General Relativity: An Introduction for Physicists · Problem 15.10

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · cosmological-field-equations-density-parameters

Problem

In terms of the dimensionless density parameters, show that the two cosmological field equations can be written in the forms

H^2 = H_0^2\left[\Omega_{\mathrm{m},0}\, a^{-3} + \Omega_{\mathrm{r},0}\, a^{-4} + \Omega_{\Lambda,0} + \Omega_{k,0}\, a^{-2}\right],

q = \tfrac{1}{2}(\Omega_m + 2\Omega_r - 2\Omega_\Lambda),

where

H

and

q

are the Hubble and deceleration parameters respectively, and

a = R/R_0

is the normalised scale factor.