Understanding matched filters for precision cosmology
  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.
