Physics 310 – Cosmology
Homework Set R
1. For each of the following, estimate the thermal energy kBT of the universe. Use
geff  3.36 .
(a) When primordial tritium decays (t = 17.8 y).
(b) When primordial free neutrons decay (t = 886 s).
2. For each of the following, find geff , and estimate the age of the universe in seconds.
(a) At nucleosynthesis, when k BT  80 keV .
(b) When the thermal energy is the same as the electron rest energy, k BT  mc 2 . All
particles are at the same temperature. In addition to photons and neutrinos, there
are also electrons and positrons (g = 4 extra fermions).
(c) At the electroweak scale, k BT  100 GeV . At this time, everything is at the same
temperature, and there are g = 27 total spin states for bosons and g = 90 total spin
states for fermions.
3. Electrons and their anti-particles can be created from photon collisions,   e  e ,
with a cross section of approximately
  c 
 

 E 
2
1
where   137
is the fine structure constant, and E is the typical energy of the two
photons. However, this process only occurs if the energy is sufficient to make them,
so that E  mc 2 . The real formula is more complicated, but this is a good
approximation.
(a) Assume the photons have a typical energy E  3k BT . What is the minimum
temperature required to start pair creating?
(b) What is the age of the universe at this time, in s? Use geff  3.36 .
(c) What is the approximate cross-section at this energy?
(d) What is the number density of photons at this time? Use the formula
2  3  k BT 
 k BT 

  0.24 

2
  c 
 c 
3
n
3
(e) What is the collision rate  for photon pairs? Assume the relative velocity is
approximately v  c .
(f) What is the number of collisions t ? Will this process be in thermal
equilibrium?