> differential geometry

Flat in the sense that at the largest scales the universe is extremely well-approximated by a dust that solves the Friedmann equations given an expanding Robertson-Walker background whose parameter k = 0.

If we zoom in on the “dust” we see that it is grainy/clumpy, and can resolve the grains/clumps into distant galaxies and clusters, and can in particular spot spiral structures that maintain a consistent set of shapes at various distances (or, if you like, at different solid-angle sizes, redshifts, and brightnesses of all of the gas clouds, fast pulsars, and supernovae in them). If we set k to the other allowed values, spiral galaxies at higher redshifts would distort compared to nearby spirals.

Within the clumps — clusters and galaxies themselves — the matter does not resemble a perfect fluid dust, and the background’s permitted trajectories are not those of Robertson-Walker with any value of k. The Friedmann equations thus are no help. To deal with that, we can use a Swiss Cheese model, in which we treat these large “clumps” as holes in the expanding Robertson-Walker cheese, and as necessary use Israel junctions to trace trajectories out of these holes, which are well-modelled by collapsing matter and typically an LTB metric. We can do this for select clumps, while leaving the rest as “grains” of Friedmann-solving dust.

However, unless you flush the Copernican principle (and have good evidence that it’s wise to do so), then we inhabit a nice flat universe in the sense above at length scales above Megaparsecs.

The flatness is technical and spatial. Galaxy clusters’ motion through the strongly curved spacetime gives us orientability: we can say “future” is the direction in which galaxy clusters develop greater mutual RADAR distances. But a pair of typical spiral galaxies at arbitrary RADAR distances will look like typical spiral galaxies to one another for many billions of years into the past or into the future.

Technically the flatness only applies where the expanding “cheese” metric in a Swiss Cheese model applies, and that’s far from large masses of collapsing matter.

Inside the “holes” one might find other holes and yet other holes and so on all the way down to stars and planets and people, all the non-relativistic examples of which look roughly like Schwarzschild from a great distance. In particular, the interaction between the generators of the “in-the-hole” metric and the “in-the-cheese” metric are amenable to perturbation theory in that the deviations from the respective metrics are pretty much vanishingly small. The expansion doesn’t stop galaxy clusters from collapsing inwards, and the galaxy clusters don’t stop the expansion.

Of course, if the hierarchical analytic approach turns you off so hard that you can’t bring yourself to appreciate a Swiss Cheese model as an effective description of the observed universe, you can always dive into non-perturbative inhomogeneous cosmology. In such models backreaction is the effect of inhomogeneities of matter and geometry (“… in the neighborhood of a black hole …” as you put it) on the average evolution of the cosmos. It’s been trendy for years to argue about whether backreaction is relevant (in the Wilson sense) or not, and if it is, whether it’s the traceless part or the trace of the effective stress-energy tensor that matters (pardon the pun) most. However, (a) you won’t rack up many real nerd points in HN comments sniping down comments like your parent’s because, for example, “it should be totally obvious to you that the universe is NOT flat R-W spacetime because there are procedural problems with contracting Ricci curvature functionals into a dust!”, and (b) someone who knows better, and there is allllways one of those, will just nuke you from a higher orbit.)