### 4. THE IDEA OF INFLATION

Seen with many years of hindsight, the idea of inflation is actually rather obvious. Take for example the Friedmann equation as used to analyze the flatness problem

(26)

The problem with the hot big bang model is that aH always decreases, and so is repelled away from one.

In order to solve the problem, we will clearly need to reverse this state of affairs. Accordingly, define inflation to be any epoch where > 0, an accelerated expansion. We can rewrite this in several different ways

 (27) (28) (29)

The middle definition is the one which I prefer to use, because it has the most direct geometrical interpretation. It says that the Hubble length, as measured in comoving coordinates, decreases during inflation. At any other time, the comoving Hubble length increases. This is the key property of inflation; although typically the expansion of the Universe is very rapid, the crucial characteristic scale of the Universe is actually becoming smaller, when measured relative to that expansion.

As we will see, quite a wide range of behaviours satisfy the inflationary condition. The most classic one is one we have already seen; when the equation of state is p = - , the solution is

(30)

Since the successes of the hot big bang theory rely on the Universe having a conventional (non-inflationary) evolution, we cannot permit this inflationary period to go on forever - it must come to an end early enough that the big bang successes are not threatened. Normally, then, inflation is viewed as a phenomenon of the very early Universe, which comes to an end and is followed by the conventional behaviour. Inflation does not replace the hot big bang theory; it is a bolt-on accessory attached at early times to improve the performance of the theory.