Astron. Astrophys. 364, 1-16 (2000)

5. The nonlinear regime

In 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 stable clustering ansatz gives indication on the expected amplitude of the high order correlation functions and how they scale with the two-point one. The hierarchical tree model, and more specifically the minimal tree model , allows to build a coherent set of high-order correlation functions.

5.1. The stable-clustering ansatz

The 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 x defined by:

Then, using (19), for sufficiently "large" density contrasts the PDF can be written as (Balian & Schaeffer 1989):

when

where the scaling function is the inverse Laplace transform of :

In (92) the exponent comes from the behavior of at large y. Indeed, from very general considerations (Balian & Schaeffer 1989) one expects the function defined in (18) to behave for 3D top-hat filtering as a power-law for large y:

and to display a singularity at a small negative value of y,

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-model

If 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 p-point correlation functions. We are thus forced to adopt a specific model for the correlation functions, and the one we adopt is obviously consistent with the stable-clustering ansatz.

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 p points without making any loop, is a parameter associated with the order of the correlations and the topology involved, is a particular labeling of the topology and the product is made over the links between the p points with two-body correlation functions. A peculiar case of the models described by (100) is the "minimal tree-model" (Bernardeau & Schaeffer 1992, 1999; Munshi et al. 1999) where the weights are given by:

where is a constant weight associated to a vertex of the tree topology with q outgoing lines. Then, one can derive the generating function , defined in (18), or the coefficients , from the parameters introduced in (101) which completely specify the behavior of the point correlation functions.

In this case the cumulant generating function is given for 3D filtering by,

where w corresponds to the filter choice. In the case of the minimal tree-model, where the point correlation functions are defined by the coefficients from (100) and (101), it is possible to obtain a simple implicit expression for the function (see Bernardeau & Schaeffer 1992; Jannink & des Cloiseaux 1987):

where the function is defined as the generating function for the coefficient ,

The function obviously depends on the choice of filter through the function w. For a top-hat filter, it would simply be a characteristic function normalized in such a way that

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 V in the second line of the system (103) and then to approximate by a constant . This leads to the simple system:

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-model

As 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 p only and not on their geometrical decompositions. As a consequence the projected point correlation function can be written,

where the two-dimensional point functions have the same tree-structure as the three-dimensional point correlation functions ,

with:

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 same tree-structure as their 3D counterparts . This means that we can still use the techniques developed to deal with such tree-models. In particular, in the case of the minimal tree-model (101) we will be able to take advantage of the resummation (103-104).

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 n of the power-spectrum. On the other hand, if the initial tree-model for the 3D correlation functions is the minimal tree-model (101) we can see that the projected tree-structure (116) is not an exact minimal tree-model ¹ which would be expressed in terms of a new generating function . In other words cannot be built from a tree structure whereas can, and with the same vertex generating function defined in (105).

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 z through and . Note also that in the expansion of the generating function is .

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 y. We also found that the results we obtain are very robust regarding to the number of shells used to describe the integral in : with 2 cells only the description of is already very accurate.

Fig. 4. Profile of the function as a function of the angular radius. The two plots correspond to the two singularities, for the solid line and for the dashed line.

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,

which leads to,

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.

Table 1. Values of and singularity positions of for filter.

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
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