I am answering this in a new text box.
An infinite universe is plausible. The RxR^3 topology of the universe may not have the same vacuum structure everywhere. There may be these zones where the vacuum energy Λ_0 >> Λ, where Λ_0 and Λ are the bare vacuum and the broken vacuum cosmological constant. So our observable universe may have a sort of barrier, where signature of that might exist on the CMB. This is the Linde pocket universe model, and these bubbles are due to the physics of Coleman --- called bubble nucleation. There some good reasons to think this is the case.
However, we can assume the vacuum is the same for the entire R^3. Any observer looking out will see galaxies and objects further out move at increasing velocity, indeed faster than light. If the space is infinite the velocity as one looks out to infinity becomes infinite. I outline how this works below. So if that happens then any photon emitted sufficiently far out is not just red shifted to the IR or into the radio wave band. It red shifts beyond the horizon length ~ 10^{10} light years. Again I illustrate what the horizon is below. So this quells the Olber's paradox, for anything sufficiently far out is redshifted to such low frequencies that it is not very observable. Further. anything that far out emits photons along our light cone from behind the CMB opaque region. This means such photons are swamped by that boundary This would not be the case if these quanta are in the form of neutrinos. In principle with neutrinos we could observe the universe at a time far earlier than the CMB. Gravitons similarly could permit us to observe right to the quantum gravity event. These gravitons would be red shifted into long wave length gravity waves which might perturb the CMB in so called B-modes.
The problem with an infinite universe with a single vacuum is that it means our past light cone extends infinitely outwards. In order to have a finite time it means the initial inflationary period involved an infinite expansion. That is a bit of trouble. So to prevent some problems the Linde pocket universe idea (eternal inflation etc) is a better candidate. However, the whole R^3 contains an infinite number of these pockets expanding out at an extremely rapid rate. So in some sense we have pushed the problem out to another level. So this R^3 might have started out as a three sphere S^3 where a point was removed and that topological information is involved with quantum information of these bubbles that are finite in number. Further, for reasons I will go into right now, that huge vacuum these bubbles are contained in may run down, which would also mean the creation of pocket cosmologies is finite and of brief duration.
The expansion of the universe is described by a scale factor a(t). Given a radial distance r the scale factor a(t) gives a new radial distance r'(t) = a(t)r. I will use Newtonian mechanics and gravity, for it turns out that this gives the same thing as general relativity for flat space, but curved spacetime. General relativity is somewhat complicated to work with. We have for Newtonian mechanics with gravity that the kinetic energy of a moving object is (1/2)mv^2 and that the potential energy is -GMm/r. The total energy is the sum of these. The velocity is determined for our situation by the scale factor so that r(t) = a(t)r and v(t) = (da/dt)r. A little bit of calculus is entering in here. So the total energy we can set to zero, and we have
(a')^2 = 2GM/a, a' = da/dt
We then have M = (4π/3)d (ar)^3, for d = density of matter in a spherical region of radius r' = a(t)r. We then write this dynamical equation as
a'^2 = 8π Gd a^2/3,
where the Hubble factor is H = a'/a. Now if I assume that the density is constant then this is a differential equation a' = Ka, for H = sqrt{8πGd/3}, and the solution is
a(t) = (1/H}exp(Ht).
So the scale factor expands exponentially. This is approximately a de Sitter spacetime configuration.
The Hubble factor for small time gives v = Hr, for r a small radius out related to a time t. For v = c one can compute the radius where that occurs and we have r = c/H which is the cosmological horizon distance
R = 1/sqrt{8πG/3c^2} = sqrt{3//\}
Where /\ is the cosmological constant. This horizon distance is about 10 billion light years.
This event horizon is not a barrier to our ability to observe things. It is similar to the event horizon of a black hole, but it is analogous to looking out into the exterior world from inside a black hole. It is a barrier to our ability to send a signal to anything beyond this distance. A galaxy with a z > 1 is beyond this event horizon, and the CMB has z ~ 1000. What happens with galaxies disappearing is that they will accelerate away and become highly red shifted. In about 10 billion years all galaxies outside our local group will be red shifted out of the optical band. An intelligent life form could observe other galaxies if they use IR or microwave instruments. The CMB will recede into the radio wave band and to long wavelength frequencies.
LC
