One aspect of the Horizon Problem is that the observed equal CBR temperature from opposite directions of the sky is a mystery. If we choose two points, say A and B, at opposite ends of a arbitrary diameter of the observable universe, at the present time the distance between A and B would be D = D(t(adsbygoogle = window.adsbygoogle || []).push({}); _{0}) (= approx. 93 Gly) (see https://en.wikipedia.org/wiki/Observable_universe), where t_{0}is the current age of the universe (= approx 13.8 Gy)

If a(t) is the scale factor at time t, and a(t_{0∫}) = 1, then D= 2* c* ∫SUB][/SUB]^{t}(dt/a(t)).

(See https://www.physicsforums.com/threa...e-of-observable-universe.827068/#post-5194532)

Since the maximum distance light can travel since t=0 and t_{0}is D/2 (= approx. 46.5 Gly), nothing originating from A could ever reach B (or vise versa) at any time in the history of the universe, unlesscosmic inflationcan produce a form for a(t) that allows this to happen.

I have three questions:

(1) What are the values of the various components of Ω (= 1 by definiton) during inflation, say between t_{s}(start of inflation) and t_{e}(end of inflation)?

(2) What is the form of a(t) that results from cosmic inflation?

(3) How does the math show that with this a(t) the correponding value for D(t_{0}) would be sufficiently small for light to travel between A and B during the time interval between t = 0 and t = t_{0}?

NOTE:

I looked the related threads listed below, and none appear to answer the questions I am asking.

"Explanation on how inflation solves the horizon and flatness problem"

"Horizon problem - why do we need inflation?"

"How inflation solves the horizon problem?

"Inflation and particle horizon"

"How inflation solves the horizon problem"

This thread shows an interpretation that would be an explanation is it were OK, but I think there is a problem about constant energy density during inflation that I can't post there because the thread is closed. From https://en.wikipedia.org/wiki/Inflation_(cosmology) :

During inflation, the energy density in the inflation field is roughly constant. However, the energy density in everything else, including inhomogeneities, curvature, anisotropies, exotic particles, and standard-model particles is falling, and through sufficient inflation these all become negligible. This leaves the Universe flat and symmetric, and (apart from the homogeneous inflation field) mostly empty, at the moment inflation ends and reheating begins.

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# What is math showing inflation solves horizon problem?

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