The cosmological analysis of X-ray cluster surveys: VI. Inference based on analytically simulated observable diagrams

The number density of galaxy clusters across mass and redshift has been established as a powerful cosmological probe. Cosmological analyses with galaxy clusters traditionally employ scaling relations. However, many challenges arise from this approach as the scaling relations are highly scattered, may be ill-calibrated, depend on the cosmology, and contain many nuisance parameters with low physical significance. In this paper, we use a simulation-based inference method utilizing artificial neural networks to optimally extract cosmological information from a shallow X-ray survey of galaxy clusters, solely using count rates (CR), hardness ratios (HR), and redshifts. This procedure enables us to conduct likelihood-free inference of cosmological parameters $\Omega_{\mathrm{m}}$ and $\sigma_8$. We analytically generate simulations of galaxy cluster distribution in a CR, HR space in multiple redshift bins based on totally random combinations of cosmological and scaling relation parameters. We train Convolutional Neural Networks (CNNs) to retrieve the cosmological parameters from these simulations. We then use neural density estimation (NDE) neural networks to predict the posterior probability distribution of $\Omega_{\mathrm{m}}$ and $\sigma_8$ given an input galaxy cluster sample. The 1 $\sigma$ errors of our density estimator on one of the target testing simulations are 1000 deg$^2$: 15.2% for $\Omega_{\mathrm{m}}$ and 10.0% for $\sigma_8$; 10000 deg$^2$: 9.6% for $\Omega_{\mathrm{m}}$ and 5.6% for $\sigma_8$. We also compare our results with Fisher analysis. We demonstrate, as a proof of concept, that it is possible to calculate cosmological predictions of $\Omega_{\mathrm{m}}$ and $\sigma_8$ from a galaxy cluster population without explicitly computing cluster masses and even, the scaling relation coefficients, thus avoiding potential biases resulting from such a procedure. [abridged]