Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems General Relativity: An Introduction for Physicists · Problem 15.23
Cosmological models — Hobson, Efstathiou & Lasenby
Physics · flat-lemaitre-no-radiation-scale-factor
Problem For a spatially flat Lema\^itre model containing no radiation, show that
H 0 t = 2 3 ∣ Ω Λ , 0 ∣ ∫ 0 a 3 ∣ Ω Λ , 0 ∣ / ( 1 − Ω Λ , 0 ) d y 1 ± y 2 . H_0 t = \frac{2}{3\sqrt{|\Omega_{\Lambda,0}|}}\int_0^{\sqrt{a^3|\Omega_{\Lambda,0}|/(1-\Omega_{\Lambda,0})}} \frac{dy}{\sqrt{1\pm y^2}}. H 0 t = 3 ∣ Ω Λ , 0 ∣ 2 ∫ 0 a 3 ∣ Ω Λ , 0 ∣/ ( 1 − Ω Λ , 0 ) 1 ± y 2 d y . a ( t ) = ( 1 − Ω Λ , 0 ∣ Ω Λ , 0 ∣ ) 1 / 3 { sinh 2 / 3 ( 3 2 Ω Λ , 0 H 0 t ) if Ω Λ , 0 > 0 , sin 2 / 3 ( 3 2 ∣ Ω Λ , 0 ∣ H 0 t ) if Ω Λ , 0 < 0. a(t) = \left(\frac{1-\Omega_{\Lambda,0}}{|\Omega_{\Lambda,0}|}\right)^{1/3}\begin{cases}\sinh^{2/3}\!\left(\tfrac{3}{2}\sqrt{\Omega_{\Lambda,0}}\,H_0 t\right) & \text{if } \Omega_{\Lambda,0} > 0,\\[4pt] \sin^{2/3}\!\left(\tfrac{3}{2}\sqrt{|\Omega_{\Lambda,0}|}\,H_0 t\right) & \text{if } \Omega_{\Lambda,0} < 0.\end{cases} a ( t ) = ( ∣ Ω Λ , 0 ∣ 1 − Ω Λ , 0 ) 1/3 ⎩ ⎨ ⎧ sinh 2/3 ( 2 3 Ω Λ , 0 H 0 t ) sin 2/3 ( 2 3 ∣ Ω Λ , 0 ∣ H 0 t ) if Ω Λ , 0 > 0 , if Ω Λ , 0 < 0. Browse the problem library