### Citation:

### Date Published:

February 1, 1995### Abstract:

We present an analytical model for the nonspherical collapse ofoverdense regions out of a Gaussian random field of initial cosmologicalperturbations. The collapsing region is treated as an ellipsoid ofconstant density, acted upon by the quadrupole tidal shear from thesurrounding matter. The dynamics of the ellipsoid is set by theellipsoid self-gravity and the external quadrupole shear. Both forcesare linear in the coordinates and therefore maintain homogeneity of theellipsoid at all times. The amplitude of the external shear is evolvedinto the nonlinear regime in thin spherical shells that are allowed tomove only radially according to mass interior to them. The fulldynamical equations then reduce to a set of nine second-order ordinarydifferential equations, which reproduce the linear regime behavior butcan be evolved past turnaround, well into the nonlinear regime. Wedescribe how the initial conditions can be drawn in the appropriatecorrelated way from a random field of initial density perturbations. Themodel is applied to a restricted set of initial conditions that are moresuitable to the above approximations; most notable we focus on theproperties of rare high-density peaks (greater than of approximately = 2sigma). By considering many random realization of the initialconditions, we calculate the distributions of shapes and angular momentaacquired by objects through the coupling of their quadruole moment tothe tidal shear. The average value of the spin parameter, (mean value ofa lambda) approximately = 0.04, is found to be only weakly dependent onthe system mass, the mean cosmological density, or the initial powerspectrum of perturbations, in agreement with N-body simulations. For thecold dark matter power spectrum, most objects evolve from aquasi-spherical initial state to a pancake or filament and then tocomplete virialization. Low-spin objects tend to be more spherical. Theevolution history of shapes is primarily induced by the external shearand not by the initial triaxially of the objects. The statisticaldistribution of the triaxial shapes of collapsing region can be used totest cosmological models against galaxy surveys on large scales.