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*Astron. Astrophys. 337, 655-670 (1998)*
## 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
in the limit . 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 (or
equivalently ) 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 (resp.
) as the mean density contrast over a sphere of
radius *R* (resp. ) centered on a point O
(resp. O'), and we note the vector
. Then, the conditional probability
to get a density contrast
knowing that we have a density contrast
in the first sphere is simply a gaussian:
where we defined:
and is the top-hat window function (see for
instance Bardeen et al.1986 for properties of gaussian random fields).
Thus, the mean value of is
which only depends on the distance
from the point O where the density contrast is
constrained to be 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 . If
we consider now the limit (i.e. the
normalization of the power-spectrum goes to 0) at fixed
, the mean value does
not change but . Thus, in this limit the
profile of the density fluctuation centered on O becomes spherically
symmetric (), 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 which are not spherically
symmetric, but these areas correspond by definition to density
contrasts which tend to 0 as and they do not
influence the shape of the functions 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
given by while
. Thus, our approach explains the results of the
rigorous calculation of the generating functions
in the limit from the
exact equations of motion, and why they depend simply on the spherical
dynamics. Moreover, our model is not restricted a priori to
and the comparison with numerical simulations
shows it gives reasonable results up to .
© European Southern Observatory (ESO) 1998
Online publication: August 27, 1998
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