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In Chapter 5 of Baumann's cosmology book where he discusses structure formation starting from Newtonian perturbation theory, Baumann mentions at the beginning that

Newtonian gravity is a good approximation when all velocities are small compared to the speed of light and the departures from a flat spacetime geometry are small. The latter is satisfied on scales smaller than the Hubble radius, $H^{-1}$.

I am confused by the bolded statement. It makes sense that on very large scales there should be significant departures from flat spacetime since the primordial power spectrum of the gravitational potential $P_\Phi\propto k^{-3}$ (roughly) so on large scales there should be significant deviations from flat spacetime. It also makes sense that the Newtonian approximation breaks down above the Hubble radius since Newtonian gravity assumes instantaneous propagation of gravitational interactions which can't happen if the signal originates from outside the Hubble horizon and hasn't had enough time to propagate.

But what I'm fuzzy on is how we know that there isn't some radius $r_*<H^{-1}$ where departures from a flat spacetime geometry start to become significant or even how we would define "departures from a flat spacetime geometry"

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    $\begingroup$ Not very flat in the vicinity of a black hole, and there are plenty of those around. Still, there is definitely a velocity dispersion scale for matter of $\approx 10^{-3} c$, which implies a pretty flat geometry. I've never seen a good explanation for this, nor of the associated fractal dimension of matter clustering of ~1. $\endgroup$
    – John Doty
    Commented Jun 17, 2023 at 0:10
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    $\begingroup$ No time for full answer right now, so quick notes: (1) The power spectrum has nothing to do with curvature being relevant on large scales (perturbation amplitudes scale as $\sqrt{k^3 P}$). Curvature becomes important as you approach the Hubble scale even in a homogeneous universe. (2) The curvature is large near black holes, but those aren't relevant in cosmological calculations. It's small in all other contexts, even in the largest galaxies or clusters. $\endgroup$
    – Sten
    Commented Jun 17, 2023 at 0:53
  • $\begingroup$ Somebody has measured the primordial gravitational wave spectrum? Does anybody have a citation? $\endgroup$
    – FlatterMann
    Commented Jun 17, 2023 at 0:54
  • $\begingroup$ Have you looked at Lineweaver & Davis's papers (on various levels of sophistication, but whose titles all contain the phrase "Expanding Confusion") about the relation between spatial expansion and the Hubble Sphere? $\endgroup$
    – Edouard
    Commented Jun 17, 2023 at 4:24

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