Notes on bias and covariance matrix of the angular power spectrum on small sky maps

TL;DR

This paper investigates how small sky maps impact the measurement of the CMB angular power spectrum, emphasizing the importance of accounting for systematic effects and providing numerical estimates of cosmic variance and covariance.

Contribution

It introduces a detailed analysis of bias and covariance in small sky maps for CMB measurements, including numerical calculations and comparisons with simulations.

Findings

Small systematic effects significantly influence power spectrum measurements.

Numerical estimates of cosmic variance and covariance are provided for various map geometries.

Comparisons with simulations validate the analytical approaches.

Abstract

We compute the effects induced by the use of small CMB maps on the measurement of the $\cl{l}$ coefficients of the angular power spectrum and show that small systematic effects have to be taken into account. We also compute numerically the cosmic variance and covariance of the $\cl{l}$ spectrum for various spherical cap like maps. Comparisons with simulations are presented. The calculations are done using the standard method based on the spherical harmonic transform or using the temperature angular correlation spectrum.