We investigate the error implied by the use of the Zel ’ dovich approximation to set up the initial conditions at a finite redshift z _ { i } in numerical simulations .
Using a steepest-descent method developed in a previous work ( [ Valageas ( 2002a ) ] ) we derive the probability distribution { \cal P } ( \delta _ { R } ) of the density contrast in the quasi-linear regime .
This also provides its dependence on the redshift z _ { i } at which the simulation is started .
Thus , we find that the discrepancy with the exact pdf ( defined by the limit z _ { i } \rightarrow \infty ) is negligible after the scale factor has grown by a factor a / a _ { i } \ga 5 , for scales which were initially within the linear regime with \sigma _ { i } \la 0.1 .
This shows that the use of the Zel ’ dovich approximation to implement the initial conditions is sufficient for practical purposes since these are not very severe constraints .