\documentstyle[12pt]{article}
\headheight 40pt
\headsep 30pt
\textwidth 6.5in
\textheight 8.7in
\oddsidemargin 0pt
\topmargin -45pt
\renewcommand{\baselinestretch}{1.1}
%\pagestyle{empty}
\def\bib{\parskip=0pt\par\noindent\hangindent\parindent
\parskip =2ex plus .5ex minus .1ex}
\newcounter{question}
\newcounter{subquestion}
\renewcommand{\theequation}{\arabic{question}.\arabic{equation}}
\newcommand{\sect}[1]{
\addtocounter{question}{1}
\setcounter{equation}{0}
\setcounter{subquestion}{0}
\vskip6truept
\noindent
{\bf \arabic{question}. #1}
\vskip4truept}
\newcommand{\subsect}[1]{
\addtocounter{subquestion}{1}
\vskip6truept
\noindent
{\bf (\alph{subquestion}) #1}
\vskip4truept}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\nn}{\nonumber \\}
\newcommand{\cm}{{\rm cm}}
\newcommand{\eV}{{\rm eV}}
\newcommand{\gm}{{\rm gm}}
\newcommand{\km}{{\rm km}}
\newcommand{\kms}{{\rm km}\,{\rm s}^{-1}}
\newcommand{\GeV}{{\rm GeV}}
\newcommand{\MeV}{{\rm MeV}}
\newcommand{\Mpc}{{\rm Mpc}}
\begin{document}
% Include Rokicki's epsf.sty file for Encapsulated PostScript graphics
\input{epsf.sty}
\vskip6truept
\begin{center}
{\bf ASTR 5720 Galaxies \& Cosmology Fall 1997. Problem Set 4.\\
Due Tue 28 Oct}
\end{center}
\vskip4truept
\sect{Horizon Problem}
{\it Estimate} approximately how many $e$-foldings of inflation are required to
solve the horizon problem, that is, to ensure that regions of the CMB
$180^\circ$ apart today were once in causal contact.
Your answer will depend on the characteristic energy scale at which inflation
occured.
Evaluate your answer for the case where the energy scale is the Planck energy
$10^{19} \, \GeV$.
[Hint:
Assume the Universe is matter-dominated now, and was radiation-dominated
before $z_{{\rm eq}}$, the redshift of matter-radiation equality,
which you know from Problem Set 2.]
\vskip6truept
\sect{Free Streaming of Neutrinos}
\begin{itemize}
\item[(a)]
Do massive neutrinos become nonrelativistic when the Universe is
matter-dominated or radiation-dominated?
Consider in particular the astrophysically interesting case of a
Mixed Dark Matter (MDM) Universe with $\Omega_M = 1$ and
$\Omega_\nu \approx 0.25$ in one massive neutrino type
(the tau neutrino, presumably).
\item[(b)]
Derive an expression,
as an integral involving conformal time $\tau$ and cosmic scale factor $a$,
for the comoving distance travelled by a massive, freely streaming neutrino
(which may be relativistic, nonrelativistic, or anything in between).
[Hint:
The proper velocity $v$ of a particle,
measured in the comoving frame, is $v/c = dx/d\tau$ (why?).
Recall that in general the proper momentum $p \equiv m v \gamma$
(where $\gamma \equiv [1 - (v/c)^2]^{-1/2}$ is the Lorentz factor)
of a freely streaming particle, measured in the comoving frame,
declines as $p \propto a^{-1}$.]
\item[(c)]
Evaluate the comoving distance in (b) for the case of a massive neutrino
of the kind given by (a).
Assume a neutrino with typical initial momentum $p c \approx kT_\nu$,
assume a flat, radiation-dominated Universe for simplicity,
and truncate the integral where the Universe becomes matter-dominated
(are these approximations reasonable?).
Express your answer in today's units of $h^{-1} \Mpc$
(assume the Universe today is matter-dominated).
\end{itemize}
\sect{Recombination}
\begin{itemize}
\item[(a)]
Write down the Saha equation which relates the number densities of
electrons, protons, and neutral H atoms
in a neutral gas of these particles plus photons in thermodynamic equilibrium.
Assume for simplicity that all H atoms are in their ground states,
and ignore He.
Hence obtain an expression for the ionization fraction $x \equiv n_p / n_b$,
where $n_b = n_p + n_{{\rm H}}$ is the baryon density.
Write your answer in terms of temperature $T$ and the baryon-to-photon ratio
$\eta \equiv n_b / n_\gamma$.
\item[(b)]
At what redshift $z_R$ is the ionization fraction $x = 0.1$,
which, roughly speaking, defines the epoch of Recombination?
For definiteness, take $\Omega_b h^2 = 0.01$
(the preferred primordial nucleosynthesis value).
\item[(c)]
Is the assumption of thermodynamic equilibrium during recombination justified?
The recombination rate
(recombinations per proton per unit time)
is $\Gamma = n_e \langle \sigma v \rangle$,
where $n_e$ is the proper electron density, and
the thermally averaged recombination cross-section is
$\langle \sigma v \rangle \approx
1.4 \times 10^{-13} (T/ 1\, \eV)^{-1/2} \cm^3 \, {\rm s}^-1$.
Determine the ionization fraction $x$ and temperature $T$
at which recombination freezes out,
when the recombination rate equals the expansion rate $\Gamma = H$.
[Hint:
You should find $x \ll 1$,
in which case your expression for $x$ from (a) simplifies.]
\end{itemize}
\vskip6truept
\noindent
{\bf The last question is NOT FOR CREDIT}
\vskip4truept
\sect{Neutron-Proton Ratio}
Write down the equations which govern the evolution of the numbers
of neutrons and protons at temperatures $\sim 10$-$0.1 \, \MeV$ in the early
Universe.
Assume that the relevant processes are the weak interaction processes
\be
n + \nu_e \leftrightarrow p + e
\ , \quad
n + \bar e \leftrightarrow p + \bar\nu_e
\ .
\ee
Derive integral expressions for the form of the relevant cross-sections,
which are related by unitarity to the neutron decay rate
\be
n \rightarrow p + e + \bar \nu_e
\ee
with a measured $e$-folding life of $887 \pm 2$ seconds in a vacuum.
The neutron proton mass difference is $m_n - m_p = 1.293 \, \MeV$.
[Hint:
This problem is covered in Kolb \& Turner (1990) ``The Early Universe'', \S4.2.
Assume that the $n$ and $p$'s are all in their ground states,
and are infinitely massive (hence no recoil in an interaction)
compared to the $e$, $\bar e$, $\nu_e$, and $\bar\nu_e$'s,
which are in thermodynamic equilibrium.
Don't forget to include the relevant phase space blocking factors
$1 - {\cal N}$ in the final states of the light fermions.]
\end{document}