## 2. Quasi-linear regime: density contrastAs 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 . ## 2.1. Critical universe:## 2.1.1. Lagrangian point of viewWe 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 The function is defined by the dynamics of the spherical model (Peebles 1980): This definition of breaks down for
, with , where
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 with .
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
smaller than , hence we
restrict ourselves to . Now, we define the
Lagrangian probability distribution : if we
choose at random a spherical region of mass where 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: where as usual is the amplitude of the
density fluctuations at scale We can also characterize the probability distribution by the generating function (and by ) which we define by: where . The function generates the series of the moments of : In the highly non-linear regime considered by VS the function was assumed to be scale-invariant, so that it did not depend on , but here this is not the case. If the usual series of the cumulants is generated by : The relation (6) can be written in terms of the linear density contrast : In the quasi-linear regime, that is for , and , we can use the saddle point method to get: where is the limit of for . If we define and , we obtain 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 or which represents the spherical dynamics appears naturally in the expression of . We shall come back to this point in Sect. 4. We can note that our model implies in addition a specific dependence of on (or ), which could be computed from (9). ## 2.1.2. Eulerian point of viewFor practical purposes, one is in fact more interested in the
Eulerian properties of the density field, where the filtering scale is
a length scale with . Thus the mass embedded in cells of
scale where is the mass fraction obtained in the linearly extrapolated universe. From (12) and (5) we have: as in VS. Of course we recover all the mass of the universe . 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: (but in the linear regime, and , its normalization tends to unity). We can still define a generating function , as in (6), which leads to: Using , with , we define: Hence . In the quasi-linear regime, that is for , and , the saddle point method leads to: where is the limit of for . Thus, once again we recover the result derived rigorously by Bernardeau (1994a). Note that we should have modified 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 ) and a change of the shape of for density contrasts much larger than as , this does not modify the function we obtained. If the power-spectrum is a power-law , (16) can be written: 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
, like a CDM power-spectrum, an easier and still
reasonable approximation is to use (20) where ## 2.2. Open universe:## 2.2.1. Lagrangian point of viewIn 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 . Thus, (4) becomes: In the specific case where , one defines: and which is the growing mode of the linear approximation normalized so that . Then, the function is given by: and In the case (), large overdensities have already collapsed, and we are left with: This simple form for leads to: which provides a convenient estimation for . As for a critical universe we can still define a generating function which allows us to recover the results of Bernardeau (1994a). ## 2.2.2. Eulerian point of viewNaturally, we can obtain an approximation for the probability
distribution of the density contrast within
cells of scale As we shall see in the next section, this very simple formula provides in fact a good approximation to for all values of of interest (even for ) due to the weak dependence of on . As for a critical universe we can also define a generating function 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 , time ## 2.3. Comparison with numerical resultsFrom the results of previous paragraphs, since the functions
obtained with our prescription in the limit
are exactly those derived by a rigorous
calculation we can expect that the probability distribution
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
We can see that our approach leads indeed to satisfactory results up to given its extreme simplicity. As was noticed by Bernardeau (1992), the - 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 . However, as we can see on Fig. 2, our prescription leads to a sharp peak for very underdense regions which increases with 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 . 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
would be to define the latter from the
generating function which one would take
identical to for any :
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 , and , in
other ranges of 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 and
although it completely fails for underdense
regions. Of course the agreement with numerical results improves for
smaller , as shown on Fig. 1 and
Fig. 2, and becomes excellent in the limit .
We only show on Fig. 3 the largest values of
where our approximation still makes some sense,
in order to present its range of validity and to show clearly for
which values of (underdensities) it breaks down
first. As Protogeros et al.(1997) noticed, the problem becomes
increasingly severe as
© European Southern Observatory (ESO) 1998 Online publication: August 27, 1998 |