General Relativity: An Introduction for Physicists · Problem 16.16

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · gauge-invariant-curvature-perturbation

Problem

Show that the gauge-invariant Fourier curvature perturbation

\zeta_k \equiv \Phi_k + H\frac{\delta\phi_k}{\dot{\phi}_0}

satisfies the equation of motion

\ddot{\zeta}_k + \left(\frac{\dot{\phi}_0^2}{H} + \frac{2\ddot{\phi}_0}{\dot{\phi}_0} + 3H\right)\dot{\zeta}_k + \frac{k^2}{R^2}\zeta_k = 0.

Defining the new variables

d\eta = c\,dt/R

and

\xi_k = \alpha\zeta_k

, where

\alpha = R\dot{\phi}_0/H

, show further that

\xi_k'' + \left(k^2 - \frac{\alpha''}{\alpha}\right)\xi_k = 0,

where a prime denotes

d/d\eta

and the 'effective mass'

m_\xi^2 = -\alpha''/\alpha

is given by

m_\xi^2 = \frac{\phi_0'^2}{2} - \frac{2\phi_0'\phi_0''}{RH} - \frac{\phi_0'^4}{2R^2H^2} - \frac{\phi_0'''}{\phi_0'}

= -2R^2H^2\left(1 + \frac{\dot{\phi}_0^2}{2H^2} + \frac{\dot{\phi}_0^4}{4H^4} + \frac{\dot{\phi}_0\ddot{\phi}_0}{H^3} + \frac{3\ddot{\phi}_0}{2H\dot{\phi}_0} + \frac{\dddot{\phi}_0}{2H^2\dot{\phi}_0}\right).