Infinite universe, infinite volume from the beginning?

[mister i]

TL;DR

If the universe were spatially infinite today, continuous expansion would seem to imply that it must always have been infinite. Would this mean that even immediately after the Big Bang (for example just after the Planck time) the total spatial volume of the universe would already have been infinite?

If the universe is spatially infinite today, it seems unavoidable that it has always been infinite. A finite space cannot become infinite through continuous expansion.

In the FLRW framework, physical distances scale as

d_phys = a(t) · d_comoving

If the comoving spatial manifold is infinite, multiplying it by any finite scale factor still gives an infinite spatial extent. The total spatial volume would therefore be infinite at every cosmic time.

This leads to a somewhat uncomfortable implication. If the universe is indeed spatially infinite, then even an instant after the Big Bang — for example just after the Planck time — the total spatial volume of the universe must already have been infinite.

So the usual mental picture of the universe beginning in an extremely small state seems misleading, at least in the case of an infinite universe.

Is this interpretation correct within standard cosmology, or is there some subtle point about the Big Bang limit a(t) → 0 that avoids the conclusion that the universe had infinite spatial volume essentially from the very beginning?

 

[Ibix]

So the usual mental picture of the universe beginning in an extremely small state seems misleading, at least in the case of an infinite universe.

Not misleading, just wrong. You are correct that if the universe is closed and finite then it always was; if it is flat or open and infinite then it always was.

Sources that talk about "when the universe was the size of a grapefruit" or whatever are either talking about a closed universe, talking about the observable universe (which is finite in any FLRW model), or wrong.

 

[Jaime Rudas]

Is this interpretation correct within standard cosmology, or is there some subtle point about the Big Bang limit a(t) → 0 that avoids the conclusion that the universe had infinite spatial volume essentially from the very beginning?

Complementing what @Ibix mentioned, it might be worth mentioning that the expansion of the universe does not necessarily imply an increase in its volume but, rather, an increase in distances.