The perturbed FLRW metric on all scales: Newtonian limit and top-hat
  collapse
  The applicability of a linearized perturbed FLRW metric to the late, lumpy
universe has been subject to debate. We consider in an elementary way the
Newtonian limit of the Einstein equations with this ansatz for the case of
structure formation in late-time cosmology, on small and large scales, and
argue that linearizing the Einstein tensor produces only a small error down to
arbitrarily small, decoupled scales (e.g. Solar system scales). On subhorizon
patches, the metric scale factor becomes a coordinate choice equivalent to
choosing the spatial curvature, and not a sign that the FLRW metric cannot
perturbatively accommodate very different local physical expansion rates of
matter; we distinguish these concepts, and show that they merge on large scales
for the Newtonian limit to be globally valid. Furthermore, on subhorizon
scales, a perturbed FLRW metric ansatz does not already imply assumptions on
isotropy, and effects beyond an FLRW background, including those potentially
caused by non-linearities of general relativity (GR), may be encoded into
non-trivial boundary conditions. The corresponding cosmologies have already
been developed in a Newtonian setting by Heckmann and Sch\"ucking and none of
these boundary conditions can explain the accelerated expansion of the
universe. Our analysis of the field equations is confirmed on the level of
solutions by an example of pedagogical value, comparing a collapsing top-hat
overdensity (embedded into a cosmological background) treated in such
perturbative manner to the corresponding exact solution of GR, where we find
good agreement even in the regimes of strong density contrast.
