Search

We investigate large-scale structure formation of collisionless dark matter in the phase space description based on the Vlasov (or collisionless Boltzmann) equation whose nonlinearity is induced solely by gravitational interaction according to the Poisson equation. Determining the time evolution of density and peculiar velocity demands solving the full Vlasov hierarchy for the moments of the phase space distribution function. In the presence of long-range interaction no consistent truncation of the hierarchy is known apart from the pressureless fluid (dust) model, which is incapable of describing virialization due to the occurrence of shell-crossing singularities and the inability to generate vorticity and higher cumulants like velocity dispersion. Our goal is to find a simple ansatz for the phase space distribution function that approximates the full Vlasov distribution function without pathologies in a controlled way and therefore can serve as theoretical TV-body double and as a replacement for the dust model. We argue that the coarse-grained Wigner probability distribution obtained from a wave function fulfilling the Schrödinger-Poisson equation (SPE) is the sought-after function. We show that its evolution equation approximates the Vlasov equation and therefore also the dust fluid equations before shell crossing, but cures the shell-crossing singularities and is able to describe regions of multistreaming and virialization. This feature was already employed in cosmological simulations of large-scale structure formation by Widrow and Kaiser (1993). The coarse-grained Wigner ansatz allows us to calculate all higher moments from density and velocity analytically, thereby incorporating nonzero higher cumulants in a self-consistent manner. On this basis we are able to show that the Schrödinger method automatically closes the corresponding hierarchy such that it suffices to solve the SPE in order to directly determine density and velocity and all higher cumulants. [ABSTRACT FROM AUTHOR]

We chart a path toward solving for the nonlinear gravitational dynamics of cold dark matter by relying on a semiclassical description using the propagator. The evolution of the propagator is given by a Schrödinger equation, where the small parameter ℏ acts as a softening scale that regulates singularities at shell-crossing. The leading-order propagator, called free propagator, is the semiclassical equivalent of the Zel'dovich approximation, that describes inertial particle motion along straight trajectories. At next-to-leading order, we solve for the propagator perturbatively and obtain, in the classical limit the displacement field from second-order Lagrangian perturbation theory (LPT). The associated velocity naturally includes an additional term that would be considered as third order in LPT. We show that this term is actually needed to preserve the underlying Hamiltonian structure, and ignoring it could lead to the spurious excitation of vorticity in certain implementations of second-order LPT. We show that for sufficiently small ℏ the corresponding propagator solutions closely resemble LPT, with the additions that spurious vorticity is avoided and the dynamics at shell-crossing is regularized. Our analytical results possess a symplectic structure that allows us to advance numerical schemes for the large-scale structure. For times shortly after shell-crossing, we explore the generation of vorticity, which in our method does not involve any explicit multistream averaging, but instead arises naturally as a conserved topological charge. [ABSTRACT FROM AUTHOR]