Solving the Vlasov equation in two spatial dimensions with the Schrödinger method

Abstract

We demonstrate that the Vlasov equation describing collisionless self-gravitating matter may be solved with the so-called Schr\"odinger method (ScM). With the ScM, one solves the Schr\"odinger-Poisson system of equations for a complex wave function in d dimensions, rather than the Vlasov equation for a 2d-dimensional phase space density. The ScM also allows calculating the d-dimensional cumulants directly through quasi-local manipulations of the wave function, avoiding the complexity of 2d-dimensional phase space. We perform for the first time a quantitive comparison of the ScM and a conventional Vlasov solver in d=2 dimensions. Our numerical tests were carried out using two types of cold cosmological initial conditions: the classic collapse of a sine wave and those of a gaussian random field as commonly used in cosmological cold dark matter N-body simulations. We compare the first three cumulants, that is, the density, velocity and velocity dispersion, to those obtained by solving the Vlasov equation using the publicly available code ColDICE. We find excellent qualitative and quantitative agreement between these codes, demonstrating the feasibility and advantages of the ScM as an alternative to N-body simulations. We discuss, the emergence of effective vorticity in the ScM through the winding number around the points where the wave function vanishes. As an application we evaluate the background pressure induced by the non-linearity of large scale structure formation, thereby estimating the magnitude of cosmological backreaction. We find that it is negligibly small and has time dependence and magnitude compatible with expectations from the effective field theory of large scale structure.