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Gravitational Lensing on the Cosmic Microwave Background by Gravity

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Gravitational Lensing on the Cosmic Microwave Background by Gravity 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 ij ij 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. ij ~ 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 1 ~ sources along its path. It is given by = r x ( ), with r = ( ) the distance to ? E E O E E the last scattering surface and Z E i ij i j > k > k x ( )=( e e ) d e e E ? jk Ojk O Z E 1 ij i j > k l ( e e ) d( ) 2' ; (2) e e E ;j kl;j 2 O ^ 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 j ^ ^ S ( )=h (e ) (e )i ^ ^ 1 j 2 (e e =cos ) 1 2 Z Z d d ^ ^ e e 1 2 j ^ ^ ^ ^ = (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 Z 1 > 3 ^ (x;)= d k exp (ik x) (k;) (k); (4) ij ij 3 (2 ) ^ where (k) is the p olarization tensor, with ranging over the p olarization comp onents ij +; , 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 ) 1 (k;) A(k)a (k) ; (5) k 0 0 where a (k) is a zero mean random variable with auto correlation function ha (k)a (k )i = 3 3 3 0 0 (2 ) k (k + k ) , and j (x) denotes the spherical Bessel function of rst order. The 1 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 ij ^ (k )= m n +n m : (6) i j i j ij 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 ? ? ? Z E 1 ij i j > k l Ii ( e e ) d( ) e e ; x = E kl;j ? 2 O Z E IIi ij i j > k > k : (7) x =( e e ) d e e ? jk Ojk O I(I I )j I(II) Ij 1 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 2 1 1 1 9 d! A (! ) I 0 0 S ( )= d d j (! (1 ))j (! (1 )) 1 1 3 2 ! (2 ) 0 0 0 Z Z 1 0 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 2 1 1 1 1 A (! ) j (! (1 )) d! j (! ) 1 1 0 II d d d cos exp (i! cos ) S ( )=18 3 ! (2 ) ! (1 ) ! 0 0 1 0 ! Z 0 j (! (1 )) j (! ) 1 1 0 d exp [i! (cos cos cos sin sin )] 0 ! (1 ) ! 0 h ih i 2 2 2 cos (cos cos cos sin sin ) cos sin + cos sin sin cos h 2 2 2 4 2 + cos 2 cos 1 3 cos cos + cos cos i 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 T mo des. For de niteness, we will consider a scale invariant sp ectrum n =0. An upp er limit T 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 2 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 1 2 2 ^ ^ ( )= h( (e ) (e )) i = S (0) S ( ): (10) 1 2 ^ ^ (e e =cos ) 1 2 2 Figure 1 shows the ratio ( )= for a range of angular separations.
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