The missing link: a nonlinear post-Friedmann framework for small and
  large scales
  We present a nonlinear post-Friedmann framework for structure formation,
generalizing to cosmology the weak-field (post-Minkowskian) approximation,
unifying the treatment of small and large scales. We consider a universe filled
with a pressureless fluid and a cosmological constant $\Lambda$, the theory of
gravity is Einstein's general relativity and the background is the standard
flat $\Lambda$CDM cosmological model.
  We expand the metric and the energy-momentum tensor in powers of $1/c$,
keeping the matter density and peculiar velocity as exact fundamental
variables. We assume the Poisson gauge, including scalar and tensor modes up to
$1/c^4$ order and vector modes up to $1/c^5$ terms. Through a redefinition of
the scalar potentials as a resummation of the metric contributions at different
orders, we obtain a complete set of nonlinear equations, providing a unified
framework to study structure formation from small to superhorizon scales, from
the nonlinear Newtonian to the linear relativistic regime. We explicitly show
the validity of our scheme in the two limits: at leading order we recover the
fully nonlinear equations of Newtonian cosmology; when linearized, our
equations become those for scalar and vector modes of first-order relativistic
perturbation theory in the Poisson gauge. Tensor modes are nondynamical at the
$1/c^4$ order we consider: they are purely nonlinear and describe a distortion
of the spatial slices determined at this order by a constraint, quadratic in
the scalar and vector variables. The main results of our analysis are as
follows: (a) at leading order a purely Newtonian nonlinear energy current
sources a frame-dragging gravitomagnetic vector potential, and (b) in the
leading-order Newtonian regime and in the linear relativistic regime the two
scalar metric potentials are the same, while the nonlinearity of general
relativity makes them different.
