Question on the singularity theorem I have just started studying Cosmology and we have been asked to prove that in an expanding FRW Universe which obeys the strong energy condition: $$\rho + 3P &gt;0$$
Then there must exist a Big Bang singularity. I can see that this condition implies $$\ddot{a}/a \leq 0$$
which when you plot against t gives an always decreasing rate of expansion over time. This when plotted then shows that if you extend the curve far back enough will cross the t axis at some finite t-value representing a Big Bang singularity. My question is why can't the universe begin with some finite non-zero a value? is there some physical reason that we expect the Universe to begin with a=0?
It does not have to start from the big bang. There are also different universe models such as the big bounce. Where the universe has an infinite past, such that it expands and then contract.
See
https://www.quantamagazine.org/big-bounce-models-reignite-big-bang-debate-20180131/
https://www.wired.com/story/what-if-the-big-bang-was-actually-a-big-bounce/
For more detailed explanations you can look some cosmology textbooks