AST3220 - Project 2 - Solved

In addition to the solution of the analytical part, the deliverables are the following:

•    Your code in a separate file, ready to compile and run

•    All graphs and other output asked for in the following. Note that the plots must be included in your report. It is not enough that your code generates them.

•    Your report can be either a straightforward set of answers to the questions, or in the form of a paper. Either way is fine.

•    The maximum score on the project is 100 points.

In the lectures we looked at the relationship between the neutrino temperature Tν and the photon temperature T after the neutrinos decoupled. Among the assumptions we made, was that the electrons and positrons become non-relativistic as soon as the temperature drops below kBT = mec2, where me is the electron (and positron) mass. This is not strictly correct, and you will in the following look at the relation between Tν and T when the electrons and positrons are treated more accurately. The following expressions for the energy density and pressure of a gas of fermions will be useful:

 

where g is the number of internal degrees of freedom and m is the rest mass of the particle in question. In the lectures we showed that the entropy density is given by

 ,

and that entropy conservation implies a3s(T) = constant.

1)  (10 points) Show that entropy conservation together with the first Friedmann equation

 gives

,

where  and I have defined t(T0) = 0.

2)  (10 points) Show that the total entropy density for photons, electrons and positrons can be written as

 

 q

                                   where Ee =      p2c2 + m2ec4 and

 

(Note that S(x) here has nothing to do with the function we called S when we discussed distances.)

3)  (10 points) I showed in the lectures that after decoupling the neutrinos still fol-

low the Fermi-Dirac distribution with Tν ∝ 1/a. Show that entropy conservation implies

 ,

and that we therefore have

Tν = ATS1/3(x = mec2/kBT),

where A is a constant.

4)  (10 points) What is the relation between T and Tν at very high temperatures (T →

∞)? Use this to find A and show that

  .

The following integral may be useful:

 .

5)  (10 points) Show that the total energy density can be written as

 

where

 .

6)  (10 points) Show that

 

7)  (10 points) Write a code to evaluate S(x) and plot it. Use physical arguments to

find the value of S in the limits x = 0 and  1. Check that your code agrees with these limits.

8)  (30 points) Take T0 = 1011 K. Write a code to evaluate t(T) and Tν(T), and complete the table on the next page.

 

T (K)
Tν/T
t (s)
1011

6 · 1010

2 · 1010

1010

6 · 109

3 · 109

2 · 109 109

3 · 108

108

107

106
1.000
0