Time development of two and three-dimensional self-gravitating system using initial Dirac delta function distribution

It is shown that it is possible to extend perturbation calculations used in finding the solutions to the coupled Vlasov–Poisson equations, specifically the time evolution of density, for the one-dimensional gravitational gas to two and three dimensions. An example using initial Dirac delta distribution and Maxwellian velocity distribution is given. We reproduce main features of the one-dimensional model such as formation of inhomogeneities and also quantitatively and qualitatively new results in the higher dimensions demonstrating how galaxies and stars could be formed in the early universe.