TRANSIENTS FROM INITIAL CONDITIONS - A PERTURBATIVE ANALYSIS

The standard procedure to generate initial conditions in numerical sim ulations of structure formations is to use the Zel'dovich approximatio n (ZA). Although the ZA correctly reproduces the linear growing modes of density and velocity perturbations, non-linear growth is inaccurate ly represented, particularly for velocity perturbations because of the ZA failure to conserve momentum. This implies that it takes time for the actual dynamics to establish the correct statistical properties of density and velocity fields. We extend the standard formulation of no n-linear perturbation theory (PT) to include transients as non-linear excitations of decaying modes caused by the initial conditions. These new non-linear solutions interpolate between the initial conditions an d the late-time solutions given by the exact non-linear dynamics. To q uantify the magnitude of transients, we focus on higher order statisti cs of the density contrast delta and velocity divergence Theta, charac terized by the S-p and T-p parameters. These describe the non-Gaussian ity of the probability distribution through its connected moments [del ta(p)](c) = S-p[delta(2)](p-1), [Theta(p)](c) = T-p[Theta(2)](p-1). We calculate S-p(a) and T-p(a) to leading order in PT with top-hat smoot hing as a function of the scale factor a. We find that the time-scale of transients is determined, at a given order p, by the effective spec tral index n(eff). The skewness factor S-3(T-3) attains 10 per cent ac curacy only after a approximate to 6 (a approximate to 15) for n(eff) approximate to 0, whereas higher (lower) n(eff) demands more (less) ex pansion away from the initial conditions. These requirements become mu ch more stringent as p increases, always showing slower decay of trans ients for T-p than S-p. For models with density parameter Omega not eq ual 1, the conditions above apply to the linear growth factor; thus an Omega = 0.3 open model requires roughly a factor of 2 larger expansio n than a critical density model to reduce transients by the same amoun t. The predicted transients in S-p are in good agreement with numerica l simulations. More accurate initial conditions can be achieved by usi ng second-order Lagrangian PT (2LPT), which reproduces growing modes u p to second order and thus eliminates transients in the skewness param eters. We show that for p > 3 this scheme can reduce the required expa nsion by more than an order of magnitude compared to the ZA. Setting u p 2LPT initial conditions requires only minimal, inexpensive changes t o ZA codes. We suggest simple steps for its implementation.