At big bang, the scale factor $a(t)$ of the FLRW metric was zero, meaning that all points were in touch with one another for a moment. The horizon problem occurs, because two sufficiently far apart points A and B are homogenous even though light from A could never reach B even if it starts on big bang. But they were momentarily touching each other right at the big bang. Doesn’t that already solve the problem, without resorting to the inflation theory?
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1$\begingroup$ If you plug $a=0$ into the Friedmann equation, $a$ will stay at zero forever and will never become positive. As @Andrew's answer correctly points out: $a=0$ is NOT included in the Big Bang solution. The standard Big Bang narrative is that we can asymptotically approach the singularity $a=0$, but we will never get there. Any speculation about happens AT the singularity is science fiction instead of physics. $\endgroup$– MadMaxCommented yesterday
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$\begingroup$ No, that's the case of the de-sitter universe. Solutions of Friedmann equations for an early radiation dominated phase dictates $a=0$ finite time ago. Besides, your converging solution means there were no big bang at a finite time in the past. $\endgroup$– Nayeem1Commented yesterday
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1$\begingroup$ It seems that you don't get it: it has nothing to do with whether the singularity is at finite time or not. The point is that the singularity is not part of the physical world described by Friedmann equation/General relativity. $\endgroup$– MadMaxCommented yesterday
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$\begingroup$ Worth noting that $a=0$ is just like saying a gas's volume shrinks to zero at 0 Kelvin, per the ideal gas law, and all the molecules are coincident. That's not what really happens – instead, a different model takes over at those conditions $\endgroup$– RC_23Commented 6 hours ago
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2 Answers
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No, because the Big Bang is a spacelike singularity (at least for FLRW Universes that are matter or radiation dominated, which I'm going to assume since the point of the question is not to include inflation). What this means is that if you take the light cone of a point today, and follow it back in time, in the limit that it approaches the singularity where $a=0$, there are regions of space the light cone will not hit. This is why people say there are regions in the early Universe that remain causally disconnected today.
Calling a singularity a single point is maybe a little misleading. The curvature is infinite, but in some sense it still has a spatial extent like in the above sense. However, more to the point, we don't really think of the singularity as being part of the spacetime, we generally think of the spacetime as not including the singularity, so there's sort of a "hole" at $a=0$, and so we should really talk about a limit as you get closer and closer $a=0$. And in that limit, light cones from distant points in the Universe won't overlap.
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2$\begingroup$ I think you accidentally the verb in the first sentence $\endgroup$– TristanCommented 7 hours ago
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The horizon problem is that there isn't enough time between the big bang and the emission of the CMBR for the universe to homogenize, supposing that it didn't start in a homogeneous state. The goal is to explain the homogeneity as the outcome of a dynamical process instead of just postulating it, and the horizon problem shows that the most obvious mechanisms of that sort can't work.
You can't explain homogeneity by appealing to a time when $a=0$, because $a$ only has meaning in the context of a FLRW metric, and we only use a FLRW metric to describe the universe because we observe it to be homogeneous. That argument amounts to saying that the universe was homogeneous at $t=0$ because it was homogeneous at later times.
It may be that the generic initial conditions of the universe are symmetric for some reason, but one needs to work out the actual physics behind it before it can qualify as an explanation.
