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We present a non-perturbative calculation of the 1-point probability distribution function (PDF) for the spherically-averaged matter density field. The PDF is represented as a path integral and is evaluated using the saddle-point method. It factorizes into an exponent given by a spherically symmetric saddle-point solution and a prefactor produced by fluctuations. The exponent encodes the leading sensitivity of the PDF to the dynamics of gravitational clustering and statistics of the initial conditions. In contrast, the prefactor has only a weak dependence on cosmology. It splits into a monopole contribution which is evaluated exactly, and a factor corresponding to aspherical fluctuations. The latter is crucial for the consistency of the calculation: neglecting it would make the PDF incompatible with translational invariance. We compute the aspherical prefactor using a combination of analytic and numerical techniques. We demonstrate the factorization of spurious enhanced contributions of large bulk flows and their cancellation due to the equivalence principle. We also identify the sensitivity to the short-scale physics and argue that it must be properly renormalized. The uncertainty associated with the renormalization procedure gives an estimate of the theoretical error. For zero redshift, the precision varies from sub percent for moderate density contrasts to tens of percent at the tails of the distribution. It improves at higher redshifts. We compare our results with N-body simulation data and find an excellent agreement.
Ivanov et al. (Mon,) studied this question.