Shot noise in clustering power spectra

TL;DR

This paper clarifies that shot noise in clustering power spectra is an additive effect from point pairs, not true noise, and provides explicit formulas for its computation including non-Gaussian contributions.

Contribution

It demonstrates that shot noise is a known additive component arising from point pairs and derives explicit covariance formulas involving higher-order statistics.

Findings

Shot noise is an additive contribution, not noise.

Deviations from Poissonian shot noise reflect physical two-point statistics.

Explicit covariance formulas depend on two-, three-, and four-point statistics.

Abstract

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.