General Relativity: An Introduction for Physicists · Problem 16.7

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · hamilton-jacobi-formalism-inflation

Problem

In the slow-roll approximation, show that

\dot{H} = -\tfrac{1}{2}\dot{\phi}^2.

Assuming that

\dot{\phi}

varies monotonically with

t

throughout the period of inflation, show that

\dot{\phi} = -2H'(\phi),

where

H

is now considered as a function of

\phi

, and hence that we may write the cosmological field equation as

[H'(\phi)]^2 - \tfrac{3}{2}H^2(\phi) = -\tfrac{1}{2}V(\phi).

This is known as the Hamilton--Jacobi formalism for inflation.