General Relativity: An Introduction for Physicists · Problem 16.11

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · scalar-field-fluctuation-density-contrast

Problem

A fluctuation

\delta\phi

in the scalar inflation field leads to a local delay of the end of inflation by

\delta t \sim \delta\phi/\dot{\phi}

. Assuming that the density of the universe after inflation decreases as

t^{-2}

, show that the fluctuation in the scalar field leads to a relative density contrast at the end of inflation given by

\frac{\delta\rho}{\rho} \sim \frac{H\,\delta\phi}{\dot{\phi}}.

Assuming the root mean square (rms) scalar field perturbation to be

\delta\phi_{\rm rms} \sim H/(2\pi)

, show that

\left(\frac{\delta\rho}{\rho}\right)_{\rm rms} \sim \frac{H^2}{2\pi\dot{\phi}}.