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 _ { linear } , and its logarithmic derivative with respect to the filtering scale - ( n _ { linear } +3 ) = d \log \sigma _ { 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 nonlinear regime and compare the results with the above predictions , assuming that the spectral index and the variance are adjustable parameters , n _ { eff } and \sigma _ { eff } \equiv \sigma , where \sigma is the true , nonlinear variance .
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 _ { eff } is of course equal to n _ { 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 _ { eff } and n _ { linear } increases when n _ { linear } decreases .
For initial power-spectra with n _ { linear } = -2 , -1 , 0 , +1 , we find n _ { eff } \simeq - 9 , -3 , -1 , -0.5 when \sigma ^ { 2 } \simeq 100 .

It is worth noting that - ( 3 + n _ { eff } ) is 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 _ { 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 .