Second-order perturbation theory in an expanding universe: Spherical harmonics expansion

The formalism of gravitational instability in an expanding universe is developed here to second order in terms of the spherical harmonics representation and is applied to study the evolution of single localized perturbations to an otherwise flat Einstein-de Sitter universe. A simple analytical form of the radial dependence of the various harmonics is used to model the structure expected for a scale-free Gaussian perturbation field. The general formalism is applied to study the dynamics of the monopole terms, i.e., the spherically averaged variables, of a proto-object made of monopole and quadrupole terms only. Our main result is that at the inner part, i.e., the core, the dynamics is dominated by the monopole term, and it reproduces the exact spherical top-hat model. However, the outer region, i.e., the halo, is dominated by the quadrupole terms, and its dynamics differs significantly from the top-hat model. The l = 2 terms lead to the formation of a secondary maximum in the run of the density (averaged over spherical shells) versus radius. This is a pure second-order effect whose very clear signature is that it occurs at a radius of the transition to the nonlinear regime. Indeed, such a secondary maximum has been observed in the outer parts of real clusters of galaxies and in N-body numerical simulations. Another pure second-order effect discovered here is the case of a pure quadrupole primordial perturbation. For such a perturbation the vanishing monopole does not grow in the linear regime, but is unstable to second order. We find a considerable growth for typical values of parameters, and conjecture that this is the main dynamical factor in the formation of filaments in the large-scale structure of the universe.