Ranku said:
I realized just now that the question I should really be asking is would the proper distance between two antipodal points in the last scattering surface be lesser than the horizon distance, if no inflation occurred? If so, can we deduce whether inflation occurred or not by observing if last scattering surface is greater or lesser than the horizon?
I'm not sure I understand your intent, since this looks to me like exactly the same question as before.
For the purposes of this topic, the particle horizon and the last scattering surface are pretty much the same thing. Where they are is determined solely by what happens during regular expansion.
So, 1) You can't make them be in different places by adding or removing inflation; 2) Since they're closely related, what you're asking here boils down to: can a (positive) distance be more than twice the same distance.
I suspect your overarching reasoning here goes as follows (please, clarify otherwise):
- we have these regions that look connected
- the expansion model suggests they should be disconnected
- inflation spread them apart
Therefore removing inflation should mean they'd be closer (and possibly connected), as there would have been nothing to increase their separation before BB.
In which case the error would be in that the first two steps cannot be changed, as they are fixed by observation. These regions definitely look connected, and the BB expansion definitely would have them disconnected.
What you can change is the hypothetical third step, but all it does is provide a reason for steps one and two to not be at odds. I.e. flipping inflation on/off means that you see the same problem either way, but you either have or don't have a possible solution.
It's an issue with logic. It's like saying: I see a blue cow. Cows are not blue. This is the cow colour problem. To attempt to solve the problem I hypothesise somebody painted the cow. Now, does it make any sense to propose that maybe nobody painted the cow, since then the cow wouldn't be blue, thus solving the problem?
You could assume the third step to be true in our universe, and ask what would change in some hypothetical universe, where inflation didn't occur, but expansion proceeded normally (i.e. you relax the first step). In such universe, the distant regions would look like the expansion model alone suggests - they wouldn't look causally connected. But they definitely do.
The cow wouldn't be blue in the absence of the sneaky bovine painter. But it definitely is.