%From: Dr. David Goodstein <dg@caltech.edu>
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\def\lya{Ly$\alpha$~}
\def\Lya{Ly$\alpha$~}
\def\HI{\rm H\,I~}
\def\zr{z_{\rm reion}}
\def\beq{\begin{equation}}
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\def\Omm{{\Omega_m}}
\def\gtrsim{>}
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\def\Ommz{{\Omega_m^{\,z}}}
\def\Omr{{\Omega_r}}
\def\Omk{{\Omega_k}}
\def\Oml{{\Omega_{\Lambda}}}
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\begin{document}
\hfill {\it David Goodstein's Cosmology Book}\bigskip\bigskip

\centerline{\it{\Large {The Reionization of Cosmic Hydrogen}}}
\centerline{\it{\Large {by the First Galaxies}}}
\bigskip\bigskip


\def\Ref#1{\noindent\hangindent=30pt\hangafter=1\nobreak
\frenchspacing #1 \par}
\def\sm{\smallskip}

{\large
% author 1
{\bf Abraham Loeb}

%affiliation 1
Department of Astronomy,
Harvard University, 60 Garden St., Cambridge MA, 02138
}

%{\bf Key Words: dark ages, first stars, high-redshift galaxies, 
%reionization, intergalactic medium}

%\section*{Contents}
%
%\noindent
%1. Introduction
%
%1.1 Observing our past
%
%1.2 The expanding universe
%
%\noindent
%2. Galaxy Formation
%
%2.1. Growth of linear perturbations
%
%2.2. Halo properties
%
%2.3. Formation of the first stars
%
%2.4. Gamma-ray Bursts: probing the first stars one star at a time
%
%2.5. Supermassive black holes
%
%2.6. The epoch of reionization
%
%2.7. Post-reionization suppression of low-mass galaxies
%
%\noindent
%3. Probing the Diffuse Intergalactic Hydrogen
%
%3.1. Lyman-alpha absorption
%
%3.2. 21-cm absorption or emission
%
%\noindent
%4. Conclusions


\section*{Abstract}
Cosmology is by now a mature experimental science.  We are privileged
to live at a time when the story of genesis (how the Universe started and
developed) can be critically explored by direct observations.  Looking deep
into the Universe through powerful telescopes, we can see images of the
Universe when it was younger because of the finite time it takes light to
travel to us from distant sources.

Existing data sets include an image of the Universe when it was 0.4 million
years old (in the form of the cosmic microwave background), as well as
images of individual galaxies when the Universe was older than a billion
years.  But there is a serious challenge: in between these two epochs was a
period when the Universe was dark, stars had not yet formed, and the cosmic
microwave background no longer traced the distribution of matter.  And this
is precisely the most interesting period, when the primordial soup evolved
into the rich zoo of objects we now see.  
%How can astronomers see this dark yet crucial time?

%The situation is similar to having a photo album of a person that contains
%the first ultra-sound image of him/her as an unborn baby and some
%additional photos as a teenager and an adult. We are currently searching
%for the missing pages in between that will tell us how the Universe evolved
%during its infancy to eventually make galaxies like our own Milky-Way.

The observers are moving ahead along several fronts.  The first involves
the construction of large infrared telescopes on the ground and in space,
that will provide us with new photos of the first galaxies.  Current plans
include ground-based telescopes which are 24-42 meter in diameter, and
NASA's successor to the Hubble Space Telescope, called the James Webb Space
Telescope.  In addition, several observational groups around the globe are
constructing radio arrays that will be capable of mapping the
three-dimensional distribution of cosmic hydrogen in the infant
Universe. These arrays are aiming to detect the long-wavelength (redshifted
21-cm) radio emission from hydrogen atoms.
%Coincidentally, this long wavelength (or low frequency) overlaps
%with the band used for radio and television broadcasting, and so these
%telescopes include arrays of regular radio antennas that one can find in
%electronics stores. 
The images from these antenna arrays will reveal how the non-uniform
distribution of neutral hydrogen evolved with cosmic time and eventually
was extinguished by the ultra-violet radiation from the first galaxies.
Theoretical research has focused in recent years on predicting the expected
signals for the above instruments and motivating these ambitious
observational projects.


\section{Introduction}

\subsection{Observing our past}

When we look at our image reflected off a mirror at a distance of 1
meter, we see the way we looked 6.7 nanoseconds ago, the light travel
time to the mirror and back. If the mirror is spaced $10^{19}~{\rm cm}
\simeq 3~$pc away, we will see the way we looked twenty one years
ago. Light propagates at a finite speed, and so by observing distant
regions, we are able to see what the Universe looked like in the past,
a light travel time ago (Figure~\ref{fig:z}). The statistical
homogeneity of the Universe on large scales guarantees that what we
see far away is a fair statistical representation of the conditions
that were present in in our region of the Universe a long time ago.

\begin{figure}
\epsfxsize=7cm \epsfbox{history.eps}
\caption{Cosmic archaeology of the observable volume of the Universe,
in comoving coordinates (which factor out the cosmic expansion). The
outermost observable boundary ($z=\infty$) marks the comoving distance
that light has travelled since the Big Bang.  Future observatories aim
to map most of the observable volume of our Universe, and improve
dramatically the statistical information we have about the density
fluctuations within it. Existing data on the CMB probes mainly a very
thin shell at the hydrogen recombination epoch ($z\sim 10^3$, beyond
which the Universe is opaque), and current large-scale galaxy surveys
map only a small region near us at the center of the diagram. The
formation epoch of the first galaxies that culminated with hydrogen
reionization at a redshift $z\sim 10$ is shaded grey. Note that the
comoving volume out to any of these redshifts scales as the distance
cubed. {\bf Figure credit:} Loeb, A., {\it ``How Did the First Stars and
Galaxies Form?''}, Princeton University Press (2010).}
\label{fig:z}       % Give a unique label
\end{figure}

This fortunate situation makes cosmology an empirical science. We do
not need to guess how the Universe evolved. Using telescopes we can
simply see how it appeared at earlier cosmic times. In principle, this
allows the entire 13.7 billion year cosmic history of our universe to
be reconstructed by surveying the galaxies and other sources of light
to large distances (Figure~\ref{fig:history}). Since a greater
distance means a fainter flux from a source of a fixed luminosity, the
observation of the earliest sources of light requires the development
of sensitive instruments and poses challenges to observers.

\begin{figure}
\epsfxsize=10cm \epsfbox{SciRev2bnd.eps}
\caption{Overview of cosmic history, with the age of the universe
shown on the top axis and the corresponding redshift on the bottom
axis. Yellow represents regions where the hydrogen is ionized, and
gray, neutral regions. Stars form in galaxies located within dark
matter concentrations whose typical mass grows with time, starting
with $\sim 10^5 M_{\odot}$ (red circles) for the host of the first
star, rising to $10^7$--$10^9 M_{\odot}$ (blue circles) for the
sources of reionization, and reaching $\sim 10^{12} M_{\odot}$ (green
circles) for present-day galaxies like our own Milky Way. Astronomers
probe the evolution of the cosmic gas using the absorption of
background light (dotted lines) by atomic hydrogen along the line of
sight. The classical technique uses absorption by the Lyman-$\alpha$
resonance of hydrogen of the light from bright quasars located within
massive galaxies, while a new type of astronomical observation will
use the 21-cm line of hydrogen with the cosmic microwave background as
the background source. {\bf Figure credit:} Barkana, R., \& Loeb, A.,
{\it Rep. Prog. Phys.} {\bf 70}, 627 (2007).}
\label{fig:history}
\end{figure}

As the universe expands, photon wavelengths get stretched as well.
%, so that the spectrum of light we observe today is shifted from the
The factor by which the observed wavelength is increased (i.e. shifted
towards the red) relative to the emitted one is denoted by $(1+z)$, where
$z$ is the cosmological redshift.  Astronomers use the known emission
patterns of hydrogen and other chemical elements in the spectrum of each
galaxy to measure $z$.  This then implies that the universe has expanded by
a factor of $(1+z)$ in linear dimension since the galaxy emitted the
observed light, and cosmologists can calculate the corresponding distance
and cosmic age for the source galaxy. Large telescopes have allowed
astronomers to observe faint galaxies that are so far away that we see them
more than twelve billion years back in time. Thus, we know directly that
galaxies were in existence as early as 500 million years after the Big
Bang, at a redshift of $z \sim 10$ or higher.

We can in principle image the Universe only if it is transparent. Earlier
than $400\,000$ years after the big bang, the cosmic hydrogen was broken
into its constituent electrons and protons (i.e. ``ionized'') and the
Universe was opaque to scattering by the free electrons in the dense
plasma. Thus, telescopes cannot be used to electromagnetically image the
infant Universe at earlier times (or redshifts $\ga 10^3$). The earliest
possible image of the Universe was recorded by the COBE and WMAP
satellites, which measured the temperature distribution of the cosmic
microwave background (CMB) on the sky.
%(Figure~\ref{fig:CMB}).

%\begin{figure}
%\epsfxsize=10cm \epsfbox{wmap.eps}
%\caption{Image of the Universe when it first became transparent, 400
%thousand years after the Big Bang, taken over five years by the {\it
%Wilkinson Microwave Anisotropy Probe} (WMAP) satellite ({\bf Image
%credit:} WMAP/NASA, http://map.gsfc.nasa.gov/).  Slight density
%inhomogeneities at the level of one part in $\sim 10^5$ in the
%otherwise uniform early Universe imprinted hot and cold spots in the
%temperature map of the cosmic microwave background on the sky. The
%fluctuations are shown in units of $\mu$K, with the unperturbed
%temperature being 2.73 K.  The same primordial inhomogeneities seeded
%the large-scale structure in the present-day Universe.  The existence
%of background anisotropies was predicted in a number of theoretical
%papers, three decades before the technology for taking this image
%became available [Sachs, R.K. \& Wolfe, A.M.,  {\it Astrophys. J.}
%\bf{147}, 73 (1967); Rees, M. J. \& Sciama D.W.,  {\it Nature}
%\bf{217}, 511 (1968); Silk, J., {\it Astrophys. J.} \bf{151}, 459
%(1968); Sunyaev, R.A. \& Zeldovich Y.B.,  {\it APSS} \bf{7}, 3 (1970),
%and Peebles, P.J.E. \& Yu, J.T., {\it Astrophys. J.} \bf{162}, 815
%(1970)].}
%\label{fig:CMB}
%\end{figure}

The CMB, the relic radiation from the hot, dense beginning of the universe,
is indeed another major probe of observational cosmology. The universe
cools as it expands, so it was initially far denser and hotter than it is
today. For hundreds of thousands of years the cosmic gas consisted of a
plasma of free protons and electrons, and a slight mix of light nuclei,
sustained by the intense thermal motion of these particles. Just like the
plasma in our own Sun, the ancient cosmic plasma emitted and scattered a
strong field of visible and ultraviolet photons. As mentioned above, about
$400\,000$ years after the Big Bang the temperature of the universe dipped
for the first time below a few thousand degrees Kelvin. The protons and
electrons were now moving slowly enough that they could attract each other
and form hydrogen atoms, in a process known as cosmic recombination. With
the scattering of the energetic photons now much reduced, the photons
continued traveling in straight lines, mostly undisturbed except that
cosmic expansion has redshifted their wavelength into the microwave regime
today. The emission temperature of the observed spectrum of these CMB
photons is the same in all directions to one part in $100\,000$,
%(Figure~\ref{fig:CMB}), 
which reveals that conditions were nearly uniform in the early
universe.

It was just before the moment of cosmic recombination (when matter started
to dominate in energy density over radiation) that gravity started to
amplify the tiny fluctuations in temperature and density observed in the
CMB data. Regions that started out slightly denser than average began to
develop a greater density contrast with time because the gravitational
forces were also slightly stronger than average in these
regions. Eventually, after hundreds of millions of years, the overdense
regions stopped expanding, turned around, and eventually collapsed to make
bound objects such as galaxies. The gas within these collapsed objects
cooled and fragmented into stars. This process, however, would have taken
too long to explain the abundance of galaxies today, if it involved only
the observed cosmic gas. Instead, gravity is strongly enhanced by the
presence of dark matter -- an unknown substance that makes up the vast
majority (83\%) of the cosmic density of matter. The motion of stars and
gas around the centers of nearby galaxies indicates that each is surrounded
by an extended mass of dark matter, and so dynamically-relaxed dark matter
concentrations are generally referred to as ``halos''.

According to the standard cosmological model, the dark matter is cold
(abbreviated as CDM), i.e., it behaves as a collection of collisionless
particles that started out at matter domination with negligible thermal
velocities and have evolved exclusively under gravitational forces. The
model explains how both individual galaxies and the large-scale patterns in
their distribution originated from the small initial density
fluctuations. On the largest scales, observations of the present galaxy
distribution have indeed found the same statistical patterns as seen in the
CMB, enhanced as expected by billions of years of gravitational
evolution. On smaller scales, the model describes how regions that were
denser than average collapsed due to their enhanced gravity and eventually
formed gravitationally-bound halos, first on small spatial scales and later
on larger ones. In this hierarchical model of galaxy formation, the small
galaxies formed first and then merged or accreted gas to form larger
galaxies. At each snapshot of this cosmic evolution, the abundance of
collapsed halos, whose masses are dominated by dark matter, can be computed
from the initial conditions using numerical simulations. The common
understanding of galaxy formation is based on the notion that stars formed
out of the gas that cooled and subsequently condensed to high densities in
the cores of some of these halos.

Gravity thus explains how some gas is pulled into the deep potential wells
within dark matter halos and forms the galaxies. One might naively expect
that the gas outside halos would remain mostly undisturbed. However,
observations show that it has not remained neutral (i.e., in atomic form)
but was largely ionized by the UV radiation emitted by the galaxies. The
diffuse gas pervading the space outside and between galaxies is referred to
as the intergalactic medium (IGM). For the first hundreds of millions of
years after cosmological recombination, the so-called cosmic ``dark ages'',
the universe was filled with diffuse atomic hydrogen.  As soon as galaxies
formed, they started to ionize diffuse hydrogen in their vicinity. Within
less than a billion years, most of the IGM was re-ionized.  We have not yet
imaged the cosmic dark ages before the first galaxies had formed.  One of
the frontiers in current cosmological studies aims to study the cosmic
epoch of reionization and the first generation of galaxies that triggered
it.

\subsection{The expanding universe}

The modern physical description of the Universe as a whole can be traced
back to Einstein, who assumed for simplicity the so-called ``cosmological
principle'': that the distribution of matter and energy is homogeneous and
isotropic on the largest scales. Today isotropy is well established for the
distribution of faint radio sources, optically-selected galaxies, the X-ray
background, and most importantly the cosmic microwave background
(hereafter, CMB). The constraints on homogeneity are less strict, but a
cosmological model in which the Universe is isotropic but significantly
inhomogeneous in spherical shells around our special location, is also
excluded.

In General Relativity, the metric for a space-time which is spatially
homogeneous and isotropic is the Friedman-Robertson-Walker metric, which
can be written in the form \beq \label{RW}
ds^2=c^2dt^2-a^2(t)\left[\frac{dr^2}{1-k\,r^2}+r^2
\left(d\theta^2+\sin^2\theta\,d\phi^2\right)\right]\ , \eeq where $c$ is
the speed of light, $a(t)$ is the cosmic scale factor which describes
expansion in time $t$, and $(r,\theta,\phi)$ are spherical comoving
coordinates. The constant $k$ determines the geometry of space; it is
positive in a closed Universe, zero in a flat Universe (Euclidean space),
and negative in an open Universe. Observers at rest remain at rest, at
fixed $(r,\theta,\phi)$, with their physical separation increasing with
time in proportion to $a(t)$. A given observer sees a nearby observer at
physical distance $D$ receding at the Hubble velocity $H(t)D$, where the
Hubble constant at time $t$ is $H(t)=d\,a(t)/dt$. Light emitted by a source
at time $t$ is observed at $t=0$ with a redshift $z=1/a(t)-1$, where we set
$a(t=0) \equiv 1$ for convenience.

The Einstein field equations of General Relativity yield the Friedmann
equation \beq H^2(t)=\frac{8 \pi G}{3}\rho-\frac{k}{a^2}\ , \eeq which
relates the expansion of the Universe to its matter-energy content. For
each component of the energy density $\rho$, with an equation of state
$p=p(\rho)$, the density $\rho$ varies with $a(t)$ according to the
thermodynamic relation \beq d (\rho c^2 r^3)=-p d(r^3)\ . \eeq With the
critical density \beq \rho_c(t) \equiv \frac{3 H^2(t)}{8 \pi G} \eeq
defined as the density needed for $k=0$, we define the ratio of the total
density to the critical density as \beq \Omega \equiv \frac{\rho}{\rho_c}\
. \eeq With $\Omm$, $\Oml$, and $\Omr$ denoting the present contributions
to $\Omega$ from matter (including cold dark matter as well as a
contribution $\Omega_b$ from ordinary matter [``baryons''] made of protons
and neutrons), vacuum energy (cosmological constant), and radiation,
respectively, the Friedmann equation becomes \beq \frac{H(t)}{H_0}= \left[
\frac{\Omm} {a^3}+ \Oml+ \frac{\Omr}{a^4}+ \frac{\Omk}{a^2}\right]\ , \eeq
where we define $H_0$ and $\Omega_0=\Omm+\Oml+\Omr$ to be the present
values of $H$ and $\Omega$, respectively, and we let \beq \Omk \equiv
-\frac{k}{H_0^2}=1-\Omega_m. \eeq In the particularly simple Einstein-de
Sitter model ($\Omm=1$, $\Oml=\Omr=\Omk=0$), the scale factor varies as
$a(t) \propto t^{2/3}$. Even models with non-zero $\Oml$ or $\Omk$ approach
the Einstein-de Sitter scaling-law at high redshift, i.e.\ when $(1+z) \gg
|\Omm^{-1}-1|$ (as long as $\Omr$ can be neglected). In this high-$z$
regime the age of the Universe is
\begin{equation}
t\approx {2\over 3 H_0 \sqrt{\Omega_m}} \left(1+z\right)^{-3/2}\
\approx 10^9 {\rm yr} \left({1+z\over 7}\right)^{-3/2} .
\end{equation}

Recent observations confine the standard set of cosmological parameters to
a relatively narrow range. In particular, we seem to live in a universe
dominated by a cosmological constant ($\Lambda$) and cold dark matter, or
in short a $\Lambda$CDM cosmology (with $\Omk$ so small that it is usually
assumed to equal zero) with an approximately scale-invariant primordial
power spectrum of density fluctuations, i.e., $n \approx 1$ where the
initial power spectrum is $P(k)=\vert \delta_{\bf k} \vert^2\propto k^n$ in
terms of the wavenumber $k$ of the Fourier modes $\delta_{\bf k}$ (see \S
\ref{sec:lin} below). Also, the Hubble constant today is written as
\begin{equation}
H_0=100h \mbox{ km s}^{-1}\mbox{Mpc}^{-1},
\end{equation} 
in terms of $h$, and the overall normalization of the power spectrum
is specified in terms of $\sigma_8$, the root-mean-square amplitude of
mass fluctuations in spheres of radius $8\ h^{-1}$ Mpc. For example,
the best-fit cosmological parameters matching the WMAP data together
with large-scale surveys of galaxies and supernovae are
$\sigma_8=0.81$, $n=0.96$, $h=0.72$, $\Omega_m=0.28$,
$\Omega_\Lambda=0.72$ and $\Omega_b=0.046$.
%
%A different cosmological parameter set, also based on the CMB data together
%with other large-scale structure measurements is: $\sigma_8=0.785$,
%$n=0.957$, $h=0.723$ $\Omega_m=0.253$, $\Omega_\Lambda=0.747$, and
%$\Omega_b=0.0425$. The difference between these two parameter sets roughly
%represents current $1-\sigma$ parameter uncertainties.


\section{Galaxy Formation}

\subsection{Growth of linear perturbations}

\label{sec:lin}

As noted in the Introduction, observations of the CMB show that the
universe at cosmic recombination (redshift $z\sim 10^3$) was remarkably
uniform apart from spatial fluctuations in the energy density and in the
gravitational potential of roughly one part in $\sim 10^5$. The primordial
inhomogeneities in the density distribution grew over time and eventually
led to the formation of galaxies as well as galaxy clusters and large-scale
structure. In the early stages of this growth, as long as the density
fluctuations on the relevant scales were much smaller than unity, their
evolution can be understood with a linear perturbation analysis.

As before, we distinguish between fixed and comoving coordinates. Using
vector notation, the fixed coordinate ${\bf r}$ corresponds to a comoving
position $\xb=\rb/a$. In a homogeneous Universe with density $\rho$, we
describe the cosmological expansion in terms of an ideal pressureless fluid
of particles each of which is at fixed $\xb$, expanding with the Hubble
flow $\vb=H(t) \rb$ where $\vb=d\rb/dt$. Onto this uniform expansion we
impose small perturbations, given by a relative density perturbation \beq
\delta(\xb)=\frac{\rho(\rb)}{\bar{\rho}}-1\ , \eeq where the mean fluid
density is $\bar{\rho}$, with a corresponding peculiar velocity $\ub \equiv
\vb - H \rb$. Then the fluid is described by the continuity and Euler
equations in comoving coordinates: \beqa \frac{\partial
\delta}{\partial t}+\frac{1}{a}{\bf \nabla} \cdot \left[(1+\delta)
\ub\right] &=& 0 \\ \frac{\partial \ub}{\partial t}+H\ub+\frac{1}{a}(\ub
\cdot {\bf \nabla}) \ub&=&-\frac{1}{a}{\bf \nabla}\phi\ .  \eeqa The
potential $\phi$ is given by the Poisson equation, in terms of the density
perturbation: \beq \nabla^2\phi=4 \pi G \bar{\rho} a^2 \delta\ . \eeq This
fluid description is valid for describing the evolution of collisionless
cold dark matter particles until different particle streams cross.  This
``shell-crossing'' typically occurs only after perturbations have grown to
become non-linear, and at that point the individual particle trajectories
must in general be followed. Similarly, baryons can be described as a
pressureless fluid as long as their temperature is negligibly small, but
non-linear collapse leads to the formation of shocks in the gas.

For small perturbations $\delta \ll 1$, the fluid equations can be
linearized and combined to yield \beq \frac{\partial^2\delta}{\partial
t^2}+2 H \frac{\partial\delta}{\partial t}=4 \pi G \bar{\rho} \delta\
. \eeq This linear equation has in general two independent solutions, only
one of which grows with time. Starting with random initial conditions, this
``growing mode'' comes to dominate the density evolution. Thus, until it
becomes non-linear, the density perturbation maintains its shape in
comoving coordinates and grows in proportion to a growth factor $D(t)$. The
growth factor in the matter-dominated era is given by \beq D(t)
\propto \frac{\left(\Oml a^3+\Omk a+\Omm\right)^{1/2}}{a^{3/2}}\int_0^a
\frac{a'^{3/2}\, da'}{\left(\Oml a'^3+\Omk a'+\Omm\right)^{3/2}}\ , \eeq
where we neglect $\Omr$ when considering halos forming in the
matter-dominated regime at $z \ll 10^4$. In the Einstein-de Sitter model
(or, at high redshift, in other models as well) the growth factor is simply
proportional to $a(t)$.

The spatial form of the initial density fluctuations can be described in
Fourier space, in terms of Fourier components \beq \delta_\kb = \int d^3x\,
\delta(x) e^{-i \kb \cdot \xb}\ .\eeq Here we use the comoving wave-vector
$\kb$, whose magnitude $k$ is the comoving wavenumber which is equal to
$2\pi$ divided by the wavelength. The Fourier description is particularly
simple for fluctuations generated by inflation. Inflation generates
perturbations given by a Gaussian random field, in which different
$\kb$-modes are statistically independent, each with a random phase. The
statistical properties of the fluctuations are determined by the variance
of the different $\kb$-modes, and the variance is described in terms of the
power spectrum $P(k)$ as follows: \beq \left<\delta_{\kb} \delta_{{\bf
k'}}^{*}\right>=\left(2 \pi\right)^3 P(k)\, \delta^{(3)} \left(\kb-{\bf
k'}\right)\ , \eeq where $\delta^{(3)}$ is the three-dimensional Dirac
delta function.  The gravitational potential fluctuations are sourced by
the density fluctuations through Poisson's equation.

In standard models, inflation produces a primordial power-law spectrum
$P(k) \propto k^n$ with $n \sim 1$. Perturbation growth in the
radiation-dominated and then matter-dominated Universe results in a
modified final power spectrum, characterized by a turnover at a scale
of order the horizon $cH^{-1}$ at matter-radiation equality, and a
small-scale asymptotic shape of $P(k) \propto k^{n-4}$. The overall
amplitude of the power spectrum is not specified by current models of
inflation, and it is usually set by comparing to the observed CMB
temperature fluctuations or to local measures of large-scale
structure.

Since density fluctuations may exist on all scales, in order to
determine the formation of objects of a given size or mass it is
useful to consider the statistical distribution of the smoothed
density field.  Using a window function $W(\rb)$ normalized so that
$\int d^3r\, W(\rb)=1$, the smoothed density perturbation field, $\int
d^3r \delta(\xb) W(\rb)$, itself follows a Gaussian distribution with
zero mean. For the particular choice of a spherical top-hat, in which
$W=1$ in a sphere of radius $R$ and is zero outside, the smoothed
perturbation field measures the fluctuations in the mass in spheres of
radius $R$. The normalization of the present power spectrum is often
specified by the value of $\sigma_8 \equiv \sigma(R=8 h^{-1} {\rm
Mpc})$. For the top-hat, the smoothed perturbation field is denoted
$\delta_R$ or $\delta_M$, where the mass $M$ is related to the
comoving radius $R$ by $M=4 \pi \rho_m R^3/3$, in terms of the current
mean density of matter $\rho_m$. The variance $\langle \delta_M
\rangle^2$ is \beq \sigma^2(M)= \sigma^2(R)=
\int_0^{\infty}\frac{dk}{2 \pi^2} \,k^2 P(k) \left[\frac{3
j_1(kR)}{kR} \right]^2\ ,\label{eqsigM}\eeq where $j_1(x)=(\sin x-x
\cos x)/x^2$. The function $\sigma(M)$, plotted in Figure \ref{sigM},
plays a crucial role in estimates of the abundance of collapsed
objects.

\begin{figure}
\epsfxsize=7cm \epsfbox{sigM.eps}
\caption{The root-mean-square amplitude of linearly-extrapolated
density fluctuations $\sigma$ as a function of mass $M$ (in solar
masses $M_\odot$, within a spherical top-hat filter) at different
redshifts $z$. Halos form in regions that exceed the background
density by a factor of order unity. This threshold is only surpassed
by rare (many-$\sigma$) peaks for high masses at high redshifts.
{\bf Figure credit:} Loeb, A., {\it ``How Did the First Stars and Galaxies
Form?''}, Princeton University Press (2010).
%When discussing the abundance of halos, we will
%factor out the linear growth of perturbations and use the function
%$\sigma(M)$ at $z=0$.
}
\label{sigM}
\end{figure}

Different physical processes contributed to the perturbation growth. In the
absence of other influences, gravitational forces due to density
perturbations imprinted by inflation would have driven parallel
perturbation growth in the dark matter, baryons and photons. However, since
the photon sound speed is of order the speed of light, the radiation
pressure produced sound waves on a scale of order the cosmic horizon and
suppressed sub-horizon perturbations in the photon density. The baryonic
pressure similarly suppressed perturbations in the gas below the (much
smaller) so-called baryonic {\it Jeans} scale. Since the formation of
hydrogen at recombination had decoupled the cosmic gas from its mechanical
drag on the CMB, the baryons subsequently began to fall into the
pre-existing gravitational potential wells of the dark matter.

Spatial fluctuations developed in the gas temperature as well as in the gas
density. Both the baryons and the dark matter were affected on small scales
by the temperature fluctuations through the gas pressure.  Compton heating
due to scattering of the residual free electrons (constituting a fraction
$\sim10^{-4}$) with the CMB photons remained effective, keeping the gas
temperature fluctuations tied to the photon temperature fluctuations, even
for a time after recombination.  The growth of linear perturbations can be
calculated with the standard CMBFAST code (http://www.cmbfast.org), after a
modification to account for the fact that the speed of sound of the gas
also fluctuates spatially.

%\begin{figure}
%\epsfxsize=10cm \epsfbox{binfall9.eps}
%\caption{Power spectra of density and temperature fluctuations vs.\
%comoving wavenumber, at redshifts 1200, 800, 400, and 200 (from Barkana \&
%Loeb 2007). We consider fluctuations in the CDM density (solid curves),
%baryon density (dotted curves), baryon temperature (short-dashed curves),
%and photon temperature (long-dashed curves).}
%\label{fig:photons}
%\end{figure}

%The magnitude of the fluctuations in the CDM and baryon densities, and in
%the baryon and photon temperatures, is shown in Figure~\ref{fig:photons},
%in terms of the dimensionless combination $[k^3 P(k)/(2 \pi^2)]^{1/2}$,
%where $P(k)$ is the corresponding power spectrum of fluctuations in terms
%of the comoving wavenumber $k$ of each Fourier mode. 
After recombination, two main drivers affect the baryon density and
temperature fluctuations, namely, the thermalization with the CMB and
the gravitational force that attracts the baryons to the dark matter
potential wells. 
%As shown in the figure, 
The density perturbations in
all species grow together on scales where gravity is unopposed,
outside the horizon (i.e., at $k \lesssim 0.01$ Mpc$^{-1}$ at $z \sim
1000$). At $z=1200$ the perturbations in the baryon-photon fluid
oscillate as acoustic waves on scales of order the sound horizon ($k
\sim 0.01~{\rm Mpc^{-1}}$), while smaller-scale perturbations in both
the photons and baryons are damped by photon diffusion and the drag of
the diffusing photons on the baryons. On sufficiently small scales the
power spectra of baryon density and temperature roughly assume the
shape of the dark matter fluctuations (except for the gas-pressure
cutoff at the very smallest scales), due to the effect of
gravitational attraction on the baryon density and of the resulting
adiabatic expansion on the gas temperature. After the mechanical
coupling of the baryons to the photons ends at $z \sim 1000$, the
baryon density perturbations gradually grow towards the dark matter
perturbations because of gravity. Similarly, after the thermal
coupling ends at $z \sim 200$, the baryon temperature fluctuations are
driven by adiabatic expansion towards a value of 2/3 of the density
fluctuations. 
%As the figure shows, 
By $z=200$ the baryon infall into the dark matter potentials is well
advanced and adiabatic expansion is becoming increasingly important in
setting the baryon temperature.


\subsection{Halo properties}

The small density fluctuations evidenced in the CMB grow over time as
described in the previous subsection, until the perturbation $\delta$
becomes of order unity, and the full non-linear gravitational problem
must be considered. A dark matter halo forms in a region where 
matter relaxes dynamically to a make a stable object that is much
denser than the mean cosmic density and is held together by its own
self-gravity.  The dynamical collapse of a dark matter halo can be
solved analytically only in cases of particular symmetry. If we
consider a region which is much smaller than the horizon $cH^{-1}$,
then the formation of a halo can be formulated as a problem in
Newtonian gravity, in some cases with minor corrections coming from
General Relativity. The simplest case is that of spherical symmetry,
with an initial ($t=t_i\ll t_0$) top-hat of uniform overdensity
$\delta_i$ inside a sphere of radius $R$. Although this model is
restricted in its direct applicability, the results of spherical
collapse have turned out to be surprisingly useful in understanding
the properties and distribution of halos in models based on cold dark
matter.

The collapse of a spherical top-hat perturbation is described by the
Newtonian equation (with a correction for the cosmological constant) \beq
\frac{d^2r}{dt^2}=H_0^2 \Oml\, r-\frac{GM}{r^2}\ , \eeq where $r$ is the
radius in a fixed (not comoving) coordinate frame, $H_0$ is the present-day
Hubble constant, $M$ is the total mass enclosed within radius $r$, and the
initial velocity field is given by the Hubble flow $dr/dt=H(t) r$. The
enclosed $\delta$ grows initially as $\delta_L=\delta_i D(t)/D(t_i)$, in
accordance with linear theory, but eventually $\delta$ grows above
$\delta_L$. If the mass shell at radius $r$ is bound (i.e., if its total
Newtonian energy is negative) then it reaches a radius of maximum expansion
and subsequently collapses. As demonstrated in the previous section, at the
moment when the top-hat collapses to a point, the overdensity predicted by
linear theory is $\delta_L\, = 1.686$ in the Einstein-de Sitter model, with
only a weak dependence on $\Omm$ and $\Oml$. Thus a tophat collapses at
redshift $z$ if its linear overdensity extrapolated to the present day
(also termed the critical density of collapse) is \beq \delta_{\rm
crit}(z)=\frac{1.686}{D(z)}\ ,
\label{deltac} \eeq where we set $D(z=0)=1$.

Even a slight violation of the exact symmetry of the initial perturbation
can prevent the tophat from collapsing to a point. Instead, the halo
reaches a state of virial equilibrium by violent relaxation (phase
mixing). Using the virial theorem $U=-2K$ to relate the potential energy
$U$ to the kinetic energy $K$ in the final state (implying that the virial
radius is half the turnaround radius - where the kinetic energy vanishes),
the final overdensity relative to the critical density at the collapse
redshift is $\Delta_c=18\pi^2 \simeq 178$ in the Einstein-de Sitter model,
modified in a Universe with $\Omm+\Oml=1$ to the fitting formula \beq
\Delta_c=18\pi^2+82 d-39 d^2\ , \eeq where $d\equiv \Ommz-1$ is evaluated
at the collapse redshift, so that \beq \Ommz=\frac{\Omm (1+z)^3}{\Omm
(1+z)^3+\Oml+\Omk (1+z)^2}\ .
\label{Ommz} \eeq

\begin{figure}
\epsfysize=7cm \epsfbox{ST1.eps}
\epsfysize=7cm \epsfbox{ST.eps}\\
\caption{{\it Left panel:} The mass fraction incorporated into dark matter
halos per logarithmic bin of halo mass $(M^2dn/dM)/\rho_m$, as a
function of $M$ at different redshifts $z$. Here
$\rho_m=\Omega_m\rho_c$ is the present-day matter density, and
$n(M)dM$ is the comoving density of halos with masses between $M$ and
$M+dM$. The halo mass distribution was calculated based on an improved
version of the Press-Schechter formalism for ellipsoidal collapse
[Sheth, R. K., \& Tormen, G. {\it Mon. Not. R. Astron. Soc.} {\bf
329}, 61 (2002)] that fits better numerical simulations. {\it Right panel:}
Number density of halos per logarithmic bin of halo mass, $Mdn/dM$ (in
units of comoving Mpc$^{-3}$), at various redshifts.  
{\bf Figure credit:} Loeb, A., {\it ``How Did the First Stars and Galaxies
Form?''}, Princeton University Press (2010).}
\label{ST}
\end{figure}

A halo of mass $M$ collapsing at redshift $z$ thus has a virial radius \beq
r_{\rm vir}=0.784 \left(\frac{M}{10^8\ h^{-1} \ M_{\sun} }\right)^{1/3}
\left[\frac{\Omm} {\Ommz}\ \frac{\Delta_c} {18\pi^2}\right]^{-1/3} \left
(\frac{1+z}{10} \right)^{-1}\ h^{-1}\ {\rm kpc}\ , \label{rvir}\eeq and a
corresponding circular velocity, \beq V_c=\left(\frac{G M}{r_{\rm
vir}}\right)^{1/2}= 23.4 \left( \frac{M}{10^8\ h^{-1} \ M_{\sun}
}\right)^{1/3} \left[\frac {\Omm} {\Ommz}\ \frac{\Delta_c}
{18\pi^2}\right]^{1/6} \left( \frac{1+z} {10} \right)^{1/2}\ {\rm km\
s}^{-1}\ . \label{Vceqn} \eeq In these expressions we have assumed a
present Hubble constant written in the form $H_0=100\, h\mbox{ km
s}^{-1}\mbox{Mpc}^{-1}$. We may also define a virial temperature \beq
\label{tvir} T_{\rm vir}=\frac{\mu m_p V_c^2}{2 \kB}=1.98\times 10^4\
\left(\frac{\mu}{0.6}\right) \left(\frac{M}{10^8\ h^{-1} \ M_{\sun}
}\right)^{2/3} \left[ \frac {\Omm} {\Ommz}\ \frac{\Delta_c}
{18\pi^2}\right]^{1/3} \left(\frac{1+z}{10}\right)\ {\rm K} \ , \eeq where
$\mu$ is the mean molecular weight and $m_p$ is the proton mass. Note that
the value of $\mu$ depends on the ionization fraction of the gas; for a
fully ionized primordial gas $\mu=0.59$, while a gas with ionized hydrogen
but only singly-ionized helium has $\mu=0.61$. The binding energy of the
halo is approximately\footnote{The coefficient of $1/2$ in
equation~(\ref{Ebind}) would be exact for a singular isothermal sphere with
$\rho(r)\propto 1/r^2$.} \beq \label{Ebind} E_b= {1\over 2}
\frac{GM^2}{r_{\rm vir}} = 5.45\times 10^{53} \left(\frac{M}{10^8\ h^{-1} \
M_{\sun} }\right)^{5/3} \left[ \frac {\Omm} {\Ommz}\ \frac{\Delta_c}
{18\pi^2}\right]^{1/3} \left(\frac{1+z}{10}\right) h^{-1}\ {\rm erg}\
. \eeq Note that the binding energy of the baryons is smaller by a factor
equal to the baryon fraction $\Omega_b/\Omm$.

Although spherical collapse captures some of the physics governing the
formation of halos, structure formation in cold dark matter models
proceeds hierarchically. At early times, most of the dark matter is in
low-mass halos, and these halos continuously accrete and merge to form
high-mass halos (see Figure \ref{ST}). Numerical simulations of
hierarchical halo formation indicate a roughly universal
spherically-averaged density profile for the resulting halos, though
with considerable scatter among different halos. The typical profile
has the form \beq \rho(r)=\frac{3 H_0^2} {8 \pi G} (1+z)^3
\frac{\Omm}{\Ommz} \frac{\delta_c} {\cN x (1+\cN x)^2}\ ,
\label{NFW} \eeq where $x=r/r_{\rm vir}$, and the characteristic density
$\delta_c$ is related to the concentration parameter $\cN$ by \beq
\delta_c=\frac{\Delta_c}{3} \frac{\cN^3} {\ln(1+\cN)-\cN/(1+\cN)} \ . \eeq
The concentration parameter itself depends on the halo mass $M$, at a given
redshift $z$.

\subsection{Formation of the first stars}

Theoretical expectations for the properties of the first galaxies are based
on the standard cosmological model outlined in the Introduction. The
formation of the first bound objects marked the central milestone in the
transition from the initial simplicity (discussed in the previous
subsection) to the present-day complexity. Stars and accreting black holes
output copious radiation and also produced explosions and outflows that
brought into the IGM chemical products from stellar nucleosynthesis and
enhanced magnetic fields. However, the formation of the very first stars,
in a universe that had not yet suffered such feedback, remains a
well-specified problem for theorists.

%\begin{figure}
%%\epsfxsize=5cm \epsfbox{Fig2a.eps,Fig2b.eps}
%\epsfxsize=7cm \epsfbox{Fig2b.eps}
%\caption{Collapse and fragmentation of a primordial cloud of gas (from
%Bromm \& Loeb 2004).  Shown is the projected gas density at a redshift
%$z\simeq 21.5$, briefly after gravitational runaway collapse has commenced
%in the center of the cloud.  
%%{\it Top:} The coarse-grained morphology in a
%%box with linear physical size of 23.5~pc.  At this time in the unrefined
%%simulation, a high-density clump has formed with an initial mass of $\sim
%%10^{3}M_{\odot}$.  {\it Right:} 
%The refined morphology is plotted in a simulation box with linear
%physical size of 0.5~pc.  The central density peak, vigorously gaining mass
%by accretion, defines the seed of a metal-free (Population III) star.
%Searches for metal-poor stars are underway in the halo of the Milky Way
%galaxy, an environment less crowded by metal-rich (Population I,II) stars
%than the core of our galaxy. The goal of these searches is to constrain
%theoretical calculations (such as the one shown here) for the formation of
%the first stars.  }
%\label{2ab}
%\end{figure}

Stars form when large amounts of matter collapse to high
densities. However, the process can be stopped if the pressure exerted by
the hot intergalactic gas prevents outlying gas from falling into dark
matter concentrations. As the gas falls into a dark matter halo, it forms
shocks due to converging supersonic flows and in the process heats up and
can only collapse further by first radiating its energy away. This
restricts this process of collapse to very large clumps of dark matter that
are around $100\,000$ times the mass of the Sun. Inside these clumps, the
shocked gas loses energy by emitting radiation from excited molecular
hydrogen that formed naturally within the primordial gas mixture of
hydrogen and helium.

The first stars are expected to have been quite different from the stars
that form today in the Milky Way. The higher pressure within the primordial
gas due to the presence of fewer cooling agents suggests that fragmentation
only occurred into relatively large units, in which gravity could overcome
the pressure. Due to the lack of carbon, nitrogen, and oxygen -- elements
that would normally dominate the nuclear energy production in modern
massive stars -- the first stars must have condensed to extremely high
densities and temperatures before nuclear reactions were able to heat the
gas and balance gravity. These unusually massive stars produced high
luminosities of UV photons, but their nuclear fuel was exhausted after 2--3
million years, resulting in a huge supernova or in collapse to a black
hole. The heavy elements which were dispersed by the first supernovae in
the surrounding gas, enabled the enriched gas to cool more effectively and
fragment into lower mass stars. Simple calculations indicate that a carbon
or oxygen enrichment of merely $\la 10^{-3}$ of the solar abundance is
sufficient to allow solar mass stars to form. These second-generation
``low-metallicity'' stars are long-lived and could in principle be
discovered in the halo of the Milky Way galaxy, providing fossil record of
the earliest star formation episode in our cosmic environment.

Advances in computing power have made possible detailed numerical
simulations of how the first stars formed. These simulations begin in
the early universe, in which dark matter and gas are distributed
uniformly, apart from tiny variations in density and temperature that
are statistically distributed according to the patterns observed in
the CMB. In order to span the vast range of scales needed to simulate
an individual star within a cosmological context, the adopted codes
zoom in repeatedly on the densest part of the first collapsing cloud
that is found within the simulated volume. The simulation follows
gravity, hydrodynamics, and chemical processes in the primordial gas,
and resolves a scale that is $>10$ orders of magnitudes smaller than
that of the simulated box.  In state-of-the-art simulations, the
resolved scale is approaching the scale of the proto-star.  The
simulations have established that the first stars formed within halos
containing $\sim 10^5 M_{\odot}$ in total mass, and indicate that the
first stars most likely weighed tens to hundreds of solar masses each.

To estimate {\it when}\/ the first stars formed we must remember that the
first $100\,000$ solar mass halos collapsed in regions that happened to
have a particularly high density enhancement very early on. There was
initially only a small abundance of such regions in the entire universe, so
a simulation that is limited to a small volume is unlikely to find such
halos until much later. Simulating the entire universe is well beyond the
capabilities of current simulations, but analytical models predict that the
first observable star in the universe probably formed 30 million years
after the Big Bang, less than a quarter of one percent of the Universe's
total age of 13.7 billion years.

\begin{figure}
\epsfxsize=7cm \epsfbox{jwst.ps}
\caption{A full scale model of the {James Webb Space Telescope}
(JWST), the successor to the {Hubble Space Telescope}.  JWST includes
a primary mirror 6.5 meters in diameter, and offers instrument
sensitivity across the infrared wavelength range of 0.6--28$\mu$m
which will allow detection of the first galaxies.  The size of the Sun
shield (the large flat screen in the image) is 22 meters$\times$10
meters (72 ft$\times$29 ft).  The telescope will orbit 1.5 million
kilometers from Earth at the Lagrange L2 point.  {\bf Image credit:}
JWST/NASA (http://www.jwst.nasa.gov/).}
\label{jwst}
\end{figure}

Although stars were extremely rare at first, gravitational collapse
increased the abundance of galactic halos and star formation sites with
time (Figure~\ref{fig:history}). Radiation from the first stars is expected
to have eventually dissociated all the molecular hydrogen in the
intergalactic medium, leading to the domination of a second generation of
larger galaxies where the gas cooled via radiative transitions in atomic
hydrogen and helium. Atomic cooling occurred in halos of mass above
$\sim10^8 M_{\odot}$, in which the infalling gas was heated above 10,000 K
and became ionized. The first galaxies to form through atomic cooling are
expected to have formed around redshift 45, and such
galaxies were likely the main sites of star formation by the time
reionization began in earnest. As the IGM was heated above 10,000 K by
reionization, its pressure jumped and prevented the gas from accreting into
newly forming halos below $\sim10^9 M_{\odot}$. The first Milky-Way-sized
halo $M = 10^{12} M_{\odot}$ is predicted to have formed 400 million years
after the Big Bang, but such halos have become typical
galactic hosts only in the last five billion years.

Hydrogen is the most abundant element in the Universe, The prominent
Lyman-$\alpha$ spectral line of hydrogen (corresponding to a transition
from its first excited level to its ground state) provides an important
probe of the condensation of primordial gas into the first galaxies.
Existing searches for Lyman-$\alpha$ emission have discovered galaxies
robustly out to a redshift $z\sim 7$ with some unconfirmed candidate
galaxies out to $z\sim 10$. The spectral break owing to Lyman-$\alpha$
absorption by the IGM allows to identify high-redshifts galaxies
photometrically.  Existing observations provide only a preliminary glimpse
into the formation of the first galaxies.

Within the next decade, NASA plans to launch an infrared space telescope
({\it JWST}\/; Figure~\ref{jwst}) that will image some of the earliest
sources of light (stars and black holes) in the Universe. In parallel,
there are several initiatives to construct large-aperture infrared
telescopes on the ground with the same goal in mind.

The next generation of ground-based telescopes will have a diameter of
twenty to thirty meters (Figure~\ref{gmt}). Together with {\it JWST}\/
(which will not be affected by the atmospheric background) they will be
able to image and make spectral studies of the early galaxies. Given that
these galaxies also create the ionized bubbles around them by their UV
emission, during reionization the locations of galaxies should correlate
with bubbles within the neutral hydrogen.  Within a decade it should be
possible to explore the environmental influence of individual galaxies by
using these telescopes in combination with 21-cm probes of reionization.

\begin{figure}
\epsfysize=3.5cm \epsfbox{elt.ps}
\epsfysize=3.5cm \epsfbox{gmt.ps}
\epsfysize=3.5cm \epsfbox{tmt.ps}\\
\caption{Artist's conception of the designs for three future giant
telescopes that will be able to probe the first generation of galaxies
from the ground: the {European Extremely Large Telescope} (EELT, left),
the {Giant Magellan Telescope} (GMT, middle), and the {Thirty Meter
Telescope} (TMT, right).  {\bf Image credits:}
http://www.eso.org/sci/facilities/eelt/, http://www.gmto.org/, and
http://www.tmt.org/.  }
\label{gmt}
\end{figure}

\subsection{Gamma-ray Bursts: probing the first stars one 
star at a time}

So far, to learn about diffuse IGM gas pervading the space outside and
between galaxies, astronomers routinely study its absorption signatures in
the spectra of distant quasars, the brightest long-lived astronomical
objects. Quasars' great luminosities are believed to be powered by
accretion of gas onto black holes weighing up to a few billion times the
mass of the Sun that are situated in the centers of massive galaxies. As
the surrounding gas spirals in toward the black hole sink, the viscous
dissipation of heat makes the gas glow brightly into space, creating a
luminous source visible from afar.

Over the past decade, an alternative population of bright sources at
cosmological distances was discovered, the so-called afterglows of {\it
Gamma-Ray Bursts} (GRBs). These events are characterized by a flash of
high-energy ($>0.1$ MeV) photons, typically lasting 0.1--100 seconds, which
is followed by an afterglow of lower-energy photons over much longer
timescales. The afterglow peaks at X-ray, UV, optical and eventually radio
wavelengths on time scales of minutes, hours, days, and months,
respectively.  The central engines of GRBs are believed to be associated
with the compact remnants (neutron stars or stellar-mass black holes) of
massive stars. Their high luminosities make them detectable out to the edge
of the visible Universe. GRBs offer the opportunity to detect the most
distant (and hence earliest) population of massive stars, the so-called
Population~III (or Pop~III), one star at a time. In the
hierarchical assembly process of halos that are dominated by cold dark
matter (CDM), the first galaxies should have had lower masses (and lower
stellar luminosities) than their more recent counterparts. Consequently,
the characteristic luminosity of galaxies or quasars is expected to decline
with increasing redshift. GRB afterglows, which already produce a peak flux
comparable to that of quasars or starburst galaxies at $z\sim 1-2$, are
therefore expected to outshine any competing source at the highest
redshifts, when the first dwarf galaxies formed in the Universe.

%\begin{figure}
%\epsfxsize=10cm \epsfbox{GRB.ps}
%\caption{Illustration of a long-duration gamma-ray burst in the popular
%``collapsar'' model. 
%(from NASA E/PO)
%The collapse of the core of a massive star (which lost its hydrogen
%envelope) to a black hole generates two opposite jets moving out at a speed
%close to the speed of light. The jets drill a hole in the star and shine
%brightly towards an observer who happens to be located within with the
%collimation cones of the jets. The jets emanating from a single massive
%star are so bright that they can be seen across the Universe out to the
%epoch when the first stars formed (for more information see
%http://swift.gsfc.nasa.gov/). }
%\label{grb}
%\end{figure}

GRBs, the electromagnetically-brightest explosions in the Universe, should
be detectable out to redshifts $z>10$. High-redshift GRBs can be identified
through infrared photometry, based on the Lyman-$\alpha$ break induced by
absorption of their spectrum at wavelengths below $1.216\, \mu {\rm m}\,
[(1+z)/10]$. Follow-up spectroscopy of high-redshift candidates can then be
performed on a 10-meter-class telescope. GRB afterglows offer the
opportunity to detect stars as well as to probe the metal enrichment level
of the intervening IGM. Recently, the {\it Swift} satellite has
detected a GRB originating at $z\simeq 8.2$, thus demonstrating the
viability of GRBs as probes of the early Universe.

Another advantage of GRBs is that the GRB afterglow flux at a given
observed time lag after the $\gamma$-ray trigger is not expected to fade
significantly with increasing redshift, since higher redshifts translate to
earlier times in the source frame, during which the afterglow is
intrinsically brighter. For standard afterglow
lightcurves and spectra, the increase in the luminosity distance with
redshift is compensated by this cosmological time-stretching effect
as shown in Figure~\ref{fig:GRB}.

\begin{figure}
\epsfxsize=8cm \epsfbox{fig1val.eps}
\caption{GRB afterglow flux as a function of time since the
$\gamma$-ray trigger in the observer frame. The flux (solid curves) is
calculated at the redshifted Lyman-$\alpha$ wavelength. The dotted
curves show the planned detection threshold for the {\it James Webb
Space Telescope} ({\it JWST}), assuming a spectral resolution $R=5000$
with the near infrared spectrometer, a signal to noise ratio of 5 per
spectral resolution element, and an exposure time equal to $20\%$ of
the time since the GRB explosion.  Each set of curves shows a sequence
of redshifts, namely $z=5$, 7, 9, 11, 13, and 15, respectively, from
top to bottom. {\bf Figure credit:} Barkana, R., \& Loeb, A., {\it
Astrophys. J.} \bf{601}, 64 (2004).}
\label{fig:GRB}
\end{figure}

GRB afterglows have smooth (broken power-law) continuum spectra unlike
quasars which show strong spectral features (such as broad emission lines
or the so-called ``blue bump'') that complicate the extraction of IGM
absorption features. In particular, the extrapolation into the spectral
regime marked by the IGM Lyman-$\alpha$ absorption during the epoch of
reionization is much more straightforward for the smooth UV spectra of GRB
afterglows than for quasars with an underlying broad Lyman-$\alpha$
emission line. However, the interpretation may be complicated by the
presence of damped Lyman-$\alpha$ absorption by dense neutral hydrogen in
the immediate environment of the GRB within its host galaxy. Since GRBs
originate from the dense environment of active star formation, such damped
absorption is expected and indeed has been seen, including in the most
distant GRB at $z=8.2$.


\subsection{Supermassive black holes}

The fossil record in the present-day Universe indicates that every bulged
galaxy hosts a supermassive black hole (BH) at its center. This conclusion
is derived from a variety of techniques which probe the dynamics of stars
and gas in galactic nuclei.  The inferred BHs are dormant or faint most of
the time, but occasionally flash in a short burst of radiation that lasts
for a small fraction of the age of the Universe. The short duty cycle
accounts for the fact that bright quasars are much less abundant than their
host galaxies, but it begs the more fundamental question: {\it why is the
quasar activity so brief?}  A natural explanation is that quasars are
suicidal, namely the energy output from the BHs regulates their own growth.

Supermassive BHs make up a small fraction, $< 10^{-3}$, of the total mass
in their host galaxies, and so their direct dynamical impact is limited to
the central star distribution where their gravitational influence
dominates. Dynamical friction on the background stars keeps the BH close to
the center. Random fluctuations in the distribution of stars induces a
Brownian motion of the BH. This motion can be described by the same Langevin
equation that captures the motion of a massive dust particle as it responds
to random kicks from the much lighter molecules of air around it.  The
characteristic speed by which the BH wanders around the center is small,
$\sim (m_\star/M_{\rm BH})^{1/2}\sigma_\star$, where $m_\star$ and $M_{\rm
BH}$ are the masses of a single star and the BH, respectively, and
$\sigma_\star$ is the stellar velocity dispersion. Since the random force
fluctuates on a dynamical time, the BH wanders across a region that is
smaller by a factor of $\sim (m_\star/M_{\rm BH})^{1/2}$ than the region
traversed by the stars inducing the fluctuating force on it.

The dynamical insignificance of the BH on the global galactic scale is
misleading. The gravitational binding energy per rest-mass energy of
galaxies is of order $\sim (\sigma_\star/c)^2< 10^{-6}$.  Since BH are
relativistic objects, the gravitational binding energy of material that
feeds them amounts to a substantial fraction its rest mass energy. Even if
the BH mass amounts to a fraction as small as $\sim 10^{-4}$ of the
baryonic mass in a galaxy, and only a percent of the accreted rest-mass
energy is deposited into the gaseous environment of the BH, this slight
deposition can unbind the entire gas reservoir of the host galaxy. This
order-of-magnitude estimate explains why quasars may be short lived.  As
soon as the central BH accretes large quantities of gas so as to
significantly increase its mass, it releases large amounts of energy and
momentum that could suppress further accretion onto it. In short, the BH
growth might be {\it self-regulated}.

The principle of {\it self-regulation} naturally leads to a correlation
between the final BH mass, $M_{\rm bh}$, and the depth of the gravitational
potential well to which the surrounding gas is confined. The latter can be
characterized by the velocity dispersion of the associated stars, $\sim
\sigma_\star^2$. Indeed a correlation between $M_{\rm bh}$ and
$\sigma_\star^4$ is observed in the present-day Universe.
%The observed power-law relation between $M_{\rm bh}$ and $\sigma_\star$ can
%be generalized to a correlation between the BH mass and the circular
%velocity of the host halo, $v_c$, which in turn can be related to the halo
%mass, $M_{\rm halo}$, and redshift, $z$
%\begin{eqnarray}
%\label{eq:1}
%\nonumber M_{\rm bh}(M_{\rm halo},z) &=&\mbox{const} \times v_c^5\\
%&&\hspace{-25mm}= \epsilon_{\rm o} M_{\rm halo} \left(\frac{M_{\rm
%halo}}{10^{12}M_{\odot}}\right)^{\frac{2}{3}}
%[\zeta(z)]^\frac{5}{6}(1+z)^\frac{5}{2},
%\end{eqnarray}
%where $\epsilon_{\rm o}\approx 10^{-5.7}$ is a constant, and as before
%$\zeta\equiv [(\Omega_m/\Omega_m^z)(\Delta_c/18\pi^2)]$, $\Omega_m^z \equiv
%[1+(\Omega_\Lambda/\Omega_m)(1+z)^{-3}]^{-1}$,
%$\Delta_c=18\pi^2+82d-39d^2$, and $d=\Omega_m^z-1$.  
If quasars shine near
their Eddington limit as suggested by observations of low and high-redshift
quasars, then 
%the above value of $\epsilon_{\rm o}$ implies that 
a fraction of $\sim 5$--$10\%$ of the energy released by the quasar
over a galactic dynamical time needs to be captured in the surrounding
galactic gas in order for the BH growth to be self-regulated.  With
this interpretation, the $M_{\rm bh}$--$\sigma_\star$ relation
reflects the limit introduced to the BH mass by self-regulation;
deviations from this relation are inevitable during episodes of BH
growth or as a result of mergers of galaxies that have no cold gas in
them.  A physical scatter around this upper envelope could also result
from variations in the efficiency by which the released BH energy
couples to the surrounding gas.

%\begin{figure}
%\epsfxsize=12.5cm \epsfbox{ti.eps}
%\caption{Simulation images of a merger of galaxies resulting in quasar
%activity that eventually shuts-off the accretion of gas onto the black hole
%(from Di Matteo et al. 2005). The upper (lower) panels show a sequence of
%snapshots of the gas distribution during a merger with (without) feedback
%from a central black hole. The temperature of the gas is color coded.}
%\label{merger}
%\end{figure}

Various prescriptions for self-regulation were sketched in the
literature. These involve either energy or momentum-driven winds, with the
latter type being a factor of $\sim v_c/c$ ~less efficient.  The quasar
remains active during the dynamical time of the initial gas reservoir,
$\sim 10^7$ years, and fades afterwards due to the dilution of this
reservoir.  The BH growth may resume if the cold gas reservoir is
replenished through a new merger.  Following early analytic work, extensive
numerical simulations demonstrated that galaxy mergers do produce the
observed correlations between black hole mass and spheroid
properties. Because of the limited resolution near the galaxy nucleus,
these simulations adopt a simple prescription for the accretion flow that
feeds the black hole. The actual feedback in reality may depend crucially
on the geometry of this flow and the physical mechanism that couples the
energy or momentum output of the quasar to the surrounding gas.

The inflow of cold gas towards galaxy centers during the growth phase of
the BH would naturally be accompanied by a burst of star formation.  The
fraction of gas that is not consumed by stars or ejected by
supernova-driven winds, will continue to feed the BH. It is therefore not
surprising that quasar and starburst activities co-exist in Ultra Luminous
Infrared Galaxies, and that all quasars show broad metal lines indicating 
pre-enrichment of the surrounding gas with heavy elements.
%Applying a similar self-regulation principle to the stars, leads to the
%expectation that the ratio between the mass of the BH and the mass in stars
%is independent of halo mass (as observed locally but increases with
%redshift as $\propto \xi(z)^{1/2}(1+z)^{3/2}$. A consistent trend has
%indeed been inferred in an observed sample of gravitationally-lensed
%quasars \cite{Rix99}.

The upper mass of galaxies may also be regulated by the energy output from
quasar activity. This would account for the fact that cooling flows are
suppressed in present-day X-ray clusters, and that massive BHs and stars in
galactic bulges were already formed at $z\sim 2$. 
%At some epoch, the quasar energy output may have led to the
%extinction of cold gas in these galaxies and the suppression of further
%star formation in them, leading to an apparent ``anti-hierarchical'' mode
%of galaxy formation where massive spheroids formed early and did not make
%new stars at late times. In the course of subsequent merger events, the
%cores of the most massive spheroids acquired an envelope of collisionless
%matter in the form of already-formed stars or dark matter , without the
%proportional accretion of cold gas into the central BH. The upper limit on
%the mass of the central BH and the mass of the spheroid is caused by the
%lack of cold gas and cooling flows in their X-ray halos. 
In the cores of cooling X-ray clusters, there is often an active central BH
that supplies sufficient energy to compensate for the cooling of the
gas. The primary physical process by which this energy couples to the gas
is still unknown.

The quasars discovered so far at $z\sim 6$ mark the early growth of the
most massive BHs and galactic spheroids. The BHs powering these bright
quasars possess a mass of a few billion solar masses.  A quasar radiating
at its Eddington limiting luminosity, $L_E=1.4\times 10^{47}~{\rm
erg~s^{-1}}(M_{\rm bh}/10^9M_\odot)$, with a radiative efficiency,
$\epsilon_{\rm rad}=L_{E}/{\dot M}c^2$, for converting accreted mass into
radiation, would grow exponentially in mass as a function of time $t$,
$M_{\rm bh} =M_{\rm seed}\exp\{t/t_E\}$ from its initial seed mass $M_{\rm
seed}$, on a time scale, $t_E=4.1\times 10^7~{\rm yr} (\epsilon_{\rm
rad}/0.1)$. Thus, the required growth time in units of the Hubble time
$t_{\rm hubble}= 10^9~{\rm yr}[(1+z)/7]^{-3/2}$ is
\begin{equation}
{t_{\rm growth}\over t_{\rm hubble}}=0.7 \left({\epsilon_{\rm rad} \over
10\%}\right) \left({1+z\over 7}\right)^{3/2}\ln \left({{M_{\rm
bh}/10^9M_\odot} \over M_{\rm seed}/100M_\odot}\right) ~.
\end{equation}
The age of the Universe at $z\sim 6$ provides just sufficient time to grow
a BH with $M_{\rm bh}\sim 10^9M_\odot$ out of a stellar mass seed with
$\epsilon_{\rm rad}=10 \%$. The growth time is shorter for smaller
radiative efficiencies or a higher seed mass.

\subsection{The epoch of reionization}

Given the understanding described above of how many galaxies formed at
various times, the course of reionization can be determined universe-wide
by counting photons from all sources of light. Both stars and black holes
contribute ionizing photons, but the early universe is dominated by small
galaxies which in the local universe have central black holes that are
disproportionately small, and indeed quasars are rare above redshift
6. Thus, stars most likely dominated the production of ionizing UV photons
during the reionization epoch [although high-redshift galaxies should have
also emitted X-rays from accreting black holes and accelerated particles in
collisionless shocks]. Since most stellar ionizing photons are only
slightly more energetic than the 13.6 eV ionization threshold of hydrogen,
they are absorbed efficiently once they reach a region with substantial
neutral hydrogen). This makes the IGM during reionization a two-phase
medium characterized by highly ionized regions separated from neutral
regions by sharp ionization fronts.
% (see Figure~\ref{fig:rei}).

%\begin{figure}
%\epsfxsize=8cm \epsfbox{SciRev1.eps}
%%\hfill
%%\epsfxsize=5cm \epsfbox{image_yz_13.892_enhLoRes.ps}
%\caption{The spatial structure of cosmic reionization. The illustration
%shows how regions with large-scale overdensities form large concentrations
%of galaxies (dots) whose ionizing photons produce enormous joint ionized
%bubbles (upper left). At the same time, galaxies are rare within
%large-scale voids, in which the IGM is still mostly neutral (lower
%right). 
%%A numerical simulation of reionization (right panel) indeed
%%displays such variation in the sizes of ionized bubbles (orange), shown
%%overlayed on the density distribution (green).
%}
%\label{fig:rei}
%\end{figure}

We can obtain a first estimate of the requirements of reionization by
demanding one stellar ionizing photon for each hydrogen atom in the
IGM. If we conservatively assume that stars within the early galaxies
were similar to those observed locally, then each star produced $\sim
4000$ ionizing photons per baryon. Star formation is observed today to
be an inefficient process, but even if stars in galaxies formed out of
only $\sim10\%$ of the available gas, it was still sufficient to
accumulate a small fraction (of order $0.1\%$) of the total baryonic
mass in the universe into galaxies in order to ionize the entire
IGM. More accurate estimates of the actual required fraction account
for the formation of some primordial stars (which were massive,
efficient ionizers, as discussed above), and for recombinations of
hydrogen atoms at high redshifts and in dense regions.

From studies of quasar absorption lines at $z\sim 6$ we know that the IGM
is highly ionized a billion years after the big bang. There are hints,
however, that some large neutral hydrogen regions persist at these early
times and so this suggests that we may not need to go to much higher
redshifts to begin to see the epoch of reionization.  We now know that the
universe could not have fully reionized earlier than an age of 300 million
years, since WMAP observed the effect of the freshly created plasma at
reionization on the large-scale polarization anisotropies of the CMB and
this limits the reionization redshift; an earlier reionization, when the
universe was denser, would have created a stronger scattering signature
that would be inconsistent with the WMAP observations. In any case, the
redshift at which reionization ended only constrains the overall cosmic
efficiency of ionizing photon production. In comparison, a detailed picture
of reionization as it happens will teach us a great deal about the
population of young galaxies that produced this cosmic phase transition.
A key point is that the spatial distribution of ionized bubbles is
determined by clustered groups of galaxies and not by individual
galaxies. At such early times galaxies were strongly clustered even on very
large scales (up to tens of Mpc), and these scales therefore dominate the
structure of reionization. The basic idea is simple. At high redshift,
galactic halos are rare and correspond to rare, high density peaks. As an
analogy, imagine searching on Earth for mountain peaks above 5000
meters. The 200 such peaks are not at all distributed uniformly but instead
are found in a few distinct clusters on top of large mountain ranges. Given
the large-scale boost provided by a mountain range, a small-scale crest
need only provide a small additional rise in order to become a 5000 meter
peak. The same crest, if it formed within a valley, would not come anywhere
near 5000 meters in total height. Similarly, in order to find the early
galaxies, one must first locate a region with a large-scale density
enhancement, and then galaxies will be found there in abundance.

The ionizing radiation emitted from the stars in each galaxy initially
produces an isolated ionized bubble. However, in a region dense with
galaxies the bubbles quickly overlap into one large bubble, completing
reionization in this region while the rest of the universe is still
mostly neutral. Most importantly, since the abundance of rare density
peaks is very sensitive to small changes in the density threshold,
even a large-scale region with a small enhanced density (say, 10\%
above the mean density of the universe) can have a much larger
concentration of galaxies than in other regions (e.g., a 50\%
enhancement). On the other hand, reionization is harder to achieve in
dense regions, since the protons and electrons collide and recombine
more often in such regions, and newly-formed hydrogen atoms need to be
reionized again by additional ionizing photons. However, the overdense
regions end up reionizing first since the number of ionizing sources
in these regions is increased so strongly. The large-scale topology of
reionization is therefore inside out, with underdense voids reionizing
only at the very end of reionization, with the help of extra ionizing
photons coming in from their surroundings (which have a higher density
of galaxies than the voids themselves). This is a key prediction
awaiting observational testing.

Detailed analytical models that account for large-scale variations in the
abundance of galaxies confirm that the typical bubble size starts well
below a Mpc early in reionization, as expected for an individual galaxy,
rises to 5--10 Mpc during the central phase (i.e., when the universe is
half ionized), and then by another factor of $\sim$5 towards the end of
reionization. These scales are given in comoving units that scale with the
expansion of the universe, so that the actual sizes at a redshift $z$ were
smaller than these numbers by a factor of $(1+z)$. Numerical simulations have
only recently begun to reach the enormous scales needed to capture this
evolution. Accounting precisely for gravitational evolution on a wide range
of scales but still crudely for gas dynamics, star formation, and the
radiative transfer of ionizing photons, the simulations confirm that the
large-scale topology of reionization is inside out, and that this topology
can be used to study the abundance and clustering of the ionizing sources.
%(Figures~\ref{fig:rei}).

\begin{figure}
\begin{center}
\epsfxsize=7cm \epsfbox{twostep_xi.eps}
\caption{Probability distribution of $x_i$ at redshifts $z=8$, 9, 10,
and 11, based on existing data on the CMB polarization anisotropies
and the Lyman-$\alpha$ forest.  {\bf Figure credit:} Pritchard, J. R.,
Loeb, A.. \& Wyithe, S., {\it Mon. Not. R. Astr. Soc.}, in press (2010);
http://arxiv.org/abs/0908.3891.  }
\label{fig:twostep_xi}
\end{center}
\end{figure}

The characteristic observable size of the ionized bubbles at the end
of reionization can be calculated based on simple considerations that only
depend on the power-spectrum of density fluctuations and the redshift. As
the size of an ionized bubble increases, the time it takes a 21-cm photon
emitted by hydrogen to traverse it gets longer. At the same time, the
variation in the time at which different regions reionize becomes smaller
as the regions grow larger. Thus, there is a maximum size above which the
photon crossing time is longer than the cosmic variance in ionization
time. Regions bigger than this size will be ionized at their near side by
the time a 21-cm photon will cross them towards the observer from their far
side. They would appear to the observer as one-sided, and hence signal the
end of reionization. These considerations imply a characteristic size for
the ionized bubbles of $\sim 10$ physical Mpc at $z\sim 6$ (equivalent to
70 Mpc today).  This result implies that future radio experiments should be
tuned to a characteristic angular scale of tens of arcminutes for an
optimal detection of 21-cm brightness fluctuations near the end of
reionization (see \S \ref{sec:21-cmAtomic}).

Existing data on the polarization anisotropies of the CMB as well as
the Lyman-$\alpha$ forest can be used to derive a probability distribution
for the hydrogen ionization fraction ($x_i$) as a function of
redshift.  Figure \ref{fig:twostep_xi} shows this likelihood
distribution in four redshift bins of interest to upcoming
observations.  Although there is considerable uncertainty in $x_i$ at
each redshift, it is evident from existing data that hydrogen is
highly ionized by $z=8$ (at least to $x_i\gtrsim0.8$).

To produce one ionizing photon per baryon requires a minimum comoving
density of Milky-Way (so-called Population II) stars of,
\begin{equation}
\rho_\star\approx 1.7\times 10^6f_{\rm esc}^{-1}~M_\odot~{\rm Mpc}^{-3},
\label{presec}
\end{equation}
or equivalently, a cosmological density parameter in stars of
$\Omega_\star\sim 1.25\times 10^{-5}f_{\rm esc}^{-1}$. More typically,
the threshold for reionization involves at least a few ionizing
photons per proton (with the right-hand-side being $\sim 10^{-6}~{\rm
  cm^{-3}}$), since the recombination time at the mean density is
comparable to the age of the Universe at $z\sim 10$.

For the local mass function of (Population II) stars at solar
metallicity, the star formation rate per unit comoving volume that is
required for balancing recombinations in an already ionized IGM, is
given by
\begin{equation}
\dot{\rho_\star}\approx 2\times 10^{-3} f_{\rm esc}^{-1}C
\left({1+z\over 10}\right)^3~ M_\odot~{\rm yr^{-1}~Mpc^{-3}},
\label{postsec}
\end{equation}
where $C=\langle n_e^2\rangle/\langle n_H\rangle^2$ is the
volume-averaged clumpiness factor of the electron density up to some
threshold overdensity of gas which remains neutral. Current
state-of-the-art surveys (HST WFC3/IR) are only sensitive to the
bright end of the luminosity function of galaxies at $z>6$ and hence
provide a lower limit on the production rate of ionizing photons
during reionization.

\subsection{Post-reionization suppression of low-mass galaxies}

After the ionized bubbles overlapped in each region, the ionizing
background increased sharply, and the IGM was heated by the ionizing
radiation to a temperature $T_{\rm IGM}\gtrsim 10^4$ K. Due to the
substantial increase in the IGM pressure, the smallest mass scale into
which the cosmic gas could fragment, the so-called Jeans mass,
increased dramatically, changing the minimum mass of forming galaxies.

Gas infall depends sensitively on the Jeans mass. When a halo more massive
than the Jeans mass begins to form, the gravity of its dark matter
overcomes the gas pressure. Even in halos below the Jeans mass, although
the gas is initially held up by pressure, once the dark matter collapses
its increased gravity pulls in some gas. Thus, the Jeans mass is generally
higher than the actual limiting mass for accretion. Before reionization,
the IGM is cold and neutral, and the Jeans mass plays a secondary role in
limiting galaxy formation compared to cooling. After reionization, the
Jeans mass is increased by several orders of magnitude due to the
photoionization heating of the IGM, and hence begins to play a dominant
role in limiting the formation of stars.  Gas infall in a reionized and
heated Universe has been investigated in a number of numerical simulations.
Three dimensional numerical simulations found a significant suppression of
gas infall in even larger halos ($V_c \sim 75\ {\rm km\ s}^{-1}$), but this
was mostly due to a suppression of late infall at $z\la 2$.

When a volume of the IGM is ionized by stars, the gas is heated to a
temperature $T_{\rm IGM}\sim 10^4$ K. If quasars dominate the UV background
at reionization, their harder photon spectrum leads to $T_{\rm IGM}>
2\times 10^4$ K. Including the effects of dark matter, a given temperature
results in a linear Jeans mass corresponding to a halo circular velocity of
\beq V_J\approx 80 \left(\frac{T_{\rm IGM}}{1.5\times 10^4 {\rm
K}}\right)^{1/2}\ {\rm km\ s}^{-1}. \eeq In halos with a circular
velocity well above $V_J$, the gas fraction in infalling gas equals
the universal mean of $\Omega_b/\Omega_m$, but gas infall is
suppressed in smaller halos.  A simple estimate of the limiting
circular velocity, below which halos have essentially no gas infall,
is obtained by substituting the virial overdensity for the mean
density in the definition of the Jeans mass. The resulting estimate is
\beq V_{\rm lim}=34 \left(\frac{T_{\rm IGM}}{1.5\times 10^4 {\rm
K}}\right)^{1/2}\ {\rm km\ s}^{-1}. \eeq This value is in rough
agreement with the numerical simulations mentioned before.  

Although the Jeans mass is closely related to the rate of gas infall at a
given time, it does not directly yield the total gas residing in halos at a
given time. The latter quantity depends on the entire history of gas
accretion onto halos, as well as on the merger histories of halos, and an
accurate description must involve a time-averaged Jeans mass. 
The gas content of halos in simulations is well fit by an expression which
depends on the filtering mass, a particular time-averaged Jeans mass.

The reionization process was not perfectly synchronized throughout the
Universe. Large-scale regions with a higher density than the mean tended to
form galaxies first and reionized earlier than underdense regions. The
suppression of low-mass galaxies by reionization is therefore modulated by
the fluctuations in the timing of reionization.  Inhomogeneous reionization
imprint a signature on the power-spectrum of low-mass galaxies. Future
high-redshift galaxy surveys hoping to constrain inflationary parameters
must properly model the effects of reionization; conversely, they will also
place new constraints on the thermal history of the IGM during
reionization.

\section{Probing the Diffuse Intergalactic Hydrogen}

\label{sec:absorb}

\subsection{Lyman-alpha absorption}

Resonant Lyman-$\alpha$ absorption has thus far proved to be the best probe
of the state of the IGM. The optical depth to absorption by a uniform
intergalactic medium is \beqa \tau_{s}&=&{\pi e^2 f_\alpha \lambda_\alpha
n_{\HI}(z) \over m_e cH(z)} \label{G-P} \\ \nonumber &\approx& 6.45\times
10^5 x_{\HI} \left({\Omega_bh\over 0.0315}\right)\left({\Omega_m\over
0.3}\right)^{-1/2} \left({1+z\over 10}\right)^{3/2}\ , \eeqa where
$H\approx 100h~{\rm km~s^{-1}~Mpc^{-1}} \Omega_m^{1/2} (1+z)^{3/2}$ is the
Hubble parameter at redshift $z$; $f_\alpha=0.4162$ and
$\lambda_\alpha=1216$\AA\, are the oscillator strength and the wavelength
of the Lyman-$\alpha$ transition; $n_{\HI}(z)$ is the neutral hydrogen
density at $z$ (assuming primordial abundances); $\Omega_m$ and $\Omega_b$
are the present-day density parameters of all matter and of baryons,
respectively; and $x_{\HI}$ is the average fraction of neutral hydrogen. In
the second equality we have implicitly considered high redshifts.

%\begin{figure}
%\epsfxsize=12cm \epsfbox{White.eps}
%\caption{Using Lyman-$\alpha$ absorption in quasar spectra to probe
%the ionization state of the IGM. This figure shows the observed
%spectrum of a $z = 6.28$ quasar (solid curve), and the expected
%unabsorbed emission (dashed curve), based on an average over many
%quasars seen at lower redshifts. The unabsorbed emission is a sum of
%smooth emission (the ``continuum'', dotted curve) plus emission
%features from atomic resonances (``emission lines''). {\bf Figure
%Credit:} White, R. L., et al., {\it Astron. J.} \bf{126}, 1 (2003).}
%\label{fig:white}
%\end{figure}

Lyman-$\alpha$ absorption is thus highly sensitive to the presence of even
trace amounts of neutral hydrogen. The lack of full absorption in quasar
spectra then implies that the IGM has been very highly ionized during much
of the history of the universe, from at most a billion years after the big
bang to the present time. At redshifts approaching six, however, the
optical depth increases, and the observed absorption becomes very strong.
%An example of this is shown in Figure~\ref{fig:white}, where an observed
%quasar spectrum is compared to the unabsorbed expectation for the same
%quasar. The prominent Lyman-$\alpha$ emission line, which is produced by
%radiating hot gas near the quasar itself, is centered at a wavelength of
%8850\AA, which for the redshift (6.28) of this quasar corresponds to a
%rest-frame 1216\AA. Above this wavelength, the original emitted quasar
%spectrum is seen, since photons emitted with wavelengths higher than
%1216\AA\ redshift to higher wavelengths during their journey toward us and
%never encounter resonance lines of hydrogen atoms. Shorter-wavelength
%photons, however, redshift until they hit the local 1216\AA\ and are then
%absorbed by any existing hydrogen atoms. 
The difference between the
unabsorbed expectation and the actual observed spectrum can be used to
measure the amount of absorption, and thus to infer the atomic hydrogen
density. 
%For this particular quasar, this difference is very large (i.e.,
%the observed flux is near zero) just to the blue of the Lyman-$\alpha$
%emission line.

\begin{figure}
\epsfxsize=10cm \epsfbox{Fan.eps}
\caption{Spectra of 19 quasars with redshifts $5.74<z<6.42$ from the
{\it Sloan Digital Sky Survey}. For some of the highest-redshift
quasars, the spectrum shows no transmitted flux shortward of the
Lyman-$\alpha$ wavelength at the quasar redshift (the so-called
``Gunn-Peterson trough''), indicating a non-negligible neutral
fraction in the IGM. {\bf Figure credit:} Fan, X., et al.  {\it
Astron. J.} \bf{125}, 1649 (2005).  }
\label{fig:qsos}
\end{figure}

Several quasars beyond $z\sim6.1$ show in their spectra a strong (so-called
``Gunn-Peterson'') trough, a blank spectral region at wavelengths shorter
than \lya at the quasar redshift (Figure~\ref{fig:qsos}). The detection
of Gunn-Peterson troughs indicates a rapid change in the neutral content of
the IGM at $z\sim6$, and hence a rapid change in the intensity of the
background ionizing flux. However, even a small atomic hydrogen fraction of
$\sim 10^{-3}$ would still produce nearly complete \lya absorption.

While only resonant \lya absorption is important at moderate
redshifts, the damping wing of the \lya line plays a significant role
when neutral fractions of order unity are considered at $z \gtrsim 6$.
The scattering cross-section of the \lya resonance line by neutral
hydrogen is given by
\begin{equation}
\sigma_\alpha(\nu) = {3 \lambda_\alpha^2 \Lambda_\alpha^2 \over 8\pi}
{(\nu/\nu_\alpha)^4\over
4\pi^2(\nu-\nu_\alpha)^2+(\Lambda_\alpha^2/4)(\nu/\nu_\alpha)^6}\ ,
\label{eq:sig}
\end{equation}
where $\Lambda_\alpha=(8\pi^2 e^2
f_\alpha/3m_ec\lambda_\alpha^2)=6.25\times 10^8~{\rm s^{-1}}$ is the
\lya ($2p\rightarrow 1s$) decay rate, $f_\alpha=0.4162$ is the
oscillator strength, and $\lambda_\alpha=1216$\AA\, and
$\nu_\alpha=(c/\lambda_\alpha)=2.47\times 10^{15}~{\rm Hz}$ are the
wavelength and frequency of the \lya line. The term in the numerator
is responsible for the classical Rayleigh scattering.

Although reionization is an inhomogeneous process, we consider here a
simple illustrative case of instantaneous reionization. Consider a source
at a redshift $z_s$ beyond the redshift of reionization, $\zr$, and the
corresponding scattering optical depth of a uniform, neutral IGM of
hydrogen density $n_{\rm H,0}(1+z)^3$ between the source and the
reionization redshift. The optical depth is a function of the observed
wavelength $\lambda_{\rm obs}$,
%Defining,
%\begin{equation}
%\Delta \lambda= \lambda- \lambda_\alpha(1+z_s),
%\end{equation}
%we get
\begin{equation}
\tau(\lambda_{\rm obs})=\int_{\zr}^{z_s} dz\, {cdt\over dz}\, n_{\rm
H,0} (1+z)^3 \sigma_\alpha\left[\nu_{\rm obs}(1+z)\right]\ ,
\end{equation}
where $\nu_{\rm obs}=c/\lambda_{\rm obs}$ and for a flat Universe
with $(\Omega_m+\Omega_\Lambda)=1$,
\begin{equation}
{dt\over dz} = \left[(1+z)H(z)\right]^{-1}=H_0^{-1} \times
\left[\Omega_m(1+z)^5+\Omega_\Lambda(1+z)^2
%+(1-\Omega_m-\Omega_\Lambda)(1+z)^4
\right]^{-1/2}\ .
\end{equation}

At wavelengths longer than \lya at the source, the optical depth
obtains a small value; these photons redshift away from the line
center along its red wing and never resonate with the line core on
their way to the observer.  Considering only the regime in which
$\vert\nu-\nu_\alpha\vert \gg \Lambda_\alpha$, we may ignore the
second term in the denominator of equation~(\ref{eq:sig}). This leads
to an analytical result for the red damping wing of the Gunn-Peterson
trough,
\begin{equation}
\tau(\lambda_{\rm obs})=\tau_s \left(\Lambda\over
4\pi^2\nu_\alpha\right) {\tilde \lambda}_{\rm obs}^{3/2}\left[
I({\tilde\lambda}_{\rm obs}^{-1}) -
I([(1+\zr)/(1+z_s)]{\tilde\lambda}_{\rm obs}^{-1})\right]\ ,
\label{eq:shift}
\end{equation}
an expression valid for ${\tilde\lambda}_{\rm obs}\geq 1$, where
$\tau_s$ is given in equation~(\ref{G-P}), and we also define
\begin{equation}
{\tilde \lambda}_{\rm obs}\equiv {\lambda_{\rm obs}\over
(1+z_s)\lambda_\alpha}
\end{equation}
and
\begin{equation}
I(x)\equiv {x^{9/2}\over 1-x}+{9\over 7}x^{7/2}+{9\over 5}x^{5/2}+ 3
x^{3/2}+9 x^{1/2}-{9\over 2} \ln\left[ {1+x^{1/2}\over 1-x^{1/2}}
\right]\ .
\end{equation}

%At wavelengths shorter than 912\AA, the photons are absorbed when
%they photoionize atoms of hydrogen or helium. The bound-free
%absorption cross-section from the ground state of a hydrogenic ion
%with nuclear charge $Z$ and an ionization threshold $h\nu_0$, is given
%by 
%\begin{equation}
%\sigma_{bf}(\nu)= {6.30\times 10^{-18}\over Z^2}~{\rm cm^2}\times
%\left({\nu_0\over
%\nu}\right)^4{e^{4-(4\tan^{-1}\epsilon)/\epsilon}\over 1 -
%e^{-2\pi/\epsilon}}~~~~{\rm for}~~\nu\geq\nu_0\ ,
%\end{equation}
%where $\epsilon\equiv \sqrt{{(\nu/\nu_0)}-1}$.  For neutral hydrogen, $Z=1$
%and $\nu_{{\rm H},0}= (c/\lambda_c) =3.29\times 10^{15}$ Hz ($h\nu_{\rm
%H,0}=13.60$ eV); for singly-ionized helium, $Z=2$ and $\nu_{\rm He~II, 0}=
%1.31\times 10^{16}~{\rm Hz}$ ($h\nu_{\rm He~II, 0}=54.42$ eV). The
%cross-section for neutral helium is more complicated; when averaged over
%its narrow resonances it can be fitted to an accuracy of a few percent up
%to $h\nu=50$ keV by the fitting function 
%\begin{equation} 
%\sigma_{bf,{\rm
%He~I}}(\nu)=9.492\times 10^{-16}~{\rm cm^2}\, \times
%\left[(x-1)^2+4.158\right]y^{-1.953}\left(1+ 0.825
%y^{1/4}\right)^{-3.188}\ , 
%\end{equation} 
%where $x\equiv[(\nu/3.286\times 10^{15}~{\rm Hz}) -0.4434]$, $y\equiv
%x^2+4.563$, and the threshold for ionization is $\nu_{\rm He~I,
%0}=5.938\times 10^{15}~{\rm Hz}$ ($h\nu_{\rm He~I,0}=24.59$ eV).

%For rough estimates, the average photoionization cross-section for a
%mixture of hydrogen and helium with cosmic abundances can be
%approximated in the range of $54<h\nu \lesssim 10^3$ eV as
%$\sigma_{bf}\approx \sigma_0 (\nu/\nu_{\rm H,0})^{-3}$, where
%$\sigma_0\approx 6\times 10^{-17}~{\rm cm^2}$.
%The redshift factor in the cross-section then cancels exactly the
%redshift evolution of the gas density and the resulting optical depth
%depends only on the elapsed cosmic time, $t(\zr)-t(z_s)$. At high
%redshifts this yields
%\begin{eqnarray}
%\tau_{bf}(\lambda_{\rm obs})&=&\int_{\zr}^{z_s} dz {cdt\over dz}
%n_{0} (1+z)^3 \sigma_{\rm bf}\left[\nu_{\rm obs}(1+z)\right]
%%={2cn_{H,0}\sigma_0\over
%%3H_0\sqrt{\Omega_m}}\left[{1\over (1+\zr)^{3/2}}-{1\over
%%(1+z_s)^{3/2}}\right]
%\nonumber \\ &\approx&
%1.5\times 10^2 \left({\Omega_bh\over 0.03}\right) \left({\Omega_m\over
%  0.3}\right)^{-1/2}\left({\lambda\over 100{\mbox{\AA}}}\right)^{3}\times
% \left[{1\over (1+\zr)^{3/2}}
%-{1\over (1+z_s)^{3/2}}\right]\ . \label{eq:bf}
%\end{eqnarray}
%The bound-free optical depth only becomes of order unity in the
%extreme ultraviolet (UV) to soft X-rays, around $h\nu \sim 0.1$ keV, a
%regime which is unfortunately difficult to observe due to Galactic
%absorption.


\subsection{21-cm absorption or emission}
\label{sec:21-cmAtomic}


\subsubsection{The spin temperature of the 21-cm transition of hydrogen}


The ground state of hydrogen exhibits hyperfine splitting owing to the
possibility of two relative alignments of the spins of the proton and the
electron. The state with parallel spins (the triplet state) has a slightly
higher energy than the state with anti-parallel spins (the singlet
state). The 21-cm line associated with the spin-flip transition from the
triplet to the singlet state is often used to detect neutral hydrogen in
the local universe. At high redshift, the occurrence of a neutral
pre-reionization IGM offers the prospect of detecting the first sources of
radiation and probing the reionization era by mapping the 21-cm emission
from neutral regions. While its energy density is estimated to be only a
$1\%$ correction to that of the CMB, the redshifted 21-cm emission should
display angular structure as well as frequency structure due to
inhomogeneities in the gas density field, hydrogen ionized fraction, and
spin temperature. Indeed, a full mapping of the distribution of H~I as a
function of redshift is possible in principle.

The basic physics of the hydrogen spin transition is determined as
follows. The ground-state hyperfine levels of hydrogen tend to thermalize
with the CMB background, making the IGM unobservable. If other processes
shift the hyperfine level populations away from thermal equilibrium, then
the gas becomes observable against the CMB in emission or in
absorption. The relative occupancy of the spin levels is usually described
in terms of the hydrogen spin temperature $T_S$, defined by \beq
\frac{n_1}{n_0}=3\, \exp\left\{-\frac{T_*}{T_S}\right\}\ , \eeq where $n_0$
and $n_1$ refer respectively to the singlet and triplet hyperfine levels in
the atomic ground state ($n=1$), and $T_*=0.068$ K is defined by $k_B
T_*=E_{21}$, where the energy of the 21 cm transition is $E_{21}=5.9 \times
10^{-6}$ eV, corresponding to a frequency of 1420 MHz. In the presence of
the CMB alone, the spin states reach thermal equilibrium with $T_S=T_{\rm
CMB}=2.725 (1+z)$ K on a time-scale of $T_*/(T_{\rm CMB} A_{10}) \simeq 3
\times 10^5 (1+z)^{-1}$ yr, where $A_{10}=2.87 \times 10^{-15}$ s$^{-1}$ is
the spontaneous decay rate of the hyperfine transition. This time-scale is
much shorter than the age of the universe at all redshifts after
cosmological recombination.

The IGM is observable when the kinetic temperature $T_k$ of the gas
differs from $T_{\rm CMB}$ and an effective mechanism couples $T_S$ to
$T_k$. Collisional de-excitation of the triplet level dominates at
very high redshift, when the gas density (and thus the collision rate)
is still high, but once a significant galaxy population forms in the
universe, the spin temperature is affected also by an indirect
mechanism that acts through the scattering of Lyman-$\alpha$
photons. Continuum UV photons produced by early radiation sources
redshift by the Hubble expansion into the local Lyman-$\alpha$ line at
a lower redshift. These photons mix the spin states via the
Wouthuysen-Field process whereby an atom initially in the $n=1$ state
absorbs a Lyman-$\alpha$ photon, and the spontaneous decay which
returns it from $n=2$ to $n=1$ can result in a final spin state which
is different from the initial one. Since the neutral IGM is highly
opaque to resonant scattering, and the Lyman-$\alpha$ photons receive
Doppler kicks in each scattering, the shape of the radiation spectrum
near Lyman-$\alpha$ is determined by $T_k$, and the resulting spin
temperature (assuming $T_S \gg T_*$) is then a weighted average of
$T_k$ and $T_{\rm CMB}$: \beq T_S=\frac{T_{\rm CMB} T_k (1+x_{\rm
tot}) }{T_k + T_{\rm CMB} x_{\rm tot}}\ , \eeq where $x_{\rm tot} =
x_{\alpha} + x_c$ is the sum of the radiative and collisional
threshold parameters. These parameters are
\begin{equation}
x_{\alpha} = {{P_{10} T_\star}\over {A_{10} T_{\rm CMB}}}\ ,
\end{equation}
and
\begin{equation}
x_c = {{4 \kappa_{1-0}(T_k)\, n_H T_\star}\over {3 A_{10} T_{\rm CMB}}}\ ,\
\end{equation} where $P_{10}$ is the indirect de-excitation rate of the
triplet $n=1$ state via the Wouthuysen-Field process, related to the total
scattering rate $P_{\alpha}$ of Lyman-$\alpha$ photons by $P_{10}=4
P_{\alpha}/27$. Also, the atomic coefficient $\kappa_{1-0}(T_k)$ is
tabulated as a function of $T_k$. The coupling of the spin temperature to
the gas temperature becomes substantial when $x_{\rm tot} \gtrsim 1$; in
particular, $x_{\alpha} = 1$ defines the thermalization rate of
$P_{\alpha}$: \beq P_{\rm th} \equiv \frac{27 A_{10} T_{\rm CMB}}{4 T_*}
\simeq 7.6 \times 10^{-12}\, \left(\frac{1+z}{10}\right)\ {\rm s}^{-1}\
. \eeq

A patch of neutral hydrogen at the mean density and with a uniform
$T_S$ produces (after correcting for stimulated emission) an optical
depth at a present-day (observed) wavelength of $21 (1+z)$ cm,
\beq \tau(z) = 9.0
\times 10^{-3} \left(\frac{T_{\rm CMB}} {T_S} \right) \left (
\frac{\Omega_b h} {0.03} \right) \left(\frac{\Omm}{0.3}\right)^ {-1/2}
\left(\frac{1+z}{10}\right)^{1/2}\ , \eeq assuming a high redshift
$z\gg1$. The observed spectral intensity $I_{\nu}$ relative to the CMB at a
frequency $\nu$ is measured by radio astronomers as an effective
brightness temperature $T_b$ of blackbody emission at this frequency,
defined using the Rayleigh-Jeans limit of the Planck radiation
formula: $I_{\nu} \equiv 2 k_B T_b \nu^2 / c^2 $.

The brightness temperature through the IGM is $T_b=T_{\rm CMB}
e^{-\tau}+T_S (1-e^{-\tau})$, so the observed differential antenna
temperature of this region relative to the CMB is \beqa T_b&=&(1+z)^{-1}
(T_S-T_{\rm CMB}) (1-e^{-\tau}) \nonumber \\ &\simeq& 28\, {\rm mK}\,
\left( \frac{\Omega_b h} {0.033} \right) \left(\frac{\Omm}{0.27}\right)^
{-1/2} \left( \frac{1+z} {10} \right)^{1/2} \left( \frac{T_S-T_{\rm CMB}}
{T_S} \right)\ , \eeqa where $\tau \ll 1$ is assumed and $T_b$ has been
redshifted to redshift zero. Note that the combination that appears in
$T_b$ is
\begin{equation}
{T_S - T_{\rm CMB} \over T_S} = {x_{\rm tot}\over 1+ x_{\rm tot}}
\left(1 - {T_{\rm CMB}\over T_k} \right)\ .
\end{equation}
In overdense regions, the observed $T_b$ is proportional to the
overdensity, and in partially ionized regions $T_b$ is proportional to the
neutral fraction. Also, if $T_S \gg T_{\rm CMB}$ then the IGM is observed
in emission at a level that is independent of $T_S$. On the other hand, if
$T_S \ll T_{\rm CMB}$ then the IGM is observed in absorption at a level
that is enhanced by a factor of $T_{\rm CMB} / T_S$. As a result, a number
of cosmic events are expected to leave observable signatures in the
redshifted 21-cm line, as discussed below in further detail.

\begin{figure}
\epsfxsize=8cm \epsfbox{tempZlong.eps}
\caption{{\em Top panel: }Evolution with redshift $z$ of the CMB
temperature $T_{\rm CMB}$ (dotted curve),the gas kinetic temperature
$T_k$ (dashed curve), and the spin temperature $T_S$ (solid curve).
{\em Middle panel: }Evolution of the gas fraction in ionized regions
$x_i$ (solid curve) and the ionized fraction outside these regions
(due to diffuse X-rays) $x_e$ (dotted curve). {\em Bottom panel: }
Evolution of mean 21 cm brightness temperature $T_b$.  The horizontal
axis at the top provides the observed photon frequency at the
different redshifts shown at the bottom.  Each panel shows curves for
three models in which reionization is completed at different
redshifts, namely $z=6.47$ (thin curves), $z=9.76$ (medium curves),
and $z=11.76$ (thick curves). {\bf Figure credit:} Pritchard, J., 
\& Loeb, A., {\it Phys. Rev.} {\bf D78}, 3511 (2008). }
\label{fig:tempZlong}
\end{figure}

Figure \ref{fig:tempZlong} illustrates the mean IGM evolution for three
examples in which reionization is completed at different redshifts, namely
$z=6.47$ (thin curves), $z=9.76$ (medium curves), and $z=11.76$ (thick
curves).  The top panel shows the global evolution of the CMB temperature
$T_{\rm CMB}$ (dotted curve), the gas kinetic temperature $T_k$ (dashed
curve), and the spin temperature $T_S$ (solid curve).  The middle panel
shows the evolution of the ionized gas fraction and the bottom panel
presents the mean 21 cm brightness temperature, $T_b$.

\subsubsection{A handy tool for studying cosmic reionization}

The prospect of studying reionization by mapping the distribution of
atomic hydrogen across the universe using its prominent 21-cm spectral
line has motivated several teams to design and construct arrays of
low-frequency radio telescopes; the Low Frequency Array
(http://www.lofar.org/), the Murchison Wide-Field Array ({\it
http://www.mwatelescope.org/}), PAPER ({\it
http://arxiv.org/abs/0904.1181}), GMRT ({\it
http://arxiv.org/abs/0807.1056}), 21CMA ({\it
http://21cma.bao.ac.cn/}), and ultimately the Square Kilometer Array
({\it http://www.skatelescope.org}) will search over the next decade
for 21-cm emission or absorption from $z\sim 6.5$--15, redshifted and
observed today at relatively low frequencies which correspond to
wavelengths of 1.5 to 4 meters.

The idea is to use the resonance associated with the hyperfine splitting in
the ground state of hydrogen. While the CMB spectrum peaks at a wavelength
of 2 mm, it provides a still-measurable intensity at meter wavelengths that
can be used as the bright background source against which we can see the
expected 1\% absorption by neutral hydrogen along the line of sight. The
hydrogen gas produces 21-cm absorption if its spin temperature is colder
than the CMB and excess emission if it is hotter. Since the CMB covers the
entire sky, a complete three-dimensional map of neutral hydrogen can in
principle be made from the sky position of each absorbing gas cloud
together with its redshift $z$.  Different observed wavelengths slice the
Universe at different redshifts, and ionized regions are expected to appear
as cavities in the hydrogen distribution, similar to holes in swiss cheese.
Because the smallest angular size resolvable by a telescope is proportional
to the observed wavelength, radio astronomy at wavelengths as large as a
meter has remained relatively undeveloped. Producing resolved images even
of large sources such as cosmological ionized bubbles requires telescopes
which have a kilometer scale. It is much more cost-effective to use a large
array of thousands of simple antennas distributed over several kilometers,
and to use computers to cross-correlate the measurements of the individual
antennas and combine them effectively into a single large telescope. The
new experiments are being placed mostly in remote sites, because the cosmic
wavelength region overlaps with more mundane terrestrial
telecommunications.

In approaching redshifted 21-cm observations, although the first inkling
might be to consider the mean emission signal in the bottom panel of
Figure~\ref{fig:tempZlong}, the signal is orders of magnitude fainter than
foreground synchrotron emission from relativistic electrons in the magnetic
field of our own Milky Way as well as other galaxies (see
Figure~\ref{fig:Analytic}). Thus cosmologists have focused on the expected
characteristic variations in $T_b$, both with position on the sky and
especially with frequency, which signifies redshift for the cosmic
signal. The synchrotron foreground is expected to have a smooth frequency
spectrum, and so it is possible to isolate the cosmological signal by
taking the difference in the sky brightness fluctuations at slightly
different frequencies (as long as the frequency separation corresponds to
the characteristic size of ionized bubbles). The 21-cm brightness
temperature depends on the density of neutral hydrogen. As explained in the
previous subsection, large-scale patterns in the reionization are driven by
spatial variations in the abundance of galaxies; the 21-cm fluctuations
reach $\sim$5 mK (root mean square) in brightness temperature
%(Figure~\ref{fig:Mellema}) 
on a scale of 10 comoving Mpc. While detailed maps will be difficult
to extract due to the foreground emission, a statistical detection of
these fluctuations is expected to be well within the capabilities of
the first-generation experiments now being built. Current work
suggests that the key information on the topology and timing of
reionization can be extracted statistically.

%\begin{figure}
%\epsfxsize=16cm \epsfbox{slice_f250_kz75_notitleLoRes.eps}
%\caption{Close-up of cosmic evolution during the epoch of reionization, as
%revealed in a predicted 21-cm map of the IGM based on a numerical
%simulation (from Mellema et al. 2006). This map is constructed from slices
%of the simulated cubic box of side 150 Mpc (in comoving units), taken at
%various times during reionization, which for the parameters of this
%particular simulation spans a period of 250 million years from redshift 15
%down to 9.3. The vertical axis shows position $\chi$ in units of Mpc/h
%(where $h=0.7$). This two-dimensional slice of the sky (one linear
%direction on the sky versus the line-of-sight or redshift direction) shows
%$\log_{10}(T_b)$, where $T_b$ (in mK) is the 21-cm brightness temperature
%relative to the CMB. Since neutral regions correspond to strong emission
%(i.e., a high $T_b$), this slice illustrates the global progress of
%reionization and the substantial large-scale spatial fluctuations in
%reionization history. Observationally it corresponds to a narrow strip half
%a degree in length on the sky observed with radio telescopes over a
%wavelength range of 2.2 to 3.4 m (with each wavelength corresponding to
%21-cm emission at a specific redshift slice).}
%\label{fig:Mellema}
%\end{figure}

While numerical simulations of reionization are now reaching the
cosmological box sizes needed to predict the large-scale topology of the
ionized bubbles, they do this at the price of limited small-scale
resolution (see Figure \ref{fig:Zahn}). These simulations cannot yet follow
in any detail the formation of individual stars within galaxies, or the
feedback that stars produce on the surrounding gas, such as photo-heating
or the hydrodynamic and chemical impact of supernovae, which blow hot
bubbles of gas enriched with the chemical products of stellar
nucleosynthesis. Thus, the simulations cannot directly predict whether the
stars that form during reionization are similar to the stars in the Milky
Way and nearby galaxies or to the primordial $100 M_{\odot}$ stars. They
also cannot determine whether feedback prevents low-mass dark matter halos
from forming stars. Thus, models are needed that make it possible to vary
all these astrophysical parameters of the ionizing sources and to study the
effect on the 21-cm observations.

\begin{figure}
\epsfxsize=12cm \epsfbox{21cm.eps}
\caption{Map of the fluctuations in the 21 cm brightness temperature
on the sky, $\Delta T_b$ (mK), based on a numerical simulation which
follows the dynamics of dark matter and gas in the IGM as well as the
radiative transfer of ionizing photons from galaxies. The panels show
the evolution of the signal in a slice of 140 comoving Mpc on a side,
in three snapshots corresponding to the simulated volume being 25, 50,
and 75 \% ionized. Since neutral regions correspond to strong
emission (i.e., a high $T_b$), the 21-cm maps illustrate the global
progress of reionization and the substantial large-scale spatial
fluctuations in the reionization history.  {\bf Figure credit:} Trac,
H., Cen, R., \& Loeb, A., {\it Astrophys. J.}  {\bf 689}, L81 (2009).
}
%\begin{figure}
%\epsfxsize=12cm \epsfbox{ZahnFig10.eps}
%\caption{Maps of the 21-cm brightness temperature comparing results of a
%numerical simulation and of two simpler numerical schemes, at three
%different redshifts (from Zahn et al. 2006). Each map is 65.6 Mpc/$h$ on a
%side, with a depth (0.25 Mpc/$h$) that is comparable to the frequency
%resolution of planned experiments. The ionized fractions are $x_{\rm
%i}=0.13$, 0.35 and 0.55 for $z=8.16$, 7.26 and 6.89 (top to bottom),
%respectively. All three maps show a very similar large-scale ionization
%topology. \emph{Left column:} Numerical simulation, showing the ionized
%bubbles (black) produced by the ionizing sources (blue dots) that form in
%the simulation. \emph{Middle column:} Numerical scheme that applies an
%analytical model to the final distribution of ionizing sources that form in
%the simulation. \emph{Right column:} Numerical scheme that applies the
%analytical model to the linear density fluctuations that are the initial
%conditions of the simulation.}
\label{fig:Zahn}
\end{figure}

The theoretical expectations presented here for reionization and for the
21-cm signal are based on rather large extrapolations from observed
galaxies to deduce the properties of much smaller galaxies that formed at
an earlier cosmic epoch. Considerable surprises are thus possible, such as
an early population of quasars or even unstable exotic particles that
emitted ionizing radiation as they decayed. In any case, the forthcoming
observational data in 21-cm cosmology should make the next few years a very
exciting time.

At high redshifts prior to reionization, spatial perturbations in the
thermodynamic gas properties are linear and can be predicted precisely
(see section~\ref{sec:lin}). Thus, if the gas is probed with the 21-cm
technique then it becomes a promising tool of fundamental, precision
cosmology, able to probe the primordial power spectrum of density
fluctuations imprinted in the very early universe, perhaps in an era
of cosmic inflation. The 21-cm fluctuations can be measured down to
the smallest scales where the baryon pressure suppresses gas
fluctuations, while the CMB anisotropies are damped on small scales
(through the so-called Silk damping). This difference in damping
scales can be seen by comparing the baryon-density and
photon-temperature power spectra.  Since the 21-cm technique is also
three-dimensional (while the CMB yields a single sky map), there is a
much large potential number of independent modes probed by the 21-cm
signal: $N_{\rm 21-cm}\sim 3 \times 10^{16}$ compared to $N_{\rm cmb}
\sim 2\times 10^7$. This larger number should provide a measure of
non-Gaussian deviations to a level of $\sim N_{\rm 21 cm}^{-1/2}$,
constituting a test of the inflationary origin of the primordial
inhomogeneities which are expected to possess non-Gaussian deviations
$\gtrsim 10^{-6}$.

The 21cm fluctuations are expected to simply trace the primordial
power-spectrum of matter density perturbations (which is shaped by the
initial conditions from inflation and the dark matter) either before the
first population of galaxies had formed (at redshifts $z>25$) or after
reionization ($z<6$) -- when only dense pockets of self-shielded hydrogen
(such as damped Lyman-$\alpha$ systems) survive.  During the epoch of
reionization, the fluctuations are mainly shaped by the topology of ionized
regions, and thus depend on uncertain astrophysical details involving star
formation.  However, even during this epoch, the imprint of peculiar
velocities (which are induced gravitationally by density fluctuations), can
in principle be used to separate the implications for fundamental physics
from the astrophysics.

\begin{figure}
\epsfxsize=8cm \epsfbox{powerZlong_mid.eps}
\caption{Predicted redshift evolution of the angle-averaged amplitude
of the 21-cm power spectrum ($|\bar{\Delta}_{T_b}|=[k^3P_{\rm
21-cm}(k)/2\pi^2]^{1/2}$) at comoving wavenumbers $k=0.01$ (solid
curve), 0.1 (dotted curve), 1.0 (short dashed curve), 10.0 (long
dashed curve), and 100.0${\rm Mpc}^{-1}$ (dot-dashed curve). In the
model shown, reionization is completed at $z=9.76$.  The horizontal
axis at the top shows the observed photon frequency at the different
redshifts.  The diagonal straight (red) lines show various factors of
suppression for the synchrotron Galactic foreground, necessary to
reveal the 21-cm signal.  {\bf Figure credit:} Pritchard, J.R. \& Loeb
A., {\it Phys. Rev} {\bf D78}, 3511 (2008).}
\label{fig:Analytic}
\end{figure}

Peculiar velocities imprint a particular form of anisotropy in
the 21-cm fluctuations that is caused by gas motions along the line of
sight. This anisotropy, expected in any measurement of density that is
based on a spectral resonance or on redshift measurements, results from
velocity compression. Consider a photon traveling along the line of sight
that resonates with absorbing atoms at a particular point. In a uniform,
expanding universe, the absorption optical depth encountered by this photon
probes only a narrow strip of atoms, since the expansion of the universe
makes all other atoms move with a relative velocity that takes them outside
the narrow frequency width of the resonance line. If there is a density
peak, however, near the resonating position, the increased gravity will
reduce the expansion velocities around this point and bring more gas into
the resonating velocity width. This effect is sensitive only to the
line-of-sight component of the velocity gradient of the gas, and thus
causes an observed anisotropy in the power spectrum even when all physical
causes of the fluctuations are statistically isotropic. This anisotropy is
particularly important in the case of 21-cm fluctuations. When all
fluctuations are linear, the 21-cm power spectrum takes the form \beq
P_{\rm 21-cm}({\bf k}) = \mu^4 P_{\rho}(k) + 2 \mu^2 P_{\rho - {\rm iso}}
(k) + P_{\rm iso}\ , \eeq where $\mu = \cos \theta$ in terms of the angle
$\theta$ between the wave-vector ${\bf k}$ of a given Fourier mode and the
line of sight, $P_{\rm iso}$ is the isotropic power spectrum that would
result from all sources of 21-cm fluctuations without velocity compression,
$P_{\rho}(k)$ is the 21-cm power spectrum from gas density fluctuations
alone, and $P_{\rho - {\rm iso}} (k)$ is the Fourier transform of the
cross-correlation between the density and all sources of 21-cm
fluctuations. The three power spectra can also be denoted $P_{\mu^4}(k)$,
$P_{\mu^2}(k)$, and $P_{\mu^0}(k)$, according to the power of $\mu$ that
multiplies each term.  
%The prediction for these power spectra at high
%redshift ($z > 20$), neglecting the effects of any stellar radiation, are
%shown in Figure~\ref{21-cmP}.  
At these redshifts, the 21-cm fluctuations probe the infall of the
baryons into the dark matter potential wells. The power spectrum shows
remnants of the photon-baryon acoustic oscillations on large scales,
and of the baryon pressure suppression on small scales.

%\begin{figure}
%\epsfxsize=10cm \epsfbox{binfallA8a.eps}
%\caption{Power spectra of 21-cm brightness fluctuations versus comoving
%wavenumber (from Barkana \& Loeb 2005c). Shown are the three power spectra
%that are separately observable, $P_{\mu^4}$ (upper panel), $P_{\mu^2}$
%(middle panel), and $P_{\mu^0}$ (lower panel). Each case shows redshifts
%200, 150, 100, 50 (solid curves, from bottom to top), 35, 25, and 20
%(dashed curves, from top to bottom).}
%\label{21-cmP}
%\end{figure}

Once stellar radiation becomes significant, many processes can contribute
to the 21-cm fluctuations. The contributions include fluctuations in gas
density, temperature, ionized fraction, and \Lya flux. These processes can
be divided into two broad categories: The first, related to {\it
``physics''}, consists of probes of fundamental, precision cosmology, and
the second, related to {\it ``astrophysics''}, consists of probes of
stars. Both categories are interesting -- the first for precision measures
of cosmological parameters and studies of processes in the early universe,
and the second for studies of the properties of the first
galaxies. However, the astrophysics depends on complex non-linear processes
(collapse of dark matter halos, star formation, supernova feedback), and
must be cleanly separated from the physics contribution, in order to allow
precision measurements of the latter.  As long as all the fluctuations are
linear, the anisotropy noted above allows precisely this separation of the
{\it fundamental physics} from the {\it astrophysics} of the 21-cm
fluctuations. In particular, the $P_{\mu^4}(k)$ is independent of the
effects of stellar radiation, and is a clean probe of the gas density
fluctuations. Once non-linear terms become important, there arises a
significant mixing of the different terms; in particular, this occurs on
the scale of the ionizing bubbles during reionization.

The 21-cm fluctuations are affected by fluctuations in the Lyman-$\alpha$
flux from stars, a result that yields an indirect method to detect and
study the early population of galaxies at $z \sim 20$. The fluctuations are
caused by biased inhomogeneities in the density of galaxies, along with
Poisson fluctuations in the number of galaxies. Observing the power-spectra
of these two sources would probe the number density of the earliest
galaxies and the typical mass of their host dark matter halos. Furthermore,
the enhanced amplitude of the 21-cm fluctuations from the era of \Lya
coupling improves considerably the practical prospects for their
detection. Precise predictions account for the detailed properties of all
possible cascades of a hydrogen atom after it absorbs a photon. Around the
same time, X-rays may also start to heat the cosmic gas, producing strong
21-cm fluctuations due to fluctuations in the X-ray flux.

In difference from interferometric arrays, single dipole experiments
which integrate over most of the sky, can search for the global
(spectral) 21-cm signal shown in Figure \ref{fig:tempZlong}. Examples
of such experiments are CoRE or EDGES ({\it
http://www.haystack.mit.edu/ast/arrays/Edges/}). Rapid reionization
histories which span a redshift range $\Delta z\lesssim 2$ can be
constrained, provided that local foregrounds (see Figure \ref{xx2})
can be well modelled by low-order polynomials in
frequency. Observations in the frequency range 50-100 MHz can
potentially constrain the Lyman-$\alpha$ and X-ray emissivity of the
first stars forming at redshifts $z\sim 15$--25, as illustrated in
Figure \ref{xx3}.

%\begin{figure}
%\epsfxsize=7cm \epsfbox{pedplot_nu.eps}
%\caption{Evolution of the global (spectral) 21-cm signal for different
%scenarios.  {\em Solid blue curve: } no stars; {\em solid red curve:
%}$T_S\gg T_\gamma$; {\em black dotted curve: }no heating; {\em black
%dashed curve: }no ionization; {\em black solid curve: }full
%calculation.  {\bf Figure credit:} Pritchard, J., \& Loeb, A.
%Phys. Rev. {\bf D}, in press (2010). }
%\label{xx1}       % Give a unique label
%\end{figure}

\begin{figure}
\epsfysize=5cm \epsfbox{plot_example_execute.eps}
\epsfysize=5cm \epsfbox{plot24.eps}
%\includegraphics[scale=0.25,angle=90]{plot_example_execute.eps}
%\includegraphics[scale=0.25,angle=90]{plot24.eps}
\caption{{\em Left panel: }Radio map of the sky at 100 MHz. 
{\em Right panel: } Ideal dipole
response averaged over 24 hours.
{\bf Figure credits:} Pritchard, J., \& Loeb, A.
Phys. Rev. {\bf D}, in press (2010); de Oliveira-Costa, A. et al.
Mon. Not. R. Astron. Soc. {\bf 388}, 247 (2008).
}
\label{xx2}       % Give a unique label
\end{figure}

\begin{figure}
\epsfxsize=7cm \epsfbox{param_comp.eps}
\caption{Dependence of global 21-cm signal on the X-ray (top panel)
and Lyman-$\alpha$ (bottom panel) emissivity of stars.  Each case
depicts examples with the characteristic emissivity reduced or
increased by a factor of up to 100.  {\bf Figure credit:} Pritchard,
J., \& Loeb, A.  Phys. Rev. {\bf D}, in press (2010).  }
\label{xx3}       % Give a unique label
\end{figure}


\section{Epilogue}

The initial conditions of our Universe can be summarized on a single sheet
of paper. Yet the Universe is full of complex structures today, such as
stars, galaxies and groups of galaxies.  This chapter discussed the standard
theoretical model for how complexity emerged from the simple initial state
of the Universe through the action of gravity. In order to test and inform
the related theoretical calculations, large-aperture telescopes and arrays
of radio antennae are currently being designed and constructed.

The actual transition from simplicity to complexity has not been observed
as of yet.  The simple initial conditions were already traced in maps of
the microwave background radiation, but the challenge of detecting the
first generation of galaxies defines one of the exciting frontiers in the
future of cosmology.  Once at hand, the missing images of the infant
Universe might potentially surprise us and revise our current ideas.

\subsection*{Acknowledgements}

I thank my collaborators on the topics covered by this chapter: Dan
Babich, Rennan Barkana, Volker Bromm, Steve Furlanetto, Zoltan Haiman,
Joey Munoz, Jonathan Pritchard, Hy Trac, Stuart Wyithe, and Matias
Zaldarriaga.

\bigskip
\bigskip
\bigskip

\noindent
{\bf Further Reading}: Loeb, A., {\it ``How Did the First Stars and
Galaxies Form?''}, Princeton University Press (2010).

\end{document}

\subsection*{Glossary}

\noindent
{\bf Big Bang:} The point in time when the currently observed expansion of
the Universe began. 

\noindent
{\bf Hubble Parameter:} The ratio between the recession speed of distant
cosmological sources and their distance, first measured by Edwin Hubble.

\noindent
{\bf Cosmic Background Radiation:} The relic radiation left over from the
hot and dense beginning of the Universe. The Universe became nearly
transparent to this radiation as soon as hydrogen formed out of free
electrons and protons, 0.4 million years after the big bang. Prior to that
time, scattering of the radiation on free electrons made the Universe
opaque.

\noindent
{\bf Baryons:} The family of sub-atomic particles which are made of three
quarks, including the proton and the neutron which make up atomic
nuclei. This term is used in cosmology to denote ordinary (visible) matter.

\noindent
{\bf Dark Matter:} A hypothetical form of matter that does not interact
with electromagnetic radiation, and whose presence is inferred from its
gravitational effects on visible matter.

\noindent
{\bf Redshift:} The stretching of a photon's wavelength by the cosmic
expansion of space (resulting in the shift of optical photons towards the
red). The photon wavelength provides a yardstick that measures the
so-called scale factor, $a=(1+z)^{-1}$, by which space was stretched
between some early (emission) time to the present (observation) time.  By
observing a series of known spectral lines from distant sources, it is
possible to measure the source redshift from the factor by which the
wavelengths of these lines wer increased relative to the values measured in
the laboratory.

\noindent
{\bf Hydrogen:} The most abundant element in the Universe, which is made of
an electron in a bound orbit around a proton.

\noindent
{\bf Ionization:} The freeing of an electron out of its bound orbit around
a nucleus. Ionization can be caused by the absorption of a photon (with
energy greater than the binding energy of the electron), or as a result of
an energetic collision with another particle. Cosmic hydrogen was ionized
during the early hot phase of the Universe, and then formed abundantly at
$z=10^3$ when the cosmic temperature first dipped below a few thousand
degrees K.  The term `re-ionization' is used to denote the subsequent
process by which cosmic hydrogen was ionized once again hundreds of million
of years later. Today most of the cosmic gas is ionized.

\noindent
{\bf Galaxies:} Condensations of matter in the Universe which make stars
out of gas. The inner stellar core of galaxies is often surrounded by a
dark matter `halo', whose presence is inferred through its gravitational
effects.

\noindent
{\bf Power spectrum:} The square of the amplitude of the Fourier
decomposition of the distribution of some quantity (such as the matter
density or the radiation brightness) in the Universe.  The power spectrum,
$P({\bf k})$, is a function of the wavevector ${\bf k}$ of the Fourier
modes.

\noindent
{\bf Parsec:} A unit of length, equal to $3\times 10^{18}~{\rm cm}$,
abbreviated as `pc'. A kilo-parsec (kpc) is a thousand pc, a mega-parsec
(Mpc) is a million pc, and a giga-parsec (Gpc) is a billion pc.

\noindent

\vfill\eject

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%\subsection*{Biographical Sketch}
%
%{\bf Abraham (Avi) Loeb} is a theoretical physicist working on astrophysics
%and cosmology. He is currently a professor of astronomy and the director of
%the Institute for Theory and Computation (ITC) at Harvard University. Loeb
%was born in Israel in 1962 and took part in the national Talpiot program
%before receiving a BSc, MSc, and PhD degrees (in Plasma Physics) at age 24
%from the Hebrew University in Jerusalem.
%
%Between 1988-1993, he was long-term member at the Institute for Advanced
%Study in Princeton, where he started to work in theoretical
%astrophysics. In 1993 he moved to Harvard University as an assistant
%professor in the department of astronomy, where he was tenured three years
%later. He was given a number of awards including the Guggenheim Fellowship
%in 2002. He also holds a visiting professorship at the Weizmann Institute
%of Science. He is broadly regarded as an authority on studies of the first
%stars and of supermassive black holes in galaxies. His published work
%includes nearly 300 papers in refereed journals, as well as a book and a
%patent.
%
%Prof. Loeb has worked on broad range of research areas in astrophysics and
%cosmology, including the first stars, the epoch of reionization, the
%formation and evolution of massive black holes, gravitational lensing,
%gamma-ray bursts, and 21-cm cosmology. Some of his papers are considered as
%pioneering in areas that have become by now the focus of established
%communities of astrophysicists. In particular, Loeb was among the first
%theorists to trigger the current research on the first stars and
%quasars. In a series of papers with his students and postdocs, he addressed
%how and when the first stars and black holes formed and what effects they
%had on the young universe.  In 2006 Prof. Loeb was featured in a cover
%story of TIME magazine on the first stars and in a Scientific American
%article on the dark ages of the Universe. In 2008 Prof. Loeb was featured
%in a cover story of Smithsonian magazine on black holes and in a cover story
%of Astronomy magazine on the future collision between the Milky-Way and
%Andromeda.



