General Relativity: An Introduction for Physicists · Problem 15.2

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · matter-radiation-equality-redshift

Problem

Suppose that the present-day energy densities of radiation and matter (in the form of dust) are

\rho_r(t_0)c^2

and

\rho_m(t_0)c^2

respectively. Show that the energy densities of the two components were equal at a redshift

z_{\mathrm{eq}}

given by

1 + z_{\mathrm{eq}} = \frac{\rho_m(t_0)}{\rho_r(t_0)}.

What assumptions underlie this result? Hence show that

1 + z_{\mathrm{eq}} = \frac{3c^2 \Omega_{\mathrm{m},0} H_0^2}{8\pi G a T_0^4},

where

a

is the reduced Stefan--Boltzmann constant and

T_0

is the present-day temperature of the cosmic microwave background. Show that for our universe

z_{\mathrm{eq}} \approx 5000

. What was the temperature of the CMB radiation at this epoch?