General Relativity: An Introduction for Physicists · Problem 15.14

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · cubic-equation-turning-point

Problem

Show that the substitution

x = [\Omega_{\Lambda,0}/(4\Omega_{\mathrm{m},0})]^{1/3}

reduces the final cubic equation in Exercise 15.13 to

x^3 - \frac{3x}{4} + \frac{\Omega_{\mathrm{m},0} - 1}{4\Omega_{\mathrm{m},0}} = 0.

By using the standard formulae for the roots of a cubic, or otherwise, verify the results (15.18--15.20).