Read Olbers' paradox first, or none of this will make much sense...

It would appear that having a "Big Bang" event would resolve Olbers' paradox: looking at stars further and further away means looking backwards in time, so after a certain distance no stars can be seen, and the paradox vanishes.

But models with a "Big Bang" are still of bounded (but with no boundary conditions) universes. And a bounded universe doesn't resolve Olbers' paradox. Adding the "Big Bang" event just means this bounded universe evolved to its present state (a bad way to pick up chicks (and, presumably, pick up studs) is to say at parties things like "Big Band cosmology throws away anistropy in the time dimension", so don't do that). So when we look further and further away, we're just wrapping around (á la PACMAN). The universe we're looking back at is smaller and smaller the further we get away, but eventually we'll still hit a star. In fact, we'll probably hit a star sooner rather than later.

So this (on its own) affords no solution to Olbers' paradox.

The statement that ""Big Bang" are still of bounded (but with no boundary conditions) universes" is not correct.
The inital singularity in time is independant
from whether the universe is open or closed, bounded or unbounded. The geometry of the universe is
a function of what is in the universe. You can have any
combination (open-closed, bounded-unbounded)
with or without a big bang.

Ah, but the big bang imposes a finite age on the Universe. So, while it's true that every line of sight will still end on a star, the majority of these lines of sight will be longer (in light years) than the Universe is old (in years). Thus, a big bang does
resolve Olbers' paradox. Plus, there's that whole redshift thing.

Well, according to the Big Bang theory, you don't need to get a line of sight to a star - the light of the decoupling should fry you - it occurred everywhere at about the same time, so when you get back to looking at that time, you're looking at a big hydrogen emission line, which is HOT.

Of course, this IS what happens when we look at the cosmic microwave background - we're just not fried by it, due to that whole redshift thing.

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