Fisher for complements: extracting cosmology and neutrino mass from the counts-in-cells PDF

We comprehensively analyse the cosmology dependence of counts-in-cells statistics. We focus on the shape of the one-point probability distribution function (PDF) of the matter density field at mildly non-linear scales. Based on large-deviation statistics, we parametrize the cosmology dependence of the matter PDF in terms of the linear power spectrum, the growth factor, the spherical collapse dynamics, and the non-linear variance. We extend our formalism to include massive neutrinos, finding that the total matter PDF is highly sensitive to the total neutrino mass M-nu and can disentangle it from the clustering amplitude sigma(8). Using more than a million PDFs extracted from the Quijote simulations, we determine the response of the matter PDF to changing parameters in the nu Lambda CDM model and successfully cross-validate the theoretical model and the simulation measurements. We present the first nu Lambda CDM Fisher forecast for the matter PDF at multiple scales and redshifts, and its combination with the matter power spectrum. We establish that the matter PDF and the matter power spectrum are highly complementary at mildly non-linear scales. The matter PDF is particularly powerful for constraining the matter density Omega(m), clustering amplitude sigma(8) and the total neutrino mass M-nu. Adding the mildly non-linear matter PDF to the mildly non-linear matter power spectrum improves constraints on Omega(m) by a factor of 5 and sigma(8) by a factor of 2 when considering the three lowest redshifts. In our joint analysis of the matter PDF and matter power spectrum at three redshifts, the total neutrino mass is constrained to better than 0.01 eV with a total volume of 6(Gpc h(-1))(3). We discuss how density-split statistics can be used to translate those encouraging results for the matter PDF into realistic observables in galaxy surveys.