 60 e-foldings before the ending of inflation. In the ordinary ‘‘slow-roll’’ inflation with Lagrangian P  X  V, since cs  1;

cV  1;

 0;

A. DBI model DBI inflation [8,9,30,31] is motivated by the brane inflation scenario [32 –36] in warped compactifications. The effective Lagrangian is q

























P
X;   f
1 1  2f
X f
1  V
; (4.2) where f is the warp factor f   4 , and depends on flux number. The value of speed of sound cs , as well as other two parameters cf (f substituting for V in (3.15) represents warp factor f
) and are as follows: q
























cs  1  _ 2 f
; (4.3) q
























cf  1  _ 2 f
; 3   _ 6 f2
1  _ 2 f
5=2 : 8

(4.1)

it turns out that the modification is zero on the level of first order of the slow-roll parameter. Let us turn to the models with significant non-Gaussianity.

(4.4)

(4.5)

We find cf coincides with cs in the DBI model, and thus it becomes as small as the speed of sound. The leading correction terms with respect to cs coming from (B1) are

 H 4 l4s 1 H4 a 2 b 2 2i

k1 
k2   ka1 kb1 ka2 kb2 
k~1  k~2 k23 Q 4 32
2 c4s Mpl c4s
2k3i    1 k k kk iH3  24 5 120 1 6 2 720 1 72 4 2 2 p


0


ka1 2
kb2 2  ka1 kb1 ka2 kb2 
k~1  k~2  K cf cs K K   1 k k k2 k3 k3 k1 k1 k2 k3 120 8 3 24 1 2 K K6 K5    1 2 1 k k k2 k3 k3 k1 k1 k2 k3  6iH2 2 0

ka1 2
kb1 2 k2 k3 2 2 1 2 6 2

ka1 2
kb2 2 K cf K3 K4   1 k k kk 4 c2s H2 a 2  ka1 kb1 ka2 kb2 k23 2 3 6 1 4 2 24 1 52  i

k1 
kb1 2  2
K K Mpl K     1 1 k 24k21 k22 k23 5  2k22 k23 2 3 6 14 perm ; K K K

hk1 k2 k3 i  i
23 
k~1 k~2 k~3 

where a and b denote the first and second component of the k vector, respectively, K  k1 k2 k3 and perm denotes all the other terms obtained by rotating the index (1, 2, 3). 4 Here we have used the relation _ 20 ’ 0 due to a small speed of sound. In this phase, we have power spectrum of  perturbation Pk 

H4 N4

2e ; 2 _2 4 0 4

(4.6)

background explorer (COBE) normalization, Pk 23 2 . Using the 1010 , we find 1014 and _ 0 107 Mpl 1=2 1=4 limit of small speed of sound, 0 _ 0 Mpl . As a matter of result, only the first term in (4.6) with the following shape dominates, H4 l4s a 2 b 2

k 
k2   ka1 kb1 ka2 kb2 
k~1  k~2 k23 32c6s 1   1 k k kk 24 5 120 1 6 2 720 1 72 perm; K K K (4.8)

A1   (4.7)

where Ne  60 is the e-folding between the horizon crossing and the end of inflation. According to cosmic

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NONCOMMUTATIVE GEOMETRY MODIFIED NON- . . .

PHYSICAL REVIEW D 77, 063523 (2008) 1 fNL  0:02

0.6

H 4 l4s ; c6s

(4.13)

where the results are evaluated in the particular frame we used above. We find the noncommutative correction of nonGaussianity could be large if cs  1. For example, for Hubble constant H  105 Mpl , the string length scale ls  1 1 and the speed of sound cs  0:1, then fNL  2. 104 Mpl Comparing to the dominant fNL of the commutative case ((5.10) in [12]),

0.4 0.2 0 −0.2 −0.4 1 1 0.8

0.5

c 0:32c2 ; fNL s

0.6 0.4 0

FIG. 1 (color online).

0.2 0

we find the correction is more sensitive to the speed of sound cs . The present observation imposes a bound on equilateral form of fNL , 256 < fNL < 332, and future observation of Planck can detect jfNL j * 5, thus making the correction within the sensibility of these observations.

The shape of A1 =k1 k2 k3 .

We parametrize hk1 k2 k3 i as 1 hk1 k2 k3 i 
27 
k~1 k~2 k~3 
Pk 2 Q 3 A1 ; (4.9) i ki where Pk is the primordial power spectrum [10] Pk 

(4.14)

1 H2 : 2 c
8 Mpl s 2

We now consider the correction in the K-inflation [10,11] model which also has small speed of sound. The Lagrangian of the power law K-inflation is of the form

(4.10)

The shapes of A1 =k1 k2 k3 as a function of x1  k1 =k3 and x2  k2 =k3 are drawn in Fig. 1. In drawing the figures, we omit the coefficient of A1 . And we have chosen a particular frame in which the three k~ modes are in the x  y plane and k~3 is along the x-axis. We find this shape is distinct from the shape of Ac [12]. The non-Gaussianity of CMB in the Wilkinson Microwave Anisotropy Probe (WMAP) observations is analyzed by assuming the following ansatz:   g  35fNL g2 ;

B. K-inflation

(4.11)

where g represents the Gaussian part of , and fNL is an estimator of non-Gaussianity. The three-point function of  can be factorized as   3 7 3 ~ k 2 ~ ~ hk~1 k~2 k~3 i 
2 
k1 k2 k3   fNL
P  10 P 3 k (4.12) Qi i3 : i ki Despite the difference between the shape of (4.6) and 1 that of (4.12), we set k1  k2  k3  k to calculate fNL which represents the size of correction of non-Gaussianity deriving from A1 . We have

P
X;  

4 4  3 1
X X2 ; 9 2  2

(4.15)

where  is a constant. In the inflationary solution, X remains constant as X0 

2 : 4  3

(4.16)

The solution leads to the scale factor a of a  t2=3

(4.17)

for any 0 <  < 23 . And the speed of sound is c2s 

 : 8  3

(4.18)

In order to get small speed of sound, we focus on the region   1. The power spectrum in the limit of small  is [10] Pk

 3 1 2 H2 k  ; 2 2 cs 3 8 Mpl k1

(4.19)

where H is taken to be the Hubble scale at the time of the horizon exit for the perturbations currently at our horizon, and k1 is the associated comoving wave number. Using (4.18) as well as the fact that data determines Pk  109 at the horizon crossing, we get

063523-7

KEJIE FANG, BIN CHEN, AND WEI XUE 2 109 : H 2  4p3

23=2 82 Mpl

PHYSICAL REVIEW D 77, 063523 (2008)

(4.20)

It follows from (4.20) that the tilt satisfies ns  1  3    ;

(4.21)

using the central value of the spectral index in the WMAP results [2]. We also find cV  1 and  0 in this model, where we choose V  2 . The dominant terms in the three-point function are those with most cs in the denominator in each kind of coefficients,

1 1 and thus c2s  480 which allows us to determine   60

 H 4 l4s 1 H4 a 2 b 2 2i

k1 
k2   ka1 kb1 ka2 kb2 
k~1  k~2 k23 Q 4 32
2 c4s Mpl c4s
2k3i    2 1 k k kk iH3 Mpl 24 5 120 1 6 2 720 1 72  2 2

ka1 2
kb2 2  ka1 kb1 ka2 kb2 
k~1  k~2  K cs 0 K K   1 k1 k2 k2 k3 k3 k1 k1 k2 k3 120 8 3 24 K K6 K5   4  Mpl 1 k k k2 k3 k3 k1 k1 k2 k3 6 iH2 2

ka1 2
kb1 2 
k~2  k~3  2 2 1 2 K 0 K3 K4    1 k k kk 2

ka1 2
kb2 2  ka1 kb1 ka2 kb2 k23 2 3 6 1 4 2 24 1 52 perm : K K K

hk1 k2 k3 i  i
23 
k~1 k~2 k~3 

(4.22)

Following the same parametrization as (4.9), we get A1  

A2 

2 4 H 3 Mpl ls

32c4s 0

  H 4 l4s a 2 b 2 a b a b ~ ~2 k2 24 1 120 k1 k2 720 k1 k2 perm;

k 
k   k k k k 
k  k 1 2 1 1 2 2 3 K7 32c6s 1 K6 K5



ka1 2
kb2 2



ka1 kb1 ka2 kb2 
k~1

(4.23)

  1 k1 k2 k2 k3 k3 k1 k1 k2 k3 ~  k2  8 3 24 120 perm; K K6 K5

(4.24)

  4 l4  H 2 Mpl s a 2
kb 2 
k~  k~  2 1 2 k1 k2 k2 k3 k3 k1 6 k1 k2 k3

k 2 3 1 1 K 64c2s 20 K3 K4   1 k k kk 2

ka1 2
kb2 2  ka1 kb1 ka2 kb2 k23 2 3 6 1 4 2 24 1 52 perm: K K K

A3  

The shape of A1 =k1 k2 k3 is the same as that of the DBI case. The shape of A2 =k1 k2 k3 and A3 =k1 k2 k3 are drawn in Figs. 2 and 3, respectively. In the equilateral triangle configuration k1  k2  k3  1 , f2 , f3 , k, the correction of fNL is divided into fNL NL NL 1 fNL

(4.25)

1 0.5 0

H 4 l4  0:02 6 s ; cs

(4.26)

−0.5 −1

2  fNL

3 fNL

2 H3 l4s Mpl 0:02 0 c4s

 0:25

4 H 2 l4s Mpl

20 c2s

;

(4.27)

−1.5 −2 1 1

:

(4.28)

0.8

0.5

0.6 0.4 0

We estimate the size of 0 through the Friedmann equation

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FIG. 2 (color online).

0.2 0

The shape of A2 =k1 k2 k3 .

NONCOMMUTATIVE GEOMETRY MODIFIED NON- . . .

PHYSICAL REVIEW D 77, 063523 (2008)

8 6 4 2 0 −2 1 1 0.8

0.5

0.6 0.4 0

FIG. 3 (color online).

0.2 0

The shape of A3 =k1 k2 k3 .

2 H 2  E; 3Mpl

(4.29)

4 4  3 1
X 3X2 ; 9 2 2

(4.30)

where E  2XP;X  P  M2

and find 0 12p

3plc2 H . Using (4.18) and (4.20), we find s p


2 1 fNL  12 3c4s fNL ; (4.31) and 3 1 : fNL  5400c8s fNL

c2s

(4.32)

1 480 ,

1  these two are much smaller than fNL . Since 4 1 Using (4.18) and (4.20) and ls  10 Mpl , we find 1 180; fNL

(4.33)

which does not depend on cs due to the cancellation between H 4 and c6s . This property is different from the case of the DBI model. The dominant non-Gaussianity in K-inflation without considering the noncommutative effect is c fNL

0:26c2 s 125:

(4.35)

Using the upper bound on the equilateral form of fNL , fNL < 332, we get 1 : ls < 104 Mpl

ACKNOWLEDGMENTS

(4.34)

Since cs is fixed in K-inflation, it is easy to set an upper bound on ls . Adding (4.26) and (4.34) together, we get the total non-Gaussianity in K-inflation, t 4 fNL  180 1016 l4s Mpl 125:

framework. The corrections could be large in the models with a small speed of sound and a relatively low string scale. We test our result in two particular models, the DBI model and K-inflation. We find that the correction of fNL can reach O
1 or even bigger, and thus is observable within the sensibility of future experiments. Our study also shows that the noncommutative corrections are more sensitive to the speed of sound than the usual nonGaussianity estimator. This could be a clue to distinguish the different contributions to the non-Gaussianity. And also it indicates that in the inflation models with large speed of sound, the noncommutative correction to the nonGaussianity is small. Moreover, the shape of the corrections are different from the commutative case, which can be used as another distinguishing estimator. From our study, it turns out that the noncommutative corrections become significant when the string scale is relatively low. This could be just an illusion, given the noncommutative scale may not be string scale. In fact, the leading order correction terms are actually proportional to
12 2 , which could be nontrivially related to string scale. If we have a relatively low noncommutative scale, namely, we assume a little larger noncommutativity, the correction could be larger. In string theory, the noncommutative scale depends also on the background B12 field on the D-brane. In a sense, the corrections tell us the information of the string scale and also of the background field. On the other hand, without assuming another scale, one can make the natural choice (1.5). Although it is not easy to explain the great difference between the Planck scale and the string scale in the perturbative string theory, it is not hard to tune the string length scale all the way up to 1018 cm in type I compactifications and nonperturbative heterotic string theory. More interestingly, in a recent phenomenology study, the string scale is fit by WMAP data to be around 105  104 Mpl [23,24,37]. Doubtlessly, a future experiment will bring forward more data to test this tentative choice.

The work was partially supported by NSFC Grant No. 10405028, No. 10535060, NKBRPC (No. 2006CB805905), and the Key Grant Project of Chinese Ministry of Education (No. 305001). K. Fang thanks S.-H. Henry Tye for helpful suggestions and Gary Shiu for discussion. APPENDIX A: NONCOMMUTATIVE CORRECTION OF THIRD ORDER PERTURBATION

(4.36)

V. CONCLUSION We studied the noncommutative corrections of nonGaussianities of primordial perturbation in a general

The noncommutative correction of the third order action of perturbation, which is of order O

 except the terms proportional to , is

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KEJIE FANG, BIN CHEN, AND WEI XUE

PHYSICAL REVIEW D 77, 063523 (2008)

      Z P;X0 1 1 V00 1 1 2 1  S  d4 xa3
12 2  1 ’ _ P  1 ’ @1 @i ’@22 @i ’  2 @1 @2 @i ’@1 @2 @i ’  H@21 @i ’@i ’_ ;X 0 2 2 2 8 2V0 cV a a _ 0 cs _ 1 ’_ a_ 2 @2 ’@ _ 2 ’_ a_ 2 @i ’@ _ i ’_  2H a_ 2 @1 ’@ _ 1’  H@22 @i ’@i ’_ H 2 @21 @i ’@i ’ H 2 @22 @i ’@i ’ a_ 2 @1 ’@ _ 2 @1 ’@1 ’
H a _ 2 @2 ’@2 ’
H a _ 2 @i ’@i ’  @21 ’@  2Ha_ 2 @2 ’@ _ 2 ’  2H a_ 2 @i ’@ _ i ’
H a _ 22 ’_ @1 @2 ’@ _ 1 @2 ’_ _ 2 ’_ 2  Haa_ ’_ @21 ’_  Haa_ ’_ @22 ’_ 2H@21 ’@ 
Haa _ 22 ’ 2H@21 ’@22 ’_ 2H 2 aa_ ’_ @21 ’ 2H2 aa_ ’_ @22 ’ 2 2 _ 21 ’_ ’ aa@ _ 22 ’_ ’ a2 a_ 2 ’ ’ a_ 2 ’
2@  4H@1 @2 ’@ _ 1 @2 ’  4H 2 @21 ’@22 ’ 4H 2 @1 @2 ’@1 @2 ’ aa@  1 ’ 2@1 ’    1 P;X0 1 _ ’_ 2 
@’2 
@21 ’_ @22 ’_  2H@21 ’  2H@22 ’ 2Haa_ ’  aa_ ’ _   1 Haa
_ 2 _ 0 c2s     n
n  1_ 0 1 n
n  1_ 0 1 2 
@’2 
2
aa 2’ 2 ’  aa@ 2 ’  P _ _ _  1
’ _ _  a a@  1  P;X0 ;X0 1 2 c2V c2V 420 220 _ 2 ’_ 3  aa_ ’_ 2 @21 ’  aa_ ’_ 2 @22 ’  2 Haa_ _ 0 ’_ 2
@21 ’_ @22 ’_  2H@21 ’
’@ _ 21 ’@22 ’  ’@ _ 1 @2 ’@1 @2 ’
aa X0

 2H@22 ’ 2Haa_ ’: _

(A1)

APPENDIX B: NONCOMMUTATIVE CORRECTION OF THREE-POINT FUNCTION The correction of the three-point function of  is

hk~1 k~2 k~3 i  i
23 3
k~1 k~2 k~3 

H4 l4s 1 4 Q
2k3i  32
2 c4s Mpl

      1 i 1 k k kk 2cs H4 2  1

ka1 2
kb2 2  ka1 kb1 ka2 kb2 
k~1  k~2 k23 3 24 5 120 1 6 2 720 1 72 K cs cs K K   i 1 i 1 k k kk  k23 24 1 4 2 96 1 52 8  720k21 k22 k23 cs K 7 cs K 3 K K      1 k1 1 k1 k2 k1 k2 a 2 b 2 2 2 i 2 i ~ ~ ~ ~ 6 24 5

k1 
k1   
k1  k2 k2 k3 24 5 120 6
k1  k2 k3 2 cs cs K 3 K4 K K K     i 1 k 1 1 k  24k21 k22 k23 c3s 5 k22 k23 cs i 4 3 12 14 12ik21 k22 k23 cs 5 ik22 k23 cs 2 3 6 14 K K K K K K     i 1 k k k k i 2 k k k k k k1 k2 k3 2 3 2 3 1 2 2 3 3 k1 ~2  k~3  2 3 12 6 
k k21
k~2  k~3  cs K 3 cs K K4 K3 K4 K5      1 1 k 1 k1 k2 k1 k2 2
ka1 ka2 kb1 kb2 k~1  k~2  24ik21 k22 k23 cs 5  2ik22 k23 cs 2 3 6 14 k23 ics K K K K2 K3 K   1 i 1 k 24 5 120 16  4ic3s k21 k22 k23 3 2

ka1 2
kb2 2
kb1 2
ka2 2 k22 k23 cs K K K     i 1 k 1 k k kk 2 2 3 2 2 33  4ka1 kb1 ka2 kb2 k22 k23 24 5 120 16
k~2  k~3 k21 ics cs K K K K K     48 120k2 k1 k2 k1 k2 2 2 2 3 8 k k c  12 24  ik21 k22 k23 cs 
ka1 2
kb1 2  5 ik 1 2 3 s K3 K4 K6 K5 K     4 k  2k kk 4 k 2ik22 k23 cs 
ka1 2
kb1 2  3  6 2 4 1  24 1 52  2ik21 k22 k23 c3s 3  6 24 K K K K K  _       1 i 1 k k k2 k3 k3 k1 k1 k2 k3 120 ncs H 3 2  1 0

ka1 2
kb2 2  ka1 kb1 ka2 kb2 
k~1  k~2  3 8 3 24 1 2 0 cV cs K K6 K5     i 1 k i 1 k k k2 k3 k3 k1 kkk 6 1 24 3  k21 k22 24 5 120 36  4 2 2 1 2 3 cs cs K K K K K

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PHYSICAL REVIEW D 77, 063523 (2008)

     i 1 k k kk i 1 k k k2 k3 k3 k1 k1 k2 k3

ka1 2
kb1 2  
k~1  k~2 k22 6 2 3 6 1 4 3 24 1 53
k~1  k~2  2 2 1 2 cs K cs K K K3 K4 K        1 k 1 k k kk 1 k  k21 k22 ics 2 3 6 34 2k22 ics 1 2 3 2 1 33 
ka1 ka2 kb1 kb2 k~1  k~2  k21 k22 ics 2 3 6 34 K K K K K K K       1 k k k k k k k k k k k k k 1 k 2 3 3 1 1 1 22 3  k22 k23 ic3s 2k21 ics 2 2 3 2 2 23  ics K 1 2 K K K2 K K K K     i 1 k k kk i 1 k k kk 2 3 6 2 4 3 24 2 53  4ka1 kb1 ka2 kb2 k21 2 3 6 2 4 3 24 2 53 2

ka1 2
kb2 2
kb1 2
ka2 2 k21 cs K cs K K K K K    4 k  2k kk 4 k1 k2  2k3 k k  2k2 k3  2k1 k3  2 1 2  ik21 k22 cs

ka1 2
kb1 2  3  6 2 4 3  24 2 53  ik21 k22 c3s K K K K2 K3 K    kkk 2 k  2k1  2k3 k k k2 k3  2k1 k3 kkk 6 1 24 3 2ik22 cs

ka1 2
kb1 2   2 2 1 2 6 1 24 3 K K K2 K3 K  _2     2 k2  2k3 2k2 k3 n
n  1 1  cs H 2 2  1 02 3   2ik21 k22 c3s 2 K 2 cV K K 0       1 1 k i 1 k k k2 k3 k3 k1 k1 k2 k3 8k21 k22 k23 ic3s 3

ka1 2
kb1 2  3k22 k23 ics 2 3 6 14 k~2  k~3 6 2 2 1 2 cs K K K K K3 K4     1 k k kk i 1 k k kk  2
k~1  k~2 k23 ics 1 2 2 2 1 32 2

ka1 2
kb2 2  ka1 kb1 ka2 kb2 k23 2 6 1 4 2 24 1 52 K cs K 3 K K K K      2 4 cs H 1 1 k 1  (B1)

ka1 2
kb1 2  24k21 k22 k23 ics 5  2k22 k23 ics 2 3 6 14 4k21 k22 k23 ic3s 3 perm ; 2 K K Mpl
K K where a and b denote the first and second component of the k vector, respectively, K  k1 k2 k3 and perm denotes all the other terms obtained by rotating the index (1, 2, 3). All the background value and Hubble constant are calculated at the horizon crossing.

[1] A. H. Guth, Phys. Rev. D 23, 347 (1981); A. D. Linde, Phys. Lett. B 108, 389 (1982); A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982). [2] D. N. Spergel et al., Astrophys. J. Suppl. Ser. 170, 377 (2007). [3] J. Maldacena, J. High Energy Phys. 05 (2003) 013. [4] V. Acquaviva, N. Bartolo, S. Matarrese, and A. Riotto, Nucl. Phys. B667, 119 (2003). [5] D. Seery and J. E. Lidsey, J. Cosmol. Astropart. Phys. 06 (2005) 003. [6] D. Seery and J. E. Lidsey, J. Cosmol. Astropart. Phys. 09 (2005) 011. [7] N. Arkani-Hameda, P. Creminellia, S. Mukohyamaa, and M. Zaldarriagaa, J. Cosmol. Astropart. Phys. 04 (2004) 001. [8] E. Silverstein and D. Tong, Phys. Rev. D 70, 103505 (2004). [9] M. Alishahiha, E. Silverstein, and D. Tong, Phys. Rev. D 70, 123505 (2004). [10] J. Garriga and V. F. Mukhanov, Phys. Lett. B 458, 219 (1999). [11] C. Armendariz-Picon, T. Damour, and V. Mukhanov, Phys. Lett. B 458, 209 (1999). [12] X. Chen, M. Huang, S. Kachru, and G. Shiu, J. Cosmol. Astropart. Phys. 01 (2007) 002.

[13] D. Babich, P. Creminelli, and M. Zaldarriaga, J. Cosmol. Astropart. Phys. 08 (2004) 009. [14] P. Creminelli, J. Cosmol. Astropart. Phys. 10 (2003) 003. [15] A. Connes, Noncommutative Geometry (Academic, New York, 1994). [16] N. Seiberg and E. Witten, J. High Energy Phys. 09 (1999) 032. [17] N. Seiberg, L. Susskind, and N. Toumbas, J. High Energy Phys. 06 (2000) 021. [18] N. Seiberg, L. Susskind, and N. Toumbas, J. High Energy Phys. 06 (2000) 044. [19] J. Gomis and T. Mehen, Nucl. Phys. B591, 265 (2000). [20] M. Chaichian, A. Demichev, P. Presnajder, and A. Tureanu, Eur. Phys. J. C 20, 767 (2001). [21] L. Alvarez-Gaume, J. L. F. Barbon, and R. Zwicky, J. High Energy Phys. 05 (2001) 057. [22] R. Brandenberger and P. Ho, Phys. Rev. D 66 023517 (2002). [23] Q. G. Huang and M. Li, J. High Energy Phys. 06 (2003) 014; J. Cosmol. Astropart. Phys. 11 (2003) 001. [24] S. Tsujikawa, R. Maartens, and R. Brandenberger, Phys. Lett. B 574, 141 (2003). [25] C.-S. Chu, B. R. Greene, and G. Shiu, Mod. Phys. Lett. A 16, 2231 (2001).

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[26] R. Easther, B. R. Greene, W. H. Kinney, and G. Shiu, Phys. Rev. D 64, 103502 (2001). [27] R. Easther, B. R. Greene, W. H. Kinney, and G. Shiu, Phys. Rev. D 67, 063508 (2003). [28] A. Mazumdar and M. M. Sheikh-Jabbari, Phys. Rev. Lett. 87, 011301 (2001). [29] F. Lizzi, G. Mangano, G. Miele, and M. Peloso, J. High Energy Phys. 06 (2002) 049. [30] X. Chen, Phys. Rev. D 71, 063506 (2005). [31] X. Chen, J. High Energy Phys. 08 (2005) 045. [32] S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L.

[33] [34] [35]

[36] [37]

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McAllister, and S. P. Trivedi, J. Cosmol. Astropart. Phys. 10 (2003) 013. G. R. Dvali and S. H. H. Tye, Phys. Lett. B 450, 72 (1999). G. R. Dvali, Q. Shafi, and S. Solganik, arXiv:hep-th/ 0105203. C. P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh, and R. J. Zhang, J. High Energy Phys. 07 (2001) 047. G. Shiu and S. H. H. Tye, Phys. Lett. B 516, 421 (2001). D.-j. Liu and X.-z. Li, Phys. Lett. B 600, 1 (2004).