Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems General Relativity: An Introduction for Physicists · Problem 15.17
Cosmological models — Hobson, Efstathiou & Lasenby
Physics · dust-friedmann-closed-parametric-solution
Problem For a dust-only Friedmann model with Ω m , 0 > 1 \Omega_{\mathrm{m},0} > 1 Ω m , 0 > 1 a = Ω m , 0 2 ( Ω m , 0 − 1 ) ( 1 − cos ψ ) , t = Ω m , 0 2 H 0 ( Ω m , 0 − 1 ) 3 / 2 ( ψ − sin ψ ) . a = \frac{\Omega_{\mathrm{m},0}}{2(\Omega_{\mathrm{m},0}-1)}(1-\cos\psi), \qquad t = \frac{\Omega_{\mathrm{m},0}}{2H_0(\Omega_{\mathrm{m},0}-1)^{3/2}}(\psi - \sin\psi). a = 2 ( Ω m , 0 − 1 ) Ω m , 0 ( 1 − cos ψ ) , t = 2 H 0 ( Ω m , 0 − 1 ) 3/2 Ω m , 0 ( ψ − sin ψ ) . a ( t ) a(t) a ( t ) a max = Ω m , 0 Ω m , 0 − 1 , t max = π 2 H 0 Ω m , 0 ( Ω m , 0 − 1 ) 3 / 2 , a_{\max} = \frac{\Omega_{\mathrm{m},0}}{\Omega_{\mathrm{m},0}-1}, \qquad t_{\max} = \frac{\pi}{2H_0}\frac{\Omega_{\mathrm{m},0}}{(\Omega_{\mathrm{m},0}-1)^{3/2}}, a m a x = Ω m , 0 − 1 Ω m , 0 , t m a x = 2 H 0 π ( Ω m , 0 − 1 ) 3/2 Ω m , 0 , t 0 t_0 t 0 t 0 = Ω m , 0 2 H 0 ( Ω m , 0 − 1 ) 3 / 2 [ cos − 1 ( 2 Ω m , 0 − 1 ) − 2 Ω m , 0 Ω m , 0 − 1 ] < 2 3 H 0 . t_0 = \frac{\Omega_{\mathrm{m},0}}{2H_0(\Omega_{\mathrm{m},0}-1)^{3/2}}\left[\cos^{-1}\!\left(\frac{2}{\Omega_{\mathrm{m},0}}-1\right) - \frac{2}{\Omega_{\mathrm{m},0}}\sqrt{\Omega_{\mathrm{m},0}-1}\right] < \frac{2}{3H_0}. t 0 = 2 H 0 ( Ω m , 0 − 1 ) 3/2 Ω m , 0 [ cos − 1 ( Ω m , 0 2 − 1 ) − Ω m , 0 2 Ω m , 0 − 1 ] < 3 H 0 2 . Browse the problem library