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 \sigma _ { L } \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } % \raise 2.0 pt \hbox { $ \mathchar 318 $ } } 1 , where \sigma _ { L } is the linearly-evolved rms fluctuation on the smoothing scale .
In redshift space , reasonable agreement is possible only for \sigma _ { L } \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } % \raise 2.0 pt \hbox { $ \mathchar 316 $ } } 0.4 .
Numerical simulations also yield a simple empirical relation between the real-space PDF and redshift-space PDF : we find that for \sigma < 1 , the redshift space PDF , P \left [ \delta _ { ( z ) } \right ] , is , to a good approximation , a simple rescaling of the real space PDF , P \left [ \delta \right ] , i.e. , P \left [ \delta / \sigma \right ] d \left [ \delta / \sigma \right ] = P \left [ \delta _ { ( z ) } / % \sigma _ { ( z ) } \right ] d \left [ \delta _ { ( z ) } / \sigma _ { ( z ) } \right ] , where \sigma and \sigma _ { ( 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 \Lambda CDM models for \sigma as large as 0.8 .