An analytical model for the triaxial collapse of cosmological perturbations

We present an analytical model for the non-spherical collapse of overdenseregions out of a Gaussian random field of initial cosmological perturbations.The collapsing region is treated as an ellipsoid of constant density, actedupon by the quadrupole tidal shear from the surrounding matter. The dynamics ofthe ellipsoid is set by the ellipsoid self-gravity and the external quadrupoleshear. Both forces are linear in the coordinates and therefore maintainhomogeneity of the ellipsoid at all times. The amplitude of the external shearis evolved into the non-linear regime in thin spherical shells that are allowedto move only radially according to the mass interior to them. We describe howthe initial conditions can be drawn in the appropriate correlated way from arandom field of initial density perturbations. By considering many randomrealizations of the initial conditions, we calculate the distribution of shapesand angular momenta acquired by objects through the coupling of theirquadrupole moment to the tidal shear. The average value of the spin parameter,0.04, is found to be only weakly dependent on the system mass, the meancosmological density, or the initial power spectrum of perturbations, inagreement with N-body simulations. For the cold dark matter power spectrum,most objects evolve from a quasi-spherical initial state to a pancake orfilament and then to complete virialization. Low-spin objects tend to be morespherical. The evolution history of shapes is primarily induced by the externalshear and not by the initial triaxiality of the objects. The statisticaldistribution of the triaxial shapes of collapsing regions can be used to testcosmological models against galaxy surveys on large scales.Comment: 42 pages, Tex, followed by 10 uuencoded figure