Have there been other Big Bangs before?
Do we have any way to tell if we are in the time frame of the first big bang or have there been others?
First of all we have to make some definitions clear. The big bang theory is a model in which all the matter of the Universe was initially concentrated and then got into expansion. It is the moment of creation of the universe. So therefore we cannot speculate on what was there before since by definition there was no before. You can think of it as if time started to exist at the time of the big bang, so there was nothing before that. So the answer to your first question is that we are in the time frame of the first big bang, by definition of the theory itself.
But you are right, there are some cosmology models that describe the Universe as collapsing and then going through another big bang, and so on. Other models however say that the Universe will expand forever. The parameter that enables us to decide which of these models to adopt is the density of the Universe. From the theory of gravitation, it was established that there is a critical density value for the Universe. If the actual density of the Universe is exactly the critical density, then the Universe is said to be flat and will keep on expanding. If the density is larger than the critical density, the Universe is closed (it will undergo a 'big crunch', and another big bang, and so on ...). This is because in that case there would be enough matter in the Universe such that the gravity would be strong enough to overcome the expansion. Finally, if the density is smaller than the critical density, the Universe will expand forever (this is called an open Universe).
Astronomical observations of distant galaxies, indicate that the density of the Universe is probably equal to the critical density, therefore the Universe is likely to expand forever, and never go through another big bang.
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