Introduction to Cosmology Ryden · Problem 5.4

Model Universes

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

Problem

In a negatively curved universe containing only matter (Ω0<1\Omega_0 < 1, κ=−1\kappa = -1), show that the present age of the universe is given by the formula H0t0=11−Ω0−Ω02(1−Ω0)3/2cosh⁡−1 ⁣(2−Ω0Ω0).(5.118)H_0 t_0 = \frac{1}{1 - \Omega_0} - \frac{\Omega_0}{2(1-\Omega_0)^{3/2}} \cosh^{-1}\!\left(\frac{2 - \Omega_0}{\Omega_0}\right). \tag{5.118} Assuming H0=68H_0 = 68 km s−1^{-1} Mpc−1^{-1}, plot t0t_0 as a function of Ω0\Omega_0 in the range 0≤Ω0≤10 \leq \Omega_0 \leq 1.

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