Covariance matrices for halo number counts and correlation functions
  We study the mean number counts and two-point correlation functions, along
with their covariance matrices, of cosmological surveys such as for clusters.
In particular, we consider correlation functions averaged over finite redshift
intervals, which are well suited to cluster surveys or populations of rare
objects, where one needs to integrate over nonzero redshift bins to accumulate
enough statistics. We develop an analytical formalism to obtain explicit
expressions of all contributions to these means and covariance matrices, taking
into account both shot-noise and sample-variance effects. We compute low-order
as well as high-order (including non-Gaussian) terms. We derive expressions for
the number counts per redshift bins both for the general case and for the small
window approximation. We estimate the range of validity of Limber's
approximation and the amount of correlation between different redshift bins. We
also obtain explicit expressions for the integrated 3D correlation function and
the 2D angular correlation. We compare the relative importance of shot-noise
and sample-variance contributions, and of low-order and high-order terms. We
check the validity of our analytical results through a comparison with the
Horizon full-sky numerical simulations, and we obtain forecasts for several
future cluster surveys.
