Index: /trunk/Cosmo/RadioBeam/sensfgnd21cm.tex
===================================================================
--- /trunk/Cosmo/RadioBeam/sensfgnd21cm.tex	(revision 4010)
+++ /trunk/Cosmo/RadioBeam/sensfgnd21cm.tex	(revision 4011)
@@ -39,7 +39,7 @@
 \newcommand{\dang}{d_A}
 \newcommand{\hub}{ h_{70} }
-\newcommand{\hubb}{ h_{100} }
-
-\newcommand{\etaHI}{ \eta_{\tiny HI} }
+\newcommand{\hubb}{ h }    % h_100
+
+\newcommand{\etaHI}{ n_{\tiny HI} }
 \newcommand{\fHI}{ f_{H_I}(z)}
 \newcommand{\gHI}{ g_{H_I}}
@@ -52,4 +52,13 @@
 \newcommand{\citep}[1]{ (\cite{#1}) } 
 %% \newcommand{\citep}[1]{ { (\tt{#1}) } } 
+
+%%% 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}}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -100,5 +109,5 @@
    }
 
-   \date{Received June 15, 2011; accepted xxxx, 2011}
+   \date{Received July 15, 2011; accepted xxxx, 2011}
 
 % \abstract{}{}{}{}{} 
@@ -116,17 +125,19 @@
 instruments 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, 
+ { For each configuration, we determine instrument response by computing the (u,v) or 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. }
+   LSS power spectrum, after separation of 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 beamns and a surface coverage of 
-  $\lesssim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ }
+   { We show that a radio instrument with few hundred simultaneous beams and a collecting area of 
+  $\lesssim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ and will be 
+  competitive with optical surveys. }
 
    \keywords{ Cosmology:LSS --
-                 Cosmology:Dark energy 
+                 Cosmology:Dark energy -- Radio interferometer -- 21 cm 
                }
 
@@ -171,9 +182,10 @@
 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 \citep{baorss}     %  CHECK/FIND baorss baolya references
-or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars \cite{baolya}. 
+\citep{eisenstein.05}  \citep{percival.07}  \citep{percival.10}  and 2dGFRS  \citep{cole.05}  optical galaxy surveys.
+
+Ongoing \citep{eisenstein.11}   or future surveys \citep{lsst.science}  
+plan to measure precisely the BAO scale in the redshift range 
+$0 \lesssim z \lesssim 3$, using either optical observation of galaxies   %  CHECK/FIND baorss baolya references
+or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars \citep{baolya},\citep{baolya2}. 
 Mapping matter distribution using 21 cm emission of neutral hydrogen appears as 
 a very promising technique to map matter distribution up to redshift $z \sim 3$, 
@@ -186,10 +198,8 @@
 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. 
+is described in section 4, as well as a simple component separation method. The performance of this 
+method using two different sky models is 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}
+of merit for typical 21 cm intensity mapping survey are discussed in section 5.
 
 
@@ -207,19 +217,19 @@
 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 
+($D \lesssim 100 \, \mathrm{m}$). 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 
+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.   
 
 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 
+typically few $\mathrm{Gpc^3}$. 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 ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$). 
+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 
@@ -230,12 +240,12 @@
 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} } 
+S_{lim} = \frac{ \sqrt{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 
+where $t_{int}$ is the total integration time and $\delta \nu$ is the detection frequency band. In table 
+\ref{slims21} (left)  we have computed the sensitivity for 6 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:
+$S_{21}$ 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 
@@ -244,9 +254,9 @@
  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 ? } 
+% {\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 
+$200 \, \mathrm{km/s}$. 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
@@ -254,6 +264,6 @@
 
 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 
+3D distribution of neutral hydrogen \citep{chang.08} \citep{ansari.08} \citep{seo.10}. 
+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 
@@ -272,5 +282,5 @@
 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).
+(radiation power/unit solid angle/unit surface/unit frequency) 
 can be converted to brightness temperature using the well known black body Rayleigh-Jeans approximation:
 $$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} $$ 
@@ -326,31 +336,34 @@
 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}  
+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}
 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}
+neutral hydrogen number density relative fluctuations can be written as, and the corresponding
+21 cm emission temperature can be written as:
+\begin{eqnarray}
+\frac{ \delta \etaHI}{\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) \\
+ \TTlamz  &  = & \bar{T}_{21}(z) \times \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z)  
+\end{eqnarray}
 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}:
+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. 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}. 
+of Lyman-$\alpha$ absorption indicate a factor 3 increase in the neutral hydrogen 
+fraction at $z=1.5$ in the intergalactic medium \citep{wolf.05}, 
+compared to the its present day value $\gHI(z=1.5) \sim 0.025$. 
 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 } } 
+ P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
+ \bar{T}_{21}(z)  & \simeq & 0.077  \, \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}
 
@@ -363,12 +376,12 @@
 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) }  
+\theta_k = \frac{2 \pi}{k^{comov} \, \dang(z) \, (1+z) }  
 \hspace{3mm} 
-k^{comov} = \frac{2 \pi}{ \mathrm{ang.sc}  \, \dang(z) \, (1+z) }  
+k^{comov} = \frac{2 \pi}{ \theta_\mathrm{scale}  \, \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 ? } 
+% {\color{red} Faut-il developper completement le calcul en annexe ? } 
 
 \begin{table}
@@ -393,6 +406,8 @@
 
 \begin{figure}
-\centering
-\includegraphics[width=0.5\textwidth]{Figs/pk21cmz12.pdf}
+\vspace*{-15mm}
+\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 $\gHI=2\%$}
@@ -402,8 +417,12 @@
 
 \section{interferometric observations and P(k) measurement sensitivity }
-
+\label{pkmessens}
 \subsection{Instrument response}
+\label{instrumresp}
+We introduce briefly here the principles of interferometric observations and the definition of 
+quantities useful for our calculations. Interested reader may refer to \citep{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  the direction. In radio astronomy
+$I(\vec{\Theta},\lambda)$ 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 and bear no information:
@@ -425,7 +444,7 @@
 corresponds to the receiver intensity response:
 \begin{equation}
-L(\vec{\Theta}), \lambda = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) 
+L(\vec{\Theta}), \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) 
 \end{equation} 
-The visibility signal between two receivers corresponds to the time averaged correlation between 
+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 position $\vec{r_1}$ and 
@@ -471,10 +490,12 @@
 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 
+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 
+Several beams can be formed using different combination of the correlations from a set of  
 antenna pairs.  
 
@@ -490,5 +511,14 @@
 Obviously, different weighting schemes can be used, changing 
 the effective beam shape and thus the response ${\cal R}_{w}(u,v,\lambda)$
-and the noise behaviour.
+and the noise behaviour. 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, ${\cal R}_{norm}(u,v,\lambda) \lesssim 1$ as:
+\begin{equation}
+{\cal R}_{norm}(u,v,\lambda) = {\cal R}(u,v,\lambda) / \mathrm{Max_{(u,v)}} \left[ {\cal R}(u,v,\lambda) \right] 
+\end{equation} 
+This normalized  instrument response can be used to compute the effective instrument beam, 
+in particular in section \ref{recsec}.  
 
 \begin{figure}
@@ -504,4 +534,5 @@
 
 \subsection{Noise power spectrum}
+\label{instrumnoise}
 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 
@@ -510,18 +541,29 @@
 corresponds also to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated 
 noise. If the receiver has an effective area $A \simeq \pi D^2/4$ 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 $\sim A/\lambda^2$. 
+corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. The noise spectral density, in the angular frequencies plane (per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$), 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}
+
 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{2 \, \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, 
+We can characterize the sky temperature measurement by a radio instrument by the noise 
+spectral power density in the angular frequencies plane $P_{noise}(u,v)$ in units of $\mathrm{Kelvin^2}$  
+per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$. 
+For an interferometer made of identical receiver elements, several ($n$) receiver pairs 
+might have the same baseline. The noise power density in the corresponding $(u,v)$ plane area 
+is then reduced by a factor $1/n$. More generally, we cam write the instrument  noise 
+spectral power density using the instrument response defined in section \ref{instrumresp} :
+\begin{equation}
+P_{noise}(u,v) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(u,v,\lambda) } 
+\end{equation} 
+
+When the intensity maps are projected in a three dimensional 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:
 \begin{eqnarray}
@@ -529,12 +571,15 @@
 \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)  } \\
+\delta u , \delta v & \rightarrow & \delta k_\perp = \frac{ \delta u \, , \, \delta 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: 
+If we consider a uniform noise spectral density in the $(u,v)$ plane corresponding to the 
+equation \ref{eq:pnoisepairD} above,  the three dimensional projected noise spectral density 
+can then be written as: 
 \begin{equation}
 P_{noise}(k) = 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} 
 
@@ -542,57 +587,64 @@
 $t_{int}$ 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 
+which is 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 determined 
+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 pointing $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area. 
+Each sky direction or pixel 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  
+\end{equation}
+
+It is important to note that any real instrument 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 
+a maximum angular frequency $u_{max},v_{max}$. 
+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) = 2 \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  
+ 
+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 of 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  
+\label{eq:pnoiseNbeam}
 \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. 
+
+The expression above (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 $q \times q$ square.   
 
 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
+observations provide information up to $u_{max},v_{max} \lesssim 2 \pi D / \lambda $. This value of 
+$u_{max},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 
+\begin{equation}
+k^{comov}_{\perp}  =  \frac{(u,v)}{(1+z) \dang}  \hspace{8mm} 
+k^{comov}_{\perp \, max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} 
+\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 $k^{comov}$  is also shown as a function 
-of redshift, for an instrument with $D=100 \mathrm{m}$ maximum extent. In order 
+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:
+the instrument noise level with $k^{comov}_{\perp}$, we have taken the effective $k^{comov}_{ max}  $ 
+as half of the maximum transverse $k^{comov}_{\perp \, max} $ of \mbox{eq. \ref{kperpmax}}: 
 \begin{equation}
 k^{comov}_{ max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}} 
@@ -607,6 +659,6 @@
 }
 \vspace*{-40mm}
-\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) }
+\caption{Minimal noise level for a 100 beams instrument with \mbox{$\Tsys=50 \mathrm{K}$} 
+as a function of redshift (top).  Maximum $k$ value for  a 100 meter diameter primary antenna (bottom) }
 \label{pnkmaxfz}
 \end{figure}
@@ -614,26 +666,27 @@
 
 \subsection{Instrument configurations and noise power spectrum}
-
+\label{instrumnoise}
 We have numerically computed the instrument response ${\cal R}(u,v,\lambda)$ 
 with uniform weights in the $(u,v)$ plane for several instrument configurations:
 \begin{itemize}
-\item[{\bf a} :] A packed array of $n=121 \, D_{dish}=5 \mathrm{m}$ dishes, arranged in 
+\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 
+\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 
+$80 \times 80  \, \mathrm{m^2}$. The array layout for this configuration is
+shown in figure \ref{figconfab}. 
+\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 
+four array corners. This array layout is also shown figure \ref{figconfab}. 
+\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 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 $z=1$. 
+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 polarisation, as in the (e) configuration.
@@ -643,6 +696,6 @@
 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 $z=1$. 
+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 polarisation. As for the (f) configuration,  
@@ -652,5 +705,5 @@
 from different cylinders are used.
 \end{itemize}
-The array layout for configurations (b) and (c) are shown in figure \ref{figconfab}. 
+
 \begin{figure}
 \centering
@@ -667,7 +720,8 @@
 
 We have used simple triangular shaped dish response in the $(u,v)$ plane. 
-However, we have introduced a fill factor or illumination efficiency 
+However, we have introduced a filling factor or illumination efficiency 
 $\eta$, relating the effective dish diameter $D_{ill}$ to the 
-mechanical dish size $D^{ill} = \eta \, D_{dish}$. 
+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 (u,v,\lambda) & = & \bigwedge_{[\pm 2 \pi D^{ill}/ \lambda]}(\sqrt{u^2+v^2})  \\
@@ -685,5 +739,8 @@
 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.
+the results obtained here for the noise spectral power density  would not change significantly.
+The instrument responses shown here correspond to fixed pointing toward the zenith, which 
+is the case for a transit type telescope.
+
 \begin{equation}
  {\cal L}_\Box(u,v,\lambda)  = 
@@ -705,6 +762,7 @@
 \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 
+\caption{(u,v) plane coverage (non normalized instrument response ${\cal R}(u,v,\lambda)$ 
+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 row of 16 dishes each, 
 (c) 129 dishes arranged as above, single D=65 meter diameter, with 100 beams. 
@@ -714,11 +772,11 @@
  
 \begin{figure*}
-\vspace*{-10mm}
+\vspace*{-25mm}
 \centering
 \mbox{
-\hspace*{-10mm}
-\includegraphics[width=\textwidth]{Figs/pkna2h.pdf}
+\hspace*{-20mm}
+\includegraphics[width=1.15\textwidth]{Figs/pkna2h.pdf}
 }
-\vspace*{-10mm}
+\vspace*{-40mm}
 \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.}
@@ -728,4 +786,5 @@
 
 \section{ Foregrounds and Component separation }
+\label{foregroundcompsep}
 Reaching the required sensitivities is not the only difficulty of observing the large 
 scale structures in 21 cm. Indeed, the synchrotron emission of the 
@@ -736,8 +795,9 @@
 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. However, any real radio instrument has a beam shape which changes with 
+emissions.
+However, any real radio instrument has a beam shape which changes with 
 frequency: this instrumental effect significantly increases the difficulty and complexity of this component separation 
-technique. The effect of frequency dependent beam shape is often referred to as {\em 
-mode mixing} \citep{morales.09}.
+technique. The effect of frequency dependent beam shape is some time referred to as {\em 
+mode mixing}. See for example \citep{morales.06}, \citep{bowman.07}. 
 
 In this section, we present a short description of the foreground emissions and 
@@ -745,5 +805,5 @@
 range. We present also a simple component separation method to extract the LSS signal and 
 its performance. We show in particular the effect of the instrument response on the recovered 
-power spectrum, and possible way of getting around this difficulty. The results presented in this section concern the 
+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.  
@@ -753,7 +813,8 @@
 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 $\alpha= 10:00 \mathrm{h} , \delta=+10 \mathrm{deg.}$,
+$90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10:00 \, \mathrm{h} , \delta=+10 \, \mathrm{deg.}$,
 and  covering 128 MHz in frequency. The sky cube characteristics (coordinate range, size, resolution) 
-used in the simulations is given in the table below:
+used in the simulations is given in the table \ref{skycubechars}.
+\begin{table}
 \begin{center}
 \begin{tabular}{|c|c|c|}
@@ -775,8 +836,14 @@
 \hline
 \end{tabular} \\[1mm]
-Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$   \\
+\end{center}
+\caption{
+Sky cube characteristics for the simulation performed in this paper. 
+Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$   
 $ 1800 \times 600 \times 256 \simeq 123 \, 10^6$ cells 
-\end{center}
-
+}
+\label{skycubechars}
+\end{table}
+%%%%
+\par 
 Two different methods have been used to compute the sky temperature data cubes.
 We have used the Global Sky Model (GSM) \citep{gsm.08} tools to generate full sky 
@@ -788,11 +855,11 @@
 LSS power spectrum.
 
-We have thus also created 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 \cite{nvss.98}. The sky temperature cube in this model (Model-II/Haslam+NVSS) 
+We have thus made also 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) 
 has been computed through the following steps:
 
 \begin{enumerate}
-\item The Galactic synchrotron emission is modeled power law with spatially 
+\item The Galactic synchrotron emission is modeled as a power law with spatially 
 varying spectral index. We assign a power law index $\beta = -2.8  \pm 0.15$ to each sky direction.
 $\beta$ has a gaussian distribution centered at -2.8 and with standard 
@@ -802,5 +869,5 @@
 $$ T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 MHz}\right)^\beta $$
 %%
-\item A two dimensional $T_{nvss}(\alpha,\delta)$sky brightness temperature at 1.4 GHz is computed 
+\item A two dimensional $T_{nvss}(\alpha,\delta)$ 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 
@@ -813,5 +880,5 @@
 \item The sky brightness temperature data cube is obtained through the sum of
 the two contributions, Galactic synchrotron and resolved radio sources: 
-$$ T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{sync}(\alpha, \delta, \nu) $$ 
+$$ T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu) $$ 
 \end{enumerate}
 
@@ -823,6 +890,7 @@
 $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the 
 sky cube angular and frequency resolution defined above.  The mass fluctuations has been 
-converted into temperature through a factor $0.13 \mathrm{mK}$, corresponding to a hydrogen 
-fraction $0.008 \times (1+0.6)$.  The total sky brightness temperature is then computed as the sum 
+converted into temperature through a factor $0.13 \, \mathrm{mK}$, corresponding to a hydrogen 
+fraction $0.008 \times (1+0.6)$, using equation \ref{eq:tbar21z}.  
+The total sky brightness temperature is then computed as the sum 
 of foregrounds and the LSS 21 cm emission:
 $$  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm} 
@@ -831,4 +899,7 @@
 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. 
@@ -838,4 +909,5 @@
 
 \begin{table}
+\centering
 \begin{tabular}{|c|c|c|}
 \hline
@@ -850,15 +922,15 @@
 \end{tabular}
 \caption{ Mean temperature and standard deviation for the different sky brightness 
-data cubes computed for this study}
+data cubes computed for this study (see table \ref{skycubechars} for sky cube resolution and size).}
 \label{sigtsky}
 \end{table}
 
 we have computed the power spectrum for the 21cm-LSS sky temperature cube, as well 
-as for the radio foreground temperature cubes computed using our two foreground 
+as for the radio foreground temperature cubes obtained from the two 
 models. We have also computed the power spectrum on sky brightness temperature
-cubes, as measured by a perfect instrument having a 25 arcmin gaussian beam.
+cubes, as measured by a perfect instrument having a 25 arcmin (FWHM) gaussian beam.
 The resulting computed power spectra are shown on figure \ref{pkgsmlss}. 
-The GSM model has more large scale power compared to our simple model, while
-it lacks power at higher spatial frequencies. The mode mixing due to 
+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 
 frequency dependent response will thus be stronger in Model-II (Haslam+NVSS) 
 case. It can also be seen that the radio foreground power spectrum is more than
@@ -872,5 +944,6 @@
 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. 
+for the instrumental response.  The observed structures have thus a scale dependent damping
+according to the instrument response, while the instrument noise is flat (white noise or scale independent). 
 
 \begin{figure}
@@ -891,9 +964,9 @@
 \centering
 \mbox{
-\hspace*{-10mm}
+% \hspace*{-10mm}
 \includegraphics[width=0.9\textwidth]{Figs/compmapgsm.pdf}
 }
 \caption{Comparison of GSM map (top) and Model-II sky map at 884 MHz (bottom). 
-The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin gaussian beam.}
+The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin (FWHM) gaussian beam.}
 \label{compgsmmap}
 \end{figure*}
@@ -901,8 +974,8 @@
 \begin{figure}
 \centering
-\vspace*{-20mm}
+\vspace*{-25mm}
 \mbox{
-\hspace*{-20mm}
-\includegraphics[width=0.7\textwidth]{Figs/pk_gsm_lss.pdf}
+\hspace*{-15mm}
+\includegraphics[width=0.65\textwidth]{Figs/pk_gsm_lss.pdf}
 }
 \vspace*{-40mm}
@@ -910,5 +983,5 @@
 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 beam.}
+as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam.}
 \label{pkgsmlss}
 \end{figure}
@@ -917,9 +990,9 @@
 
 \subsection{ Instrument response and LSS signal extraction }
-
-The observed data cube is obtained from the sky brightness temperature 3D map 
-$T_{sky}(\alpha, \delta, \nu)$ by applying the frequency dependent instrument response
+\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
 ${\cal R}(u,v,\lambda)$. 
-As a simplification, we have considered that the instrument response is independent 
+we have considered the simple case where  the instrument response constant throughout the survey area, or independent 
 of the sky direction.
 For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$ : 
@@ -927,12 +1000,14 @@
 \item Apply a 2D Fourier transform to compute sky angular Fourier amplitudes
 $$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(u, v, \lambda_k)$$ 
-\item Apply instrument response in the angular wave mode plane 
-$$  {\cal T}_{sky}(u, v, \lambda_k)  \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda) $$
+\item Apply instrument response in the angular wave mode plane. We use here the normalized instrument response
+$ {\cal R}(u,v,\lambda_k)  \lesssim 1$. 
+$$  {\cal T}_{sky}(u, v, \lambda_k)  \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda_k) $$
 \item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map, 
 without instrumental (electronic/$\Tsys$) white noise:
 $$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda)   
 \rightarrow \mathrm{Inv-2D-FFT} \rightarrow T_{mes1}(\alpha, \delta, \lambda_k) $$ 
-\item Add white noise (gaussian fluctuations) to obtain the measured sky brightness temperature 
-$T_{mes}(\alpha, \delta, \nu_k)$. We have also considered that the system temperature and thus the 
+\item Add white noise (gaussian fluctuations) to the pixel map temperatures to obtain 
+the measured sky brightness temperature $T_{mes}(\alpha, \delta, \nu_k)$. 
+We have also considered that the system temperature and thus the 
 additive white noise level was independent of the frequency or wavelength.   
 \end{enumerate} 
@@ -940,22 +1015,28 @@
 The results shown here correspond to the (a) instrument configuration, a packed array of 
 $11 \times 11 = 121$ 5 meter diameter dishes, with a white noise level corresponding 
-to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500 kHz$ 
+to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500$ kHz 
 cell.
 
-Our simple component separation procedure is described below:
+A brief description of the simple component separation procedure that we have applied is given here:
 \begin{enumerate}
-\item The measured sky brightness temperature is first corrected for the frequency dependent 
-beam effects through a convolution by a virtual, frequency independent beam. We assume 
-that we have a perfect knowledge of the intrinsic instrument response. 
+\item The measured sky brightness temperature is first {\em corrected} for the frequency dependent 
+beam effects through a convolution by a virtual, frequency independent beam. This {\em correction}
+corresponds to a smearing or degradation of the angular resolution. We assume 
+that we have a perfect knowledge of the intrinsic instrument response, up to a threshold numerical level 
+of about $ \gtrsim 1 \%$ for  ${\cal R}(u,v,\lambda)$. We recall that this is the normalized instrument response,
+${\cal R}(u,v,\lambda) \lesssim 1$. 
 $$  T_{mes}(\alpha, \delta, \nu) \longrightarrow T_{mes}^{bcor}(\alpha,\delta,\nu) $$ 
-The virtual target instrument has a beam width larger to the worst real instrument beam,
+The virtual target instrument has a beam width larger than the worst real instrument beam,
 i.e at the lowest observed frequency.  
  \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. $b$ is the power law index and  $10^a$ 
-is the brightness temperature at the reference frequency $\nu_0$:
+ is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in 
+the $\log10 ( T ) , \log10 (\nu)$ plane and we show here the results for a pure power law (P1) 
+or modified power law (P2):
 \begin{eqnarray*} 
 P1 & :  & \log10 ( T_{mes}^{bcor}(\nu) ) = a + b \log10 ( \nu / \nu_0 ) \\
 P2 & :  & \log10 ( T_{mes}^{bcor}(\nu) ) = a + b \log10 ( \nu / \nu_0 ) + c \log10 ( \nu/\nu_0 ) ^2 
 \end{eqnarray*}
+where $b$ is the power law index and  $T_0 = 10^a$ is the brightness temperature at the 
+reference frequency $\nu_0$:
 \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.
@@ -964,25 +1045,48 @@
 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 gaussian frequency independent 
-beam is shown in orange (solid line), and the extracted power spectrum, after beam correction 
+21 cm LSS power spectrum, as seen by a perfect instrument with a 25 arcmin (FWHM) 
+gaussian frequency independent beam is shown in orange (solid line), 
+and the extracted power spectrum, after beam {\em correction} 
 and foreground separation with second order polynomial fit (P2) is shown in red (circle markers).
 We have also represented the obtained power spectrum without applying the beam correction (step 1 above),
 or with the first order polynomial fit (P1). 
 
-It can be seen that a precise knowledge of the instrument beam and the beam correction 
-is a key ingredient for recovering the 21 cm LSS power spectrum. It is also worthwhile 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 model, a stronger beam correction 
+Figure \ref{extlssmap} shows a comparison of  the original 21 cm brightness temperature map at 884 MHz 
+with the recovered 21 cm map, after subtraction of 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. This is mostly 
+due to the damping of the large scale ($k \lesssim 0.04 h \mathrm{Mpc^{-1}} $) due the poor interferometer 
+response at large angle    ($\theta \gtrsim 4^\circ $). 
+
+We have shown that it should be possible to measure the red shifted 21 cm emission fluctuations in the 
+presence of the strong radio continuum signal, provided that this latter has a smooth frequency dependence.
+However, a rather  precise knowledge of the instrument beam and the beam {\em correction} 
+or smearing procedure described here  are key ingredient for recovering the 21 cm LSS power spectrum. 
+It is also important to note that while it is enough to correct the beam to the lowest resolution instrument beam 
+($\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 the Model-II to reduce 
-significantly the ripples from bright radio sources. The effect of mode mixing is reduced for 
-an instrument with smooth (gaussian) beam, compared to the instrument response 
-${\cal R}(u,v,\lambda)$ used here.
-
-Figure \ref{extlssratio} shows the overall {\em transfer function} for 21 cm LSS power 
-spectrum measurement. We have shown (solid line, orange) the ratio of measured LSS power spectrum
-by a perfect instrument $P_{perf-obs}(k)$, with a gaussian beam of $\sim$ 36 arcmin, respectively $\sim$ 30 arcmin,
-in the absence of any foregrounds or instrument noise, to the original 21 cm power spectrum $P_{21cm}(k)$. 
-The ratio of the recovered LSS power spectrum $P_{ext}(k)$ to $P_{perf-obs}(k)$ is shown in red, and the 
-ratio of the recovered spectrum to  $P_{21cm}(k)$  is shown in black (thin line). 
+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 
+${\cal R}(u,v,\lambda)$ used for the results shown in figure \ref{extlsspk}.
+
+\begin{figure*}
+\centering
+\vspace*{-25mm}
+\mbox{
+\hspace*{-20mm}
+\includegraphics[width=1.15\textwidth]{Figs/extlsspk.pdf}
+}
+\vspace*{-35mm}
+\caption{Recovered power spectrum of the 21cm LSS temperature fluctuations, separated from the 
+continuum radio emissions at $z \sim 0.6$, for the instrument configuration (a), $11\times11$ 
+packed array interferometer.
+Left: GSM/Model-I , right: Haslam+NVSS/Model-II. black curve shows the residual after foreground subtraction,
+corresponding to the 21 cm signal, WITHOUT applying the beam correction. 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/yellow curve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency independent gaussian beam. }
+\label{extlsspk}
+\end{figure*}
+
 
 \begin{figure*}
@@ -990,65 +1094,480 @@
 \vspace*{-20mm}
 \mbox{
-\hspace*{-20mm}
-\includegraphics[width=1.1\textwidth]{Figs/extlsspk.pdf}
+\hspace*{-25mm}
+\includegraphics[width=1.20\textwidth]{Figs/extlssmap.pdf}
 }
-\vspace*{-30mm}
-\caption{Power spectrum of the 21cm LSS temperature fluctuations, separated from the 
-continuum radio emissions at $z \sim 0.6$. 
-Left: GSM/Model-I , right: Haslam+NVSS/Model-II.  }
-\label{extlsspk}
+\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.
+Notice the difference between the temperature color scales (mK)  for the top and bottom maps. }
+\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.
+We expect damping at small scales, or larges $k$, due to the finite instrument size, but also at 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.
+Ideally, one has to define a power spectrum measurement response or {\it transfer function} in the 
+radial direction,  ($\lambda$ or redshift, $TF(k_\parallel)$) and in the transverse plane ( $TF(k_\perp)$ ). 
+The real transverse plane transfer function might even be anisotropic. 
+
+However, in the scope of the present study, we define an overall transfer function $TF(k)$ as the ratio of the 
+recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$:
+\begin{equation}
+TF(k) = P_{21}^{rec}(k) / P_{21}(k) 
+\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.
+The orange/yellow curve shows the ratio $P_{21}^{smoothed}(k)/P_{21}(k)$ of the computed to the original 
+power spectrum, if the original LSS temperature cube is smoothed by the frequency independent target beam 
+FWHM=30' for the GSM simulations (left), 36' for Model-II (right). This orange/yellow 
+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 recovered power spectrum suffers also significant damping at large scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}, $ due to poor interferometer 
+response at large angles ($ \theta \gtrsim 4^\circ-5^\circ$), as well as to the filtering of radial or longitudinal Fourier modes along 
+the frequency or redshift direction ($k_\parallel$) by the component separation algorithm. 
+The red curve shows the ratio of $P(k)$ computed on the recovered or extracted 21 cm LSS signal, to the original 
+LSS temperature cube $P_{21}^{rec}(k)/P_{21}(k)$ and corresponds to the transfer function $TF(k)$ defined above, 
+for $z=0.6$ and instrument setup (a). 
+The black (thin line) curve shows the ratio of recovered to the smoothed 
+power spectrum $P_{21}^{rec}(k)/P_{21}^{smoothed}(k)$. This latter ratio (black curve) exceeds one for $k \gtrsim 0.2$, which is 
+due to the noise or system temperature. It should stressed that the simulations presented in this section were 
+focused on the study of the radio foreground effects and have been carried intently 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 a the analytical form:
+\begin{equation}
+TF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right) 
+\label{eq:tfanalytique}
+\end{equation} 
+
+We have performed  simulation of 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 larger redshifts, in particular for $z \gtrsim 2$. 
+
+We have determined the transfer function parameters of eq. \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 smoothed transfer functions are shown on  figure \ref{tfpkz0525}. 
+
+\begin{table}[hbt]
+\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 
+$k_A$ & 0.006 & 0.005 & 0.004 & 0.0035 & 0.003 \\
+$k_B$ & 0.038 & 0.019 & 0.012 & 0.0093 & 0.008 \\
+$k_C$ & 0.16   & 0.08   & 0.05   & 0.038   & 0.032 \\
+\hline 
+\end{tabular}
+\caption{Value of the parameters for the transfer function (eq. \ref{eq:tfanalytique}) at different redshift 
+for instrumental setup (e), $20\times20$ packed array interferometer.  }
+\label{tab:paramtfk}
+\end{table}
 
 \begin{figure*}
 \centering
-\vspace*{-20mm}
+\vspace*{-30mm}
 \mbox{
 \hspace*{-20mm}
-\includegraphics[width=1.1\textwidth]{Figs/extlssratio.pdf}
+\includegraphics[width=1.15\textwidth]{Figs/extlssratio.pdf}
 }
-\vspace*{-30mm}
-\caption{Power spectrum of the 21cm LSS temperature fluctuations, separated from the 
-continuum radio emissions at $z \sim 0.6$. 
+\vspace*{-35mm}
+\caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $TF(k) = P_{21}^{rec}(k) / P_{21}(k) $, at $z \sim 0.6$,  for the instrument configuration (a), $11\times11$ packed array interferometer.
 Left: GSM/Model-I , right: Haslam+NVSS/Model-II.  }
 \label{extlssratio}
 \end{figure*}
 
-\section{ BAO scale determination and constrain on dark energy parameters}
+
+\begin{figure}
+\centering
+\vspace*{-25mm}
+\mbox{
+\hspace*{-10mm}
+\includegraphics[width=0.55\textwidth]{Figs/tfpkz0525.pdf}
+}
+\vspace*{-30mm}
+\caption{Fitted/smoothed  transfer function 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: 
+%% 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}
+
+In section \ref{pkmessens}, 
+The impact of the various telescope configurations on the sensitivity for 21 cm 
+power spectrum measurement has been discussed in section \ref{pkmessens}. 
+Fig.~\ref{powerfig} shows the noise power spectra, and allows us to rank visually 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 (sec. \ref{recsec})
+that subtraction of continuum radio emissions, Galactic synchrotron and radio sources, 
+has also 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}
+
+In order 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}.
+
+
+
+To this end, we  generated reconstructed power spectra $P^{rec}(k)$ for
+ slices of 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 equations \ref{eq:pk21z} and \ref{eq:tbar21z}, 
+damped by the transfer function $TF(k)$ (Eq. \ref{eq:tfanalytique} , table \ref{tab:paramtfk})
+and a white noise component $P_{noise}$ calculated according to the equation \ref{eq:pnoiseNbeam}, 
+established in section \ref{instrumnoise} with $N=400$:
+\begin{equation}
+ P^{rec}(k) = P_{21}(k) \times TF(k) + P_{noise} 
+\end{equation}
+where the different terms ($P_{21}(k) , TF(k), P_{noise}$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}. $P_{ref}(\kperp,\kpar)$ is the 
+envelop 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 have been 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 
+which are: 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 two dimensional 
+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/non-linear transition. 
+This limit is established on the basis of the criterion discussed in  
+\citep{blake.03}. 
+In practice, we used for the redshifts 
+$z=0.5,\,\, 1.0$ and  $1.5$ respectively $k_{max}= 0.145 \hMpcm,\,\, 0.18\hMpcm$ 
+and $0.23 \hMpcm$. 
+ 
+Figure \ref{fig:fitOscill} shows the result of the fit for 
+one of theses simulations. 
+Figure \ref{fig:McV2} histograms the recovered values of  $\koperp$ and $\kopar$
+for 100 simulations.
+The widths of the two distributions give an estimate 
+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 into account the possible 
+ignorance  of the signal shape and the uncertainties in the 
+computation of the noise power spectrum. 
+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, on 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 cylinders  
+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 $\kopar$ 
+respectively, perpendicular and parallel 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 cylinder array $b$. 
+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 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
+\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 
+ $\Delta z=0.5$ slice with the solid angle $\Omega_{tot}$ = 1 $\pi$ sr. 
+The number of mode $N_{\delta k}$ 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 
+\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 
+$\Tsys = 50 \mathrm{K}$ and one year total observation time to survey $\Omega_{tot}$ = 1 $\pi$ sr.
+\item {\it Noise with transfer function}: we take into account of the interferometer and radio foreground
+subtraction represented as the measured P(k) transfer function $T(k)$ (section \ref{tfpkdef}), as 
+well as instrument noise $P_{noise}$.
 \end{itemize}
 
+\begin{table}
+\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}
+\caption{Instrument or electronic noise spectral power $P_{noise}$ for a $N=400$ dish interferometer with $\Tsys=50$ K and $t_{obs} =$ 1 year to survey $\Omega_{tot} = \pi$ sr }
+\label{tab:pnoiselevel} 
+\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 give 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 which are 3 months, 3 months, 6 months, 1 year and 1 year respectively for redshift 0.5, 1.0, 1.5, 2.0 and 2.5.
+
+\begin{table*}[ht]
+\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 & $\sigma(\koperp)/\koperp$  (\%) & 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   & $\sigma(\koperp)/\koperp$  (\%) & 2.3 & 1.8 & 2.2 & 2.4 & 2.8\\
+ (3-months/redshift)& $\sigma(\kopar)/\kopar$  (\%) & 4.1 & 3.1  & 3.6 & 4.3 & 4.4\\
+ \hline
+ \bf   Noise with Transfer Function  & $\sigma(\koperp)/\koperp$  (\%) & 3.0 & 2.5 & 3.5 & 5.2 & 6.5 \\
+ (3-months/redshift)& $\sigma(\kopar)/\kopar$  (\%) & 4.8 & 4.0 & 6.2 & 9.3 & 10.3\\
+ \hline
+ \bf  Optimized survey & $\sigma(\koperp)/\koperp$  (\%)   & 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}
+\caption{Sensitivity on the measurement of $\koperp$ and $\kopar$ as a 
+function of the redshift $z$ for various simulation configuration. 
+$1^{\rm st}$ row: simulations without noise with pure cosmic variance; 
+$2^{\rm nd}$ 
+row: simulations with electronics noise for a telescope with dishes; 
+$3^{\rm th}$ row: simulations 
+with same electronics noise and with correction with the transfer function ; 
+$4^{\rm th}$ row: optimized survey with a total observation time of 3 years (3 months, 3 months, 6 months, 1 year and 1 year respectively for redshift 0.5, 1.0, 1.5, 2.0 and 2.5 ).}
+\label{tab:ErrorOnK}
+\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 $d_H(z)/a_s$.
+$a_s \sim 105  h^{-1} \mathrm{Mpc}$ is the acoustic horizon comoving size at recombination, 
+$d_T(z) = (1+z) \dang$ is the comoving angular distance and $d_H=c/H(z)$ is 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 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 = \Omega_{\Lambda}^0 \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 
+Tab.~\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.
+The Planck Fisher matrix is
+obtained for the 8 parameters (assuming a flat universe): 
+$\Omega_m$, $\Omega_b$, $h$, $w_0$, $w_a$,
+$\sigma_8$, $n_s$ (spectral index of the primordial power spectrum) and
+$\tau$  (optical depth to the last-scatter surface).
+
+
+For an optimized project over a redshift range, $0.25<z<2.75$, with a total 
+observation time of 3 years, the packed 400-dish interferometer array has a 
+precision of  12\% on $w_0$ and 48\% on $w_a$. 
+The  Figure of Merit, 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 2010's. This BAO project based on \HI intensity 
+mapping is clearly competitive with the current generation of optical 
+surveys such as SDSS-III \citep{sdss3}. 
+
+
+\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 $(w_0,w_a)$ 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 though {\it Intensity Mapping} is a novel and complementary 
+approach to optical surveys to study the statistical properties of the large scale structures in the universe
+up to redshifts $z \lesssim 3$. A radio instrument with 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 few hundred receiver elements spread over an hectare or a hundred beam 
+focal plane array with a $\sim 100$ meter primary reflector will have the required sensitivity to measure 
+the 21 cm power spectrum. A method to compute the instrument response for interferometers  
+has been developed and we have  computed the noise power spectrum for various telescope configurations.
+The Galactic synchrotron and radio sources are a thousand time brighter than the redshifted 21 cm signal, 
+making the measurement of this latter signal a major scientific and technical challenge. We have 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 have been 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 has allowed to define and 
+compute the overall  {\it transfer function} or {\it response function} for the measurement of the 21 cm 
+power spectrum. 
+Finally, we have 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. Such a radio survey 
+could be carried using the current technology and would be comptetitive with the ongoing or planned 
+optical surveys for dark energy,  with a fraction of their cost.
+ 
 % \begin{acknowledgements}
 % \end{acknowledgements}
 
-%%% Quelques figures pour illustrer les resultats attendus 
-
-
-
-% \caption{Comparison of the original simulated LSS (frequency plane) and the recovered LSS.
-% Color scale in mK }  \label{figcompexlss}
-
-% \caption{Comparison of the original simulated foreground  (frequency plane) and 
-% the recovered foreground map. Color scale in Kelvin }  \label{figcompexfg}
-
-% \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}
-
-
-% \caption{Recovered LSS power spectrum, after component separation - - GSM (Global Sky Model) foreground sky cube}
-% \label{figexlsspk}
-
 \bibliographystyle{aa}
 
@@ -1057,8 +1576,26 @@
 %%%
 \bibitem[Ansari et al. (2008)]{ansari.08} Ansari R., J.-M. Le Goff, C. Magneville, M. Moniez, N. Palanque-Delabrouille, J. Rich, 
-    V. Ruhlmann-Kleider, \& C. Y\`eche , 2008 , ArXiv:0807.3614
-
+    V. Ruhlmann-Kleider, \& C. Y\`eche , 2008 , arXiv:0807.3614
+
+%%%% References extraites de la section fournie par C. Yeche 
+\bibitem[Abdala \& Rawlings(2005)]{SKA} Abdalla, F.B. \& Rawlings, S.  2005,  \mnras, 360,  27     
+
+\bibitem[Albrecht et al.(2006)]{DETF}  Albrecht, A., Bernstein, G., Cahn, R. {\it et al.} (Dark Energy Task Force) 2006, arXiv:astro-ph/0609591 
+	
+\bibitem[Barkana \& Loeb(2007)]{h1temp} Barkana, R., and Loeb, A. 2007, Rep. Prog. Phys, 70 627 
+
+\bibitem[Binney \& Merrifield(1998)]{binneymerrifield} Binney, J. \&  Merrifield, M. 1998, Galactic Astronomy. 
+(Princeton Univ. Press, Princeton)
+
+\bibitem[Blake and Glazebrook(2003)]{blake.03} Blake, C. \& Glazebrook, K. 2003, \apj, 594, 665;
+Glazebrook, K. \&  Blake, C. 2005 \apj, 631, 1
+
+% Galactic astronomy, emission HI d'une galaxie
+\bibitem[Binney \& Merrifield (1998)]{binney.98} Binney J. \& Merrifield M. , 1998 {\it Galactic Astronomy} Princeton University Press 
 % MWA description 
 \bibitem[Bowman et al. (2007)]{bowman.07} Bowman, J. D., Barnes, D.G., Briggs, F.H. et al 2007, \aj, 133, 1505-1518  
+
+%% Soustraction avant plans ds MWA 
+\bibitem[Bowman et al. (2009)]{bowman.07} Bowman, J. D., Morales, M., Hewitt, J.N., 2009, \apj, 695, 183-199  
 
 %  Intensity mapping/HSHS 
@@ -1078,4 +1615,7 @@
 \bibitem[Eisentein et al. (2005)]{eisenstein.05}  Eisenstein D. J., Zehavi, I., Hogg, D.W. {\it et al.}, (the SDSS Collaboration) 2005,  \apj, 633, 560
 
+% SDSS-III description 
+\bibitem[Eisentein et al. (2011)]{eisenstein.11}  Eisenstein D. J., Weinberg, D.H., Agol, E. {\it et al.}, 2011, arXiv:1101.1529 
+
 %   21 cm emission for mapping matter distribution  
 \bibitem[Furlanetto et al. (2006)]{furlanetto.06} Furlanetto, S., Peng Oh, S. \&  Briggs, F. 2006, \physrep, 433, 181-301
@@ -1088,13 +1628,26 @@
 \bibitem[Hinshaw et al. (2008)]{hinshaw.08}  Hinshaw, G., Weiland, J.L., Hill, R.S.  {\it et al.} 2008, arXiv:0803.0732) 
 
+% Distribution des radio sources
+\bibitem[Jackson(2004)]{jackson.04} Jackson, C.A. 2004, \na, 48, 1187  
+
 % HI mass in galaxies 
 \bibitem[Lah et al. (2009)]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur et al.  2009,  \mnras 
 ( astro-ph/0907.1416) 
 
+% LSST Science book
+\bibitem[LSST.Science]{lsst.science} 
+{\it LSST Science book}, LSST Science Collaborations, 2009, arXiv:0912.0201  
+
+% Foret Ly alpha - 1 
+\bibitem[McDonald et al. (2006)]{baolya} McDonald P., Seljak, U. and Burles, S.  {\it 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
 
-%  Papier sur le traitement des obseravtions radio / mode mixing - REFERENCE A CHERCHER
-\bibitem[Morales et al. (2009)]{morales.09} Morales, M and other 2009, arXiv:0999.XXXX
+%  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, 767-773
 
 %  Global Sky Model Paper 
@@ -1108,7 +1661,13 @@
 \bibitem[Percival et al. (2007)]{percival.07}   Percival, W.J., Nichol, R.C., Eisenstein, D.J. {\it et al.}, (the SDSS Collaboration) 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. {\it et al.},  2010, \mnras 401, 2148-2168   
+
 %% LOFAR description 
 \bibitem[Rottering et a,. (2006)]{rottgering.06} Rottgering H.J.A., Braun, r., Barthel, P.D. {\it et al.}  2006, arXiv:astro-ph/0610596
 %%%%
+
+%% SDSS-3
+\bibitem[SDSS-III(2008)]{sdss3} SDSS-III 2008, http://www.sdss3.org/collaboration/description.pdf 
 
 %  Frank H. Briggs, Matthew Colless, Roberto De Propris, Shaun Ferris, Brian P. Schmidt, Bradley E. Tucker
@@ -1119,6 +1678,13 @@
 { \tt http://www.skatelescope.org/pages/page\_sciencegen.htm } 
 
+% 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, 164-173 
+
 % FFT telescope 
 \bibitem[Tegmark \& Zaldarriaga (2008)]{tegmark.08} Tegmark, M. \& Zaldarriaga, M. 2008, arXiv:0802.1710 
+
+%  Thomson-Morane livre interferometry 
+\bibitem[Radio Astronomy (1998)]{radastron} Thompson, A.R., Moran, J.M., Swenson, G.W, {\it Interferometry and 
+Synthesis in Radio Astronomy}, John Wiley \& sons, 2nd Edition 2001 
 
 % Lyman-alpha, HI fraction 
