Publication : t96/081

Abstract:
Perturbation theory makes it possible to calculate the probability distribution function (PDF) of the large scale density field in the small variance limit, $\sigma \ll 1$. For top hat smoothing and scale-free Gaussian initial fluctuations, the result depends only on the linear variance, $\sigma_{\rm linear}$, and its logarithmic derivative with respect to the filtering scale $-(n_{\rm linear}+3)=d\log\sigma_{\rm linear}^2/d\log \ell$ (Bernardeau 1994a). In this paper, we measure the PDF and its low-order moments in scale-free simulations evolved well into the non-linear regime and compare the results with the above predictions, assuming that the spectral index and the variance are {\it adjustable} parameters, $n_{\rm eff}$ and $\sigma_{\rm eff}\equiv \sigma$. With these additional degrees of freedom, results from perturbation theory provide a good fit of the PDFs, even in the highly nonlinear regime. The value of $n_{\rm eff}$ is of course equal to $n_{\rm linear}$ when $\sigma \ll 1$, and it decreases with increasing $\sigma$. A nearly flat plateau is reached when $\sigma \gg 1$. In this regime, the difference between $n_{\rm eff}$ and $n_{\rm linear}$ increases when $n_{\rm linear}$ decreases. For initial power-spectra with $n_{\rm linear}=-2,-1,0,+1$, we find $n_{\rm eff} \simeq -9,-3,-1,-0.5$ when $\sigma^2 \simeq 100$. It is worth noting that $-(3+n_{\rm eff})$ is {\it different} from the logarithmic derivative of the nonlinear variance with respect to the filtering scale. Consequently, it is not straightforward to determine the nonlinearly evolved PDF from arbitrary (scale-dependent) initial conditions, such as Cold Dark Matter, although we propose a simple method that makes this feasible. Thus, estimates of the variance (using, for example, the prescription proposed by Hamilton et al. 1991) and of $n_{\rm eff}$ as functions of scale for a given power spectrum makes it possible to calculate the local density PDF at any time from the initial conditions.

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