horizon problem
Suppose you found that two thermostats on opposite ends of a huge warehouse read exactly the same temperature, to four decimal places — yet you knew the warehouse had only existed for a few seconds, far too little time for heat or any signal to travel from one end to the other. You would be puzzled: how did they agree so perfectly if they never had a chance to talk? This is the shape of the horizon problem, one of the puzzles that the plain hot Big Bang could not explain.
When we look at the cosmic microwave background, the relic glow from when the universe was 380,000 years old, we find it has almost exactly the same temperature in every direction — 2.725 degrees above absolute zero, the same to about one part in 100,000 across the whole sky. The trouble is that two patches of sky in opposite directions are so far apart that, in the standard hot Big Bang, light (and therefore any influence, since nothing travels faster) had never had time to pass between them. They lay outside each other's 'horizon.' So how did they end up at the identical temperature, as if they had carefully equalized? Regions that never interacted should not match.
Cosmic inflation offers a clean answer. Before inflation, the entire region that became our observable universe was a tiny patch, small enough that all of it was in close contact and had time to even out to one temperature. Then inflation stretched that already-uniform patch to enormous size, flinging its parts far beyond each other's horizons but carrying the agreed-upon uniformity with them. So the matching temperatures are a memory of an early intimacy, frozen in by a burst of expansion. The horizon problem is one of inflation's chief motivations and successes — though it is worth noting it is a problem of explanation, not a contradiction in observations.
Pick two patches of the microwave sky 180 degrees apart, on exactly opposite sides of the sky. In the plain hot Big Bang their light reaches us from regions that, at the moment that light set out, were separated by far more than light could have crossed in the universe's whole age up to then. They had never been in causal contact — yet their temperatures agree to within a few parts in 100,000.
Opposite sides of the sky match in temperature, though they should never have been in contact.
The horizon problem is not an observed contradiction but a puzzle of explanation: the plain Big Bang can fit the uniform sky only by assuming it was uniform to begin with. Inflation explains why that uniformity is natural rather than a lucky coincidence.