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\begin{document}

\begin{center}
{\bf ASTR 5770 Cosmology Spring 2006.  Problem Set 2.  Due Mon 27 Feb}
\end{center}
\vskip4pt

% The numbering is bad because it produces and uneven distribution of points:
% 1 gets (2.5 + 2.5 + 2.5 + 2.5) + 5 + 5 + 5 = 25
% 2 gets 5
% 3 gets 10
%
% The concept questions are harder than they look.
% They involve real mathematics, not just concepts.

\sect{Temperature}

\subsect{Concept questions}

Explain why each of the following is true:
\begin{enumerate}
\item
A distribution of relativistic particles initially in TE
retains its TE distribution in an FRW universe
as long as the particles remain relativistic,
even if the particles stream freely without interacting.
\item
The comoving entropy density of an FRW universe is constant,
$a^3 s = \mbox{constant}$.
\item
To a good approximation the entropy of our Universe,
to the extent that it is FRW,
is that of relativistic particles.
[Hint:
The entropy per particle
$s/n = ( \rho + {\cal P} - \mu n ) / ( n T )$
is always a number of order unity,
as opposed to being huge or tiny.
For relativistic bosons with zero chemical potential,
the entropy per particle is
$s/n = 2 \, \pi^4 / \left[ 45 \, \zeta(3) \right] = 3.6$,
while for relativistic fermions with zero chemical potential,
the entropy per particle is $7/6$ of this,
$s/n = 7 \, \pi^4 / \left[ 135 \, \zeta(3) \right] = 4.2$.
For non-relativistic Boltzmann species, the entropy per particle is
$s/n = (5/2) + \ln \left\{ g [ m T / (2 \pi) ]^{3/2} / n \right\}$,
which remains modest even if the argument of the logarithm is huge.]
\item
To a good approximation the relativistic particles
that dominate the entropy of an FRW universe
have zero chemical potential.
\end{enumerate}

\subsect{A relation between Fermi-Dirac and Bose-Einstein integrals}

Prove that
\begin{equation}
  \int_0^\infty {x^{n-1} \, dx \over e^x + 1}
  =
  \left( 1 - 2^{1-n} \right)
  \int_0^\infty {x^{n-1} \, dx \over e^x - 1}
  \ .
\end{equation}
[Hint:
Use the fact that
$(e^x + 1) (e^x - 1) = (e^{2x} - 1)$.]

Hence argue that the ratio of energy densities
of relativistic fermionic ($F$) to relativistic bosonic ($B$) species
at the same temperature is
\begin{equation}
  {\rho_F \over \rho_B} = \frac{7}{8}
  \ .
\end{equation}

\subsect{Neutrino temperature}

Weak interactions were fast enough to keep neutrinos
in TE with photons, electrons, and positrons
up to just before $e \bar{e}$ annihilation,
but then neutrinos decoupled.
Argue that conservation of comoving entropy implies
\begin{eqnarray}
  a^3 T^3 \, \Bigl( g_\gamma + \frac{7}{8} g_e \Bigr)
  &=&
  T_\gamma^3 \, g_\gamma
\\
  a^3 T^3 \, g_\nu
  &=&
  T_\nu^3 \, g_\nu
\end{eqnarray}
where the left hand sides refer to quantities
before $e \bar{e}$ annihilation,
and the right hand sides to present day quantities.
Deduce the ratio of neutrino to photon temperatures at the present time
\begin{equation}
  {T_\nu \over T_\gamma}
  \ .
\end{equation}
[Hint:
The proper entropy density of each relativistic species is
$s = ( \rho + {\cal P} ) / T = (4/3) \rho / T$.]

\subsect{Temperature}

Use entropy conservation to argue that the ratio of
the photon temperature $T$
in the early Universe to the photon (CMB) temperature $T_\gamma$ today is
\begin{equation}
  {T \over T_\gamma}
  =
  ( 1 + z ) \left( {g_\circledast \over g_\ast} \right)^{1/3}
\end{equation}
where
\begin{equation}
  g_\ast = g_B + \frac{7}{8} g_F
\end{equation}
was the effective number of relativistic species.
What is the arithmetic value of $g_\circledast$,
the effective number of relativistic species at the present time?
[Hint:
Don't forget to take into account the fact that
the neutrino temperature today differs from the photon temperature.]

\sect{Maximum mass of the dark matter particle}

Dodelson Ch.~3 Exercise 13.
Because of unitarity,
the annihilation cross-section of a particle $X$ of mass $m_X$
must satisfy $\lambda \la 1 / m_X^2$.
The observational constraint $\Omega_X \la 0.3$
imposes a maximum possible mass on the dark matter particle.
Approximately what is this maximum mass?
Express your answer in eV units
(i.e.~keV, MeV, GeV, TeV, as appropriate).
You may quote (a corrected version of) Dodelson equation~(3.60)
\begin{equation}
  \Omega_X h^2 =
  {0.331 \times 10^{-41} \  {\rm m^2} \over \lambda}
  \left( {a_f \over 10} \right)
  \left( {g_\ast \over 100} \right)^{-1/2}
  \ .
\end{equation}

\sect{Boltzmann equation}

The proper number density $n$ of a species
at time $t$ and comoving position $\bx$ in an FRW universe
is the integral of the occupation number $f$ over proper momenta $\bp$
\begin{equation}
  n(t,x) = \int f(t,x,p) \, {d^3 p \over (2 \pi)^3}
  \ .
\end{equation}
Integrate
\begin{equation}
  {d f \over d t}
  =
  {\partial f \over \partial t}
  +
  \dot \bx .
  {\partial f \over \partial \bx}
  +
  \dot \bp .
  {\partial f \over \partial \bp}
\end{equation}
over momenta to show that
\begin{equation}
  \int {d f \over d t} \, {d^3 p \over (2 \pi)^3}
  =
  a^{-3} {d a^3 n \over dt}
  \ .
\end{equation}
[Hint:
In an FRW Universe,
$\partial f / \partial \bx = 0$,
and $\bp \propto a^{-1}$.]

\end{document}
