General Relativity: An Introduction for Physicists · Problem 15.22

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · lemaitre-model-inflection-point

Problem

For a Lema\^itre model containing no radiation, show that at the point of inflection of the

a(t)

-curve the value of the normalised scale factor is

a_* = \left(\frac{\Omega_{\mathrm{m},0}}{2\Omega_{\Lambda,0}}\right)^{1/3},

and calculate

a_*

for our universe. Show further that, in the vicinity of the point of inflection, the scale factor obeys the equation

\dot{a}^2 \approx H_0^2\left[\Omega_{k,0} + 3\Omega_{\Lambda,0}\, a_*^2 + 3\Omega_{\Lambda,0}(a - a_*)^2\right]

and that this has the solution

a(t) = a_* + a_*\left[1 + \tfrac{1}{3}\Omega_{k,0}\left(\tfrac{1}{4}\Omega_{\Lambda,0}\Omega_{\mathrm{m},0}^2\right)^{-1/3}+\right]^{1/2}\sinh\left[H_0(3\Omega_{\Lambda,0})^{1/2}(t - t_*)\right].