general relativity - The Big Bang in an infinite universe

If the universe is spatially infinite (and assuming, if it makes a difference, that we don't have eternal inflation), what actually happened 13.7 billion years ago? Was the energy density infinite (or "very large") at each point in R³? Or did R³ itself collapse to some other structure?

Answer

The laws of physics - field theory and/or general relativity - are local (and in string theory, they're approximately local). So it doesn't make much difference whether a slice of the Universe is compact or noncompact. Near the Big Bang, the proper distances were shrunk almost to zero. But in cosmology, it still makes sense to use the coordinates in which the geometry of the slice is $$ds^2 = a(t)^2 (dx^2+dy^2+dz^2)$$ where $a(t)$ is an overall rescaling factor that goes to zero, $a(t)\to 0$, for $t\to 0$. The parameter $t$ may also be reparameterized so that the $t=|r|$ lines are null which is useful to clarify the causal structure of the spacetime.

At any rate, near $t=0$, physics works "locally" in the coordinates $(x,y,z)$ above, regardless of the fact that $a(t)$ goes to zero. Objects moving along lines with variable $t$ but fixed $(x,y,z)$ correspond to static objects. You shouldn't imagine that objects are getting further in the $(x,y,z)$ coordinates near $t=0$ just in order to keep a proper distance fixed. No real objects are trying to keep their proper distance fixed as the Universe expands; the expansion of the proper distances is very real and you shouldn't try to deny it.

At any nonzero $t$, the $R^3$ stays the very same $R^3$ (and $S^3$ or $H^3$ would stay the same manifolds as well), and the laws of physics can't be blindly extrapolated to the strict $t=0$ point, anyway. So it makes no physical sense to ask what was "before" the Universe was an $R^3$. There was just a singularity at $t=0$; at every moment that physics can discuss, the geometry was $R^3$.

Eternal inflation has some new things to say - which would be relevant if eternal inflation were right - about these matters but you explicitly asked this topics to be avoided.