Transients from Zel'dovich initial conditions

We investigate the error implied by the use of the Zel'dovich approximation to set up the initial conditions at a finite redshift $\zi$ in numerical simulations. Using a steepest-descent method developed in a previous work (paper II) we derive the probability distribution $P(\delta_R)$ of the density contrast in the quasi-linear regime. This also provides its dependence on the redshift $\zi$ at which the simulation is started. Thus, we find that the discrepancy with the exact pdf (defined by the limit $\zi \to \infty$) is negligible after the scale factor has grown by a factor $a/\ai \ga 5$, for scales which were initially within the linear regime with $\sigma_{\rm i} \la 0.1$. This shows that the use of the Zel'dovich approximation to implement the initial conditions is quite sufficient for practical purposes since these are not very severe constraints.