Likelihood-free Forward Modeling for Cluster Weak Lensing and Cosmology

Likelihood-free inference provides a rigorous approach to performing Bayesian analysis using forward simulations only. The main advantage of likelihood-free methods is their ability to account for complex physical processes and observational effects in forward simulations. Here we explore the potential of likelihood-free forward modeling for Bayesian cosmological inference using the redshift evolution of the cluster abundance combined with weak-lensing mass calibration. We use two complementary likelihood-free methods, namely Approximate Bayesian Computation (ABC) and Density-Estimation Likelihood-Free Inference (DELFI), to develop an analysis procedure for the inference of the cosmological parameters (Ω_m, σ ₈) and the mass scale of the survey sample. Adopting an eROSITA-like selection function and a 10% scatter in the observable-mass relation in a flat ΛCDM cosmology with Ω_m = 0.286 and σ ₈ = 0.82, we create a synthetic catalog of observable-selected Navarro-Frenk-White clusters in a survey area of 50 deg². The stacked tangential shear profile and the number counts in redshift bins are used as summary statistics for both methods. By performing a series of forward simulations, we obtain convergent solutions for the posterior distribution from both methods. We find that ABC recovers broader posteriors than DELFI, especially for the Ω_m parameter. For a weak-lensing survey with a source density of n _g = 20 arcmin^-2, we obtain posterior constraints on ${S}_{8}={\sigma }_{8}{\left({{\rm{\Omega }}}_{{\rm{m}}}/0.3\right)}^{0.3}$ of 0.836 ± 0.032 and 0.810 ± 0.019 from ABC and DELFI, respectively. The analysis framework developed in this study will be particularly powerful for cosmological inference with ongoing cluster cosmology programs, such as the XMM-XXL survey and the eROSITA all-sky survey, in combination with wide-field weak-lensing surveys.