Gravitational Lensing on the Cosmic Microwave Background by Gravity

Home , Cosmic variance, Gravitational lens

Gravitational lensing on the Cosmic Microwave Background by

gravity waves

Silvia Mollerach

Departamento de Astronomia y Astrofsica, Universidad de Valencia, E-46100 Burjassot,

Valencia, Spain

Abstract

We study the e ect of a sto chastic background of gravitational waves on the

gravitational lensing of the Cosmic Microwave Background (CMB) radiation.

It has b een shown that matter density inhomogeneities pro duce a smo othing

of the acoustic p eaks in the angular p ower sp ectrum of the CMB anisotropies.

A gravitational wave background gives rise to an additional smo othing of

the sp ectrum. For the most simple case of a gravitational wave background

arising during a p erio d of in ation, the e ect results to b e three to four orders

of magnitude smaller than its scalar counterpart, and is thus undetectable.

It could play a more relevant role in mo dels where a larger background of

gravitational waves is pro duced.

98.79.Vc,04.25.Nx,98 .80 .-k

Typ eset using REVT X

E 1

The gravitational lensing e ect of matter density inhomogeneities on the Cosmic Mi-

crowave Background (CMB) radiation has b een the sub ject of several studies [1{8]. It is

well known that the de ections undergone by the photons along their path since last scatter-

ing can mo dify the pattern of the observed anisotropies. The e ect is to smo oth the acoustic

or Doppler p eaks in the angular sp ectrum. Although the e ect has b een found to b e small,

it should b e observable in small angle high accuracy observations [6,8]. It has recently b een

p ointed out [9] that a sto chastic background of gravitational waves also contributes to the

gravitational lensing of the CMB radiation.

Many scenarios of the early universe mayhave pro duced a sto chastic background of grav-

itational waves, as for instance a p erio d of in ation, phase transitions leading to top ological

defects [10], or bubbles nucleated in a rst order phase transition [11]. In many in ationary

mo dels, the background of gravitational waves gives a substantial contribution to the CMB

anisotropies at large angular scales [12]. These anisotropies arise due to the redshift of the

photons induced by the time variation of the graviational waves amplitude along the photon

paths.

In this pap er wewant to quantify the e ect on the CMB anisotropies induced by the grav-

itational lensing of photons from a gravitational wave background. We consider a p erturb ed

at Rob ertson{Walker spacetime describ ed in the Poisson gauge by

h i

2 2 2 > i j

; (1) ds = a ( ) (1+2')d + (1 2') + dx dx

where is the conformal time, in the absence of vector p erturbations. ' is the p eculiar

gravitational p otential and denotes the tensor (transverse and traceless) p erturbation.

The gravitational lensing e ect on photons is describ ed by the angular displacement

that measures the di erence between the angular direction on the sky from which a given

photon arrives to the observer and the one it would have had in the absence of lensing

sources along its path. It is given by = r x ( ), with r = ( ) the distance to

? E E O E

the last scattering surface and

i ij i j > k > k

x ( )=( e e ) d e e

? jk Ojk

ij i j > k l

( e e ) d( ) 2' ; (2) e e

E ;j

kl;j

where e is a unit vector denoting the direction of arrival of the photons, and the integration

is along the photon background geo desics parametrized by . The subscript O denotes quan-

tities evaluated at the observation p oint and E at the emission (or last scattering surface).

The term including ' corresp onds to the displacement due to scalar density p erturbations

that has b een considered in some previous studies [6{8] and has an observable e ect on small

angular scales, while the rest describ es the e ect of the gravitational wave background.

It has b een shown by the studies of scalar gravitational lensing that the e ect on the CMB

anisotropies can b e obtained from the auto correlation function of the transverse displacement

^ ^

S ( )=h (e ) (e )i

^ ^ 1 j 2

(e e =cos )

1 2

Z Z

d d

^ ^

e e

1 2

^ ^ ^ ^

= (e e cos )h (e ) (e )i; (3)

1 2 1 j 2

4 2 2

where we have taken the mean over all directions separated by an angle . Once this

correlation function is known, we can compute the e ect on the temp erature correlation

function using the metho ds develop ed in ref. [5,6,8] for small angular scales, or that in ref.

[7] that apply to arbitrary angular scales.

The gravitational wave background in a at universe can b e decomp osed as

> 3

(x;)= d k exp (ik x) (k;) (k); (4)

ij ij

(2 )

where (k) is the p olarization tensor, with ranging over the p olarization comp onents

+; , and (k;) are the corresp onding amplitudes. The time evolution of the amplitude

during the matter dominated era can b e written as

3j (k )

(k;) A(k)a (k) ; (5)

where a (k) is a zero mean random variable with auto correlation function ha (k)a (k )i =

3 3 3 0

(2 ) k (k + k ) , and j (x) denotes the spherical Bessel function of rst order. The

sp ectrum of the gravitational wave background dep ends on the pro cesses by which it was

generated.

For a wave propagating in the direction k, de ning a right-handed triad given by

^ ^

(k; m; n), the p olarization tensor can be written as

(k )= m m n n

i j i j

(k )= m n +n m : (6)

i j i j

In order to compute the auto correlation function of the angular displacement S ( ) in-

i Ii IIi

duced by the gravitational wave background, we split x = x + x with

? ? ?

ij i j > k l Ii

( e e ) d( ) e e ; x =

kl;j ?

IIi ij i j > k > k

: (7) x =( e e ) d e e

? jk Ojk

I(I I )j I(II)

I II I(I I ) II

Wethus obtain S ( )=S ( )+S ( ), with S ( )= r hx x i,ashx x i =0. The

E ?j

? ?j ?

I II

expressions for S ( ) and S ( ) can b e obtained replacing in eq. (3), the ab ove expressions

Ii IIi

for x and x . Parametrizing the photon geo desics by ( )=( ), so that

O O E

? ?

= 0 and =1, and x = e( ), we obtain

O E O E

Z Z Z

1 1 1

9 d! A (! )

I 0 0

S ( )= d d j (! (1 ))j (! (1 ))

1 1

2 ! (2 )

0 0 0

Z Z

d cos d exp (i! [ cos (cos cos cos sin sin )])

1 0

2 2 2 2

(1 cos + cos sin sin cos cos cos sin sin )

2 2 2 4 2 3

2 cos 1 3 cos cos + cos cos + 2 cos sin sin cos cos

2 2 2 4

+ cos + cos sin (1 cos ) ; (8) 3

where we have de ned ! k ( ), and

O E

Z Z Z Z

1 1 1 1

A (! ) j (! (1 )) d! j (! )

1 1

0 II

d d d cos exp (i! cos ) S ( )=18

! (2 ) ! (1 ) !

0 0 1 0

j (! (1 )) j (! )

1 1

d exp [i! (cos cos cos sin sin )]

! (1 ) !

h ih i

2 2 2

cos (cos cos cos sin sin ) cos sin + cos sin sin cos

2 2 2 4 2

+ cos 2 cos 1 3 cos cos + cos cos

3 2 2 2 4

+ 2 cos sin sin cos cos + cos + cos sin (1 cos ) ; (9)

The integrals over in the previous equations can b e p erformed analytically in terms of

Bessel functions, we do not show the result here. We will instead directly show the result

of the numerical integration over the remaining variables for the case of a background of

gravitational waves generated during an in ationary p erio d. In this case, the sp ectrum is

nearly scale invariant and prop ortional to the Hubble constant during in ation. Di erent

in ationary mo dels predict slightly di erent sp ectral tensor index n and amplitude of tensor

mo des. For de niteness, we will consider a scale invariant sp ectrum n =0. An upp er limit

to the amplitude is set by the COBE measurement of CMB anisotropies at large scales. We

2 3 11 2

can thus write the sp ectrum as A (k )=(2 ) = 6 10 T (k ), were we have included the

transfer function for gravitons T (k ) that takes into account the di erence in the evolution

for mo des that entered the horizon during the radiation and matter dominated eras and can

2 2

be tted by T (k ) = 1+1:34(k=k )+2:5(k=k ) [14], where k is the scale that entered

eq eq eq

the horizon at the equality time.

A useful quantity is the disp ersion of the di erence of the photons displacement in two

directions de ned by

2 2

^ ^

( )= h( (e ) (e )) i = S (0) S ( ): (10)

1 2 ^ ^

(e e =cos )

1 2

Figure 1 shows the ratio ( )= for a range of angular separations. As the gravitational

wave background and the p eculiar gravitational p otential are two uncorrelated elds, the

gravitational lensing displacements that they pro duce are also uncorrelated. Thus, we can

describ e the total e ect of gravitational lensing as the sum of the scalar and tensor contri-

2 2 2

butions, i. e. S ( ) = S ( )+S ( ) and ( ) = ( )+ ( ), where S ( ) and

TOT S T T

TOT S T

2 2

( ) are the ones computed in this pap er and S ( ) and ( ) have b een computed by

T S

several authors [6{8].

It has b een shown that the temp erature auto correlation function including the e ects of

gravitational lensing C ( ) can b e obtained from that in the absence of gravitational lensing

C ( )ifS()isknown. For small angular scales the gravitational lensing e ect is to smo oth

the auto correlation function as if the smo othing was pro duced by a Gaussian antenna of

width ( ).

Comparing the results in Fig. 1 with the corresp onding ones for scalar p erturbations

in refs. [6{8], we see that the disp ersion of the graviational lensing displacements induced

by gravitational waves is 3 to 4 orders of magnitude smaller than the corresp onding scalar 4

ones. We thus exp ect that ( ) ' ( ) and that the e ect of the gravitational lens-

TOT S

ing by the gravitational waves background be undetectable at small scales. This result is

essentially due to the fact that the amplitude of the gravitational waves decreases during

the matter dominated era for wavelengths smaller than the Hubble radius, and thus their

contribution at small scales is supressed. A recent analysis of the e ect of a sto chastic back-

ground of gravitational waves on multiple images and weak gravitational lensing has found

a comparable supression with resp ect to the corresp onding scalar p erturbations lensing [13].

We could wonder if the very large scale mo des, that are the larger amplitude ones, can

lead to an observable e ect at large angular scales. This is not the case as the e ects of

gravitational lensing are not evident at scales for which the angular sp ectrum is smo oth.

An estimation of this e ect using the metho d prop osed in ref. [7] with the correlation S ( )

computed in this pap er shows that the e ect on the temp erature anisotropy correlation

function at large angular scales can be of the same order of magnitude than that at small

angular scales (on the contrary, for scalar p erturbations the e ect at large angular scales is

much smaller than the small scales one). This is however to o small to b e detectable, b esides

the fact that cosmic variance at large angular scales make small variations in the predicted

sp ectrum untestable.

The results discussed ab ovehave b een obtained for a gravitational wave background pro-

duced during a p erio d of in ation. There are however other scenarios of the early universe

in which a larger background of gravitational waves is exp ected. This is the case for example

for mo dels with cosmic strings [10] or rst order phase transitions [11]. The metho d devel-

op ed in this pap er can be applied to any of these cases just by replacing the corresp onding

sp ectra A(! ) in eqs. (8) and (9). It should b e taken into account that also the temp erature

anisotropies and the scalar gravitational lensing disp ersion may be di erent in alternative

theories.

ACKNOWLEDGMENTS

It is a pleasure to thank M. Portilla and S. Matarrese for useful comments and sugges-

tions. Iwould liketoacknowledge the Vicerrectorado de investigacion de la Universidad de

Valencia for nancial supp ort, and the Theory Division at CERN for hospitality. 5

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Figure 1: ( )= vs. for a range of in units of radians. 6