Power spectrum and correlation function errors: Poisson vs. Gaussian shot noise

Poisson distributed shot noise is normally considered in the Gaussian limit in cosmology. However, if the shot noise is large enough and the correlation function/power spectrum conspires, the Gaussian approximation mis-estimates the errors and their covariance significantly. The power spectrum, even for initially Gaussian densities, acquires cross correlations which can be large, while the change in the correlation function error matrix is diagonal except at zero separation. Two- and three-dimensional power law correlation function and power spectrum examples are given. These corrections appear to have a large effect when applied to galaxy clusters, e.g., for SZ selected galaxy clusters in two dimensions. This can increase the error estimates for cosmological parameter estimation and consequently affect survey strategies, as the corrections are minimized for surveys which are deep and narrow rather than wide and shallow. In addition, a rewriting of the error matrix for the power spectrum/correlation function is given which eliminates most of the Bessel function dependence (in two dimensions) and all of it (in three dimensions), which makes the calculation of the error matrix more tractable. This applies even when the shot noise is in the (usual) Gaussian limit.