Towards including super-sample covariance in the unbinned likelihood for cluster abundance cosmology

The measurement of the abundance of galaxy clusters in the Universe is a sensitive probe of cosmology, which depends on both the expansion history of the Universe and the growth of structure. Density fluctuations across the finite survey volume add noise to this measurement, this is often referred to as super-sample covariance (SSC). For an unbinned cluster analysis, such noise has not been included in the cluster likelihood, since the effect of SSC was small compared to the Poisson shot-noise for samples of a few hundred clusters. For upcoming large cluster surveys such as the Rubin LSST, which will deliver catalogs of tens of thousands of clusters, this effect will no longer be negligible. In this paper, we propose a new hybrid likelihood based on the Gauss-Poisson Compound model (GPC), by using infinitesimal mass bins and standard redshift bins. This likelihood has the advantages of an unbinned Poisson likelihood whilst successfully incorporating the effects of SSC. Using a simulated dark matter halo catalog, we find that the hybrid likelihood, accounting for both Poisson noise and SSC, increases the dispersion of the parameter posteriors by 20 percent when using 100,000 clusters compared to the standard unbinned likelihood, based on Poisson statistics only.