Shot noise in clustering power spectra

We show that the ‘shot noise’ bias in angular clustering power spectra observed from discrete samples of points is not noise, but rather a known additive contribution that naturally arises due to degenerate pairs of points. In particular, we show that the true shot noise contribution cannot have a ‘non-Poissonian’ value, even though all point processes with non-trivial two-point statistics are non-Poissonian. Apparent deviations from the ‘Poissonian’ value can arise when significant correlations or anti-correlations are localised on small spatial scales. However, such deviations always correspond to a physical difference in two-point statistics, not a difference in noise. In the context of simulations, if clustering is treated as the tracer of a discretised underlying density field, any sub- or super-Poissonian sampling of the tracer induces such small-scale modifications and vice versa; we show this explicitly using recent innovations in angular power spectrum estimation from discrete catalogues. Finally, we show that the full covariance of clustering power spectra can also be computed explicitly: it depends only on the two-, three- and four-point statistics of the point process. The usual Gaussian covariance approximation appears as one term in the shot noise contribution, which is non-diagonal even for Gaussian random fields and Poisson sampling.