Index: trunk/Cosmo/RadioBeam/radutil.cc
===================================================================
--- trunk/Cosmo/RadioBeam/radutil.cc	(revision 4011)
+++ trunk/Cosmo/RadioBeam/radutil.cc	(revision 4013)
@@ -46,4 +46,4 @@
   double cc=zz*zz/sqrt(OmegaMatter_*zz*zz*zz+OmegaLambda_);
   cc *= ((h100_/0.7)*(OmegaBaryons_/0.044)*(fracHI_/0.01));
-  return (cc*0.054);
+  return (cc*0.059);
 }
Index: trunk/Cosmo/RadioBeam/sensfgnd21cm.tex
===================================================================
--- trunk/Cosmo/RadioBeam/sensfgnd21cm.tex	(revision 4011)
+++ trunk/Cosmo/RadioBeam/sensfgnd21cm.tex	(revision 4013)
@@ -28,4 +28,9 @@
 \usepackage{color}
 
+%% Commande pour les references 
+\newcommand{\citep}[1]{ (\cite{#1}) } 
+%% \newcommand{\citep}[1]{ { (\tt{#1}) } } 
+
+%% Definitions diverses
 \newcommand{\HI}{$\mathrm{H_I}$ } 
 \newcommand{\kb}{k_B}         % Constante de Boltzmann 
@@ -39,10 +44,10 @@
 \newcommand{\dang}{d_A}
 \newcommand{\hub}{ h_{70} }
-\newcommand{\hubb}{ h }    % h_100
+\newcommand{\hubb}{ h_{100} }    % h_100
 
 \newcommand{\etaHI}{ n_{\tiny HI} }
 \newcommand{\fHI}{ f_{H_I}(z)}
-\newcommand{\gHI}{ g_{H_I}}
-\newcommand{\gHIz}{ g_{H_I}(z)}
+\newcommand{\gHI}{ f_{H_I}}
+\newcommand{\gHIz}{ f_{H_I}(z)}
 
 \newcommand{\vis}{{\cal V}_{12} }
@@ -50,6 +55,9 @@
 \newcommand{\LCDM}{$\Lambda \mathrm{CDM}$ }
 
-\newcommand{\citep}[1]{ (\cite{#1}) } 
-%% \newcommand{\citep}[1]{ { (\tt{#1}) } } 
+\newcommand{\lgd}{\mathrm{log_{10}}}
+
+%% Definition fonction de transfer 
+\newcommand{\TrF}{\mathrm{Tr}}
+
 
 %%% Definition pour la section sur les param DE par C.Y
@@ -126,5 +134,5 @@
   % methods heading (mandatory)
  { 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, 
+plane  coverage using visibilities. The (u,v) plane response is 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. }
@@ -135,5 +143,5 @@
   % conclusions heading (optional), leave it empty if necessary 
    { 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 
+  $\sim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ and will be 
   competitive with optical surveys. }
 
@@ -152,18 +160,18 @@
 % {\color{red} \large \it Jim ( + M. Moniez ) }   \\[1mm] 
 The study of the statistical properties of Large Scale Structure (LSS) in the Universe and their evolution 
-with redshift is one the major tools in observational cosmology. Theses structures are usually mapped through 
-optical observation of galaxies which are used as tracers of the underlying matter distribution. 
+with redshift is one the major tools in observational cosmology. These structures are usually mapped through 
+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 tracer with Total Intensity Mapping, has been proposed in recent years \citep{peterson.06} \citep{chang.08}. 
+(\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
 \citep{furlanetto.06} \citep{tegmark.08} and is being used in projects such as LOFAR \citep{rottgering.06} or
 MWA \citep{bowman.07} to observe reionisation  at redshifts z $\sim$ 10. 
 
-Evidences in favor of the acceleration of the expansion of the universe have been 
-accumulated over the last twelve years, thank to the observation of distant supernovae, 
+Evidence in favor of the acceleration of the expansion of the universe have been 
+accumulated over the last twelve years, thanks to the observation of distant supernovae, 
 CMB anisotropies and detailed analysis of the LSS.  
 A cosmological Constant ($\Lambda$) or new cosmological 
 energy density called {\em Dark Energy} has been advocated as the origin of this acceleration. 
-Dark Energy is considered as one the most intriguing puzzles in Physics and Cosmology. 
+Dark Energy is considered as one of the most intriguing puzzles in Physics and Cosmology. 
 % Constraining the properties of this new cosmic fluid, more precisely 
 % its equation of state is central to current cosmological researches. 
@@ -174,19 +182,21 @@
 
 BAO are features imprinted  in the distribution of galaxies, due to the frozen 
-sound waves which were present in the photons baryons plasma prior to recombination 
+sound waves which were present in the photon-baryon plasma prior to recombination 
 at z $\sim$ 1100. 
-This scale, which can be considered as a standard ruler with a comoving 
-length  of $\sim 150 Mpc$.
-Theses features have been first observed in the CMB anisotropies
-and are usually referred to as {\em acoustic pics} \citep{mauskopf.00} \citep{hinshaw.08}. 
+This scale can be considered as a standard ruler with a comoving 
+length  of $\sim 150 \mathrm{Mpc}$.
+These features have been first observed in the CMB anisotropies
+and are usually referred to as {\em acoustic peaks} \citep{mauskopf.00} \citep{hinshaw.08}. 
 The BAO modulation has been subsequently observed in the distribution of galaxies 
 at low redshift ( $z < 1$) in the galaxy-galaxy correlation function by the SDSS 
-\citep{eisenstein.05}  \citep{percival.07}  \citep{percival.10}  and 2dGFRS  \citep{cole.05}  optical galaxy surveys.
+\citep{eisenstein.05}  \citep{percival.07}  \citep{percival.10}, 2dGFRS  \citep{cole.05}  as well as 
+WiggleZ \citep{blake.11} 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 
+$0 \lesssim z \lesssim 3$, using either optical observation of galaxies  
+or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars 
+\citep{baolya},\citep{baolya2}. 
+Radio observation of the 21 cm emission of neutral hydrogen appears as 
 a very promising technique to map matter distribution up to redshift $z \sim 3$, 
 complementary to optical surveys, especially in the optical redshift desert range 
@@ -217,5 +227,5 @@
 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 ($k_{\mathrm{BAO}}$) 
+(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 
@@ -226,10 +236,11 @@
 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 $\mathrm{Gpc^3}$. 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 $\Delta z \gtrsim 1$.
+typically few $\mathrm{Gpc^3}$. Moreover, stringent constraint on DE parameters can only be 
+obtained when comparing the distance or Hubble parameter measurements with 
+DE models as a function of redshift, which requires a significant survey depth $\Delta z \gtrsim 1$.
 
 Radio instruments intended for BAO surveys must thus have large instantaneous field
-of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$). 
+of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$)
+to explore large redshift domains.
 
 Although the application of 21 cm radio survey to cosmology, in particular LSS mapping has been 
@@ -237,5 +248,5 @@
 the method envisaged has been mostly through the detection of galaxies as \HI compact sources. 
 However, extremely large radio telescopes are required to detected \HI sources at cosmological distances. 
-The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the of two polarisations
+The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the two polarisations
 of a radio instrument characterized by an effective collecting area $A$, and system temperature $\Tsys$ can be written as 
 \begin{equation}
@@ -243,14 +254,14 @@
 \end{equation} 
 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 
+\ref{slims21} (left)  we have computed the sensitivity for 6 different sets of instrument effective area and system 
 temperature, with a total integration time of 86400 seconds (1 day) over a frequency band of 1 MHz. 
 The width of this frequency band is well adapted  to 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
+dispersion of few 100 km/s. These detection limits should be compared with the expected 21 cm brightness
 $S_{21}$ of compact sources which can be computed using the expression below (e.g.\cite{binney.98}) :
 \begin{equation}
  S_{21}  \simeq  0.021 \mathrm{\mu Jy} \, \frac{M_{H_I} }{M_\odot}   \times 
-\left( \frac{ 1\, \mathrm{Mpc}}{\dlum} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v} 
+\left( \frac{ 1\, \mathrm{Mpc}}{\dlum(z)} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v}  (1+z)
 \end{equation}
- where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum$ is the luminosity distance and $\sigma_v$ 
+ where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ is the luminosity distance and $\sigma_v$ 
 is the source velocity dispersion. 
 % {\color{red} Faut-il developper le calcul en annexe ? } 
@@ -264,9 +275,9 @@
 
 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} \citep{seo.10}. 
+3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \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 
-\HI mass $ \gtrsim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow, 
+would correspond to a cell with a volume of $\sim 10^3 \mathrm{Mpc^3}$, containing ten to hundred galaxies 
+and a total \HI mass $ \sim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow, 
 the observed frequency $\nu$ would be translated to the emission redshift $z$ through 
 the well known relation:
@@ -281,5 +292,5 @@
 The large scale distribution of the neutral hydrogen, down to angular scales of $\sim 10 \mathrm{arc.min}$ 
 can then be observed without the detection of individual compact \HI sources, using the set of sky brightness 
-map as a function frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$ 
+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:
@@ -307,11 +318,11 @@
 \hline
 $z$ &  $\dlum \mathrm{(Mpc)}$ & $S_{21}  \mathrm{( \mu Jy)} $ \\
-\hline
-0.25 &  1235   & 140 \\
-0.50 &  2800   & 27 \\
-1.0   &  6600   & 4.8 \\
-1.5   &  10980 & 1.74 \\
-2.0   &  15710  & 0.85 \\
-2.5  &  20690 & 0.49 \\
+\hline      % dernier chiffre : sans le facteur (1+z)
+0.25 &  1235   & 175 \\ % 140 
+0.50 &  2800   & 40   \\  % 27 
+1.0   &  6600   & 9.6  \\  % 4.8 
+1.5   &  10980 & 3.5  \\  % 1.74 
+2.0   &  15710  & 2.5  \\ % 0.85 
+2.5  &  20690 &  1.7  \\  % 0.49 
 \hline
 \end{tabular}
@@ -330,5 +341,5 @@
   \frac{c}{H(z)} \, (1+z)^2 \times  \etaHI (\vec{\Theta}, z)  
 \end{equation}
-where $A_{21}=1.87 \, 10^{-15} \mathrm{s^{-1}}$ is the spontaneous 21 cm emission 
+where $A_{21}=2.85 \, 10^{-15} \mathrm{s^{-1}}$ \citep{lang.99} 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.
@@ -344,7 +355,7 @@
 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)  
+\etaHI (\vec{\Theta}, z(\lambda) ) & = & \gHIz \times \Omega_B  \frac{\rho_{crit}}{m_{H}}  \times 
+\left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) + 1 \right) \\
+ \TTlamz  &  = & \bar{T}_{21}(z) \times \left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z)  + 1 \right) 
 \end{eqnarray}
 where $\Omega_B, \rho_{crit}$ are respectively the present day mean baryon cosmological
@@ -355,12 +366,13 @@
 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 
+The neutral hydrogen fraction is expected to increase with redshift, as gas is used 
+in star formation during galaxy formation and evolution. Study of Lyman-$\alpha$ absorption 
+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 $\gHI(z=1.5) \sim 0.025$. 
+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 \citep{wyithe.07} :
 \begin{eqnarray}
  P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
- \bar{T}_{21}(z)  & \simeq & 0.077  \, \mathrm{mK}  
+ \bar{T}_{21}(z)  & \simeq & 0.084  \, \mathrm{mK}  
 \frac{ (1+z)^2 \, \hubb }{\sqrt{ \Omega_m (1+z)^3 + \Omega_\Lambda } } 
  \dfrac{\Omega_B}{0.044}  \,  \frac{\gHIz}{0.01} 
@@ -368,5 +380,5 @@
 \end{eqnarray}
 
-The table \ref{tabcct21} below shows the mean 21 cm brightness temperature for the 
+The table \ref{tabcct21} shows the mean 21 cm brightness temperature for the 
 standard \LCDM cosmology and either a constant \HI mass fraction $\gHI = 0.01$, or 
 linearly increasing  $\gHI \simeq 0.008 \times (1+z) $. Figure \ref{figpk21} shows the 
@@ -376,11 +388,11 @@
 shown for the standard WMAP \LCDM cosmology, according to the relation:
 \begin{equation}
-\theta_k = \frac{2 \pi}{k^{comov} \, \dang(z) \, (1+z) }  
+\theta_k = \frac{2 \pi}{k \, \dang(z) \, (1+z) }  
 \hspace{3mm} 
-k^{comov} = \frac{2 \pi}{ \theta_\mathrm{scale}  \, \dang(z) \, (1+z) }  
+k = \frac{2 \pi}{ \theta_k  \, \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. 
+where $k$ 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 ? } 
 
@@ -390,9 +402,9 @@
 \hline 
 \hline
-   & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\
+ z  & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\
 \hline 
-(a) $\bar{T}_{21}$ (mK) & 0.08 & 0.1 & 0.13 & 0.16 & 0.18 & 0.2 & 0.21 \\
+(a) $\bar{T}_{21}$ & 0.085 & 0.107 & 0.145 & 0.174 & 0.195 & 0.216 & 0.234 \\
 \hline
-(b) $\bar{T}_{21}$ (mK)  & 0.08 & 0.12 & 0.21 & 0.32 & 0.43 & 0.56 & 0.68 \\
+(b) $\bar{T}_{21}$  & 0.085 & 0.128 & 0.232 & 0.348 & 0.468 & 0.605 & 0.749 \\
 \hline
 \hline  
@@ -406,5 +418,5 @@
 
 \begin{figure}
-\vspace*{-15mm}
+\vspace*{-11mm}
 \hspace{-5mm}
 \includegraphics[width=0.57\textwidth]{Figs/pk21cmz12.pdf}
@@ -426,9 +438,10 @@
 $I(\vec{\Theta},\lambda)$ 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:
+amplitudes. However, for most sources, the phases vary randomly with a spatial correlation 
+length significantly smaller than the instrument resolution.
 \begin{eqnarray}
 & & 
 I(\vec{\Theta},\lambda)  =  | A(\vec{\Theta},\lambda) |^2  \hspace{2mm} , \hspace{1mm} I \in \mathbb{R}, A \in \mathbb{C} \\
-& & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  =  \delta(   \vec{\Theta} - \vec{\Theta '} )  I(\vec{\Theta},\lambda)
+& & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  = 0 \hspace{2mm} \mathrm{for}   \hspace{1mm} \vec{\Theta} \ne \vec{\Theta '}   I(\vec{\Theta},\lambda)
 \end{eqnarray} 
 A single receiver can be  characterized by its angular complex amplitude response $B(\vec{\Theta},\nu)$ and 
@@ -440,14 +453,14 @@
 \end{equation} 
 We have set the electromagnetic (EM) phase origin at the center of the coordinate frame and 
-the EM wave vector is related to the wavelength $\lambda$ through the usual 
+the EM wave vector is related to the wavelength $\lambda$ through the usual equation
 $ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta},\lambda)$ 
 corresponds to the receiver intensity response:
 \begin{equation}
-L(\vec{\Theta}), \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) 
+L(\vec{\Theta}, \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) 
 \end{equation} 
 The visibility signal of two receivers corresponds to the time averaged correlation between 
 signals from two receivers. If we assume a sky signal with random uncorrelated phase, the 
 visibility $\vis$ signal from two identical receivers, located at the position $\vec{r_1}$ and 
-$\vec{r_2}$ can simply be written as a function their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
+$\vec{r_2}$ can simply be written as a function of their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
 \begin{equation}
 \vis(\lambda) = < s_1(\lambda) s_2(\lambda)^* > = \iint d \vec{\Theta} \, \, I(\vec{\Theta},\lambda) L(\vec{\Theta},\lambda) 
@@ -455,10 +468,10 @@
 \end{equation} 
 This expression can be simplified if we consider receivers with narrow field of view 
-($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \mathrm{deg.} $ ), 
+($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \, \mathrm{deg.} $ ), 
 and coplanar in respect to their common axis.
 If we introduce two {\em Cartesian} like angular coordinates $(\alpha,\beta)$ centered at 
 the common receivers axis, the visibilty would be written as the 2D Fourier transform
 of the product of the sky intensity and the receiver beam, for the angular frequency 
-\mbox{$(u,v)_{12} = 2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$}:
+\mbox{$(u,v)_{12} = 2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$}:
 \begin{equation}
 \vis(\lambda) \simeq  \iint d\alpha d\beta \, \, I(\alpha, \beta) \,  L(\alpha, \beta) 
@@ -481,5 +494,5 @@
 The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular 
 wave number domain located around 
-$(u, v)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$. The weight function is 
+$(u, v)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is 
 given by the receiver beam Fourier transform. 
 \begin{equation}
@@ -504,5 +517,5 @@
 the instrument response is simply the Fourier transform of the beam. 
 For a single dish with multiple receivers, either as a Focal Plane Array (FPA) or 
-a multi horn system, each beam (b) will have its own response 
+a multi-horn system, each beam (b) will have its own response 
 ${\cal R}_b(u,v,\lambda)$. 
 For an interferometer, we can compute a raw instrument response 
@@ -529,5 +542,5 @@
 }
 \vspace*{-15mm}
-\caption{Schematic view of the $(u,v)$ plane coverage by interferometric measurement}
+\caption{Schematic view of the $(u,v)$ plane coverage by interferometric measurement.}
 \label{figuvplane}
 \end{figure}
@@ -536,5 +549,5 @@
 \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 
+bandwidth $\delta \nu$ centered on $\nu_0$, with an integration time $t_{int}$, characterized by a system temperature 
 $\Tsys$. The uncertainty or fluctuations of this measurement due to the receiver noise can be written as 
 $\sigma_{noise}^2 = \frac{2 \Tsys^2}{t_{int} \, \delta \nu}$. This term 
@@ -553,10 +566,10 @@
 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  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$:
-We can characterize the sky temperature measurement by a radio instrument by the noise 
+We can characterize the sky temperature measurement with 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 
+is then reduced by a factor $1/n$. More generally, we can write the instrument  noise 
 spectral power density using the instrument response defined in section \ref{instrumresp} :
 \begin{equation}
@@ -585,5 +598,6 @@
 
 $P_{noise}(k)$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
-$t_{int}$ in second, $\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and 
+$t_{int}$ 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 $\mathrm{km/s/Mpc}$. 
 
@@ -596,5 +610,5 @@
 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. 
+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 
 time $t_{int} = t_{obs}/N_{point} $. Using equation \ref{ctepnoisek} and the previous expression
@@ -607,5 +621,5 @@
 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 
-a maximum angular frequency $u_{max},v_{max}$. 
+a certain 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 
@@ -617,5 +631,5 @@
 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. 
+thanks to the  increase of per pointing integration time:
 
 \begin{equation}
@@ -624,5 +638,5 @@
 \end{equation}
 
-The expression above (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$ 
+This expression (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$ 
 identical receivers with a  total collection area $\sim D^2$. Such an array could be made for example 
 of $N=q \times q$ {\it small dishes}, each with diameter $D/q$, arranged as $q \times q$ square.   
@@ -631,8 +645,8 @@
 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{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}} 
+$k^{\perp}_{max}$:
+\begin{equation}
+k^{\perp}  =  \frac{(u,v)}{(1+z) \dang}  \hspace{8mm} 
+k^{\perp}_{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} 
 \label{kperpmax}
 \end{equation}   
@@ -642,11 +656,11 @@
 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 
+year. The maximum comoving wave number $k_{max}$  is also shown as a function 
 of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. 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 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}} 
+to take into account the radial component of $\vec{k}$ and the increase of 
+the instrument noise level with $k^{\perp}$, we have taken the effective $k_{ max}  $ 
+as half of the maximum transverse $k^{\perp} _{max}$ of \mbox{eq. \ref{kperpmax}}: 
+\begin{equation}
+k_{max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}} 
 \end{equation}
 
@@ -676,9 +690,9 @@
 in 8 rows, each with 16 dishes. These 128 dishes are spread over an area 
 $80 \times 80  \, \mathrm{m^2}$. The array layout for this configuration is
-shown in figure \ref{figconfab}. 
+shown in figure \ref{figconfbc}. 
 \item [{\bf c} :] An array of $n=129  \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged 
  over an area $80 \times 80  \, \mathrm{m^2}$. This configuration has in 
 particular 4 sub-arrays of packed 16 dishes ($4\times4$), located in the 
-four array corners. This array layout is also shown figure \ref{figconfab}. 
+four array corners. This array layout is also shown figure \ref{figconfbc}. 
 \item [{\bf d} :] A single dish instrument, with diameter $D=75 \, \mathrm{m}$, 
 equipped with a 100 beam focal plane receiver array. 
@@ -716,5 +730,5 @@
 \caption{ Array layout for configurations (b) and (c) with 128 and 129  D=5 meter 
 diameter dishes. }
-\label{figconfab}
+\label{figconfbc}
 \end{figure}
 
@@ -723,5 +737,5 @@
 $\eta$, relating the effective dish diameter $D_{ill}$ to the 
 mechanical dish size $D^{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales 
-as $\eta^2$ or $eta_x \eta_y$. 
+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})  \\
@@ -762,10 +776,10 @@
 \includegraphics[width=0.90\textwidth]{Figs/uvcovabcd.pdf}
 }
-\caption{(u,v) plane coverage (non normalized instrument response ${\cal R}(u,v,\lambda)$ 
+\caption{(u,v) plane coverage (raw 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. 
-color scale : black $<1$, blue, green, yellow, red $\gtrsim 80$ }
+of $11 \times 11$, (b) 128 dishes arranged in 8 row 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. 
+(color scale : black $<1$, blue, green, yellow, red $\gtrsim 80$) }
 \label{figuvcovabcd}
 \end{figure*}
@@ -789,5 +803,5 @@
 Reaching the required sensitivities is not the only difficulty of observing the large 
 scale structures in 21 cm. Indeed, the synchrotron emission of the 
-Milky Way and the extra galactic radio sources are a thousand time brighter than the 
+Milky Way and the extra galactic radio sources are a thousand times brighter than the 
 emission of the neutral hydrogen distributed in the universe. Extracting the LSS signal
 using Intensity Mapping, without identifying the \HI point sources is the main challenge 
@@ -813,7 +827,7 @@
 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 $\alpha= 10:00 \, \mathrm{h} , \delta=+10 \, \mathrm{deg.}$,
-and  covering 128 MHz in frequency. The sky cube characteristics (coordinate range, size, resolution) 
-used in the simulations is given in the table \ref{skycubechars}.
+$90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10\mathrm{h}00\mathrm{m} , \delta=+10 \, \mathrm{deg.}$, and  covering 128 MHz in frequency. We have selected this particular area of the sky to in order to minimize 
+the Galactic synchrotron foreground. The sky cube characteristics (coordinate range, size, resolution) 
+used in the simulations are given in the table \ref{skycubechars}.
 \begin{table}
 \begin{center}
@@ -875,5 +889,5 @@
 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 follow:
+contribution of the radiosources to the sky temperature is computed as follows:
 $$ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 MHz}\right)^{\beta_{src}} $$
 %%
@@ -884,10 +898,23 @@
 
  The 21 cm temperature fluctuations due to neutral hydrogen in large scale structures
-$T_{lss}(\alpha, \delta, \nu)$  has been computed using the SimLSS software package 
-\footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }.  
-{\color{red}: CMV, please add few line description of SimLSS}. 
-We have generated the mass fluctuations $\delta \rho/\rho$ at $z=0.6$, in cells of size 
-$1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the 
-sky cube angular and frequency resolution defined above.  The mass fluctuations has been 
+$T_{lss}(\alpha, \delta, \nu)$  have been computed using the 
+SimLSS \footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }  software package: 
+%
+complex normal Gaussian fields were first generated in Fourier space.
+The amplitude of each mode was then multiplied by the square root
+of the power spectrum $P(k)$ at $z=0$ computed according to the parametrization of 
+\citep{eisenhu.98}. We have used the standard cosmological parameters, 
+ $H_0=71 \mathrm{km/s/Mpc}$, $\Omega_m=0.27$, $\Omega_b=0.044$,
+$\Omega_\lambda=0.73$ and $w=-1$.
+An inverse FFT was then performed to compute the matter density fluctuations
+in the linear regime, 
+$\delta \rho / \rho$ in a box of $3420 \times 1140 \times 716  \, \mathrm{Mpc^3}$ and evolved
+to redshift $z=0.6$. 
+The size of the box is about 2500 $\mathrm{deg^2}$  in the transverse direction and
+$\Delta z \simeq 0.23$ in the longitudinal direction.  
+The size of the cells is  $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the 
+sky cube angular and frequency resolution defined above.  
+
+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}.  
@@ -940,5 +967,5 @@
 
 It should also be noted that in section 3, we presented the different instrument 
-configuration noise level after {\em correcting or deconvolving} the instrument response. The LSS 
+configuration noise levels after {\em correcting or deconvolving} the instrument response. The LSS 
 power spectrum is recovered unaffected in this case, while the noise power spectrum 
 increases at high k values (small scales). In practice, clean deconvolution is difficult to 
@@ -994,5 +1021,5 @@
 $T_{sky}(\alpha, \delta, \nu)$ by applying the frequency or wavelength dependent instrument response
 ${\cal R}(u,v,\lambda)$. 
-we have considered the simple case where  the instrument response constant throughout the survey area, or independent 
+We have considered the simple case where  the instrument response is constant throughout the survey area, or independent 
 of the sky direction.
 For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$ : 
@@ -1021,5 +1048,5 @@
 \begin{enumerate}
 \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}
+beam effects through a convolution by a fiducial 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 
@@ -1031,9 +1058,9 @@
  \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 $\log10 ( T ) , \log10 (\nu)$ plane and we show here the results for a pure power law (P1) 
+the $\lgd ( T ) , \lgd (\nu)$ 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 
+P1 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) \\
+P2 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) + c \, \lgd ( \nu/\nu_0 ) ^2 
 \end{eqnarray*}
 where $b$ is the power law index and  $T_0 = 10^a$ is the brightness temperature at the 
@@ -1055,5 +1082,5 @@
 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 
+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 $). 
@@ -1112,11 +1139,11 @@
 The sky reconstruction and the component separation introduce additional filtering and distortions.
 Ideally, one has to define a power spectrum measurement response or {\it transfer function} in the 
-radial direction,  ($\lambda$ or redshift, $TF(k_\parallel)$) and in the transverse plane ( $TF(k_\perp)$ ). 
+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 $TF(k)$ as the ratio of the 
+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 $P_{21}(k)$:
 \begin{equation}
-TF(k) = P_{21}^{rec}(k) / P_{21}(k) 
+\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) 
 \end{equation}
 
@@ -1131,15 +1158,15 @@
 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, 
+LSS temperature cube $P_{21}^{rec}(k)/P_{21}(k)$ and corresponds to the transfer function $\TrF(k)$ 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 
+due to the noise or system temperature. It should be stressed that the simulations presented in this section were 
 focused on the study of the radio foreground effects and have been carried 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) 
+This transfer function is well represented by the analytical form:
+\begin{equation}
+\TrF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right) 
 \label{eq:tfanalytique}
 \end{equation} 
@@ -1157,4 +1184,5 @@
 
 \begin{table}[hbt]
+\begin{center}
 \begin{tabular}{|c|ccccc|}
 \hline 
@@ -1167,4 +1195,5 @@
 \hline 
 \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.  }
@@ -1180,5 +1209,5 @@
 }
 \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.
+\caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $, 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}
@@ -1194,5 +1223,5 @@
 }
 \vspace*{-30mm}
-\caption{Fitted/smoothed  transfer function obtained for the recovered 21 cm power spectrum at different redshifts, 
+\caption{Fitted/smoothed  transfer function $\TrF(k)$ obtained for the recovered 21 cm power spectrum at different redshifts, 
 $z=0.5 , 1.0 , 1.5 , 2.0 , 2.5$ for the instrument configuration (e), $20\times20$ packed array interferometer. }
 \label{tfpkz0525}
@@ -1221,8 +1250,7 @@
 \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 
+Fig. \ref{figpnoisea2g} 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
@@ -1252,11 +1280,11 @@
 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})
+damped by the transfer function $\TrF(k)$ (Eq. \ref{eq:tfanalytique} , table \ref{tab:paramtfk})
 and a white noise component $P_{noise}$ calculated according to the 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} 
+ P^{rec}(k) = P_{21}(k) \times \TrF(k) + P_{noise} 
 \end{equation}
-where the different terms ($P_{21}(k) , TF(k), P_{noise}$depend on the slice redshift.  
+where the different terms ($P_{21}(k) , \TrF(k), P_{noise}$) depend on the slice redshift.  
 The expected 21 cm power spectrum $P_{21}(k)$ has been generated according to the formula:
 %\begin{equation}
@@ -1299,9 +1327,9 @@
  
 Figure \ref{fig:fitOscill} shows the result of the fit for 
-one of theses simulations. 
+one of these 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.
+of the statistical errors.
 
 In addition, in the fitting procedure, both the parameters modeling the 
@@ -1503,5 +1531,5 @@
 (Eq. \ref{eq:dTdH}) by:
 \begin{equation}
-\Omega_\Lambda = \Omega_{\Lambda}^0 \exp \left[ 3  \int_0^z   
+\Omega_\Lambda \rightarrow \Omega_{\Lambda} \exp \left[ 3  \int_0^z   
 \frac{1+w(z^\prime)}{1+z^\prime } dz^\prime  \right]
 \end{equation}
@@ -1551,5 +1579,5 @@
 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 
+focal plane array with a $\sim 100 \, \mathrm{meter}$ 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.
@@ -1558,11 +1586,11 @@
 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 
+requires a very good knowledge of the instrument response. Our method has allowed us to define and 
 compute the overall  {\it transfer function} or {\it response function} for the measurement of the 21 cm 
 power spectrum. 
-Finally, we have used the computed noise power spectrum and  P(k) 
+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 
+could be carried using the current technology and would be competitive with the ongoing or planned 
 optical surveys for dark energy,  with a fraction of their cost.
  
@@ -1591,4 +1619,7 @@
 Glazebrook, K. \&  Blake, C. 2005 \apj, 631, 1
 
+% WiggleZ BAO observation 
+\bibitem[Blake et al. (2011)]{blake.11} Blake, Davis, T., Poole, G.B.  {\it et al.}  2011, \mnras  (arXiv/1105.2862)
+
 % Galactic astronomy, emission HI d'une galaxie
 \bibitem[Binney \& Merrifield (1998)]{binney.98} Binney J. \& Merrifield M. , 1998 {\it Galactic Astronomy} Princeton University Press 
@@ -1610,11 +1641,11 @@
 
 %  Parametrisation P(k) 
-\bibitem[Eisentein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, ApJ 496:605-614 (astro-ph/9709112)
+\bibitem[Eisenstein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, ApJ 496:605-614 (astro-ph/9709112)
 
 % SDSS first BAO observation 
-\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
+\bibitem[Eisenstein 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 
+\bibitem[Eisenstein 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  
@@ -1634,4 +1665,7 @@
 \bibitem[Lah et al. (2009)]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur et al.  2009,  \mnras 
 ( astro-ph/0907.1416) 
+
+% Livre Astrophysical Formulae de Lang
+\bibitem[Lang (1999)]{radastron} Lang, K.R. {\it Astrophysical Formulae}, Springer, 3rd Edition 1999 
 
 % LSST Science book
@@ -1685,5 +1719,5 @@
 
 %  Thomson-Morane livre interferometry 
-\bibitem[Radio Astronomy (1998)]{radastron} Thompson, A.R., Moran, J.M., Swenson, G.W, {\it Interferometry and 
+\bibitem[Thompson, Moran \& Swenson (2001)]{radastron} Thompson, A.R., Moran, J.M., Swenson, G.W, {\it Interferometry and 
 Synthesis in Radio Astronomy}, John Wiley \& sons, 2nd Edition 2001 
 
