Cosmological fluid dynamics in the Schrödinger formalism

Abstract

We investigate the dynamics of a cosmological dark matter fluid in the Schrödinger formulation, seeking to evaluate the approach as a potential tool for theorists. We find simple wave-mechanical solutions of the equations for the cosmological homogeneous background evolution of the dark matter field, and use them to obtain a piecewise analytic solution for the evolution of a compensated spherical overdensity. We analyse this solution from a 'quantum mechanical' viewpoint, and establish the correct boundary conditions satisfied by the wavefunction. Using techniques from multiparticle quantum mechanics, we establish the equations governing the evolution of multiple fluids and then solve them numerically in such a system. Our results establish the viability of the Schrödinger formulation as a genuine alternative to standard methods in certain contexts, and a novel way to model multiple fluids.

PyUltraLight: a pseudo-spectral solver for ultralight dark matter dynamics

“…The Madelung equations resemble the continuity and Euler equations of classical fluid dynamics, with the addition of a 'quantum pressure' term accounting for resistance against gravitational collapse. The Madelung formalism is discussed in detail in [43,[46][47][48]. Because this hydrodynamical formulation defines the fluid velocity as the gradient of the phase of the field ψ, problems arise when ψ = 0, where the phase is not well defined.…”

Gravitational collapse in the Schrödinger-Poisson system

“…Hence, it is expected that the Schrödinger-Poisson capture the effects of shell crossing better than the fluid representation. This is also supported by the relation between the Wigner transform of two-point functions and the Boltzmann equation discussed in the introduction and the original work [15,41,42], as well as the general arguments provided in [34]. The 1+1 dimensional particle distribution function is defined using the Wigner transformation…”

Using numerical methods from nonlocal optics to simulate the dynamics of N -body systems in alternative theories of gravity

“…At large scales, matter can be modeled as a perfect fluid [21], hence, from the full Vlasov-Poisson system, it is possible to obtain a set of hydrodynamic equations capable of describing the dynamics of the density ρ and velocity v of the fluid…”

“…The benefit in beginning with this hydrodynamic approach is that, by applying the Madelung transformation [22], this set of equations is transformed into a Schrödinger-Newton system. This correspondence is normally used to establish a hydrodynamic analogy for quantum mechanics [21], however, in this work we use this approach in the opposite direction. This transformation is achieved by writing: ψ = √ ρe iΦ/ν , where ν work as an analogous of the reduced Planck constant, , ρ = |ψ| 2 and v = ∇Φ.…”