SELF-SIMILAR PERTURBATIONS OF A FRIEDMANN UNIVERSE

We analyze spherically symmetric solutions to the Einstein equations which are self-similar in the sense that all dimensionless variables are functions only of z = r/t, where r is the comoving radial coordinate and t is cosmic time, and which have equation of state of the form p = αμ with 0 ≤ α ≤ 1. We focus on solutions which are asymptotically k = 0 Friedmann at large values of z and which have a finite but perturbed density at the origin. Such solutions represent nonlinear density fluctuations which grow at the same rate as the universe's particle horizon, and they nearly all have weak discontinuities where the fluid velocity equals the sound speed √α. The overdense solutions only span a narrow range of parameters: the amount by which the density at the origin exceeds the Friedmann value must lie within narrow bands, and only one solution per band is analytic at the sonic point. The highly overdense solutions resemble static isothermal gas spheres just inside the sonic point. The underdense solutions can have arbitrarily low density at the origin but exhibit a unique relationship between their amplitude and scale; they could be relevant to the existence of large-scale voids, since self-similar perturbations might arise naturally in a universe with general initial conditions.