Covariance of cross-correlations: towards efficient measures for large-scale structure
Abstract
We study the covariance of the cross-power spectrum (CP) of different tracers for large-scale structure. We use the counts-in-cells framework to derive expressions for the full non-Gaussian covariance, including all contributions from the discreteness of matter. We pay attention to the assumed sampling distribution: besides the usual Poisson model, we also consider a toy-model where one tracer is a sub-sample of the other. This is instructive, since it is likely that not all galaxies are equally good tracers of the mass -- in particular those hosted in the same halo. We then compare the efficiency of the CP with the simple auto-statistic and find that the CP approach can out perform the standard auto-spectrum, provided one is cross-correlating a high-density sample with a rare sample. We then test the theory by measuring the fractional errors in the mass-mass, halo-mass, and halo-halo power spectra from the zHORIZON-I simulations, total volume ~100 [Gpc/h]^3. Good agreement is found on large-scales k<0.07 h/Mpc and there is no obvious advantage gained from the different estimators, since fractional errors all scale simply with the number of modes and survey volume. On smaller scales, there is an increase in the errors for all spectra. This can be attributed to increased importance of Poisson sampling fluctuations and the generation of non-Gaussian error terms. However, for cluster studies, there is a factor ~2 advantage to be gained from using the CP approach. All of the analysis was repeated in configuration space, and the main difference is that, on very large scales, there is a factor ~2 improvement in the S/N for this method. This work points the way towards the design of improved estimators and is expected to be of most use in studies of primordial non-Gaussianity. (Abridged)Keywords
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