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. 3 (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