Extracting cosmological information from the abundance of galaxy clusters with simulation-based inference

The abundance of galaxy clusters as a function of mass and redshift is a well-established and powerful cosmological probe. Cosmological analyses based on galaxy cluster number counts have traditionally relied on explicitly computed likelihoods, which are often challenging to develop with the required accuracy and expensive to evaluate. In this work, we implement an alternative approach based on simulation-based inference (SBI) methods that relies solely on synthetic galaxy cluster catalogues generated under a given model. These catalogues are much easier to produce than it is to develop and validate a likelihood. We validate this approach in the context of the galaxy cluster survey of the upcoming Simons Observatory for a setup in which we can also evaluate an exact explicit likelihood. We find that our SBI-based approach yields cosmological parameter posterior means that are within $0.2\,\sigma$ of those obtained with the explicit likelihood and with biases smaller than $0.1\,\sigma$. We also introduce and validate a procedure to assess the goodness of fit using only synthetic catalogues similar to those used for training. This demonstrates, for the first time, that a galaxy cluster number count cosmological analysis can be performed fully without resorting to a likelihood at any stage. Finally, we apply our SBI-based approach to the real Planck MMF3 cosmology sample, obtaining cosmological parameter constraints that are within $0.1\,\sigma$ of their likelihood-based counterparts. This constitutes the first SBI-based number count cosmological analysis of a real galaxy cluster catalogue.