As far as we can tell, the expansion of the Universe started many billions
of years ago from a very hot, very small state. From that hot, small
state, it mushroomed and evolved into the Universe we know today.
Cosmologists call that process of expansion the Big Bang
because at some phases, especially in the beginning, the process was
rather like an explosion.
Much of understanding the Big Bang is extrapolating between knowledge
of particle physics today, and projections from the mathematical model
of an expanding universe in general relativity.
The Einstein equations give us a mathematical model for describing how
fast the Universe would expanding at what size and time, given the energy
density of matter and radiation at that time. We base our guesses about
the matter and radiation density of the early Universe based on the
ancient light reaching us from the past in our night skies, and what
we have learned about elementary particle physics, through theory and
experiment.
