General Relativity: An Introduction for Physicists · Problem 16.14

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · scalar-field-perturbation-energy-momentum

Problem

For the scalar-field perturbation

\phi(t) \to \phi_0(t) + \delta\phi(t,\vec{x}),

show that, to first order in

\delta\phi

, the perturbed parts of the scalar-field energy–momentum tensor are given by

\delta T^0{}_i = \dot{\phi}_0\,\partial_i(\delta\phi),

\delta T^0{}_0 = -\dot{\phi}_0^2\Phi + \dot{\phi}_0\,\dot{\delta\phi} + V'\delta\phi,

\delta T^i{}_i = \dot{\phi}_0^2\Phi - \dot{\phi}_0\,\dot{\delta\phi} + V'\delta\phi,

where

V' = dV/d\phi_0

and the remaining components either follow from symmetry or are zero.