Imagine an eruption of energy/mass $E$ from a singularity $O$, as in a Big Bang. After the energy/mass $E$ is all at more than a distance $d$ from $O$, is it for some value of $d$ possible that there could be a new eruption of energy/mass from $O$, i.e. a new Big Bang? If yes, is there an upper limit to the number of succeeding Big Bangs?
There are several scenarios that point out cyclic cosmologies. The first one is called Matter bounce cosmology which is based on the idea that that the universe originated from a cosmological bounce in which quantum fluctuations develop into a scale-invariant spectrum of curvature perturbations. The bounce is realized beyond Einstein's General Relativity and introduces new forms of matter which violates key energy conditions, since these two aspects allows to avoid the cosmological singularity. Another one is the so-called Ekpyrotic Cosmology which represents our universe as a Dp-brane which arises in string theory. There are two branes separated by a 5-dimensional bulk spacetime (one of those branes is our universe), which contains an attractive force called quitessence which causes the two branes to approach, collide, move through and bounce off each other. The collision seen from an observer living in one of the branes look like a big bang. This collisions and separation of the branes happen infinitely many times.
For more information check the following arxiv articles:
- Lehners, J., Ekpyrotic and Cyclic Cosmology, arXiv:0806.1245v2
- R. Brandenberger, The matter bounce alternative to in iflationary cosmology," arXiv:1206.4196v1.