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The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the late 1960s, primarily by Charles Misner. It points out that different regions of the universe have not "contacted" each other because of the great distances between them, but nevertheless they have the same temperature and other physical properties. This should not be possible, given that the transfer of information (or energy, heat, etc.) can occur, at most, at the speed of light. The horizon problem may have been answered by inflationary theory, and is one of the reasons for that theory's formation.
When one looks out into the night sky, distances also correspond to time into the past. A galaxy measured at ten billion light years in distance appears to us as it was ten billion years ago, because the light has taken that long to travel to the viewer. If one were to look at a galaxy ten billion light years away in one direction, say "west", and another in the opposite direction, "east", the total distance between them is twenty billion light years. This means that the light from the first has not yet reached the second, because the 13.8 billion years that the universe has existed simply isn't a long enough time to allow it to occur. In a more general sense, there are portions of the universe that are visible to us, but invisible to each other, outside each other's respective particle horizons.
In standard physical theories, no information can travel faster than the speed of light. In this context, "information" means "any sort of physical interaction". For instance, heat will naturally flow from a hotter area to a cooler one, and in physics terms this is one example of information exchange. Given the example above, the two galaxies in question cannot have shared any sort of information; they are not in "causal contact". One would expect, then, that their physical properties would be different, and more generally, that the universe as a whole would have varying properties in different areas.
Contrary to this expectation, the universe is in fact extremely isotropic, which also implies homogeneity. The cosmic microwave background radiation (CMB), which fills the universe, is almost precisely the same temperature everywhere in the sky, about 2.728 +/- 0.004 K. The differences in temperature are so slight that it has only recently become possible to develop instruments capable of making the required measurements. This presents a serious problem; if the universe had started with even slightly different temperatures in different areas, then there would simply be no way it could have evened itself out to a common temperature by this point in time.
The magnitude of this problem is quite large. According to the Big Bang model, as the density of the universe dropped while it expanded, it eventually reached a point where photons in the "mix" of particles were no longer immediately impacting matter; they "decoupled" from the plasma and spread out into the universe as a burst of light. This is thought to have occurred about 300,000 years after the Big Bang. The volume of any possible information exchange at that time was 900,000 light years across, using the speed of light and the rate of expansion of space in the early universe. Instead, the entire sky has the same temperature, a volume 1088 times larger.
Inflationary theory allows for a solution to the problem (along with several others such as the flatness problem) by postulating a short 10−32 second period of exponential expansion (dubbed "inflation") within the first minute or so of the history of the universe. During inflation, the universe would have increased in size by an enormous factor.
If correct, inflation solves the horizon problem by suggesting that prior to the inflationary period the entire universe was causally connected, and it was during this period that the physical properties evened out. Inflation then expanded it rapidly, freezing in these properties all over the sky; at this point the universe would be forced to be almost perfectly homogeneous, as the information needed to change it from that state was no longer causally connected. In the modern era distant areas in the sky appear to be unconnected causally, but in fact were much closer together in the past.
One consequence of cosmic inflation is that the anisotropies in the Big Bang are reduced but not entirely eliminated. Differences in the temperature of the cosmic background are smoothed by cosmic inflation, but they still exist. The theory predicts a spectrum for the anisotropies in the microwave background which is mostly consistent with observations from WMAP and COBE.