Abstract
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 optimization bias. Both aspects arise from the fact that the signal-to-noise is constructed through an optimization 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 optimization 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.
Highlights
First proposed in the galaxy cluster context more than two decades ago (Haehnelt & Tegmark 1996; Herranz et al 2002; Melin et al 2006), matched filters have become a standard tool with which to detect and characterise galaxy clusters from CMB observations via their thermal Sunyaev-Zeldovich signature
The signal-to-noise can be employed as a mass proxy in a cluster number counts analysis, in which the cluster abundance across a number of observables is used to constrain cosmology (e.g., Planck 2015 results XXIV 2016; Bocquet et al 2019; Zubeldia & Challinor 2019; see Allen et al 2011 for a review)
In any bin more clusters are found if the optimisation bias is accounted for, since the cluster abundance decreases with qopt and the optimisation bias boosts the signal-tonoise at given mass and redshift
Summary
First proposed in the galaxy cluster context more than two decades ago (Haehnelt & Tegmark 1996; Herranz et al 2002; Melin et al 2006), matched filters have become a standard tool with which to detect and characterise galaxy clusters from CMB observations via their thermal Sunyaev-Zeldovich (tSZ) signature. A layer of additive ‘noise’ or ‘observational scatter’ connects the true signalto-noise with the observed one This observational scatter is typically assumed to be Gaussian-distributed and to have a standard deviation equal to unity and a mean equal to either the true-signal-to-noise (Planck and ACT; see, e.g., Hasselfield et al 2013; Planck 2015 results XXIV 2016) or a corrected version of it (SPT; see, e.g., Bocquet et al 2019). After a brief review of how matched filters are used in practice in the tSZ context (Section 3.1), in Section 3.2 we use a set of MC mock observations in order to study the statistics of the observational scatter of the signal-to-noise, quantifying its departure from Gaussianity, its variance, and its mean (the latter as quantified through the optimisation bias).
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