Self-similar spherical voids in an expanding universe

The authors derive similarity solutions which describe the evolution of spherically symmetric voids in a perturbed Einstein-de Sitter universe filled with cold, collisionless matter. The character of a solution depends upon the profile of the initial density deficit. Gradual perturbations give rise to holes within which the density rises smoothly to the background value. Steep perturbations result in voids bounded by overdense shells with sharp edges, i.e. collisionless gravitational shocks.