Likelihood-free Forward Modeling for Cluster Weak Lensing and Cosmology

Likelihood-free inference provides a rigorous approach to preform Bayesian analysis using forward simulations only. The main advantage of likelihood-free methods is its 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 inference of the cosmological parameters $(\Omega_\mathrm{m},\sigma_8)$ and the mass scale of the survey sample. Adopting an eROSITA-like selection function and a 10-percent scatter in the observable-mass relation in a flat $\Lambda$CDM cosmology with $\Omega_\mathrm{m}=0.286$ and $\sigma_8=0.82$, we create a synthetic catalog of observable-selected NFW clusters in a survey area of 50 deg$^2$. 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 $\Omega_\mathrm{m}$ parameter. For a weak-lensing survey with a source density of $n_\mathrm{g}=20$ arcmin$^{-2}$, we obtain posterior constraints on $S_8=\sigma_8(\Omega_\mathrm{m}/0.3)^{0.3}$ of $0.836 \pm 0.032$ and $0.810 \pm 0.019$ 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.