PeterDonis said:
Can you be more specific about what you're referring to here? Are you saying that an inflating spacetime has to have a past singularity? Or just that an inflating spacetime has to start from some non-inflating background?
The model of inflation unambiguously predicts a past singularity (this is distinct from the Big Bang singularity, but related).
One way to understand this past singularity is that as inflation progresses, the universe becomes exponentially more dilute. This is why inflation explains the horizon problem: it makes a region much larger than the observable universe almost perfectly uniform by its nature. If you instead run time the other way, and ask what inflation looks like into the past, then the answer you get is if there are
any contents in the universe
at all during inflation, even a single photon, will result in a past singularity. Even just a slight uneveness in the inflaton field itself will cause this. Thus you are saddled with two possibilities:
1. Inflation is perfectly finely-tuned, being perfectly uniform and with no other matter (including photons) in the universe.
2. There is a past singularity in the model (at an unknown time).
Generally the first possibility is considered absurd enough to discount entirely, leaving the past singularity. That past singularity has the same general class of possibilities as the Big Bang singularity. One possibility is that there are some unknown physics that resolve the singularity. Another is that there was an event that occurred that isn't captured by simply extrapolating inflation back in time.
This line of argument is why some physicists claim that inflation
doesn't solve certain problems it claims to solve. The hope of people advocating for inflation models is that the uncertainties about how inflation began are somehow easier to explain than the Big Bang singularity. This isn't proven, but it's reasonable on the surface: the event that started inflation would have had to occur over a much smaller region of space, lending hope that it's easier to come up with a physical process that will do that. At this point, though, exactly what would explain Inflation's past singularity is speculation.