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Cosmic structure from the path integral of classical mechanics and its comparison to standard perturbation theory

Marvin Sipp, Hannes Heisler, Matthias Bartelmann

TL;DR

This work reframes cosmic structure formation within a path-integral, phase-space framework (RKFT) and identifies that previous deviations from Eulerian SPT arose from inconsistent initial-condition sampling. By adopting a Gaussian initial phase-space density, the authors reproduce linear SPT at tree level and obtain exact one-loop agreement for density, momentum-divergence, and stress-tensor cumulants, demonstrating that spurious velocity dispersion from earlier sampling was responsible for the discrepancies. The results imply that the full phase-space description, aligned with Vlasov-Poisson dynamics, requires nonperturbative treatment to capture small-scale physics beyond one loop. Consequently, proper initial-condition treatment is crucial for perturbative approaches, and extending to higher loops or fully nonperturbative RKFT/Vlasov formulations is a natural direction for future work.

Abstract

We investigate cosmic structure formation in the framework of a path-integral formulation of an $N$-particle ensemble in phase space, dubbed Resummed Kinetic Field Theory (RKFT), up to one-loop perturbative order. In particular, we compute power spectra of the density contrast, the divergence and curl of the momentum density and arbitrary $n$-point cumulants of the stress tensor. In contrast to earlier works, we propose a different method of sampling initial conditions, with a Gaussian initial phase-space density. Doing so, we exactly reproduce the corresponding results from Eulerian standard perturbation theory (SPT) at one-loop order, showing that formerly found deviations can be fully attributed to inconsistencies in the previous sampling method. Since, in contrast to SPT, the full phase-space description does not assume a truncation of the Vlasov hierarchy, our findings suggest that nonperturbative techniques are required to accurately capture the physics of cosmic structure formation.

Cosmic structure from the path integral of classical mechanics and its comparison to standard perturbation theory

TL;DR

This work reframes cosmic structure formation within a path-integral, phase-space framework (RKFT) and identifies that previous deviations from Eulerian SPT arose from inconsistent initial-condition sampling. By adopting a Gaussian initial phase-space density, the authors reproduce linear SPT at tree level and obtain exact one-loop agreement for density, momentum-divergence, and stress-tensor cumulants, demonstrating that spurious velocity dispersion from earlier sampling was responsible for the discrepancies. The results imply that the full phase-space description, aligned with Vlasov-Poisson dynamics, requires nonperturbative treatment to capture small-scale physics beyond one loop. Consequently, proper initial-condition treatment is crucial for perturbative approaches, and extending to higher loops or fully nonperturbative RKFT/Vlasov formulations is a natural direction for future work.

Abstract

We investigate cosmic structure formation in the framework of a path-integral formulation of an -particle ensemble in phase space, dubbed Resummed Kinetic Field Theory (RKFT), up to one-loop perturbative order. In particular, we compute power spectra of the density contrast, the divergence and curl of the momentum density and arbitrary -point cumulants of the stress tensor. In contrast to earlier works, we propose a different method of sampling initial conditions, with a Gaussian initial phase-space density. Doing so, we exactly reproduce the corresponding results from Eulerian standard perturbation theory (SPT) at one-loop order, showing that formerly found deviations can be fully attributed to inconsistencies in the previous sampling method. Since, in contrast to SPT, the full phase-space description does not assume a truncation of the Vlasov hierarchy, our findings suggest that nonperturbative techniques are required to accurately capture the physics of cosmic structure formation.

Paper Structure

This paper contains 21 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Basic setup and notation
  4. Path-integral approach to kinetic theory
  5. Macroscopic reformulation
  6. Application to cosmology
  7. Initial conditions in earlier works
  8. Cumulants and shot noise
  9. Inconsistencies
  10. Perfectly cold initial conditions
  11. Tree-level spectra
  12. One-loop spectra
  13. Summary, conclusions and outlook
  14. Derivation of the macroscopic action
  15. Free cumulants
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Tree-level power spectra for the density contrast (left panel) and the divergence of the momentum density (right panel). Solid orange lines represent the results from the newly proposed initial conditions \ref{['sec:new_ICs']}, while dashed blue lines correspond to the initial conditions \ref{['sec:old_ICs']} used in previous works. For reference, the corresponding freely evolved power spectra are shown as thin lines of the same style. For dimensional consistency, note the relation \ref{['eq:velocity_relation']}.
  • Figure 2: One-loop power spectra of the density contrast (blue), divergence of the momentum density (green), and the absolute value of its curl (red). The linear power spectrum of $\delta$ (or, equivalently, $\nabla\cdot\vec{\Pi}$) is shown as a dashed orange line. For dimensional consistency, note the relation \ref{['eq:velocity_relation']}.