Gravitational Lensing Effect on Cosmic Microwave Background Polarization

TL;DR

This work analyzes how gravitational lensing by the universe’s matter distribution alters the CMB polarization and temperature–polarization cross-spectra. Extending prior temperature-focused lensing formalisms, it derives compact, numerically efficient expressions for the lensed spectra and demonstrates that lensing induces $E$-$B$ mixing, generating a nonzero $C_{Bl}$ from the intrinsic $E$; these results are implemented in the CMBFAST code as version 2.4. Using a concordance cosmology, the authors show that polarization is more sensitive to lensing than temperature, with changes up to about ten percent near $l\sim 1000$, and quantify the observability of the induced $B$-mode with Planck as marginal, while highlighting the potential of ground-based experiments to map the lensing signal and constrain the matter power spectrum across large redshift ranges. The study emphasizes incorporating lensing into high-precision CMB analyses and demonstrates a pathway to extracting information about structure formation from polarization data.

Abstract

We investigate the effect of gravitational lensing by matter distribution in the universe on the cosmic microwave background (CMB) polarization power spectra and temperature-polarization cross-correlation spectrum. As in the case of temperature spectrum gravitational lensing leads to smoothing of narrow features and enhancement of power on the damping tail of the power spectrum. Because acoustic peaks in polarization spectra are narrower than in the temperature spectrum the smoothing effect is significantly larger and can reach up to 10\% for $l<1000$ and even more above that. A qualitatively new feature is the generation of $B$ type polarization even when only $E$ is intrinsically present, such as in the case of pure scalar perturbations. This may be directly observed with Planck and other future small scale polarization experiments. The gravitational lensing effect is incorporated in the new version (2.4) of CMBFAST code.

Figures (2)

• Figure 1: The upper panel shows the two functions $\sigma_0(\theta)$ and $\sigma_2(\theta)$ for the cosmic concordance model discussed in the text. The lower panels show ${1 \over 2}[W_{1 l}^{l^{\prime}} \pm W_{2 l}^{l^{\prime}}]$ for $l=2000$. The Window function for the mixing between $E$ and $B$ is is not well localized in $l$ space and is rapidly oscillating as a function of $l$.
• Figure 2: The upper panel shows the $T$, $E$ and $C$ power spectra. Dashed (solid) lines correspond to the lensed (unlensed) spectra. The bottom left panel shows the relative difference between the lensed and unlensed spectra for $T$ and $E$ ($\delta C_l/C_l \equiv (C_l^{lensed}-C_l^{unlensed})/C_l^{unlensed}$) and shows both suppression of oscillations and enhancement of power on small scales. The bottom right panel shows the $B$ type polarization induced by lensing. We include the $E$ and $B$ spectra for inflationary model where scalars and tensors produce equal amount of power in the temperature on COBE scales ($T/S=1$).