# Weak lensing effect on CMB in the presence of a dipole anisotropy

## Abstract

We investigate weak lensing effect on cosmic microwave background (CMB) in the presence of a superhorizon mode which leads to a dipole anisotropy in the CMB power spectrum and in large scale structures. The approach of flat-sky approximation is considered. We determine the functions \(\sigma _0^2\) and \(\sigma _2^2\) that appear in expressions of the lensed CMB power spectrum in the presence of the dipole anisotropy. We determine the correction to B-mode power spectrum which is found to be appreciable at low multipoles (*l*), around \(15\%\) at \(l=10\). However, the temperature and E-mode power spectrum are not altered significantly.

## 1 Introduction

In unveiling the dynamics and contents of the universe, cosmic microwave background radiation has been one of the essential tools. CMB constrains cosmological parameters, and it provides a path in developing modern cosmology. The precision level of the experiments for the CMB spectrum and hence the accuracy of prediction of a theoretical model has been increasing year by year. The CMB anisotropy can be determined by the Physics of last scattering surface (LSS) and the medium effects due to the propagation of photons from LSS to the observer. The primary contribution to the anisotropy is due to the strength of the metric perturbations at LSS. Some other effects such as ISW effect, reionization give additional modifications. The lensing effect is associated with the geodesics of photons from LSS to us. It can be considered as a secondary contribution in modifying CMB anisotropy, and it must be taken into account to reliably predict the CMB signal. The lensing gives a very small contribution on the reionization and the ISW signal as these are associated with the large scale. However, on degree-scale of primary acoustic peaks, weak lensing effect is expected at the order of arc minute on the CMB spectrum [1].

*l*and the alignment of quadrupole and octopole axes. It has also been studied in Refs. [3, 17, 18]. The CMB spectrum in this model has been investigated in Refs. [19, 20, 21]. We assume a long wavelength mode of the form [3, 18]

In this paper, we study the weak lensing effect on the CMB spectrum with the flat-sky approximation in the model given in Eq. (1). The contribution of the standard weak lensing effect to the CMB power spectrum, without including the inhomogeneous term given in Eq. (1), has been computed in Refs. [22, 23, 24]. In Sect. 2, we generalize the two-point correlation function due to the weak lensing by including the super horizon mode. In this section, we also calculate the expectation value of a term which arises due to the isotropic part of lensing potential. In Sect. 3, we estimate the modification to this term in presence of the inhomogeneous term. In Sect. 4, we obtain the expressions for the CMB power spectrum. In Sect. 5, we conclude.

## 2 Gravitational lensing effect on two-point correlation functions in the presence of dipole anisotropy

*x*-axis. In the similar way, we obtain the following relation for \(C_{Q}(\theta )\) and \(C_{U}(\theta )\),

## 3 Dipole anisotropy correction

*x*-direction. We keep \( \psi ^d\) in generalized form of 3D. Now, we redefine the momentum

*k*in terms of

*l*, by \(\vec {l}= \chi ^* \vec {k}\), \(\int d^3 k {{{\varvec{\psi }}}}^d(k) \rightarrow \int d^3l {{{\varvec{\psi }}}}^d(l)\) and \(\frac{\eta _0-\eta }{\chi ^*} = \frac{\chi }{\chi ^*} =\omega \). In this definition,

*x*and

*y*and in the flat sky approximation these are equivalent to \(\frac{\partial }{\ \partial {n_x}}\) and \(\frac{\partial }{\ \partial {n_y}}\) respectively. Here \(n_x\) and \(n_y\) are x and y components of \(\hat{n}\), the unit vector in the direction of observation from us. The two points of observation could be anywhere on the sphere, however, we consider these points on the great circle in the \(x-z\) plane. The unit direction \(\hat{n}\), which is given by

*x*–

*y*plane), we can simplify this quantity as,

## 4 CMB power spectrum

We note that in Eq. (37), the modified power spectra are determined by functions \(W_{1l}^{l'}\), \(W_{2l}^{l'}\) and \(W_{3l}^{l'}\) which depend on \(\sigma _{0T}^2\) and \(\sigma _{2T}^2\). Since theoretically, the primordial \(C_{\tilde{B} l}\) has nearly zero contribution, the correction to \(C_{\tilde{E} l}\) is proportional to \(W_{1l}^{l'}+ W_{2l}^{l'}\) and the only contribution to \(C_{B l}\) comes from the correction which is proportional to \(W_{1l}^{l'}- W_{2l}^{l'}\). In Fig. 2, on the left panel, we plot \(\sigma _0^2\), \(\sigma _2^2\) which are due to standard weak lensing. The function \(\sigma _d^2\) due to dipole distribution is also plotted. We emphasize here that the term \(\sigma _d\) is due to superhorizon mode in gravitational potential perturbation. As mentioned in the introduction, such a term can explain the observed dipole anisotropy in CMB power spectrum. One of the mechanism to produce such a large scale mode is multi-field inflation. For example, the curvaton model can explain the observed deviation in power spectrum [3]. The deviation of background value of scalar field across the observable Universe plays important role in generating power asymmetry. Fitting the observed values leads to the constraint \(|\kappa _s^3 \beta | \lesssim 1.26 \times 10^{-5} H_0^3\), where, \(H_0\) is the Hubble constant [3, 18]. We considered \(\beta =1\) and \(\kappa = 0.046\) which satisfies \(|\kappa _s^3 \beta | \le 1.26 \times 10^{-5} H_0^3\) or \(|\kappa ^3 \beta | \le 10^{-4}\) [18]. This provides an estimate of maximum possible change that can arise due to the dipole. We observe a reasonable value of \(\sigma _d^2\) comparable to \(\sigma _0^2\). In Fig. 3, on left panel, we plot the fractional changes \( \left( \frac{\Delta C_{El}}{C_{El}}, \frac{\Delta C_{Tl}}{C_{Tl}}\right) \) due to weak lensing with dipole distribution. These are very small due to presence of the leading order terms \(\tilde{C}_{El}\) and \(\tilde{C}_{Tl}\). The term \(W_{1l}^{l'}+ W_{2l}^{l'}\) is oscillatory as well as roughly symmetric (see right panel of Fig. 4). Therefore, we do not get much contribution to the \(C_{El}\) due to the dipole distribution. This can be seen on right panel of Fig. 3 where we plot \(({\Delta C^d_{El}-\Delta C^s_{El}})/{\Delta C^s_{El}}\) and \(({\Delta C^d_{Tl}-\Delta C^s_{Tl}})/{\Delta C^s_{Tl}}\), where \(\Delta C^d_{El}, \Delta C^d_{Tl}\) are changes due to weak lensing in the presence of dipole and \(\Delta C^s_{El}, \Delta C^s_{Tl}\) are changes due to standard lensing. Similarly, \(W_{1l}^{l'}\) forms a roughly symmetric pattern (see Fig. 5) and it does not give much change to \(C_{Tl}\) in the standard weak lensing modification due to dipole anisotropy. The fractional changes observed in lensing for \(C_{El}\) and \(C_{Tl}\) are shown in the left panel of Fig. 3. Those values at \(l=10\) are around \(10^{-3}\) and \(10^{-7}\) respectively. Fractional contributions from dipole are shown in right panel of the plot. Those values are approximately 0.05 and \( 10^{-3}\) respectively. So, in total, dipole gives very tiny change in these spectra, e.g. for \(C_{El}\) it is \(\sim 10^{-5}\).

The contribution of \(\sigma _d^2\) appears in the B-mode spectrum. The dipole distribution marginally decreases its value due to the presence of function \(W_{1l}^{l'}- W_{2l}^{l'}\). In the right panel of Fig. 2, we plot the *B*-mode power spectrum including the dipole contribution. We also show its absolute fractional change on right side vertical axis in this plot. The standard weak lensing result is shown for comparison. In plotting the lensed \(C_{Bl}\), we use unlensed \(\tilde{C}_{Bl}=0\) and \(\tilde{C}_{El}\) from CAMB package. We restrict this curve for \(l>10\) for which the Flat sky approximation is expected to be reliable [26]. We find that the dipole contribution leads to a small downward shift of the spectrum at low values of *l*. The shift is large at low *l* and roughly 1% at *l* of order 40. Hence it is small but not negligible. At higher values \(l\ge 40\), the fraction change becomes smaller and becomes \(\sim 0.1\%\) at \(l\sim 100\). In Fig. 4, on the left panel, we note that \(W_{1l}^{l'}- W_{2l}^{l'}\) is oscillatory and mainly lie in positive-value quadrant. Due to this asymmetry, dipole distribution leads to a modification of \(C_{Bl}\).

## 5 Conclusions

We generalized the formalism of weak lensing by adding a superhorizon mode in the gravitational potential. Such a mode leads to a dipole modulation in the CMB power spectrum as well as a dipole anisotropy in the large scale structures. We estimated the effect of lensing on the CMB power spectrum in this model. We followed the approach of the flat sky approximation. Weak lensing mixes E and B mode and hence we necessarily find some contribution to \(C_{Bl}\) due to weak lensing. The correction to \(C_{Bl}\) is proportional to \(\tilde{C}_{El}\). We observed that adding dipole anisotropy changes \(C_{Bl}\) for the lower range of *l*. However, dipole anisotropy does not lead to an appreciable change in \(C_{El}\), since the correction is much smaller than the leading order term \(\tilde{C}_{El}\). We did not even observe much change for lower *l* in the standard weak lensing modification due to the dipole anisotropy, since \(W_{1l}^{l'}+ W_{2l}^{l'}\) has an approximately symmetric pattern. The same argument applies to \( C_{Tl}\). In the case of \(C_{Bl}\) we obtain a correction since in this case the leading order \(\tilde{C}_{Bl}\) is zero and \(W_{1l}^{l'}- W_{2l}^{l'}\) is not symmetric. For this case we computed the maximum possible correction to \(C_{Bl}\) by fixing the direction of dipole along the line on which, the projections of the observation points lie. A more reliable calculation would use the spherical harmonics approach. Flat sky approximation and the spherical harmonics approach deviate below \(l=10\) [26]. Thus, our calculation is reliable for \(l>10\), where we still find an observable correction to \(C_{Bl}\). Due to dipole term, the low multipoles get a significant correction for the B-mode. We may detect the resulting lensing effect as the polarization noise in CMB observations planned for near future is expected to be around one-fifth of lensing-B mode spectrum [27]. Our calculation would be useful in extracting the information about B-mode from the future experiments as long as dipole asymmetry persists in observations.

## Notes

### Acknowledgements

The work of Prabhakar Tiwari is supported by NSFC Grants 11720101004 and 11673025, and the National Key Basic Research and Development Program of China (No. 2018YFA0404503). The work of Pankaj Jain is supported by a grant from the Science and Engineering Research Board (SERB), Government of India.

## References

- 1.A. Lewis, A. Challinor, Phys. Rept.
**429**, 1 (2006). https://doi.org/10.1016/j.physrep.2006.03.002. arXiv:astro-ph/0601594 ADSCrossRefGoogle Scholar - 2.C. Gordon, W. Hu, D. Huterer, T.M. Crawford, Phys. Rev. D
**72**, 103002 (2005). https://doi.org/10.1103/PhysRevD.72.103002. arXiv:astro-ph/0509301 ADSCrossRefGoogle Scholar - 3.A.L. Erickcek, M. Kamionkowski, S.M. Carroll, Phys. Rev. D
**78**, 123520 (2008). https://doi.org/10.1103/PhysRevD.78.123520. arXiv:0806.0377 [astro-ph]ADSCrossRefGoogle Scholar - 4.C.L. Bennett, et al. [WMAP Collaboration], Astrophys. J. Suppl.
**208**, 20 (2013). https://doi.org/10.1088/0067-0049/208/2/20. arXiv:1212.5225 [astro-ph.CO]ADSCrossRefGoogle Scholar - 5.M. Tegmark, A. de Oliveira-Costa, A. Hamilton, Phys. Rev. D
**68**, 123523 (2003). https://doi.org/10.1103/PhysRevD.68.123523. arXiv:astro-ph/0302496 - 6.C.J. Copi, D. Huterer, G.D. Starkman, Phys. Rev. D
**70**, 043515 (2004). https://doi.org/10.1103/PhysRevD.70.043515. arXiv:astro-ph/0310511 ADSCrossRefGoogle Scholar - 7.K. Land, J. Magueijo, Phys. Rev. Lett.
**95**, 071301 (2005). https://doi.org/10.1103/PhysRevLett.95.071301. arXiv:astro-ph/0502237 ADSCrossRefGoogle Scholar - 8.P. Vielva, E. Martinez-Gonzalez, R.B. Barreiro, J.L. Sanz, L. Cayon, Astrophys. J.
**609**, 22 (2004). https://doi.org/10.1086/421007. arXiv:astro-ph/0310273 ADSCrossRefGoogle Scholar - 9.P. Mukherjee, Y. Wang, Astrophys. J.
**613**, 51 (2004). https://doi.org/10.1086/423021. arXiv:astro-ph/0402602 ADSCrossRefGoogle Scholar - 10.M. Cruz, E. Martinez-Gonzalez, P. Vielva, L. Cayon, Mon. Not. R. Astron. Soc.
**356**, 29 (2005). https://doi.org/10.1111/j.1365-2966.2004.08419.x. arXiv:astro-ph/0405341 ADSCrossRefGoogle Scholar - 11.H.K. Eriksen, F.K. Hansen, A.J. Banday, K.M. Gorski, P.B. Lilje, Astrophys. J.
**605**, 14 (2004). Erratum: [Astrophys. J.**609**, 1198 (2004)]. https://doi.org/10.1086/382267. arXiv:astro-ph/0307507 ADSCrossRefGoogle Scholar - 12.P.A.R. Ade, et al. [Planck Collaboration], Astron. Astrophys. A
**571**, 23 (2014). https://doi.org/10.1051/0004-6361/201321534. arXiv:1303.5083 [astro-ph.CO] - 13.P. Jain, J.P. Ralston, Mod. Phys. Lett. A
**14**, 417 (1999). https://doi.org/10.1142/S0217732399000481. arXiv:astro-ph/9803164 ADSCrossRefGoogle Scholar - 14.J.P. Ralston, P. Jain, Int. J. Mod. Phys. D
**13**, 1857 (2004). https://doi.org/10.1142/S0218271804005948. arXiv:astro-ph/0311430 ADSCrossRefGoogle Scholar - 15.D. Hutsemékers, A&A
**332**, 410–428 (1998)ADSGoogle Scholar - 16.P. Jain, G. Narain, S. Sarala, Mon. Not. R. Astron. Soc.
**347**, 394 (2004). https://doi.org/10.1111/j.1365-2966.2004.07169.x. arXiv:astro-ph/0301530 ADSCrossRefGoogle Scholar - 17.A.R. Liddle, M. Cortês, Phys. Rev. Lett.
**111**(11), 111302 (2013). https://doi.org/10.1103/PhysRevLett.111.111302. arXiv:1306.5698 [astro-ph.CO]ADSCrossRefGoogle Scholar - 18.S. Ghosh, Phys. Rev. D
**89**, 063518 (2014). https://doi.org/10.1103/PhysRevD.89.063518. arXiv:1309.6547 [astro-ph.CO]ADSCrossRefGoogle Scholar - 19.R. Kothari, S. Ghosh, P.K. Rath, G. Kashyap, P. Jain, Mon. Not. R. Astron. Soc.
**460**(2), 1577 (2016). https://doi.org/10.1093/mnras/stw1039. arXiv:1503.08997 [astro-ph.CO]ADSCrossRefGoogle Scholar - 20.S. Ghosh, R. Kothari, P. Jain, P.K. Rath, JCAP
**1601**(01), 046 (2016). https://doi.org/10.1088/1475-7516/2016/01/046. arXiv:1507.04078 [astro-ph.CO]ADSCrossRefGoogle Scholar - 21.S. Ghosh, P. Jain, arXiv:1807.02359 [astro-ph.CO]
- 22.A. Blanchard, J. Schneider, Astron. Astrophys.
**184**, 1 (1987)ADSGoogle Scholar - 23.U. Seljak, Astrophys. J.
**463**, 1 (1996). https://doi.org/10.1086/177218. arXiv:astro-ph/9505109 ADSCrossRefGoogle Scholar - 24.M. Zaldarriaga, U. Seljak, Phys. Rev. D
**58**, 023003 (1998). https://doi.org/10.1103/PhysRevD.58.023003. arXiv:astro-ph/9803150 ADSCrossRefGoogle Scholar - 25.B. Yu, T. Lu, Astrophys. J.
**698**, 1771 (2009). https://doi.org/10.1088/0004-637X/698/2/1771. arXiv:0903.4519 [astro-ph.CO]ADSCrossRefGoogle Scholar - 26.W. Hu, Phys. Rev. D
**62**, 043007 (2000). https://doi.org/10.1103/PhysRevD.62.043007. arXiv:astro-ph/0001303 ADSCrossRefGoogle Scholar - 27.A. Challinor, et al. [CORE Collaboration], JCAP
**1804**(04), 018 (2018). https://doi.org/10.1088/1475-7516/2018/04/018. arXiv:1707.02259 [astro-ph.CO]CrossRefGoogle Scholar

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