General Relativity: An Introduction for Physicists · Problem 15.19

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · radiation-only-friedmann-non-flat

Problem

For a radiation-only Friedmann model with Ωr,0≠1\Omega_{\mathrm{r},0} \neq 1, show that a(t)=(2H0Ωr,01/2 t)1/2(1+1−Ωr,02Ωr,01/2H0t)1/2.a(t) = \left(2H_0\Omega_{\mathrm{r},0}^{1/2}\, t\right)^{1/2}\left(1 + \frac{1-\Omega_{\mathrm{r},0}}{2\Omega_{\mathrm{r},0}^{1/2}}H_0 t\right)^{1/2}. Hence, for Ωr,0>1\Omega_{\mathrm{r},0} > 1, show that the a(t)a(t)-curve has a maximum at amax⁡=(Ωr,0Ωr,0−1)1/2,tmax⁡=1H0Ωr,01/2(Ωr,0−1),a_{\max} = \left(\frac{\Omega_{\mathrm{r},0}}{\Omega_{\mathrm{r},0}-1}\right)^{1/2}, \qquad t_{\max} = \frac{1}{H_0}\frac{\Omega_{\mathrm{r},0}^{1/2}}{(\Omega_{\mathrm{r},0}-1)}, and that the age t0t_0 of such a universe is given by t0=1H01Ωr,01/2+1<12H0.t_0 = \frac{1}{H_0}\frac{1}{\Omega_{\mathrm{r},0}^{1/2}+1} < \frac{1}{2H_0}.

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