General Relativity: An Introduction for Physicists · Problem 15.3

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · radiation-dominated-temperature-evolution

Problem

Show that in the early, radiation-dominated, phase of the universe the temperature

T

of the radiation satisfies the equation

\left(\frac{\dot{T}}{T}\right)^2 = \frac{8\pi G a T^4}{3},

where the dot denotes differentiation with respect to the cosmic time

t

and

a

is the reduced Stefan--Boltzmann constant. Hence show that

T = \left(\frac{3c^2}{32\pi G a}\right)^{1/4} t^{-1/2} \approx 1.5\times 10^{10}\left(\frac{t}{\mathrm{s}}\right)^{-1/2}\mathrm{K},

and that the cosmic time at matter--radiation equality is

t_{\mathrm{eq}} \approx 16\,000

years.