Covariance of cross-correlations: towards efficient measures for large-scale structure

We study the covariance of the cross-power spectrum of different tracers for the largescale structure. We develop the counts-in-cells framework for the multi-tracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show, that for the usual auto-power statistic, besides the off-diagonal covariance generated through gravitational mode-coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as k ≫ kf = 2π/L. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (S/N ), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, then the cross-power approach can out perform the auto-power one by factors of a few. We confront the theory with measurements of the mass-mass, halo-mass, and halo-halo power spectra from a large ensemble of N–body simulations. We show that there is a significant S/N advantage to be gained from using the cross-power approach when studying the bias of rare haloes. The analysis is repeated in configuration space and again S/N improvement is found. We estimate the covariance matrix for these samples, and find strong off-diagonal contributions. The covariance depends on halo mass, with higher mass samples having stronger covariance. In agreement with theory, we show that the covariance is suppressed for the cross-power. This work points the way towards improved estimators for studying the clustering of tracers as a function of their physical properties.