source: Sophya/trunk/Cosmo/RadioBeam/sensfgnd21cm.tex@ 4069

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%     BAORadio : LAL/UPS, Irfu/SPP
%  21cm LSS P(k) sensitivity and foreground substraction 
%  R. Ansari, C. Magneville, J. Rich, C. Yeche et al 
%     2010 - 2011
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%% Definitions diverses
\newcommand{\HI}{$\mathrm{H_I}$ } 
\newcommand{\kb}{k_B}         % Constante de Boltzmann 
\newcommand{\Tsys}{T_{sys}}      % instrument noise (system) temperature    
\newcommand{\TTnu}{ T_{21}(\vec{\Theta} ,\nu)  } 
\newcommand{\TTnuz}{ T_{21}(\vec{\Theta} ,\nu(z))  } 
\newcommand{\TTlam}{ T_{21}(\vec{\Theta} ,\lambda)  } 
\newcommand{\TTlamz}{ T_{21}(\vec{\Theta} ,\lambda(z))  } 

\newcommand{\dlum}{d_L}
\newcommand{\dang}{d_A}
\newcommand{\hub}{ h_{70} }
\newcommand{\hubb}{ h_{100} }    % h_100

\newcommand{\etaHI}{ n_{\tiny HI} }
\newcommand{\fHI}{ f_{H_I}(z)}
\newcommand{\gHI}{ f_{H_I}}
\newcommand{\gHIz}{ f_{H_I}(z)}

\newcommand{\vis}{{\cal V}_{12} }
 
\newcommand{\LCDM}{$\Lambda \mathrm{CDM}$ }

\newcommand{\lgd}{\mathrm{log_{10}}}

%% Definition fonction de transfer 
\newcommand{\TrF}{\mathbf{T}}
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%%% Definition pour la section sur les param DE par C.Y
\def\Mpc{\mathrm{Mpc}}
\def\hMpcm{\,h \,\Mpc^{-1}}
\def\hmMpc{\,h^{-1}\Mpc}
\def\kperp{k_\perp}
\def\kpar{k_\parallel}
\def\koperp{k_{BAO\perp }}
\def\kopar{k_{BAO\parallel}}

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\begin{document}
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   \title{21 cm observation of LSS at z $\sim$ 1 }

   \subtitle{Instrument sensitivity and foreground subtraction}

   \author{
          R. Ansari
          \inst{1} \inst{2}
          \and
          J.E. Campagne \inst{2}  
         \and
          P.Colom  \inst{3} 
          \and
          J.M. Le Goff \inst{4}
          \and
          C. Magneville \inst{4}
          \and
          J.M. Martin  \inst{5} 
          \and
          M. Moniez \inst{2}  
        \and 
          J.Rich \inst{4}
          \and 
          C.Y\`eche \inst{4}
          }

   \institute{
     Universit\'e Paris-Sud, LAL, UMR 8607, CNRS/IN2P3,  F-91405 Orsay, France 
     \email{ansari@lal.in2p3.fr}
    \and
    CNRS/IN2P3, Laboratoire de l'Acc\'el\'erateur Lin\'eaire (LAL) 
    B.P. 34, 91898 Orsay Cedex, France
    \and
    LESIA, UMR 8109, Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon Cedex, France
    % \thanks{The university of heaven temporarily does not
    %                 accept e-mails}
    \and 
    CEA, DSM/IRFU, Centre d'Etudes de Saclay, F-91191 Gif-sur-Yvette, France 
    \and 
    GEPI, UMR 8111, Observatoire de Paris, 61 Ave de l'Observatoire, 75014 Paris, France
  }

   \date{Received August 5, 2011; accepted December 22, 2011}

% \abstract{}{}{}{}{} 
% 5 {} token are mandatory
 
  \abstract
  % context heading (optional)
  % {} leave it empty if necessary  
   { Large scale structures (LSS) in the universe can be traced using the neutral atomic hydrogen \HI through its 21 
cm emission. Such a 3D matter distribution map can be used to test the cosmological model and to constrain the dark energy 
properties or its equation of state. A novel approach, called intensity mapping, can be used to map the \HI distribution, 
using radio interferometers with a large instantaneous field of view and waveband.}
 % aims heading (mandatory)
  {We study the sensitivity of different radio interferometer configurations, or multi-beam 
instruments for observing LSS and baryon acoustic oscillations (BAO) in 21 cm, and we 
discuss the problem of foreground removal. }
  % methods heading (mandatory)
 { For each configuration, we determined instrument response by computing the $(\uv)$ or Fourier angular frequency 
plane  coverage using visibilities. The $(\uv)$ plane response determines the noise power spectrum, 
hence the instrument sensitivity for LSS P(k) measurement. We also describe a simple foreground subtraction method
of separating LSS 21 cm signal from the foreground due to the galactic synchrotron and radio source emission. }
  % results heading (mandatory)
   { We have computed the noise power spectrum for different instrument configurations, as well as the extracted 
   LSS power spectrum, after separating the 21cm-LSS signal from the foregrounds. We have also obtained 
  the uncertainties on the dark energy parameters for an optimized 21 cm BAO survey.}
  % conclusions heading (optional), leave it empty if necessary 
   { We show that a radio instrument with few hundred simultaneous beams and a collecting area of 
  \mbox{$\sim 10000 \, \mathrm{m^2}$} will be able to  detect BAO signal at redshift z $\sim 1$ and will be 
  competitive with optical surveys. }

   \keywords{ large-scale structure of Universe --
                 dark energy -- Instrumentation: interferometers -- 
                 Radio lines; galaxies -- Radio continuum: general }

   \maketitle
%
%________________________________________________________________
% {\color{red} \large \bf A discuter : liste des auteurs, plans du papier et repartition des taches 
% Toutes les figures sont provisoires }

\section{Introduction}

% {\color{red} \large \it Jim ( + M. Moniez ) }   \\[1mm] 
The study of the statistical properties of large scale structures (LSS) in the Universe and of their evolution 
with redshift is one of the major tools in observational cosmology. These structures are usually mapped through 
optical observation of galaxies that are used as tracers of the underlying matter distribution. 
An alternative and elegant approach for mapping the matter distribution, which uses neutral atomic hydrogen 
(\HI) as a tracer with intensity mapping, has been proposed in recent years (\cite{peterson.06}; \cite{chang.08}). 
Mapping the matter distribution using \HI 21 cm emission as a tracer has been extensively discussed in the literature
(\cite{furlanetto.06}; \cite{tegmark.09}) and is being used in projects such as LOFAR \citep{rottgering.06} or
MWA \citep{bowman.07} to observe reionization  at redshifts z $\sim$ 10. 

Evidence of the acceleration in the expansion of the universe has
accumulated over the last twelve years, thanks to the observation of
distant supernovae and cosmic microwave background (CMB) anisotropies and to detailed analysis of the LSS.  
A cosmological constant ($\Lambda$) or new cosmological 
energy density called {\em dark energy} has been advocated as the origin of this acceleration. 
dark energy is considered as one of the most intriguing puzzles in physics and cosmology. 
% Constraining the properties of this new cosmic fluid, more precisely 
% its equation of state is central to current cosmological researches. 
Several cosmological probes can be used to constrain the properties of this new cosmic fluid, 
more precisely its equation of state: the Hubble diagram, or the luminosity distance as a function 
of redshift of supernovae as standard candles, galaxy clusters, weak shear observations 
and baryon acoustic oscillations (BAO). 

BAO are features imprinted  in the distribution of galaxies, due to the frozen 
sound waves that were present in the photon-baryon plasma prior to recombination 
at \mbox{$z \sim 1100$}. 
This scale can be considered as a standard ruler with a comoving 
length  of \mbox{$\sim 150 \, \mathrm{Mpc}$}, and
these features have been first observed in the CMB anisotropies
and are usually referred to as {\em acoustic peaks} (\cite{mauskopf.00}; \cite{larson.11}). 
The BAO modulation has been subsequently observed in the distribution of galaxies 
at low redshift ( $z < 1$) in the galaxy-galaxy correlation function by the SDSS 
(\cite{eisenstein.05};  \cite{percival.07};  \cite{percival.10}), 2dGFRS  \cite{cole.05}, 
as well as WiggleZ \citep{blake.11} optical galaxy surveys.

Ongoing {\changemarkb surveys, such as BOSS} \citep{eisenstein.11}  or future surveys, 
{\changemarkb such as LSST} \citep{lsst.science}, 
plan to measure the BAO scale precisely in the redshift range 
$0 \lesssim z \lesssim 3$, using either optical observation of galaxies  
or 3D mapping of Lyman $\alpha$ absorption lines toward distant quasars 
(\cite{baolya}; \cite{baolya2}). 
Radio observation of the 21 cm emission of neutral hydrogen is %  ?ENG? appears as 
a very promising technique for mapping matter distribution up to redshift $z \sim 3$, 
and it complements optical surveys, especially in the optical redshift desert range 
$1 \lesssim z \lesssim 2$, and possibly up to the reionization redshift \citep{wyithe.08}.

In section 2, we  discuss the intensity mapping and its potential for measuring the
\HI mass distribution power spectrum. The method used in this paper to characterize 
a radio instrument response and sensitivity for $P_{\mathrm{H_I}}(k)$ is presented in section 3.
We also show  the results for the 3D noise power spectrum for several  instrument configurations.
The contribution of foreground emissions due to both the galactic synchrotron and radio sources
is described in section 4, as is  a simple component separation method. The performance of this 
method using two different sky models is also presented in section 4. 
The constraints that can be obtained on the dark energy parameters and DETF figure 
of merit for typical 21 cm intensity mapping survey are discussed in section 5.


%__________________________________________________________________

\section{Intensity mapping and \HI power spectrum}

% {\color{red} \large \it  Reza (+ P. Colom ?) } \\[1mm] 

\subsection{21 cm intensity mapping}
%%% 
Most of the cosmological information in the LSS is located on large scales
($ \gtrsim 1 \mathrm{deg}$), while the interpretation on the smallest scales 
might suffer from the uncertainties on the nonlinear  clustering effects.  
The BAO features in particular are at the degree angular scale on the sky 
and thus can be resolved easily with a rather modest size radio instrument 
(diameter $D \lesssim 100 \, \mathrm{m}$). The specific BAO clustering scale ($k_{\mathrm{BAO}}$) 
can be measured both in the transverse plane (angular correlation function, $k_{\mathrm{BAO}}^\perp$) 
or along the longitudinal (line of sight or redshift $k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of 
the Hubble parameter $H(z)$ can be obtained by comparing   the longitudinal and transverse 
BAO scales. A reasonably good redshift resolution $\delta z \lesssim 0.01$ is needed to resolve 
longitudinal BAO clustering, which is a challenge for photometric optical surveys.   

To obtain a measurement of the LSS power spectrum with small enough statistical
uncertainties (sample or cosmic variance),  a large volume of the universe should be observed, 
typically a few $\mathrm{Gpc^3}$. Moreover, stringent constraint on DE parameters can only be 
obtained when comparing the distance or Hubble parameter measurements with 
DE models as a function of redshift, which requires a significant survey depth $\Delta z \gtrsim 1$.
Radio instruments intended for BAO surveys must thus have large instantaneous field
of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$)
to explore large redshift domains.

Although the application of 21 cm radio survey to cosmology, in particular LSS mapping, has been 
discussed in length in the framework of large future instruments, such as the SKA (e.g \cite{ska.science}; \cite{abdalla.05}),
the method envisaged has mostly been through the detection of galaxies as \HI compact sources. 
However, extremely large radio telescopes are required to detected \HI sources at cosmological distances. 
The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the two polarizations
of a radio instrument characterized by an effective collecting area $A$, and system temperature $\Tsys$ can be written as 
\begin{equation}
S_{lim} = \frac{ \sqrt{2} \, \kb \, \Tsys }{ A \, \sqrt{t_{int} \delta \nu} } 
\end{equation} 
where $t_{int}$ is the total integration time and $\delta \nu$ the detection frequency band. In Table 
\ref{slims21} (left)  we computed the sensitivity for six different sets of instrument effective area and system 
temperature, with a total integration time of 86400 seconds (1 day) over a frequency band of 1 MHz. 
The width of this frequency band is well adapted  to detecting an \HI source with an intrinsic velocity 
dispersion of a few 100 km/s.
These detection limits should be compared with the expected 21 cm brightness
$S_{21}$ of compact sources, which can be computed using the expression below (e.g. \cite{binney.98}):
\begin{equation}
 S_{21}  \simeq  0.021 \mathrm{\mu Jy} \, \frac{M_{H_I} }{M_\odot}   \times 
\left( \frac{ 1\, \mathrm{Mpc}}{\dlum(z)} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v}  (1+z)
\end{equation}
 where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ the luminosity distance, and $\sigma_v$ 
the source velocity dispersion.  
{\changemark The 1 MHz bandwidth mentioned above is only used for computing the
galaxy detection thresholds and does not determine the total bandwidth or frequency resolution
of an intensity mapping survey.}
% {\color{red} Faut-il developper le calcul en annexe ? } 
 
In Table \ref{slims21} (right), we show the 21 cm brightness for 
compact objects with a total \HI \, mass of $10^{10} M_\odot$ and an intrinsic velocity dispersion of 
$200 \, \mathrm{km/s}$. The luminosity distance was computed for the standard 
WMAP \LCDM universe \citep{komatsu.11}. From $10^9$ to  $10^{10} M_\odot$ of neutral gas mass 
is typical of large galaxies \citep{lah.09}. It is clear that detecting \HI sources at cosmological distances
would require collecting area in the range of \mbox{$10^6 \, \mathrm{m^2}$}. 

Intensity mapping has been suggested as an alternative and economic method of mapping the 
3D distribution of neutral hydrogen by (\cite{chang.08}; \cite{ansari.08}; \citep{seo.10}). 
{\changemark There have also been attempts to detect the 21 cm LSS signal at GBT 
\citep{chang.10} and at GMRT \citep{ghosh.11}}.
In this approach, a sky brightness map with angular resolution 
\mbox{$\sim 10-30 \, \mathrm{arc.min}$} is created for a   %% ?ENG? was created ?
wide range of frequencies. Each 3D pixel  (2 angles $\vec{\Theta}$, frequency $\nu$, or wavelength $\lambda$)  
would correspond to a cell with a volume of $\sim 10^3 \mathrm{Mpc^3}$, containing ten to a hundred galaxies 
and a total \HI mass $ \sim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow, 
the observed frequency $\nu$ would be translated into the emission redshift $z$ through 
the well known relation
\begin{eqnarray}
 z(\nu) & = & \frac{\nu_{21} -\nu}{\nu} 
\, ; \, \nu(z) = \frac{\nu_{21}}{(1+z)} 
\hspace{1mm} \mathrm{with}   \hspace{1mm}  \nu_{21} = 1420.4 \, \mathrm{MHz}  \\
 z(\lambda) & = & \frac{\lambda - \lambda_{21}}{\lambda_{21}} 
\, ; \, \lambda(z) = \lambda_{21} \times (1+z) 
\hspace{1mm} \mathrm{with}   \hspace{1mm}  \lambda_{21} = 0.211 \, \mathrm{m.} 
\end{eqnarray}
The large-scale distribution of the neutral hydrogen, down to an angular scale of \mbox{$\sim 10 \, \mathrm{arc.min}$} 
can then be observed without detecting individual compact \HI sources, using the set of sky-brightness 
maps as a function of frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$ 
(radiation power/unit solid angle/unit surface/unit frequency) 
can be converted to brightness temperature using the Rayleigh-Jeans approximation of  black body radiation law:
$$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} .$$ 
 
%%%%%%%%
\begin{table}
\caption{21 cm source brightness and detection limits. }
\label{slims21} 
\begin{center}
\begin{tabular}{|c|c|c|}
\hline 
$A (\mathrm{m^2})$ & $ T_{sys} (K) $ & $ S_{lim} \, \mathrm{\mu Jy} $ \\
\hline 
5000 & 50 & 66 \\
5000 & 25 & 33 \\
100 000 & 50 & 3.3 \\
100 000 & 25 & 1.66 \\
500 000 & 50 & 0.66 \\
500 000 & 25 & 0.33 \\
\hline  
\end{tabular}
%% 
\hspace{3mm}
%% 
\begin{tabular}{|c|c|c|}
\hline
$z$ &  $\dlum \mathrm{(Mpc)}$ & $S_{21}  \mathrm{( \mu Jy)} $ \\
\hline      % dernier chiffre : sans le facteur (1+z)
0.25 &  1235   & 175 \\ % 140 
0.50 &  2800   & 40   \\  % 27 
1.0   &  6600   & 9.6  \\  % 4.8 
1.5   &  10980 & 3.5  \\  % 1.74 
2.0   &  15710  & 2.5  \\ % 0.85 
2.5  &  20690 &  1.7  \\  % 0.49 
\hline
\end{tabular}
\end{center}
\tablefoot{Left panel: sensitivity or source detection limit for 1-day integration time (86400 s) and 1-MHz 
frequency band. Right panel: 21 cm brightness for sources containing $10^{10} M_\odot$ of \HI at different redshifts.}
\end{table}

\subsection{ \HI power spectrum and BAO}
In the absence of any foreground or background radiation 
{\changemark and assuming a high spin temperature, $\kb T_{spin} \gg h \nu_{21}$},
the brightness temperature for a given direction and wavelength $\TTlam$ would be proportional to 
the local \HI number density $\etaHI(\vec{\Theta},z)$ through the
relation {\changemarkb (\cite{field.59}; \cite{zaldarriaga.04})}:
\begin{equation}
  \TTlamz  =   \frac{3}{32 \pi}  \, \frac{h}{\kb} \,  A_{21}  \, \lambda_{21}^2 \times
  \frac{c}{H(z)} \, (1+z)^2 \times  \etaHI (\vec{\Theta}, z)  
\end{equation}
where $A_{21}=2.85 \, 10^{-15} \mathrm{s^{-1}}$ \citep{astroformul} is the spontaneous 21 cm emission 
coefficient, $h$ the Planck constant, $c$ the speed of light, $\kb$ the Boltzmann 
constant, and $H(z)$ the Hubble parameter at the emission 
redshift.
For a \LCDM universe and neglecting radiation energy density, the Hubble parameter
can be expressed as
\begin{equation}
H(z)  \simeq  \hubb  \, \left[ \Omega_m (1+z)^3 + \Omega_\Lambda \right]^{\frac{1}{2}} 
\times  100 \, \, \mathrm{km/s/Mpc.}  
\label{eq:expHz}
\end{equation}
After introducing the \HI mass fraction relative to the total baryon mass $\gHI$, the 
neutral hydrogen number density and the corresponding 21 cm emission temperature 
can be written as a function of \HI relative density fluctuations: 
\begin{eqnarray}
\etaHI (\vec{\Theta}, z(\lambda) ) & = & \gHIz \times \Omega_B  \frac{\rho_{crit}}{m_{H}}  \times 
\left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) + 1 \right) \\
 \TTlamz  &  = & \bar{T}_{21}(z) \times \left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z)  + 1 \right) 
\end{eqnarray}
where $\Omega_B$ and $\rho_{crit}$ are the present-day mean baryon cosmological
and critical densities, respectively, $m_{H}$ the hydrogen atom mass, and 
$\frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}}$ the \HI density fluctuations. 

The present-day neutral hydrogen fraction $\gHI(0)$ present in local galaxies has been 
measured to be $\sim 1\%$ of the baryon density \citep{zwann.05}
$$ \Omega_{H_I} \simeq 3.5 \, 10^{-4} \sim 0.008 \times \Omega_B .$$
The neutral hydrogen fraction is expected to increase with redshift, as gas is used 
in star formation during galaxy formation and evolution. Study of Lyman-$\alpha$ absorption 
indicates a factor 3 increase in the neutral hydrogen 
fraction at $z=1.5$ in the intergalactic medium \citep{wolf.05}, 
compared to its current value $\gHI(z=1.5) \sim 0.025$. 
The 21 cm brightness temperature and the corresponding power spectrum can be written as 
(\cite{madau.97}; \cite{zaldarriaga.04}; \cite{barkana.07})
\begin{eqnarray}
 P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
 \bar{T}_{21}(z)  & \simeq & 0.084  \, \mathrm{mK}  
\frac{ (1+z)^2 \, \hubb }{\sqrt{ \Omega_m (1+z)^3 + \Omega_\Lambda } } 
 \dfrac{\Omega_B}{0.044}  \,  \frac{\gHIz}{0.01} \, .
\label{eq:tbar21z}
\end{eqnarray}

Table \ref{tabcct21} shows the mean 21 cm brightness temperature for the 
standard \LCDM cosmology and either a constant \HI mass fraction $\gHI = 0.01$, or 
linearly increasing  $\gHI \simeq 0.008 \times (1+z) $. Figure \ref{figpk21} shows the 
21 cm emission power spectrum at several redshifts, with a constant neutral fraction at 2\%
($\gHI=0.02$). The matter power spectrum has been computed using the 
\cite{eisenhu.98} parametrization. The correspondence  with the angular scales is also 
shown for the standard WMAP \LCDM cosmology, according to the relation
\begin{equation}
\theta_k = \frac{2 \pi}{k \, \dang(z) \, (1+z) }  
\hspace{3mm} , \hspace{3mm}
k = \frac{2 \pi}{ \theta_k  \, \dang(z) \, (1+z) }  \hspace{5mm} ,
\end{equation}
where $k$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance.
{ \changemark The matter power spectrum $P(k)$ has been measured using 
galaxy surveys, for example by SDSS and 2dF at low redshift $z \lesssim 0.3$
(\cite{cole.05}; \cite{tegmark.04}). The 21 cm brightness power spectra $P_{T_{21}}(k)$
shown here are comparable to the power spectrum measured from the galaxy surveys, 
once the mean 21 cm temperature conversion factor $\left( \bar{T}_{21}(z)  \right)^2$,
redshift evolution, and different bias factors have been accounted for. }
% It should be noted that the maximum transverse $k^{comov} $ sensitivity range 
% for an instrument corresponds approximately to half of its angular resolution. 
% {\color{red} Faut-il developper completement le calcul en annexe ? } 

\begin{table}
\caption{21 cm brightness temperature (mK) at different redshifts. }
\label{tabcct21} 
% \begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline 
\hline
 z  & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\
\hline 
(a) $\bar{T}_{21}$ & 0.085 & 0.107 & 0.145 & 0.174 & 0.195 & 0.216 & 0.234 \\
\hline
(b) $\bar{T}_{21}$  & 0.085 & 0.128 & 0.232 & 0.348 & 0.468 & 0.605 & 0.749 \\
\hline
\hline  
\end{tabular}
%\end{center}
\tablefoot{ Mean 21 cm brightness temperature in mK  for the 
standard \LCDM cosmology as a function of redshift:
\tablefoottext{a}{Constant \HI mass fraction \mbox{$\gHIz=0.01$}}
\tablefoottext{b}{Linearly increasing mass fraction \mbox{$\gHIz=0.008(1+z)$} }
}
\end{table}

\begin{figure}
\vspace*{-5mm}
\hspace{-5mm}
\includegraphics[width=0.57\textwidth]{Figs/pk21cmz12.pdf}
\vspace*{-10mm}
\caption{\HI 21 cm emission power spectrum at redshifts z=1 (blue) and z=2 (red), with 
neutral gas fraction $\gHI=2\%$}
\label{figpk21}
\end{figure}


\section{interferometric observations and P(k) measurement sensitivity }
\label{pkmessens}
\subsection{Instrument response}
\label{instrumresp}
We briefly introduce here the principles of interferometric observations and the definition of 
quantities useful for our calculations. The interested reader may refer to \cite{radastron} for a detailed 
and complete presentation of observation methods and signal processing in radio astronomy.  
In astronomy we are usually interested in measuring the sky emission intensity,
$I(\vec{\Theta},\lambda)$ in a given wave band, as a function of the sky direction. In radio astronomy
and interferometry in particular, receivers are sensitive to the sky emission complex 
amplitudes. However, for most sources, the phases vary randomly with a spatial correlation 
length significantly smaller than the instrument resolution,
\begin{eqnarray}
& & 
I(\vec{\Theta},\lambda)  =  | A(\vec{\Theta},\lambda) |^2  \hspace{2mm} , \hspace{1mm} I \in \mathbb{R}, A \in \mathbb{C} \\
& & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  = 0 \hspace{2mm} \mathrm{for}   \hspace{1mm} \vec{\Theta} \ne \vec{\Theta ' \, .}  
\end{eqnarray} 
A single receiver can be  characterized by its angular complex amplitude response $B(\vec{\Theta},\nu)$ and 
its position $\vec{r}$ in a reference frame. The waveform complex amplitude $s$ measured by the receiver, 
for each frequency can be written as a function of the electromagnetic wave vector 
$\vec{k}_{EM}(\vec{\Theta}, \lambda) $:
\begin{equation}
s(\lambda)  =  \iint d \vec{\Theta} \, \, \, A(\vec{\Theta},\lambda) B(\vec{\Theta},\lambda) e^{i ( \vec{k}_{EM} . \vec{r} )} \\
\end{equation} 
We set the electromagnetic (EM) phase origin at the center of the coordinate frame, and 
the EM wave vector is related to the wavelength $\lambda$ through the usual equation
$ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta},\lambda)$ 
corresponds to the receiver intensity response:
\begin{equation}
L(\vec{\Theta}, \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) \, .
\end{equation} 
The visibility signal of two receivers corresponds to the time-averaged correlation between 
signals from two receivers. If we assume a sky signal with random uncorrelated phase, the 
visibility $\vis$ signal from two identical receivers, located at the positions $\vec{r_1}$ and 
$\vec{r_2}$, can simply be written as a function of their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
\begin{equation}
\vis(\lambda) = < s_1(\lambda) s_2(\lambda)^* > = \iint d \vec{\Theta} \, \, I(\vec{\Theta},\lambda) L(\vec{\Theta},\lambda) 
e^{i ( \vec{k}_{EM} . \vec{\Delta r} ) }
\end{equation} 
This expression can be simplified if we consider receivers with a narrow field of view 
($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \, \mathrm{deg.} $ ), 
and coplanar with respect to their common axis.
If we introduce two cartesian-like angular coordinates $(\alpha,\beta)$ centered on
the common receivers axis, the visibilty would be written as the 2D Fourier transform
of the product of the sky intensity and the receiver beam, for the angular frequency 
\mbox{$(\uv)_{12} = ( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$}:
\begin{equation}
\vis(\lambda) \simeq  \iint d\alpha d\beta \, \, I(\alpha, \beta) \,  L(\alpha, \beta) 
\exp \left[ i 2 \pi \left( \alpha \frac{\Delta x}{\lambda} + \beta  \frac{\Delta y}{\lambda} \right) \right]
\end{equation} 
where $(\Delta x, \Delta y)$ are the two receiver distances on a plane perpendicular to 
the receiver axis. The $x$ and $y$ axes in the receiver plane are taken parallel to the 
two $(\alpha, \beta)$ angular planes. 
Furthermore, we introduce the conjugate Fourier variables $(\uv)$ and the Fourier transforms
of the sky intensity and the receiver beam:
\begin{center}
\begin{tabular}{ccc}
$(\alpha, \beta)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & $(\uv)$ \\
$I(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal I}(\uv, \lambda)$ \\
$L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(\uv, \lambda)$ \, .\\
\end{tabular} 
\end{center}

The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular 
wave number domain located around 
$(\uv)_{12}=( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is 
given by the receiver-beam Fourier transform
\begin{equation}
\vis(\lambda) \simeq  \iint \dudv \, \, {\cal I}(\uv, \lambda) \, {\cal L}(\uvu - \frac{\Delta x}{\lambda} , \uvv - \frac{\Delta y}{\lambda} , \lambda) \, .
\end{equation} 

\noindent A single receiver instrument would measure the total power integrated in a spot centered on the 
origin in the $(\uv)$ or the angular wave-mode plane. The shape of the spot depends on the receiver 
beam pattern, but its extent would be $\sim 2 \pi D / \lambda$, where $D$ is the receiver physical 
size. 

The correlation signal from a pair of receivers would measure the integrated signal on a similar 
spot, located around the central angular wave-mode  $(\uv)_{12}$, determined by the relative 
position of the two receivers (see figure \ref{figuvplane}).
In an interferometer with multiple receivers, the area covered by different receiver pairs in the 
$(\uv)$ plane might overlap, and some pairs might measure the same area (same base lines). 
Several beams can be formed using different combinations of the correlations from a set of  
antenna pairs.  

An instrument can thus be characterized by its $(\uv)$ plane coverage or response 
${\cal R}(\uv,\lambda)$. For a single dish with a single receiver in the focal plane, 
the instrument response is simply the Fourier transform of the beam. 
For a single dish with multiple receivers, either as a focal plane array (FPA) or 
a multi-horn system, each beam (b) will have its own response 
${\cal R}_b(\uv,\lambda)$. 
For an interferometer, we can compute a raw instrument response 
${\cal R}_{raw}(\uv,\lambda)$, which corresponds to $(\uv)$ plane coverage by all 
receiver pairs  with uniform weighting. 
Obviously, different weighting schemes can be used, changing 
the effective beam shape, hence the response ${\cal R}_{w}(\uv,\lambda)$
and the noise behavior. If the same Fourier angular frequency mode is measured 
by several receiver pairs, the raw instrument response might then be larger 
that unity. This non-normalized instrument response is used to compute the projected 
noise power spectrum in the following section (\ref{instrumnoise}). 
We can also define a  normalized instrument response, ${\cal R}_{norm}(\uv,\lambda) \lesssim 1$ as
\begin{equation}
{\cal R}_{norm}(\uv,\lambda) = {\cal R}(\uv,\lambda) / \mathrm{Max_{(\uv)}} \left[ {\cal R}(\uv,\lambda) \right] \, .
\end{equation} 
This normalized  instrument response is the basic ingredient for computing the effective 
instrument beam, in particular in section \ref{recsec}.  

{\changemark Detection of the reionization at 21 cm has been an active field 
in the last decade, and different groups have built 
instruments to detect a reionization signal around 100 MHz: LOFAR
\citep{rottgering.06}, MWA (\cite{bowman.07}; \cite{lonsdale.09}), and PAPER \citep{parsons.10}. 
Several authors have studied the instrumental noise 
and statistical uncertainties when measuring the reionization signal power spectrum, and 
the methods presented here to compute the instrument response 
and sensitivities are similar to the ones developed in these publications 
(\cite{morales.04}; \cite{bowman.06}; \cite{mcquinn.06}). }

\begin{figure}
% \vspace*{-2mm}
\centering
\mbox{
\includegraphics[width=0.5\textwidth]{Figs/uvplane.pdf}
}
\vspace*{-15mm}
\caption{Schematic view of the $(\uv)$ plane coverage by interferometric measurement.}
\label{figuvplane}
\end{figure}

\subsection{Noise power spectrum computation}
\label{instrumnoise}
We consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency 
bandwidth $\delta \nu$ centered on $\nu_0$, with an integration time $t_{int}$, characterized by a system temperature 
$\Tsys$. The uncertainty or fluctuations of this measurement due to the receiver noise can be written as 
$\sigma_{noise}^2 = \frac{2 \Tsys^2}{t_{int} \, \delta \nu}$. This term also
corresponds to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated 
noise. If the receiver has an effective area $A \simeq \pi D^2/4$ or $A \simeq D_x D_y$, the measurement 
corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. 
The noise's spectral density, in the angular frequency plane (per unit area of angular frequency
\mbox{$\delta \uvu \times \delta \uvv$}), corresponding to a visibility 
measurement from a pair of receivers can be written as
\begin{eqnarray}
P_{noise}^{\mathrm{pair}} & = & \frac{\sigma_{noise}^2}{ A / \lambda^2 }  \\
P_{noise}^{\mathrm{pair}} & \simeq & \frac{2 \, \Tsys^2 }{t_{int}  \, \delta \nu} \, \frac{ \lambda^2 }{ D^2 } 
\hspace{5mm} \mathrm{units:} \, \mathrm{K^2 \times rad^2}  \, .
\label{eq:pnoisepairD}
\end{eqnarray}

We can characterize the sky temperature measurement with a radio instrument by the noise's 
spectral power density in the angular frequencies plane $P_{noise}(\uv)$ in units of $\mathrm{Kelvin^2}$  
per unit area of angular frequencies  $\delta \uvu \times \delta \uvv$. 
For an interferometer made of identical receiver elements, several ($n$) receiver pairs 
might have the same baseline. The noise power density in the corresponding $(\uv)$ plane area 
is then reduced by a factor $1/n$. More generally, we can write the instrument  noise 
spectral power density using the instrument response defined in section \ref{instrumresp} as
\begin{equation}
P_{noise}(\uv) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(\uv,\lambda) }  \hspace{4mm} .
\label{eq:pnoiseuv}
\end{equation} 

When the intensity maps are projected in a 3D box in the universe and the 3D power spectrum 
$P(k)$ is computed, angles are translated into comoving transverse distances, 
and frequencies or wavelengths into comoving radial distance, using the following relations
{\changemarkb (e.g. chap. 13 of \cite{cosmo.peebles}; \cite{cosmo.rich})} :
{ \changemark
\begin{eqnarray}
\alpha , \beta & \rightarrow &  \ell_\perp = \ell_x, \ell_y = (1+z) \, \dang(z) \,  \alpha,\beta  \\
\uv & \rightarrow & k_\perp = k_x, k_y = 2 \pi \frac{ \uvu , \uvv }{  (1+z) \, \dang(z)  }   \label{eq:uvkxky} \\
\delta \nu & \rightarrow & \delta \ell_\parallel = (1+z) \frac{c}{H(z)} \frac{\delta \nu}{\nu} 
  = (1+z) \frac{\lambda}{H(z)} \delta \nu \\
% \delta \uvu , \delta \uvv & \rightarrow & \delta k_\perp = 2 \pi \frac{ \delta \uvu \, , \, \delta \uvv }{  (1+z) \, \dang(z)  } \\
\frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = \delta k_z =
2 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
 =  \frac{H(z)}{c} \frac{1}{(1+z)^2} \, \frac{\nu_{21}}{\delta \nu}
\end{eqnarray}
}
{ \changemark
A brightness measurement at a  point $(\uv,\lambda)$, covering 
the 3D spot  $(\delta \uvu, \delta \uvv, \delta \nu)$, would correspond 
to a cosmological power spectrum measurement at a transverse wave mode $(k_x,k_y)$
defined by the equation \ref{eq:uvkxky}, measured at a redshift given by the observation frequency.
The measurement noise spectral density given by the Eq. \ref{eq:pnoisepairD} can then be 
translated into a 3D noise power spectrum, per unit of spatial frequencies 
$ \delta k_x \times \delta k_y \times \delta k_z / 8 \pi^3 $ (units: $ \mathrm{K^2 \times Mpc^3}$) :  

\begin{eqnarray}
(\uv , \lambda) & \rightarrow & k_x(\uvu),k_y(\uvv), z(\lambda) \\
P_{noise}(k_x,k_y, z)  & = & P_{noise}(\uv) 
 \frac{ 8 \pi^3 \delta \uvu \times \delta \uvv }{\delta k_x \times \delta k_y \times \delta k_z} \\
                       & = &   \left( 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \right) 
   \, \frac{1}{{\cal R}_{raw}} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  
\label{eq:pnoisekxkz} 
\end{eqnarray} 

It is worthwhile to note that the ``cosmological'' 3D noise power spectrum does not depend 
anymore on the individual measurement bandwidth.
In the following paragraph, we will first consider an ideal instrument 
with uniform $(\uv)$ coverage
in order to establish the general noise power spectrum behavior for cosmological 21 cm surveys.
The numerical method used to compute the 3D noise power spectrum is then presented in section 
\ref{pnoisemeth}.
}
 
\subsubsection{Uniform $(\uv)$ coverage}
{ \changemarkb We consider here an instrument with uniform $(\uv)$ plane coverage  (${\cal R}(\uv)=1$), 
and measurements at regularly spaced frequencies centered on a central frequency $\nu_0$ or redshift $z(\nu_0)$.
The noise's spectral power  density from equation (\ref{eq:pnoisekxkz}) would then be  
constant, independent of $(k_x, k_y, \ell_\parallel(\nu))$. Such a noise power spectrum thus  corresponds 
to a 3D white noise, with a uniform noise spectral density:}
\begin{equation}
P_{noise}(k_\perp, \ell_\parallel(\nu) ) = P_{noise} = 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  
\label{ctepnoisek}
\end{equation} 
%
where $P_{noise}$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
$t_{int}$ is the integration time expressed in second, 
$\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and 
 $H(z)$ in $\mathrm{km/s/Mpc}$. 

The statistical uncertainties of matter or \HI distribution power spectrum estimate decreases
with the number of observed Fourier modes, a number that is proportional to the volume of the universe 
being observed (sample variance).  As the observed volume is proportional to the 
surveyed solid angle, we  consider the survey of a fixed
fraction of the sky, defined by  total solid angle $\Omega_{tot}$, performed during a given
total observation time $t_{obs}$. 
A single-dish instrument with diameter $D$ would have an instantaneous field of view 
$\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and would require 
a number of pointings  $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area. 
Each sky direction or patch of size $\Omega_{FOV}$ will be observed during an integration 
time $t_{int} = t_{obs}/N_{point} $. Using equation \ref{ctepnoisek} and the previous expression
for the integration time, we can compute a simple expression
for the noise spectral power density by a single-dish instrument of diameter $D$:
\begin{equation}
P_{noise}^{survey}(k) = 2 \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  \hspace{2mm} .
\end{equation}

It is important to note that any real instrument does not have a flat 
response in the $(\uv)$ plane, and the observations provide no information above 
a certain maximum angular frequency $\uvu_{max},\uvv_{max}$. 
One has to take into account either a damping of the observed sky power 
spectrum or an increase in the noise spectral density if 
the observed power spectrum is corrected for damping. The white-noise 
expressions given below should thus be considered as a lower limit or floor of the 
instrument noise spectral density. 
 
For a single-dish instrument of diameter $D$ equipped with a multi-feed or 
phase-array receiver system, with $N$ independent beams on sky, 
the noise spectral density decreases by a factor $N$, 
thanks to the  increase in per pointing integration time:

\begin{equation}
P_{noise}^{survey}(k) = \frac{2}{N} \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  \hspace{2mm} .
\label{eq:pnoiseNbeam}
\end{equation}
%
This expression (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$ 
identical receivers with a  total collection area $\sim D^2$. Such an array could be made for example 
of $N=q \times q$ {\it small dishes}, each with diameter $D/q$, arranged as a $q \times q$ square.   

For a single dish of diameter $D$, or an interferometric instrument with maximal extent $D$, 
observations provide information up to $\uvu_{max},\uvv_{max} \lesssim D / \lambda $. This value of 
$\uvu_{max},\uvv_{max}$ would be mapped to a maximum transverse cosmological wave number
$k_{\perp}^{max}$:
\begin{equation}
k_{\perp}^{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} \hspace{3mm} .
\label{kperpmax}
\end{equation}   
%
Figure \ref{pnkmaxfz} shows the evolution of the noise spectral density $P_{noise}^{survey}(k)$
as a function of redshift, for a radio survey of the sky, using an instrument with $N=100$
beams and a system noise temperature $\Tsys = 50 \mathrm{K}$.
The survey is supposed to cover a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$, in one 
year. The maximum comoving wave number $k^{max}$  is also shown as a function 
of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. 
To take the radial component of $\vec{k}$ and the increase of 
the instrument noise level with $k_{\perp}$ into account, we have taken the effective $k_{ max} $ 
as half of the maximum transverse $k_{\perp} ^{max}$ of \mbox{Eq. \ref{kperpmax}}: 
\begin{equation}
k_{max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}}  \hspace{3mm} .
\end{equation}

\begin{figure}
\vspace*{-25mm}
\centering
\mbox{
\hspace*{-10mm}
\includegraphics[width=0.65\textwidth]{Figs/pnkmaxfz.pdf}
}
\vspace*{-40mm}
\caption{Top: minimal noise level for a 100-beam instrument with \mbox{$\Tsys=50 \mathrm{K}$} 
as a function of redshift (top), for a one-year survey of a quarter of the sky. Bottom:
maximum $k$ value for 21 cm LSS power spectrum measurement by  a 100-meter diameter 
primary antenna. }
\label{pnkmaxfz}
\end{figure}
 
\subsubsection{3D noise power spectrum computation}  
\label{pnoisemeth}
{ \changemark
We describe here the numerical method used to compute the 3D noise power spectrum, for a given instrument 
response, as presented in section \ref{instrumnoise}. The noise power spectrum is a good indicator to compare 
sensitivities for different instrument configurations, although the noise realization for a real instrument would not be 
isotropic. 
\begin{itemize}
\item We start by a 3D Fourier coefficient grid, with the two first coordinates the transverse angular wave modes,
and the third the frequency $(k_x,k_y,\nu)$. The grid is positioned at the mean redshift $z_0$ for which 
we want to compute $P_{noise}(k)$. For the results at redshift \mbox{$z_0=1$} discussed in section  \ref{instrumnoise}, 
the grid cell size and dimensions have been chosen to represent a box in the universe  
\mbox{$\sim 1500 \times 1500 \times 750 \, \mathrm{Mpc^3}$}, 
with \mbox{$3\times3\times3 \, \mathrm{Mpc^3}$} cells.
This corresponds to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given 
the small angular extent, we have neglected the curvature of redshift shells. 
\item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. 
The coordinates $(k_x,k_y)$ are converted to the $(\uv)$ angular frequency coordinates 
using equation (\ref{eq:uvkxky}), and the 
angular diameter distance $\dang(z)$ for \LCDM model with standard WMAP parameters \citep{komatsu.11}. 
The noise variance is taken proportional to $P_{noise}$ 
\begin{equation}
\sigma_{re}^2=\sigma_{im}^2 \propto \frac{1}{{\cal R}_{raw}(\uv,\lambda)} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \hspace{2mm} .
\end{equation}
\item An FFT is then performed in the frequency or redshift direction to obtain the noise Fourier 
complex coefficients ${\cal F}_n(k_x,k_y,k_z)$ and the power spectrum is estimated as 
\begin{equation}
\tilde{P}_{noise}(k) = < | {\cal F}_n(k_x,k_y,k_z) |^2 >  \hspace{2mm}  \mathrm{for} \hspace{2mm} 
  \sqrt{k_x^2+k_y^2+k_z^2} = k  \hspace{2mm} .
\end{equation}
Noise samples corresponding to small instrument response, typically less than 1\% of the 
maximum instrument response, are ignored when calculating  $\tilde{P}_{noise}(k)$. 
However, we require a significant fraction, typically 20\% to 50\% of all possible modes 
$(k_x,k_y,k_z)$ measured for a given $k$ value. 
\item the above steps are repeated $\sim$ 50 times to decrease the statistical fluctuations 
from random generations. The averaged computed noise power spectrum is normalized using 
equation \ref{eq:pnoisekxkz}  and the instrument and survey parameters: 
{\changemarkb system temperature $\Tsys= 50 \mathrm{K}$, 
individual receiver size $D^2$ or $D_x D_y$ and the integration time $t_{int}$. 
This last parameter is obtained through the relation 
$t_{int} = t_{obs} \Omega_{FOV}  / \Omega_{tot}$ using the total survey duration 
$t_{obs}=1 \mathrm{year}$, the instantaneous field of view 
$\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and the total sky coverage 
$\Omega_{tot} = \pi$ srad. }
\end{itemize} 

It should be noted that it is possible to obtain a  good approximation of the noise 
power spectrum shape by neglecting the redshift or frequency dependence of the 
instrument response function and $\dang(z)$ for a small redshift interval around $z_0$,
using a fixed  instrument response ${\cal R}(\uv,\lambda(z_0))$ and 
a constant radial distance $\dang(z_0)\times(1+z_0)$:
\begin{equation}
\tilde{P}_{noise}(k) = < |  {\cal F}_n (k_x,k_y,k_z) |^2 > \simeq < P_{noise}(\uv, k_z) > 
\end{equation}
The approximate power spectrum obtained through this simplified and much faster 
method is shown as dashed curves on figure \ref{figpnoisea2g} for few instrument 
configurations. 
}

\subsection{Instrument configurations and noise power spectrum}
\label{instrumnoise}
We have numerically computed the instrument response ${\cal R}(\uv,\lambda)$ 
with uniform weights in the $(\uv)$ plane for several instrument configurations:
\begin{itemize}
\item[{\bf a} :] A packed array of $n=121 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in 
a square $11 \times 11$ configuration ($q=11$). This array covers an area of 
$55 \times 55 \, \mathrm{m^2}$ \, .
\item [{\bf b} :] An array of $n=128  \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged 
in eight rows, each with 16 dishes. These 128 dishes are spread over an area 
$80 \times 80  \, \mathrm{m^2}$. The array layout for this configuration is
shown in figure \ref{figconfbc}. 
\item [{\bf c} :] An array of $n=129  \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged 
 over an area $80 \times 80  \, \mathrm{m^2}$. This configuration has in 
particular four subarrays of packed 16 dishes ($4\times4$), located in the 
four array corners. This array layout is also shown in figure \ref{figconfbc}. 
\item [{\bf d} :] A single-dish instrument, with diameter $D=75 \, \mathrm{m}$, 
equipped with a 100 beam focal plane receiver array. 
\item[{\bf e} :] A packed array of $n=400 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in 
a square $20 \times 20$ configuration ($q=20$). This array covers an area of 
$100 \times 100 \, \mathrm{m^2}$ 
\item[{\bf f} :] A packed array of four cylindrical reflectors, each 85 meter long and 12 meter 
wide. The focal line of each cylinder is equipped with 100 receivers, each 
$2 \lambda$ long, corresponding to $\sim 0.85 \, \mathrm{m}$ at $z=1$. 
This array covers an area of $48 \times 85 \, \mathrm{m^2}$, and have 
a total of $400$ receivers per polarization, as in the (e) configuration.
We computed the noise power spectrum for {\em perfect} 
cylinders, where all receiver pair correlations are used (fp), or for 
an imperfect instrument, where only correlations between receivers 
from different cylinders are used.
\item[{\bf g} :] A packed array of eight cylindrical reflectors, each 102 meters long and 12 meters 
wide. The focal line of each cylinder is equipped with 120 receivers, each 
$2 \lambda$ long, corresponding to $\sim 0.85 \, \mathrm{m}$ at $z=1$. 
This array covers an area of $96 \times 102 \, \mathrm{m^2}$ and has 
a total of 960  receivers per polarization. As for the (f) configuration,  
we have computed the noise power spectrum for {\em perfect} 
cylinders, where all receiver pair correlations are used (gp), or for 
an imperfect instrument, where only correlations between receivers 
from different cylinders are used.
\end{itemize}

\begin{figure}
\centering
\vspace*{-15mm}
\mbox{
\hspace*{-10mm}
\includegraphics[width=0.5\textwidth]{Figs/configab.pdf}
}
\vspace*{-15mm}
\caption{ Array layout for configurations (b) and (c) with 128 and 129  D=5 meter 
diameter dishes. }
\label{figconfbc}
\end{figure}

We used simple triangular shaped dish response in the $(\uv)$ plane;
however, we did introduce a filling factor or illumination efficiency 
$\eta$, relating the effective dish diameter $D_{ill}$ to the 
mechanical dish size $D_{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales 
as $\eta^2$ or $\eta_x \eta_y$. 
\begin{eqnarray}
{\cal L}_\circ (\uv,\lambda) & = & \bigwedge_{[\pm  \eta D_{dish}/ \lambda]}(\sqrt{u^2+v^2})  \\
 L_\circ (\alpha,\beta,\lambda) & = & \left[ \frac{ \sin (\pi (D^{ill}/\lambda) \sin \theta ) }{\pi (D^{ill}/\lambda) \sin \theta} \right]^2 
\hspace{4mm} \theta=\sqrt{\alpha^2+\beta^2}
\end{eqnarray}
For the multidish configuration studied here, we have taken the illumination efficiency factor
{\bf $\eta = 0.9$}. 

For the receivers along the focal line of cylinders, we assumed that the 
individual receiver response in the $(\uv)$ plane corresponds to a 
rectangular antenna. The illumination efficiency factor was taken 
equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$ 
along the cylinder length. {\changemark  We used a double triangular 
response function in the $(\uv)$ plane for each of the receiver elements along the cylinder:
\begin{equation}
 {\cal L}_\Box(\uv,\lambda)  = 
\bigwedge_{[\pm \eta_x D_x / \lambda]} (\uvu ) \times
\bigwedge_{[\pm \eta_y D_y / \lambda ]} (\uvv )  
\end{equation}
}

\noindent It should be noted that the small angle approximation 
used here for the expression of visibilities is not valid for the receivers along 
the cylinder axis. However, some preliminary numerical checks indicate that 
the results for the noise spectral power density  would not change significantly.
The instrument responses shown here correspond to a fixed pointing toward the zenith, which 
is the case for a transit type telescope.

Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}(\uv,\lambda)$ 
for the four configurations (a,b,c,d) with $\sim 100$ receivers per 
polarization. 
{\changemark Using the numerical method sketched in section \ref{pnoisemeth}, we 
computed the 3D noise power spectrum for each of the eight instrument configurations presented
here, with a system noise temperature $\Tsys = 50 \mathrm{K}$, for a one year survey 
of a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$ at a mean redshift $z_0=1, \nu_0=710 \mathrm{MHz}$.}
The resulting noise spectral power densities are shown in figure 
\ref{figpnoisea2g}. The increase of $P_{noise}(k)$ at low $k^{comov} \lesssim 0.02$ 
is due to our having ignored all auto-correlation measurements.  
It can be seen that an instrument with $100-200$ beams and $\Tsys = 50 \mathrm{K}$ 
should have enough sensitivity to map LSS in 21 cm at redshift z=1.

\begin{figure*}
\centering
\mbox{
% \hspace*{-10mm}
\includegraphics[width=\textwidth]{Figs/uvcovabcd.pdf}
}
\caption{Raw instrument response ${\cal R}_{raw}(\uv,\lambda)$ or the $(\uv)$ plane coverage
at 710 MHz (redshift $z=1$) for four configurations.
(a) 121 $D_{dish}=$ 5-meter diameter dishes arranged in a compact, square array 
of $11 \times 11$, (b) 128 dishes arranged in 8 rows of 16 dishes each (fig. \ref{figconfbc}), 
(c) 129 dishes arranged as shown in figure \ref{figconfbc}, (d) single D=75 meter diameter, with 100 beams. 
The common color scale (1 \ldots 80) is shown on the right. }
\label{figuvcovabcd}
\end{figure*}
 
\begin{figure*}
\vspace*{-10mm}
\centering
\mbox{
% \hspace*{-5mm}
\includegraphics[width=\textwidth]{Figs/pkna2h.pdf}
}
\vspace*{-20mm}
\caption{P(k) 21 cm LSS power spectrum at redshift $z=1$ with $\gHI=2\%$ 
and the noise power spectrum for several interferometer configurations
 ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400, and 960 receivers. The noise power spectrum has been 
computed for all configurations assuming a survey of a quarter of the sky over one year, 
with a system temperature $\Tsys = 50 \mathrm{K}$. }
\label{figpnoisea2g}
\end{figure*}


\section{ Foregrounds and component separation }
\label{foregroundcompsep}
Reaching the required sensitivities is not the only difficulty of observing the 
LSS in 21 cm. Indeed, the synchrotron emission of the 
Milky Way and the extragalactic radio sources are a thousand times brighter than the 
emission of the neutral hydrogen distributed in the universe. Extracting the LSS signal
using intensity mapping, without identifying the \HI point sources is the main challenge 
for this novel observation method. Although this task might seem impossible at first, 
it has been suggested that the smooth frequency dependence of the synchrotron
emissions can be used to separate the faint LSS signal from the Galactic and radio source 
emissions. {\changemark Discussion of contribution of different sources 
of radio foregrounds for measurement of reionization through redshifted 21 cm,
as well as foreground subtraction using their smooth frequency dependence, can 
be found in (\cite{shaver.99}; \cite{matteo.02};\cite{oh.03}).}
However, any real radio instrument has a beam shape that changes with 
frequency, and this instrumental effect significantly increases the difficulty and complexity of this component separation 
technique. The effect of frequency dependent beam shape is sometimes referred to as {\em 
mode mixing},  {\changemark and its impact on foreground subtraction 
has been discussed for example in \cite{morales.06}.}

In this section, we present a short description of the foreground emissions and 
the simple models we used for computing the sky radio emissions in the GHz frequency 
range. We also present a simple component-separation method to extract the LSS signal and 
its performance. {\changemark The analysis presented here follows a similar path to 
a detailed foreground subtraction study carried out for MWA at $\sim$ 150 MHz by \cite{bowman.09}. } 
We computed in particular, the effect of the instrument response on the recovered 
power spectrum. The results presented in this section concern the 
total sky emission and the LSS 21 cm signal extraction in the $z \sim 0.6$ redshift range,
corresponding to the central frequency $\nu \sim 884$ MHz.  
 
\subsection{ Synchrotron and radio sources }
We modeled the radio sky in the form of 3D maps (data cubes) of sky temperature 
brightness $T(\alpha, \delta, \nu)$ as a function of two equatorial angular coordinates $(\alpha, \delta)$ 
and the frequency $\nu$. Unless otherwise specified, the results presented here are based on simulations of 
$90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10\mathrm{h}00\mathrm{m} , \delta=+10 \, \mathrm{deg.}$, and  covering 128 MHz in frequency. We have selected this particular area of the sky  in order to minimize 
the Galactic synchrotron foreground. The sky cube characteristics (coordinate range, size, resolution) 
used in the simulations are given in the Table \ref{skycubechars}.
\begin{table}
\caption{
Sky cube characteristics for the simulations described in this paper.  }
\label{skycubechars}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline 
 & range & center  \\
\hline 
Right ascension & 105 $ < \alpha < $ 195 deg. &  150 deg.\\
Declination & -5 $ < \delta < $ 25 deg. & +10 deg. \\
Frequency & 820 $ < \nu < $ 948 MHz & 884 MHz \\
Wavelength & 36.6 $ < \lambda < $ 31.6 cm & 33.9 cm \\
Redshift & 0.73 $ < z < $ 0.5 & 0.61 \\
\hline 
\hline
& resolution & N-cells \\
\hline
Right ascension & 3 arcmin & 1800 \\
Declination & 3 arcmin & 600 \\
Frequency & 500 kHz ($d z \sim 10^{-3}$) & 256 \\
\hline
\end{tabular} 
\end{center}
\tablefoot{ Cube size: $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$; 
$1800 \times 600 \times 256 \simeq 123 \times 10^6$ cells }
\end{table}
%%%%
\par 
Two different methods were used to compute the sky temperature data cubes.
We used the global sky model (GSM) \citep{gsm.08} tools to generate full sky 
maps of the emission temperature at different frequencies, from which we 
extracted the brightness temperature cube for the region defined above 
(Model-I/GSM $T_{gsm}(\alpha, \delta, \nu)$).
Because the GSM maps have an intrinsic resolution of $\sim$ 0.5 degree, it is 
difficult to have reliable results for the effect of point sources on the reconstructed 
LSS power spectrum.

We have thus also made a simple sky model using the Haslam Galactic synchrotron map
at 408 MHz \citep{haslam.82} and the NRAO VLA Sky Survey (NVSS) 1.4 GHz radio source 
catalog \citep{nvss.98}. The sky temperature cube in this model (Model-II/Haslam+NVSS) 
was computed through the following steps:

\begin{enumerate}
\item The Galactic synchrotron emission is modeled as a power law with a spatially 
varying spectral index. We assign a power law index $\beta = -2.8  \pm 0.15$ to each sky direction,
where $\beta$ has a Gaussian distribution centered on -2.8 with a standard 
deviation $\sigma_\beta = 0.15$. {\changemark The 
diffuse radio background spectral index has been measured, for example, by 
\citep{platania.98} or \citep{rogers.08}.}
The synchrotron contribution to the sky temperature for each cell is then 
obtained  through the formula:
\begin{equation}
 T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 \, \mathrm{MHz}}\right)^\beta 
\end{equation}
%%
\item A 2D $T_{nvss}(\alpha,\delta)$ sky brightness temperature at 1.4 GHz is computed 
by projecting the radio sources in the NVSS catalog to a grid with the same angular resolution as 
the sky cubes. The source brightness in Jansky is converted to temperature taking the 
pixel angular size into account ($ \sim 21 \mathrm{mK/mJy}$ at 1.4 GHz and $3' \times 3'$ pixels).  
A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source 
map. We have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the 
contribution of the radiosources to the sky temperature is computed as:
\begin{equation}
 T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}}  
\end{equation}
%%
\item The sky brightness temperature data cube is obtained through the sum of
the two contributions, Galactic synchrotron and resolved radio sources: 
\begin{equation}
 T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu) 
\end{equation}
\end{enumerate}

 The 21 cm temperature fluctuations due to neutral hydrogen in LSS
$T_{lss}(\alpha, \delta, \nu)$ were computed using the 
SimLSS\footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }  software package, where
%
complex normal Gaussian fields were first generated in Fourier space.
The amplitude of each mode was then multiplied by the square root
of the power spectrum $P(k)$ at $z=0$ computed according to the parametrization of 
\citep{eisenhu.98}. We used the standard cosmological parameters, 
 $H_0=71 \, \mathrm{km/s/Mpc}$, $\Omega_m=0.264$, $\Omega_b=0.045$,
$\Omega_\lambda=0.73$ and $w=-1$ \citep{komatsu.11}.
An inverse FFT was then performed to compute the matter density fluctuations $\delta \rho / \rho$ 
in the linear regime, 
in a box of $3420 \times 1140 \times 716  \, \mathrm{Mpc^3}$, and evolved
to redshift $z=0.6$. 
The size of the box is about 2500 $\mathrm{deg^2}$  in the transverse direction and
$\Delta z \simeq 0.23$ in the longitudinal direction.  
The size of the cells is  $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the 
sky cube angular and frequency resolution defined above.  
{\changemarkb 
We did not take the curvature of redshift shells into account when 
converting SimLSS euclidean coordinates to angles and frequency coordinates 
of the sky cubes analyzed here. This approximate treatment causes distortions visible at large angles $\gtrsim 10^\circ$. 
These angular scales correspond to small wave modes $k \lesssim 0.02 \mathrm{h \, Mpc^{-1}}$ and
 are excluded for results presented in this paper.
}  

The mass fluctuations have been converted into temperature using equation \ref{eq:tbar21z}, 
and a neutral  hydrogen fraction \mbox{$0.008 \times (1+0.6)$}, leading to a mean temperature of 
$0.13 \, \mathrm{mK}$. 
The total sky brightness temperature is computed as the sum 
of foregrounds and the LSS 21 cm emission:
\begin{equation}
  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm} 
T_{sky} = T_{gsm}+T_{lss} 
\end{equation}

Table \ref{sigtsky} summarizes the mean and standard deviation of the sky brightness 
temperature $T(\alpha, \delta, \nu)$ for the different components computed in this study.
It should be noted that the standard deviation depends on the map resolution, and the values given 
in Table \ref{sigtsky}  correspond to sky cubes computed here, with $\sim 3$ arc minute
angular and 500 kHz frequency resolutions (see Table \ref{skycubechars}).
Figure \ref{compgsmmap} shows the comparison of the GSM temperature map at 884 MHz 
with Haslam+NVSS map, smoothed with a 35 arcmin Gaussian beam. 
Figure \ref{compgsmhtemp} shows the comparison of the sky cube temperature distribution 
for Model-I/GSM and Model-II. There is good agreement between the two models, although 
the mean temperature for Model-II is slightly higher ($\sim 10\%$) than Model-I.

\begin{table}
\caption{Mean temperature and standard deviation for different sky cubes.}
\label{sigtsky}
\centering
\begin{tabular}{|c|c|c|}
\hline
 & mean (K) & std.dev (K) \\
\hline 
Haslam & 2.17 & 0.6 \\
NVSS & 0.13 & 7.73 \\
Haslam+NVSS & 2.3 & 7.75 \\
(Haslam+NVSS)*Lobe(35') & 2.3 & 0.72 \\
GSM & 2.1 & 0.8 \\
\hline
\end{tabular}
% \tablefoot{See table \ref{skycubechars} for sky cube resolution and size.}
\end{table}

We computed the power spectrum for the 21cm-LSS sky temperature cube, as well 
as for the radio foreground temperature cubes obtained from the two 
models. We also computed the power spectrum on sky brightness temperature
cubes, as measured by a perfect instrument having a 25 arcmin (FWHM) Gaussian beam.
The resulting computed power spectra are shown in figure \ref{pkgsmlss}. 
The GSM model has more large-scale power compared to our simple Haslam+NVSS model, 
while it lacks power at higher spatial frequencies. The mode mixing due to a
frequency-dependent response will thus be stronger in Model-II (Haslam+NVSS) 
case. It can also be seen that the radio foreground's power spectrum is more than
$\sim 10^6$ times higher than the 21 cm signal from LSS. This corresponds 
to the factor $\sim 10^3$ of the sky brightness temperature fluctuations (\mbox{$\sim$ K}),
compared to the mK LSS signal.  

{ \changemark In contrast to most similar studies, where it is assumed that bright sources 
can be nearly perfectly subtracted, our aim was to compute also their 
effect in the foreground subtraction process. 
The GSM model lacks the angular resolution needed to correctly compute 
the effect of bright compact sources for 21 cm LSS observations and 
the mode mixing due to the frequency dependence of the instrumental response, 
while Model-II provides a reasonable description of these compact sources. Our simulated 
sky cubes have an angular resolution $3'\times3'$, well below the typical 
$15'$ resolution of the instrument configuration considered here. 
However, Model-II might lack spatial structures on large scales, above a degree, 
compared to Model-I/GSM, and the frequency variations as a simple power law
might not be realistic enough. The differences for the two sky models can be seen
in their power spectra shown in figure \ref{pkgsmlss}. The smoothing or convolution with 
a 25' beam has negligible effect on the GSM power spectrum, since it originally lacks 
structures below 0.5 degree. By using 
these two models, we explored some of the systematic uncertainties 
related to foreground subtraction.}

It should also be noted that in section 3, we presented the different instrument 
configuration noise levels after {\em correcting or deconvolving} the instrument response. The LSS 
power spectrum is recovered unaffected in this case, while the noise power spectrum 
increases at high k values (small scales). In practice, clean deconvolution is difficult to 
implement for real data and the power spectra presented in this section are NOT corrected 
for the instrumental response.  The observed structures thus have a scale-dependent damping
according to the instrument response, while the instrument noise is flat (white noise or scale-independent). 

\begin{figure}
\centering
\vspace*{-10mm}
\mbox{
\hspace*{-20mm}
\includegraphics[width=0.6\textwidth]{Figs/comptempgsm.pdf}
}
\vspace*{-10mm}
\caption{Comparison of GSM (black) and Model-II (red) sky cube temperature distribution.
The Model-II (Haslam+NVSS),
has been smoothed with a 35 arcmin Gaussian beam. }
\label{compgsmhtemp}
\end{figure}

\begin{figure*}
\centering
\mbox{
% \hspace*{-10mm}
\includegraphics[width=0.9\textwidth]{Figs/compmapgsm.pdf}
}
\caption{Comparison of GSM (top) and Model-II (bottom) sky maps at 884 MHz. 
The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin (FWHM) Gaussian beam.}
\label{compgsmmap}
\end{figure*}

\begin{figure}
\centering
% \vspace*{-25mm}
\mbox{
\hspace*{-6mm}
\includegraphics[width=0.52\textwidth]{Figs/pk_gsm_lss.pdf}
}
\vspace*{-5mm}
\caption{Comparison of the 21cm LSS power spectrum at $z=0.6$ with \mbox{$\gHI\simeq1.3\%$} (red, orange) 
with the radio foreground power spectrum.
The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple
model based on Haslam+NVSS (Model-II, black). The curves with circle markers show the power spectrum 
as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam. 
}
\label{pkgsmlss}
\end{figure}



\subsection{ Instrument response and LSS signal extraction }
\label{recsec}
The {\it observed} data cube is obtained from the sky brightness temperature 3D map 
$T_{sky}(\alpha, \delta, \nu)$ by applying the frequency or wavelength dependent instrument response
${\cal R}(\uv,\lambda)$. 
We have considered the simple case where  the instrument response is constant throughout the survey area, or independent 
of the sky direction.
For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$: 
\begin{enumerate}
\item Apply a 2D Fourier transform to compute sky angular Fourier amplitudes
$$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(\uv, \lambda_k) \hspace{2mm} .$$ 
\item Apply instrument response in the angular wave mode plane. We use here the normalized instrument response
$ {\cal R}(\uv,\lambda_k)  \lesssim 1$
$$  {\cal T}_{sky}(\uv, \lambda_k)  \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda_k) \hspace{1mm} . $$
\item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map
without instrumental (electronic/$\Tsys$) white noise:
$$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda)   
\rightarrow \mathrm{Inv-2D-FFT} \rightarrow T_{mes1}(\alpha, \delta, \lambda_k) $$ 
\item Add white noise (Gaussian fluctuations) to the pixel map temperatures to obtain 
the measured sky brightness temperature $T_{mes}(\alpha, \delta, \nu_k)$. 
{\changemark The white noise hypothesis is reasonable at this level, since $(\uv)$ 
dependent instrumental response has already been applied.} 
We also considered that the system temperature, and thus the 
additive white noise level, was independent of the frequency or wavelength.   
\end{enumerate} 
The LSS signal extraction performance obviously depends on the white noise level. 
The results shown here correspond to the (a) instrument configuration, a packed array of 
$11 \times 11 = 121$ dishes  (5 meter diameter), with a white noise level corresponding 
to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500$ kHz 
cell. \\[1mm]

The different steps in the simple component separation procedure that has been applied are 
briefly described here.
\begin{enumerate}
\item The measured sky brightness temperature is first {\em corrected} for the frequency dependent 
beam effects through a convolution by a fiducial frequency independent beam ${\cal R}_f(\uv)$ This {\em correction}
corresponds to a smearing or degradation of the angular resolution
\begin{eqnarray*} 
 {\cal T}_{mes}(u, v, \lambda_k) & \rightarrow & {\cal T}_{mes}^{bcor}(u, v, \lambda_k) \\  
 {\cal T}_{mes}^{bcor}(u, v, \lambda_k)  & = & 
{\cal T}_{mes}(u, v, \lambda_k) \times \sqrt{ \frac{{\cal R}_f(\uv)}{{\cal R}(\uv,\lambda)} } \\
{\cal T}_{mes}^{bcor}(u, v, \lambda_k)  & \rightarrow & \mathrm{2D-FFT} \rightarrow  T_{mes}^{bcor}(\alpha,\delta,\lambda) \hspace{2mm} . 
\end{eqnarray*}
{\changemark 
The virtual target beam ${\cal R}_f(\uv)$  has a lower resolution than the worst real instrument beam,
i.e at the lowest observed frequency.
No correction has been  applied for modes with ${\cal R}(\uv,\lambda) \lesssim 1\%$, as a first 
attempt to represent imperfect knowledge of the instrument response.
We recall that this is the normalized instrument response,
${\cal R}(\uv,\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$ 
also has a numerical upper bound $\sim 100$. }
\item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
 is fitted to the beam-corrected brightness temperature. The parameters were obtained 
using a linear $\chi^2$ fit in the $\lgd ( T ) , \lgd (\nu)$ plane. 
We show here the results for a pure power law (P1), as well as a modified power law (P2):
\begin{eqnarray*} 
P1 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) \\
P2 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) + c \, \lgd ( \nu/\nu_0 ) ^2 
\end{eqnarray*}
where $b$ is the power law index and  $T_0 = 10^a$ the brightness temperature at the 
reference frequency $\nu_0$.

{\changemark Interferometers have a poor response at small $(\uv)$ corresponding to baselines 
smaller than interferometer element size. The zero-spacing baseline, the $(\uv)=(0,0)$ mode,  represents 
the mean temperature for a given frequency plane and cannot be measured with interferometers. 
We used a simple trick to make the power-law fitting procedure applicable,
by setting the mean value of the temperature for 
each frequency plane according to a power law with an index close to the synchrotron index 
($\beta\sim-2.8$). And we checked that the results are not too sensitive to the 
arbitrarily fixed mean temperature power law parameters. } 

\item The difference between the beam-corrected sky temperature and the fitted power law
$(T_0(\alpha, \delta), b(\alpha, \delta))$ is our extracted 21 cm LSS signal.
\end{enumerate}

Figure \ref{extlsspk} shows the performance of this procedure at a redshift $\sim 0.6$, 
for the two radio sky models used here: GSM/Model-I and Haslam+NVSS/Model-II. The 
21 cm LSS power spectrum, as seen by a perfect instrument with a 25 arcmin (FWHM) 
Gaussian frequency independent beam is shown, as well as  
the extracted power spectrum, after beam {\em correction} 
and foreground separation with second order polynomial fit (P2).
We have also represented the obtained power spectrum without applying the beam correction (step 1 above),
or with the first-order polynomial fit (P1). 

Figure \ref{extlssmap} shows a comparison of  the original 21 cm brightness temperature map at 884 MHz 
with the recovered 21 cm map, after subtracting the radio continuum component. It can be seen that structures 
present in the original map have been correctly recovered, although the amplitude of the temperature 
fluctuations on the recovered map is significantly smaller (factor $\sim 5$) than in the original map. 
{\changemark This is mostly due to the damping of the large-scale power ($k \lesssim 0.1 h \mathrm{Mpc^{-1}} $) 
due to the foreground subtraction procedure (see figure \ref{extlssratio}).}

We have shown that it should be possible to measure the red-shifted 21 cm emission fluctuations in the 
presence of the strong radio continuum signal, provided that the latter has a smooth frequency dependence.
However, a rather  precise knowledge of the instrument beam and the beam {\em correction} 
or smearing procedure described here  are key ingredients for recovering the 21 cm LSS power spectrum. 
It is also important to note that, while it is enough to correct the beam to the lowest resolution instrument beam 
($\sim 30'$ or $D \sim 50$ meter @ 820 MHz) for the GSM sky model, a stronger beam correction 
has to be applied ($\sim 36'$ or $D \sim 40$ meter @ 820 MHz) for Model-II to reduce 
significantly the ripples from bright radio sources. 
We have also applied the same procedure to simulate observations and LSS signal extraction for an instrument 
with a frequency-dependent Gaussian beam shape. The mode mixing effect is greatly reduced for
such a smooth beam, compared to the more complex instrument response 
${\cal R}(\uv,\lambda)$ used for the results shown in figure \ref{extlsspk}.

\begin{figure*}
\centering
% \vspace*{-25mm}
\mbox{
% \hspace*{-20mm}
\includegraphics[width=\textwidth]{Figs/extlsspk.pdf}
}
% \vspace*{-10mm}
\caption{Recovered power spectrum of the 21cm LSS temperature fluctuations, separated from the 
continuum radio emissions at $z \sim 0.6$, \mbox{$\gHI\simeq1.3\%$}, for the instrument configuration (a), $11\times11$ 
packed array interferometer.
Left: GSM/Model-I , right: Haslam+NVSS/Model-II. The black curve shows the residual after foreground subtraction,
corresponding to the 21 cm signal, WITHOUT applying the beam correction. The red curve shows the recovered 21 cm 
signal power spectrum, for P2 type fit of the frequency dependence of the radio continuum, and violet curve is the P1 fit (see text). The orange curve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency-independent Gaussian beam. }
\label{extlsspk}
\end{figure*}


\begin{figure*}
\centering
\vspace*{-20mm}
\mbox{
\hspace*{-25mm}
\includegraphics[width=1.20\textwidth]{Figs/extlssmap.pdf}
}
\vspace*{-25mm}
\caption{Comparison of the original 21 cm LSS temperature map @ 884 MHz ($z \sim 0.6$), smoothed
with 25 arc.min (FWHM) beam (top), and the recovered LSS map, after foreground subtraction for Model-I (GSM) (bottom),  for the instrument configuration (a), $11\times11$ packed array interferometer. }
\label{extlssmap}
\end{figure*}

\subsection{$P_{21}(k)$ measurement transfer function} 
\label{tfpkdef}
The recovered red shifted 21 cm emission power spectrum $P_{21}^{rec}(k)$ suffers a number of distortions, mostly damping,
 compared to the original $P_{21}(k)$ due to  the instrument response and the component separation procedure.
{\changemarkb 
We recall that we have neglected the curvature of redshift or frequency shells
in this numerical study, which affect our result at large angles $\gtrsim 10^\circ$. 
The results presented here and our conclusions are thus restricted to the wave-mode range 
$k \gtrsim 0.02 \mathrm{h \, Mpc^{-1}}$. 
} 
We expect damping on small scales, or large $k$, due to the finite instrument size, but also on large scales, small $k$,
if total power measurements (auto-correlations) are not used in the case of interferometers. 
The sky reconstruction and the component separation introduce additional filtering and distortions.
The real transverse plane transfer function might even be anisotropic. 

However, within the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the 
recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$
{\changemarkb , similar to the one defined by \cite{bowman.09}, equation (23):}
\begin{equation}
\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) \hspace{3mm} .
\end{equation}

Figure \ref{extlssratio} shows this overall transfer function for the simulations and component 
separation performed here, around $z \sim 0.6$, for the instrumental setup (a), 
a filled array of 121 $D_{dish}=5$ m dishes. {\changemark This transfer function has been obtained after averaging the reconstructed  
$ P_{21}^{rec}(k)$ for several realizations (50) of the LSS temperature field.
The black curve shows the ratio $\TrF(k)=P_{21}^{beam}(k)/P_{21}(k)$ of the computed to the original 
power spectrum, if the original LSS temperature cube is smoothed by the frequency independent 
target beam FWHM=30'. This black curve shows the damping effect due to the finite instrument size at 
small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}, \theta \lesssim 1^\circ$).  
The transfer function for the GSM foreground model (Model-I) and  the $11\times11$ filled array 
interferometer (setup (a)) is represented, as well as  the transfer function for a D=55 meter 
diameter dish. The transfer function for the Model-II/Haslam+NVSS and the setup (a) filled interferometer 
array is also shown. The recovered power spectrum also suffers significant damping on large 
scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}$, mostly due to the filtering of radial or 
longitudinal Fourier modes along the frequency or redshift direction ($k_\parallel$) 
by the component separation algorithm. We were able to remove the ripples on the reconstructed 
power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$36' 
target beam resolution, compared to $\sim$30' for the GSM model. This explains the lower transfer function
obtained for Model-II on small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}$). }

 It should be stressed that the simulations presented in this section were 
focused on the study of the radio foreground effects and have been carried 
intentionally with a very low instrumental noise level of 
$0.25$ mK per pixel, corresponding to several years of continuous 
observations ($\sim 10$ hours per $3' \times 3'$ pixel).
%
This transfer function is well represented by the analytical form:
\begin{equation}
\TrF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right)  \hspace{1mm} .
\label{eq:tfanalytique}
\end{equation} 

We simulated observations and radio foreground subtraction using 
the procedure described here for different redshifts and instrument configurations, in particular 
for the (e) configuration with 400 five-meter dishes. As the synchrotron and radio source strength 
increases quickly with decreasing frequency, we have seen that recovering the 21 cm LSS signal 
becomes difficult for higher redshifts, in particular for $z \gtrsim 2$. 

We have determined the transfer function parameters of equation (\ref{eq:tfanalytique}) $k_A, k_B, k_C$ 
for setup (e) for three redshifts, $z=0.5, 1 , 1.5$, and then extrapolated the value of the parameters 
for redshift $z=2, 2.5$. The value of the parameters are grouped in Table \ref{tab:paramtfk}, 
and the corresponding transfer functions are shown in Fig. \ref{tfpkz0525}. 

\begin{table}[hbt]
\caption{Transfer function parameters.} 
\label{tab:paramtfk}
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline 
\hspace{2mm} z \hspace{2mm} & \hspace{2mm} 0.5 \hspace{2mm}  & \hspace{2mm} 1.0 \hspace{2mm} &
\hspace{2mm} 1.5 \hspace{2mm} & \hspace{2mm} 2.0 \hspace{2mm}  & \hspace{2mm} 2.5 \hspace{2mm} \\
\hline 
$k_A \, (\mathrm{Mpc^{-1}})$ & 0.006 & 0.005 & 0.004 & 0.0035 & 0.003 \\
$k_B \, (\mathrm{Mpc^{-1}})$ & 0.038 & 0.019 & 0.012 & 0.0093 & 0.008 \\
$k_C \, (\mathrm{Mpc^{-1}})$ & 0.16   & 0.08   & 0.05   & 0.038   & 0.032 \\
\hline 
\end{tabular}
\end{center}
\tablefoot{ The transfer function parameters, $(k_A,k_B,k_C)$  (Eq. \ref{eq:tfanalytique}) 
at different redshifts and for instrumental setup (e), $20\times20$ packed array interferometer,
are given in $\mathrm{Mpc^{-1}}$ unit, and not in $\mathrm{h \, Mpc^{-1}}$. } 
\end{table}

\begin{figure}
\centering
% \vspace*{-25mm}
\mbox{
% \hspace*{-10mm}
\includegraphics[width=0.5\textwidth]{Figs/extlssratio.pdf}
}
% \vspace*{-30mm}
\caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 
21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $ (transfer function), at $z \sim 0.6$ 
for the instrument configuration (a), $11\times11$ packed array interferometer. 
The effect of a frequency-independent Gaussian beam of $\sim 30'$ is shown in black.
The transfer function $\TrF(k)$  for the instrument configuration (a), $11\times11$ packed array interferometer,
for the GSM/Model-I is shown in red, and in orange for Haslam+NVSS/Model-II. The transfer function 
for a D=55 meter diameter dish for the GSM model is also shown as the dashed red curve. }
\label{extlssratio}
\end{figure}


\begin{figure}
\centering
% \vspace*{-25mm}
\mbox{
% \hspace*{-10mm}
\includegraphics[width=0.5\textwidth]{Figs/tfpkz0525.pdf}
}
%\vspace*{-30mm}
\caption{Fitted/smoothed  transfer function $\TrF(k)$ obtained for the recovered 21 cm power spectrum at different redshifts, 
$z=0.5 , 1.0 , 1.5 , 2.0 , 2.5$ for the instrument configuration (e), $20\times20$ packed array interferometer. }
\label{tfpkz0525}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \section{ BAO scale determination and constrain on dark energy parameters}
% {\color{red} \large \it  CY ( + JR  )  } \\[1mm] 
%% We compute  reconstructed LSS-P(k) (after component separation) at different z's 
%% and determine BAO scale as a function of redshifts.
%% Method: 
%% \begin{itemize}
%% \item Compute/guess the overall transfer function for several redshifts (0.5 , 1.0 1.5 2.0 2.5 ) \\
%% \item Compute / guess the instrument noise level for the same redshit values
%% \item Compute the observed P(k) and extract $k_{BAO}$ , and the corresponding error 
%% \item Compute the DETF ellipse with different priors
%% \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Figures et texte fournis par C. Yeche - 10 Juin 2011 %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Sensitivity to cosmological parameters}
\label{cosmosec}

The impact of the various telescope configurations on the sensitivity for 21 cm 
power spectrum measurement has been discussed in Sec. \ref{pkmessens}. 
Figure \ref{figpnoisea2g} shows the noise power spectra and allows us to visually rank 
the configurations in terms of instrument noise contribution to P(k) measurement. 
The differences in $P_{noise}$ will translate into differing precisions
in the reconstruction of the BAO peak positions and in
the estimation of cosmological parameters. In addition, we have seen (Sect. \ref{recsec})
that subtraction of continuum radio emissions, Galactic synchrotron, and radio sources
also has an effect on the measured 21 cm power spectrum. 
In this paragraph, we present our method and the results for the precisions on the estimation 
of dark energy parameters through a radio survey of the redshifted 21 cm emission of LSS, 
with an instrumental setup similar to the (e) configuration (sec. \ref{instrumnoise}), 400 five-meter diameter 
dishes, arranged into a filled $20 \times 20$ array. 


\subsection{BAO peak precision}

To estimate the precision with which BAO peak positions can be
measured, we  used a method similar to the one established in 
\citep{blake.03} and \citep{glazebrook.05}.
%
To this end, we  generated reconstructed power spectra $P^{rec}(k)$ for
 slices of the  Universe with a quarter-sky coverage and a redshift depth,
 $\Delta z=0.5$ for  $0.25<z<2.75$. 
The peaks in the generated spectra were then determined by a 
fitting procedure and the reconstructed peak positions compared with the 
generated peak positions.
The reconstructed power spectrum used in the simulation is  
the sum of the expected \HI signal term, corresponding to Eqs. \ref{eq:pk21z} and \ref{eq:tbar21z}, 
damped by the transfer function $\TrF(k)$ (Eq. \ref{eq:tfanalytique} , Table \ref{tab:paramtfk})
and a white noise component $P_{noise}$ calculated according to the Eq. \ref{eq:pnoiseNbeam}, 
established in section \ref{instrumnoise} with $N=400$:
\begin{equation}
 P^{rec}(k) = P_{21}(k) \times \TrF(k) + P_{noise} 
\end{equation}
where the different terms ($P_{21}(k) , \TrF(k), P_{noise}$) depend on the slice redshift.  
The expected 21 cm power spectrum $P_{21}(k)$ has been generated according to the formula
%\begin{equation}
\begin{eqnarray}
\label{eq:signal}
\frac{P_{21}(\kperp,\kpar)}{P_{ref}(\kperp,\kpar)} = 
1\; + 
\hspace*{40mm}
\nonumber
\\ \hspace*{20mm}
A\, k \exp \bigl( -(k/\tau)^\alpha\bigr)
\sin\left( 2\pi\sqrt{\frac{\kperp^2}{\koperp^2} + 
\frac{\kpar^2}{\kopar^2}}\;\right)
\end{eqnarray}
%\end{equation}
where $k=\sqrt{\kperp^2 + \kpar^2}$, the parameters $A$, $\alpha$, and $\tau$ 
are adjusted to the  formula presented in 
\citep{eisenhu.98}, and $P_{ref}(\kperp,\kpar)$ is the 
envelope curve of the HI power spectrum without baryonic oscillations. 
The parameters $\koperp$ and $\kopar$ 
are the inverses of the oscillation periods in k-space. 
The following values were used for these 
parameters for the results presented here: $A=1.0$, $\tau=0.1 \, \hMpcm$, 
$\alpha=1.4$, and $\koperp=\kopar=0.060 \, \hMpcm$.

Each simulation is performed for a given set of parameters:
the system temperature $\Tsys$, an observation time 
$t_{obs}$, an average redshift, and a redshift depth $\Delta z=0.5$. 
Then,  each simulated  power spectrum  is fitted with a 2D
normalized function $P_{tot}(\kperp,\kpar)/P_{ref}(\kperp,\kpar)$, which is 
the sum of the signal power spectrum damped by the transfer function and the 
noise power spectrum  multiplied by a 
linear term,  $a_0+a_1k$. The upper limit $k_{max}$ in $k$ of the fit 
corresponds to the approximate position of the linear/nonlinear transition. 
This limit is established on the basis of the criterion discussed in  
\citep{blake.03}. 
In practice, we used $k_{max}= 0.145 \hMpcm,\,\, 0.18\hMpcm$, 
and $0.23 \hMpcm$ for the redshifts $z=0.5,\,\, 1.0$, and  $1.5$, respectively. 
 
Figure \ref{fig:fitOscill} shows the result of the fit for one of these simulations. 
Figure \ref{fig:McV2} histogram show the recovered values of  $\koperp$ and $\kopar$
for 100 simulations.
The widths of the two distributions give an estimate 
of the statistical errors.

In addition, in the fitting procedure, both the parameters modeling the 
signal $A$, $\tau$, $\alpha$, and the parameter correcting the noise power 
spectrum $(a_0,a_1)$ are floated to take the possible 
ignorance  of the signal shape and the uncertainties in the 
computation of the noise power spectrum into account.
In this way, we can correct possible imperfections, and the 
systematic uncertainties are directly propagated to statistical errors 
on the relevant parameters  $\koperp$ and $\kopar$. By subtracting the 
fitted noise contribution to each simulation, the baryonic oscillations 
are clearly observed, for instance, in Fig.~\ref{fig:AverPk}.

 
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm]{Figs/FitPk.pdf}
\caption{1D projection of the power spectrum for one simulation. 
The \HI power spectrum is divided by an envelop curve $P(k)_{ref}$ 
corresponding to the power spectrum without baryonic oscillations. 
The dots represents one simulation for a "packed" array of dishes  
with a system temperature,$T_{sys}=50$K, an observation time, 
$T_{obs}=$ 1 year, 
a solid angle of $1\pi sr$,
an average redshift, $z=1.5$ and a redshift depth, $\Delta z=0.5$. 
The solid line is the result of the fit to the data.}
\label{fig:fitOscill}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
%\includegraphics[width=\textwidth]{McV2.eps}
\includegraphics[width=9.0cm]{Figs/McV2.pdf}
\caption{ Distributions of the reconstructed 
wavelength  $\koperp$ and $\kopar$ perpendicular and parallel, 
respectively, to the line of sight
for simulations as in Fig. \ref{fig:fitOscill}. 
The fit by a Gaussian of the distribution (solid line) gives the 
width of the distribution,  which represents the statistical error 
expected on these parameters.}
\label{fig:McV2}
\end{center}
\end{figure}


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8.5cm]{Figs/AveragedPk.pdf}
\caption{1D projection of the power spectrum averaged over 100 simulations
of the packed dish array. 
The simulations are performed for the following conditions: a system 
temperature $T_{sys}=50$K, an observation time $T_{obs}=1$ year, 
a solid angle of $1 \pi sr$,
an average redshift $z=1.5$, and a redshift depth $\Delta z=0.5$. 
The \HI power spectrum is divided by an envelop curve $P(k)_{ref}$ 
corresponding to the power spectrum without baryonic oscillations, 
and the background estimated by a fit is subtracted. The errors are 
the RMS of the 100 distributions for each $k$ bin, and the dots are 
the mean of the distribution for each $k$ bin. }
\label{fig:AverPk}
\end{center}
\end{figure}




%\subsection{Results}

In our comparison of the various configurations, we have considered 
the following cases for $\Delta z=0.5$ slices with $0.25<z<2.75$.
\begin{itemize}
\item {\it Simulation without electronics noise}: the statistical errors on the power 
spectrum are directly related to the number of modes in the surveyed volume $V$ corresponding to 
the $\Delta z=0.5$ slice with the solid angle $\Omega_{tot}$ = 1 $\pi$ sr. 
The number of modes $N_{\delta k}$ in the wave number interval $\delta k$ can be written as
\begin{equation}
V  =  \frac{c}{H(z)} \Delta z  \times (1+z)^2 \dang^2  \Omega_{tot} \hspace{10mm}
N_{\delta k}  =  \frac{ V }{4 \pi^2} k^2 \delta k \hspace{3mm} .
\end{equation}   
\item {\it Noise}: we add the instrument noise as a constant term $P_{noise}$ as described in Eq. 
\ref {eq:pnoiseNbeam}. Table \ref{tab:pnoiselevel} gives the white noise level for an $N=400$ dish interferometer 
with $\Tsys = 50 \mathrm{K}$ and one year total observation time to survey $\Omega_{tot}$ = 1 $\pi$ sr.
\item {\it Noise with transfer function}: we consider the interferometer response and radio foreground
subtraction represented as the measured P(k) transfer function $T(k)$ (section \ref{tfpkdef}), as 
well as the instrument noise $P_{noise}$.
\end{itemize}

\begin{table}
\caption{Noise spectral power.} 
\label{tab:pnoiselevel} 
\begin{tabular}{|l|ccccc|}
\hline
z & \hspace{1mm} 0.5 \hspace{1mm} &  \hspace{1mm} 1.0 \hspace{1mm} &
\hspace{1mm} 1.5 \hspace{1mm} &  \hspace{1mm} 2.0 \hspace{1mm} & \hspace{1mm} 2.5 \hspace{1mm} \\
\hline 
$P_{noise} \, \mathrm{mK^2 \, (Mpc/h)^3}$ &  8.5 & 35 & 75 & 120 & 170 \\
\hline 
\end{tabular}
\end{table}

Table \ref{tab:ErrorOnK} summarizes the result. The errors both on $\koperp$ and $\kopar$
decrease as a function of redshift for simulations without electronic noise because the volume 
of the universe probed is larger. Once we apply the electronics noise, each slice in redshift gives
comparable results.  Finally, after applying the full reconstruction of the interferometer, the best 
accuracy is obtained for the first slices in redshift around 0.5 and 1.0 for an identical time of 
observation. We can optimize the survey by using a different observation time for each 
slice in redshift. Finally, for a 3-year survey we can split in five observation periods 
with durations that are three months, three months, six months, one year and one year 
for redshift 0.5, 1.0, 1.5, 2.0, and 2.5, respectively (Table  \ref{tab:ErrorOnK}, 4$^{\rm th}$ row). 

\begin{table*}[ht]
\caption{Sensitivity on $\mathbf{k}_{BAO}$ measurement.}
\label{tab:ErrorOnK}
\begin{center}
\begin{tabular}{lc|c c c c c }
\multicolumn{2}{c|}{$\mathbf z$ }& \bf 0.5 & \bf 1.0 &  \bf 1.5 & \bf 2.0 & \bf 2.5 \\
\hline\hline
\bf No noise, pure cosmic variance & $\sigma(\koperp)/\koperp$  (\%) & 1.8 & 0.8 & 0.6 & 0.5 &0.5\\
 & $\sigma(\kopar)/\kopar$  (\%) & 3.0 & 1.3 & 0.9 &  0.8 & 0.8\\
 \hline
 \bf  Noise without transfer function (a)  & $\sigma(\koperp)/\koperp$  (\%) & 2.3 & 1.8 & 2.2 & 2.4 & 2.8\\
 (3-months/redshift bin)& $\sigma(\kopar)/\kopar$  (\%) & 4.1 & 3.1  & 3.6 & 4.3 & 4.4\\
 \hline
 \bf   Noise with transfer function (a)  & $\sigma(\koperp)/\koperp$  (\%) & 3.0 & 2.5 & 3.5 & 5.2 & 6.5 \\
 (3-months/redshift bin)& $\sigma(\kopar)/\kopar$  (\%) & 4.8 & 4.0 & 6.2 & 9.3 & 10.3\\
 \hline
 \bf  Optimized survey (b) & $\sigma(\koperp)/\koperp$  (\%)   & 3.0 & 2.5 & 2.3 &  2.0 &  2.7\\
 (Observation time :  3 years)& $\sigma(\kopar)/\kopar$  (\%) & 4.8 & 4.0 & 4.1 &  3.6  & 4.3 \\
 \hline
\end{tabular}
\end{center}
\tablefoot{Relative errors on $\koperp$ and $\kopar$ measurements are given 
as a function of the redshift $z$ for various simulation configurations: \\
\tablefoottext{a}{simulations with electronics noise, without ($2^{\rm nd}$ row) and with ($3^{\rm rd}$ row) the transfer function;  } \\
\tablefoottext{b}{optimized survey, simulations with electronic noise and the transfer function} 
}
\end{table*}%



\subsection{Expected sensitivity on $w_0$  and $w_a$}

\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{Figs/dist.pdf}
\caption{
The two ``Hubble diagrams'' for BAO experiments.
The four falling curves give the angular size of the acoustic horizon
(left scale) and the four 
rising curves give the redshift interval of the acoustic horizon (right scale).
The solid lines are for 
$(\Omega_M,\Omega_\Lambda,w)=(0.27,0.73,-1)$,
the dashed  for 
$(1,0,-1)$
the dotted for 
$(0.27,0,-1)$, and
the dash-dotted  for 
$(0.27,0.73,-0.9)$,
The error bars on the solid curve correspond to the four-month run
(packed array)
of Table \ref{tab:ErrorOnK}.
 }
\label{fig:hubble}
\end{center}
\end{figure}


The observations give the \HI power spectrum in
angle-angle-redshift space rather than in real space.
The inverse of the peak positions  in the observed power spectrum therefore  
gives the angular and redshift intervals corresponding to the
sonic horizon.
The peaks in the angular spectrum are proportional to 
$d_T(z)/a_s$ and those in the redshift spectrum to $d_H(z)/a_s$, where 
$a_s \sim 105  h^{-1} \mathrm{Mpc}$ is the acoustic horizon comoving size at recombination, 
$d_T(z) = (1+z) \dang$ is the comoving angular distance and $d_H=c/H(z)$ the Hubble distance
(see Eq. \ref{eq:expHz}):
\begin{equation}
d_H = \frac{c}{H(z)} =  \frac{c/H_0}{\sqrt{\Omega_\Lambda+\Omega_m (1+z)^3} }   \hspace{5mm}
d_T = \int_0^z d_H(z) dz 
\label{eq:dTdH}
\end{equation}
The quantities $d_T$, $d_H$, and $a_s$ all depend on
the cosmological parameters.
Figure \ref{fig:hubble} gives the angular and redshift intervals
as a function of redshift for four cosmological models.
The error bars on the lines for
$(\Omega_M,\Omega_\Lambda)=(0.27,0.73)$
correspond to the expected errors 
on the peak positions
taken from Table \ref{tab:ErrorOnK}
for the four-month runs with the packed array.
We see that with these uncertainties, the data would be able to 
measure $w$ at better than the 10\% level.


To estimate the sensitivity 
to parameters describing the dark energy equation of 
state, we follow the procedure explained in 
\citep{blake.03}. We can introduce the equation of 
state of dark energy, $w(z)=w_0 + w_a\cdot z/(1+z)$, by 
replacing $\Omega_\Lambda$ in the definition of $d_T (z)$ and $d_H (z)$,
(Eq. \ref{eq:dTdH}) by
\begin{equation}
\Omega_\Lambda \rightarrow \Omega_{\Lambda} \exp \left[ 3  \int_0^z   
\frac{1+w(z^\prime)}{1+z^\prime } dz^\prime  \right]
\end{equation}
where $\Omega_{\Lambda}^0$ is the present-day dark energy fraction with 
respect to the critical density. 
Using the relative errors on  $\koperp$ and  $\kopar$ given in 
Table \ref{tab:ErrorOnK}, we can compute the Fisher matrix for  
five cosmological parameter: $(\Omega_m, \Omega_b, h, w_0, w_a)$. 
Then, the combination of this BAO Fisher 
matrix with the Fisher matrix obtained for Planck mission allows us to 
compute the errors on dark energy parameters.
{\changemark We used the Planck Fisher matrix, computed for the 
Euclid proposal \citep{laureijs.09}, for the 8 parameters: 
$\Omega_m$, $\Omega_b$, $h$, $w_0$, $w_a$,
$\sigma_8$, $n_s$ (spectral index of the primordial power spectrum) and
$\tau$  (optical depth to the last-scatter surface), 
assuming a flat universe. }

For an optimized project over a redshift range, $0.25<z<2.75$, with a total 
observation time of three years, the packed 400-dish interferometer array has a 
precision of  12\% on $w_0$ and 48\% on $w_a$. 
The  figure of merit (FOM), the inverse of the area in the 95\% confidence level 
contours, is 38.
Finally, Fig.~\ref{fig:Compw0wa} 
shows a comparison of different BAO projects, with a set of priors on 
$(\Omega_m, \Omega_b, h)$ corresponding to the expected precision on 
these parameters in early 2010s. {\changemark The confidence contour 
level in the plane $(w_0,w_a)$  were obtained by marginalizing 
over all the other parameters.} This BAO project based on \HI intensity 
mapping is clearly competitive with the current generation of optical 
surveys such as SDSS-III \citep{eisenstein.11}. 


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.55\textwidth]{Figs/Ellipse21cm.pdf}
\caption{$1\sigma$ and $2\sigma$ confidence level contours  in the 
parameter plane $(w_0,w_a)$, marginalized over all the other parameters, 
for two BAO projects:   SDSS-III (LRG) project 
(blue dotted line), 21 cm project with HI intensity mapping (black solid line).}
\label{fig:Compw0wa}
\end{center}
\end{figure}

\section{Conclusions}
The 3D mapping of redshifted 21 cm emission through {\it intensity mapping} is a novel and complementary 
approach to optical surveys for studying the statistical properties of the LSS in the universe
up to redshifts $z \lesssim 3$. A radio instrument with a large instantaneous field of view 
(10-100 deg$^2$) and large bandwidth ($\gtrsim 100$ MHz) with $\sim 10$ arcmin resolution is needed 
to perform a cosmological neutral hydrogen survey over a significant fraction of the sky. We have shown that 
a nearly packed interferometer array with a few hundred receiver elements spread over an hectare or a hundred beam 
focal plane array with a $\sim \hspace{-1.5mm} 100  \, \mathrm{meter}$ primary reflector will have the required sensitivity to measure 
the 21 cm power spectrum. A method of computing the instrument response for interferometers  
was developed, and we  computed the noise power spectrum for various telescope configurations.
The Galactic synchrotron and radio sources are a thousand times brighter than the redshifted 21 cm signal, 
making the measurement of the latter signal a major scientific and technical challenge. 
We also studied  the performance of a simple foreground subtraction method through realistic models of the sky 
emissions in the GHz domain and simulation of interferometric observations. 
We were able to show that the cosmological 21 cm signal from the LSS should be observable, but 
requires a very good knowledge of the instrument response. Our method allowed us to define and 
compute the overall  {\it transfer function} or {\it response function} for the measurement of the 21 cm 
power spectrum. 
Finally, we used the computed noise power spectrum and  $P(k)$ 
measurement response function to estimate 
the precision on the determination of dark energy parameters, for a 21 cm BAO survey. This radio survey 
could be carried using the current technology and would be competitive with the ongoing or planned 
optical surveys for dark energy,  with a fraction of their cost.
 
% \begin{acknowledgements}
% \end{acknowledgements}

\bibliographystyle{aa}

\begin{thebibliography}{}

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%  WMAP CMB 7 years power spectrum 2011 
% \bibitem[Hinshaw et al. (2008)]{hinshaw.08}  Hinshaw, G., Weiland, J.L., Hill, R.S.  {\it et al.} 2008, arXiv:0803.0732) 
\bibitem[Larson et al. 2011]{larson.11}  Larson, D.  et al. 2011,  (WMAP) \apjs, 192, 16 

%% Description MWA 
\bibitem[Lonsdale et al. 2009]{lonsdale.09}  Lonsdale C.J., Cappallo  R.J., Morales M.F. et al. 2009,
IEEE Proceeding, 97, 1497 (arXiv:0903.1828)

% Planck Fischer matrix, computed for EUCLID 
\bibitem[Laureijs 2009]{laureijs.09} Laureijs, R. 2009, ArXiv:0912.0914

% Temperature du 21 cm 
\bibitem[Madau et al. 1997]{madau.97} Madau, P., Meiksin, A. and Rees, M.J., 1997, \apj, 475, 429 
 
% Foret Ly alpha - 1 
\bibitem[McDonald et al. 2006]{baolya} McDonald P., Seljak, U. and Burles, S. et al. 2006, \apjs, 163, 80

% Foret Ly alpha - 2 , BAO from Ly-a
\bibitem[McDonald \& Eisenstein 2007]{baolya2} McDonald P., Eisenstein, D.J.  2007, Phys Rev D 76, 6, 063009

%  Boomerang 2000, Acoustic pics 
\bibitem[Mauskopf et al. 2000]{mauskopf.00} Mauskopf, P. D., Ade, P. A. R., de Bernardis, P. {\it et al.}  2000, \apjl, 536, 59

%% PNoise and cosmological parameters with reionization
\bibitem[McQuinn et al. 2006]{mcquinn.06}  McQuinn M., Zahn O., Zaldarriaga M., Hernquist L.,  Furlanetto S.R.
2006, \apj, 653, 815

%  Papier sur la mesure de sensibilite P(k)_reionisation 
\bibitem[Morales \& Hewitt 2004]{morales.04}  Morales M. \& Hewitt J., 2004, \apj, 615, 7

%  Papier sur le traitement des observations radio / mode mixing 
\bibitem[Morales et al. (2006)]{morales.06} Morales, M., Bowman, J.D., Hewitt, J.N., 2006, \apj, 648, 767

%% Foreground removal using smooth frequency dependence 
\bibitem[Oh \& Mack 2003]{oh.03} Oh S.P. \& Mack K.J., 2003, \mnras, 346, 871-877

%  Global Sky Model Paper 
\bibitem[Oliveira-Costa et al. 2008]{gsm.08} de Oliveira-Costa, A., Tegmark, M., Gaensler, B.~M. et al. 2008, 
\mnras, 388, 247

%%  Description+ resultats PAPER - arXiv:0904.2334
\bibitem[Parsons et al. 2010]{parsons.10}  Parsons A.R.,Backer D.C.,Bradley  R.F. et al.  2010, \aj, 139,  1468

% Livre Cosmo de Peebles 
\bibitem[Peebles 1993]{cosmo.peebles} Peebles, P.J.E., 1993, Principles of Physical Cosmology, 
(Princeton University Press)

% Original CRT HSHS paper (Moriond Cosmo 2006 Proceedings) 
\bibitem[Peterson et al. 2006]{peterson.06} Peterson, J.B.,  Bandura, K., \& Pen, U.-L. 2006, arXiv:0606104  

% Synchrotron index =-2.8 in the freq range 1.4-7.5 GHz 
\bibitem[Platania et al. 1998]{platania.98}   Platania P., Bensadoun M., Bersanelli M. et al. 1998,  \apj, 505, 473

% SDSS BAO 2007 
\bibitem[Percival et al. 2007]{percival.07}   Percival, W.J., Nichol, R.C., Eisenstein, D.J. et al. 2007, \apj,  657, 645

% SDSS BAO 2010  - arXiv:0907.1660 
\bibitem[Percival et al. 2010]{percival.10}   Percival, W.J., Reid, B.A., Eisenstein, D.J. et al. 2010, \mnras, 401, 2148

% Livre Cosmo de Jim Rich 
\bibitem[Rich 2001]{cosmo.rich} James Rich,  2001, Fundamentals of Cosmology (Springer)

% Radio spectral index between 100-200 MHz 
\bibitem[Rogers \& Bowman 2008]{rogers.08} Rogers, A.E.E. \& Bowman, J. D. 2008, \aj 136, 641

%% LOFAR description 
\bibitem[Rottering et al. 2006]{rottgering.06} Rottgering H.J.A., Braun, r., Barthel, P.D. et al. 2006, arXiv:astro-ph/0610596
%%%%

%% SDSS-3
%  \bibitem[SDSS-III 2008]{sdss3} SDSS-III 2008, http://www.sdss3.org/collaboration/description.pdf 

% Reionisation: Can the reionization epoch be detected as a global signature in the cosmic background?
\bibitem[Shaver  et al. 1999]{shaver.99} Shaver P.A., Windhorst R. A., Madau P., de Bruyn A.G. \aap, 345, 380

%  Frank H. Briggs, Matthew Colless, Roberto De Propris, Shaun Ferris, Brian P. Schmidt, Bradley E. Tucker

% Papier 21cm-BAO Fermilab  ( arXiv:0910.5007)
\bibitem[Seo et al 2010]{seo.10} Seo, H.J. Dodelson, S., Marriner, J. et al.  2010, \apj, 721, 164

% Mesure P(k) par SDSS 
\bibitem[Tegmark et al.  2004]{tegmark.04}  Tegmark M., Blanton  M.R, Strauss M.A. et al. 2004, \apj, 606, 702

% FFT telescope   % arXiv:0802.1710 
\bibitem[Tegmark \& Zaldarriaga 2009]{tegmark.09} Tegmark, M. \& Zaldarriaga, M., 2009, \prd, 79, 8, 083530 

%  Thomson-Morane livre interferometry 
\bibitem[Thompson, Moran \& Swenson (2001)]{radastron} Thompson, A.R., Moran, J.M., Swenson, 2001, G.W, Interferometry and 
Synthesis in Radio Astronomy (John Wiley \& sons, 2nd Edition)

% Lyman-alpha, HI fraction 
\bibitem[Wolf et al. 2005]{wolf.05} Wolfe, A. M., Gawiser, E. \& Prochaska, J.X. 2005  \araa, 43, 861

%  BAO à 21 cm et reionisation  %  http://fr.arxiv.org/abs/0709.2955, 
\bibitem[Wyithe et al. 2008]{wyithe.08} Wyithe, S., Loeb, A. \& Geil, P. 2008, \mnras, 383, 1195 

%% Papier fluctuations 21 cm par Zaldarriaga et al 
\bibitem[Zaldarriaga et al. 2004]{zaldarriaga.04}  Zaldarriaga, M., Furlanetto, S.R., Hernquist, L., 2004, \apj, 608, 622

%% Today HI cosmological density 
\bibitem[Zwaan et al. 2005]{zwann.05} Zwaan, M.A., Meyer, M.J., Staveley-Smith, L., Webster, R.L. 2005, \mnras, 359, L30

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