GRAVITATIONAL LENSING EFFECT ON COSMIC MICROWAVE BACKGROUND ANISOTROPIES: A POWER SPECTRUM APPROACH

TL;DR

This work presents a power-spectrum framework to quantify gravitational lensing effects on CMB anisotropies across cosmologies, including non-flat geometries and nonlinear evolution. By evolving the gravitational potential power spectrum $P_ ho(k, au)$ and applying Limber's equation, it derives the lensing dispersion $S(\theta)$ and its impact on the observed CMB power spectrum $C_l$, showing that lensing mainly redistributes power and smooths acoustic peaks rather than altering the overall pattern. Using observational constraints such as $P_ ho(k, au)$ from large-scale structure and ellipticity correlations $p(\theta)$, the study finds lensing-induced changes are small on degree scales but become relevant at arcminute scales, potentially erasing peaks for realistic spectra. The results provide a robust postprocessing tool for CMB analyses and clarify why certain earlier, simplified models overestimated the lensing impact on large-angle CMB features. The methodology enables model-by-model lensing calculations for precision cosmology in upcoming experiments.

Abstract

The effect of gravitational lensing on cosmic microwave background (CMB) anisotropies is investigated using the power spectrum approach. The lensing effect can be calculated in any cosmological model by specifying the evolution of gravitational potential. Previous work on this subject is generalized to a non-flat universe and to a nonlinear evolution regime. Gravitational lensing cannot change the gross distribution of CMB anisotropies, but it may redistribute the power and smooth the sharp features in the CMB power spectrum. The magnitude of this effect is estimated using observational constraints on the power spectrum of gravitational potential from galaxy and cluster surveys and also using the limits on correlated ellipticities in distant galaxies. For realistic CMB power spectra the effect on CMB multipole moments is less then a few percent on degree angular scales, but gradually increases towards smaller scales. On arcminute angular scales the acoustic oscillation peaks may be partially or completely smoothed out because of the gravitational lensing.

Figures (3)

• Figure 1: $\sigma(\theta)/\theta$ versus $\theta$ for 3 different values of $\Omega_{m0}$ and $\Omega_{v0}$ using the power spectrum with $\Omega_{m0}=0.25$ and $\sigma_8=0.8$. Thick lines are the result of a full nonlinear calculation, while the thin lines give the corresponding linear case. Also indicated are the 90% c.l. upper limits from ellipticity correlations of distant galaxies, as derived from observations by Fahlman et al. (1994) (A) and Mould et al. (1994) (B).
• Figure 2: CMB anisotropy power spectrum $l(l+1)C_l$ versus $l$ with lensing (dashed lines) and without lensing (solid lines). Upper curves are for adiabatic CDM model with $h=0.5$, $\Omega_{m0}=0.4$ and $\Omega_{v0}=0.6$, lower curves are for adiabatic CDM model with $h=0.5$, $\Omega_{m0}=1$ and $\Omega_{v0}=0$. Both models are normalized to COBE. Lensing smoothes the sharp features in the power spectrum, but leaves the overall shape unchanged. The two models show a typical range of the lensing effect on CMB.
• Figure 3: Comparison between several approximations for calculating the lensing effect on the CMB anisotropies in the COBE normalized CDM model with $h=0.5$ and $\Omega_{m0}h=0.5$. Both nonlinear and linear isotropic approximations gives results that are almost indistinguishable from the fully nonlinear and nonisotropic calculation over this angular range, while $\epsilon={\rm const}$ approximation gives reliable results only over a limited range of $l$ and cannot be used for an accurate calculation of the lensing effect.