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I'm trying to figure out how inflation (just deSitter) solves the horizon problem, but I am stuck. I understand the solution in terms of conformal coordinates, allowing for a negative conformal time let's the lightcones of CMB intersect. Fine. But how do I see "physically" what is going on?

In most reviews I studied they compare some (comoving) scale L to the comoving Hubble scale 1/(H a(t)) (a(t) being the scale factor, here a~exp(H t)), and since this Hubble radius shrinks down, the horizon problem is no more.

BUT: I don't get why we compare the scale L to the Hubble radius in the first place. None of my reviews provide a proper meaning of 1/Ha (well, besides some handwaving scaling arguments...), so this seems fishy to me. If I try to do it the way I thought it was right, comparing the scale to the integral over 1/a from the beginning of inflation to time t, it comes out wrong, since this integral still increases with time, i.e. the "horizon" does not shrink down.

Any help very much appreciated!

Regards,

torus