  We present a first principle approach to obtain analytical predictions for
spherically-averaged cosmic densities in the mildly non-linear regime that go
well beyond what is usually achieved by standard perturbation theory. A large
deviation principle allows us to compute the leading-order cumulants of average
densities in concentric cells. In this symmetry, the spherical collapse model
leads to cumulant generating functions that are robust for finite variances and
free of critical points when logarithmic density transformations are
implemented. They yield in turn accurate density probability distribution
functions (PDFs) from a straightforward saddle-point approximation valid for
all density values. Based on this easy-to-implement modification, explicit
analytic formulas for the evaluation of the one- and two-cell PDF are provided.
The theoretical predictions obtained for the PDFs are accurate to a few percent
compared to the numerical integration, regardless of the density under
consideration and in excellent agreement with N-body simulations for a wide
range of densities. This formalism should prove valuable for accurately probing
the quasi-linear scales of low redshift surveys for arbitrary primordial power
spectra.
