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The horizon problem is usually stated as follows: Causally disconnected parts of the universe that we see today have the same temperature, so how could that be so if no information travels faster than the speed of light.

But (without assuming inflation) wasn't the universe causally connected right after the big bag? And that those parts that are disconnected today only became disconnected due to expansion?

Also, if temperature irregularities existed but to a very minute level, then why do we expect that the universe without inflation must look different than the homogeneous universe (on large scales) we see today?

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Assuming that the FLRW metric is a valid description of the early stages in the expansion of the universe, then for any two simultaneous spacetime points the proper distance between them is given by:

$$ D(t) = R(t) \chi $$

for some constant $\chi$, where $R(t)$ is the scale factor. The scale factor goes to zero at $t = 0$, but $R(t) > 0$ for any $t > 0$.

It is certainly true that as $t \rightarrow 0$ then $R(t) \rightarrow 0$ so at $t = 0$, the moment of the Big Bang itself, all separations are zero i.e. every spacetime point is in the same place. However this is a singular point and we can't do any calculations here. We have to restrict ourselves to non-zero values of $t$.

Assuming that the universe is infinite we can choose any two spacetime points, so we can make $\chi$ arbitrarily large. This means that for every value of $t > 0$ we can find two points where $D(t) > ct$ so the two points are causally disconnected. This means that the whole universe was not causally connected for any $t > 0$. Hence the horizon problem.

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  • $\begingroup$ Thanks John for the answer. So, how does inflation solve this? And does it abandon the infinite universe assumption? $\endgroup$ – stupidity Feb 21 '14 at 17:39
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    $\begingroup$ Inflation solves this by allowing the universe to expand faster than the speed of light. We still can't see the whole universe, hence references to the observable universe, but it does mean that regions that should not have any past in common actually do. $\endgroup$ – jazzwhiz Feb 21 '14 at 20:42
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    $\begingroup$ @stupidity: As jazzwhiz says. At any time we can choose $\chi$ small enough that the points are causally connected, so there will be a causally connected region. If $\dot{R}(t)$ is constant this patch grows linearly and today will be smaller than the observable universe. Inflation makes the causal patch grow exponentially so it can be (much!) larger than the observable universe. $\endgroup$ – John Rennie Feb 22 '14 at 7:19
  • $\begingroup$ Non-sequitur. How did those causally dis-connected regions of space achieve different temperatures in the first place? If energy/matter was distributed evenly at $t=0$ and at $t=1\times 10^{-44}\space s$, and if space expanded at exactly the same rate everywhere, then how does a volume achieve a different temperature than some, other non-connected volume? What mechanism do you have that either distributes the mass-energy unevenly across the universe or makes each volume grow at a different rate? $\endgroup$ – Quarkly Jan 29 '19 at 17:15

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