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\begin{document}

\begin{center}
{\bf ASTR 5770 Cosmology Spring 2006.  Problem Set 3.  Due Mon 13 Mar}
\end{center}
\vskip4pt

\sect{Boltzman equation in the synchronous gauge (Dodelson Ch.~4 Ex.~2)}

\subsect{Equation of motion}
Scalar fluctuations in the synchronous gauge are described by the metric
\begin{eqnarray}
  g_{00} &=& -1
\nonumber
\\
  g_{0i} &=& 0
\nonumber
\\
  g_{ij} &=&
  a^2 \bigl[ \delta_{ij} ( 1 + 2 \Phi ) + 2 \nabla_i \nabla_j h \bigr]
\end{eqnarray}
where $\nabla_i \equiv \partial / \partial x^i$ signifies
the spatial derivative with respect to comoving position $x^i$.
Show that the equation of motion, for either massive or massless particles,
is, to linear order in the small
quantities $\Phi$ and $h$,
\begin{equation}
  {1 \over p} {d p \over d t}
  =
  - \left( 
  {\dot a \over a}
  +
  {\partial \Phi \over \partial t}
  +
  \hat p^i \hat p^j \nabla_i \nabla_j {\partial h \over \partial t}
  \right)
  \ .
\end{equation}
Hint:
Start from the Lagrangian form of the equation of motion
given on page 4.7 of the notes
\begin{equation}
  {d P_\kappa \over d \lambda}
  =
  {P^\mu P^\nu \over 2} {\partial g_{\mu\nu} \over \partial x^\kappa}
  \ .
\end{equation}

\subsect{LHS of the Boltzmann equation}

Argue that the change from conformal Newtonian to synchronous gauge
is accomplished, for either massive or massless particles, by replacing
\begin{equation}
\label{cn2sync}
  {\partial \Phi \over \partial t}
  + {E \hat p^i \over a p} {\partial \Psi \over \partial x^i}
  \rightarrow
  {\partial \Phi \over \partial t}
  +
  \hat p^i \hat p^j \nabla_i \nabla_j {\partial h \over \partial t}
\end{equation}
on the left hand side of the Boltzmann equation.
[The LHS of the Boltzmann equation
is given, in the conformal Newtonian gauge,
by the LHS of Dodelson equation (4.68),
or by the equation on page 4.9 of the notes.]

\subsect{Collision term on the RHS of the Boltzmann equation}

Why is the collision term on the RHS of the Boltzmann equation
not modified by the change from conformal Newtonian to synchronous gauge?

\subsect{Boltzmann equation for photons}

Fourier transform, $\nabla_j \rightarrow i k_j$,
define $\mu \equiv \hat \bk . \hat \bp$,
and convert to conformal time,
$d t \rightarrow a \, d \eta$.
Conclude that the Boltzmann equation for photons is
\begin{equation}
  {\partial \Theta \over \partial \eta}
  +
  i k \mu \Theta
  +
  {\partial \Phi \over \partial \eta}
  -
  k^2 \mu^2
  {\partial h \over \partial \eta}
  =
  n_e \sigma_{\rm T} a
  \left[
  - \Theta + \Theta_0 - \frac{1}{2} \Theta_2 {\cal P}_2 ( \mu )
  + i \mu v_b
  \right]
  \ .
\end{equation}
[Hint:
You are welcome to quote the Boltzmann equation for photons
derived in the conformal Newtonian gauge in the book or in the notes.
All you have to do is apply the replacement of equation~(\ref{cn2sync}).]

\sect{Spherical harmonics of the photon distribution}

Start with the Boltzmann evolution equation for photons
\begin{equation}
  {\partial \Theta \over \partial \eta}
  +
  i k \mu \Theta
  +
  {\partial \Phi \over \partial \eta}
  +
  i k \mu
  \Psi
  =
  n_e \sigma_{\rm T} a
  \left[
  - \Theta + \Theta_0 - \frac{1}{2} \Theta_2 {\cal P}_2 ( \mu )
  + i \mu v_b
  \right]
  \ .
\end{equation}
Turn this into a set of coupled differential equations for the harmonics
$\Theta_\ell(\eta,\mu)$, Dodelson eq.~(4.99), by
integrating over $i^\ell \int {\cal P}_\ell(\mu) \, d o / 4 \pi$
to obtain
\begin{equation}
  {\partial \Theta_\ell \over \partial \eta}
  +
  {k \over 2 \ell + 1}
  \left[ (\ell+1) \Theta_{\ell+1} - \ell \Theta_{\ell-1} \right]
  + \delta_{\ell 0} {\partial \Phi \over \partial \eta}
  - \delta_{\ell 1} k {\Psi \over 3}
  =
  n_e \sigma_{\rm T} a \left(
  - \Theta_\ell
  + \delta_{\ell 0} \Theta_0
  + \delta_{\ell 2} {\Theta_2 \over 10}
  - \delta_{\ell 1} {v_b \over 3}
  \right)
  .
\end{equation}
Hint:
The Lagrange Polynomials ${\cal P}_\ell$ satisfy the orthogonality relation
\begin{equation}
  \int {\cal P}_{\ell}(\mu) {\cal P}_{\ell^\prime}(\mu) {d o \over 4\pi}
  =
  \int_{-1}^1 {\cal P}_{\ell}(\mu) {\cal P}_{\ell^\prime}(\mu) {d \mu \over 2}
  = \delta_{\ell\ell^\prime} {1 \over 2 \ell + 1}
  \ .
\end{equation}
You will need to use relations (C.1) and (C.3) on p.~418 of Dodelson,
\begin{equation}
  {\cal P}_0(\mu) = 1
  \ , \quad
  {\cal P}_1(\mu) = \mu
  \ , \quad
  \mu {\cal P}_\ell(\mu)
  =
  {1 \over 2 \ell + 1} \left[
  (\ell + 1) {\cal P}_{\ell+1}(\mu) + \ell {\cal P}_{\ell-1}(\mu)
  \right]
  \ .
\end{equation}

\sect{WIMP velocity at matter-radiation equality}

\subsect{Redshift of matter-radiation equality}

At what redshift were the energy densities of matter
(non-relativistic particles)
and radiation (relativistic particles) equal?
What was the photon temperature at that time, in ${\rm eV}$?
[Hint:
The energy density in relativistic particles today is
(units energy/volume)
\begin{equation}
  \rho
  =
  g_0 {\pi^2 (k T_0)^4 \over 30 \, c^3 \hbar^3}
  \ .
\end{equation}
You may assume that the effective energy-weighted number
of relativistic species today is
$g_0 = 2 + 6 (7/8) (4/11)^{4/3} = 3.36$.]

\subsect{Neutrino mass}

Observational evidence for neutrino oscillations
indicates that at least one neutrino type
has mass $\ga 0.05 \  {\rm eV}$?
Would such a neutrino be relativistic or non-relativistic today?
Does this change your result for the redshift of matter-radiation equality?

\subsect{WIMP velocity (Dodelson Ch.~4 Ex.~9)}

In a FRW Universe,
the proper momentum of a non-interacting particle evolves as $p \propto a^{-1}$.
It follows that the temperature of
non-relativistic non-interacting particles evolves as $T \propto a^{-2}$,
as opposed to relativistic particles,
whose temperature evolves as $T \propto a^{-1}$.
Estimate roughly
the typical thermal velocity of a non-baryonic dark matter particle
of mass $m_X \approx 100 \  {\rm GeV}$
at the time of matter-radiation equality.

\end{document}
