Density split statistics: Cosmological constraints from counts and lensing in cells in DES Y1 and SDSS data

We derive cosmological constraints from the probability distribution function (PDF) of evolved large-scale matter density fluctuations. We do this by splitting lines of sight by density based on their count of tracer galaxies, and by measuring both gravitational shear around and counts-in-cells in overdense and underdense lines of sight, in Dark Energy Survey (DES) First Year and Sloan Digital Sky Survey (SDSS) data. Our analysis uses a perturbation theory model [O. Friedrich et al., Phys. Rev. D 98, 023508 (2018)] and is validated using $N$-body simulation realizations and log-normal mocks. It allows us to constrain cosmology, bias and stochasticity of galaxies with respect to matter density and, in addition, the skewness of the matter density field. From a Bayesian model comparison, we find that the data weakly prefer a connection of galaxies and matter that is stochastic beyond Poisson fluctuations on $\le 20\mathrm{arcmin}$ angular smoothing scale. The two stochasticity models we fit yield DES constraints on the matter density ${\mathrm{\Omega }}_m=0.26_{-0.03}^{+0.04}$ and ${\mathrm{\Omega }}_m=0.28_{-0.04}^{+0.05}$ that are consistent with each other. These values also agree with the DES analysis of galaxy and shear two-point functions (3x2pt, DES Collaboration et al.) that only uses second moments of the PDF. Constraints on ${\sigma }_8$ are model dependent (${\sigma }_8=0.97_{-0.06}^{+0.07}$ and $0.80_{-0.07}^{+0.06}$ for the two stochasticity models), but consistent with each other and with the 3 x 2pt results if stochasticity is at the low end of the posterior range. As an additional test of gravity, counts and lensing in cells allow to compare the skewness $S_3$ of the matter density PDF to its $\mathrm{\Lambda }\mathrm{CDM}$ prediction. We find no evidence of excess skewness in any model or data set, with better than 25 per cent relative precision in the skewness estimate from DES alone.