The Real and Redshift Space Density Distribution Function 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 $S_p$, in both real and redshift space. We compare our results with numerical simulations, focusing on the $\Omega =1$ 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 $σ_L \simgt 1$, where ${\sigma }_L$ is the linearly-evolved rms fluctuation on the smoothing scale. In redshift space, reasonable agreement is possible only for $σ_L \simlt 0.4$. Numerical simulations also yield a simple empirical relation between the real-space PDF and redshift-space PDF: we find that for $σ\simlt 1$, the redshift space PDF, P[δ_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 $\sigma$ 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 Lambda-CDM model. It breaks down for SCDM at $\sigma \approx 1$, but provides a good fit to the Λ-CDM models for $\sigma$ as large as 0.8.

9 pages, latex, 12 figures added (26 total), minor changes to conclusions, to appear in MNRAS

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