Gravitational instability via the Schrödinger equation

Abstract

We explore a novel approach to the study of large-scale structure formation in whichself-gravitating cold dark matter (CDM) is represented by a complex scalar field whosedynamics are governed by coupled Schrödinger and Poisson equations. We show that, in thequasi-linear regime, the Schrödinger equation can be reduced to the free-particle Schrödingerequation. We advocate using the free-particle Schrödinger equation as the basis of a newapproximation method—the free-particle approximation—that is similar in spirit to thesuccessful adhesion model. In this paper we test the free-particle approximation byappealing to a planar collapse scenario and find that our results are in excellentagreement with those of the Zeldovich approximation, provided care is takenwhen choosing a value for the effective Planck constant in the theory. We alsodiscuss how extensions of the free-particle approximation are likely to require theinclusion of a time-dependent potential in the Schrödinger equation. Since theSchrödinger equation with a time-dependent potential is typically impossible to solveexactly, we investigate whether standard quantum-mechanical approximationtechniques can be used, in a cosmological setting, to obtain useful solutions of theSchrödinger equation. In this paper we focus on one particular approximation method:time-dependent perturbation theory (TDPT). We elucidate the properties ofperturbative solutions of the Schrödinger equation by considering a simple example: thegravitational evolution of a plane-symmetric density fluctuation. We use TDPT tocalculate an approximate solution of the relevant Schrödinger equation and showthat this perturbative solution can be used to successfully follow gravitationalcollapse beyond the linear regime, but there are several pitfalls to be avoided.