  Pinning down the total neutrino mass and the dark energy equation of state is
a key aim for upcoming galaxy surveys. Weak lensing is a unique probe of the
total matter distribution whose non-Gaussian statistics can be quantified by
the one-point probability distribution function (PDF) of the lensing
convergence. We calculate the convergence PDF on mildly non-linear scales from
first principles using large-deviation statistics, accounting for dark energy
and the total neutrino mass. For the first time, we comprehensively validate
the cosmology-dependence of the convergence PDF model against large suites of
simulated lensing maps, demonstrating its percent-level precision and accuracy.
We show that fast simulation codes can provide highly accurate covariance
matrices, which can be combined with the theoretical PDF model to perform
forecasts and eliminate the need for relying on expensive N-body simulations.
Our theoretical model allows us to perform the first forecast for the
convergence PDF that varies the full set of $\Lambda$CDM parameters. Our Fisher
forecasts establish that the constraining power of the convergence PDF compares
favourably to the two-point correlation function for a Euclid-like survey area
at a single source redshift. When combined with a CMB prior from Planck, the
PDF constrains both the neutrino mass $M_\nu$ and the dark energy equation of
state $w_0$ more strongly than the two-point correlation function.
