Introduction to Cosmology Ryden · Problem 5.8

Model Universes

Physics — Astrophysics · big-bounce-lambda-only-positively-curved-universe

Problem

Consider an expanding, positively curved universe containing only a cosmological constant (

\Omega_0 = \Omega_{\Lambda,0} > 1

). Show that such a universe underwent a ``Big Bounce'' at a scale factor

a_{\rm bounce} = \left(\frac{\Omega_0 - 1}{\Omega_0}\right)^{1/2}, \tag{5.120}

and that the scale factor as a function of time is

a(t) = a_{\rm bounce}\cosh[\sqrt{\Omega_0}\,H_0(t - t_{\rm bounce})], \tag{5.121}

where

t_{\rm bounce}

is the time at which the Big Bounce occurred. What is the time

t_0 - t_{\rm bounce}

that has elapsed since the Big Bounce, expressed as a function of

H_0

and

\Omega_0