Unbiased methods for removing systematics from galaxy clustering measurements

TL;DR

This paper analyzes three methods for removing observational systematics from galaxy clustering measurements, identifying biases and variances, and providing corrections to enable unbiased, high-precision cosmological analyses.

Contribution

It offers a detailed mathematical assessment of template subtraction, basic mode projection, and extended mode projection, revealing biases and variances, and proposing corrections for unbiased galaxy clustering estimates.

Findings

Template subtraction is biased without correction.

Basic mode projection is bias-free.

Extended mode projection introduces bias, requiring correction.

Abstract

Measuring the angular clustering of galaxies as a function of redshift is a powerful method for extracting information from the three-dimensional galaxy distribution. The precision of such measurements will dramatically increase with ongoing and future wide-field galaxy surveys. However, these are also increasingly sensitive to observational and astrophysical contaminants. Here, we study the statistical properties of three methods proposed for controlling such systematics -- template subtraction, basic mode projection, and extended mode projection -- all of which make use of externally supplied template maps, designed to characterise and capture the spatial variations of potential systematic effects. Based on a detailed mathematical analysis, and in agreement with simulations, we find that the template subtraction method in its original formulation returns biased estimates of the galaxy angular clustering. We derive closed-form expressions that should be used to correct results for this shortcoming. Turning to the basic mode projection algorithm, we prove it to be free of any bias, whereas we conclude that results computed with extended mode projection are biased. Within a simplified setup, we derive analytical expressions for the bias and discuss the options for correcting it in more realistic configurations. Common to all three methods is an increased estimator variance induced by the cleaning process, albeit at different levels. These results enable unbiased high-precision clustering measurements in the presence of spatially-varying systematics, an essential step towards realising the full potential of current and planned galaxy surveys.