Galaxy cluster matter profiles: I. Self-similarity, mass calibration, and observable-mass relation validation employing cluster mass posteriors

We present a study of the weak lensing inferred matter profiles $\Delta\Sigma(R)$ of 698 South Pole Telescope thermal Sunyaev-Zel'dovich effect selected and MCMF optically confirmed galaxy clusters in the redshift range $0.25 <z< 0.94$ that have associated weak gravitational lensing shear profiles from the Dark Energy Survey. Rescaling these profiles to account for the mass dependent size and the redshift dependent density produces average rescaled matter profiles $\Delta\Sigma(R/R_\mathrm{200c})/(\rho_\mathrm{crit}R_\mathrm{200c})$ with a lower dispersion than the unscaled $\Delta\Sigma(R)$ versions, indicating a significant degree of self-similarity. Galaxy clusters from hydrodynamical simulations also exhibit matter profiles that suggest a high degree of self-similarity, with RMS variation among the average rescaled matter profiles with redshift and mass falling by a factor of approximately six and 23, respectively, compared to the unscaled average matter profiles. We employed this regularity in a new Bayesian method for weak lensing mass calibration that employs the so-called cluster mass posterior $P(M_\mathrm{200c}|\hat{\zeta}, \hat{\lambda}, z)$, which describes the individual cluster masses given their tSZE and optical observables. We validated the method using realistic mock datasets and present observable-mass relation constraints for the SPT$\times$DES sample. We present new validation tests of the observable-mass relation that indicate the underlying power-law form and scatter are adequate to describe the real cluster sample but that also suggest a redshift variation in the intrinsic scatter of the $\lambda$-mass relation may offer a better description. In addition, the average rescaled matter profiles offer high signal-to-noise ratio constraints on the shape of real cluster matter profiles, which are in good agreement with available hydrodynamical $\Lambda$CDM simulations.