Publication : t96/081

Extended perturbation theory for the local density distribution function

Colombi S. (Canadian Institute for Theoretical Astrophysics (CITA), University of Toronto, 60 St. George Street, Toronto, M5S 3H8 Ontario, CANADA)
Bernardeau F. (CEA, DSM, SPhT (Service de Physique Théorique), F-91191 Gif-sur-Yvette, FRANCE)
Bouchet F.R. (Institut d\'Astrophysique de Paris (IAP-CNRS), 98 bis Bld Arago, F-75014 Paris, FRANCE)
Hernquist L. (Board of Studies in Astronomy and Astrophysics, Univ. of Californie, Santa Cruz, CA 95064, USA)
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.
Année de publication : 1997
Revue : Mon. Not. R. Astron. Soc. 287 241-252 (1997)
Langue : Anglais

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