N ov 2 00 6 Gravitational instability via the Schrödinger equation

We explore a novel approach to the study of large-scale structure formation in which self-gravitating cold dark matter (CDM) is represented by a complex scalar field whose dynamics are governed by coupled Schrödinger and Poisson equations. We show that, in the quasi-linear regime, the Schrödinger equation can be reduced to the free-particle Schrödinger equation. We advocate using the free-particle Schrödinger equation as the basis of a new approximation method-the free-particle approximation-that is similar in spirit to the successful adhesion model. In this paper we test the free-particle approximation by appealing to a planar collapse scenario and find that our results are in excellent agreement with those of the Zeldovich approximation, provided care is taken when choosing a value for the effective Planck constant in the theory. We also discuss how extensions of the free-particle approximation are likely to require the inclusion of a time-dependent potential in the Schrödinger equation. Since the Schrödinger equation with a time-dependent potential is typically impossible to solve exactly, we investigate whether standard quantum-mechanical approximation techniques can be used, in a cosmological setting, to obtain useful solutions of the Schrödinger equation. In this paper we focus on one particular approximation method: time-dependent perturbation theory (TDPT). We elucidate the properties of perturbative solutions of the Schrödinger equation by considering a simple example: the gravitational evolution of a plane-symmetric density fluctuation. We use TDPT to calculate an approximate solution of the relevant Schrödinger equation and show that this perturbative solution can be used to successfully follow gravitational collapse beyond the linear regime, but there are several pitfalls to be avoided.