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.

 

[javisot]

The two points mentioned by Jaime and Ibix are very important. The expansion of the universe is a property of spacetime itself (it's not a receding "edge of the universe"), and it's crucial to understand the difference between the universe and the observable universe.

To add a third point, the observable universe is finite by definition, but the best measurements we can make in our observable universe have enough uncertainty to determine whether "the universe" is finite or infinite. No one knows the size of "the universe".

 

[timmdeeg]

To add a third point, the observable universe is finite by definition, but the best measurements we can make in our observable universe have enough uncertainty to determine whether "the universe" is finite or infinite. No one knows the size of "the universe".

Just to add a forth point, that's because we will never be able to to measure spacial flatness zero in the strict euclidean sense.

 

[Hornbein]

Just to add a forth point, that's because we will never be able to to measure spacial flatness zero in the strict euclidean sense.

If the Universe is hyperbolic then it is infinite. Maybe someday data will confirm this is the case.

 

[pinball1970]

If the Universe is hyperbolic then it is infinite. Maybe someday data will confirm this is the case.

Is the data not pointing to a flat universe?
Ω = 1 or very close to?

 

[Hornbein]

Is the data not pointing to a flat universe?
Ω = 1 or very close to?

The data weakly indicates a hyperbolic universe with a spherical universe also possible. Somehow a flat universe is more popular.

It's possible that more accurate data will give us an answer. Only if the universe is flat does it seem impractical for the data to ever prove anything.

 

[Jaime Rudas]

Is the data not pointing to a flat universe?
Ω = 1 or very close to?

Yes, that's what they currently point to because the best measurement of $\Omega_{\kappa}$ is $0.0007 \pm 0.0019$, but in the future, it could well be $0.00070 \pm 0.00019$, which would already point to a hyperbolic universe.

 

[pinball1970]

Yes, that's what they currently point to because the best measurement of $\Omega_{\kappa}$ is $0.0007 \pm 0.0019$, but in the future, it could well be $0.00070 \pm 0.00019$, which would already point to a hyperbolic universe.

Will measurements ever be able to a stage to find that curvature? I assume you mean distant galaxies or will that be something else?

 

[PeroK]

The data weakly indicates a hyperbolic universe with a spherical universe also possible. Somehow a flat universe is more popular.

It's possible that more accurate data will give us an answer. Only if the universe is flat does it seem impractical for the data to ever prove anything.

You would also have to justify infinite homogeneity and isotropy.

 

[mister i]

In the case of an infinite universe, the BB singularity seems even more difficult to understand: in a time interval dt the universe would go from zero volume to infinite volume?

 

[PeroK]

In the case of an infinite universe, the BB singularity seems even more difficult to understand: in a time interval dt the universe would go from zero volume to infinite volume?

A singularity is where the mathematics breaks down. It is not a physical state. And it is not part of the mathematical model, as such. Zero volume to infinite volume is, indeed, a singularity.

PS $dt$ is not a time interval. It is a mathematical abstraction, called a differential (or infinitesimal).

 

[pinball1970]

In the case of an infinite universe, the BB singularity seems even more difficult to understand: in a time interval dt the universe would go from zero volume to infinite volume?

Not what I get.

An infinite region of space is infinite by definition. Why would it be zero volume?

 

[Jaime Rudas]

Will measurements ever be able to a stage to find that curvature?

Yes, when the uncertainty margins give us only positive or only negative values, we could be reasonably sure that the curvature is negative or positive (assuming homogeneity and isotropy, as @PeroK correctly points out).

I assume you mean distant galaxies or will that be something else?

The measurements I cited were made from the cosmic microwave background and baryon acoustic oscillations.

 

[Jaime Rudas]

In the case of an infinite universe, the BB singularity seems even more difficult to understand: in a time interval dt the universe would go from zero volume to infinite volume?

No, that's not the case. According to the Big Bang theory, if the universe is infinite, it has always been so.

 

[Ibix]

In the case of an infinite universe, the BB singularity seems even more difficult to understand: in a time interval dt the universe would go from zero volume to infinite volume?

The point about the singularity is that it's where the model no longer works. The singularity doesn't have a volume because it isn't part of the model (in particular, it's not part of the spacetime manifold, which is the geometric structure that provides notions like volume) - it's the edge of the map. It isn't expected to be a thing in reality. It's a breakdown of the model.

We expect a quantum theory of gravity will deviate from general relativity somewhere between here and the singularity, and hopefully will explain what's actually there.

 

[Ibix]

The data weakly indicates a hyperbolic universe with a spherical universe also possible. Somehow a flat universe is more popular.

I think the fact that it seems so close to flat that we can't tell the difference even when looking on scales of multiple billion light years makes it attractive to consider the case. Also, as I understand it inflation theory (which solves the thermodynamic equilibrium problem) drives the universe towards flatness. So it's popular for reasons unrelated to curvature measurement.

 

[PeterDonis]

The data weakly indicates a hyperbolic universe with a spherical universe also possible.

Which means a flat universe, in between the two, is also possible.

 

[Jaime Rudas]

Which means a flat universe, in between the two, is also possible.

Yes, it's possible that it's flat, but we cannot determine that with certainty.

 

[PeterDonis]

Yes, it's possible that it's flat, but we cannot determine that with certainty.

Yes, agreed. But @Hornbein left out that possibility, which seems odd, since "flat", as I said, is in between "hyperbolic" and "spherical", so as long as the latter two are both within the error bars, so is "flat".