Wouldn’t the universe at the end of the inflationary period still be only very small? Surely, the inflationary theory doesn’t posit that the universe grew to billions of light years across in 10^32 seconds?
First, that's 10 to the MINUS 32. ie 10^ -32. Humongous difference. 64 zeros (64 powers of 10!) :)
Second, if I recall correctly (too lazy to check right now), I believe the theory states that the universe grew to the size of a grapefruit following the near-infinitely brief supposed inflationary period.