Understanding matched filters for precision cosmology

TL;DR

This paper investigates the statistical properties of matched filter detection signals in cosmology, revealing a positive bias and non-Gaussianity that affect cluster counts, especially for future surveys like SO.

Contribution

It provides a detailed analysis of the non-Gaussianity and bias in the signal-to-noise observable from matched filters, including an approximate correction formula.

Findings

Signal-to-noise is approximately Gaussian for values above 6.

Optimization bias can significantly affect future cluster counts.

Bias impact is negligible for Planck but notable for the Simons Observatory.

Abstract

Matched filters are routinely used in cosmology in order to detect galaxy clusters from mm observations through their thermal Sunyaev-Zeldovich (tSZ) signature. In addition, they naturally provide an observable, the detection signal-to-noise or significance, which can be used as a mass proxy in number counts analyses of tSZ-selected cluster samples. In this work, we show that this observable is, in general, non-Gaussian, and that it suffers from a positive bias, which we refer to as optimisation bias. Both aspects arise from the fact that the signal-to-noise is constructed through an optimisation operation on noisy data, and hold even if the cluster signal is modelled perfectly well, no foregrounds are present, and the noise is Gaussian. After reviewing the general mathematical formalism underlying matched filters, we study the statistics of the signal-to-noise with a set Monte Carlo mock observations, finding it to be well-described by a unit-variance Gaussian for signal-to-noise values of 6 and above, and quantify the magnitude of the optimisation bias, for which we give an approximate expression that may be used in practice. We also consider the impact of the bias on the cluster number counts of Planck and the Simons Observatory (SO), finding it to be negligible for the former and potentially significant for the latter.