Multi-Scale Perturbation Theory I: Methodology and Leading-Order Bispectrum Corrections in the Matter-Dominated Era

TL;DR

This paper introduces a two-parameter perturbation theory framework that effectively models nonlinear density contrasts and provides leading-order bispectrum corrections in the matter-dominated era, aligning with standard second-order perturbation results on small scales.

Contribution

It presents a novel perturbation approach that combines Newtonian and cosmological perturbation theories to accurately capture relativistic effects in the matter bispectrum.

Findings

Leading-order gravitational potentials match second-order perturbation theory.

Dark matter bispectrum shows differences on Hubble scales.

Framework simplifies inclusion of relativistic effects in cosmological models.

Abstract

Two-parameter perturbation theory is a scheme tailor-made to consistently include nonlinear density contrasts on small scales ($<100\; \mathrm{Mpc}$), whilst retaining a traditional approach to cosmological perturbations in the long-wavelength universe. In this paper we study the solutions that arise from this theory in a spatially-flat dust-filled cosmology, and what these imply for the bispectrum of matter. This is achieved by using Newtonian perturbation theory to model the gravitational fields of nonlinear structures in the quasi-linear regime, and then using the resulting solutions as source terms for the cosmological equations. We find that our approach results in the leading-order part of the cosmological gravitational potentials being identical to those that result from standard cosmological perturbation theory at second-order, while the dark matter bispectrum itself yields some differences on Hubble scales. This demonstrates that our approach is sufficient to capture most leading-order relativistic effects, but within a framework that is far easier to generalize. We expect this latter property to be particularly useful for calculating leading-order relativistic corrections to the matter power spectrum, as well as for deriving predictions for relativistic effects in alternative theories of gravity.