A&A 387, 759-777 (2002)

DOI: 10.1051/0004-6361:20020399

## On the evolution of aspherical perturbations in the universe: An analytical model

**A. Del Popolo**

^{1, 2, 3}^{1}Dipartimento di Matematica, Università Statale di Bergamo, via dei Caniana 2, 24129 Bergamo, Italy

e-mail: adelpop@unibg.it

^{2}Feza Gürsey Institute, PO box 6 Çengelköy, Istanbul, Turkey

^{3}Bo aziçi University, Physics Department, 80815 Bebek, Istanbul, Turkey

(Received 30 October 2001 / Accepted 18 February 2002 )

** Abstract **

I study the role of shear fields
on the evolution of density perturbations
by using an analytical approximate solution for the equations of
motion of homogeneous ellipsoids embedded in a homogeneous
background.
The equations of motion of a homogeneous ellipsoid (Icke 1973;
White & Silk 1979, hereafter WS) are modified in order to take
account of the tidal field, as done in Watanabe (1993) and then are
integrated analytically, similar to what was done in WS. The
comparison of the analytical solution with numerical simulations
shows that it is a good approximation of the numerical one. This
solution is used to study the evolution of the configuration of
the ellipsoids, to calculate the evolution of the density contrast
and that of the axial peculiar velocity of the ellipsoids for
several values of the amplitude of the external tidal field,
and is compared again with numerical simulations. In order to
calculate the evolution of the density contrast at turn-around and
collapse velocity at the epoch of collapse, as a function of the
ratio of the initial value of the semi-axes, I use the previously-obtained
approximate solution to modify the analytical model
proposed by Barrow & Silk (1981) for the ellipsoid evolution in
the non-linear regime. The density contrast at turn-around and the
collapse velocity are found to be reduced with respect to that
found by means of the spherical model. The reduction increases
with increasing strength of the external tidal field and with
increasing initial asymmetry of the ellipsoids. These last
calculations are also compared with numerical solutions and they
are again in good agreement with the numerical ones.

**Key words:**cosmology: theory

**--**large-scale structure of Universe

**--**galaxies: formation

SIMBAD Objects

**©**

*ESO 2002*