An Analytic Expression for the Growth Function in a Flat Universe with a Cosmological Constant

An analytic expression is given for the growth function for linear perturbations in a low-density universe made at by a cosmological constant. The result involves elliptic integrals but is otherwise straight-forward. Subject headings: cosmology: theory { large-scale structure of the universe In linear cosmological perturbation theory, the evolution of perturbations in the absence of pressure is reduced to the superposition of two modes with xed time dependence and arbitrary spatial dependence (Peebles 1980). The behavior of these modes in time is a function of the background cosmological parameters. The functions of time that describe this behavior are called the growth functions and denoted D1 and D2. As shown by Heath (1977), the growth functions in cosmologies where pressure uctuations are negligible can be written as D1(a) / H(a) Z a 0 da a03H(a0)3 ; (1) D2(a) / H(a); (2) H(a) = p 0a 3 + Ra 2 + : (3) Here, a is the expansion scale factor of the universe, chosen so that a = 1 today; this will be our time coordinate. The redshift z is then a 1 1. 0 is the density of non-relativistic matter in the universe today in units of the critical density. = =3H 2 0 is the cosmological constant relative to the present-day Hubble constant H0. R is the curvature term and is equal to 1 0 . We assume that relativistic matter is a negligible contributor to the density of the universe. Finally, H(a) is proportional to the Hubble constant at epoch a, scaled so that H = 1 today. We are interested in the behavior of D1(a), as this is the mode whose amplitude grows in time. However, the overall normalization of D1 is merely a convention. The analytic form for D1 in the = 0 case (Weinberg 1972; Groth & Peebles 1975; Edwards & Heath 1976) is widely known, but no analytic solution for the case of 6= 0 has been presented in the literature. Instead, workers integrate equation (1) numerically or use the approximations given by Lahav et al. (1991) or Carroll et al. (1992). However, the integral can be done in terms of elliptic