  We study the covariance of the cross-power spectrum of different tracers for
the large-scale 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>>k_f=2 pi/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 (SN), 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 SN 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 SN
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 clustering studies.

Astrophysics