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%TITLES AND AUTHORS
\title[The properties of Ly$\alpha$-emitting galaxies in 
hierarchical galaxy formation models]
{The properties of Ly$\alpha$ emitting galaxies in 
hierarchical galaxy formation models}
 
 
\author[Le Delliou  et al.]{
M. Le Delliou\thanks{E-mail: delliou@alfmail.cii.fc.ul.pt (MLeD)},
C. G. Lacey,
C. M. Baugh
and S. L. Morris\\
Institute for Computational Cosmology, Department of Physics,
University of Durham, South Road, Durham, DH1 3LE, UK. 
}
 

\begin{document}

\maketitle
  
\begin{abstract}
We present detailed predictions for the properties of
Ly$\alpha$-emitting galaxies in the framework of the $\Lambda$CDM
cosmology, calculated using the semi-analytical galaxy formation model
{\tt GALFORM}. We explore a model which assumes a top-heavy IMF
in starbursts, and which has previously been shown to explain the
sub-mm number counts and the luminosity function of Lyman-break
galaxies at high redshift.  We show that this model, with the
simple assumption that a fixed fraction of Ly$\alpha$ photons escape
from each galaxy, is remarkably successful at explaining the observed
luminosity function of Ly$\alpha$ emitters over the redshift range
$3<z<6.6$. We also examine the distribution of Ly$\alpha$
equivalent widths and the broad-band continuum magnitudes of emitters,
which are in good agreement with the available observations. We look
more deeply into the nature of Ly$\alpha$ emitters, presenting
predictions for fundamental properties such as the stellar mass 
and radius of the emitting galaxy and the mass of the host dark
matter halo. The model predicts that the clustering of Ly$\alpha$
emitters at high redshifts should be strongly biased relative to the
dark matter, in agreement with observational estimates. We also
present predictions for the luminosity function of Ly$\alpha$ emitters
at $z>7$, a redshift range which is starting to be be probed by
near-IR surveys and using new instruments such as DAzLE.
\end{abstract}

\begin{keywords}
galaxies:evolution -- galaxies:formation -- galaxies:high-redshift --
galaxies:luminosity function -- cosmology:theory
\end{keywords}


\section{Introduction }

After an unpromising start, searches for Ly$\alpha$ emission are now
proving to be a powerful means of detecting star-forming galaxies at
high redshift \citep[e.g.][]{Hu98, Pascarelle98, Kudritzki00},
competing in observing efficiency with techniques such as broad-band
searches for Lyman-break galaxies.  The next generation of
near-infrared instrumentation \citep[e.g.][]{DAZLE} will in principle
allow Ly$\alpha$ emitting galaxies to be found up to $z \sim 20$,
permitting a probe of the star formation history of the Universe
before the epoch when reionization is thought to have taken place.

There are in fact a number of different mechanisms which can produce
Ly$\alpha$ emission from high redshift objects. (1) Gas in galaxies
which is photo-ionized by young stars will emit Ly$\alpha$ as
hydrogen atoms recombine; this was originally proposed as a signature
of primeval galaxies by \citet{Partridge67}. (2) Gas can alternatively
be ionized by radiation from an active galactic nucleus (AGN). 
%this
%can produce Ly$\alpha$ emission either from the AGN itself (as in
%quasars) or from an extended gaseous halo around it. 
(3) Intergalactic gas clouds are predicted to emit Ly$\alpha$
recombination radiation due to ionization of the gas by the
intergalactic ultraviolet background \citep[e.g.][]{Hogan87,
  Cantalupo05}. (4) Gas within a dark matter halo which is cooling and
collapsing to form a galaxy may radiate much of the gravitational
collapse energy by collisionally-excited Ly$\alpha$ emission
\citep[e.g.][]{Haiman00, Fardal01}. (5) Finally, Ly$\alpha$ can also
be emitted from gas which has been shock heated by galactic winds or
by jets in radio galaxies \citep[e.g.][]{McCarthy87}. The majority of
high-redshift Ly$\alpha$ emitters (LAEs) detected so far are compact,
and appear to be individual galaxies in which the Ly$\alpha$ emission
is powered by photoionization of gas by young stars. Ly$\alpha$
surveys have also found another class of emitter, the so-called
Ly$\alpha$ blobs, in which the Ly$\alpha$ emission is much more
extended than individual galaxies, and may be powered partly by AGNs
or gas cooling \citep{Steidel00, Bower04, Matsuda04}. We will be
focusing in this paper on Ly$\alpha$ emission powered by young stars,
and so will not consider the Ly$\alpha$ blobs further.

To date, there has been relatively little theoretical work on trying
to predict the properties of star-forming Ly$\alpha$-emitting
galaxies within a realistic galaxy formation framework.
%Most work has 
%focused upon following the propagation of Ly$\alpha$ photons 
%through the interstellar medium (ISM ref).
\citet{Haiman99} made predictions for the number of emitters based on
the halo mass function and using ad-hoc assumptions linking Ly$\alpha$
emission to halo mass, while \citet{Barton04} made predictions for
very high redshifts ($z>7$) based on a gas-dynamical simulation.
\citet{Furlanetto05} used gas-dynamical simulations to calculate
Ly$\alpha$ emission both from star-forming objects and from the
intergalactic medium in the redshift range $0<z<5$.  However, the
first calculation of the abundance of Ly$\alpha$ emitters based on a
detailed hierarchical galaxy formation model was that of
\citet[][hereafter Paper I]{LeD05}.  In Paper~I, we used the {\tt
GALFORM} semi-analytical galaxy formation model to predict the abundance
of star-forming Ly$\alpha$ emitters as a function of redshift in the
cold dark matter (CDM) model. The {\tt GALFORM} model computes the
assembly of dark matter halos by mergers, and the growth of galaxies
both by cooling of gas in halos and by galaxy mergers. It calculates
the star formation history of each galaxy, including both quiescent
star formation in galaxy disks and also bursts triggered by galaxy mergers,
as well as the feedback effects of galactic winds driven by supernova
explosions. In Paper~I, we found that a very simple model, in which a
fixed fraction of Ly$\alpha$ photons escape from each galaxy,
regardless of its other properties, gave a surprisingly good match to
the total numbers of Ly$\alpha$ emitters detected in different surveys
over a range of redshifts.  We also explored the impact of varying
certain parameters in the model, such as the redshift of reionization
of the intergalactic medium, on the abundance of emitters.

In this paper, we explore in more detail the fiducial model of Paper~I
(based on an $\Omega_{m}=0.3$, spatially flat, $\Lambda$CDM model with
a reionization redshift of $10$). We use the full capability of the
{\tt GALFORM} model to predict a wide range of galaxy properties,
connecting various observables to Ly$\alpha$ emission. The galaxy
formation model we use is the same as that proposed by
\citet{Baugh05}. A critical assumption of this model is that
stars formed in starbursts have a top-heavy initial mass function
(IMF), while stars formed quiescently in galactic disks have a solar
neighbourhood IMF. We showed in \citeauthor{Baugh05} that, within the
framework of $\Lambda$CDM, the top-heavy IMF is essential for matching
the counts and redshifts of sub-millimetre galaxies and the luminosity
function of Lyman break galaxies at $z=3$ (once dust extinction is
included), while remaining consistent with galaxy properties in the
local universe such as the optical and far-IR luminosity functions and
galaxy gas fractions and metallicities. More detailed comparisons of
this model with observations of Lyman-break galaxies and of galaxy
evolution in the IR will be presented in Lacey \etal (2005a, 2005b, in
preparation).  The assumption of a top-heavy IMF is controversial,
but underpins the success of the model in explaining the 
high-redshift sub-mm and Lyman-break galaxies. It is therefore
important to test this model against as many observables as
possible. \citet{Nagashima05a} showed that a top-heavy IMF seems to be
required to explain the metal content of the hot intracluster gas in
galaxy clusters, and \citet{Nagashima05b} showed that a similar
top-heavy IMF also seems to be necessary to explain the observed
abundances of $\alpha$-elements (such as Mg) in the stellar
populations of elliptical galaxies.   In the present paper, we
explore the predictions of the \citet{Baugh05} model for the
properties of Ly$\alpha$-emitting galaxies and compare them with
observational data. We emphasize that our aim here is to explore in
detail a particular galaxy formation model which has been shown to
satisfy a wide range of other observational constraints, rather than
to conduct a survey of Ly$\alpha$ predictions for different model
parameters.


In Section~\ref{sec:GALFORM}, we give an outline of the {\tt GALFORM}
model, focusing on how the predictions we present later on are
calculated. Section~\ref{sec:lf-evoln} examines the evolution of the
Ly$\alpha$ luminosity function, and compares the model predictions
with observational data over the redshift range $3 \lsim z \lsim
7$. In Section~\ref{sec:obs-props}, we compare a selection of observed
properties of Ly$\alpha$ emitters with the model predictions. In
Section~\ref{sec:phys-props}, we look at some other predictions of the
model, most of which  cannot currently be compared directly with
observations. Section~\ref{sec:high-z} extends the predictions for the
Ly$\alpha$ luminosity function to $z> 7$. We present our
conclusions in Section~\ref{sec:conc} .
  

\section{Galaxy formation model}
\label{sec:GALFORM}

We use the semi-analytical model of galaxy formation, {\tt GALFORM},
to predict the Ly$\alpha$ emission and many other properties of
galaxies as a function of redshift.  The general methodology and
approximations behind the {\tt GALFORM} model are set out in detail in
\citet{Cole00}. The particular model that we use in this paper is the
same as that described by \citet{Baugh05}.  The background cosmology
is a cold dark matter universe with a cosmological constant
($\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, $\Omega_{b}=0.04$, $h
\equiv H_0/100\kms\Mpc^{-1}=0.7$, $\sigma_8=0.93$).
%The model is similar in many respects to that
%used by Benson et al. (2003), but has a number of important
%differences that are responsible for the model reproducing the
%luminosity function of Lyman-break galaxies and the number counts of
%sources at submillimetre wavelengths, at the same time as giving a
%reasonable match to the present day galaxy population. The key changes
%are set out in Baugh et al. (2005). 
Below we review the physics behind the particular model predictions
that we highlight in this paper.

The {\tt GALFORM} model follows the main processes which shape the
formation and evolution of galaxies. These include: (i) the collapse
and merging of dark matter halos; (ii) the shock-heating and radiative
cooling of gas inside dark halos, leading to formation of galaxy
disks; (iii) quiescent star formation in galaxy disks; (iv) feedback
both from supernova explosions and from photo-ionization of the IGM; (v)
chemical enrichment of the stars and gas; (vi) galaxy mergers driven
by dynamical friction within common dark matter halos, leading to
formation of stellar spheroids, and also triggering bursts of star
formation.  The end product of the calculations is a prediction of the
number of galaxies that reside within dark matter haloes of different
masses. The model predicts the stellar and cold gas masses of the
galaxies, along with their star formation and merger histories, and
their sizes and metallicities.

The prescriptions and parameters for the different processes which we
use in this paper are identical to \citet{Baugh05}.  Feedback is
treated in a similar way to \citet{Benson03}: energy injection by
supernovae reheats some of the gas in galaxies and returns it to the
halo, but also ejects some gas from halos as a ``superwind'' - the
latter is essential for reproducing the observed cutoff at the bright
end of the present-day galaxy luminosity function. We also include
feedback from photo-ionization of the IGM: following reionization
(i.e. for $z<z_{\rm reion}$), we assume that gas cooling in halos with
circular velocities $V_c < 60\, \kms$ is completely suppressed. We
assume in this paper that reionization occurs at $z_{\rm reion}=10$,
chosen to be intermediate between the low value $z\sim 6$ suggested by
measurements of the Gunn-Peterson trough in quasars \citep{Becker01}
and the high value $z\sim 20$ suggested by the WMAP measurement of
polarization of the microwave background \citep{Kogut03}. Our model
has two different IMFs: quiescent star formation in galactic disks is
assumed to produce stars with a solar neighbourhood IMF (we use the
\citet{Kennicutt83} paramerization, with slope $x=0.4$ below $1
M_{\odot}$ and $x=1.5$ above), whereas bursts of star formation
triggered by galaxy mergers are assumed to form stars with a
top-heavy, flat IMF with slope $x=0$ (where the Salpeter slope is
$x=1.35$).  In either case, the IMF covers the mass range $0.15 <m<
120 \Msol$. As mentioned in the Introduction, the choice of a flat IMF
in bursts is essential for the model to reproduce the observed counts
of galaxies at sub-mm wavelengths. The parameters for star formation
in disks and for triggering bursts and morphological transformations
in galaxy mergers are given in \citet{Baugh05}.

The sizes of galaxies are computed as in \citet{Cole00}: gas which
cools in a halo is assumed to conserve its angular momentum as it
collapses, forming a rotationally-supported galaxy disk; the radius of
this disk is then calculated from its angular momentum, including the
gravity of the disk, spheroid (if any) and dark halo. Galaxy spheroids
are built up both from pre-existing stars in galaxy mergers, and from
the stars formed in bursts triggered by these mergers; the radii of
spheroids formed in mergers are computed using an energy conservation
argument. In calculating the sizes of disks and spheroids, we include
the adiabatic contraction of the dark halo due to the gravity of the
baryonic components.

Given the star formation and metal enrichment history of a galaxy,
{\tt GALFORM} computes the spectrum of the integrated stellar
population using a population synthesis model based on the Padova
stellar evolution tracks \citep[see][for
details]{Granato00}. Broad-band magnitudes are then computed by
redshifting the galaxy spectrum and convolving it with the filter
response functions. We include extinction of the stellar continuum by
dust in the galaxy; this is computed based on a two-phase model of the
dust distribution, in which stars are born inside giant molecular
clouds and then leak out into a diffuse dust medium \citep[see][for
more details]{Granato00}. The optical depth for dust extinction of the
diffuse component is calculated from the mass and metallicity of the
cold gas and the sizes of the disk and bulge. We note that the
extinction predicted by our model in which the stars and dust are
mixed together is very different from what one obtains if all of the
dust is in a foreground screen (as is commonly assumed in other
theoretical models). Finally, we also include the effects on the
observed stellar continuum of absorption and scattering of radiation
by intervening neutral hydrogen along the line of sight to the galaxy;
we calculate this IGM attenuation using the formula of
\citet{Madau95}, which is based on the observed statistics of neutral
hydrogen absorbers seen in quasar spectra.

We compute the Ly$\alpha$ luminosities of galaxies by the following
procedure: (i) The model calculates the integrated stellar spectrum of
the galaxy as described above, based on its star formation history,
and including the effects of the distribution of stellar metallicities
and of variations in the IMF. (ii) We compute the rate of production
of Lyman continuum (Lyc) photons by integrating over the stellar
spectrum, and assume that all of these ionizing photons are absorbed
by neutral hydrogen within the galaxy. We assume photoionization
equilibrium applies within each galaxy, producing Ly$\alpha$ photons
according to case~B recombination \citep[e.g.][]{Osterbrock}. We note
that for solar metallicity, 11 times as many Lyc and Ly$\alpha$
photons are produced per unit mass of stars formed for our top-heavy
(burst) IMF as compared to our solar neighbourhood (disk) IMF. (iii)
The observed Ly$\alpha$ flux or luminosity of a galaxy depends on the
fraction $f_{\rm esc}$ of Ly$\alpha$ photons which escape from the
galaxy. Ly$\alpha$ photons are resonantly scattered by neutral
hydrogen, and absorbed by dust. Early estimates of this process
\citep[e.g.][]{Charlot91} showed that only a tiny fraction of
Ly$\alpha$ photons should escape from a static neutral galaxy ISM if
even a tiny amount of dust is present. Many star-forming galaxies are
nonetheless observed to have significant Ly$\alpha$ luminosities
\citep[e.g.][]{Kunth98, Pettini01}, and this is generally ascribed to
the presence of galactic winds in these systems, which allow
Ly$\alpha$ photons to escape after many fewer resonant
scatterings. Radiative transfer calculations of Ly$\alpha$ through
winds have shown that this process can explain the asymmetric
Ly$\alpha$ line profiles which are typically observed
\citep[e.g.][]{Ahn04}. The effects of radiative transfer of
Ly$\alpha$ through clumpy dust and gas have been considered by
\citet{Neufeld91} and \citet{Hansen05}.

Calculating Ly$\alpha$ escape fractions from first principles is
clearly very complicated, and so we instead adopt a simpler
approach. In Paper~I, we found that assuming a fixed escape fraction
$f_{\rm esc}$ for each galaxy, regardless of its dust properties,
resulted in a surprisingly good agreement between the predicted number
counts of emitters and the available observations. In that paper, we
chose $f_{\rm esc}=0.02$ to match the number counts at $z\approx 3$ at
a flux $f\approx 2\times 10^{-17} \ergs$, and we use the same value of
$f_{\rm esc}$ in this paper. Although this extreme simplification of a
constant escape fraction may seem implausible, it does give a
reasonably good match to the observed Ly$\alpha$ luminosity functions
and equivalent widths at different redshifts, as we show in the next
sections.

Our calculations do not include any attenuation of the Ly$\alpha$ flux
from a galaxy by propagation through the IGM. Ly$\alpha$ photons can
be scattered out of the line-of-sight by any neutral hydrogen in the
IGM close to the galaxy. If the emitting galaxy is at a redshift
before reionization, when the IGM was still mostly neutral, this could
in principle strongly suppress the observed Ly$\alpha$ flux
\citep{Miralda98}. However, various effects can greatly reduce the
amount of attenuation: ionization of the IGM around the galaxy
\citep{Madau00, Haiman02}, clearing of the IGM by galactic winds,
gravitational infall of the IGM towards the galaxy, and redshifting of
the Ly$\alpha$ emission by scattering in a wind \citep{Santos04a}. In
any case, since measurements of Gunn-Peterson absorption in quasars
show that reionization must have occured at $z\gsim 6.5$, attenuation
of Ly$\alpha$ fluxes by the IGM should not affect our predictions for
$z\lsim 6.5$, but only our predictions for very high redshifts given
in Section~\ref{sec:high-z}.


%% include graphics the MODERN way
%% Fig 1
\begin{figure}

\includegraphics[width=8.5cm]{figs/lfcum_lowz}

\caption
{ The predicted evolution with redshift of the cumulative Ly$\alpha$
luminosity function, defined as the comoving number density of
galaxies with Ly$\alpha$ luminosities brighter than
$L_{Ly\alpha}$. The model predictions are shown for selected redshifts
in the interval $z=0$ to $z=7$.  }

\label{fig:LF_lowz} % \label must come AFTER \caption command

\end{figure}





\section{Evolution of the Ly$\alpha$ luminosity function}
\label{sec:lf-evoln}

A basic prediction of our model is the evolution of the luminosity
function of Ly$\alpha$ emitters with redshift. This depends on the
distribution of star formation rates in quiescent and starburst
galaxies (with solar neighbourhood and top-heavy IMFs respectively),
and on the metallicity with which the stars are formed. Paper~I showed
predictions for the cumulative number counts of emitters per unit
redshift as a function of observed Ly$\alpha$ flux. Here we focus on a
closely related quantity, the cumulative space density of emitters as
a function of Ly$\alpha$ luminosity at different redshifts.
Fig.\ref{fig:LF_lowz} shows the cumulative luminosity function of
Ly$\alpha$ emitters predicted by our standard model for a set of
redshifts over the interval $z=0-7$. The model luminosity function
initially gets brighter with increasing redshift, peaking at $z=3$,
before declining again in number density at even higher redshifts. The
increase in the luminosity function from $z=0$ to $z\sim 3$ is driven
both by the increase in galaxy star formation rates, and by the
increasing fraction of star formation occuring in bursts (which have a
top-heavy IMF). As shown in Fig.1 in \citet{Baugh05}, the model
predicts that the fraction of all star formation occuring in bursts
increases from $\sim 5\%$ at $z=0$ to $50\%$ at $z\sim 3.5$ and then
to $\sim 80\%$ at $z\gsim 6$.

Some data are given in Table~\ref{tab:data}:


%% Table 1
\begin{table*}

\begin{tabular}
{llllllllllll}
\hline 
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11)\\
z&
\( \Delta z \)&
\( f \) &
\( \frac{d^{2}N}{dzd\Omega }(>f) \)&
\( \Delta \left( \frac{d^{2}N}{dzd\Omega }\right)  \) 
&
Nobj&
Area &
Fcorr &
method &
confirmation &
ref.\\
\hline
\( 2.42 \)&
\( 0.14 \)&
\( 20 \)&
\( 0.33 \)&
\( 0.04 \)&
\( 58 \)&
\( 1200 \)&
\( 0.65 \)&
NBF &
EW/colour &
%\cite{Stiavelli et al. 01}
Sti01\\
\hline 
\( 3.09 \)&
\( 0.07 \)&
\( 2 \)&
\( 2.3 \)$^*$&
\( 0.3 \)&
\( 12 \)$^*$&
\( 78 \)&
\( 0.94 \)&
NBF&
EW/colour &
%\cite{Steidel et al. 00}
Ste00\\
\( 3.13 \) &
\( 0.04 \)&
\( 2 \)&
\( 3.8 \) &
\( 1.3 \)&
\( 8 \)&
\( 49 \)&
\( 0.7 \)&
NBF &
spec on 10 &
%\cite{Kudritzki et al. 00}
K00\\
\( 3.43 \)&
\( 0.06 \)&
\( 1.5 \)&
\( 3.5 \)&
\( 0.9 \)&
\( 16 \)&
\( 75 \)&
\( 0.87 \)&
NBF &
spec on 15 &
%\cite{Hu et al. 98}\\
H98\\
%\( 3.43 \)& \( 0.063 \)& \( 2 \)& \( 3.5 \)& \( 1.1 \)& \( 10 \)& \(
%46 \)& \( 0\% \)& NBF& \cite{Cowie & Hu 98}\\ % superceded by Hu etal 98
\( 3.72 \)&
\( 0.23 \)&
\( 6.4 \)&
\( 0.26 \)&
\( 0.09 \)&
\( 8 \)&
\( 130 \)&
\( 0.35 \)&
NBF &
colours &
%\cite{Fujita et al. 2003}\\
F03\\
\hline
\( 4.39 \)&
\( 0.07 \)&
\( 2.6 \)&
\( 0.97 \)&
\( 0.11 \)&
\( 75 \)&
\( 1100 \)&
\( 0.33 \)&
NBF &
spec on 3 &
%\cite{Rhoads et al. 00}
R00\\
\( 4.54 \)&
\( 0.06 \)&
\( 1.5 \)&
\( 1.3 \)&
\( 0.9 \)&
\( 2 \)&
24 &
\( 0.67 \)&
NBF &
spec on 3 &
%\cite{Hu et al. 98}
H98\\
\( 4.79 \)&
\( 0.08 \)&
\( 0.5 \)&
\( 0.46 \)&
\( 0.07 \)&
\( 41 \)&
\( 1100 \)&
\( 0.8 \)&
NBF &
\( - \) &
%\cite{Shimasaku et al. 04}
S04\\
\( 4.86 \)&
\( 0.06 \)&
\( 0.5 \)&
\( 0.52 \)&
\( 0.09 \)&
\( 34 \)&
\( 1100 \)&
\( 0.8 \)&
NBF &
spec on 5 &
%\cite{Shimasaku et al. 03}
S03\\
\( 4.86 \)&
\( 0.06 \)&
\( 0.3 \)&
\( 1.6 \)&
\( 0.2 \)&
\( 52 \)&
\( 540 \)&
\( 0.6 \)&
NBF &
colours &
%\cite{Ouchi et al. 03}
O03\\
\hline
\( 5.1 \)&
\( 1.0 \)&
\( 0.012 \)$^{**}$&
\( 48 \)$^{**}$&
\( 48 \)&
\( 1 \)&
\( 0.02 \)$^{**}$&
\( - \)  &
LS &
\( - \) &
%\cite{Santos et al. 04}
Sa04\\
\( `` \)&
\( `` \)&
\( 0.037 \)$^{**}$&
\( 30 \)$^{**}$&
\( 15 \)&
\( 4 \)&
\( 0.14 \)$^{**}$&
\( - \) &
\( `` \)&
\( - \) &
``\\
\( `` \)&
\( `` \)&
\( 0.12 \)$^{**}$&
\( 4.0 \)$^{**}$&
\( 2.3 \)&
\( 3 \)&
\( 0.75 \)$^{**}$&
\( - \) &
\( `` \)&
\( - \) &
``\\
\( `` \)&
\( `` \)&
\( 0.37 \)$^{**}$&
\( 0.89 \)$^{**}$&
\( 0.51 \)&
\( 3 \)&
\( 3.4 \)$^{**}$&
\( - \) &
\( `` \)&
\( - \) &
``\\
\( `` \)&
\( `` \)&
\( 1.2 \)$^{**}$&
\( 0.14 \)$^{**}$&
\( 0.14 \)&
\( 1 \)&
\( 7.5 \)$^{**}$&
\( - \) &
\( `` \)&
\( - \) &
``\\
\( 5.3 \)&
\( 1.0 \)&
\( 2? \)&
\( 2.3 \)&
\( 1.0 \)&
\( 5 \)&
\( 2.2 \)&
\( - \)&
LS &
\( - \) &
%\cite{Dawson et al. 01}
D01\\
%\( 5.7 \)&\( 0.12 \)&\( 1.5 \)&\( 0.20 \)&\( 0.06 \)&\( 13 \)&
%\( 550\)&\( 0.72 \)&CCS&\cite{Rhoads & Malhotra 01}\\ % superseded by
%Rhoads etal 2003
\( 5.7 \)&
\( 0.13 \)&
\( 1.5 \)&
\( 0.14 \)&
\( 0.04 \)&
\( 13 \)&
\( 710 \)&
\( 0.75 \)&
NBF &
spec on 4 &
%\cite{Rhoads et al. 03}
R03\\
\hline
\( 6.56 \)&
\( 0.10 \)&
\( 0.6 \)$^{**}$&
\( 20 \)$^{**}$&
\( 20 \)&
\( 1 \)&
\( 0.46 \)$^{**}$&
\( 1 \)&
NBF &
spec on 1 &
%\cite{Hu et al. 02}
H02\\
% \( 6.56 \)& \( 0.10 \)& \( 2.7 \)& \tiny\(<0.0036\)& ? & \( 0 \)& ?
% & n/a & ? & ? & \cite{Hu et al. 02}\\ % Hu etal upper limit
\( 6.56 \)&
\( 0.11 \)&
\( 0.9 \)&
\( 0.18 \)&
\( 0.05 \)&
\( 16 \)&
\( 810 \)&
\( 0.22 \)&
NBF &
spec on 9 &
%\cite{Kodaira et al. 03}
K03\\
\hline
\end{tabular}

\caption{
\label{tab:data}
Data Compilation. The data are divided into unit redshift intervals:
the following symbols are used to denote data from each redshift
interval in the figures ($\blacksquare$ :[$z<3$], $\blacktriangledown$
:[$3<z<4$], $\blacktriangle$ :[$4<z<5$], {\LARGE $\circ$}:[$z=5.1$],
{\LARGE $\bullet$}:[$5<z<6$], {{\bf X}}:[$6<z<7$]).  {\bf Col.1:}
redshift; {\bf Col.2:} redshift interval; {\bf Col.3:} Ly-$\alpha$
flux (in $10^{-17} {\rm ergs\, cm}^{-2} {\rm s}^{-1}$); {\bf Col.4:}
cumulative counts per unit solid angle per unit redshift (in ${\rm
arcmin}^{-2}$); {\bf Col.5:} Poisson error on counts (in ${\rm
arcmin}^{-2}$); {\bf Col.6:} number of Ly-$\alpha$ emitters; {\bf
Col.7:} area of survey (in ${\rm arcmin}^{2}$); {\bf Col.8:} factor
applied to correct for contamination by low-z interlopers; {\bf
Col.9:} method (NBF=narrow band filter, LS=long-slit spectroscopy);
{\bf Col.10:} method used to reject or correct for low-z interlopers
(EW=equivalent width, spec on N = follow-up spectroscopy of N
objects); {\bf Col.11:} reference (D01: Dawson \etal 2001; F03: Fujita
\etal 2003; H98: Hu \etal 1998; H02: Hu \etal 2002; K03: Kodaira \etal
2003; K00: Kudritzki \etal 2000; O03: Ouchi \etal 2003; R00: Rhoads
\etal 2000; R03: Rhoads \etal 2003; Sa04: Santos \etal 2004; S03:
Shimasaku \etal 2003; S04: Shimasaku \etal 2004; Ste00: Steidel \etal
2000; Sti01: Stiavelli \etal 2001) }

\footnotesize
\noindent{$^*$ corrected for factor 6 overdensity
~~$^{**}$ corrected for gravitational lensing}

\end{table*}



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\end{document}





















\begin{tabular}
{llllllllllll}
\hline 
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11)\\
z &
\( \Delta z \)&
\( f \) &
\( \frac{d^{2}N}{dzd\Omega }(>f) \)&
\( \Delta \left( \frac{d^{2}N}{dzd\Omega }\right)  \) &
Nobj & Area & Fcorr & method & confirmation & ref.\\
\hline
 2.42 & 0.14 & 20 & 0.33 & 0.04 & 58 & 1200 &
 0.65 & NBF & EW/colour & Sti01\\
\hline 
 3.09 &  0.07 & 2 & $2.3^*$ & 0.3 & $12^*$ & 78 &
 0.94 &
NBF & EW/colour & Ste00\\
$3.13$ &
$0.04$&
$2$&
$3.8$ &
$1.3$&
$8$&
$49$&
$0.7$&
NBF &
spec on 10 &
K00\\
$3.43$&
$0.06$&
$1.5$&
$3.5$&
$0.9$&
$16$&
$75$&
$0.87$&
NBF &
spec on 15 &
H98\\
$3.72$&
$0.23$&
$6.4$&
$0.26$&
$0.09$&
$8$&
$130$&
$0.35$&
NBF &
colours &
%\cite{Fujita et al. 2003}\\
F03\\
\hline
$4.39$&
$0.07$&
$2.6$&
$0.97$&
$0.11$&
$75$&
$1100$&
$0.33$&
NBF &
spec on 3 &
%\cite{Rhoads et al. 00}
R00\\
$4.54$&
$0.06$&
$1.5$&
$1.3$&
$0.9$&
$2$&
24 &
$0.67$&
NBF &
spec on 3 &
%\cite{Hu et al. 98}
H98\\
$4.79$&
$0.08$&
$0.5$&
$0.46$&
$0.07$&
$41$&
$1100$&
$0.8$&
NBF &
$-$ &
%\cite{Shimasaku et al. 04}
S04\\
$4.86$&
$0.06$&
$0.5$&
$0.52$&
$0.09$&
$34$&
$1100$&
$0.8$&
NBF &
spec on 5 &
%\cite{Shimasaku et al. 03}
S03\\
$4.86$&
$0.06$&
$0.3$&
$1.6$&
$0.2$&
$52$&
$540$&
$0.6$&
NBF &
colours &
%\cite{Ouchi et al. 03}
O03\\
\hline
$5.1$&
$1.0$&
$0.012$$^{**}$&
$48$$^{**}$&
$48$&
$1$&
$0.02$$^{**}$&
$-$  &
LS &
$-$ &
%\cite{Santos et al. 04}
Sa04\\
$``$&
$``$&
$0.037$$^{**}$&
$30$$^{**}$&
$15$&
$4$&
$0.14$$^{**}$&
$-$ &
$``$&
$-$ &
``\\
$``$&
$``$&
$0.12$$^{**}$&
$4.0$$^{**}$&
$2.3$&
$3$&
$0.75$$^{**}$&
$-$ &
$``$&
$-$ &
``\\
$``$&
$``$&
$0.37$$^{**}$&
$0.89$$^{**}$&
$0.51$&
$3$&
$3.4$$^{**}$&
$-$ &
$``$&
$-$ &
``\\
$``$&
$``$&
$1.2$$^{**}$&
$0.14$$^{**}$&
$0.14$&
$1$&
$7.5$$^{**}$&
$-$ &
$``$&
$-$ &
``\\
$5.3$&
$1.0$&
$2?$&
$2.3$&
$1.0$&
$5$&
$2.2$&
$-$&
LS &
$-$ &
%\cite{Dawson et al. 01}
D01\\
%$5.7 \)&\( 0.12 \)&\( 1.5 \)&\( 0.20 \)&\( 0.06 \)&\( 13 \)&
%\( 550\)&\( 0.72 \)&CCS&\cite{Rhoads & Malhotra 01}\\ % superseded by
%Rhoads etal 2003
\( 5.7 \)&
\( 0.13 \)&
\( 1.5 \)&
\( 0.14 \)&
\( 0.04 \)&
\( 13 \)&
\( 710 \)&
\( 0.75 \)&
NBF &
spec on 4 &
%\cite{Rhoads et al. 03}
R03\\
\hline
\( 6.56 \)&
\( 0.10 \)&
\( 0.6 \)$^{**}$&
\( 20 \)$^{**}$&
\( 20 \)&
\( 1 \)&
\( 0.46 \)$^{**}$&
\( 1 \)&
NBF &
spec on 1 &
%\cite{Hu et al. 02}
H02\\
% \( 6.56 \)& \( 0.10 \)& \( 2.7 \)& \tiny\(<0.0036\)& ? & \( 0 \)& ?
% & n/a & ? & ? & \cite{Hu et al. 02}\\ % Hu etal upper limit
\( 6.56 \)&
\( 0.11 \)&
\( 0.9 \)&
\( 0.18 \)&
\( 0.05 \)&
\( 16 \)&
\( 810 \)&
\( 0.22 \)&
NBF &
spec on 9 &
%\cite{Kodaira et al. 03}
K03\\
\hline
\end{tabular}
