∞ Was the 'big-bang' the beginning of the universe, or just the end of some other epoch of cosmological history?
There is excellent evidence from a wealth of astronomical data that 13.8 billions years ago, the observable universe was a nearly uniform expanding fireball of radiation—the 'big bang.' Yet there is also compelling evidence that prior to this fireball was a period of exponential 'inflationary' expansion, probably driven by 'vacuum energy'—an un-dilutable substance like the 'dark energy' we observe cosmologically, but much higher in energy. Such an inflationary period resolves a number of perplexing questions about the big bang, and also makes detailed predictions about the cosmic microwave background and other observables, many of which have been confirmed. If such an inflationary epoch occurred, a wide-open question is for how long it occurred—only a very short epoch is necessary, but generic models of inflation often have an immense, or even infinite, inflationary epoch prior to our big bang.
∞ Did the universe as a whole have a beginning?
If there was an extremely long inflationary epoch before our big-bang fireball formed, one can ask whether the Universe as a whole ever actually started. Presently, there is no consensus on this question. Singularity theorems have been proven that suggest an initial 'edge' to any inflating region, but 'past-eternal' cosmological models have been constructed by myself and collaborators (as well as other cosmologists) that evade these theorems and appear perfectly well-defined.
∞ Is the universe infinite in space? in time? in spacetime?
If inflation continues forever into the future ('everlasting' or 'eternal' inflation), then it would create an unbounded spacetime volume, and an infinite amount of time for things to happen. A very interesting implication of Einstein's general relativity applied to this sort of spacetime is that the question of whether the universe is spatially infinite or finite can be ambiguous: there can be equally valid ways to 'break up' spacetime into space and time, some of which will make the Universe spatially infinite, and some of which will make it spatially finite. Arguably, however, the most 'natural' way of breaking it in most such models makes the Universe infinite—in fact, it looks like many infinite 'pocket' universes patched together!
∞ (How) can we make predictions in an infinite universe?
If the Universe is infinite in space, time, or spacetime, it raises a number of very thorny conceptual issues for cosmology. For example, any physical configuration (or set of observed data) will be replicated many times through the Universe, along with all sorts of variations on this data. In this sense, 'everything' is predicted to happen. So to what should we compare our astronomical observations? One approach is to consider what sorts of observations are most common. This, however, requires counting different sorts of observations, which turns out to be ambiguous in an infinite spacetime as well. This very challenging 'measure problem' has been a focus of intense effort in recent years by a part of the cosmological community.
∞ How can we observationally test theories of the ultra-early- or ultra-large-scale universe?
Although regions far prior to, or outside of, our observable universe might seem to just be ad-hoc speculative hypotheses, in many versions they fit squarely into science, as they are natural implications of theories (such as inflation) that are supported by data within the observed universe. However, not all inflation models are eternal. For this and other reasons, it would be great to have evidence within the observed universe for the particular sorts of processes that would create a really big 'multiverse' of which we are a part. A number of proposals have been made for such evidence, including one by myself and collaborators that we could search for relic evidence of collisions between 'bubble universes.'
∞ What happens when our 'universe' collides with another?
If our observable universe resides within a 'bubble' universe in an inflationary background, our bubble would collide with many others. Some of these collisions might in principle exist within the region of the Universe we can observe, and leave 'bruises' or imprints on our cosmological initial conditions that would appear as a disk of influence on the microwave background sky. My group and others have worked to evaluate the probability of such collisions, what form a post-collision spacetime would take, and exactly what signature would appear in, for example, the cosmic microwave background. We have also begun to compare these predictions with data from the WMAP and PLANCK microwave background satellites.