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If the universe is infinitely large, then any two arbitrarily distant points must have been arbitrarily close together at some earlier point in time. Doesn't that mean that the expansion rate of the universe must approach infinity as you go back in time? With each successively earlier time interval, there is less and less time for any two nearby points to reach a given distance from each other, so the expansion rate would have to get faster the farther back you go, without limit.

A finite universe, if I understand correctly, has no such requirement. Because there is always an upper limit to how far apart any two points can be, there's no need for the expansion rate to go to infinity.

Doesn't this result in radically different expansion profiles for a finite versus an infinite universe?

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The Friedman equations, based on the field equations of General Relativity with the assumptions of homogeneity and isotropy, provide the expression for the expansion rate of the Universe, (Hubble Parameter) as a function of the scale factor "a".

$$H(a)=H_0 \sqrt{\frac{\Omega_{R_0}}{a^4}+\frac{\Omega_{M_0}}{a^3}+\frac{\Omega_{K_0}}{a^2}+\Omega_{\Lambda_0}}$$

This expression works for infinite universes $\rightarrow \Omega_{K_0} \geq 0$

And also for finite universes $\rightarrow \Omega_{K_0} < 0$

$H_0$ is the current Hubble parameter. The Omegas are respectively the current density ratios of radiation, matter, curvature and dark energy.

$H_0>0, \quad \Omega_{R_0}>0 ,\quad \Omega_{M_0}>0, \quad \Omega_{\Lambda_0}>0 $

$\Omega_{K_0}=1-\Omega_{R_0}-\Omega_{M_0}-\Omega_{\Lambda_0}$

It is observed that for both, finite and infinite universes:

$a \to 0 \Rightarrow H \to \infty$

As you well presupposed.

That means that at the instant when a=0 General Relativity is not applicable, it would be necessary to have a quantum theory of gravity to study the beginning. Unfortunately no one has yet developed such a quantum theory of gravity.

Best regards.

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The instant of the big bang is singular. That is there is an "infinity/infinity" problem if you want to describe the state of the universe at that point.

There are ways around this problem, perhaps that state at time=0 never existed, there was just an initial state of an infinitely large chaotic space. Perhaps there was inflation continuing exponentially for an infinite amount of time into the past. Perhaps there is some way in which finite can become infinite in the initial $10^{-44}$ seconds of the universe. The best we can can do is describe the universe at times t>0 and at these time, to the best of out knowledge, the Universe seems infinite in space.

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