## 5. The nonlinear regimeIn the nonlinear regime, there exist no derivations from first
principles of the behavior of the high order correlation functions of
the cosmic density field. However, hints of its behavior can be found.
The ## 5.1. The stable-clustering ansatzThe stable clustering ansatz (Peebles 1980) simply states that the high order correlation functions should compensate, in virialized objects, the expansion of the Universe. It gives not only the growth factor of the two-point correlation, but also a scaling relation between the high-order correlation functions. Expressed in terms of the coefficient -hence of the generating function defined in (18)- it means that they are independent of time and scale. As a consequence, the knowledge of the evolution of the power-spectrum , or of the two-point correlation function , is sufficient to obtain the full PDF of the local density contrast. This property has been checked in numerical simulations by several authors (Valageas et al. 2000; Bouchet et al. 1991; Colombi et al. 1997; Munshi et al. 1999). In particular, the statistics of the counts-in-cells measured in numerical simulations provide an estimate of the generating function for 3D top-hat filters. More precisely, in the highly non-linear regime one considers the
variable Then, using (19), for sufficiently "large" density contrasts the PDF can be written as (Balian & Schaeffer 1989): where the scaling function is the inverse Laplace transform of : In (92) the exponent comes from
the behavior of at large and to display a singularity at a small negative value of where we neglected less singular terms (note that this behavior has indeed been observed in the quasilinear regime, Bernardeau 1992). Taking advantage of these assumptions, one obtains (Balian & Schaeffer 1989), with . Hence, using (91) we see that the density probability distribution shows a power-law behavior from up to with an exponential cutoff above . It implies in particular that the function measured in numerical simulations can give rise to constraints on the cumulant generating function from the inverse relation, Note that depends on the power-spectrum and, in the absence of a reliable theory for describing the nonlinear regime, it has to be obtained from numerical simulations. This is the case in particular for the evolution of the two-point correlation function, or equivalently of the power-spectrum. To this order we use the analytic formulae obtained by Peacock & Dodds (1996) from fits to N-body simulations. Note that the relation (93) holds independently of the stable-clustering ansatz. However, if the latter is not realized the generating function depends on time (and scale). Then, most of the results we obtain in the next sections still hold but one needs to take into account the evolution with redshift of . That would be necessary in particular if one wants to describe the transition from the quasilinear regime to the strongly nonlinear regime. In the following we assume that the stable-clustering ansatz is valid, so that is time-independent. As mentioned above this is consistent with the results of numerical simulations. ## 5.2. Minimal tree-modelIf one is interested in the statistics of the top-hat filtered convergence, it is reasonable to assume that (Valageas 2000a,b), In case of the aperture mass statistics however the filtering
scheme is too intricate (with both positive and negative weights) to
make such an assumption, and in particular the resulting values for
depend crucially on the geometrical
dependences of the A popular model for the point correlation functions in the non-linear regime is to consider a "tree-model" (Schaeffer 1984; Groth & Peebles 1977) where is expressed in terms of products of as: where is a particular
tree-topology connecting the where is a constant weight
associated to a vertex of the tree topology with In this case the cumulant generating function is given for 3D filtering by, where where the function is defined as the generating function for the coefficient , The function obviously depends on
the choice of filter through the function A simple "mean field" approximation which provides very good
results in case of top-hat filter (Bernardeau & Schaeffer 1992) is
to integrate over the volume
Then, the singularity of , see (95), corresponds to the point where the vanishes. Note that is regular at this point and that the singularity is simply brought about by the form of the implicit system (108) as observed in Bernardeau & Schaeffer (1992). Making the approximation (107-108) for both and leads to the approximation (99) which is thus natural for the minimal tree model. In the case of a compensated filter such a simple mean field approximation however cannot be done. It is in particular due to the fact that the weights given to then strongly depend on the radius distance. Before we go to this point we need first to take into account the projection effects. ## 5.3. Projection effects for the minimal tree-modelAs noted in Tóth et al. (1989) and analyzed in detail in Valageas (2000b), we know that the tree structure assumed for the 3D correlation functions is preserved (except for one final integration along the line-of-sight) for the projected density. Indeed, inserting (100) in (24) we obtain: where we noted . It can be noted
that in the small angle approximation the weight applied to each
diagram depends on their order where the two-dimensional point functions have the same tree-structure as the three-dimensional point correlation functions , Here is the Bessel function of order 0. For convenience we also note the angular average of , which, expressed in terms of the power spectrum gives, Thus, we see that the correlation functions of the projected
density itself do not show an exact
tree-structure as the underlying 3D correlation functions
. Nevertheless, as seen in (110)
they are given by one simple integration along the line-of-sight of
the 2D point functions
which exhibit the Once again, it is interesting to note that for power-law spectra, , the angular and the redshift dependences of can be factorized so that the correlation functions of the projected density itself now exhibit a (new) tree-structure. Then, the 2-point function reads, while the high-order point functions follow the tree-structure (100) with the projected weights : Note that the relation depends
on the slope As a consequence, in the nonlinear regime, the relation (34) is to be used with, where is the 3D vertex
generating function. Note that this function depends on We have computed the resulting shape of the generating function in such a model for various cases. For comparison with the previous quasi-linear case we assume here the power spectrum to follow a power law behavior with index . The vertex generating function is assumed to be given by with (in the parameterization of
we followed the traditional
notation and used as a simple free
parameter. It is not to be confused with the local convergence). In
Fig. 4 we present typical profiles obtained for
. We see that it is regular in
. In particular it does not exhibit
discontinuities nor abnormal behavior near the singular values of
The choice of the value of in (119) relies a priori on numerical results. The EPT (Colombi et al. 1997) or more convincingly the HEPT (Scoccimarro & Frieman 1999) provide however a convenient frame which can be used to predict the value of . This can be done for instance by identifying the predicted values for the skewness both from the form (119) and HEPT. Indeed in our model we have, whereas, in HEPT, is related to the initial power spectrum index, In the numerical applications presented in the following we will use this scheme. In particular, leads to . On the other hand, at the angular scale which we consider below the local slope of the linear power-spectrum is which leads to . The properties of we get in this regime are very similar to those obtained for the quasilinear regime. In particular we found that the function exhibits 2 singular points on the real axis. As for the quasilinear regime this behavior is due to the implicit equation in and not to peculiar choice of the vertex generating function. In Table 1 we summarize the parameters that describe the singularities of in different regimes. It appears, as expected, that the singularities are closer to the origin. This is to be expected since the nonlinearities contained in are stronger in the nonlinear regime compared to the quasilinear regime.
Similarly to the quasilinear regime it is also possible to define an effective vertex generating function from which can be built and which reproduces its singular points, The result given here has been obtained for in Eq. (119). © European Southern Observatory (ESO) 2000 Online publication: December 15, 2000 |