# Temperature and Time

By

Dr Lim Zhi Han

●

Published 2022-10-09

In this section, we find a relation between the temperature of the Universe and time. We assume here that the Universe started off with a (infinitely) hot Big Bang and time starts ticking at the moment of the Big Bang. In 1948, Ralph Alpher, Hans Bethe and George Gamow published a paper to discuss a model of a hot and dense expanding Universe where a burst of nuclear reactions occured, thus explaining the origins of chemical elements. As an interesting side note, while Alpher did most of the work, his supervisor Gamow included Bethe as a co-author without Bethe’s knowledge, thinking that it is a good idea for a paper on cosmic beginnings be created by \alpha (Alpher), \beta (Bethe) and \gamma (Gamow). The model proposed in the \alpha - \beta -\gamma paper was subsequently corrected and modified but the key ideas remained. The early Universe was hot and dense, comprising of photons and relativistic particles in constant collision with each other. The mean free path of the particles was short and the Universe was in thermodynamic equilibrium. The radiation (photons and relativistic particles) is described by a blackbody spectrum with a characteristic temperature. The energy per unit volume having wavelengths between \lambda and \lambda +d\lambda is given by

\begin{equation}
u_\lambda d\lambda = \frac{8 \pi hc \lambda^{-5}}{\exp(\frac{hc}{\lambda kT})-1}d\lambda
\end{equation}

We had seen in the previous chapter that the energy density of the radiation field scales inversely to the fourth power of the cosmological scale factor.

\begin{equation}
\rho_r \propto S^{-4}
\end{equation}

Thus the radiation energy density present (denoted by a subscript 0) is related to an earlier value by

\begin{equation}
S_0^4 u_0 d \lambda_0 = S^4u_\lambda d\lambda
\end{equation}

Since the wavelength is proportional to the scale factor due to cosmological redshift,

u_0d\lambda_0 = \frac{S^4}{S^4_0} \frac{8 \pi hc S^{-5}S^5_0 \lambda^{-5}_0}{\exp(\frac{S_0hc}{S\lambda_0kT})-1} SS^{-1}_0d \lambda_0

\begin{align}
\hspace*{-0.3cm}= \frac{8 \pi hc / \lambda^5_0}{\exp(\frac{hc}{\lambda_0k} \frac{S_0}{ST})-1}d\lambda_0
\end{align}

which implies that

\begin{equation}
ST = S_0T_0
\end{equation}

or

\begin{equation}
T \propto S^{-1}
\end{equation}

Hence if one solves the Friedmann equation in the radiation era, one will have the relation between temperature and time. For the case of a flat, radiation-dominated Universe¹,

\begin{equation}
S \propto t^{1/2}.
\end{equation}

Therefore

\begin{equation}
T \propto t^{-1/2}.
\end{equation}

The above temperature–time relation holds for closed and open radiation-dominated Universe as well. The interested reader may work this out yourself!

\begin{equation}
\rho_{rad} = \frac{aT^4}{c^2}
\end{equation}

\begin{align}

T(t) \space = \space & \left( \frac{3c^2}{32 \pi Ga} \right) ^{1/4}t^{-1/2} \nonumber \\

 = \space & \space  (1.52 \times 10^{10}\text{Ks}^{1/2})t^{-1/2}
\end{align}