How inflation solves the horizon problem

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Inflation, specifically in de Sitter space, addresses the horizon problem by allowing light cones of the Cosmic Microwave Background (CMB) to intersect. The Hubble radius, which roughly sets the interaction length, becomes crucial during inflation as it remains nearly constant while comoving scales shrink. This means that as the universe expands, any length scale L will eventually be smaller than the Hubble radius, allowing for interactions that would otherwise be impossible. The comparison of scale L to the Hubble radius is justified because it effectively defines the limits of causal contact. Overall, the dynamics of inflation ensure that the horizon problem is resolved by maintaining a consistent Hubble scale throughout the process.
torus
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Hi,
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
 
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The reason why the scale L is compared to the Hubble radius is that the Hubble radius sets (somewhat roughly) the possible interaction length: on longer scales, speed of light limitations prevent any interaction.

Inflation solves this particular issue because during inflation, the Hubble scale was nearly constant, such that if we take any length scale L today, and consider that L scales as a*L going into the past, at some point during inflation a*L < c/H(a) (since H is nearly a constant during inflation, and a*L decreases monotonically into the past).

Does that help?
 
Well, the question is exactly: Why does the Hubble radius set the interaction length?
Sure, it has the right dimension, but we could still multiply by a or something. Or it could be an integral.
 
torus said:
Well, the question is exactly: Why does the Hubble radius set the interaction length?
Sure, it has the right dimension, but we could still multiply by a or something. Or it could be an integral.
Well, technically it is an integral, but the result is no more than a factor of a few from the Hubble radius. And, if I remember correctly, the result is precisely the Hubble radius in de Sitter space.
 
Okay, I finally found a way to make it clear for myself: The comoving distance a lightray travels from a_1 to a_2 is given by
\int_{a_1}^{a_2} \frac{1}{aH} d\ln a = \frac{1}{H a_1} - \frac{1}{H a_2}
so let's say in the first half of inflation: a_1 = a_i, a_2=a_f/2 with a_f/a_i being the huge factor of like 60 e-folds, we have the length 1/Ha_i, whereas in the second half (a_1=a_f/2, a_2 = a_f) we get 1/Ha_f, so at the end of inflation, a light ray can travel less far compared to the initial time, since H stays constant the entire time. The same argument does not hold for matter or radiation dominated, since we do not get the huge 1/Ha_i but rather something small like a_i.

This looks fine, thank you for your help, Chalnoth!

Regards,
torus
 
https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
Why was the Hubble constant assumed to be decreasing and slowing down (decelerating) the expansion rate of the Universe, while at the same time Dark Energy is presumably accelerating the expansion? And to thicken the plot. recent news from NASA indicates that the Hubble constant is now increasing. Can you clarify this enigma? Also., if the Hubble constant eventually decreases, why is there a lower limit to its value?

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