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Astron. Astrophys. 337, 655-670 (1998)

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2. Quasi-linear regime: density contrast

As VS showed, it is possible to extend the usual PS approach to obtain an approximation of the density field and the counts-in-cells it implies, in addition to the mass function of just-virialized objects one generally considers. Although VS focused on the highly non-linear regime (using the stable-clustering ansatz), we shall here first consider the simpler case of the quasi-linear regime [FORMULA].

2.1. Critical universe: [FORMULA]

2.1.1. Lagrangian point of view

We first consider a Lagrangian point of view, which is well suited to PS-like approaches since the fundamental hypothesis of such approximations is that it is possible to follow the evolution of fluid elements recognized in the early linear universe (one usually only considers overdensities) up to the non-linear regime. Thus, the usual PS prescription assumes i) that the dynamics of these objects is given by the spherical model and ii) that their keep their identity throughout their evolution. Hence, to any piece of matter identified in the early linear universe one can assign at a later time a specified radius and density. Of course, this cannot be exact as all particles cannot follow simultaneously a spherical dynamics, and this approach cannot take into account mergings which change the number of objects and destroy their identity. However, we shall see below that it can provide a reasonable estimate of the early density field.

In this Lagrangian approach, we consider the evolution of "objects" of mass M, identified in the early linear universe. In other words, the filtering scale is the mass and not the radius. According to the spherical model, particles which were initially embedded in a spherical region of space characterized by the density contrast [FORMULA] ([FORMULA] is the density contrast given by the linear theory at any considered time: [FORMULA] where [FORMULA] is the scale factor) find themselves in a spherical region of density contrast [FORMULA] at the same scale M, given at any epoch by:

[EQUATION]

The function [FORMULA] is defined by the dynamics of the spherical model (Peebles 1980):

[EQUATION]

and

[EQUATION]

This definition of [FORMULA] breaks down for [FORMULA], with [FORMULA], where [FORMULA] becomes infinite. This is usually cured by a virialization prescription: the halo is assumed to virialize in a finite radius taken for instance as one half its radius of maximum expansion, so that [FORMULA] with [FORMULA]. However, we shall not consider this modification in this section, as we focus here on the quasi-linear regime where most of the mass is embedded within regions of space with a density contrast [FORMULA] smaller than [FORMULA], hence we restrict ourselves to [FORMULA]. Now, we define the Lagrangian probability distribution [FORMULA]: if we choose at random a spherical region of mass M its density contrast is between [FORMULA] and [FORMULA] with probability [FORMULA] (here the index m does not stand for a given mass scale, its use is only to distinguish the Lagrangian quantities used here, where we follow matter elements, from their Eulerian counterparts whi ch we shall introduce in the next sections). Within the framework of the spherical model, this probability distribution is simply given by:

[EQUATION]

where [FORMULA] is the probability distribution relative to the linear density contrast, or the early universe. We shall assume that the initial density fluctuations are gaussian, so that:

[EQUATION]

where as usual [FORMULA] is the amplitude of the density fluctuations at scale M given by the linear theory at the considered time ([FORMULA] for an initial power-spectrum which is a power-law: [FORMULA]). We can note that the probability distribution [FORMULA] is correctly normalized: [FORMULA] by definition. However, the mean value of [FORMULA] is not equal to 0. This is quite natural since particles get embedded within increasingly high overdensities (while the density contrast cannot be smaller than -1), and at late times one expects that all the mass will be part of overdense virialized halos. However, this late stage is not described by the model introduced in this section, which does not include virialization and where half of the mass remains at any time within underdensities (this is the well-known normalization problem of the PS prescription by a factor 2).

We can also characterize the probability distribution [FORMULA] by the generating function [FORMULA] (and by [FORMULA]) which we define by:

[EQUATION]

where [FORMULA]. The function [FORMULA] generates the series of the moments of [FORMULA]:

[EQUATION]

In the highly non-linear regime considered by VS the function [FORMULA] was assumed to be scale-invariant, so that it did not depend on [FORMULA], but here this is not the case. If [FORMULA] the usual series of the cumulants is generated by [FORMULA]:

[EQUATION]

The relation (6) can be written in terms of the linear density contrast [FORMULA]:

[EQUATION]

[EQUATION]

In the quasi-linear regime, that is for [FORMULA], [FORMULA] and [FORMULA], we can use the saddle point method to get:

[EQUATION]

where [FORMULA] is the limit of [FORMULA] for [FORMULA]. If we define [FORMULA] and [FORMULA], we obtain

[EQUATION]

This is exactly the result derived rigorously by Bernardeau (1994a) from the equations of motion, when the matter is described by a pressure-less fluid. Hence our approach should give a good description of the early stages of gravitational clustering, since it is leads to the right limit for the cumulants in the linear regime, which was not obvious at first sight. Moreover, it provides some hindsight into the result of the exact calculation, as the function [FORMULA] or [FORMULA] which represents the spherical dynamics appears naturally in the expression of [FORMULA]. We shall come back to this point in Sect. 4. We can note that our model implies in addition a specific dependence of [FORMULA] on [FORMULA] (or [FORMULA]), which could be computed from (9).

2.1.2. Eulerian point of view

For practical purposes, one is in fact more interested in the Eulerian properties of the density field, where the filtering scale is a length scale R. For instance, a convenient way to describe the density fluctuations is to consider the counts-in-cells: one divides the universe into cells of scale R (and volume V) and defines the probability distribution [FORMULA] of the density contrast within these cells. Hence, we need to relate this Eulerian description to the Lagrangian model we developed in the previous section. As was done in VS, we shall use:

[EQUATION]

with [FORMULA]. Thus the mass embedded in cells of scale R with a density contrast larger than [FORMULA] is taken equal to the mass formed by particles which are located within spherical regions of scale M with a density contrast larger than [FORMULA]. Using our Lagrangian model, we get:

[EQUATION]

where [FORMULA] is the mass fraction obtained in the linearly extrapolated universe. From (12) and (5) we have:

[EQUATION]

with:

[EQUATION]

Finally, we obtain:

[EQUATION]

as in VS. Of course we recover all the mass of the universe [FORMULA]. In fact, half of the mass is in overdensities and the other half in underdensities, as was the case in the Lagrangian description. However, in general this probability distribution is not correctly normalized: [FORMULA] (but in the linear regime, [FORMULA] and [FORMULA], its normalization tends to unity).

We can still define a generating function [FORMULA], as in (6), which leads to:

[EQUATION]

Using [FORMULA], with [FORMULA], we define:

[EQUATION]

Hence [FORMULA]. In the quasi-linear regime, that is for [FORMULA], [FORMULA] and [FORMULA], the saddle point method leads to:

[EQUATION]

where [FORMULA] is the limit of [FORMULA] for [FORMULA]. Thus, once again we recover the result derived rigorously by Bernardeau (1994a). Note that we should have modified [FORMULA] in (17) since it is not correctly normalized, however if this modification only consists of a multiplication factor (which must go to unity in the limit [FORMULA]) and a change of the shape of [FORMULA] for density contrasts much larger than [FORMULA] as [FORMULA], this does not modify the function [FORMULA] we obtained. If the power-spectrum is a power-law [FORMULA], (16) can be written:

[EQUATION]

[EQUATION]

If the power-spectrum is not a power-law, our method is still valid and we have to use (16). However, for a very smooth [FORMULA], like a CDM power-spectrum, an easier and still reasonable approximation is to use (20) where n is the local slope of the power-spectrum at scale R. We shall compare this approximation to numerical results in Sect. 2.3. The fact that we recover the exact generating function [FORMULA] for [FORMULA] suggests again that our prescription could provide reasonable results in the early linear universe when [FORMULA]. We shall see below that it is indeed the case, by a comparison with numerical simulations. We can note that our probability distributions (20) look rather different from those obtained by Bernardeau (1994a) since the high-density cutoff is usually different from a simple exponential. However, they both agree with numerical results for [FORMULA]. In fact, neither of these approaches should be used for large density contrasts [FORMULA] where shell-crossing and virialization play an important role.

2.2. Open universe: [FORMULA]

2.2.1. Lagrangian point of view

In the case of an open universe, we can still apply the method described previously for a critical universe but there is now an additional time dependence in the relation [FORMULA]. Thus, (4) becomes:

[EQUATION]

In the specific case where [FORMULA], one defines:

[EQUATION]

and

[EQUATION]

which is the growing mode of the linear approximation normalized so that [FORMULA]. Then, the function [FORMULA] is given by:

[EQUATION]

and

[EQUATION]

In the case [FORMULA] ([FORMULA]), large overdensities have already collapsed, and we are left with:

[EQUATION]

This simple form for [FORMULA] leads to:

[EQUATION]

which provides a convenient estimation for [FORMULA]. As for a critical universe we can still define a generating function [FORMULA] which allows us to recover the results of Bernardeau (1994a).

2.2.2. Eulerian point of view

Naturally, we can obtain an approximation for the probability distribution [FORMULA] of the density contrast within cells of scale R in a fashion similar to what we did for a critical universe from the Lagrangian probability [FORMULA]. In the case of a power-spectrum which is a power-law, and in the limit [FORMULA], we get for instance:

[EQUATION]

As we shall see in the next section, this very simple formula provides in fact a good approximation to [FORMULA] for all values of [FORMULA] of interest (even for [FORMULA]) due to the weak dependence of [FORMULA] on [FORMULA]. As for a critical universe we can also define a generating function [FORMULA] and recover the results of Bernardeau (1994a).

We can note that Protogeros & Scherrer (1997) obtained similar results with an approach close to ours. They considered "local Lagrangian approximations" where the density contrast at the Lagrangian point [FORMULA], time t, is related to its initial value by [FORMULA]. They used several approximations for [FORMULA] including the simplified spherical collapse model (24) and obtained the probability distribution (26), which they modified by introducing an ad-hoc multiplicative function [FORMULA] within the relation [FORMULA] in order to normalize properly [FORMULA]. However, our model differs from theirs by some aspects. Thus, our Lagrangian probability distribution is defined from the start with respect to a given mass scale M (which might be seen as a "smoothing" scale). This appears naturally in our approach, and it ensures we always work with well-defined quantities. Indeed, for a power-spectrum which is a pure power-law with [FORMULA] the "unsmoothed" density field is not a function but a distribution. In fact, one cannot characterize a point by a finite density, without specifying the scale over which this density is realised. Then, the change from the Lagrangian to the Eulerian view-point is quite natural, and it provides an Eulerian distribution function which differs from the Lagrangian one in a very simple and physical manner and which depends on the power-spectrum. No smoothing procedure needs to be applied in fine in order to compare with observations: a filtering scale (M or R) is always automatically included in our approach. Our approximation also shows clearly the dependence on [FORMULA] of the functions [FORMULA] and [FORMULA]. Finally, one can note that contrary to Protogeros & Scherrer (1997) we did not normalize our probability distribution [FORMULA]. In fact, we think such a procedure is somewhat artificial and may lead to an even worse approximation. Indeed, if the normalization problem comes mainly from a specific interval of the density contrast where our approximation is very bad, a simple normalization procedure will not give very accurate results in this interval (since the starting values have no relation with the correct ones) while it will destroy our predictions in the interval where they were fine. Thus, one can fear such "cure" may in fact spread errors over all density contrasts. We shall come back to this point below, but we can already note that for [FORMULA] the probability distribution (26) cannot be meaningfully normalized since [FORMULA].

2.3. Comparison with numerical results

From the results of previous paragraphs, since the functions [FORMULA] obtained with our prescription in the limit [FORMULA] are exactly those derived by a rigorous calculation we can expect that the probability distribution [FORMULA] we get should provide a good approximation in the linear regime. Thus, Fig. 1 and Fig. 2 present a comparison of our approximation with the results of numerical simulations, taken from Bernardeau (1994a) and Bernardeau & Kofman (1995) in the case of a CDM initial power-spectrum in a critical universe. We display our predictions for a critical universe (relation (20)) and an "empty" universe (relation (26)), for a power-spectrum which is a power-law with n given by the local slope of the actual power-spectrum.

[FIGURE] Fig. 1. The probability distribution of the density contrast [FORMULA]. The solid lines present the prediction of our prescription for a critical universe and a power-spectrum which is a power-law (relation (16) or (20)), for various [FORMULA] and n, while the dashed-lines show the corresponding curves for an empty universe [FORMULA] (relation (26)). The data points are taken from Bernardeau (1994a) and Bernardeau & Kofman (1995) and correspond to a numerical simulation with a CDM initial power-spectrum in a critical universe (thus n is the local slope of the power-spectrum). The density fluctuation [FORMULA] and n were measured in the simulation.

[FIGURE] Fig. 2. The probability distribution of the density contrast [FORMULA] as in Fig. 1 but for two different values of [FORMULA].

We can see that our approach leads indeed to satisfactory results up to [FORMULA] given its extreme simplicity. As was noticed by Bernardeau (1992), the [FORMULA] - dependence of the probability distribution is very weak (the dashed lines are very close to the solid lines on the figures), which means that the simple expression (26) provides a reasonable fit for all cosmological models up to [FORMULA]. However, as we can see on Fig. 2, our prescription leads to a sharp peak for very underdense regions [FORMULA] which increases with [FORMULA] but does not appear in the numerical results. This defect is also related to the fact that our probability distribution is not correctly normalized to unity. This problem is due to the expansion of underdense regions, which according to the spherical dynamics grow faster than the average expansion of the universe so that in our present model these areas occupy after some time a volume which is larger than the total volume which is available, which leads to a probability which is too large. Indeed, within our approach underdense regions can expand without any limit while in reality this growth is constrained by the fact that on large scales we must recover the average expansion [FORMULA]. Thus, underdensities join together after some time and their mutual influence alters their dynamics, which we did not take into account.

A simple way to normalize correctly the probability distribution [FORMULA] would be to define the latter from the generating function [FORMULA] which one would take identical to [FORMULA] for any [FORMULA]: this is the method used successfully by Bernardeau (1994a). However, there is a priori no fundamental reason why this should be a particularly efficient procedure (except from the mere constatation that it works). In fact, as we can see on Fig. 2 we can expect most of the problem to come from the peak which develops at low density contrasts, so that one should keep the consequences of the evolution of [FORMULA], and [FORMULA], in other ranges of [FORMULA] and simply disregard the predictions obtained in the vicinity of this peak or introduce a specific modification for this interval. This is even clearer on Fig. 3 where we can see that our approximation can still provide reasonable predictions for [FORMULA] and [FORMULA] although it completely fails for underdense regions. Of course the agreement with numerical results improves for smaller [FORMULA], as shown on Fig. 1 and Fig. 2, and becomes excellent in the limit [FORMULA]. We only show on Fig. 3 the largest values of [FORMULA] where our approximation still makes some sense, in order to present its range of validity and to show clearly for which values of [FORMULA] (underdensities) it breaks down first. As Protogeros et al.(1997) noticed, the problem becomes increasingly severe as n gets larger. However, our predictions work better than those used by these authors because we did not normalize our probability distribution (see their Fig. 2). This was expected since we noticed earlier that [FORMULA] for [FORMULA], so that it cannot even be normalized, but this does not prevent our approximation to provide very good results for low [FORMULA], and when [FORMULA] we recover the gaussian on any finite interval of the density contrast which does not include [FORMULA].

[FIGURE] Fig. 3. The probability distribution of the density contrast [FORMULA]. The solid lines present the prediction of our prescription for a critical universe and a power-spectrum which is a power-law (relation (16) or (20)), for various [FORMULA] and n, while the dashed-lines show the corresponding curves for an empty universe [FORMULA] (relation (26)). The data points are taken from Protogeros et al.(1997).

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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