The real- and redshift-space density distribution functions for large-scale structure in the spherical collapse approximation

Abstract

We use the spherical collapse (SC) approximation to derive expressions for the smoothed redshift-space probability distribution function (PDF), as well as the p-order hierarchical amplitudes Sp, in both real and redshift space. We compare our results with numerical simulations, focusing on the Ω = 1 standard CDM model, where redshift distortions are strongest. We find good agreement between the SC predictions and the numerical PDF in real space even for σL ≳ 1, where σL is the linearly evolved rms fluctuation on the smoothing scale. In redshift space, reasonable agreement is possible only for σL ≲ 0.4. Numerical simulations also yield a simple empirical relation between the real-space PDF and the redshift-space PDF: we find that for σL < 1, the redshift-space PDF, [Pδ(z)], is, to a good approximation, a simple rescaling of the real-space PDF, P[δ], i.e., P[δ/σ]d[δ/σ] = P[δ(z)/σ(z)]d[δ (z)/σ(z)], where σ and σ(z) are the real-space and redshift-space rms fluctuations, respectively. This result applies well beyond the validity of linear perturbation theory, and it is a good fit for both the standard CDM model and the ACDM model. It breaks down for SCDM at σL ≈ 1, but provides a good fit to the ACDM models for σL as large as 0.8.