The perturbed FLRW metric on all scales: Newtonian limit and top-hat collapse.

The applicability of a linearized perturbed FLRW metric ansatz to the late, lumpy universe has been subject to debate. One question is: Does the appearance of the scale factor imply physical or artifactual cosmological effects on small scales? Considering the Newtonian limit in the pressureless, spatially flat case, we argue that neither is the case. Instead, locally, the metric scale factor can be understood as a pure gauge degree of freedom, undetermined by the Einstein equations. This restores consistency with the fact that in Newtonian cosmology the scale factor is similarly introduced as a coordinate choice. We illustrate this with a 'top-hat Einstein-Straus' toy model possessing two physical expansion rates, either of which may serve as the scale factor: a collapsing core surrounded by a vacuum shell in turn embedded into an expanding universe. We recover the weak-field limit of the exact solution from the perturbed FLRW solutions for both choices of scale factor under explicit coordinate changes in all three regions. This also serves as a simple example where a single perturbed FLRW metric does apply simultaneously to a cosmological system and a small subsystem that has decoupled from the expansion. The standard scale factor emerges only on large scales in the Newtonian limit, which remains valid under mild assumptions.