General Relativity: An Introduction for Physicists · Problem 16.17

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · inflation-mode-equation-solution

Problem

Consider the equation of motion ξk′′+[k2−2(ηend−η)2]ξk=0.\xi_k'' + \left[k^2 - \frac{2}{(\eta_{\rm end}-\eta)^2}\right]\xi_k = 0. Show that the unique solution (up to a phase factor) that tends to (16.51) for small η\eta is given by ξk=12k3i+k(ηend−η)ηend−η e−ikη.\xi_k = \frac{1}{\sqrt{2k^3}}\frac{i+k(\eta_{\rm end}-\eta)}{\eta_{\rm end}-\eta}\,e^{-ik\eta}.

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