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

TL;DR

This paper investigates how convolutional neural networks can optimally extract cosmological parameters from X-ray selected galaxy cluster surveys by leveraging field-level information. Using 3LPT-based Pinocchio mocks to generate $32{,}768$ realizations that jointly sample cosmology and the $M-L_{X}$ relation, the authors train a 3D CNN on a cube-embedded overdensity field and compare its performance to a traditional approach based on cluster abundance and the power spectrum. They show that field-level inference combined with abundance improves $Ω_{ m m}$ and $σ_8$ constraints by about $10 ext{--}20 ext{%}$ relative to summary statistics, with even larger gains (up to $ ext{>}50 ext{%}$) when cluster luminosities are included as CNN inputs. The work highlights the potential of ML-based field-level cosmology, while outlining future steps toward Bayesian posteriors, multi-fidelity training, and realistic survey applications before deployment on actual data from surveys like eROSITA.

Abstract

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 $Ω_{\rm m}$ and $σ_8$ 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 $50\%$.

Figures (12)

• Figure 1: X-ray luminosity functions of the mock galaxy cluster catalogues, compared to the estimate from REFLEX-II boehringer14. The dark shaded area correspond to $95.4\%$ interval of the region spanned by the $4096$ luminosity functions obtained from the mocks, when only the cosmological parameters are varied, i.e. fixing the $M-L_{\rm X}$ relation parameters to the fiducial values. The lighter shaded area shows how this changes when the full set of $32768$ mocks, spanning the spread in the $M-L_{\rm X}$ relation parameters, is considered. X-ray luminosities are computed in the ROSAT $[0.1-2.4]$ keV band. The step line represents the reference cosmology model P18 and the dashed line corresponds to the fit using the extended Schechter function from \ref{['eq:extendedSchechter']}. Filled dots with error bars show the observed REFLEX-II luminosity function.
• Figure 4: General architecture of the deep neural networks used for parameter estimation. A set of feature extraction networks extracts in parallel features for each of the considered observational statistics. The CNN used to extract features at the field level takes as input a tensor of shape ($64^3$, channels). The CNN module is an encoder that consists of five consecutive convolution blocks with $N_{\rm conv\_per\_block}$ convolution layers each and $N_{\rm chs\_first}$ output channels for the first block. Each subsequent block multiplies the number of output channels by two with respect to the previous one. For cluster abundance and power spectrum, the extraction network consists of two fully connected (FC) layers with $N_{\rm fc\_units}$ neurons per layer. The extracted features are concatenated into a one-dimensional array that feeds a final regression block that outputs the target parameters. Different combinations of input observational statistics can be chosen to produce different inference models. The output of the network is a vector of size $N_{\rm params}$, corresponding to the $5+4$ cosmological and $M-L_{\rm X}$ relation parameters. A more detailed description of the architecture is given in \ref{['subsec:architecture']}.
• Figure 5: Test set predictions of the neural network compared to the true target parameters, for the case of the CA+CNN model. Each panel focuses on a different parameter. We omit the parameter $\sigma_{\rm tot}$ since it is poorly constrained. We have added the derived parameter $S_8$.
• Figure 6: $R^2$ score evaluated on the test set for each target parameter. We have also included the derived parameter $S_8$. Each colour represents a different combination of statistics. We exclude $\sigma_{\rm tot}$, since it is unconstrained in all the cases considered.
• Figure 7: Mean absolute relative error evaluated on the test set for each target parameter. We have also included the derived parameter $S_8$. Each colour represents a different combination of statistics. We exclude $\sigma_{\rm tot}$, since it is unconstrained in all the cases considered.