Questions about the topology of the universe

[dougy]

In the absence of a cosmological constant, there is a critical density (in the FLRW model) at which the universe expands asymptotically to zero velocity. If the density of the universe (without a cosmological constant) is above that critical density, at some point the expansion reverses and galaxies start moving towards each other. If the density of the universe is below that critical density, the universe keeps expanding forever.

Then it is said that if the universe has the critical density it is infinite and flat (zero curvature), if it has a greater density it is finite and spherical (positive curvature), and if it has a lower density it is infinite and hyperbolic (negative curvature).

I have some questions about the above.

Why cannot a universe with positive curvature be infinite? Is that because the bigger a spherical universe the smaller its curvature, and so that the infinite flat universe is the limit of a spherical universe as its curvature tends to zero?

Once again assuming no cosmological constant, why in a spherical universe the expansion reverses while in a hyperbolic universe the expansion keeps going forever? Why couldn't galaxies always keep moving away from each other in a spherical universe, and why couldn't they move towards each other at some point in a hyperbolic universe?

A spherical universe is one in which light sent in one direction eventually 'circles' the universe and returns to its starting point. Why couldn't galaxies always keep moving away from each other in such a universe?

 

[Chalnoth]

I think this statement comes from assuming that the homogeneity and isotropy of the universe is globally-accurate, even outside of our observable universe, and that there's no non-trivial topology (an example of non-trivial topology is the old video game asteroids, where when the ship exits the right side of the screen it reappears on the left, so that you have a flat but finite "universe").

With those assumptions, a closed universe is a hypersphere, with finite size. Flat and open universes are infinite.

But I don't think there's any reason to take this kind of model seriously outside of our observable universe. There could be very large-scale variations in curvature, for example, that would make it so that the locally-measured curvature has literally nothing to do with whether the universe beyond our horizon is infinite or not.

As for why galaxies can't keep moving away from one another forever in the spherical universe, such a universe is expanding slowly enough that the gravitational attraction of the galaxies (absent a cosmological constant) eventually stops the expansion and the universe starts to recollapse.

 

[dougy]

As for why galaxies can't keep moving away from one another forever in the spherical universe, such a universe is expanding slowly enough that the gravitational attraction of the galaxies (absent a cosmological constant) eventually stops the expansion and the universe starts to recollapse.

That's precisely the core of my question: why does the fact a universe expands slowly enough that the gravitational attraction eventually takes over imply that this universe is spherical? Where would be the contradiction in a spherical universe where galaxies are moving too fast for the gravitational attraction to take over?

Assuming no cosmological constant and a homogeneous and isotropic universe, it seems like the statements "the universe is spherical" and "the density of the universe is strong enough for it to recollapse" are equivalent, as well as the statements "the universe is hyperbolic" and "the density of the universe is low enough for it to keep expanding forever", but I don't get why.

It feels like "the universe is spherical" is a reformulation of "the density of the universe is strong enough for it to recollapse", just like "space expands" is a reformulation of "galaxies move away from each other at a velocity proportional to their distance". And if it is not merely a reformulation, then I don't see why galaxies couldn't keep moving away forever in a spherical universe. And the reason I don't think this is a reformulation is that in a spherical universe light can travel in one direction and go back to its starting point, which cannot happen in a hyperbolic universe, so there is a physical difference between the two besides the differing rate of expansion.

 

[PeterDonis]

why does the fact a universe expands slowly enough that the gravitational attraction eventually takes over imply that this universe is spherical?

Because in General Relativity, the geometry of spacetime and the presence of matter and energy are connected; they are not independent things. In your intuition, those two things are independent; but your intuition is not correct when GR is taken into account. The geometry of spatial slices of constant "comoving" time is part of the geometry of spacetime, so it is connected to how much matter and energy is present.