Lensing of the CMB: Non Gaussian aspects

TL;DR

The paper addresses non-Gaussian signatures generated in the CMB by gravitational lensing on small angular scales and derives a direct link between the tail of the lensed power and the deflection-angle spectrum through $\langle T(\mathbf{l}_1) T(\mathbf{l}_2)\rangle = (2\pi)^2 \delta^D(\mathbf{l}_{12}) \sigma_S C^{\delta\delta}_{l_1}/2$. It develops a comprehensive framework—via toy cluster lensing and large-scale structure in the small-angle limit—to show how lensing induces a characteristic non-Gaussian pattern, including a strong correlation between small-scale power and the large-scale gradient, and provides explicit expressions for the three- and four-point functions. The authors introduce estimators based on specific quadrilateral configurations of the four-point function and demonstrate how to maximize signal-to-noise for lensing detection, including cross-correlation tests with the large-scale gradient to distinguish lensing from intrinsic CMB or other secondaries. The work bridges prior analyses by unifying the lensing-induced non-Gaussianities into a consistent set of statistics (three- and four-point functions) and highlights practical pathways for using small-scale CMB data to constrain the matter power spectrum and cosmological parameters via lensing.

Abstract

We study the generation of CMB anisotropies by gravitational lensing on small angular scales. We show these fluctuations are not Gaussian. We prove that the power spectrum of the tail of the CMB anisotropies on small angular scales directly gives the power spectrum of the deflection angle. We show that the generated power on small scales is correlated with the large scale gradient. The cross correlation between large scale gradient and small scale power can be used to test the hypothesis that the extra power is indeed generated by lensing. We compute the three and four point function of the temperature in the small angle limit. We relate the non-Gaussian aspects presented in this paper as well as those in our previous studies of the lensing effects on large scales to the three and four point functions. We interpret the statistics proposed in terms of different configurations of the four point function and show how they relate to the statistic that maximizes the S/N.

Figures (10)

• Figure 1: In the upper panel we show a cluster lensing a background gradient. The bottom panel shows the temperature measured for a fixed $\theta_x$ as a function of $\theta_y$ in the presence and absence of the cluster. Points with $\theta_y>0$ get deflected to a smaller $\theta_y$ in the lens plane and thus for a positive gradient they will have a lower temperature in the lensed example than in the unlensed one. The opposite is true if $\theta_y<0$.
• Figure 2: Temperature profile of a CMB gradient lensed by a cluster. We took $T_{y0}\delta\theta= 13 \mu K$. The gradient part has been subtracted out for clarity. The cluster profile was cut-off at $4\ {\rm arcmin}$.
• Figure 3: The panel (a) shows the lensed and unlensed temperature power spectra together with $\sigma_S C_l^{\delta \delta}/2$. Panel (b) the power spectra of the derivative of the unlensed CMB field, (c) the spectra of $\delta {\hbox{\boldmath{$\theta$}}}$ and (d) the cumulative power in the CMB derivative.
• Figure 4: Example of the generation of power by gravitational lensing in one dimension. The upper panel shows the unlensed temperature and the result of filtering the lensed $T$. The middle panel shows the high $l$ power generated by lensing. In the bottom we show the square of the small scale power and the square of the large scale gradient (arbitrarily scaled).
• Figure 5: The upper panel on the left shows the lensed temperature field. The upper right panel shows the high passed temperature. Bottom left has gradient square and bottom right smoothed square of high passed temperature.