Index: /trunk/Cosmo/RadioBeam/sensfgnd21cm.tex
===================================================================
--- /trunk/Cosmo/RadioBeam/sensfgnd21cm.tex	(revision 4042)
+++ /trunk/Cosmo/RadioBeam/sensfgnd21cm.tex	(revision 4043)
@@ -66,6 +66,8 @@
 
 % Commande pour marquer les changements du papiers pour le referee
-\def\changemark{\bf }
-% \def\changemark{ }
+% \def\changemark{\bf }
+\def\changemark{}
+\def\changemarkb{\bf }
+
 
 %%% Definition pour la section sur les param DE par C.Y
@@ -92,7 +94,7 @@
           \inst{1} \inst{2}
           \and
-          J.E. Campagne \inst{3}
+          J.E. Campagne \inst{2}  
          \and
-          P.Colom  \inst{5} 
+          P.Colom  \inst{3} 
           \and
           J.M. Le Goff \inst{4}
@@ -102,5 +104,5 @@
           J.M. Martin  \inst{5} 
           \and
-          M. Moniez \inst{3}
+          M. Moniez \inst{2}  
         \and 
           J.Rich \inst{4}
@@ -110,11 +112,11 @@
 
    \institute{
-     Universit\'e Paris-Sud, LAL, UMR 8607, F-91898 Orsay Cedex, France 
-   \and 
-    CNRS/IN2P3,  F-91405 Orsay, France \\
+     Universit\'e Paris-Sud, LAL, UMR 8607, 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,
+    CNRS/IN2P3, Laboratoire de l'Acc\'el\'erateur Lin\'eaire (LAL) 
     B.P. 34, 91898 Orsay Cedex, France
+    \and
+    LESIA, UMR 8109, Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon Cedex, France
     % \thanks{The university of heaven temporarily does not
     %                 accept e-mails}
@@ -123,5 +125,5 @@
     \and 
     GEPI, UMR 8111, Observatoire de Paris, 61 Ave de l'Observatoire, 75014 Paris, France
-   }
+  }
 
    \date{Received August 5, 2011; accepted xxxx, 2011}
@@ -142,14 +144,14 @@
   % methods heading (mandatory)
  { For each configuration, we determine instrument response by computing the $(\uv)$ or Fourier angular frequency 
-plane  coverage using visibilities. The $(\uv)$ plane response is the noise power spectrum, 
+plane  coverage using visibilities. The $(\uv)$ plane response determines the 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 
+   { We have computed the noise power spectrum for different instrument configurations as well as the extracted 
    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 beams and a collecting area of 
-  $\sim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ and will be 
+  \mbox{$\sim 10000 \, \mathrm{m^2}$} will be able to  detect BAO signal at redshift z $\sim 1$ and will be 
   competitive with optical surveys. }
 
@@ -171,6 +173,6 @@
 optical observation of galaxies which are used as a tracer of the underlying matter distribution. 
 An alternative and elegant approach for mapping the matter distribution, using neutral atomic hydrogen 
-(\HI) as a tracer with 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
+(\HI) as a tracer with intensity mapping has been proposed in recent years (\cite{peterson.06} , \cite{chang.08}). 
+Mapping the matter distribution using \HI 21 cm emission as a tracer has been extensively discussed in literature
 \citep{furlanetto.06} \citep{tegmark.09} 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. 
@@ -191,7 +193,7 @@
 BAO are features imprinted  in the distribution of galaxies, due to the frozen 
 sound waves which were present in the photon-baryon plasma prior to recombination 
-at z $\sim$ 1100. 
+at \mbox{$z \sim 1100$}. 
 This scale can be considered as a standard ruler with a comoving 
-length  of $\sim 150 \mathrm{Mpc}$.
+length  of \mbox{$\sim 150 \mathrm{Mpc}$}.
 These features have been first observed in the CMB anisotropies
 and are usually referred to as {\em acoustic peaks} (\cite{mauskopf.00}, \cite{larson.11}). 
@@ -201,8 +203,9 @@
 WiggleZ \citep{blake.11} optical galaxy surveys.
 
-Ongoing \citep{eisenstein.11}   or future surveys \citep{lsst.science}  
+Ongoing {\changemarkb surveys such as BOSS} \citep{eisenstein.11}  or future surveys 
+{\changemarkb such as LSST} \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  
-or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars 
+or through 3D mapping of Lyman $\alpha$ absorption lines toward distant quasars 
 \citep{baolya},\citep{baolya2}. 
 Radio observation of the 21 cm emission of neutral hydrogen appears as 
@@ -236,6 +239,6 @@
 and thus can be resolved easily with a rather modest size radio instrument 
 (diameter $D \lesssim 100 \, \mathrm{m}$). The specific BAO clustering scale ($k_{\mathrm{BAO}}$) 
-can be measured both in the transverse plane (angular correlation function, ($k_{\mathrm{BAO}}^\perp$) 
-or along the longitudinal (line of sight or redshift ($k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of 
+can be measured both in the transverse plane (angular correlation function, $k_{\mathrm{BAO}}^\perp$) 
+or along the longitudinal (line of sight or redshift $k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of 
 the Hubble parameter $H(z)$ can be obtained by comparing   the longitudinal and transverse 
 BAO scales. A reasonably good redshift resolution $\delta z \lesssim 0.01$ is needed to resolve 
@@ -284,11 +287,11 @@
 WMAP \LCDM universe \citep{komatsu.11}. $10^9 - 10^{10} M_\odot$ 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 $10^6 \mathrm{m^2}$. 
+would require collecting area in the range of \mbox{$10^6 \, \mathrm{m^2}$}. 
 
 Intensity mapping has been suggested as an alternative and economic method to map the 
-3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} \citep{seo.10}. 
-{\changemark There have been attempts to detect the 21 cm LSS signal at GBT 
+3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} and \citep{seo.10}. 
+{\changemark There have also been attempts to detect the 21 cm LSS signal at GBT 
 \citep{chang.10} and at GMRT \citep{ghosh.11}}.
-In this approach, sky brightness map with angular resolution $\sim 10-30 \, \mathrm{arc.min}$ is made for a 
+In this approach, sky brightness map with angular resolution \mbox{$\sim 10-30 \, \mathrm{arc.min}$} 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 $\sim 10^3 \mathrm{Mpc^3}$, containing ten to hundred galaxies 
@@ -304,13 +307,16 @@
 \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 $\sim 10 \mathrm{arc.min}$ 
+The large scale distribution of the neutral hydrogen, down to angular scale of \mbox{$\sim 10 \, \mathrm{arc.min}$} 
 can then be observed without the detection of individual compact \HI sources, using the set of sky brightness 
 map as a function of frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$ 
 (radiation power/unit solid angle/unit surface/unit frequency) 
-can be converted to brightness temperature using the well known black body Rayleigh-Jeans approximation:
+can be converted to brightness temperature using the Rayleigh-Jeans approximation of  black body radiation law:
 $$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} $$ 
  
 %%%%%%%%
 \begin{table}
+\caption{Sensitivity or source detection limit for 1 day integration time (86400 s) and 1 MHz 
+frequency band (left). 21 cm brightness for $10^{10} M_\odot$ \HI for different redshifts (right)  }
+\label{slims21} 
 \begin{center}
 \begin{tabular}{|c|c|c|}
@@ -341,7 +347,4 @@
 \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}
@@ -358,5 +361,6 @@
 where $A_{21}=2.85 \, 10^{-15} \mathrm{s^{-1}}$ \citep{astroformul} 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.
+constant and $H(z)$ is the Hubble parameter at the emission 
+redshift {\changemarkb (\cite{field.59} , \cite{zaldarriaga.04})}.
 For a \LCDM universe and neglecting radiation energy density, the Hubble parameter
 can be expressed as:
@@ -387,5 +391,5 @@
 compared to 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 
-(\cite{barkana.07} and \cite{madau.97}) :
+(\cite{madau.97}, \cite{zaldarriaga.04}), \cite{barkana.07}) :
 \begin{eqnarray}
  P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
@@ -420,5 +424,9 @@
 
 \begin{table}
-\begin{center}
+\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} 
+% \begin{center}
 \begin{tabular}{|l|c|c|c|c|c|c|c|}
 \hline 
@@ -432,13 +440,9 @@
 \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{center}
 \end{table}
 
 \begin{figure}
-\vspace*{-11mm}
+\vspace*{-4mm}
 \hspace{-5mm}
 \includegraphics[width=0.57\textwidth]{Figs/pk21cmz12.pdf}
@@ -516,5 +520,5 @@
 The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular 
 wave number domain located around 
-$(\uv)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is 
+$(\uv)_{12}=( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is 
 given by the receiver beam Fourier transform. 
 \begin{equation}
@@ -586,5 +590,6 @@
 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 $\sim A/\lambda^2$. The noise spectral density, in the angular frequencies plane (per unit area of angular frequencies  $\delta \uvu \times \uvv$), corresponding to a visibility 
+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  
+\mbox{$\delta \uvu \times \delta \uvv$}), corresponding to a visibility 
 measurement from a pair of receivers can be written as:
 \begin{eqnarray}
@@ -595,7 +600,4 @@
 \end{eqnarray}
 
-The sky temperature measurement can thus be characterized by the noise spectral power density in 
-the angular frequencies plane $P_{noise}^{(\uv)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$  
-per unit area of angular frequencies  $\delta \uvu \times \delta \uvv$:
 We can characterize the sky temperature measurement with a radio instrument by the noise 
 spectral power density in the angular frequencies plane $P_{noise}(\uv)$ in units of $\mathrm{Kelvin^2}$  
@@ -612,5 +614,6 @@
 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:
+and frequencies or wavelengths into comoving radial distance, using the following relations
+{\changemarkb (e.g. \cite{cosmo.peebles} chap. 13, \cite{cosmo.rich})} :
 { \changemark
 \begin{eqnarray}
@@ -620,5 +623,6 @@
   = (1+z) \frac{\lambda}{H(z)} \delta \nu \\
 % \delta \uvu , \delta \uvv & \rightarrow & \delta k_\perp = 2 \pi \frac{ \delta \uvu \, , \, \delta \uvv }{  (1+z) \, \dang(z)  } \\
-\frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = 2 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
+\frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = \delta k_z =
+2 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
  =  \frac{H(z)}{c} \frac{1}{(1+z)^2} \, \frac{\nu_{21}}{\delta \nu}
 \end{eqnarray}
@@ -631,5 +635,5 @@
 The measurement noise spectral density given by the equation \ref{eq:pnoisepairD} can then be 
 translated into a 3D noise power spectrum, per unit of spatial frequencies 
-$ \frac{\delta k_x \times \delta k_y \times \delta k_z}{8 \pi^3} $ (units: $ \mathrm{K^2 \times Mpc^3}$) :  
+$ \delta k_x \times \delta k_y \times \delta k_z / 8 \pi^3 $ (units: $ \mathrm{K^2 \times Mpc^3}$) :  
 
 \begin{eqnarray}
@@ -652,28 +656,29 @@
  
 \subsubsection{Uniform $(\uv)$ coverage}
-
-If we consider a uniform noise spectral density in the $(\uv)$ 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  
+{ \changemarkb We consider here an instrument with uniform $(\uv)$ plane coverage  (${\cal R}(\uv)=1$), 
+and measurements at regularly spaced frequencies centered on a central frequency $\nu_0$ or redshift $z(\nu_0)$.
+The noise spectral power  density from equation (\ref{eq:pnoisekxkz}) would then be  
+constant, independent of $(k_x, k_y, \ell_\parallel(\nu)$. Such a noise power spectrum corresponds thus 
+to a 3D white noise, with a uniform noise spectral density:}
+\begin{equation}
+P_{noise}(k_\perp, l_\parallel(\nu) ) = P_{noise} = 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  
 \label{ctepnoisek}
 \end{equation} 
 
-$P_{noise}(k)$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
+$P_{noise}$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
 $t_{int}$ is the integration time expressed in second, 
 $\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and 
  $H(z)$ in $\mathrm{km/s/Mpc}$. 
 
-The 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 
+The statistical uncertainties of matter or \HI distribution power spectrum estimate decreases
+with the number of observed Fourier modes; this number is proportional to the volume of the universe 
 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 
+fraction of the sky, defined by  total solid angle $\Omega_{tot}$, performed during a given
 total observation time $t_{obs}$. 
 A single dish instrument with diameter $D$ would have an instantaneous field of view 
 $\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and would require 
 a number of pointings  $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area. 
-Each sky direction or pixel of size $\Omega_{FOV}$ will be observed during an integration 
+Each sky direction or patch of size $\Omega_{FOV}$ will be observed during an integration 
 time $t_{int} = t_{obs}/N_{point} $. Using equation \ref{ctepnoisek} and the previous expression
 for the integration time, we can compute a simple expression
@@ -685,7 +690,7 @@
 It is important to note that any real instrument do not have a flat 
 response in the $(\uv)$ plane, and the observations provide no information above 
-a certain maximum angular frequency $u_{max},v_{max}$. 
+a certain maximum angular frequency $\uvu_{max},\uvv_{max}$. 
 One has to take into account either a damping of the observed sky power 
-spectrum or an increase of the noise spectral power if 
+spectrum or an increase of the noise spectral density if 
 the observed power spectrum is corrected for damping. The white noise 
 expressions given below should thus be considered as a lower limit or floor of the 
@@ -755,6 +760,6 @@
 we want to compute $P_{noise}(k)$. For the results at redshift \mbox{$z_0=1$} discussed in section  \ref{instrumnoise}, 
 the grid cell size and dimensions have been chosen to represent a box in the universe  
-\mbox{$\sim 1500 \times 1500 \times 750 \mathrm{Mpc^3}$}, 
-with $3\times3\times3 \mathrm{Mpc^3}$ cells.
+\mbox{$\sim 1500 \times 1500 \times 750 \, \mathrm{Mpc^3}$}, 
+with \mbox{$3\times3\times3 \, \mathrm{Mpc^3}$} cells.
 This correspond to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given 
 the small angular extent, we have neglected the curvature of redshift shells. 
@@ -779,5 +784,12 @@
 \item the above steps are repeated $\sim$ 50 times to decrease the statistical fluctuations 
 from random generations. The averaged computed noise power spectrum is normalized using 
-equation \ref{eq:pnoisekxkz}  and the instrument and survey parameters ($\Tsys \ldots$).
+equation \ref{eq:pnoisekxkz}  and the instrument and survey parameters: 
+{\changemarkb system temperature $\Tsys= 50 \mathrm{K}$, 
+individual receiver size $D^2$ or $D_x D_y$ and the integration time $t_{int}$. 
+This last parameter is obtained through the relation 
+$t_{int} = t_{obs} \Omega_{FOV}  / \Omega_{tot}$ using the total survey duration 
+$t_{obs}=1 \mathrm{year}$ and the instantaneous field of view 
+$\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, for a total survey sky coverage 
+of $\pi$ srad. }
 \end{itemize} 
 
@@ -786,5 +798,5 @@
 instrument response function and $\dang(z)$ for a small redshift interval around $z_0$,
 using a fixed  instrument response ${\cal R}(\uv,\lambda(z_0))$ and 
-a constant the radial distance $\dang(z_0)*(1+z_0)$. 
+a constant radial distance $\dang(z_0)*(1+z_0)$. 
 \begin{equation}
 \tilde{P}_{noise}(k) = < |  {\cal F}_n (k_x,k_y,k_z) |^2 > \simeq < P_{noise}(\uv, k_z) > 
@@ -863,5 +875,5 @@
 
 For the receivers along the focal line of cylinders, we have assumed that the 
-individual receiver response in the $(\uv)$ plane corresponds to one from a 
+individual receiver response in the $(\uv)$ plane corresponds to 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 $\eta_y = 0.8$ 
@@ -901,8 +913,8 @@
 \includegraphics[width=\textwidth]{Figs/uvcovabcd.pdf}
 }
-\caption{Raw instrument response ${\cal R}(\uv,\lambda)$ or the $(\uv)$ plane coverage
+\caption{Raw instrument response ${\cal R}_{raw}(\uv,\lambda)$ or the $(\uv)$ plane coverage
 at 710 MHz (redshift $z=1$) for four configurations.
 (a) 121 $D_{dish}=5$ meter diameter dishes arranged in a compact, square array 
-of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each (fig. \ref{figconfbc}), 
+of $11 \times 11$, (b) 128 dishes arranged in 8 rows of 16 dishes each (fig. \ref{figconfbc}), 
 (c) 129 dishes arranged as shown in figure \ref{figconfbc} , (d) single D=75 meter diameter, with 100 beams. 
 The common color scale (1 \ldots 80) is shown on the right. }
@@ -965,4 +977,10 @@
 used in the simulations are given in the table \ref{skycubechars}.
 \begin{table}
+\caption{
+Sky cube characteristics for the simulation performed in this paper. 
+Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$ ; 
+$1800 \times 600 \times 256 \simeq 123 \times 10^6$ cells 
+}
+\label{skycubechars}
 \begin{center}
 \begin{tabular}{|c|c|c|}
@@ -985,10 +1003,4 @@
 \end{tabular} \\[1mm]
 \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 
-}
-\label{skycubechars}
 \end{table}
 %%%%
@@ -1017,18 +1029,24 @@
 The synchrotron contribution to the sky temperature for each cell is then 
 obtained  through the formula:
-$$ T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 \, \mathrm{MHz}}\right)^\beta $$
+\begin{equation}
+ T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 \, \mathrm{MHz}}\right)^\beta 
+\end{equation}
 %%
 \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 
-pixel angular size into account ($ \sim 21 \mathrm{mK / mJansky}$ at 1.4 GHz and $3' \times 3'$ pixels).  
+pixel angular size into account ($ \sim 21 \mathrm{mK/mJy}$ at 1.4 GHz and $3' \times 3'$ pixels).  
 A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source 
 map; we have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the 
 contribution of the radiosources to the sky temperature is computed as follows:
-$$ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}} $$
+\begin{equation}
+ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}} 
+\end{equation}
 %%
 \item The sky brightness temperature data cube is obtained through the sum of
 the two contributions, Galactic synchrotron and resolved radio sources: 
-$$ T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu) $$ 
+\begin{equation}
+ T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu) 
+\end{equation}
 \end{enumerate}
 
@@ -1051,12 +1069,21 @@
 The size of the cells is  $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the 
 sky cube angular and frequency resolution defined above.  
-
-The mass fluctuations has been 
-converted into temperature through a factor $0.13 \, \mathrm{mK}$, corresponding to a hydrogen 
-fraction $0.008 \times (1+0.6)$, using equation \ref{eq:tbar21z}.  
-The total sky brightness temperature is then computed as the sum 
+{\changemarkb 
+We haven't taken into account the curvature of redshift shells when 
+converting SimLSS euclidean coordinates to angles and frequency coordinates 
+of the sky cubes analyzed here, which introduces distortions visible at large angles $\gtrsim 10^\circ$. 
+These angular scales, corresponding to small wave modes $k \lesssim 0.02 \mathrm{h \, Mpc^{-1}}$ 
+ are excluded for results presented in this paper.
+}  
+
+The mass fluctuations have been converted into temperature using equation \ref{eq:tbar21z}, 
+and a neutral  hydrogen fraction \mbox{$0.008 \times (1+0.6)$}, leading to a mean temperature of 
+$0.13 \, \mathrm{mK}$. 
+The total sky brightness temperature is computed as the sum 
 of foregrounds and the LSS 21 cm emission:
-$$  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm} 
-T_{sky} = T_{gsm}+T_{lss} $$ 
+\begin{equation}
+  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm} 
+T_{sky} = T_{gsm}+T_{lss} 
+\end{equation}
 
 Table \ref{sigtsky} summarizes the mean and standard deviation of the sky brightness 
@@ -1072,4 +1099,7 @@
 
 \begin{table}
+\caption{ Mean temperature and standard deviation for the different sky brightness 
+data cubes computed for this study (see table \ref{skycubechars} for sky cube resolution and size).}
+\label{sigtsky}
 \centering
 \begin{tabular}{|c|c|c|}
@@ -1084,7 +1114,4 @@
 \hline
 \end{tabular}
-\caption{ Mean temperature and standard deviation for the different sky brightness 
-data cubes computed for this study (see table \ref{skycubechars} for sky cube resolution and size).}
-\label{sigtsky}
 \end{table}
 
@@ -1161,5 +1188,5 @@
 }
 \vspace*{-5mm}
-\caption{Comparison of the 21cm LSS power spectrum at $z=0.6$ with $\gHI=1\%$ (red, orange) 
+\caption{Comparison of the 21cm LSS power spectrum at $z=0.6$ with \mbox{$\gHI\simeq1.3\%$} (red, orange) 
 with the radio foreground power spectrum.
 The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple
@@ -1198,11 +1225,12 @@
 additive white noise level was independent of the frequency or wavelength.   
 \end{enumerate} 
-The LSS signal extraction depends indeed on the white noise level. 
+The LSS signal extraction performance depends obviously on the white noise level. 
 The results shown here correspond to the (a) instrument configuration, a packed array of 
 $11 \times 11 = 121$ dishes  (5 meter diameter), with a white noise level corresponding 
 to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500$ kHz 
-cell.
-
-A brief description of the simple component separation procedure that we have applied is given here:
+cell. \\[1mm]
+
+The different steps of the simple component separation procedure that has been applied are 
+briefly described here.
 \begin{enumerate}
 \item The measured sky brightness temperature is first {\em corrected} for the frequency dependent 
@@ -1223,7 +1251,7 @@
 ${\cal R}(\uv,\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$  has also a numerical upper bound $\sim 100$. }
 \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
- is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in 
-the $\lgd ( T ) , \lgd (\nu)$ plane and we show here the results for a pure power law (P1) 
-or modified power law (P2):
+ is fitted to the beam-corrected brightness temperature. The parameters have been obtained 
+using a linear $\chi^2$ fit in the $\lgd ( T ) , \lgd (\nu)$ plane. 
+We show here the results for a pure power law (P1), as well as a modified power law (P2):
 \begin{eqnarray*} 
 P1 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) \\
@@ -1239,5 +1267,5 @@
 we have set the mean value of the temperature for 
 each frequency plane according to a power law with an index close to the synchrotron index 
-($\beta\sim-2.8$) and we have checked that results are not too sensitive to the 
+($\beta\sim-2.8$) and we have checked that the results are not too sensitive to the 
 arbitrarily fixed mean temperature power law parameters. } 
 
@@ -1284,5 +1312,5 @@
 % \vspace*{-10mm}
 \caption{Recovered power spectrum of the 21cm LSS temperature fluctuations, separated from the 
-continuum radio emissions at $z \sim 0.6, \gHI=1\%$, for the instrument configuration (a), $11\times11$ 
+continuum radio emissions at $z \sim 0.6$, \mbox{$\gHI\simeq1.3\%$}, for the instrument configuration (a), $11\times11$ 
 packed array interferometer.
 Left: GSM/Model-I , right: Haslam+NVSS/Model-II. The black curve shows the residual after foreground subtraction,
@@ -1311,13 +1339,18 @@
 The recovered red shifted 21 cm emission power spectrum $P_{21}^{rec}(k)$ suffers a number of distortions, mostly damping,
  compared to the original $P_{21}(k)$ due to  the instrument response and the component separation procedure.
+{\changemarkb 
+We remind that we have neglected the curvature of redshift or frequency shells
+in this numerical study, which affect our result at large angles $\gtrsim 10^\circ$. 
+The results presented here and our conclusions are thus restricted to wave mode range 
+$k \gtrsim 0.02 \mathrm{h \, Mpc^{-1}}$. 
+} 
 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, $\TrF(k_\parallel)$) and in the transverse plane ( $\TrF(k_\perp)$ ). 
 The real transverse plane transfer function might even be anisotropic. 
 
 However, in the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the 
-recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$:
+recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$
+{\changemarkb , similar to the one defined by \cite{bowman.09} , equation (23):}
 \begin{equation}
 \TrF(k) = P_{21}^{rec}(k) / P_{21}(k) 
@@ -1338,5 +1371,5 @@
 longitudinal Fourier modes along the frequency or redshift direction ($k_\parallel$) 
 by the component separation algorithm. We have been able to remove the ripples on the reconstructed 
-power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$37' 
+power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$36' 
 target beam resolution, compared to $\sim$30' for the GSM model. This explains the lower transfer function
 obtained for Model-II at small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}$). }
@@ -1361,7 +1394,10 @@
 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}. 
+and the corresponding transfer functions are shown on  figure \ref{tfpkz0525}. 
 
 \begin{table}[hbt]
+\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}
 \begin{center}
 \begin{tabular}{|c|ccccc|}
@@ -1376,7 +1412,4 @@
 \end{tabular}
 \end{center}
-\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}
 
@@ -1590,5 +1623,5 @@
 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:
+The number of modes $N_{\delta k}$ in the wave number interval $\delta k$ can be written as:
 \begin{equation}
 V  =  \frac{c}{H(z)} \Delta z  \times (1+z)^2 \dang^2  \Omega_{tot} \hspace{10mm}
@@ -1598,10 +1631,12 @@
 \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
+\item {\it Noise with transfer function}: we take into account the interferometer response and radio foreground
 subtraction represented as the measured P(k) transfer function $T(k)$ (section \ref{tfpkdef}), as 
-well as instrument noise $P_{noise}$.
+well as the instrument noise $P_{noise}$.
 \end{itemize}
 
 \begin{table}
+\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} 
 \begin{tabular}{|l|ccccc|}
 \hline
@@ -1612,6 +1647,4 @@
 \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}
 
@@ -1620,4 +1653,11 @@
 
 \begin{table*}[ht]
+\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; 
+$3^{\rm rd}$ row: simulations with the same electronics noise and 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}
 \begin{center}
 \begin{tabular}{lc|c c c c c }
@@ -1638,13 +1678,4 @@
 \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*}%
 
@@ -1790,4 +1821,8 @@
 
 %%%
+%%%% LSST Science book
+\bibitem[Abell et al. (2009)]{lsst.science} 
+{\it LSST Science book}, LSST Science Collaborations, Abell, P.A. {\it et al.} 2009, arXiv:0912.0201  
+
 %% reference SKA - BAO / DE en radio avec les sources 
 \bibitem[Abdalla \& Rawlings (2005)]{abdalla.05} Abdalla, F.B. \& Rawlings, S.  2005,  \mnras, 360,  27     
@@ -1820,4 +1855,10 @@
 \bibitem[Bowman et al. (2009)]{bowman.09} Bowman, J. D., Morales, M., Hewitt, J.N., 2009, \apj, 695, 183-199  
 
+%%%  SKA-Science
+\bibitem[Carilli et al. (2004)]{ska.science} 
+{\it Science with the Square Kilometre Array}, eds: C. Carilli, S. Rawlings, 
+New Astronomy Reviews, Vol.48, Elsevier, December 2004 \\
+{ \tt http://www.skatelescope.org/pages/page\_sciencegen.htm } 
+
 %  Intensity mapping/HSHS 
 \bibitem[Chang et al. (2008)]{chang.08}  Chang, T.,  Pen, U.-L., Peterson, J.B. \&  McDonald, P., 2008, \prl, 100, 091303
@@ -1845,4 +1886,6 @@
 \bibitem[Eisenstein et al. (2011)]{eisenstein.11}  Eisenstein D. J., Weinberg, D.H., Agol, E. {\it et al.}, 2011, arXiv:1101.1529 
 
+% Papier de Field sur la profondeur optique HI en 1959 
+\bibitem[Field (1959)]{field.59} Field G.B., 1959, \apj, 129, 155 
 %   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
@@ -1877,7 +1920,4 @@
 %% Description MWA 
 \bibitem[Lonsdale et al. (2009)]{lonsdale.09}  Lonsdale C.J., Cappallo R.J., Morales M.F.  {\it et al.}  2009, arXiv:0903.1828
-% LSST Science book
-\bibitem[LSST.Science]{lsst.science} 
-{\it LSST Science book}, LSST Science Collaborations, 2009, arXiv:0912.0201  
 
 % Planck Fischer matrix, computed for EUCLID 
@@ -1916,10 +1956,14 @@
 \bibitem[Parsons et al. (2009)]{parsons.09}  Parsons A.R.,Backer D.C.,Bradley  R.F. {\it et al.}  2009, arXiv:0904.2334
 
+% Livre Cosmo de Peebles 
+\bibitem[Peebles (1993)]{cosmo.peebles} Peebles, P.J.E., {\it Principles of Physical Cosmology}, 
+Princeton University Press (1993)
+
+% Original CRT HSHS paper (Moriond Cosmo 2006 Proceedings) 
+\bibitem[Peterson et al. (2006)]{peterson.06} Peterson, J.B.,  Bandura, K., \& Pen, U.-L. 2006, arXiv:0606104  
+
 % Synchrotron index =-2.8 in the freq range 1.4-7.5 GHz 
 \bibitem[Platania et al. (1998)]{platania.98}   Platania P., Bensadoun M., Bersanelli M. {\it al.} 1998,  \apj 505, 473-483
 
-% Original CRT HSHS paper (Moriond Cosmo 2006 Proceedings) 
-\bibitem[Peterson et al. (2006)]{peterson.06} Peterson, J.B.,  Bandura, K., \& Pen, U.-L. 2006, arXiv:0606104  
-
 % SDSS BAO 2007 
 \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
@@ -1927,4 +1971,7 @@
 % 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   
+
+% Livre Cosmo de Jim Rich 
+\bibitem[Rich (2001)]{cosmo.rich} James Rich,  {\it Fundamentals of Cosmology}, Springer (2001)
 
 % Radio spectral index between 100-200 MHz 
@@ -1943,9 +1990,4 @@
 %  Frank H. Briggs, Matthew Colless, Roberto De Propris, Shaun Ferris, Brian P. Schmidt, Bradley E. Tucker
 
-\bibitem[SKA.Science]{ska.science} 
-{\it Science with the Square Kilometre Array}, eds: C. Carilli, S. Rawlings, 
-New Astronomy Reviews, Vol.48, Elsevier, December 2004 \\
-{ \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 
@@ -1966,4 +2008,8 @@
 %  BAO à 21 cm et reionisation 
 \bibitem[Wyithe et al.(2008)]{wyithe.08} Wyithe, S., Loeb, A. \& Geil, P. 2008, \mnras, 383, 1195 %  http://fr.arxiv.org/abs/0709.2955, 
+
+%% Papier fluctuations 21 cm par Zaldarriaga et al 
+\bibitem[Zaldarriaga et al.(2004)]{zaldarriaga.04}  Zaldarriaga, M., Furlanetto, S.R., Hernquist, L., 2004, 
+\apj, 608, 622-635
 
 %% Today HI cosmological density 
