Ellipsoidal collapse and the redshift space probability distribution function of dark matter

We use the physics of ellipsoidal collapse to model the probability distribution function of the smoothed dark matter density field in real and redshift space. We provide a simple approximation to the exact collapse model which shows clearly how the evolution can be thought of as a modification of the spherical evolution model as well as of the Zeldovich Approximation (Zel’Dovich 1970). In essence, our model specifies how the initial smoothed overdensity and shear fields can be used to determine the shape and size of the region at later times. We use our parametrization to extend previous work on the real-space PDF so that it predicts the redshift space PDF as well. Our results are in good agreement with measurements of the PDF in simulations of clustering from Gaussian initial conditions down to scales on which the rms fluctuation is slightly greater than unity. We also show how the highly non-Gaussian non-linear redshifted density field in a numerical simulation can be transformed so that it provides an estimate of the shape of the initial real-space PDF. When applied to our simulations, our method recovers the initial Gaussian PDF, provided the variance in the nonlinear smoothed field is less than 4.