The development of structure in the expanding universe

Abstract

We develop a new model for clustering in an expanding universe, based on an application of the coagulation equation to the collision and aggregation of bound condensations. While the growth rate of clustering is determined by the rate at which density fluctuations reach the nonlinear regime, and therefore depends on the initial fluctuation spectrum, the mass spectrum rapidly approaches a self-similar limiting form. This form is determined by the tidal processes which lead to the merging of condensations, and is not dependent on initial conditions. Subject headings: cosmology-galaxies : clusters of I. INTRODUCTION AND SUMMARY In an interesting paper, Press and Schechter (1974, hereafter PS) have used a simple energy argument to derive a mass spectrum for the condensations which grow in an Einstein-de Sitter universe from a power-law spectrum of primordial density fluctuations, such that the typical density fluctuation on mass scale m satisfies bp/p oc nr y. The PS mass function evolves in a self-similar fashion (in the sense that it preserves its shape as clustering proceeds) but its form depends on the exponent 7 in the initial fluctuation spectrum. The scale mass in their solution increases steadily as perturbations on larger and larger scales become nonlinear and separate from the general expansion: for a spectrum bp/p nr^ this scale must grow as if) oc t 2IZy. This behavior is to be expected on very general grounds, independent of the details of the PS theory. Peebles (1974) and Doroshkevich and ZePdovich (1975) show that the maximum possible effective value of 7 is 7/6, since a spectrum with this index at large masses is generated in an otherwise perturbation-free medium by any small-scale lumpiness. Values of 7 less than 7/6 are to be expected if there is any process which can have generated density fluctuations on large scales. For example, 7 = J for a Poisson distribution of points or for a corresponding "white-noise" spectrum of density fluctuations. In deriving their mass spectrum, PS assume not only an Einstein-de Sitter universe, but also that evolution on different mass scales is entirely independent, and further, that all perturbations can be considered to be uniform, spherically symmetric, and isolated from each other. The assumption of an Einstein-de Sitter universe is easily relaxed (cf. White and Rees 1978), with the result that in a low density universe a mass spectrum "freezes out" as the universe becomes open (i.e., at 1 + £2 _1). The other assumptions are unavoidable, however, and seem far removed from the highly nonlinear and inhomogeneous processes which must occur in reality. In fact, the PS theory contains essentially no dynamics, and asserts that the mass function of condensed lumps in the universe can be deduced from the initial fluctuation spectrum without any consideration of intermediate evolutionary stages. The present Letter describes a novel and quite independent treatment of the growth of clustering in an expanding universe, from which we derive mass spectra which resemble those of PS but which appear to be independent of initial conditions. We apply the coagulation equation to describe the nonlinear tidal effects which lead to the amalgamation of bound lumps of material in an expanding universe. While the growth rate of clustering depends on the rate at which density fluctuations enter the nonlinear regime and therefore on the initial fluctuation spectrum, the mass function of the lumps soon approaches a self-similar limiting form which is determined by the tidal merging process and not by initial conditions. The scale mass of this spectrum grows as a power law of time while the universe is approximately Einstein-de Sitter, but the spectrum freezes out and further aggregation ceases as soon as the universe becomes effectively open. II. THE COAGULATION EQUATION At its formation, any condensation in the expanding universe has a peculiar velocity with respect to the Hubble flow resulting in a complex way from the peculiar velocities of its progenitors and from its tidal interaction with other nearby irregularities. Some position-velocity correlation is thus inevitable among newly formed units. Further amalgamation occurs when condensations pass sufficiently close to one another to undergo a strong tidal encounter. PS assumed that condensations in an overdense region always coalesce, regardless of their tidally induced velocities. We here employ the entirely different assumption that peculiar velocities are generated by perturbations over such a wide range of scales that they can be considered to be randomly directed when calculating the probability of individual close encounters. Let iV(w, t) dm be the number density of condensed lumps in the mass range (w, m + dm) at time ¿.We define an aggregation rate for particles in the mass range (w, m + dm) with particles in the range (m' y m f + dm') by N{m, t)