The first departure from the standard top-hat model considered is motivated by the fact that the gravitational growth of cosmological density fluctuations in general favors the formation of filamentary structure over spherical. Iliev and Shapiro (1999, in preparation) have calculated the collapse and post-collapse virial equilibrium of cylindrical, planar, and spheroidal top-hats, for comparison with the well-known results for the spherical top-hat model. Among other things, they find that the departure from spherical symmetry results in in an earlier collapse to infinite density than in the spherical case, which means that the critical overdensity which is obtained by extrapolating the linear solution to this collapse moment is smaller for nonspherical top-hats than the well-known result for spheres of 1.686. These results also allow a more accurate interpretation of simulations of cosmological structure formation in 2D, for comparison with 3D calculations. Martel, Shapiro, and Iliev (1999, in preparation]) considered a second kind of departure from the standard problem of the collapse of spherical top-hat density perturbations, that of the generalization from the well-known solutions for matter-dominated cosmological models to the more general models in which a nonclumping component of energy density is present (e.g. cosmological constant), in addition to the standard baryonic and dark matter. This work includes the further generalization to the case of nonspherical top-hat perturbations, as well - i.e. uniform density cylinders, sheets, and spheroids embedded in an unperturbed background universe.
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