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, yielding important information on the matter components of the Universe. Cosmological analyses with galaxy clusters traditionally employ scaling relations, which are empirical relationships between cluster masses and their observable properties. However, many challenges arise from this approach as the scaling relations are highly scattered, maybe ill-calibrated, depend on the cosmology, and contain many nuisance parameters with low physical significance. For this paper, we used a simulation-based inference method utilizing artificial neural networks to optimally extract cosmological information from a shallow X-ray survey, solely using count rates, hardness ratios, and redshifts. This procedure enabled us to conduct likelihood-free inference of cosmological parameters $ m $ and $ To achieve this, we analytically generated several datasets of 70\,000 cluster samples with totally random combinations of cosmological and scaling relation parameters. Each sample in our simulation is represented by its galaxy cluster distribution in a count rate (CR) and hardness ratio (HR) space in multiple redshift bins. We trained convolutional neural networks (CNNs) to retrieve the cosmological parameters from these distributions. We then used neural density estimation (NDE) neural networks to predict the posterior probability distribution of $ m $ and $ given an input galaxy cluster sample. Using the survey area as a proxy for the number of clusters detected for fixed cosmological and astrophysical parameters, and hence of the Poissonian noise, we analyze various survey sizes. The 1 sigma errors of our density estimator on one of the target testing simulations are 1\,000\,deg$^2$, 15.2 for $ m $ and 10.0 for $ and 10\,000\,deg$^2$, 9.6 for $ m $ and 5.6 for $ We also compare our results with a traditional Fisher analysis and explore the effect of an additional constraint on the redshift distribution of the simulated samples. We demonstrate, as a proof of concept, that it is possible to calculate cosmological predictions of $ m $ and $ 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.