The evolution of spherically symmetric configurations in the Schrödinger equation approach to cosmic structure formation

The evolution of spherically symmetric cold dark matter overdensities in an expanding Universe is studied using Schrödinger-Newton (SN) equations, which model self-gravitating collisionless matter. For doing so, the density profiles of the perturbations are ideally divided into shells, for which an explicit SN solution for a Λ=0 background can be found. Then, supposing absence of shell crossing during the whole evolution of the overdensity, the free-particle approximation is applied to each shell. This approximation, under appropriate limits, which are separately discussed, reduces either to the Zel'dovich approximation or to the adhesion one. Then the evolution of the overdensity is treated with SN equations in Zel'dovich approximation as a whole, without dividing the system into shells, obtaining results that perfectly overlap with the ones held by the shell by shell study in the Zel'dovich limit. Eventually, for a specific density profile, time dependent perturbation theory is used to refine the evolution of its shells computed in the free-particle approximation. Then it is studied the evolution of a density profile coherent with the initial conditions of the Universe which are described in literature. For this system, it is explicitly found the shell by shell exact SN solution, the SN solution in Zel'dovich approximation, and it is discussed the evolution of a mini halo placed inside it. Independently on the specific density profile considered, the exact solution prescribes that the shells of the overdensity initially expand at a slower rate than the background, then they turn around and collapse. The free-particle approximation similarly predicts that regions of the overdensity for which the density is below a critical value initially expand, then turn around and collapse; but differently, if they exist, regions whose density exceeds, at the initial time, the critical density, directly contract. In both treatments, eventually the density diverges: in the centre of symmetry of the perturbation if it is spherically symmetric, or possibly elsewhere if a test halo is added to the system. Finally, the effect on the system of a non-null cosmological constant is studied, by deriving its effect on the solution which describes a shell. For low enough cosmological constants, the evolution quantitatively resembles the one computed for the Λ=0 case.