Towards an optimal extraction of cosmological parameters from galaxy cluster surveys using convolutional neural networks

The possibility to constrain cosmological parameters from galaxy surveys using field-level machine learning methods that bypass traditional summary statistics analyses, depends crucially on our ability to generate simulated training sets. The latter need to be both realistic, as to reproduce the key features of the real data, and produced in large numbers, as to allow us to refine the precision of the training process. The analysis presented in this paper is an attempt to respond to these needs by (a) using clusters of galaxies as tracers of large-scale structure, together with (b) adopting a 3LPT code (Pinocchio) to generate a large training set of $32\,768$ mock X-ray cluster catalogues. X-ray luminosities are stochastically assigned to dark matter haloes using an empirical $M-L_X$ scaling relation. Using this training set, we test the ability and performances of a 3D convolutional neural network (CNN) to predict the cosmological parameters, based on an input overdensity field derived from the cluster distribution. We perform a comparison with a neural network trained on traditional summary statistics, that is, the abundance of clusters and their power spectrum. Our results show that the field-level analysis combined with the cluster abundance yields a mean absolute relative error on the predicted values of $\Omega_{\rm m}$ and $\sigma_8$ that is a factor of $\sim 10 \%$ and $\sim 20\%$ better than that obtained from the summary statistics. Furthermore, when information about the individual luminosity of each cluster is passed to the CNN, the gain in precision exceeds $50\%$.