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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 addot > 0 is satisfied for a much wider range of behaviours than just (quasi-)exponential expansion. A classic example is power-law inflation a propto tp for p > 1, which is an exact solution for an exponential potential

Equation 45 (45)

We can manipulate the condition for inflation as

Equations 45a

where the last manipulation uses the slow-roll approximation. The final condition is just the slow-roll condition epsilon < 1, and hence

Equation 45b

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

Equation 45c

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 phi.

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