On the joint analysis of CMB temperature and lensing-reconstruction power spectra

TL;DR

This paper rigorously analyzes the covariance between CMB lensing power spectrum and trispectrum measurements, demonstrating that they can be treated as independent for Planck-like experiments, and evaluates likelihood approximations.

Contribution

It provides a detailed covariance analysis between lensing power spectrum and trispectrum, justifies their independent treatment, and tests effective likelihood approximations.

Findings

Correlation between noise bias and sample variance is mitigated by data-dependent correction.

Sample variance of lenses is negligible for Planck-like experiments.

Gaussian likelihood with fixed covariance is a good approximation.

Abstract

Gravitational lensing provides a significant source of cosmological information in modern CMB parameter analyses. It is measured in both the power spectrum and trispectrum of the temperature fluctuations. These observables are often treated as independent, although as they are both determined from the same map this is impossible. In this paper, we perform a rigorous analysis of the covariance between lensing power spectrum and trispectrum analyses. We find two dominant contributions coming from: (i) correlations between the disconnected noise bias in the trispectrum measurement and sample variance in the temperature power spectrum; and (ii) sample variance of the lenses themselves. The former is naturally removed when the dominant N0 Gaussian bias in the reconstructed deflection spectrum is dealt with via a partially data-dependent correction, as advocated elsewhere for other reasons. The remaining lens-cosmic-variance contribution is easily modeled but can safely be ignored for a Planck-like experiment, justifying treating the two observable spectra as independent. We also test simple likelihood approximations for the deflection power spectrum, finding that a Gaussian with a parameter-independent covariance performs well.