Unbiased pseudo-Cℓpower spectrum estimation with mode projection

Abstract

With the steadily improving sensitivity afforded by current and future galaxy surveys, a robust extraction of two-point correlation function measurements may become increasingly hampered by the presence of astrophysical foregrounds or observational systematics. The concept of mode projection has been introduced as a means to remove contaminants for which it is possible to construct a spatial map reflecting the expected signal contribution. Owing to its computational efficiency compared to minimum-variance methods, the sub-optimal pseudo-Cl (PCL) power spectrum estimator is a popular tool for the analysis of high-resolution data sets. Here, we integrate mode projection into the framework of PCL power spectrum estimation. In contrast to results obtained with optimal estimators, we show that the uncorrected projection of template maps leads to biased power spectra. Based on analytical calculations, we find exact closed-form expressions for the expectation value of the bias and demonstrate that they can be recast in a form that allows a numerically efficient evaluation, preserving the favorable O(l_max^3) time complexity of PCL estimator algorithms. Using simulated data sets, we assess the scaling of the bias with various analysis parameters and demonstrate that it can be reliably removed. We conclude that in combination with mode projection, PCL estimators allow for a fast and robust computation of power spectra in the presence of systematic effects -- properties in high demand for the analysis of ongoing and future large scale structure surveys.