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The "inflationary epoch" is more or less a "fudge factor" in the Big Bang theory to explain a couple of nagging little problems. One specific problem is why the universe appears to be so isotropic and homogeneous (on very very large scales; on smaller scales, it obviously isn't, since there are stars and galaxies and empty space).

This can be explained if we assume that the early universe underwent a very rapid expansion (much faster than the speed of light). The time period in which this occurred is referred to as "the inflationary epoch".

It didn't last very long at all: currently it's theorized that it started at about t = 10-36 seconds and was over by about t = 10-32 seconds at the latest. That's just a hair under one ten-thousandth of a billionth of a billionth of a nanosecond, but in that time the universe got at least a hundred million billion billion times bigger (and possibly much more than that... the 1026 factor is necessary for the universe to appear as we see it today, but we don't know how much larger than that it really is; it's rather like testing the depth of a murky pool with a ten foot pole ... if the pole doesn't reach bottom, all you can say for sure is that the pool is at least ten feet deep).

So, why is this necessary? Because the cosmic microwave background radiation, which dates from a time very early in the universe, is the same to within a very small margin of error in all directions. For this to be true, the temperature of the early universe needs to have been the same at every point to within that same small margin of error, and there simply wasn't time for the universe to have equilibrated to that extent unless every two parts of it were initially much closer together than extrapolating back in time can possibly get them without some kind of faster-than-light expansion of space at some point.

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