Constraining Power of Wavelet vs. Power Spectrum Statistics for CMB Lensing and Weak Lensing with Learned Binning

TL;DR

This study compares the information content of standard angular power spectra against nonlinear, wavelet-based statistics (WST for auto-lensing and WPH for cross-lensing) in CMB lensing and galaxy weak-lensing contexts, using the ULAGAM simulation suite. A novel learned binning approach compresses high-dimensional summaries to a small, interpretable set while maximizing the Fisher information, and cross-validation mitigates overfitting. Key findings show that WST provides modest gains over $C_\ell$ for Planck-like data and similar gains for SO/SPT, while WPH dramatically enhances constraints for CMB×WL cross-correlations, with improvements up to factors ~2.7–3.8 in $Ω_m$ and $σ_8$ depending on the survey. The work demonstrates robust, interpretable gains from non-Gaussian statistics in cross-survey analyses and introduces a practical compression framework that preserves statistical power and resists overfitting.

Abstract

We present forecasts for constraints on the matter density ($Ω_m$) and the amplitude of matter density fluctuations at 8h$^{-1}$Mpc ($σ_8$) from CMB lensing convergence maps and galaxy weak lensing convergence maps. For CMB lensing convergence auto statistics, we compare the angular power spectra ($C_\ell$'s) to the wavelet scattering transform (WST) coefficients. For CMB lensing convergence $\times$ galaxy weak lensing convergence statistics, we compare the cross angular power spectra to wavelet phase harmonics (WPH). This work also serves as the first application of WST and WPH to these probes. For CMB lensing convergence, we find that WST and $C_\ell$'s yield similar constraints in forecasts for the $\textit{Simons}$ Observatory and the South Pole Telescope. However, WST gives a tighter constraint on $σ_8$ by a factor of $1.7$ for $\textit{Planck}$ data. When CMB lensing convergence is crossed with galaxy weak lensing convergence projected from $\textit{Euclid}$ Data Release 2 (DR2), we find that WPH outperforms cross-$C_\ell$'s by factors between $2.4$ and $3.8$ for individual parameter constraints. To compare these different summary statistics we develop a novel learned binning approach. This method compresses summary statistics while maintaining interpretability. We find this leads to improved constraints compared to more naive binning schemes for $C_\ell$'s, WST, and most significantly WPH. By learning the binning and measuring constraints on distinct data sets, our method is robust to overfitting by construction.

Figures (15)

• Figure 1: We show the average noiseless auto power spectrum of $\kappa_\text{CMB}$ (black) in comparison to noise power spectra from all CMB surveys used (dashed lines). For ACT, we only plot until $\ell=2090$ as this is the limit of our noise power spectrum data. For SPT, we plot the noise power spectra from the main and summer fields separately. The cusps (for instance at $\ell\approx2200$ for SPT) occur because our noise power spectra are linear interpolations between specified points.
• Figure 1: Constraint comparisons between surveys for all summary statistics used. The upper plots use $\kappa_\text{CMB}$ while the bottom plots use $\kappa_\text{CMB}\times\kappa_\text{WL}$.
• Figure 1: Contour convergence as fewer derivatives are used for cross-$C_\ell$'s applied to $\kappa_\text{CMB}\times\kappa_\text{WL}$. Shown are contours when 50% (blue) and 100% (orange) of the total available derivatives are used.
• Figure 1: Probability density functions of the $\chi^2$ distribution for all summary statistics used. One survey is picked at random for each summary statistic for brevity, but results for other surveys are comparable. $500$ fiducial realizations (orange) are used for the distributions after being binned with a binning learned by the other $500$ fiducial simulations. Alongside the distribution of the data, we show the theoretical $\chi^2$ distribution with $15$ degrees of freedom (black) and samples drawn from a Gaussian distribution with the same mean and covariance as the data (blue). We find no evidence of significant departures from Gaussianity for any of our statistics.
• Figure 2: We show $\kappa_\text{CMB}$ with Planck levels of noise on Planck's footprint.