Slow-roll parameter in Inflation

Question

What is the definition of slow-roll parameter? What equation that these parameters are showed up?

I read a book "Modern Cosmology" by Scott Dodelson. In chapter 6, it is just given the parameter without mention what are these parameters actually saying and how cosmologists know the field roll slowly from those parameters.

Answer 1

In order to produce an accelerating expansion (desirable for inflation and dark energy models) using a scalar field, that field must be slowly rolling. The two slow-roll parameters are defined as $$ \begin{align} \epsilon &= \frac{1}{\kappa^2} \left(\frac{V_{, \phi}}{V} \right)^2, \\ \eta &= \frac{V_{, \phi \phi}}{\kappa^2 \phi}, \end{align} $$ where $\kappa = 8 \pi G$, $\phi$ is the scalar field and $V$ its potential.

When $\epsilon \ll 1$ and $|\eta| \ll 1$, the field is rolling slowly enough for acceleration to occur.

More on the slow roll formalism can be found in this work by Liddle, Parsons and Barrow: https://arxiv.org/abs/astro-ph/9408015.