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

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.

💡 Research Summary

This paper investigates cosmic structure formation using a path‑integral formulation of classical mechanics for an N‑particle ensemble in phase space, a framework known as Resummed Kinetic Field Theory (RKFT). The authors develop the formalism up to one‑loop order and compute power spectra for the density contrast, the divergence and curl of the momentum density, as well as arbitrary n‑point cumulants of the stress tensor.

A central issue addressed is the treatment of initial conditions. Earlier applications of Kinetic Field Theory (KFT) employed “perfectly cold” initial conditions, i.e. a Dirac‑delta distribution in momentum space, which is inconsistent with the Gaussian statistics of primordial cosmological perturbations. The authors propose a new sampling method that uses a Gaussian phase‑space density for the initial particle ensemble, thereby respecting the statistical independence of positions and momenta and matching the standard cosmological power spectrum.

With this Gaussian sampling, the tree‑level density and momentum spectra derived from RKFT exactly reproduce the linear results of Eulerian Standard Perturbation Theory (SPT). More importantly, the one‑loop calculations—performed via a macroscopic reformulation of the theory that introduces the phase‑space density field f and a response field B—also match the one‑loop SPT results for all considered observables, including higher‑order stress‑tensor cumulants. This demonstrates that the previously reported small‑scale damping in RKFT spectra was solely due to the inconsistent initial‑condition prescription, not an intrinsic limitation of the formalism.

The paper outlines the underlying path‑integral construction: starting from the Hamiltonian of N particles, the authors introduce an auxiliary field χ to enforce the classical equations of motion, leading to an action S