Accurate analytic model for the weak lensing convergence one-point probability distribution function and its autocovariance

The one-point probability distribution function (PDF) is a powerful summary statistic for non-Gaussian cosmological fields, such as the weak lensing (WL) convergence reconstructed from galaxy shapes or cosmic microwave background maps. Thus far, no analytic model has been developed that successfully describes the high-convergence tail of the WL convergence PDF for small smoothing scales from first principles. Here, we present a halo-model formalism to compute the WL convergence PDF, building upon our previous results for the thermal Sunyaev-Zel'dovich field. Furthermore, we extend our formalism to analytically compute the covariance matrix of the convergence PDF. Comparisons to numerical simulations generally confirm the validity of our formalism in the non-Gaussian, positive tail of the WL convergence PDF, but also reveal the convergence PDF's strong sensitivity to small-scale systematic effects in the simulations (e.g., due to finite resolution). Finally, we present a simple Fisher forecast for a Rubin Observatory-like survey, based on our new analytic model. Considering the $\{A_s, \Omega_m, \Sigma m_\nu\}$ parameter space and assuming a Planck CMB prior on $A_s$ only, we forecast a marginalized constraint $\sigma(\Sigma m_\nu) \approx 0.08$ eV from the WL convergence PDF alone, even after marginalizing over parameters describing the halo concentration-mass relation. This error bar on the neutrino mass sum is comparable to the minimum value allowed in the normal hierarchy, illustrating the strong constraining power of the WL convergence PDF. We make our code publicly available at this https URL.