General Relativity: An Introduction for Physicists · Problem 16.3

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · scalar-field-perfect-fluid-inflation

Problem

Show that a scalar field acts like a perfect fluid with an energy density and pressure given by

\rho_\phi = \tfrac{1}{2}\dot{\phi}^2 + V(\phi) + \tfrac{1}{2}(\vec{\nabla}\phi)^2,

p_\phi = \tfrac{1}{2}\dot{\phi}^2 - V(\phi) - \tfrac{1}{6}(\vec{\nabla}\phi)^2.

Show further that if the field

\phi

is spatially constant then inflation will occur, provided that

\dot{\phi}^2 < V(\phi)

. If, in addition, the scalar field does not change with time, show that its equation of state is

p_\phi = -\rho_\phi

and that it thus acts as an effective cosmological constant.