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Phase-space perturbation theory for cosmic large-scale structure

Hannes Heisler, Marvin Sipp, Matthias Bartelmann

TL;DR

The paper develops a perturbative expansion for the full Vlasov-Poisson system of cosmic structure formation that avoids truncating the momentum-cumulant hierarchy. It linearizes via a Volterra-type integral equation with a freely evolving background and then builds higher-order corrections using a kernel-based recursion, recovering Eulerian SPT kernels for perfectly cold initial conditions. By introducing a small initial velocity dispersion, it shows that all higher momentum cumulants are dynamically generated at any order, complemented by analytical large-scale approximations and a numerical scheme. The approach provides a natural framework for background-perturbation splits and offers a path toward non-perturbative techniques beyond the traditional single-stream approximation, with connections to kinetic-field theory and related beyond-SSA methods.

Abstract

We consider a perturbative approach to the Vlasov-Poisson system for cosmic structure formation that does not rely on any truncation of the momentum-cumulant hierarchy. The generally non-trivial linear solution is computed by solving a Volterra-type integral equation and higher orders are obtained recursively. As expected, the results of Eulerian standard perturbation theory are recovered for perfectly cold initial conditions. Deviating slightly from the latter by introducing a homogeneous and isotropic initial velocity dispersion, we show that all higher momentum cumulants are generated dynamically at any perturbative order. We support our numerical solutions by an analytical large-scale approximation. Our approach serves as a basis for exploring different background-perturbation splits of the phase-space density and non-perturbative techniques.

Phase-space perturbation theory for cosmic large-scale structure

TL;DR

The paper develops a perturbative expansion for the full Vlasov-Poisson system of cosmic structure formation that avoids truncating the momentum-cumulant hierarchy. It linearizes via a Volterra-type integral equation with a freely evolving background and then builds higher-order corrections using a kernel-based recursion, recovering Eulerian SPT kernels for perfectly cold initial conditions. By introducing a small initial velocity dispersion, it shows that all higher momentum cumulants are dynamically generated at any order, complemented by analytical large-scale approximations and a numerical scheme. The approach provides a natural framework for background-perturbation splits and offers a path toward non-perturbative techniques beyond the traditional single-stream approximation, with connections to kinetic-field theory and related beyond-SSA methods.

Abstract

We consider a perturbative approach to the Vlasov-Poisson system for cosmic structure formation that does not rely on any truncation of the momentum-cumulant hierarchy. The generally non-trivial linear solution is computed by solving a Volterra-type integral equation and higher orders are obtained recursively. As expected, the results of Eulerian standard perturbation theory are recovered for perfectly cold initial conditions. Deviating slightly from the latter by introducing a homogeneous and isotropic initial velocity dispersion, we show that all higher momentum cumulants are generated dynamically at any perturbative order. We support our numerical solutions by an analytical large-scale approximation. Our approach serves as a basis for exploring different background-perturbation splits of the phase-space density and non-perturbative techniques.

Paper Structure

This paper contains 12 sections, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Vlasov equation and its moment hierarchy
  3. Perturbation theory of the Vlasov equation
  4. Linear perturbation theory
  5. Higher-order perturbation theory
  6. Graphical representation, power spectra, and loop expansion
  7. Perfectly cold initial conditions
  8. Beyond the single-stream approximation
  9. Linear perturbation theory
  10. Second-order perturbation theory
  11. Conclusions and outlook
  12. Proof of the closed-form kernels in one dimension

Figures (2)

  • Figure 1: Late-time linear density contrast with non-vanishing initial velocity dispersion, normalized to the perfectly cold case. The solid blue line corresponds to the numerical solution. Dashed grey lines indicate corrections to the LSL (black, dotted) up to seventh order in $(k/k_\sigma)^2$.
  • Figure 2: Numerical results for the time evolution of the longitudinal mode of the linear velocity dispersion relative to the background, for four different scales.