Weak Lensing Detection in CMB Maps

TL;DR

This work shows that weak gravitational lensing imprints a detectable non-Gaussian signature in CMB maps, specifically in the connected four-point function, even though the primary CMB is Gaussian and its power spectrum largely encodes standard cosmological information. By expanding the lensed temperature field and computing the four-point function, the paper links the signal to the derivative of the two-point function and to the angular correlation of the lensing displacement field, with a characteristic geometric dependence controlled by the relative patch angles. Quantitative predictions for two CDM-like models indicate the dimensionless four-point signal can be of order a few per mille (κ4 ~ 5×10^-3) and is sensitive to the power-spectrum shape and the lensing efficiency along the line of sight, suggesting this non-Gaussian statistic provides complementary cosmological constraints beyond $C_l$. The analysis also highlights practical considerations for detection, including beam smoothing, geometry-assisted strategies, and potential confounding contributions from nonlinear Doppler effects or foregrounds.

Abstract

The weak lensing effects are known to change only weakly the shape of the power spectrum of the Cosmic Microwave Background (CMB) temperature fluctuations. I show here that they nonetheless induce specific non-Gaussian effects that can be detectable with the four-point correlation function of the CMB anisotropies. The magnitude and geometrical dependences of this correlation function are investigated in detail. It is thus found to scale as the square of the derivative of the two-point correlation function and as the angular correlation function of the gravitational displacement field. It also contains specific dependences on the shape of the quadrangle formed by the four directions. When averaged at a given scale, the four-point function, that identifies with the connected part of the fourth moment of the probability distribution function of the local filtered temperature, scales as the square of logarithmic slope of its second moment, and as the variance of the gravitational magnification at the same angular scale. All these effects have been computed for specific cosmological models. It is worth noting that, as the amplitude of the gravitational lens effects has a specific dependence on the cosmological parameters, the detection of the four-point correlation function could provide precious complementary constraints to those brought by the temperature power spectrum.

Figures (11)

• Figure 1: Description of the angles intervening in the expression (\ref{['corr4']}) of the four-point correlation function. The thick solid line materializes the ${\rm d}\,C(\theta)/{\rm d}\,\theta$ factor, whereas the hatched lines represents the correlation functions of the displacement field.
• Figure 2: Geometrical representation of the terms intervening in the expression (\ref{['G1K']}).
• Figure 3: Geometrical representation of the terms intervening in the expression (\ref{['G2K']}).
• Figure 4: The function $1/C(\theta)\ {\rm d} C(\theta)/{\rm d}\theta$ as a function of the angle. The thick line is for model 1 and the thin line for model 2
• Figure 5: The functions $D_0$ (solid lines), $D_2$ (dashed lines) and $\overline{D}_0$ (long dashed lines) for model 1 (thick lines) and model 2 (thin lines).