  We consider a new variant of cosmological perturbation theory that has been
designed specifically to include non-linear density contrasts on scales 100
Mpc, while still allowing for linear fluctuations on larger scales. This theory
is used to derive the relativistic equations of Eulerian hydrodynamics in
realistic cosmological scenarios that contain radiation and a cosmological
constant, as well as matter that has been allowed to clump into galaxies and
clusters of galaxies. These equations can be used to evolve energy densities
and velocities in the presences of small-scale non-linear structures, and on
scales all the way up to the horizon and beyond. The leading-order part of
these equations reproduces the expected Newtonian equations, while subsequent
orders prescribe relativistic corrections. We demonstrate that these evolution
equations are consistent with maintaining the Einstein constraints, and hence
that the system as a whole is mathematically well posed. The relativistic
corrections that we derive are found to exhibit non-trivial interactions
between perturbations on different scales, as well as the mixing of scalar,
vector and tensor modes. They deviate from those that occur in both
post-Friedmann and standard cosmological perturbation theory approaches, and
point towards new relativistic effects that could be measurable by upcoming
ultra-large-scale surveys.
