Dipole Modulation in Tensor Modes: Signatures in CMB Polarization

In this work we consider a dipole asymmetry in tensor modes and study the effects of this asymmetry on the angular power spectra of CMB. We derive analytical expressions for the $C_{l}^{TT}$ and $C_{l}^{BB}$ in the presence of such dipole modulation in tensor modes for $l<100$. We also discuss on the amplitude of modulation term and show that the $C_{l}^{BB}$ is considerably modified due to this term.

where η is the conformal time, a(η) is the scale factor and Φ, B i and h ij are the scalar, vector and tensor perturbations of the metric. The tensor perturbations are characterized by the transverse traceless tensor h T T ij and using the Einstein equations is governed by the following equation where the prime denotes derivative with respect to conformal time. We apply the decomposition technics to the tensor modes and write h T T ij (η, x) = h T T ij (η, k) e −ik·x where x = (η 0 − η) n will be the distance from the last scattering surface and n is the direction of photon propagator. In order to calculate the CMB power spectra it is convenient to rotate the coordinate system so that the wave vector k is aligned along z axis. Hence one can write k · n = k cos θ. The tensor perturbations h T T ij (η, k) are separated into the fourier modes of two polarization states, where h where we have ignored the source term due to neutrino anisotropic stress [40]. Here and elsewhere we do not include the neutrino perturbations in our calculations. One can show that for a mixture of radiation and matter fluid the Friedmann equation gives the scale factor as [41] a(η) = a eq η η 1 2 where a eq is the value of scale factor at the time of equality and η 1 78.8 Ω −1 m with the parameter Ω m denoting the current abundance of matter. The equation (4) can be solved numerically using scale factor (5). The results are presented in Fig. 1. As we can see in Fig. 1(a), for those modes with k k eq (≈ 0.01Mpc −1 ), the numerical results are in good agreement with the analytic solution H(k, η) = sin(kη)/kη in radiation dominated era. As well as for those long wavelength modes which enter the horizon after equality the numerical solution are in agreement with the analytic solution 3j 1 (kη)/kη where j 1 is the spherical Bessel function. For reasons that will become clear later on when we will calculate the C BB l , we are interested in the modes which enter the horizon at the time of recombination η r 288 Mpc. Usually, at this time the analytical solution 3j 1 (kη)/kη is approximated as the transfer function [42][43][44]. Interestingly, as we can see in Fig. 1(b) the numerical solution of equation (4) for transfer function has a closer agrement with the analytical result sin(kη)/kη at η = η r . Moreover, our later calculations in section IV for deriving C BB l , suggest that the sin(kη)/kη solution is an appropriate transfer function at the time η = η r . A gravitational wave carrier can be modulated similar to what occurs in wave mechanics. In a simplified picture the modulation may be due to a superhorizon long wave tensor mode. The long wavelength mode can change the amplitude of gravity wave in an especial direction from one side of the sky to the other. Adopting a dipole asymmetry term to the position dependent part of the tensor mode we obtain where k L is a long mode directed along the n k which make angle θ with the z direction. Here we assume that the long mode perturbs the CMB photons at the last scattering surface (lss). As a result an observer sees a dipole asymmetry in the direction n corresponding to the amplitude A t ≡ k L x lss . In order to track the impact of such dipole asymmetry on the CMB temperature and polarization power spectra we analytically recalculate the tensor part of CMB multipoles by considering the following replacement where the angular integration over the θ will contribute corrections to the CMB power spectra.

III. DIPOLE MODULATION IN CMB TEMPERATURE POWER SPECTRUM
In the absence of the modulation, the contribution of tensor perturbations to the CMB temperature anisotropy is parameterized as [43,45] Here Θ t is the brightness function where the superscript t indicates that the CMB temperature anisotropy are due to tensor modes. The n i and n j coefficients are also the unit vectors along the photon momentum and the integral in Eq.(8) is computed along the photon trajectory from the the recombination time, η r to the present time η 0 . In Fourier space the Θ t (n) is represented in the following form where we have made use of the expansion of the exponential in terms of Legendre polynomials P l One can expand the brightness function Θ t into multipoles a t with Y lm (n) the spherical harmonic functions. Using the orthogonality of spherical harmonics and the convolutions n i n j e (+) ij = sin θ cos 2φ and n i n j e (×) ij = sin θ sin 2φ in spherical frame (θ, φ) we arrive at To reduce this expression we use the recursion and orthogonality relations for Legendre polynomials and after some straightforward calculations we find the multipoles as where the m = ±2 is appeared as a result of integration over the azimuthal angle φ. After calculating the a t l±2 coefficients, one can also take into account the angular power spectrum C T T l . Here we must distinguish between the anisotropies from scalar and tensor modes. The total angular power spectrum in general is written as , is given in the following manner Using the two point correlation function of the primordial tensor perturbation h A (i) with polarization A = +, × one can write where P t is the tensor amplitude and is set by the amplitude of scalar amplitude A s as P t = r A s . After changing the variables of integration from kη 0 to u and η/η 0 to ξ and using the fact that With r = 0.1 and A s = 2.2 × 10 −9 we numerically integrate the (17) and compare it with the results of CAMB CMB code [39]. Here we set the Planck 2013 best fit parameters [46] in CAMB. We also do not consider the effects of reionization on the temperature and polarization anisotropies and the effects of neutrino on the amplitude of tensor perturbations. Therefore we switch off both effects in the CAMB program. From Fig. 2.(a), we see a fair agreement between results of CAMB and the analytical results of (17) for l < 50. We want to extend the calculations leading to Eq. (17) to the case in which the tensor modes are modulated. To proceed, we first replace h T T ij with h T T ij (1 + A t cos θ) and then divide the multipoles into two parts a t lm + δa t lm such that the second part contains the A t h T T ij cos θ. The method of calculation of δa t lm is the same as described above for the a t lm but rather more complex, so we do not present all details. After some straightforward calculations we arrive at the following expression for δa t where x = cos θ. The integration over the x variable can be performed by using again the recurrence and orthogonality relations of Legendre polynomials (13). We find Using (15) one can also define where this expresses the contribution of dipole modulation in the tensor angular power spectrum. Therefore, using (19) we get where so that the tensor angular spectrum is given by C l . Setting A t = 0 gives rise to the unmodulated case. Here, we have not considered the dipole modulation in the scalar perturbations. Hence the total angular power spectrum is We keep the curvature perturbation ζ unmodulated hence the C T T (ζ) l spectrum is calculated using the CAMB code. The C T T (t) l and the A 2 t δ (1) l factors are also given by numerically integrating the equations (17) and (21). Then we combine the C T T (ζ) l given by CAMB with the C given by equations (17) and (21) and obtain the total angular power spectrum C T T l . In Fig. 2(b) we have shown the resulting C T T l with A t = 1 and 2. They have been compared with the total angular power spectrum derived by CAMB. As the curves depicted in Fig 2(b) clearly manifest the dipole modulation in tensor perturbations with A t ∼ 1 does not make a considerable contribution to the C T T l . For l ∼ 10 we see a small deviation from the non modulated case which falls down for l > 10. Note that these effects are one order of magnitude smaller in the C EE l and C T E l spectra.

IV. THE EFFECTS OF DIPOLE MODULATION ON C BB l
The polarization of CMB is quantified by Stokes parameters Q(n) and U (n) measured as a function of position on the sky. It is known that the combination Q(n) ± i U (n) transforms like a spin-2 variable under rotation. Hence expanding this combination in spin weighted spherical harmonics, ±2 Y lm , gives This help us to define two E-and B-modes by linear combinations of coefficients a ±2 where E-modes is invariant under the parity transformations while B-modes change sign. Usually the full sky polarization map of CMB is decomposed into E-mode and B-mode [47,48]. Physically the E-mode polarization is generated by scalar and tensor perturbations. It can be shown that the B-mode is just generated by the tensor perturbation. Therefore, the B-mode can probe the primordial gravitational wave. Any diploe modulation in tensor modes can imprint on both E mode and B-mode. However, we expect larger effects on the B-mode. In order to calculate the a E,B lm multipoles as it is convenient we define the polarization matrix in terms of Stokes Parameters P ab (n) = d 3 kP ab (k, n) Hence the coefficient a E,B lm are given by where with the auxiliary functions X lm and W lm constructed as The parameters of polarization matrix and also the CMB angular spectra are mostly derived by a hierarchy of Boltzmann equations [48,49]. Instead, we take an analytic approach proposed in [43,47] to study the CMB polarization. We compare our results with the methods implemented in Boltzmann code CAMB to check the analytical method. We then extend the analytical calculation to include the modulation in the tensor modes. The Fourier transformation of polarization matrix P ab (k, n) for tensor perturbations is analytically given by the following matrix [43] P t ab (k, n) = ∆η r 10 where H(k, η) is again the transfer function for tensor modes and ∆η r is the thickness of the last scattering sphere. Note that in this expresion we have not considered the gravitational lensing and also the reionization effect. One can easily show that in the scalar perturbations case the off diagonal components of polarization tensor vanish. However, for the tensor perturbations, the new terms supplied by gravity waves result in non-vanishing values for the Stokes parameter U has a principal role in generating the B mode polarization. Now after computing the polarization matrix (31) one can find the coefficients a B lm by using the relation (27). We defer the details of calculation to the Appendix. By using the results presented in the Appendix we can evaluate the parity independent angular power spectra C BB l as follows The transfer function is computed at the time η = η r . As we discussed in section II at this time one can approximate the transfer function by H(k, η r ) = sin(kη r )/(kη r ). Changing the integration variables to ξ and u we find We have actually found that the analytical expression (33) has a good agreement with the C BB l calculated by CAMB with ∆ξ r = 0.028 at l < 100. In Fig. 3 we see this agreement with r = 0.1 and A s = 2.2 × 10 −9 . At l < 10 the C BB l curve grows up while the analytical curve displays an opposite behavior. This is due to impact of reionization on the CMB which we have not considered in this work.
We now consider the effects of the modulation in tensor mode on the angular power spectra of CMB. Recall that to derive the multipole coefficients we need to perform the integration over all angles θ. As we discussed the modulation contributes the new factor (1 + A t cos θ) in front of the integrand. We therefore separate the multipole coefficients into a B lm + δa B lm where the δa B lm are those containing the A t cos θ term. The details of the calculation of δa B lm coefficients are presented in Appendix. Using these results one can derive where where we have changed the variables of integration to ξ and u. By considering the modulation the total BB power spectrum will be In Fig. 3 we have also plotted the total predicted BB power spectrum for A t = 0.5 and A t = 1. As we can see the C BB l is shifted above due to the modulation term in (36).

V. CONCLUSION
In this work we have studied the imprints of dipole modulation in tensor modes on the C XY l with XY = T T and BB. The modulation of tensor modes can be due to a long wavelength scalar or tensor mode which is superhorizon during inflation. Here we have modulated the tensor mode by multiplying it's amplitude by a modulated factor like (1 + sin(k · x lss )). The angular power spectra of CMB have been analytically computed in the presence of modulation factor. With modulation in tensor modes one can see a larger modification in the C BB l . The future detection of gravitational waves can constraint the amplitude of modulation. However this task needs a comprehensive study of the effects of modulation in tensor modes on the CMB temperature and polarization anisotropies. Here we have not considered the reionization and lensing effects. Either of these phenomena can change the simplified picture studied in this work.
Inserting the (29) and (30) into equations (A1) and (A2), changing the variable of integration from θ to x and integrating over the azimuthal angle φ we get and as well as a B+ l2 = − a B+ l−2 , a B× l2 = i a B+ l2 and a B× l−2 = −i a B+ l−2 . From the series expansion of plane wave in terms of Legendre polynomials (10) one can find
The angular power spectrum C BB l is given by In the case of modulation in tensor mode we have and some calculations yield