An Integral Equation for the Development of Irregularities in an Expanding Universe

A calculation of the growth of irregularities in a statistically homogeneous and isotropic, dust-filled universe is presented in the Newtonian approximation. The description employed is based upon particle distribution functions rather than the hydrodynamic approximation. An integral equation for the spectrum of small-amplitude fluctuations is derived and numerically solved for the case of a Boltzmann single-particle velocity distribution function. It is found that the density contrast of large wavelength disturbances grows monotonically with time, but the density contrast of small wavelength disturbances first decays and later grows. The dividing line between these two types of behavior is at an initial wavelength of X = [7r ( V2 /3po (t0) G] 112, where ( is the mean square peculiar velocity and po (t0) is the average density at 1o, the time at which the initial conditions are imposed. This value is related to , the Jeans critical wavelength at t8 by, X = Xt 2It, where is the ratio of specific heats.