The Parameter-Level Performance of Covariance Matrix Conditioning in Cosmic Microwave Background Data Analyses

Abstract

Empirical estimates of the band power covariance matrix are commonly used in cosmic microwave background (CMB) power spectrum analyses. While this approach easily captures correlations in the data, noise in the resulting covariance estimate can systematically bias the parameter fitting. Conditioning the estimated covariance matrix, by applying prior information on the shape of the eigenvectors, can reduce these biases and ensure the recovery of robust parameter constraints. In this work, we use simulations to benchmark the performance of four different conditioning schemes, motivated by contemporary CMB analyses. The simulated surveys measure the $TT$, $TE$, and $EE$ power spectra over the angular multipole range $300 \le \ell \le 3500$ in $\Delta \ell = 50$ wide bins, for temperature map-noise levels of $10, 6.4$ and $2\,\mu$K-arcmin. We divide the survey data into $N_{\mathrm{real}} = 30, 50,$ or 100 uniform subsets. We show the results of different conditioning schemes on the errors in the covariance estimate, and how these uncertainties on the covariance matrix propagate to the best-fit parameters and parameter uncertainties. The most significant effect we find is an additional scatter in the best-fit point, beyond what is expected from the data likelihood. For a minimal conditioning strategy, $N_{\mathrm{real}} = 30$, and a temperature map-noise level of 10$\,\mu$K-arcmin, we find the uncertainty on the recovered best fit parameter to be $\times 1.3$ larger than the apparent posterior width from the likelihood ($\times 1.2$ larger than the uncertainty when the true covariance is used). Stronger priors on the covariance matrix reduce the misestimation of parameter uncertainties to $<1\%$. As expected, empirical estimates perform better with higher $N_{\mathrm{real}}$, ameliorating the adverse effects on parameter constraints.