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 discovered it. The energy density per frequency interval dν of the black body radiation is given by
\[u_{\nu}d\nu = \frac{8\pi h_P \nu^3}{c^3} \frac{1}{e^{\frac{h_P\nu}{k_B T}}-1}d\nu\]
where hP is the Planck constant, kB is the Boltzmann constant, ν is the frequency, and T is the temperature.
(a) How is the equation for uνdν different from the equation for flux in previous worksheets?
(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.)
(c) Remind yourself how the energy density of the radiation dominated universe depends on the scale factor a.
(d) Combine the two results and 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.
(a) How is the equation for uνdν different from the equation for flux given in our previous work- sheets?
In previous worksheets, we dealt with energy per unit of time per unit of area. This equation expresses total energy density in a volume of space.
(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.)
\[u_{\nu}d\nu = \frac{8\pi h_P \nu^3}{c^3} \frac{1}{e^{\frac{h_P\nu}{k_B T}}-1}d\nu\]
We want to integrate this over all frequencies to find total energy density's relation to temperature.
\[\int_0^{\infty} u_{\nu}d\nu = \int_0^{\infty}\frac{8\pi h_P \nu^3}{c^3} \frac{1}{e^{\frac{h_P\nu}{k_B T}}-1}d\nu\]
This may be difficult to integrate, and we only care about our variable of integration, T, and u, so we can drop all other constants.
\[\int_0^{\infty} u_{\nu}d\nu \propto \int_0^{\infty}\frac{\nu^3}{e^{\frac{\nu}{T}}-1}d\nu\]
Now we will ignore the -1 in the denominator. This may seem sketchy, but the use of a Taylor Series approximation allows it.
\[u \propto \int_0^{\infty}\frac{\nu^3}{e^{\frac{\nu}{T}}}d\nu\]
This integral is solvable through integration by parts. Luckily, we have WolframAlpha to take care of that for us.
Our final result is:
\[\boxed{u \propto T^4}\]
(c) Remind yourself how the energy density of the radiation dominated universe depends on the scale factor a.
We solved this on a previous worksheet dealing with Friedmann Applications.
Our result was:
\[u\propto a^{-4}\]
(d) Combine the two results and 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.
This is simple algebra.
\[u\propto a^{-4}\text{, } u \propto T^4\]
\[a^{-4}\propto T^4\]
\[T\propto a^{-1}\]
\[\boxed{T\propto\frac{1}{a}}\]
This makes sense because In the early universe when \(a\) was very small, \(T\) would be very large.
I worked with B. Brzycki, G. Grell, and N. James on this problem.
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