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?
Answer
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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