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

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3. Quasi-linear regime: velocity divergence

The prescription we described in the previous paragraphs can also predict the statistical properties of the divergence of the velocity field. Note that we cannot get any reliable result for the shear, since our model is based on a pure spherical dynamics, so that the shear is zero for all the individual regions of matter we consider. This is clearly an important shortcoming of this simple approximation, however in the linear regime where density fluctuations are small the rotational part of the velocity field decays so that after a long time (but still in the linear regime) keeping only the growing mode the velocity can be described by its divergence, or a velocity potential. Thus, our model does not contradict a priori the properties of the linear regime, which is of course a first requisite. We define the peculiar velocity [FORMULA] by:

[EQUATION]

and the divergence [FORMULA] ([FORMULA] in comoving coordinates, and [FORMULA]) by:

[EQUATION]

We can note that in the linear regime, where we keep only the growing mode, we have:

[EQUATION]

where [FORMULA] is the growing mode of the density fluctuations and [FORMULA] (see Peebles 1980).

To use the method we described for the density contrast [FORMULA], we must now link the divergence [FORMULA] of the velocity field to the linear density contrast [FORMULA] of our Lagrangian elements of matter. One possibility is to make the approximation that the density is uniform over these individual regions, and to use the continuity equation (in physical coordinates):

[EQUATION]

An alternative is to consider the mean divergence over the Lagrangian matter element V, which can be expressed in terms of the expansion of this global volume (velocity of the outer boundary):

[EQUATION]

This last "definition" of [FORMULA] is the most natural as it does not need any additional information on the density profile. However, in both cases we obtain:

[EQUATION]

which relates [FORMULA] to [FORMULA] through the relation [FORMULA] used in the previous sections (note that the derivative in (32) is to be understood at fixed [FORMULA]: one follows the motion of a given fluid element, so that [FORMULA] is also a function of a). Then, we obtain the Lagrangian probability distribution [FORMULA] and its Eulerian counterpart, which is simply given by [FORMULA] (we introduced a negative sign because [FORMULA]). Thus the latter can be written:

[EQUATION]

We can also introduce the functions [FORMULA] and [FORMULA] (with [FORMULA]). As was the case for the density contrast we recover in the limit [FORMULA] the functions derived by Bernardeau (1994a).

If we make the approximation [FORMULA] where [FORMULA] is the function obtained in the limit [FORMULA] (see (24)) we get:

[EQUATION]

Then, if we define [FORMULA] we obtain:

[EQUATION]

with

[EQUATION]

[EQUATION]

Note that the probability distribution of the reduced variable [FORMULA] does not depend any longer on the cosmology (we only neglected the slight dependence on [FORMULA] of the function [FORMULA]).

We compare this approximation (36) to the results of numerical simulations taken from Bernardeau et al.(1997) on Fig. 4. We can see that we match the high cutoff of [FORMULA], which corresponds simply to the expansion rate of a void (zero density), but the negative tail is not well reproduced for the case of the open universe. Thus, it seems that the description of the velocity field is more sensitive on the approximations involved in our method than the density field. This may be due to the influence of the shear, which implies that the velocity can no longer be determined by a mere scalar (through the potential or the divergence). Moreover, we can note that, contrary to the Zeldovich approximation for instance, our prescription does not provide the location of particles and their velocity field from initial conditions (all regions of space cannot follow a spherical dynamics at the same time), it only gives an estimate for some probability distributions without considering the consistent dynamics of all particles simultaneously. Thus, the goal of our approach is more modest than such a global modelization and it is not entirely self-consistent. However, it appears that after accepting this shortcomings we obtain nevertheless reasonable results for the density field. In fact, as we shall see in the next section, we expect large density fluctuations to follow a dynamics close to the spherical model while the intermediate areas which connect these regions obey a complex non-spherical dynamics but as their density contrast is of the order of [FORMULA] they do not play an important role for the global shape of [FORMULA] as long as [FORMULA].

[FIGURE] Fig. 4. The probability distribution of the divergence of the velocity field [FORMULA]. The solid line is the prediction of our prescription for a critical universe while the dashed-lines corresponds to an open universe [FORMULA]. The data points are taken from Bernardeau et al.(1997) for the same conditions (filled circles for the critical universe and rectangles for the open universe).

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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