On the evolution of aspherical perturbations in the universe: an analytical model

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, similarly to what done in WS. The comparison of the analytical solution with numerical simulations shows that it is a good approximation to 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 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 ellipsoids 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.