Physics 310 – Cosmology
Homework Set T
1. There is an additional problem with neutrinos as dark matter: It turns out the Pauli
exclusion principle makes it hard to fit them in a galaxy. The computation is a bit
complex, but we can approximate it pretty well as follows:
(a) The local density of dark matter in the neighborhood of the Sun is around 400,000
GeV/c2/m3. For a neutrino of mass 10.6 eV/c 2 , what is the local number density
n of neutrinos?
(b) Suppose I placed each neutrino in a box of volume L3. What would be the size of
the box such that the density of neutrinos would match part (a)?
(c) According to quantum mechanics, to fit a particle in a box of size L would require
it to acquire a momentum of p    L . Work out the corresponding momentum.
The most convenient units for this would be eV c .
(d) What is the corresponding velocity of the neutrinos? Compare to the approximate
escape velocity of a galaxy like ours, probably around 400 km/s.
2. At the time of recombination, k BT  0.256 eV , there will be atoms at about the same
temperature as everything else.
(a) According to thermodynamics, a typical thermal velocity for a non-relativistic
particle is given by 12 mv 2  23 k BT . Estimate the typical velocity of a hydrogen
atom at this time, with an approximate mass of m  939 MeV/c 2 .
(b) Multiply this speed by the age of the universe at this time to get an approximate
distance d that an atom would move at the time of recombination.
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(c) The density of ordinary matter at this time would be about b  b 0 1  z*  .
Calculate the density of ordinary baryons at this time. b 0 can be found in
homework set O.
(d) Find the mass of a sphere of radius d (from part (b)) and density b (from part c)
to get the approximate smallest mass that will not be wiped out by wandering
baryons. Convert to M 