Blog Post 34, Worksheet 11.1, Problem 1

1. Temperature of the Universe. Remember that, although the universe today is dominated by dark energy and matter (including ordinary matter and dark matter), much earlier on it was dominated by radiation. In this exercise we study the temperature evolution of a radiation dominated universe. 

When the electromagnetic wave is in equilibrium with the environment, its spectrum is uniquely determined by the temperature of the equilibrium. This state is called the blackbody radiation. The spectrum is called the Planck spectrum, named after the physicist who discovered it. The energy density per frequency interval dv of the black body radiation is given by \[u_{\nu}d\nu = \frac{8πh_p\nu^3}{c^3}\frac{1}{e^{\frac{h_p\nu}{e_BT}}-1}d\nu\] where \(h_p\) is the Planck constant, \(k_B\) is the Boltzmann constant, \(\nu\) is the frequency, and T is the temperature. 

(a) How is the equation for \(u_{\nu}d\nu\) difference from the equation for flux given in our previous worksheets?

From previous worksheets, we know that the equation for flux as a function of frequency is as below: \[F_{\nu} = \frac{2πh\nu^3}{c^2}\frac{1}{e^{\frac{h\nu}{kT}}-1}\] Therefore, \(u_{\nu}d\nu = \frac{4}{c}F_{\nu}d\nu\).

(b) Integrate the Planck spectrum over the frequency and figure out how the energy density u of the black body radiation depends on temperature T. Namely, figure out the power n in \(u \propto T^n\). (Since only the functional form of T is important here, in this exercise you do not have to figure out the exact value of the T-independent coefficient a). 

In Worksheet 2.1, we integrated \(F_{\nu}(T)\) over the frequency to determine that \(F(T) = \sigma T^4\), or \(F_\nu \propto T^4\). As we have just shown that \(u_\nu \propto F_\nu\), we can therefore conclude that \(u \propto T^4\).

(c) Remind yourself how the energy density of the radiation dominated universe depends on the scale factor a. 

In previous worksheets, we determined that the energy density of the radiation dominated universe changes with the scale factor a by a factor of \(a^{-4}\).

(d) Combine the two results to see how the temperature T of the universe depends on the scale factor a. Explain why this result implies that the early universe is very hot. 

From parts a, b, and c above, we know both of the following: \[u \propto a^{-4}\] \[u \propto T^4\] We can combine these two proportionalities to see that: \[a^{-4} \propto T^4\] \[T \propto \frac{1}{a}\] As the scale factor increases through time with the expansion of the universe, the temperature thus decreases. Therefore, the temperature of the universe increases as you go back in time, and the early universe was very hot.