Understanding matched filters for precision cosmology

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.

Galaxy cluster SZ detection with unbiased noise estimation: an iterative approach

“…Clusters are then identified as the peaks in these signal-to-noise maps above some threshold. As shown in Zubeldia et al (2021), for values of qopt of about 5 and above, this optimal signalto-noise qopt follows approximately a unit-variance Gaussian with mean equal to (q + f ) 1/2 , where f is the number of parameters that the signal-to-noise is being maximised for, usually 3 (angular size and sky location).…”

“…This estimator is known as a multi-frequency matched filter estimator (see, e.g., Melin et al 2006), so-called because it acts directly on the multi-frequency data. Its variance is given by σ 2 y 0 = N −1 , and we note that it is unbiased only if the input θ500 is equal to its true value and if the template is placed at the cluster's true sky location (for a more detailed discussion, see Zubeldia et al 2021).…”

“…We note that our iterative approach has the advantage of completely removing the tSZ signal of the detections for noise estimation purposes in a clean, model-agnostic way. An alternative possibility could involve iteratively subtracting a cluster model from the frequency maps for each of the detections above some threshold, as proposed in Zubeldia et al (2021), instead of masking them. This approach, however, has the disadvantage of being model-dependent, removing the signal in as much as the model is a good description of cluster profile.…”

“…In addition, the corresponding values of ŷ0 and of the filter angular scale θ500 are assigned as the matched estimates for y0 and θ500, and the pixel coordinates of the peak as the sky coordinates of the detection. As was detailed in Zubeldia et al (2021), these estimates of y0, θ500 and sky location can be seen as maximum-likelihood estimates, and as such are, in general, biased (except the two sky coordinates, due to the symmetry of the problem).…”

“…The residuals are defined as ∆q = qopt −(q 2 t +3) 1/2 for the optimal signal-to-noise qopt, i.e., if the clusters are blindly detected (which is the case shown in Figure 8), and as ∆q = q − qt if the signal-to-noise is measured at the true cluster parameters (θ500 and sky location). The former definition accounts for the optimisation bias, as discussed extensively in Zubeldia et al (2021). According to our theoretical understanding of matched filters (see Sections 2 and 3), both sets of residuals should have zero mean and unity standard deviation if the noise covariance is correctly determined.…”