General Relativity: An Introduction for Physicists · Problem 15.24

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · flat-lemaitre-hubble-matter-density-evolution

Problem

Show that, in general, R¨R=H2+H˙.\frac{\ddot{R}}{R} = H^2 + \dot{H}. Hence use the cosmological field equations to show that, for a spatially flat Lema\^itre model containing no radiation, the Hubble parameter and the matter density satisfy the equations 2H˙+3H2=Λc2,2\dot{H} + 3H^2 = \Lambda c^2, 3H2−Λc2=8πGρm.3H^2 - \Lambda c^2 = 8\pi G\rho_m. Assuming Λ>0\Lambda > 0 and requiring ρm>0\rho_m > 0, thus show that H(t)=Λc23coth⁡ ⁣(32Λc23 t),H(t) = \sqrt{\frac{\Lambda c^2}{3}}\coth\!\left(\frac{3}{2}\sqrt{\frac{\Lambda c^2}{3}}\,t\right), ρm(t)=Λc28πGcosech⁡2 ⁣(32Λc23 t),\rho_m(t) = \frac{\Lambda c^2}{8\pi G}\operatorname{cosech}^2\!\left(\frac{3}{2}\sqrt{\frac{\Lambda c^2}{3}}\,t\right), and therefore find expressions for Ωm(t)\Omega_m(t) and ΩΛ(t)\Omega_\Lambda(t). Show further that t=23Htanh⁡−1 ⁣ΩΛΩΛ.t = \frac{2}{3H}\frac{\tanh^{-1}\!\sqrt{\Omega_\Lambda}}{\sqrt{\Omega_\Lambda}}. Hint: ∫a2/(a2−x2) dx=coth⁡−1(x/a)+constant\int a^2/(a^2 - x^2)\,dx = \coth^{-1}(x/a) + \text{constant}, for x2>a2x^2 > a^2.

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