5.3. The relation between inflation and slow-roll
As it happens, the applicability of the slow-roll condition is closely connected to the condition for inflation to take place, and in many contexts the conditions can be regarded as equivalent. Let's quickly see why.
The inflationary condition > 0 is satisfied for a much wider range of behaviours than just (quasi-)exponential expansion. A classic example is power-law inflation a tp for p > 1, which is an exact solution for an exponential potential
We can manipulate the condition for inflation as
where the last manipulation uses the slow-roll approximation. The final
condition is just the slow-roll condition
< 1, and hence
Inflation will occur when the slow-roll conditions are satisfied (subject
to some caveats on whether the `attractor' behaviour has been
attained. [5])
However, the converse is not strictly true, since we had to use the
SRA in the derivation. However, in practice
The last condition arises because unless the curvature of the potential
is small, the potential will not be flat for a wide enough range of
.