General Relativity: An Introduction for Physicists · Problem 16.5

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · power-law-inflation-exponential-potential

Problem

For a potential of the form

V(\phi) = V_0 \exp(-\lambda\phi),

where

\lambda

is a positive constant, show that the inflation equations can be solved exactly to give

R(t) = R_0\left(\frac{t}{t_0}\right)^{2/\lambda^2}, \qquad \phi(t) = \phi_0 + \frac{2}{\lambda}\ln\left(\sqrt{\frac{V_0}{2(6-\lambda^2)}}\,\lambda^2 t\right).

Hence show that, provided

\lambda < \sqrt{2}

, the solution corresponds to a period of inflation. Show further that the slow-roll parameters for this model are

\epsilon = \tfrac{1}{2}\eta = \tfrac{1}{2}\lambda^2

, and so the inflationary epoch never ends. This model is known as power-law inflation.