  The abundance of galaxy clusters is a sensitive probe to the amplitude of
matter density fluctuations, the total amount of matter in the Universe as well
as its expansion history. Inferring correct values and accurate uncertainties
of cosmological parameters requires accurate knowledge of cluster abundance
statistics, encoded in the likelihood function. In this paper, we test the
accuracy of cluster abundance likelihoods used in the literature, namely the
Poisson and Gaussian likelihoods as well as the more complete description of
the Gauss-Poisson Compound likelihood. This is repeated for a variety of
binning choices and analysis setups. In order to evaluate the accuracy of a
given likelihood, this work compares individual posterior covariances to the
covariance of estimators over the 1000 simulated dark matter halo catalogs
obtained from PINOCCHIO algorithm. We find that for Rubin/LSST or Euclid-like
surveys the Gaussian likelihood gives robust constraints over a large range of
binning choices. The Poisson likelihood, that does not account for sample
covariance, always underestimates the errors on the parameters, even when the
sample volume is reduced or only high-mass clusters are considered. We find no
benefit in using the more complex Gauss-Poisson Compound likelihood as it gives
essentially the same results as the Gaussian likelihood, but at a greater
computational cost. Finally, in this ideal setup, we note only a small gain on
the parameter error bars when using a large number of bins in the mass-redshift
plane.
