Radiation-Dominated Dynamics of the Universe

Relevant wiki: Cosmology

By the conservation of energy, the total amount of energy due to radiation in a fixed volume must remain constant. Naively, this gives $\rho a^3 = \text{ constant}$ . However, note that the energy of a photon goes as $E = \frac{hc}{\lambda}$ , i.e., inversely proportional to its wavelength. As the universe expands and $a(t)$ increases, $\lambda$ therefore increases as well, redshifting as it is stretched with the expanding universe. Thus the energy density falls by an additional factor of $a(t)$ due to the redshifting of the radiation, plus the factor of $a^3$ due to the volume expansion:

$\rho \propto \frac{1}{a^4}.$

Now one must only solve the first Friedmann equation given $\rho \propto \frac{1}{a^4 (t)}$ , i.e.:

$\left(\frac{\dot{a}(t)}{a(t)}\right)^2 \propto \frac{1}{a^4 (t)} .$

Rearranging, one has

$\dot{a}^2 \propto \frac{1}{a^2} \implies \dot{a} \propto \frac{1}{a}.$

Solving by separation and integration,

$\int a da \propto \int dt \implies a^2 \propto t \implies a(t) \propto t^{1/2},$

as claimed.