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