Lensed CMB power spectra from all-sky correlation functions

TL;DR

This work addresses the need for precise modeling of CMB lensing effects on power spectra to enable unbiased parameter constraints from high-precision data. It introduces a full-sky correlation-function approach that is non-perturbative in the isotropic lensing displacement and perturbative to second order in the anisotropic component, computed under the Born approximation for a linear Gaussian lensing potential. The authors demonstrate accuracy better than $0.1\%$ for $l<2500$, compare against previous harmonic perturbative results and flat-sky methods, and quantify non-linear evolution using Halofit, finding small temperature changes but substantial $B$-mode enhancements. The method is fast, scalable to MCMC analyses, and, with public CAMB-based code, provides a practical tool for precision cosmology and lensing studies of the CMB.

Abstract

Weak lensing of the CMB changes the unlensed temperature anisotropy and polarization power spectra. Accounting for the lensing effect will be crucial to obtain accurate parameter constraints from sensitive CMB observations. Methods for computing the lensed power spectra using a low-order perturbative expansion are not good enough for percent-level accuracy. Non-perturbative flat-sky methods are more accurate, but curvature effects change the spectra at the 0.3-1% level. We describe a new, accurate and fast, full-sky correlation-function method for computing the lensing effect on CMB power spectra to better than 0.1% at l<2500 (within the approximation that the lensing potential is linear and Gaussian). We also discuss the effect of non-linear evolution of the gravitational potential on the lensed power spectra. Our fast numerical code is publicly available.

Figures (6)

• Figure 1: The power spectrum of the lensing potential for a concordance $\Lambda$CDM model. The linear theory spectrum (solid) is compared with the same model including non-linear corrections (dashed) from halofitSmith:2002dz using Eq. \ref{['Tnonlin']}.
• Figure 2: The functions $\sigma^2(\beta)\equiv C_{\text{gl}}(0)-C_{\text{gl}}(\beta)$ [solid] and $C_{\text{gl},2}(\beta)$ [dashed] as a function of angular separation $\beta$ (in radians) for a typical concordance model. The results are calculated using the full-sky definitions of Eqs. \ref{['cgl_def']}, and use the linear power spectrum for $C_l^\psi$.
• Figure 3: The geometry of the weak lensing deflections (shown without curvature for clarity).
• Figure 4: Difference between the lensed and unlensed temperature, cross-correlation and $E$-polarization power spectra (top three plots), and the lensed $B$ power spectrum (bottom) for a fiducial concordance model. The unlensed model has no tensor component (so no $B$-mode power), and the lensed $B$ power spectrum shown is not highly accurate due to the neglect of non-linear evolution in the lensing potential. The magnitude of the lensing effect depends on the fluctuation amplitude in the model; here the model has curvature perturbation power $A_s = 2.5\times 10^{-9}$ on $0.05\,\text{Mpc}^{-1}$ scales and spectral index $n_s=0.99$.
• Figure 5: Comparison of our new result with the $O(C_l^\psi)$ harmonic result of Ref. Hu:2000ee (dashed) and the flat-sky non-perturbative result of Ref. Seljak:1996ve, extended to second order in $C_{\text{gl},2}$ (solid). The magnitude of the difference depends on the exact model and we have neglected non-linear contributions to the lensing potential.