Simulation-based cosmological inference from optically-selected galaxy clusters with $\texttt{Capish}$

TL;DR

Capish presents a forward-modeling SBI framework for optically-selected galaxy cluster cosmology, jointly inferring cosmology and mass–observable relations from cluster counts and mean weak-lensing masses while explicitly modeling systematics such as super-sample covariance, selection effects, and measurement noise. It generates synthetic catalogs from a halo mass function, maps halos to observed richness and lensing masses via parameterized relations, and trains neural density estimators to recover posterior distributions. Validation shows SBI posteriors align with likelihood-based results but are broader due to a more realistic forward model; Capish successfully reproduces analytic predictions and performs well on Euclid Flagship mocks when the halo mass function is matched. The framework offers a robust, flexible tool for DES, Euclid, and LSST cluster analyses and serves as a test bed for likelihood-based pipelines in upcoming wide-area surveys.

Abstract

Galaxy clusters are powerful probes of the growth of cosmic structure through measurements of their abundance as a function of mass and redshift. Extracting precise cosmological constraints from cluster surveys is challenging, as we must contend the complex relationship between richness and the underlying halo mass, selection function biases, super-sample covariance, and correlated measurement noise between mass proxies. As upcoming photometric surveys are expected to detect tens to hundreds of thousands of galaxy clusters, controlling these systematics becomes essential. In this paper, we present a forward-modelling approach using Simulation-Based Inference (SBI), which provides a natural framework for jointly modelling cluster abundance and lensing mass observables while capturing systematic uncertainties at higher fidelity than analytic likelihood methods - which rely on simplifying assumptions such as fixed covariances and Gaussianity - without requiring an explicit likelihood formulation. We introduce $\texttt{Capish}$, a Python code for generating forward-modelled galaxy cluster catalogues using halo mass functions and incorporating observational effects. We perform SBI using neural density estimation with normalizing flows, trained on abundance and mean lensing mass measurements in observed redshift-richness bins. Our forward model accounts for realistic noise, redshift uncertainties, selection functions, and correlated scatter between lensing mass and observed richness. We find good agreement with likelihood-based analyses, with broader SBI posteriors reflecting the increased realism of the forward model. We also test $\texttt{Capish}$ on cluster catalogues built from a large cosmological simulation, finding a good fit to cosmological parameters.

Figures (13)

• Figure 1: Organization of Capish, detailed in Section \ref{['sec:capish']}: The code is organized in three separate blocks: the first block [1$\star$] is dedicated to generate halo masses and true redshift from an underlying halo mass function, accounting for the effect of SSC (presented in Section \ref{['sec:capish_dm_halos']}) The second block [2$\star$] is dedicated to compute the observed richness and lensing mass for halos, along with photometric redshift (presented in Section \ref{['sec:capish_galaxy_cluster_cat']}). The third block [3$\star$] computes the summary statistics (presented in Section \ref{['sec:summary_stats']}).
• Figure 2: Upper panel: Mean Capish halo count compared to analytical prediction. Lower panel: Variance of Capish halo counts compared to Poisson-only or Poisson+SSC variance analytical prediction.
• Figure 3: Completeness of the simulated cluster catalogue as a function of halo mass for different observed richness thresholds, and intrinsic richness–mass scatter $\sigma_{\ln\lambda, \mathrm{int}}$ in Eq. \ref{['eq:sigma2_lnlambda']}. Lower richness thresholds and smaller intrinsic richness–mass scatter yield higher completeness at fixed mass, with all curves asymptoting to unity at high mass.
• Figure 4: Left: Upper panel: Capish mean mass outputs, averaged over 100 simulations, compared to the theoretical prediction for the mean mass of a stack of clusters in redshift richness bins. Lower panel: Standard deviation of the same mean masses over the 100 simulations. Right: Upper panel: Capish mean count outputs, averaged over 100 simulations, compared to the theoretical prediction for the mean cluster count of a stack of clusters in redshift richness bins. Lower panel: Standard deviation of the same counts over the 100 simulations.
• Figure 5: Probability coverage tests using the estimated posterior over 6 Capish parameters, given the observation of counts and masses (count_Nm). To prevent inaccuracies in the density estimation near prior boundaries, the neural posterior is trained on a broad prior, then restricted to a tighter one. The ECP calibration is computed for each parameter, and the TARP calibration is computed for all parameters at once.