source: Sophya/trunk/Cosmo/RadioBeam/sensfgnd21cm.tex@ 3975

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%  21cm LSS P(k) sensitivity and foreground substraction 
%  R. Ansari, C. Magneville, J. Rich, C. Yeche et al 
%     2010 - 2011
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\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} }

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

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

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\begin{document}
%
   \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{3}
         \and
          P.Colom  \inst{5} 
          \and
          J.M. Le Goff \inst{4}
          \and
          C. Magneville \inst{4}
          \and
          J.M. Martin  \inst{5} 
          \and
          M. Moniez \inst{3}
        \and 
          J.Rich \inst{4}
          \and 
          C.Y\`eche \inst{4}
          }

   \institute{
     Universit\'e Paris-Sud, LAL, UMR 8607, F-91898 Orsay Cedex, France 
   \and 
    CNRS/IN2P3,  F-91405 Orsay, France \\
     \email{ansari@lal.in2p3.fr}
    \and
    Laboratoire de lÍAcc\'el\'erateur Lin\'eaire, CNRS-IN2P3, Universit\'e Paris-Sud,
    B.P. 34, 91898 Orsay 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 December 15, 2010; accepted xxxx, 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 large instanteneous field of view and waveband}
 % aims heading (mandatory)
  { In this paper, we study the sensitivity of different radio interferometer configuration for the observation of large scale structures 
   and BAO oscillations in 21 cm and we discuss the problem of foreground removal. }
  % methods heading (mandatory)
 { For each configuration, we determine instrument response by computing the (u,v) plane (Fourier angular frequency plane) 
  coverage using visibilities. The (u,v) plane response is then used to compute the three dimensional noise power spectrum, 
hence the instrument sensitivity for LSS P(k) measurement. We describe also   a simple foreground subtraction method to 
separate LSS 21 cm signal from the foreground due to the galactic synchrotron and radio sources emission. }
  % results heading (mandatory)
   { We have computed the noise power spectrum for different instrument configuration as well as the extracted 
   LSS power spectrum, after separation of 21cm-LSS signal from the foregrounds. }
  % conclusions heading (optional), leave it empty if necessary 
   { We show that an interferometer with few hundred elements and a surface coverage of 
  $\lesssim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ }

   \keywords{ Cosmology:LSS --
                 Cosmology:Dark energy 
               }

   \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 Structure (LSS) in the Universe and their evolution 
with redshift is one the major tools in observational cosmology. Theses structures are usually mapped through 
optical observation of galaxies which are used as tracers of the underlying matter distribution. 
An alternative and elegant approach for mapping the matter distribution, using neutral atomic hydrogen 
(\HI) as tracer with Total Intensity Mapping, has been proposed in recent years \citep{peterson.06} \citep{chang.08}. 
Mapping the matter distribution using HI 21 cm emission as a tracer has been extensively discussed in literature
\citep{furlanetto.06} \citep{tegmark.08} and is being used in projects such as LOFAR \citep{rottgering.06} or
MWA \citep{bowman.07} to observe reionisation  at redshifts z $\sim$ 10. 

Evidences in favor of the acceleration of the expansion of the universe have been 
accumulated over the last twelve years, thank to the observation of distant supernovae, 
CMB anisotropies and 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 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 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 which were present in the photons baryons plasma prior to recombination 
at z $\sim$ 1100. 
This scale, which can be considered as a standard ruler with a comoving 
length  of $\sim 150 Mpc$.
Theses features have been first observed in the CMB anisotropies
and are usually referred to as {\em acoustic pics} \citep{mauskopf.00} \citep{hinshaw.08}. 
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 
\citep{eisenstein.05}  \citep{percival.07} and 2dGFRS  \citep{cole.05}  optical galaxy surveys.

Ongoing or future surveys plan to measure precisely the BAO scale in the redshift range 
$0 \lesssim z \lesssim 3$, using either optical observation of galaxies or through 3D mapping 
Lyman $\alpha$ absorption lines toward distant quasars \citep{baorss} \cite{baolya}. 
Mapping matter distribution using 21 cm emission of neutral hydrogen appears as 
a very promising technique to map matter distribution up to redshift $z \sim 3$, 
complementary to optical surveys, especially in the optical redshift desert range 
$1 \lesssim z \lesssim 2$. 

In section 2, we  discuss the intensity mapping and its potential for measurement of 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 show also  the results for the 3D noise power spectrum for several  instrument configurations.
The contribution of foreground emissions due to the galactic synchrotron and radio sources
is described in section 4, as well as a simple component separation method  The performance of this 
method using sky model or known radio sources are also presented in section 4. 
The constraints which can be obtained on the Dark Energy parameters and DETF figure 
of merit for typical 21 cm intensity mapping survey are shown in section 5.

\citep{ansari.08}


%__________________________________________________________________

\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 at large scales
($ \gtrsim 1 \mathrm{deg}$), while the interpretation at smallest scales 
might suffer from the uncertainties on the non linear clustering effects.  
The BAO features in particular are at the degree angular scale on the sky 
and thus can be resolved easily with a rather modest size radio instrument 
($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 scale. 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.   

In order 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 few $Gpc^3$. Moreover, stringent constrain on DE parameters can be obtained when 
comparing the distance or Hubble parameter measurements as a function of redshift with 
DE models, which translates into a 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 ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$). 

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}),
the method envisaged has been mostly 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 a radio instrument 
characterized by an effective collecting area $A$, and system temperature $\Tsys$ can be written as 
\begin{equation}
S_{lim} = \frac{ 2 \kb \, \Tsys }{ A \, \sqrt{t_{int} \delta \nu} } 
\end{equation} 
where $t_{int}$ is the total integration time $\delta \nu$ is the detection frequency band. In table 
\ref{slims21} (left)  we have computed the sensitivity for 4 different set 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 detection of \HI source with an intrinsic velocity 
dispersion of few 100 km/s. Theses detection limits should be compared with the expected 21 cm brightness
$S_{21}$ of compact sources which can be computed using the expression below:
\begin{equation}
 S_{21}  \simeq  0.021 \mathrm{\mu Jy} \, \frac{M_{H_I} }{M_\odot}   \times 
\left( \frac{ 1\, \mathrm{Mpc}}{\dlum} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v} 
\end{equation}
 where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum$ is the luminosity distance and $\sigma_v$ 
is the source velocity dispersion. 
{\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 is computed for the standard 
WMAP \LCDM universe. $10^9 - 10^{10} M_\odot$ of neutral gas mass 
is typical for large galaxies \citep{lah.09}. It is clear that detection of \HI sources at cosmological distances
would require collecting area in the range of $10^6 \mathrm{m^2}$. 

Intensity mapping has been suggested as an alternative and economic method to map the 
3D distribution of neutral hydrogen \citep{chang.08} \citep{ansari.08}. In this approach, 
sky brightness map with angular resolution $\sim 10-30 \mathrm{arc.min}$ is made for a 
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 $\sim 10 \mathrm{Mpc^3}$, containing hundreds of galaxies and a total 
\HI mass $ \gtrsim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow, 
the observed frequency $\nu$ would be translated to 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 angular scales of $\sim 10 \mathrm{arc.min}$ 
can then be observed without the detection of individual compact \HI sources, using the set of sky brightness 
map as a function 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 well known black body Rayleigh-Jeans approximation:
$$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} $$ 
 
%%%%%%%%
\begin{table}
\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 & 3.5 \\
100 000 & 25 & 1.7 \\
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
0.25 &  1235   & 140 \\
0.50 &  2800   & 27 \\
1.0   &  6600   & 4.8 \\
1.5   &  10980 & 1.74 \\
2.0   &  15710  & 0.85 \\
2.5  &  20690 & 0.49 \\
\hline
\end{tabular}
\caption{Sensitivity or source detection limit for 1 day integration time (86400 s) and 1 MHz 
frequency band (left). Source 21 cm brightness for $10^{10} M_\odot$ \HI for different redshifts (right)  }
\label{slims21} 
\end{center}
\end{table}

\subsection{ \HI power spectrum and BAO}
In the absence of any foreground or background radiation, 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:
\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 $A_{21}=1.87 \, 10^{-15} \mathrm{s^{-1}}$ is the spontaneous 21 cm emission 
coefficient, $h$ is the Planck constant, $c$ the speed of light, $\kb$ the Boltzmann 
constant and $H(z)$ is 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  \hub  \, \left[ \Omega_m (1+z)^3 + \Omega_\Lambda \right]^{\frac{1}{2}} 
\times  70 \, \, \mathrm{km/s/Mpc}  
\end{equation}
Introducing the \HI mass fraction relative to the total baryon mass $\gHI$, the 
neutral hydrogen number density can be written as:
\begin{equation}
\etaHI (\vec{\Theta}, z(\lambda) ) = \gHIz \times \Omega_B  \frac{\rho_{crit}}{m_{H}}  \times 
\frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) 
\end{equation}
where $\Omega_B, \rho_{crit}$ are respectively the present day mean baryon cosmological
and critical densities, $m_{H}$ is the hydrogen atom mass, and 
$\frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}}$ is the \HI density fluctuations. 

The present day neutral hydrogen fraction $\gHI(0)$ 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. Study 
of Lyman-$\alpha$ absorption indicate a factor 3 increase in the neutral hydrogen
fraction at $z=1.5$, compared to the its present day value $\gHI(z=1.5) \sim 0.025$
\citep{wolf.05}. 
The 21 cm brightness temperature and the corresponding power spectrum can be written as \citep{wyithe.07} :
\begin{eqnarray}
  \TTlamz  &  = & \bar{T}_{21}(z) \times \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z)  \\
  P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)  \\
 \bar{T}_{21}(z)  & \simeq & 0.054  \, \mathrm{mK}  
\frac{ (1+z)^2 \, \hub }{\sqrt{ \Omega_m (1+z)^3 + \Omega_\Lambda } } 
 \dfrac{\Omega_B}{0.044}  \,  \frac{\gHIz}{0.01} 
\end{eqnarray}

The table \ref{tabcct21} below 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} parametrisation. The correspondence  with the angular scales is also 
shown for the standard WMAP \LCDM cosmology, according to the relation:
\begin{equation}
\mathrm{ang.sc} = \frac{2 \pi}{k^{comov} \, \dang(z) \, (1+z) }  
\hspace{3mm} 
k^{comov} = \frac{2 \pi}{ \mathrm{ang.sc}  \, \dang(z) \, (1+z) }  
\end{equation}
where $k^{comov}$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance.
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}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline 
\hline
   & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\
\hline 
(a) $\bar{T}_{21}$ (mK) & 0.08 & 0.1 & 0.13 & 0.16 & 0.18 & 0.2 & 0.21 \\
\hline
(b) $\bar{T}_{21}$ (mK)  & 0.08 & 0.12 & 0.21 & 0.32 & 0.43 & 0.56 & 0.68 \\
\hline
\hline  
\end{tabular}
\caption{Mean 21 cm brightness temperature in mK, as a function of redshift, for the 
standard \LCDM cosmology with constant \HI mass fraction at $\gHIz$=0.01  (a) or linearly 
increasing mass fraction (b)  $\gHIz=0.008(1+z)$ }
\label{tabcct21} 
\end{center}
\end{table}

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Figs/pk21cmz12.pdf}
\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 }

\subsection{Instrument response}
In astronomy we are usually interested in measuring the sky emission intensity,
$I(\vec{\Theta},\lambda)$ in a given wave band, as a function  the 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 and bear no information:
\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}  =  \delta(   \vec{\Theta} - \vec{\Theta '} )  I(\vec{\Theta},\lambda)
\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 have 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 
$ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta})$ 
corresponds to the receiver intensity response:
\begin{equation}
L(\vec{\Theta}) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) 
\end{equation} 
The visibility signal between 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 position $\vec{r_1}$ and 
$\vec{r_2}$ can simply be written as a function 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 narrow field of view 
($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \mathrm{deg.} $ ), 
and coplanar in respect to their common axis.
If we introduce two {\em Cartesian} like angular coordinates $(\alpha,\beta)$ centered at 
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 
$2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$:
\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 $y$ axis in the receiver plane are taken parallel to the 
two $(\alpha, \beta)$ angular planes. 

Furthermore, we introduce the conjugate Fourier variables $(u,v)$ and the Fourier transforms
of the sky intensity and the receiver beam:
\begin{center}
\begin{tabular}{ccc}
$(\alpha, \beta)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & $(u,v)$ \\
$I(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal I}(u,v, \lambda)$ \\
$L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(u,v, \lambda)$ \\
\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 
$(u, v)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$. The weight function is 
given by the receiver beam Fourier transform. 
\begin{equation}
\vis(\lambda) \simeq  \iint d u d v \, \, {\cal I}(u,v, \lambda) \, {\cal L}(u - 2 \pi \frac{\Delta x}{\lambda} , v - 2 \pi \frac{\Delta y}{\lambda} , \lambda)
\end{equation} 

A single receiver instrument would measure the total power integrated in a spot centered around the 
origin in the $(u,v)$ 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  $(u, v)_{12}$ 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 
$(u,v)$ plane might overlap and some pairs might measure the same area (same base lines). 
Several beam can be formed using different combination of the correlation from different 
antenna pairs.  

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

\subsection{Noise power spectrum}
Let's consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency 
bandwidth $\delta \nu$, 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 
$\sigma_{noise}^2 = \frac{4 \Tsys^2}{t_{int} \, \delta \nu}$. This term 
corresponds also to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated 
noise. If the receiver has an effective area $A \simeq \pi D^2$ or $A \simeq 4 D_x D_y$, the measurement 
corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. 
The sky temperature measurement can thus be characterized by the noise spectral power density in 
the angular frequencies plane $P_{noise}^{(u,v)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$  
per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$:
\begin{eqnarray}
P_{noise}^{(u,v)} & = & \frac{\sigma_{noise}^2}{ A / \lambda^2 }  \\
P_{noise}^{(u,v)} & \simeq & \frac{ \Tsys^2 }{t_{int}  \, \delta \nu} \, \frac{ \lambda^2 }{ D^2 } 
\hspace{5mm} \mathrm{units:} \, \mathrm{K^2 \times rad^2} \\
\end{eqnarray}

In a given instrument configuration, if several ($n$) receiver pairs have the same baseline, 
the noise power density in the corresponding $(u,v)$ plane area is reduced by a factor $1/n$. 
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 distance scale, 
and frequencies or wavelengths into comoving radial distance, using the following relations:
\begin{eqnarray}
\delta \alpha , \beta & \rightarrow & \delta \ell_\perp = (1+z) \, \dang(z) \, \delta \alpha,\beta  \\
\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 u , v & \rightarrow & \delta k_\perp = \frac{ \delta u , v }{  (1+z) \, \dang(z)  } \\
\frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = \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}

The three dimensional projected noise spectral density can then be written as: 
\begin{equation}
P_{noise}(k) = \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  
\end{equation} 

$P_{noise}(k)$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
$t_{int}$ 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 matter or \HI distribution power spectrum determination statistical errors vary as the number of 
observed Fourier modes, which is inversely proportional to volume of the universe 
which is observed (sample variance). 

In the following, we will consider the survey of a fixed
fraction of the sky, defined by  total solid angle $\Omega_{tot}$, performed during a fixed total 
observation time $t_{obs}$. We will consider several instrument configurations, having 
comparable instantaneous bandwidth, and comparable system receiver noise $\Tsys$:  
\begin{enumerate}
\item Single dish instrument, diameter $D$ with one or several independent feeds (beams) in the focal plane
\item Filled square shaped arrays, made of $n = q \times q$ dishes of diameter $D_{dish}$ 
\item Packed or unpacked cylinder arrays
\item Semi-filled array of $n$ dishes    
\end{enumerate}

We  compute below a simple expression for the noise spectral power density for radio 
sky 3D mapping surveys. 
It is important to notice that the instruments we are considering do not have a flat 
response in the $(u,v)$ plane, and the observations provide no information above 
$u_{max},v_{max}$. One has to take into account either a damping of the 
observed sky power spectrum or an increase of the noise spectral power 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. 
  
% \noindent {\bf Single dish instrument} \\
A single dish instrument with diameter $D$ would have an instantaneous field of view 
(or 2D pixel size) $\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and would require 
a number of pointing $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area. 
The noise power spectral density  could then be written as:
\begin{equation}
P_{noise}^{survey}(k) = \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  
\end{equation}
For a single dish instrument equipped with a multi-feed or phase array receiver system, 
with $n$ independent beam on sky, the noise spectral density decreases by a factor $n$, 
thanks to the an increase of per pointing integration time. 

For a single dish of diameter $D$, or an interferometric instrument with maximal extent $D$, 
observations provide information up to $u,v_{max} \lesssim 2 \pi D / \lambda $. This value of 
$u,v_{max}$ would be mapped to a maximum transverse cosmological wave number
$k^{comov}_{\perp \, max}$:
\begin{eqnarray}
k^{comov}_{\perp} & = & \frac{(u,v)}{(1+z) \dang}  \\
k^{comov}_{\perp \, max} & \lesssim &  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} 
\end{eqnarray}   

Figure \ref{pnkmaxfz} shows the evolution of a radio 3D  temperature mapping 
$P_{noise}^{survey}(k)$ as a function of survey redshift. 
The survey is supposed to cover a quarter of sky $\Omega_{tot} = \pi \mathrm{srad}$, in one 
year. The maximum comoving wave number $k^{comov}$  is also shown as a function 
of redshift, for an instrument with $D=100 \mathrm{m}$ maximum extent. In order 
to take into account the radial component of $\vec{k^{comov}}$ and the increase of 
the instrument noise level with $k^{comov}_{\perp}$, we have taken:
\begin{equation}
k^{comov}_{ max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}} 
\end{equation}

\begin{figure}
\vspace*{-10mm}
\centering
\mbox{
\hspace*{-10mm}
\includegraphics[width=0.6\textwidth]{Figs/pnkmaxfz.pdf}
}
\vspace*{-35mm}
\caption{Minimal noise level for a 100 beam instrument as a function of redshift (top). 
 Maximum $k$ value for  a 100 meter diameter primary antenna (bottom) }
\label{pnkmaxfz}
\end{figure}


\subsection{Instrument configurations and noise power spectrum}

We have numerically computed the instrument response in the (u,v) plane for several 
instrument configurations, at redshift $z=1$.
\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 8 rows, each with 16 dishes. These 128 dishes are spread over an area 
$80 \times 80  \, \mathrm{m^2}$ 
\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 4 sub-arrays of packed 16 dishes ($4\times4$), located in the 
four array corners.
\item [{\bf d} :] A single dish instrument, with diameter $D=75 \mathrm{m}$, 
equipped with a 100 beam focal plane instrument. 
\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 4 cylindrical reflectors, each 85 meter long and 12 meter 
wide. The focal line of each cylinder is equipped with 100 receivers, each with length 
$2 \lambda$, which corresponds 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 polarisation, as in the (e) configuration.
We have computed the noise power spectrum for {\em perfect} 
cylinders, where all receiver pair correlations are used (fp), or for 
a non perfect instrument, where only correlations between receivers 
from different cylinders are used.
\item[{\bf g} :] A packed array of 8 cylindrical reflectors, each 102 meter long and 12 meter 
wide. The focal line of each cylinder is equipped with 100 receivers, each with length 
$2 \lambda$, which corresponds 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 polarisation. 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 
a non perfect instrument, where only correlations between receivers 
from different cylinders are used.
\end{itemize}
The array layout for configurations (b) and (c) are shown in figure \ref{figconfab}. 
\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{figconfab}
\end{figure}

We have used simple triangular shaped dish response in the $(u,v)$ plane. 
However, we have introduced a fill factor or illumination efficiency 
$\eta$, relating the effective dish diameter $D_{ill}$ to the 
mechanical dish size $D^{ill} = \eta \, D_{dish}$. 
\begin{eqnarray}
{\cal L}_\circ (u,v,\lambda) & = & \bigwedge_{[\pm 2 \pi D^{ill}/ \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 multi-dish configuration studied here, we have taken the illumination efficiency factor
{\bf $\eta = 0.9$}. 

For the receivers along the focal line of cylinders, we have assumed that the 
individual receiver response in the $(u,v)$ plane corresponds to one from a 
rectangular shaped antenna. The illumination efficiency factor has been taken 
equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$ 
along the cylinder length. 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 obtained here for the noise power would not be significantly changed.
\begin{equation}
 {\cal L}_\Box(u,v,\lambda)  = 
\bigwedge_{[\pm 2 \pi D^{ill}_x / \lambda]} (u ) \times
\bigwedge_{[\pm 2 \pi D^{ill}_y / \lambda ]} (v ) 
\end{equation}
Figure \ref{figuvcovabcd} for the four configurations with $\sim 100$ receivers per 
polarisation. The resulting projected noise spectral power density is shown in figure 
\ref{figpnoisea2g}. The increase of $P_{noise}(k)$ at low $k^{comov} \lesssim 0.02$ 
is due to the fact that we have ignored all auto-correlation measurements.  
It can be seen that an instrument with $100$ 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=0.90\textwidth]{Figs/uvcovabcd.pdf}
}
\caption{(u,v) plane coverage for four configurations.
(a) 121 D=5 meter diameter dishes arranged in a compact, square array 
of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each, 
(c) 129 dishes arranged as above, single D=65 meter diameter, with 100 beams. 
color scale : black $<1$, blue, green, yellow, red $\gtrsim 80$ }
\label{figuvcovabcd}
\end{figure*}
 
\begin{figure*}
\vspace*{-10mm}
\centering
\mbox{
\hspace*{-10mm}
\includegraphics[width=\textwidth]{Figs/pnoisea2g.pdf}
}
\vspace*{-10mm}
\caption{P(k) LSS power  and noise power spectrum for several interferometer 
configurations ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers.}
\label{figpnoisea2g}
\end{figure*}


\section{ Foregrounds and Component separation }

\subsection{ Synchrotron and radio sources }
% {\color{red}  \large \it  Reza (+  J.M. Martin ?) + CMV   } \\[1mm] 

Description of the radio foregrounds for LSS@21cm and the sky models used
\begin{itemize}
\item Galactic synchrotron 
\item Radio sources : spectral behavior and brightness distribution 
\item GSM global sky model (Angelica) 
\item simple sky model : Synchrotron (HASLAM/WMAP) + sources (North20 / NVSS catalogue ) 
\end{itemize}

\subsection{ LSS signal extraction }
{\color{red} \large \it  CMV + Reza  +  J.M. Martin  } \\[1mm] 
Description of the component separation method and the results 
\begin{itemize}
\item Component separation method, based on instrument response correction and frequency 
smoothness / power law 
\item Foreground power spectrum 
\item Performance of component separation : comparison of frequency slices of recovered LSS 
and foreground maps, source catalogs 
\item Performance in statistical sense (power spectrum) : comparison of recovered P(k)-LSS 
and true P(k), residual noise/systematic effect power spectrum 
\end{itemize}


\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.
We can this a large number of time ( ~ 100 \ldots 1000 ) to have the reconstructed P(k) 
with {\it realistic } errors. We can then determine the error on the reconstructed DE 
parameters 

\section{Conclusions}

% \begin{acknowledgements}
% \end{acknowledgements}

%%% Quelques figures pour illustrer les resultats attendus 



\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{Figs/compexlss.png}
\caption{Comparison of the original simulated LSS (frequency plane) and the recovered LSS.
Color scale in mK }
\label{figcompexlss}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{Figs/compexfg.png}
\caption{Comparison of the original simulated foreground  (frequency plane) and 
the recovered foreground map. Color scale in Kelvin }
\label{figcompexfg}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{Figs/pklssfg.pdf}
\caption{Comparison of the LSS power spectrum at 21 cm at 900 MHz ($z \sim 0.6$) 
and the synchrotron/radio sources - GSM (Global Sky Model) foreground sky cube}
\label{figcompexfg}
\end{figure*}


\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{Figs/exlsspk.pdf}
\caption{Recovered LSS power spectrum, after component separation - - GSM (Global Sky Model) foreground sky cube}
\label{figexlsspk}
\end{figure*}

\bibliographystyle{aa}

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%%%%

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\end{thebibliography}

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