I have very little knowledge of astrophysics, but I thought it would be an interesting idea if the universe constantly recycled itself. That is, every so often it collapsed in on itself and produced another big bang. Theoretically some parts of the universe would have already achieved escape velocity, but the rest of it is gradually gravitating towards some center of mass. Is such a thing even feasible?
It's definitely an interesting an idea that has been considered before. Hopefully the Wikipedia page on cyclic models can be a starting point of interest to you! https://en.wikipedia.org/wiki/Cyclic_model
A few decades ago, that was widely believed as the most likely state of the universe. However, it's now known that the big bang is essentially still accelerating and this is blamed on an unknown force called "dark energy". It doesn't have much effect within galaxies, but galaxies themselves are blowing away from each other at ridiculous speeds.
Well it's more that space itself is expanding. It doesn't affect small scales because other forces collapse everything back together fast enough to overcome it. If the increase in dark energy is consistent though, eventually it'll be expanding so fast not even our atoms can hold themselves together.
Not really, because you again assuming that there's an independent time-line, where a ball of universe expands and collapses.
Our current understanding of the Universe is that _time itself_ "expands and collapses". You cannot talk about time "before", because there was no time "before" the big bang, just as there's no space "outside" the ball, to talk about its boundaries.
Not sure if you were somehow subtly referencing this, but...
“There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.” ― Douglas Adams
Lee Smolin makes an intriguing suggestion with regard to how the universe reproduces in his book "The Life of the Cosmos" -- I can't say I was persuaded, but I think it's worth reading
