To begin, I'll give a quick review of the big bang cosmology. More detailed accounts can be found in any cosmology textbook. The main aim in this subsection is to set down the notation for the rest of the article.
The standard hot big bang theory, by which I mean the description of the Universe from a time of around one second onwards, is an extremely successful one, passing some crucial observational tests of which I'd highlight five.
In combination these are extremely compelling, and there is little doubt that the physical framework for describing the Universe from one second onwards is firmly in place, although the values of the various parameters required to specify the model in detail remain uncertain.
However, there is a series of questions which the standard big bang theory does not address. It does not predict the geometry of the Universe, except insofar as to indicate that a spatially-flat cosmology requires considerable fine-tuning of the initial conditions. It does not predict the relative abundances of different kinds of material - baryons, radiation, dark matter, etc - in the present Universe; these are assumed. And, most importantly of all, it does not offer an explanation for why the Universe was homogeneous to a high degree of accuracy at early times, but with sufficient irregularities within it to enable gravitational collapse to lead to structures such as galaxies and galaxy clusters. In order to address these questions, one must consider possible physical phenomena which might have taken place in the very early Universe, at epochs so early that the appropriate laws of physics are unknown. This raises the exciting possibility that an accurate determination of the properties of the present Universe might shed light on the physics of those early stages.