Weak Lensing of the CMB: Extraction of Lensing Information from the Trispectrum

TL;DR

The paper develops a full trispectrum (four-point) framework for CMB weak lensing, showing how the diagonal of the trispectrum can be accessed via the power spectrum of squared temperatures to recover the lensing deflection spectrum $C_l^{\phi\phi}$. It extends prior work by including non-Gaussian noise from lensing itself and from lensing–secondary couplings (notably with SZ and ISW) and by deriving optimal filters for temperature and polarization-based estimators, including EE and EB variants. A key result is that removing the Gaussian noise bias in the squared-temperature statistic dramatically lowers the reconstruction noise (by an order of magnitude), enabling high-S/N lensing measurements with Planck and polarization-based gains that help disentangle inflationary B-modes from lensing-induced B-modes. The study emphasizes the necessity of multifrequency SZ cleaning and high-resolution, arcminute-scale observations to realize robust, confusion-free constraints on the inflationary energy scale and to exploit CMB lensing as a powerful cosmological and astrophysical probe.

Abstract

We discuss the four-point correlation function, or the trispectrum in Fourier space, of CMB temperature and polarization anisotropies due to the weak gravitational lensing effect by intervening large scale structure. We discuss the squared temperature power spectrum as a probe of this trispectrum and, more importantly, as an observational approach to extracting the power spectrum of the deflection angle associated with the weak gravitational lensing effect on the CMB. We extend previous discussions on the trispectrum and associated weak lensing reconstruction from CMB data by calculating non-Gaussian noise contributions, beyond the previously discussed dominant Gaussian noise. Non-Gaussian noise contributions are generated by lensing itself and by the correlation between the lensing effect and other foreground secondary anisotropies in the CMB such as the Sunyaev-Zel'dovich (SZ) effect. When the SZ effect is removed from temperature maps using its spectral dependence, we find these additional non-Gaussian noise contributions to be an order of magnitude lower than the dominant Gaussian noise. If the noise-bias due to the dominant Gaussian part of the temperature squared power spectrum is removed, then these additional non-Gaussian contributions provide the limiting noise level for the lensing reconstruction. The temperature squared power spectrum allows a high signal-to-noise extraction of the lensing deflections and a confusion-free separation of the curl (or B-mode) polarization due to inflationary gravitational waves from that due to lensed gradient (or E-mode) polarization. The small angular scale temperature and polarization anisotropy measurements provide a novel approach to weak lensing studies, complementing the approach based on galaxy ellipticities.

Figures (7)

• Figure 1: Left: The lensing trispectrum configuration, following equation (\ref{['eqn:trilens']}), where the diagonal of the configuration contains information related to lensing while the sides of the quadrilateral contain information related to primary anisotropy power spectra at the last scattering surface. Right: This lensing information can be extracted by a statistic that effectively probes the diagonal, such as the power spectrum of squared temperatures discussed here. This statistic sums over all quadrilateral configurations, as indicated by thin lines, for a given length of the diagonal. As we find later, the lensing information associated with the other --- vertical in this diagram --- diagonal, proportional to $C_{|{\bf l}_1+{\bf l}_2|}^{\phi \phi}$, acts as a source of non-Gaussian noise in this extraction. The other contribution to the non-Gaussian noise, through a term proportional to $C_{|{\bf l}-{\bf l}_1+{\bf l}_2|}$, comes from interchanging ${\bf l}_1$ with ${\bf l}-{\bf l}_1$, which alters the vertical diagonal while keeping the horizontal diagonal fixed.
• Figure 2: Optimal filter function $W({\bf l}_1+{\bf l}_2, {\bf l}_1)$ for the extraction of $C_l^{\phi\phi}$ from the CMB$^2$-CMB$^2$ power spectrum as a function of $l_1$ and $l_2$. The angle between ${\bf l}_1$ and ${\bf l}_2$ is constrained by the fixed value of $l = |{\bf l}_1 + {\bf l}_2|$ in each plot. The left plot $(l=100)$ and right plot $(l=2000)$ correspond to large and small angular scales respectively. The optimal filter removes excess noise at multipoles less than $\sim$ 1500 and is weighted to exploit the sensitivity to lensing of the damping tail.
• Figure 3: The extraction of the lensing power spectrum from temperature data alone. We show the dominant Gaussian noise (top lines) and the noise associated with the extra terms in the trispectrum due to lensing alone (bottom lines). The curves are for three values of the resolution in the CMB temperature map with an effective sensitivity of 1 $\mu$K $\sqrt{\rm sec}$.
• Figure 4: The extraction of lensing power spectrum from temperature data in the presence of foregrounds (mainly the thermal SZ effect). The increase in dominant Gaussian noise (top lines) is due to the extra SZ power spectrum, which peaks at small angular scales. The dot-dashed line is the resulting non-Gaussian noise contribution due to the trispectrum formed by the SZ-lensing cross-correlation.
• Figure 5: The extraction of lensing power spectrum from polarization data. We show the associated noise due to the dominant Gaussian noise (top lines) and noise due to extra terms in the trispectrum (bottom lines). The curves are for a resolution of 5 arcmin in the CMB temperature map with an effective sensitivity of 1 $\mu$K $\sqrt{\rm sec}$. As shown, the EB estimator probes the lensing potential power spectrum to smaller angular scales than the quadratic estimator on temperature data alone or estimators based on other combinations of polarization and the temperature.