PHY 7097
September 24, 2012
Assignment #5
Reading:
Finish chapter 3 of Mukhanov.
Problems:
Due in class on Monday, 10/1/12.
(1) Recall the chemical potentials of protons, electrons and neutral Hydrogen in the classical regime,
np
e
H
µp = kB T ln 2nQp
,
µe = kB T ln 2nnQe
,
µH = kB T ln 4nnQH
−I ,
where I ≃ 13.6 eV is the atomic binding energy and the quantum concentrations are,
nQp ≡
mp kB T
2πh̄2
3
2
,
nQe ≡
me kB T
2πh̄2
3
2
,
nQH ≡
mH kB T
2πh̄2
3
2
.
Electrical neutrality requires that the number densities of protons and electrons are
equal, np = ne . The other necessary conservation law is provided by the number density
of baryons in protons and Hydrogen, np + nH ≃ 43 × nB . The current number density
of baryons is nB0 ≃ .245 m−3 and its value at redshift z is nB = (1 + z)3 nB0 . The
current photon temperature is T0 ≃ 2.725 K and its value at redshift z is T = (1+z)T0 .
(a) Assume thermal equilibrium (which is not quite valid) to estimate the redshift zrec
at which half of the protons and electrons have combined to form neutral Hydrogen.
(b) What is the ionization fraction ne /(np + nH ) at z = zrec + 50, assuming thermal
equilibrium?
(c) What is the ionization fraction at z = zrec − 50, assuming thermal equilibrium?
(2) Problem 3.8 of Mukhanov.
(3) Problem 3.10 of Mukhanov.