Introduction to Cosmology Ryden · Problem 5.3

Model Universes

Physics — Astrophysics · age-positively-curved-matter-only-universe

Problem

In a positively curved universe containing only matter (

\Omega_0 > 1

\kappa = +1

), show that the present age of the universe is given by the formula

H_0 t_0 = \frac{\Omega_0}{2(\Omega_0 - 1)^{3/2}} \cos^{-1}\!\left(\frac{2 - \Omega_0}{\Omega_0}\right) - \frac{1}{\Omega_0 - 1}. \tag{5.117}

Assuming

H_0 = 68

km s

^{-1}

Mpc

^{-1}

, plot

t_0

as a function of

\Omega_0

in the range

1 \leq \Omega_0 \leq 3