Shot noise in clustering power spectra

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.

Figures (3)

• Figure 1: Two-point statistics of tracers of a fixed density $\delta$ from a negative binomial distribution, a Poisson distribution, and a binomial distribution. Top panel: The angular power spectra $C_\ell$ of the simulated maps show the apparently different 'shot noise' in each realisation. Middle panel: The angular power spectra $C_\ell$ measured from the sampled catalogues reveal that the bias returns to the 'Poissonian' value beyond the simulated scales (hatched). Bottom panel: The angular correlations show the corresponding differences $\Delta w(\theta)$ in clustering with respect to the Poissonian case below the pixel scale (hatched).
• Figure 2: Angular power spectrum covariance from 1 000 000 Gaussian full-sky simulations with Poisson sampling of 100 points. Results are binned as indicated. Top panel: Relative error of the variance predicted by the Gaussian approximation and with the full shot noise contribution compared to the sample variance of the simulations. Bottom panel: Correlation matrix predicted when including the full shot noise contribution (upper triangle) and measured from the simulations (lower triangle). The off-diagonal contributions are not captured by the Gaussian approximation.
• Figure 3: Angular power spectrum covariance for a sample of 100 million points in a realistic setting. Results are binned as indicated. Top panel: Relative error of the variance predicted by the Gaussian approximation compared to the variance with the full shot noise contribution. Bottom panel: Correlation matrix predicted when including the full shot noise contribution.