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

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4. Spherical density fluctuations

As we showed in the previous paragraphs, the spherical dynamics provides a good approximation to the behaviour of the density field (and a reasonable description for the velocity divergence) in the quasi-linear regime, and it even gives the exact generating functions [FORMULA] in the limit [FORMULA]. This may look surprising, since all regions of space cannot follow simultaneously a spherical dynamics as we have already noticed, so that one would expect this prescription to be always somewhat different from the exact results. In fact, as we shall see below, this is due to the fact that the limit [FORMULA] (or equivalently [FORMULA]) constrains the increasingly rare density fluctuations of order unity to be spherically symmetric. Then a spherical dynamics should naturally lead to correct results. Note that the mean evolution before virialization of rare large density fluctuations was treated by Bernardeau 1994b.

Thus, let us define [FORMULA] (resp. [FORMULA]) as the mean density contrast over a sphere of radius R (resp. [FORMULA]) centered on a point O (resp. O'), and we note [FORMULA] the vector [FORMULA]. Then, the conditional probability [FORMULA] to get a density contrast [FORMULA] knowing that we have a density contrast [FORMULA] in the first sphere is simply a gaussian:

[EQUATION]

where we defined:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and [FORMULA] is the top-hat window function (see for instance Bardeen et al.1986 for properties of gaussian random fields). Thus, the mean value of [FORMULA] is [FORMULA] which only depends on the distance [FORMULA] from the point O where the density contrast is constrained to be [FORMULA] over R, which is obvious from the symmetry of the problem. However, the profile of the density fluctuation centered on O is usually not spherically symmetric, because of the fluctuations of [FORMULA]. If we consider now the limit [FORMULA] (i.e. the normalization of the power-spectrum goes to 0) at fixed [FORMULA], the mean value [FORMULA] does not change but [FORMULA]. Thus, in this limit the profile of the density fluctuation centered on O becomes spherically symmetric ([FORMULA]), and its dynamics is exactly described by the spherical model we used in the previous paragraphs. Of course, most of the matter (and space) is formed by density fluctuations [FORMULA] which are not spherically symmetric, but these areas correspond by definition to density contrasts which tend to 0 as [FORMULA] and they do not influence the shape of the functions [FORMULA] which depend on finite values of the density contrast. This can be seen for instance on (10) which shows that a finite value of y corresponds to a finite value of the density contrast [FORMULA] given by [FORMULA] while [FORMULA]. Thus, our approach explains the results of the rigorous calculation of the generating functions [FORMULA] in the limit [FORMULA] from the exact equations of motion, and why they depend simply on the spherical dynamics. Moreover, our model is not restricted a priori to [FORMULA] and the comparison with numerical simulations shows it gives reasonable results up to [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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