What do cluster counts really tell us about the Universe?
  We study the covariance matrix of the cluster mass function in cosmology. We
adopt a two-line attack: firstly, we employ the counts-in-cells framework to
derive an analytic expression for the covariance of the mass function.
Secondly, we use a large ensemble of N-body simulations in the LCDM framework
to test this. Our theoretical results show that the covariance can be written
as the sum of two terms: a Poisson term, which dominates in the limit of rare
clusters; and a sample variance term, which dominates for more abundant
clusters. Our expressions are analogous to those of Hu & Kravtsov (2003) for
multiple cells and a single mass tracer. Calculating the covariance depends on:
the mass function and bias of clusters, and the variance of mass fluctuations
within the survey volume. The predictions show that there is a strong
bin-to-bin covariance between measurements. In terms of the cross-correlation
coefficient, we find r~0.5 for haloes with M<3e14 Msol at z=0. Comparison of
these predictions with estimates from simulations shows excellent agreement. We
use the Fisher matrix formalism to explore the cosmological information content
of the counts. We compare the Poisson likelihood model, with the more realistic
likelihood model of Lima & Hu (2004), and all terms entering the Fisher
matrices are evaluated using the simulations. We find that the Poisson
approximation should only be used for the rarest objects, M>3e14 Msol,
otherwise the information content of a survey of size V~13.5 [Gpc/h]^3 would be
overestimated, resulting in errors that are ~2 times smaller. As an auxiliary
result, we show that the bias of clusters, obtained from the cluster-mass
cross-variance, is linear on scales >50 Mpc/h, whereas that obtained from the
auto-variance is nonlinear.
