Impact of Next-to-Leading Order Contributions to Cosmic Microwave Background Lensing

In this Letter we study the impact on cosmological parameter estimation, from present and future surveys, due to lensing corrections on cosmic microwave background temperature and polarization anisotropies beyond leading order. In particular, we show how post-Born corrections, large-scale structure effects, and the correction due to the change in the polarization direction between the emission at the source and the detection at the observer are non-negligible in the determination of the polarization spectra. They have to be taken into account for an accurate estimation of cosmological parameters sensitive to or even based on these spectra. We study in detail the impact of higher order lensing on the determination of the tensor-to-scalar ratio $r$ and on the estimation of the effective number of relativistic species $N_{\mathrm{eff}}$. We find that neglecting higher order lensing terms can lead to misinterpreting these corrections as a primordial tensor-to-scalar ratio of about $\mathit{O}({10}^{-3})$. Furthermore, it leads to a shift of the parameter $N_{\mathrm{eff}}$ by nearly $2\sigma$ considering the level of accuracy aimed by future S4 surveys.

Figure 1

Next-to-leading order corrections to $B$ modes. We show the correction from the post-Born second group (thin dot-dashed red), the post-Born third group (thick dot-dashed blue), and the LSS (thin orange), and the contribution coming from the rotation angle ${\beta }^{(2)}$ (thick green); negative values are dashed. All functions are normalized to the resummed first order deflection angle ${\tilde{C}}_{\ell }^{\mathcal{B}(1)}$, which is included in standard CMB codes.

Figure 2

The first order resummed lensing correction to the $B$ modes of the CMB (thick blue line) and the additional correction from the full higher order contribution ($\mathrm{post}\text{−}\mathrm{Born}+\mathrm{LSS}+\mathrm{rotation}\mathrm{contributions}$, thin red line) are compared. The gray lines refer to primordial $B$ modes with no lensing for different values of $r$, $r={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, ${10}^{-5}$, ${10}^{-6}$ from top to bottom. Negative values are dashed.

Figure 3

The bias introduced in cosmological parameter estimation if subleading CMB lensing effects are neglected. We consider multipoles up to ${\ell }_{\max }=3500$ for $TT$, $EE$, and $TE$ and ${\ell }_{\max }=1500$ for $BB$ spectra, for an ideal cosmic variance limited survey with sky coverage $f_{\mathrm{sky}}=0.75$ for $TT$, $EE$, and $TE$ and $f_{\mathrm{sky}}=0.5$ for $BB$ spectra, keeping all the parameters not shown fixed at their fiducial values.