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A new approach to gravitational clustering: a path-integral formalism and large-N expansions

P. Valageas

TL;DR

This work casts gravitational clustering in an expanding universe within a path-integral framework by deriving an action $S[f]$ that encodes Gaussian initial conditions and Vlasov-Poisson dynamics for the phase-space density $f({\bf x},{\bf p},t)$. It then develops three large-$N$ schemes—the direct steepest-descent, the 1PI effective action, and the 2PI effective action—to produce self-consistent nonlinear equations for the mean $\bar f$ and two-point correlators $G$ (and the response $R$). The analysis shows that standard perturbative results on quasi-linear scales are recovered while the large-$N$ techniques enable controlled resummations that remain well-behaved in the nonlinear regime, provided infrared cancellations are handled properly. Infrared issues favor the semiclassical/2PI approaches over the straightforward $N$-field expansion, and the framework naturally connects to BBGKY-like closures and may extend to non-Gaussian initial conditions. The methodology provides a principled, field-theoretic route to modeling structure formation with potential numerical implementations.

Abstract

We show that the formation of large-scale structures through gravitational instability in the expanding universe can be fully described through a path-integral formalism. We derive the action S[f] which gives the statistical weight associated with any phase-space distribution function f(x,p,t). This action S describes both the average over the Gaussian initial conditions and the Vlasov-Poisson dynamics. Next, applying a standard method borrowed from field theory we generalize our problem to an N-field system and we look for an expansion over powers of 1/N. We describe three such methods and we derive the corresponding equations of motion at the lowest non-trivial order for the case of gravitational clustering. This yields a set of non-linear equations for the mean $\fb$ and the two-point correlation G of the phase-space distribution f, as well as for the response function R. These systematic schemes match the usual perturbative expansion on quasi-linear scales but should also be able to handle the non-linear regime. Our approach can also be extended to non-Gaussian initial conditions and may serve as a basis for other tools borrowed from field theory.

A new approach to gravitational clustering: a path-integral formalism and large-N expansions

TL;DR

This work casts gravitational clustering in an expanding universe within a path-integral framework by deriving an action that encodes Gaussian initial conditions and Vlasov-Poisson dynamics for the phase-space density . It then develops three large- schemes—the direct steepest-descent, the 1PI effective action, and the 2PI effective action—to produce self-consistent nonlinear equations for the mean and two-point correlators (and the response ). The analysis shows that standard perturbative results on quasi-linear scales are recovered while the large- techniques enable controlled resummations that remain well-behaved in the nonlinear regime, provided infrared cancellations are handled properly. Infrared issues favor the semiclassical/2PI approaches over the straightforward -field expansion, and the framework naturally connects to BBGKY-like closures and may extend to non-Gaussian initial conditions. The methodology provides a principled, field-theoretic route to modeling structure formation with potential numerical implementations.

Abstract

We show that the formation of large-scale structures through gravitational instability in the expanding universe can be fully described through a path-integral formalism. We derive the action S[f] which gives the statistical weight associated with any phase-space distribution function f(x,p,t). This action S describes both the average over the Gaussian initial conditions and the Vlasov-Poisson dynamics. Next, applying a standard method borrowed from field theory we generalize our problem to an N-field system and we look for an expansion over powers of 1/N. We describe three such methods and we derive the corresponding equations of motion at the lowest non-trivial order for the case of gravitational clustering. This yields a set of non-linear equations for the mean and the two-point correlation G of the phase-space distribution f, as well as for the response function R. These systematic schemes match the usual perturbative expansion on quasi-linear scales but should also be able to handle the non-linear regime. Our approach can also be extended to non-Gaussian initial conditions and may serve as a basis for other tools borrowed from field theory.

Paper Structure

This paper contains 20 sections, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equations of motion
  3. Vlasov-Poisson system
  4. Initial conditions
  5. Long-wavelength divergences
  6. Functional formulation
  7. Derivation of the actions $S[f]$ and $S[f,\lambda]$, mean response functions
  8. Approximation schemes
  9. Direct steepest-descent method
  10. General formalism
  11. Application to gravitational clustering
  12. 1PI effective action
  13. 2PI effective action
  14. General formalism
  15. Comparison with the 1PI effective action approach and the BBGKY hierarchy
  16. ...and 5 more sections

Figures (1)

  • Figure 1: The two-loop 2PI vacuum diagrams of the reduced effective action ${\tilde{\Gamma}}$. Each solid line stands for a propagator $G$, each dashed line for the propagator $D$ and each full-dashed line for the mixed propagator $R$ (cross-correlation). The indices $f,\lambda$ denote the $f-$field and $\lambda-$field sides of the propagators $G$, $R$ or $D$. These indices $f,\lambda$ also correspond to the legs of the three-leg vertex $K_{\rm s}$ (big dots).