Changeset 4049 in Sophya for trunk

Timestamp:

Feb 13, 2012, 6:56:58 PM (14 years ago)

Author:

ansari

Message:

Version avec les corrections du language editor, Reza 13/02/2012

File:

• trunk/Cosmo/RadioBeam/sensfgnd21cm.tex (modified) (83 diffs)

trunk/Cosmo/RadioBeam/sensfgnd21cm.tex

@@ -18 +18 @@
 %\documentclass[letter]{aa} % for the letters 
 %
-\documentclass[structabstract]{aa}  
+\documentclass[structabstract]{aa}    % version standard, utilise pour ce papier 
 %\documentclass[traditabstract]{aa} % for the abstract without structuration 
                                    % (traditional abstract) 
@@ -27 +27 @@
 \usepackage{graphicx}
 \usepackage{color}
+
+%% \usepackage{natbib}    Probleme - pas tente de le resoudre (Reza, Jan 2012)
+%% \bibpunct{(}{)}{;}{a}{}{,} % to follow the A&A style
 
 %% Commande pour les references 
@@ -128 +131 @@
   }
 
-   \date{Received August 5, 2011; accepted xxxx, 2011}
+   \date{Received August 5, 2011; accepted December 22, 2011}
 
 % \abstract{}{}{}{}{} 
@@ -136 +139 @@
   % context heading (optional)
   % {} leave it empty if necessary  
-   { Large Scale Structures (LSS) in the universe can be traced using the neutral atomic hydrogen \HI through its 21 
-cm emission. Such a 3D matter distribution map can be used to test the Cosmological model and to constrain the Dark Energy 
-properties or its equation of state. A novel approach, called intensity mapping can be used to map the \HI distribution, 
-using radio interferometers with large instantaneous field of view and waveband.}
+   { Large scale structures (LSS) in the universe can be traced using the neutral atomic hydrogen \HI through its 21 
+cm emission. Such a 3D matter distribution map can be used to test the cosmological model and to constrain the dark energy 
+properties or its equation of state. A novel approach, called intensity mapping, can be used to map the \HI distribution, 
+using radio interferometers with a large instantaneous field of view and waveband.}
  % aims heading (mandatory)
-  { In this paper, we study the sensitivity of different radio interferometer configurations, or multi-beam 
-instruments for the observation of large scale structures and BAO oscillations in 21 cm and we discuss the problem of foreground removal. }
+  {We study the sensitivity of different radio interferometer configurations, or multi-beam 
+instruments for observing LSS 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 $(\uv)$ or Fourier angular frequency 
+ { For each configuration, we determined instrument response by computing the $(\uv)$ or Fourier angular frequency 
 plane  coverage using visibilities. The $(\uv)$ plane response determines the noise power spectrum, 
-hence the instrument sensitivity for LSS P(k) measurement. We 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. }
+hence the instrument sensitivity for LSS P(k) measurement. We also describe a simple foreground subtraction method
+of separating LSS 21 cm signal from the foreground due to the galactic synchrotron and radio source emission. }
   % results heading (mandatory)
-   { We have computed the noise power spectrum for different instrument configurations as well as the extracted 
-   LSS power spectrum, after 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.}
+   { We have computed the noise power spectrum for different instrument configurations, as well as the extracted 
+   LSS power spectrum, after separating the 21cm-LSS signal from the foregrounds. We have also obtained 
+  the uncertainties on the dark energy parameters for an optimized 21 cm BAO survey.}
   % conclusions heading (optional), leave it empty if necessary 
    { We show that a radio instrument with few hundred simultaneous beams and a collecting area of 
@@ -170 +173 @@
 
 % {\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. 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 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. 
-
-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 of the most intriguing puzzles in Physics and Cosmology. 
+The study of the statistical properties of large scale structures (LSS) in the Universe and of their evolution 
+with redshift is one of the major tools in observational cosmology. These structures are usually mapped through 
+optical observation of galaxies that are used as tracers of the underlying matter distribution. 
+An alternative and elegant approach for mapping the matter distribution, which uses neutral atomic hydrogen 
+(\HI) as a tracer with intensity mapping, has been proposed in recent years (\cite{peterson.06}; \cite{chang.08}). 
+Mapping the matter distribution using \HI 21 cm emission as a tracer has been extensively discussed in the literature
+(\cite{furlanetto.06}; \cite{tegmark.09}) and is being used in projects such as LOFAR \citep{rottgering.06} or
+MWA \citep{bowman.07} to observe reionization  at redshifts z $\sim$ 10. 
+
+Evidence of the acceleration in the expansion of the universe has
+accumulated over the last twelve years, thanks to the observation of
+distant supernovae and CMB anisotropies and to detailed analysis of the LSS.  
+A cosmological constant ($\Lambda$) or new cosmological 
+energy density called {\em dark energy} has been advocated as the origin of this acceleration. 
+dark energy is considered as one of the most intriguing puzzles in physics and cosmology. 
 % Constraining the properties of this new cosmic fluid, more precisely 
 % its equation of state is central to current cosmological researches. 
 Several cosmological probes can be used to constrain the properties of this new cosmic fluid, 
-more precisely its equation of state: The Hubble Diagram, or luminosity distance as a function 
+more precisely its equation of state: the Hubble diagram, or the luminosity distance as a function 
 of redshift of supernovae as standard candles, galaxy clusters, weak shear observations 
-and Baryon Acoustic Oscillations (BAO). 
+and baryon acoustic oscillations (BAO). 
 
 BAO are features imprinted  in the distribution of galaxies, due to the frozen 
-sound waves which were present in the photon-baryon plasma prior to recombination 
+sound waves that were present in the photon-baryon plasma prior to recombination 
 at \mbox{$z \sim 1100$}. 
 This scale can be considered as a standard ruler with a comoving 
-length  of \mbox{$\sim 150 \mathrm{Mpc}$}.
-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}). 
+length  of \mbox{$\sim 150 \, \mathrm{Mpc}$}, and
+these features have been first observed in the CMB anisotropies
+and are usually referred to as {\em acoustic peaks} (\cite{mauskopf.00}; \cite{larson.11}). 
 The BAO modulation has been subsequently observed in the distribution of galaxies 
 at low redshift ( $z < 1$) in the galaxy-galaxy correlation function by the SDSS 
-\citep{eisenstein.05}  \citep{percival.07}  \citep{percival.10}, 2dGFRS  \citep{cole.05}  as well as 
-WiggleZ \citep{blake.11} optical galaxy surveys.
-
-Ongoing {\changemarkb surveys such as BOSS} \citep{eisenstein.11}  or future surveys 
-{\changemarkb such as LSST} \citep{lsst.science} 
-plan to measure precisely the BAO scale in the redshift range 
+(\cite{eisenstein.05};  \cite{percival.07};  \cite{percival.10}), 2dGFRS  \cite{cole.05}, 
+as well as WiggleZ \citep{blake.11} optical galaxy surveys.
+
+Ongoing {\changemarkb surveys, such as BOSS} \citep{eisenstein.11}  or future surveys, 
+{\changemarkb such as LSST} \citep{lsst.science}, 
+plan to measure the BAO scale precisely in the redshift range 
 $0 \lesssim z \lesssim 3$, using either optical observation of galaxies  
-or 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 
-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 
+or 3D mapping of Lyman $\alpha$ absorption lines toward distant quasars 
+(\cite{baolya}; \cite{baolya2}). 
+Radio observation of the 21 cm emission of neutral hydrogen is %  ?ENG? appears as 
+a very promising technique for mapping matter distribution up to redshift $z \sim 3$, 
+and it complements optical surveys, especially in the optical redshift desert range 
 $1 \lesssim z \lesssim 2$, and possibly up to the reionization redshift \citep{wyithe.08}.
 
-In section 2, we  discuss the intensity mapping and its potential for measurement of the
+In section 2, we  discuss the intensity mapping and its potential for measuring of the
 \HI mass distribution power spectrum. The method used in this paper to characterize 
 a radio instrument response and sensitivity for $P_{\mathrm{H_I}}(k)$ is presented in section 3.
-We show also  the results for the 3D noise power spectrum for several  instrument configurations.
-The contribution of foreground emissions due to the galactic synchrotron and radio sources
-is described in section 4, as well as a simple component separation method. The performance of this 
+We also show  the results for the 3D noise power spectrum for several  instrument configurations.
+The contribution of foreground emissions due to both the galactic synchrotron and radio sources
+is described in section 4, as is  a simple component separation method. The performance of this 
 method using two different sky models is also presented in section 4. 
-The constraints which can be obtained on the Dark Energy parameters and DETF figure 
+The constraints that can be obtained on the dark energy parameters and DETF figure 
 of merit for typical 21 cm intensity mapping survey are discussed in section 5.
 
@@ -234 +237 @@
 \subsection{21 cm intensity mapping}
 %%% 
-Most of the cosmological information in the LSS is located at large scales
-($ \gtrsim 1 \mathrm{deg}$), while the interpretation at smallest scales 
-might suffer from the uncertainties on the non linear clustering effects.  
+Most of the cosmological information in the LSS is located on large scales
+($ \gtrsim 1 \mathrm{deg}$), while the interpretation on the smallest scales 
+might suffer from the uncertainties on the nonlinear  clustering effects.  
 The BAO features in particular are at the degree angular scale on the sky 
 and thus can be resolved easily with a rather modest size radio instrument 
@@ -246 +249 @@
 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
+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 constraint on DE parameters can only be 
+typically a few $\mathrm{Gpc^3}$. Moreover, stringent constraint on DE parameters can only be 
 obtained when comparing the distance or Hubble parameter measurements with 
 DE models as a function of redshift, which requires a significant survey depth $\Delta z \gtrsim 1$.
-
 Radio instruments intended for BAO surveys must thus have large instantaneous field
 of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$)
 to explore large redshift domains.
 
-Although the application of 21 cm radio survey to cosmology, in particular LSS mapping has been 
-discussed in length in the framework of large future instruments, such as the SKA (e.g \cite{ska.science}, \cite{abdalla.05}),
-the method envisaged has been mostly through the detection of galaxies as \HI compact sources. 
+Although the application of 21 cm radio survey to cosmology, in particular LSS mapping, has been 
+discussed in length in the framework of large future instruments, such as the SKA (e.g \cite{ska.science}; \cite{abdalla.05}),
+the method envisaged has mostly been through the detection of galaxies as \HI compact sources. 
 However, extremely large radio telescopes are required to detected \HI sources at cosmological distances. 
-The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the two polarisations
+The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the two polarizations
 of a radio instrument characterized by an effective collecting area $A$, and system temperature $\Tsys$ can be written as 
 \begin{equation}
 S_{lim} = \frac{ \sqrt{2} \, \kb \, \Tsys }{ A \, \sqrt{t_{int} \delta \nu} } 
 \end{equation} 
-where $t_{int}$ is the total integration time and $\delta \nu$ is the detection frequency band. In table 
-\ref{slims21} (left)  we have computed the sensitivity for 6 different sets of instrument effective area and system 
+where $t_{int}$ is the total integration time and $\delta \nu$ the detection frequency band. In Table 
+\ref{slims21} (left)  we computed the sensitivity for six different sets of instrument effective area and system 
 temperature, with a total integration time of 86400 seconds (1 day) over a frequency band of 1 MHz. 
-The width of this frequency band is well adapted  to detection of \HI source with an intrinsic velocity 
-dispersion of few 100 km/s.
+The width of this frequency band is well adapted  to detecting an \HI source with an intrinsic velocity 
+dispersion of a few 100 km/s.
 These detection limits should be compared with the expected 21 cm brightness
-$S_{21}$ of compact sources which can be computed using the expression below (e.g.\cite{binney.98}) :
+$S_{21}$ of compact sources, which can be computed using the expression below (e.g. \cite{binney.98}):
 \begin{equation}
  S_{21}  \simeq  0.021 \mathrm{\mu Jy} \, \frac{M_{H_I} }{M_\odot}   \times 
 \left( \frac{ 1\, \mathrm{Mpc}}{\dlum(z)} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v}  (1+z)
 \end{equation}
- where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ is the luminosity distance and $\sigma_v$ 
-is the source velocity dispersion.  
+ where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ the luminosity distance, and $\sigma_v$ 
+the source velocity dispersion.  
 {\changemark The 1 MHz bandwidth mentioned above is only used for computing the
 galaxy detection thresholds and does not determine the total bandwidth or frequency resolution
@@ -283 +285 @@
 % {\color{red} Faut-il developper le calcul en annexe ? } 
  
-In table \ref{slims21} (right), we show the 21 cm brightness for 
+In Table \ref{slims21} (right), we show the 21 cm brightness for 
 compact objects with a total \HI \, mass of $10^{10} M_\odot$ and an intrinsic velocity dispersion of 
-$200 \, \mathrm{km/s}$. The luminosity distance is computed for the standard 
-WMAP \LCDM universe \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
+$200 \, \mathrm{km/s}$. The luminosity distance was computed for the standard 
+WMAP \LCDM universe \citep{komatsu.11}. From $10^9$ to  $10^{10} M_\odot$ of neutral gas mass 
+is typical of large galaxies \citep{lah.09}. It is clear that detecting \HI sources at cosmological distances
 would require collecting area in the range of \mbox{$10^6 \, \mathrm{m^2}$}. 
 
-Intensity mapping has been suggested as an alternative and economic method to map the 
-3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} and \citep{seo.10}. 
+Intensity mapping has been suggested as an alternative and economic method of mapping the 
+3D distribution of neutral hydrogen by (\cite{chang.08}; \cite{ansari.08}; \citep{seo.10}). 
 {\changemark There have also been attempts to detect the 21 cm LSS signal at GBT 
 \citep{chang.10} and at GMRT \citep{ghosh.11}}.
-In this approach, 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 
+In this approach, a sky brightness map with angular resolution 
+\mbox{$\sim 10-30 \, \mathrm{arc.min}$} is created for a   %% ?ENG? was created ?
+wide range of frequencies. Each 3D pixel  (2 angles $\vec{\Theta}$, frequency $\nu$, or wavelength $\lambda$)  
+would correspond to a cell with a volume of $\sim 10^3 \mathrm{Mpc^3}$, containing ten to a hundred galaxies 
 and a total \HI mass $ \sim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow, 
-the observed frequency $\nu$ would be translated to the emission redshift $z$ through 
-the well known relation:
+the observed frequency $\nu$ would be translated into the emission redshift $z$ through 
+the well known relation
 \begin{eqnarray}
  z(\nu) & = & \frac{\nu_{21} -\nu}{\nu} 
@@ -306 +309 @@
  z(\lambda) & = & \frac{\lambda - \lambda_{21}}{\lambda_{21}} 
 \, ; \, \lambda(z) = \lambda_{21} \times (1+z) 
-\hspace{1mm} \mathrm{with}   \hspace{1mm}  \lambda_{21} = 0.211 \, \mathrm{m} 
+\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 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 $B_{21}$ 
+The large-scale distribution of the neutral hydrogen, down to an angular scale of \mbox{$\sim 10 \, \mathrm{arc.min}$} 
+can then be observed without detecting individual compact \HI sources, using the set of sky-brightness 
+maps as a function of frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$ 
 (radiation power/unit solid angle/unit surface/unit frequency) 
 can be converted to brightness temperature using the Rayleigh-Jeans approximation of  black body radiation law:
-$$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} $$ 
+$$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} .$$ 
  
 %%%%%%%%
@@ -348 +351 @@
 \end{tabular}
 \end{center}
-\tablefoot{The left panel shows the  sensitivity or source detection limit for 1 day integration time (86400 s) and 1 MHz 
-frequency band. The 21 cm brightness for sources containing $10^{10} M_\odot$ of \HI at different redshifts is given
-in the right panel.  }
+\tablefoot{Left panel: sensitivity or source detection limit for 1-day integration time (86400 s) and 1-MHz 
+frequency band. Right panel: 21 cm brightness for sources containing $10^{10} M_\odot$ of \HI at different redshifts.}
 \end{table}
 
 \subsection{ \HI power spectrum and BAO}
 In the absence of any foreground or background radiation 
-{\changemark and assuming high spin temperature, $\kb T_{spin} \gg h \nu_{21}$},
+{\changemark and assuming a high spin temperature, $\kb T_{spin} \gg h \nu_{21}$},
 the brightness temperature for a given direction and wavelength $\TTlam$ would be proportional to 
 the local \HI number density $\etaHI(\vec{\Theta},z)$ through the
-relation {\changemarkb (\cite{field.59} , \cite{zaldarriaga.04})}:
+relation {\changemarkb (\cite{field.59}; \cite{zaldarriaga.04})}:
 \begin{equation}
   \TTlamz  =   \frac{3}{32 \pi}  \, \frac{h}{\kb} \,  A_{21}  \, \lambda_{21}^2 \times
@@ -364 +366 @@
 \end{equation}
 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 
+coefficient, $h$ the Planck constant, $c$ the speed of light, $\kb$ the Boltzmann 
+constant, and $H(z)$ the Hubble parameter at the emission 
 redshift.
 For a \LCDM universe and neglecting radiation energy density, the Hubble parameter
-can be expressed as:
+can be expressed as
 \begin{equation}
 H(z)  \simeq  \hubb  \, \left[ \Omega_m (1+z)^3 + \Omega_\Lambda \right]^{\frac{1}{2}} 
-\times  100 \, \, \mathrm{km/s/Mpc}  
+\times  100 \, \, \mathrm{km/s/Mpc.}  
 \label{eq:expHz}
 \end{equation}
-Introducing the \HI mass fraction relative to the total baryon mass $\gHI$, the 
+After introducing the \HI mass fraction relative to the total baryon mass $\gHI$, the 
 neutral hydrogen number density and the corresponding 21 cm emission temperature 
 can be written as a function of \HI relative density fluctuations: 
@@ -382 +384 @@
  \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
-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 $\gHI(0)$ present in local galaxies has been 
-measured to be $\sim 1\%$ of the baryon density \citep{zwann.05}:
-$$ \Omega_{H_I} \simeq 3.5 \, 10^{-4} \sim 0.008 \times \Omega_B $$
+where $\Omega_B$ and $\rho_{crit}$ are the present-day mean baryon cosmological
+and critical densities, respectively, $m_{H}$ the hydrogen atom mass, and 
+$\frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}}$ the \HI density fluctuations. 
+
+The present-day neutral hydrogen fraction $\gHI(0)$ present in local galaxies has been 
+measured to be $\sim 1\%$ of the baryon density \citep{zwann.05}
+$$ \Omega_{H_I} \simeq 3.5 \, 10^{-4} \sim 0.008 \times \Omega_B .$$
 The neutral hydrogen fraction is expected to increase with redshift, as gas is used 
 in star formation during galaxy formation and evolution. Study of Lyman-$\alpha$ absorption 
-indicate a factor 3 increase in the neutral hydrogen 
+indicates a factor 3 increase in the neutral hydrogen 
 fraction at $z=1.5$ in the intergalactic medium \citep{wolf.05}, 
-compared to its present day value $\gHI(z=1.5) \sim 0.025$. 
+compared to its current value $\gHI(z=1.5) \sim 0.025$. 
 The 21 cm brightness temperature and the corresponding power spectrum can be written as 
-(\cite{madau.97}, \cite{zaldarriaga.04}), \cite{barkana.07}) :
+(\cite{madau.97}; \cite{zaldarriaga.04}); \cite{barkana.07})
 \begin{eqnarray}
  P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
  \bar{T}_{21}(z)  & \simeq & 0.084  \, \mathrm{mK}  
 \frac{ (1+z)^2 \, \hubb }{\sqrt{ \Omega_m (1+z)^3 + \Omega_\Lambda } } 
- \dfrac{\Omega_B}{0.044}  \,  \frac{\gHIz}{0.01} 
+ \dfrac{\Omega_B}{0.044}  \,  \frac{\gHIz}{0.01} \, .
 \label{eq:tbar21z}
 \end{eqnarray}
 
-The table \ref{tabcct21} shows the mean 21 cm brightness temperature for 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) $. Figure \ref{figpk21} shows the 
 21 cm emission power spectrum at several redshifts, with a constant neutral fraction at 2\%
 ($\gHI=0.02$). The matter power spectrum has been computed using the 
-\cite{eisenhu.98} parametrisation. The correspondence  with the angular scales is also 
-shown for the standard WMAP \LCDM cosmology, according to the relation:
+\cite{eisenhu.98} parametrization. The correspondence  with the angular scales is also 
+shown for the standard WMAP \LCDM cosmology, according to the relation
 \begin{equation}
 \theta_k = \frac{2 \pi}{k \, \dang(z) \, (1+z) }  
-\hspace{3mm} 
-k = \frac{2 \pi}{ \theta_k  \, \dang(z) \, (1+z) }  
+\hspace{3mm} , \hspace{3mm}
+k = \frac{2 \pi}{ \theta_k  \, \dang(z) \, (1+z) }  \hspace{5mm} ,
 \end{equation}
 where $k$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance.
 { \changemark The matter power spectrum $P(k)$ has been measured using 
 galaxy surveys, for example by SDSS and 2dF at low redshift $z \lesssim 0.3$
-(\cite{cole.05}, \cite{tegmark.04}). The 21 cm brightness power spectra $P_{T_{21}}(k)$
+(\cite{cole.05}; \cite{tegmark.04}). The 21 cm brightness power spectra $P_{T_{21}}(k)$
 shown here are comparable to the power spectrum measured from the galaxy surveys, 
 once the mean 21 cm temperature conversion factor $\left( \bar{T}_{21}(z)  \right)^2$,
-redshift evolution and different bias factors have been accounted for. }
+redshift evolution, and different bias factors have been accounted for. }
 % It should be noted that the maximum transverse $k^{comov} $ sensitivity range 
 % for an instrument corresponds approximately to half of its angular resolution. 
@@ -465 +467 @@
 \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 
+We briefly introduce here the principles of interferometric observations and the definition of 
+quantities useful for our calculations. The interested reader may refer to \cite{radastron} for a detailed 
 and complete presentation of observation methods and signal processing in radio astronomy.  
 In astronomy we are usually interested in measuring the sky emission intensity,
@@ -472 +474 @@
 and interferometry in particular, receivers are sensitive to the sky emission complex 
 amplitudes. However, for most sources, the phases vary randomly with a spatial correlation 
-length significantly smaller than the instrument resolution.
+length significantly smaller than the instrument resolution,
 \begin{eqnarray}
 & & 
 I(\vec{\Theta},\lambda)  =  | A(\vec{\Theta},\lambda) |^2  \hspace{2mm} , \hspace{1mm} I \in \mathbb{R}, A \in \mathbb{C} \\
-& & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  = 0 \hspace{2mm} \mathrm{for}   \hspace{1mm} \vec{\Theta} \ne \vec{\Theta '}  
+& & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  = 0 \hspace{2mm} \mathrm{for}   \hspace{1mm} \vec{\Theta} \ne \vec{\Theta ' \, .}  
 \end{eqnarray} 
 A single receiver can be  characterized by its angular complex amplitude response $B(\vec{\Theta},\nu)$ and 
-its position $\vec{r}$ in a reference frame. the waveform complex amplitude $s$ measured by the receiver, 
+its position $\vec{r}$ in a reference frame. The waveform complex amplitude $s$ measured by the receiver, 
 for each frequency can be written as a function of the electromagnetic wave vector 
-$\vec{k}_{EM}(\vec{\Theta}, \lambda) $ :
+$\vec{k}_{EM}(\vec{\Theta}, \lambda) $:
 \begin{equation}
 s(\lambda)  =  \iint d \vec{\Theta} \, \, \, A(\vec{\Theta},\lambda) B(\vec{\Theta},\lambda) e^{i ( \vec{k}_{EM} . \vec{r} )} \\
 \end{equation} 
-We have set the electromagnetic (EM) phase origin at the center of the coordinate frame and 
+We set the electromagnetic (EM) phase origin at the center of the coordinate frame, and 
 the EM wave vector is related to the wavelength $\lambda$ through the usual equation
 $ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta},\lambda)$ 
 corresponds to the receiver intensity response:
 \begin{equation}
-L(\vec{\Theta}, \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda) 
+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 
+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 of their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
+visibility $\vis$ signal from two identical receivers, located at the positions $\vec{r_1}$ and 
+$\vec{r_2}$, can simply be written as a function of their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
 \begin{equation}
 \vis(\lambda) = < s_1(\lambda) s_2(\lambda)^* > = \iint d \vec{\Theta} \, \, I(\vec{\Theta},\lambda) L(\vec{\Theta},\lambda) 
 e^{i ( \vec{k}_{EM} . \vec{\Delta r} ) }
 \end{equation} 
-This expression can be simplified if we consider receivers with narrow field of view 
+This expression can be simplified if we consider receivers with a narrow field of view 
 ($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \, \mathrm{deg.} $ ), 
-and coplanar in respect to their common axis.
-If we introduce two {\em Cartesian} like angular coordinates $(\alpha,\beta)$ centered at 
+and coplanar with respect to their common axis.
+If we introduce two cartesian-like angular coordinates $(\alpha,\beta)$ centered on
 the common receivers axis, the visibilty would be written as the 2D Fourier transform
 of the product of the sky intensity and the receiver beam, for the angular frequency 
@@ -512 +514 @@
 \end{equation} 
 where $(\Delta x, \Delta y)$ are the two receiver distances on a plane perpendicular to 
-the receiver axis. The $x$ and $y$ axis in the receiver plane are taken parallel to the 
+the receiver axis. The $x$ and $y$ axes in the receiver plane are taken parallel to the 
 two $(\alpha, \beta)$ angular planes. 
-
 Furthermore, we introduce the conjugate Fourier variables $(\uv)$ and the Fourier transforms
 of the sky intensity and the receiver beam:
@@ -521 +522 @@
 $(\alpha, \beta)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & $(\uv)$ \\
 $I(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal I}(\uv, \lambda)$ \\
-$L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(\uv, \lambda)$ \\
+$L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(\uv, \lambda)$ \, .\\
 \end{tabular} 
 \end{center}
@@ -528 +529 @@
 wave number domain located around 
 $(\uv)_{12}=( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is 
-given by the receiver beam Fourier transform. 
-\begin{equation}
-\vis(\lambda) \simeq  \iint \dudv \, \, {\cal I}(\uv, \lambda) \, {\cal L}(\uvu - \frac{\Delta x}{\lambda} , \uvv - \frac{\Delta y}{\lambda} , \lambda)
+given by the receiver-beam Fourier transform
+\begin{equation}
+\vis(\lambda) \simeq  \iint \dudv \, \, {\cal I}(\uv, \lambda) \, {\cal L}(\uvu - \frac{\Delta x}{\lambda} , \uvv - \frac{\Delta y}{\lambda} , \lambda) \, .
 \end{equation} 
 
-A single receiver instrument would measure the total power integrated in a spot centered around the 
-origin in the $(\uv)$ or the angular wave mode plane. The shape of the spot depends on the receiver 
+\noindent A single receiver instrument would measure the total power integrated in a spot centered on the 
+origin in the $(\uv)$ or the angular wave-mode plane. The shape of the spot depends on the receiver 
 beam pattern, but its extent would be $\sim 2 \pi D / \lambda$, where $D$ is the receiver physical 
 size. 
 
 The correlation signal from a pair of receivers would measure the integrated signal on a similar 
-spot, located around the central angular wave mode  $(\uv)_{12}$ determined by the relative 
+spot, located around the central angular wave-mode  $(\uv)_{12}$, determined by the relative 
 position of the two receivers (see figure \ref{figuvplane}).
 In an interferometer with multiple receivers, the area covered by different receiver pairs in the 
-$(\uv)$ plane might overlap and some pairs might measure the same area (same base lines). 
-Several beams can be formed using different combination of the correlations from a set of  
+$(\uv)$ plane might overlap, and some pairs might measure the same area (same base lines). 
+Several beams can be formed using different combinations of the correlations from a set of  
 antenna pairs.  
 
@@ -549 +550 @@
 ${\cal R}(\uv,\lambda)$. For a single dish with a single receiver in the focal plane, 
 the instrument response is simply the Fourier transform of the beam. 
-For a single dish with multiple receivers, either as a Focal Plane Array (FPA) or 
+For a single dish with multiple receivers, either as a focal plane array (FPA) or 
 a multi-horn system, each beam (b) will have its own response 
 ${\cal R}_b(\uv,\lambda)$. 
 For an interferometer, we can compute a raw instrument response 
-${\cal R}_{raw}(\uv,\lambda)$ which corresponds to $(\uv)$ plane coverage by all 
+${\cal R}_{raw}(\uv,\lambda)$, which corresponds to $(\uv)$ plane coverage by all 
 receiver pairs  with uniform weighting. 
 Obviously, different weighting schemes can be used, changing 
-the effective beam shape and thus the response ${\cal R}_{w}(\uv,\lambda)$
-and the noise behaviour. If the same Fourier angular frequency mode is measured 
+the effective beam shape, hence the response ${\cal R}_{w}(\uv,\lambda)$
+and the noise behavior. If the same Fourier angular frequency mode is measured 
 by several receiver pairs, the raw instrument response might then be larger 
-that unity. This non normalized instrument response is used to compute the projected 
+that unity. This non-normalized instrument response is used to compute the projected 
 noise power spectrum in the following section (\ref{instrumnoise}). 
-We can also define a  normalized instrument response, ${\cal R}_{norm}(\uv,\lambda) \lesssim 1$ as:
-\begin{equation}
-{\cal R}_{norm}(\uv,\lambda) = {\cal R}(\uv,\lambda) / \mathrm{Max_{(\uv)}} \left[ {\cal R}(\uv,\lambda) \right] 
+We can also define a  normalized instrument response, ${\cal R}_{norm}(\uv,\lambda) \lesssim 1$ as
+\begin{equation}
+{\cal R}_{norm}(\uv,\lambda) = {\cal R}(\uv,\lambda) / \mathrm{Max_{(\uv)}} \left[ {\cal R}(\uv,\lambda) \right] \, .
 \end{equation} 
-This normalized  instrument response can be used to compute the effective instrument beam, 
-in particular in section \ref{recsec}.  
-
-{\changemark Detection of the reionisation at 21 cm has been an active field 
-in the last decade and different groups have built 
-instruments to detect reionisation signal around 100 MHz: LOFAR
-\citep{rottgering.06}, MWA (\cite{bowman.07}, \cite{lonsdale.09}) and PAPER \citep{parsons.09} . 
+This normalized  instrument response is the basic ingredient for computing the effective 
+instrument beam, in particular in section \ref{recsec}.  
+
+{\changemark Detection of the reionization at 21 cm has been an active field 
+in the last decade, and different groups have built 
+instruments to detect a reionization signal around 100 MHz: LOFAR
+\citep{rottgering.06}, MWA (\cite{bowman.07}; \cite{lonsdale.09}), and PAPER \citep{parsons.10}. 
 Several authors have studied the instrumental noise 
-and statistical uncertainties when measuring the reionisation signal power spectrum;
+and statistical uncertainties when measuring the reionization signal power spectrum, and 
 the methods presented here to compute the instrument response 
 and sensitivities are similar to the ones developed in these publications 
-(\cite{morales.04}, \cite{bowman.06}, \cite{mcquinn.06}). }
+(\cite{morales.04}; \cite{bowman.06}; \cite{mcquinn.06}). }
 
 \begin{figure}
@@ -591 +592 @@
 \subsection{Noise power spectrum computation}
 \label{instrumnoise}
-Let's consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency 
+We consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency 
 bandwidth $\delta \nu$ centered on $\nu_0$, with an integration time $t_{int}$, characterized by a system temperature 
 $\Tsys$. The uncertainty or fluctuations of this measurement due to the receiver noise can be written as 
-$\sigma_{noise}^2 = \frac{2 \Tsys^2}{t_{int} \, \delta \nu}$. This term 
-corresponds also to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated 
+$\sigma_{noise}^2 = \frac{2 \Tsys^2}{t_{int} \, \delta \nu}$. This term also
+corresponds to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated 
 noise. If the receiver has an effective area $A \simeq \pi D^2/4$ or $A \simeq D_x D_y$, the measurement 
-corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. The noise spectral density, in the angular frequencies plane (per unit area of angular frequencies  
+corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. 
+The noise's spectral density, in the angular frequency plane (per unit area of angular frequency
 \mbox{$\delta \uvu \times \delta \uvv$}), corresponding to a visibility 
-measurement from a pair of receivers can be written as:
+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} 
+\hspace{5mm} \mathrm{units:} \, \mathrm{K^2 \times rad^2}  \, .
 \label{eq:pnoisepairD}
 \end{eqnarray}
 
-We can characterize the sky temperature measurement with a radio instrument by the noise 
+We can characterize the sky temperature measurement with a radio instrument by the noise's 
 spectral power density in the angular frequencies plane $P_{noise}(\uv)$ in units of $\mathrm{Kelvin^2}$  
 per unit area of angular frequencies  $\delta \uvu \times \delta \uvv$. 
@@ -613 +615 @@
 might have the same baseline. The noise power density in the corresponding $(\uv)$ plane area 
 is then reduced by a factor $1/n$. More generally, we can write the instrument  noise 
-spectral power density using the instrument response defined in section \ref{instrumresp} :
-\begin{equation}
-P_{noise}(\uv) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(\uv,\lambda) } 
+spectral power density using the instrument response defined in section \ref{instrumresp} as
+\begin{equation}
+P_{noise}(\uv) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(\uv,\lambda) }  \hspace{4mm} .
 \label{eq:pnoiseuv}
 \end{equation} 
 
-When the intensity maps are projected in a three dimensional box in the universe and the 3D power spectrum 
+When the intensity maps are projected in a 3D box in the universe and the 3D power spectrum 
 $P(k)$ is computed, angles are translated into comoving transverse distances, 
 and frequencies or wavelengths into comoving radial distance, using the following relations
-{\changemarkb (e.g. \cite{cosmo.peebles} chap. 13, \cite{cosmo.rich})} :
+{\changemarkb (e.g. chap. 13 of \cite{cosmo.peebles}; \cite{cosmo.rich})} :
 { \changemark
 \begin{eqnarray}
@@ -638 +640 @@
 A brightness measurement at a  point $(\uv,\lambda)$, covering 
 the 3D spot  $(\delta \uvu, \delta \uvv, \delta \nu)$, would correspond 
-to cosmological power spectrum measurement at a transverse wave mode $(k_x,k_y)$
+to a cosmological power spectrum measurement at a transverse wave mode $(k_x,k_y)$
 defined by the equation \ref{eq:uvkxky}, measured at a redshift given by the observation frequency.
-The measurement noise spectral density given by the equation \ref{eq:pnoisepairD} can then be 
+The measurement noise spectral density given by the Eq. \ref{eq:pnoisepairD} can then be 
 translated into a 3D noise power spectrum, per unit of spatial frequencies 
 $ \delta k_x \times \delta k_y \times \delta k_z / 8 \pi^3 $ (units: $ \mathrm{K^2 \times Mpc^3}$) :  
@@ -657 +659 @@
 In the following paragraph, we will first consider an ideal instrument 
 with uniform $(\uv)$ coverage
-in order to establish the general noise power spectrum behaviour for cosmological 21 cm surveys.
+in order to establish the general noise power spectrum behavior for cosmological 21 cm surveys.
 The numerical method used to compute the 3D noise power spectrum is then presented in section 
 \ref{pnoisemeth}.
@@ -665 +667 @@
 { \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 
+The noise's spectral power  density from equation (\ref{eq:pnoisekxkz}) would then be  
+constant, independent of $(k_x, k_y, \ell_\parallel(\nu))$. Such a noise power spectrum thus  corresponds 
 to a 3D white noise, with a uniform noise spectral density:}
 \begin{equation}
@@ -672 +674 @@
 \label{ctepnoisek}
 \end{equation} 
-
-$P_{noise}$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
+%
+where $P_{noise}$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, 
 $t_{int}$ is the integration time expressed in second, 
 $\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and 
@@ -679 +681 @@
 
 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 
+with the number of observed Fourier modes, a number that is proportional to the volume of the universe 
+being observed (sample variance).  As the observed volume is proportional to the 
 surveyed solid angle, we  consider the survey of a fixed
 fraction of the sky, defined by  total solid angle $\Omega_{tot}$, performed during a given
 total observation time $t_{obs}$. 
-A single dish instrument with diameter $D$ would have an instantaneous field of view 
+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. 
@@ -690 +692 @@
 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  
+for the noise spectral power density by a single-dish instrument of diameter $D$:
+\begin{equation}
+P_{noise}^{survey}(k) = 2 \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  \hspace{2mm} .
 \end{equation}
 
-It is important to note that any real instrument do not have a flat 
+It is important to note that any real instrument does not have a flat 
 response in the $(\uv)$ plane, and the observations provide no information above 
 a certain maximum angular frequency $\uvu_{max},\uvv_{max}$. 
 One has to take into account either a damping of the observed sky power 
-spectrum or an increase of the noise spectral density if 
-the observed power spectrum is corrected for damping. The white noise 
+spectrum or an increase in the noise spectral density if 
+the observed power spectrum is corrected for damping. The white-noise 
 expressions given below should thus be considered as a lower limit or floor of the 
 instrument noise spectral density. 
  
-For a single dish instrument of diameter $D$ equipped with a multi-feed or 
-phase array receiver system, with $N$ independent beams on sky, 
+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  
+thanks to the  increase in per pointing integration time:
+
+\begin{equation}
+P_{noise}^{survey}(k) = \frac{2}{N} \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4  \hspace{2mm} .
 \label{eq:pnoiseNbeam}
 \end{equation}
-
+%
 This expression (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$ 
 identical receivers with a  total collection area $\sim D^2$. Such an array could be made for example 
-of $N=q \times q$ {\it small dishes}, each with diameter $D/q$, arranged as $q \times q$ square.   
+of $N=q \times q$ {\it small dishes}, each with diameter $D/q$, arranged as a $q \times q$ square.   
 
 For a single dish of diameter $D$, or an interferometric instrument with maximal extent $D$, 
@@ -723 +725 @@
 $k_{\perp}^{max}$:
 \begin{equation}
-k_{\perp}^{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} 
+k_{\perp}^{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} \hspace{3mm} .
 \label{kperpmax}
 \end{equation}   
-
+%
 Figure \ref{pnkmaxfz} shows the evolution of the noise spectral density $P_{noise}^{survey}(k)$
 as a function of redshift, for a radio survey of the sky, using an instrument with $N=100$
@@ -732 +734 @@
 The survey is supposed to cover a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$, in one 
 year. The maximum comoving wave number $k^{max}$  is also shown as a function 
-of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. In order 
-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}} 
+of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. 
+To take the radial component of $\vec{k}$ and the increase of 
+the instrument noise level with $k_{\perp}$ into account, we have taken the effective $k_{ max} $ 
+as half of the maximum transverse $k_{\perp} ^{max}$ of \mbox{Eq. \ref{kperpmax}}: 
+\begin{equation}
+k_{max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}}  \hspace{3mm} .
 \end{equation}
 
@@ -748 +750 @@
 }
 \vspace*{-40mm}
-\caption{Top: minimal noise level for a 100 beams instrument with \mbox{$\Tsys=50 \mathrm{K}$} 
-as a function of redshift (top), for a one year survey of a quarter of the sky. Bottom:
-maximum $k$ value for 21 cm LSS power spectrum measurement by  a 100 meter diameter 
+\caption{Top: minimal noise level for a 100-beam instrument with \mbox{$\Tsys=50 \mathrm{K}$} 
+as a function of redshift (top), for a one-year survey of a quarter of the sky. Bottom:
+maximum $k$ value for 21 cm LSS power spectrum measurement by  a 100-meter diameter 
 primary antenna. }
 \label{pnkmaxfz}
@@ -760 +762 @@
 We describe here the numerical method used to compute the 3D noise power spectrum, for a given instrument 
 response, as presented in section \ref{instrumnoise}. The noise power spectrum is a good indicator to compare 
-sensitivities for different instrument configurations, albeit the noise realization for a real instrument would not be 
+sensitivities for different instrument configurations, although the noise realization for a real instrument would not be 
 isotropic. 
 \begin{itemize}
-\item We start by a 3D Fourier coefficient grid, with the two first coordinates being the transverse angular wave modes,
-and the third being the frequency $(k_x,k_y,\nu)$. The grid is positioned at the mean redshift $z_0$ for which 
+\item We start by a 3D Fourier coefficient grid, with the two first coordinates the transverse angular wave modes,
+and the third the frequency $(k_x,k_y,\nu)$. The grid is positioned at the mean redshift $z_0$ for which 
 we want to compute $P_{noise}(k)$. For the results at redshift \mbox{$z_0=1$} discussed in section  \ref{instrumnoise}, 
 the grid cell size and dimensions have been chosen to represent a box in the universe  
 \mbox{$\sim 1500 \times 1500 \times 750 \, \mathrm{Mpc^3}$}, 
 with \mbox{$3\times3\times3 \, \mathrm{Mpc^3}$} cells.
-This correspond to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given 
+This corresponds to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given 
 the small angular extent, we have neglected the curvature of redshift shells. 
 \item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. 
@@ -775 +777 @@
 using equation (\ref{eq:uvkxky}), and the 
 angular diameter distance $\dang(z)$ for \LCDM model with standard WMAP parameters \citep{komatsu.11}. 
-The noise variance is taken proportional to $P_{noise}$ :
-\begin{equation}
-\sigma_{re}^2=\sigma_{im}^2 \propto \frac{1}{{\cal R}_{raw}(\uv,\lambda)} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 
+The noise variance is taken proportional to $P_{noise}$ 
+\begin{equation}
+\sigma_{re}^2=\sigma_{im}^2 \propto \frac{1}{{\cal R}_{raw}(\uv,\lambda)} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \hspace{2mm} .
 \end{equation}
-\item an FFT is then performed in the frequency or redshift direction to obtain the noise Fourier 
-complex coefficients ${\cal F}_n(k_x,k_y,k_z)$ and the power spectrum is estimated as :
+\item An FFT is then performed in the frequency or redshift direction to obtain the noise Fourier 
+complex coefficients ${\cal F}_n(k_x,k_y,k_z)$ and the power spectrum is estimated as 
 \begin{equation}
 \tilde{P}_{noise}(k) = < | {\cal F}_n(k_x,k_y,k_z) |^2 >  \hspace{2mm}  \mathrm{for} \hspace{2mm} 
-  \sqrt{k_x^2+k_y^2+k_z^2} = k 
+  \sqrt{k_x^2+k_y^2+k_z^2} = k  \hspace{2mm} .
 \end{equation}
 Noise samples corresponding to small instrument response, typically less than 1\% of the 
-maximum instrument response are ignored when calculating  $\tilde{P}_{noise}(k)$. 
-However, we require to have a significant fraction, typically 20\% to 50\% of all possible modes 
+maximum instrument response, are ignored when calculating  $\tilde{P}_{noise}(k)$. 
+However, we require a significant fraction, typically 20\% to 50\% of all possible modes 
 $(k_x,k_y,k_z)$ measured for a given $k$ value. 
 \item the above steps are repeated $\sim$ 50 times to decrease the statistical fluctuations 
@@ -802 +804 @@
 
 It should be noted that it is possible to obtain a  good approximation of the noise 
-power spectrum shape, neglecting the redshift or frequency dependence of the 
+power spectrum shape by neglecting the redshift or frequency dependence of the 
 instrument response function and $\dang(z)$ for a small redshift interval around $z_0$,
 using a fixed  instrument response ${\cal R}(\uv,\lambda(z_0))$ and 
-a constant radial distance $\dang(z_0)*(1+z_0)$. 
+a constant radial distance $\dang(z_0)\times(1+z_0)$:
 \begin{equation}
 \tilde{P}_{noise}(k) = < |  {\cal F}_n (k_x,k_y,k_z) |^2 > \simeq < P_{noise}(\uv, k_z) > 
@@ -821 +823 @@
 \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 
-$55 \times 55 \, \mathrm{m^2}$ 
+$55 \times 55 \, \mathrm{m^2}$ \, .
 \item [{\bf b} :] An array of $n=128  \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged 
-in 8 rows, each with 16 dishes. These 128 dishes are spread over an area 
+in eight rows, each with 16 dishes. These 128 dishes are spread over an area 
 $80 \times 80  \, \mathrm{m^2}$. The array layout for this configuration is
 shown in figure \ref{figconfbc}. 
 \item [{\bf c} :] An array of $n=129  \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged 
  over an area $80 \times 80  \, \mathrm{m^2}$. This configuration has in 
-particular 4 sub-arrays of packed 16 dishes ($4\times4$), located in the 
-four array corners. This array layout is also shown figure \ref{figconfbc}. 
-\item [{\bf d} :] A single dish instrument, with diameter $D=75 \, \mathrm{m}$, 
+particular four subarrays of packed 16 dishes ($4\times4$), located in the 
+four array corners. This array layout is also shown in figure \ref{figconfbc}. 
+\item [{\bf d} :] A single-dish instrument, with diameter $D=75 \, \mathrm{m}$, 
 equipped with a 100 beam focal plane receiver array. 
 \item[{\bf e} :] A packed array of $n=400 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in 
 a square $20 \times 20$ configuration ($q=20$). This array covers an area of 
 $100 \times 100 \, \mathrm{m^2}$ 
-\item[{\bf f} :] A packed array of 4 cylindrical reflectors, each 85 meter long and 12 meter 
+\item[{\bf f} :] A packed array of four cylindrical reflectors, each 85 meter long and 12 meter 
 wide. The focal line of each cylinder is equipped with 100 receivers, each 
 $2 \lambda$ long, corresponding to $\sim 0.85 \, \mathrm{m}$ at $z=1$. 
 This array covers an area of $48 \times 85 \, \mathrm{m^2}$, and have 
-a total of $400$ receivers per polarisation, as in the (e) configuration.
-We have computed the noise power spectrum for {\em perfect} 
+a total of $400$ receivers per polarization, as in the (e) configuration.
+We computed the noise power spectrum for {\em perfect} 
 cylinders, where all receiver pair correlations are used (fp), or for 
-a non perfect instrument, where only correlations between receivers 
+an imperfect instrument, where only correlations between receivers 
 from different cylinders are used.
-\item[{\bf g} :] A packed array of 8 cylindrical reflectors, each 102 meter long and 12 meter 
+\item[{\bf g} :] A packed array of eight cylindrical reflectors, each 102 meters long and 12 meters 
 wide. The focal line of each cylinder is equipped with 120 receivers, each 
 $2 \lambda$ long, corresponding to $\sim 0.85 \, \mathrm{m}$ at $z=1$. 
 This array covers an area of $96 \times 102 \, \mathrm{m^2}$ and has 
-a total of 960  receivers per polarisation. As for the (f) configuration,  
+a total of 960  receivers per polarization. As for the (f) configuration,  
 we have computed the noise power spectrum for {\em perfect} 
 cylinders, where all receiver pair correlations are used (gp), or for 
-a non perfect instrument, where only correlations between receivers 
+an imperfect instrument, where only correlations between receivers 
 from different cylinders are used.
 \end{itemize}
@@ -868 +870 @@
 \end{figure}
 
-We have used simple triangular shaped dish response in the $(\uv)$ plane. 
-However, we have introduced a filling factor or illumination efficiency 
+We used simple triangular shaped dish response in the $(\uv)$ plane;
+however, we did introduce a filling factor or illumination efficiency 
 $\eta$, relating the effective dish diameter $D_{ill}$ to the 
 mechanical dish size $D_{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales 
@@ -878 +880 @@
 \hspace{4mm} \theta=\sqrt{\alpha^2+\beta^2}
 \end{eqnarray}
-For the multi-dish configuration studied here, we have taken the illumination efficiency factor
+For the multidish configuration studied here, we have taken the illumination efficiency factor
 {\bf $\eta = 0.9$}. 
 
-For the receivers along the focal line of cylinders, we have assumed that the 
+For the receivers along the focal line of cylinders, we assumed that the 
 individual receiver response in the $(\uv)$ plane corresponds to a 
-rectangular shaped antenna. The illumination efficiency factor has been taken 
+rectangular antenna. The illumination efficiency factor was taken 
 equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$ 
-along the cylinder length. {\changemark  We have used double triangular shaped
+along the cylinder length. {\changemark  We used a double triangular 
 response function in the $(\uv)$ plane for each of the receiver elements along the cylinder:
 \begin{equation}
  {\cal L}_\Box(\uv,\lambda)  = 
 \bigwedge_{[\pm \eta_x D_x / \lambda]} (\uvu ) \times
-\bigwedge_{[\pm \eta_y D_y / \lambda ]} (\uvv ) 
+\bigwedge_{[\pm \eta_y D_y / \lambda ]} (\uvv )  
 \end{equation}
 }
-It should be noted that the small angle approximation 
+
+\noindent It should be noted that the small angle approximation 
 used here for the expression of visibilities is not valid for the receivers along 
 the cylinder axis. However, some preliminary numerical checks indicate that 
-the results 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 
+the results for the noise spectral power density  would not change significantly.
+The instrument responses shown here correspond to a fixed pointing toward the zenith, which 
 is the case for a transit type telescope.
 
 Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}(\uv,\lambda)$ 
 for the four configurations (a,b,c,d) with $\sim 100$ receivers per 
-polarisation. 
-
-{\changemark Using the numerical method sketched in section \ref{pnoisemeth}, we have 
+polarization. 
+{\changemark Using the numerical method sketched in section \ref{pnoisemeth}, we 
 computed the 3D noise power spectrum for each of the eight instrument configurations presented
 here, with a system noise temperature $\Tsys = 50 \mathrm{K}$, for a one year survey 
@@ -910 +912 @@
 The resulting noise spectral power densities are shown in figure 
 \ref{figpnoisea2g}. The increase of $P_{noise}(k)$ at low $k^{comov} \lesssim 0.02$ 
-is due to the fact that we have ignored all auto-correlation measurements.  
+is due to our having ignored all auto-correlation measurements.  
 It can be seen that an instrument with $100-200$ beams and $\Tsys = 50 \mathrm{K}$ 
 should have enough sensitivity to map LSS in 21 cm at redshift z=1.
@@ -922 +924 @@
 \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 
+(a) 121 $D_{dish}=$ 5-meter diameter dishes arranged in a compact, square array 
 of $11 \times 11$, (b) 128 dishes arranged in 8 rows of 16 dishes each (fig. \ref{figconfbc}), 
-(c) 129 dishes arranged as shown in figure \ref{figconfbc} , (d) single D=75 meter diameter, with 100 beams. 
+(c) 129 dishes arranged as shown in figure \ref{figconfbc}, (d) single D=75 meter diameter, with 100 beams. 
 The common color scale (1 \ldots 80) is shown on the right. }
 \label{figuvcovabcd}
@@ -939 +941 @@
 \caption{P(k) 21 cm LSS power spectrum at redshift $z=1$ with $\gHI=2\%$ 
 and the noise power spectrum for several interferometer configurations
- ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers. The noise power spectrum has been 
+ ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400, and 960 receivers. The noise power spectrum has been 
 computed for all configurations assuming a survey of a quarter of the sky over one year, 
 with a system temperature $\Tsys = 50 \mathrm{K}$. }
@@ -946 +948 @@
 
 
-\section{ Foregrounds and Component separation }
+\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 
-Milky Way and the extra galactic radio sources are a thousand times brighter than the 
+Reaching the required sensitivities is not the only difficulty of observing the 
+LSS in 21 cm. Indeed, the synchrotron emission of the 
+Milky Way and the extragalactic radio sources are a thousand times brighter than the 
 emission of the neutral hydrogen distributed in the universe. Extracting the LSS signal
-using Intensity Mapping, without identifying the \HI point sources is the main challenge 
+using intensity mapping, without identifying the \HI point sources is the main challenge 
 for this novel observation method. Although this task might seem impossible at first, 
 it has been suggested that the smooth frequency dependence of the synchrotron
@@ -958 +960 @@
 emissions. {\changemark Discussion of contribution of different sources 
 of radio foregrounds for measurement of reionization through redshifted 21 cm,
-as well foreground subtraction using their smooth frequency dependence can 
-be found in (\cite{shaver.99}, \cite{matteo.02},\cite{oh.03}).}
-However, any real radio instrument has a beam shape 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 some time referred to as {\em 
-mode mixing}.  {\changemark The effect of the frequency dependent beam shape on foreground subtraction 
+as well as foreground subtraction using their smooth frequency dependence, can 
+be found in (\cite{shaver.99}; \cite{matteo.02};\cite{oh.03}).}
+However, any real radio instrument has a beam shape that changes with 
+frequency, and this instrumental effect significantly increases the difficulty and complexity of this component separation 
+technique. The effect of frequency dependent beam shape is sometimes referred to as {\em 
+mode mixing},  {\changemark and its impact on foreground subtraction 
 has been discussed for example in \cite{morales.06}.}
 
 In this section, we present a short description of the foreground emissions and 
-the simple models we have used for computing the sky radio emissions in the GHz frequency 
-range. We present also a simple component separation method to extract the LSS signal and 
+the simple models we used for computing the sky radio emissions in the GHz frequency 
+range. We also present a simple component-separation method to extract the LSS signal and 
 its performance. {\changemark The analysis presented here follows a similar path to 
-a detailed foreground subtraction study carried for MWA at $\sim$ 150 MHz by \cite{bowman.09}. } 
-We compute in particular the effect of the instrument response on the recovered 
+a detailed foreground subtraction study carried out for MWA at $\sim$ 150 MHz by \cite{bowman.09}. } 
+We computed in particular, the effect of the instrument response on the recovered 
 power spectrum. The results presented in this section concern the 
 total sky emission and the LSS 21 cm signal extraction in the $z \sim 0.6$ redshift range,
@@ -977 +979 @@
  
 \subsection{ Synchrotron and radio sources }
-We have modeled the radio sky in the form of three dimensional maps (data cubes) of sky temperature 
+We modeled the radio sky in the form of three 3D maps (data cubes) of sky temperature 
 brightness $T(\alpha, \delta, \nu)$ as a function of two equatorial angular coordinates $(\alpha, \delta)$ 
 and the frequency $\nu$. Unless otherwise specified, the results presented here are based on simulations of 
 $90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10\mathrm{h}00\mathrm{m} , \delta=+10 \, \mathrm{deg.}$, and  covering 128 MHz in frequency. We have selected this particular area of the sky  in order to minimize 
 the Galactic synchrotron foreground. The sky cube characteristics (coordinate range, size, resolution) 
-used in the simulations are given in the table \ref{skycubechars}.
+used in the simulations are given in the Table \ref{skycubechars}.
 \begin{table}
 \caption{
@@ -1007 +1009 @@
 \end{tabular} 
 \end{center}
-\tablefoot{ Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$ ; 
+\tablefoot{ Cube size: $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$; 
 $1800 \times 600 \times 256 \simeq 123 \times 10^6$ cells }
 \end{table}
 %%%%
 \par 
-Two different methods 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 
-maps of the emission temperature at different frequencies, from which we have 
+Two different methods were used to compute the sky temperature data cubes.
+We used the global sky model (GSM) \citep{gsm.08} tools to generate full sky 
+maps of the emission temperature at different frequencies, from which we 
 extracted the brightness temperature cube for the region defined above 
 (Model-I/GSM $T_{gsm}(\alpha, \delta, \nu)$).
-As the GSM maps have an intrinsic resolution of $\sim$ 0.5 degree, it is 
+Because the GSM maps have an intrinsic resolution of $\sim$ 0.5 degree, it is 
 difficult to have reliable results for the effect of point sources on the reconstructed 
 LSS power spectrum.
 
-We have thus made also a simple sky model using the Haslam Galactic synchrotron map
+We have thus also made a simple sky model using the Haslam Galactic synchrotron map
 at 408 MHz \citep{haslam.82} and the NRAO VLA Sky Survey (NVSS) 1.4 GHz radio source 
 catalog \citep{nvss.98}. The sky temperature cube in this model (Model-II/Haslam+NVSS) 
-has been computed through the following steps:
+was computed through the following steps:
 
 \begin{enumerate}
-\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 
+\item The Galactic synchrotron emission is modeled as a power law with a spatially 
+varying spectral index. We assign a power law index $\beta = -2.8  \pm 0.15$ to each sky direction,
+where $\beta$ has a Gaussian distribution centered on -2.8 with a standard 
 deviation $\sigma_\beta = 0.15$. {\changemark The 
-diffuse radio background spectral index has been measured  for example by 
-\cite{platania.98} or \cite{rogers.08}.}
+diffuse radio background spectral index has been measured, for example, by 
+\citep{platania.98} or \citep{rogers.08}.}
 The synchrotron contribution to the sky temperature for each cell is then 
 obtained  through the formula:
@@ -1039 +1041 @@
 \end{equation}
 %%
-\item A two dimensional $T_{nvss}(\alpha,\delta)$ sky brightness temperature at 1.4 GHz is computed 
+\item A 2D $T_{nvss}(\alpha,\delta)$ sky brightness temperature at 1.4 GHz is computed 
 by projecting the radio sources in the NVSS catalog to a grid with the same angular resolution as 
 the sky cubes. The source brightness in Jansky is converted to temperature taking the 
 pixel angular size into account ($ \sim 21 \mathrm{mK/mJy}$ at 1.4 GHz and $3' \times 3'$ pixels).  
 A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source 
-map; we have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the 
-contribution of the radiosources to the sky temperature is computed as follows:
-\begin{equation}
- T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}} 
+map. We have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the 
+contribution of the radiosources to the sky temperature is computed as:
+\begin{equation}
+ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}}  
 \end{equation}
 %%
@@ -1057 +1059 @@
 \end{enumerate}
 
- The 21 cm temperature fluctuations due to neutral hydrogen in large scale structures
-$T_{lss}(\alpha, \delta, \nu)$  have been computed using the 
-SimLSS \footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }  software package: 
+ The 21 cm temperature fluctuations due to neutral hydrogen in LSS
+$T_{lss}(\alpha, \delta, \nu)$ were computed using the 
+SimLSS\footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }  software package, where
 %
 complex normal Gaussian fields were first generated in Fourier space.
 The amplitude of each mode was then multiplied by the square root
 of the power spectrum $P(k)$ at $z=0$ computed according to the parametrization of 
-\citep{eisenhu.98}. We have used the standard cosmological parameters, 
+\citep{eisenhu.98}. We used the standard cosmological parameters, 
  $H_0=71 \, \mathrm{km/s/Mpc}$, $\Omega_m=0.264$, $\Omega_b=0.045$,
 $\Omega_\lambda=0.73$ and $w=-1$ \citep{komatsu.11}.
 An inverse FFT was then performed to compute the matter density fluctuations $\delta \rho / \rho$ 
 in the linear regime, 
-in a box of $3420 \times 1140 \times 716  \, \mathrm{Mpc^3}$ and evolved
+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
@@ -1076 +1078 @@
 sky cube angular and frequency resolution defined above.  
 {\changemarkb 
-We haven't taken into account the curvature of redshift shells when 
+We did not take the curvature of redshift shells into account when 
 converting SimLSS euclidean coordinates to angles and frequency coordinates 
 of the sky cubes analyzed here. This approximate treatment causes distortions visible at large angles $\gtrsim 10^\circ$. 
@@ -1095 +1097 @@
 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}).
+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. 
+with Haslam+NVSS map, smoothed with a 35 arcmin Gaussian beam. 
 Figure \ref{compgsmhtemp} shows the comparison of the sky cube temperature distribution 
 for Model-I/GSM and Model-II. There is good agreement between the two models, although 
@@ -1122 +1124 @@
 \end{table}
 
-we have computed the power spectrum for the 21cm-LSS sky temperature cube, as well 
+We computed the power spectrum for the 21cm-LSS sky temperature cube, as well 
 as for the radio foreground temperature cubes obtained from the two 
-models. We have also computed the power spectrum on sky brightness temperature
-cubes, as measured by a perfect instrument having a 25 arcmin (FWHM) gaussian beam.
-The resulting computed power spectra are shown on figure \ref{pkgsmlss}. 
-The GSM model has more large scale power compared to our simple Haslam+NVSS model, 
-while it lacks power at higher spatial frequencies. The mode mixing due to 
-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
-$\sim 10^6$ times higher than the 21 cm signal from large scale structures. This corresponds 
-to the factor $\sim 10^3$ of the sky brightness temperature fluctuations ($\sim$ K),
+models. We also computed the power spectrum on sky brightness temperature
+cubes, as measured by a perfect instrument having a 25 arcmin (FWHM) Gaussian beam.
+The resulting computed power spectra are shown in figure \ref{pkgsmlss}. 
+The GSM model has more large-scale power compared to our simple Haslam+NVSS model, 
+while it lacks power at higher spatial frequencies. The mode mixing due to a
+frequency-dependent response will thus be stronger in Model-II (Haslam+NVSS) 
+case. It can also be seen that the radio foreground's power spectrum is more than
+$\sim 10^6$ times higher than the 21 cm signal from LSS. This corresponds 
+to the factor $\sim 10^3$ of the sky brightness temperature fluctuations (\mbox{$\sim$ K}),
 compared to the mK LSS signal.  
 
-{ \changemark Contrary to most similar studies, where it is assumed that bright sources 
+{ \changemark In contrast to most similar studies, where it is assumed that bright sources 
 can be nearly perfectly subtracted, our aim was to compute also their 
 effect in the foreground subtraction process. 
-The GSM model lacks the angular resolution needed to compute 
-correctly the effect of bright compact sources for 21 cm LSS observations and 
+The GSM model lacks the angular resolution needed to correctly compute 
+the effect of bright compact sources for 21 cm LSS observations and 
 the mode mixing due to the frequency dependence of the instrumental response, 
 while Model-II provides a reasonable description of these compact sources. Our simulated 
 sky cubes have an angular resolution $3'\times3'$, well below the typical 
 $15'$ resolution of the instrument configuration considered here. 
-However, Model-II might lack spatial structures at large scales, above a degree, 
+However, Model-II might lack spatial structures on large scales, above a degree, 
 compared to Model-I/GSM, and the frequency variations as a simple power law
 might not be realistic enough. The differences for the two sky models can be seen
 in their power spectra shown in figure \ref{pkgsmlss}. The smoothing or convolution with 
-a 25' beam has negligible effect of the GSM power spectrum, as it originally lacks 
+a 25' beam has negligible effect on the GSM power spectrum, since it originally lacks 
 structures below 0.5 degree. By using 
-these two models, we have explored some of the systematic uncertainties 
+these two models, we explored some of the systematic uncertainties 
 related to foreground subtraction.}
 
@@ -1158 +1160 @@
 increases at high k values (small scales). In practice, clean deconvolution is difficult to 
 implement for real data and the power spectra presented in this section are NOT corrected 
-for the instrumental response.  The observed structures have thus a scale dependent damping
-according to the instrument response, while the instrument noise is flat (white noise or scale independent). 
+for the instrumental response.  The observed structures thus have a scale-dependent damping
+according to the instrument response, while the instrument noise is flat (white noise or scale-independent). 
 
 \begin{figure}
@@ -1171 +1173 @@
 \caption{Comparison of GSM (black) and Model-II (red) sky cube temperature distribution.
 The Model-II (Haslam+NVSS),
-has been smoothed with a 35 arcmin gaussian beam. }
+has been smoothed with a 35 arcmin Gaussian beam. }
 \label{compgsmhtemp}
 \end{figure}
@@ -1182 +1184 @@
 }
 \caption{Comparison of GSM (top) and Model-II (bottom) sky maps at 884 MHz. 
-The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin (FWHM) gaussian beam.}
+The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin (FWHM) Gaussian beam.}
 \label{compgsmmap}
 \end{figure*}
@@ -1198 +1200 @@
 The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple
 model based on Haslam+NVSS (Model-II, black). The curves with circle markers show the power spectrum 
-as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam. This beam has 
-negligible effect on the GSM/Model-I power spectrum, as GSM has no structures below $\sim 0.5^\circ$. 
+as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam. 
 }
 \label{pkgsmlss}
@@ -1213 +1214 @@
 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 $\lambda_k=c/\nu_k$ : 
+For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$: 
 \begin{enumerate}
 \item Apply a 2D Fourier transform to compute sky angular Fourier amplitudes
-$$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(\uv, \lambda_k)$$ 
+$$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(\uv, \lambda_k) \hspace{2mm} .$$ 
 \item Apply instrument response in the angular wave mode plane. We use here the normalized instrument response
-$ {\cal R}(\uv,\lambda_k)  \lesssim 1$. 
-$$  {\cal T}_{sky}(\uv, \lambda_k)  \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda_k) $$
-\item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map, 
+$ {\cal R}(\uv,\lambda_k)  \lesssim 1$
+$$  {\cal T}_{sky}(\uv, \lambda_k)  \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda_k) \hspace{1mm} . $$
+\item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map
 without instrumental (electronic/$\Tsys$) white noise:
 $$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda)   
 \rightarrow \mathrm{Inv-2D-FFT} \rightarrow T_{mes1}(\alpha, \delta, \lambda_k) $$ 
-\item Add white noise (gaussian fluctuations) to the pixel map temperatures to obtain 
+\item Add white noise (Gaussian fluctuations) to the pixel map temperatures to obtain 
 the measured sky brightness temperature $T_{mes}(\alpha, \delta, \nu_k)$. 
 {\changemark The white noise hypothesis is reasonable at this level, since $(\uv)$ 
 dependent instrumental response has already been applied.} 
-We have also considered that the system temperature and thus the 
-additive white noise level was independent of the frequency or wavelength.   
+We also considered that the system temperature, and thus the 
+additive white noise level, was independent of the frequency or wavelength.   
 \end{enumerate} 
-The LSS signal extraction performance depends obviously on the white noise level. 
+The LSS signal extraction performance obviously depends on the white noise level. 
 The results shown here correspond to the (a) instrument configuration, a packed array of 
 $11 \times 11 = 121$ dishes  (5 meter diameter), with a white noise level corresponding 
@@ -1237 +1238 @@
 cell. \\[1mm]
 
-The different steps of the simple component separation procedure that has been applied are 
+The different steps in the simple component separation procedure that has been applied are 
 briefly described here.
 \begin{enumerate}
 \item The measured sky brightness temperature is first {\em corrected} for the frequency dependent 
 beam effects through a convolution by a fiducial frequency independent beam ${\cal R}_f(\uv)$ This {\em correction}
-corresponds to a smearing or degradation of the angular resolution.
+corresponds to a smearing or degradation of the angular resolution
 \begin{eqnarray*} 
  {\cal T}_{mes}(u, v, \lambda_k) & \rightarrow & {\cal T}_{mes}^{bcor}(u, v, \lambda_k) \\  
  {\cal T}_{mes}^{bcor}(u, v, \lambda_k)  & = & 
 {\cal T}_{mes}(u, v, \lambda_k) \times \sqrt{ \frac{{\cal R}_f(\uv)}{{\cal R}(\uv,\lambda)} } \\
-{\cal T}_{mes}^{bcor}(u, v, \lambda_k)  & \rightarrow & \mathrm{2D-FFT} \rightarrow  T_{mes}^{bcor}(\alpha,\delta,\lambda) 
+{\cal T}_{mes}^{bcor}(u, v, \lambda_k)  & \rightarrow & \mathrm{2D-FFT} \rightarrow  T_{mes}^{bcor}(\alpha,\delta,\lambda) \hspace{2mm} . 
 \end{eqnarray*}
 {\changemark 
@@ -1255 +1256 @@
 attempt to represent imperfect knowledge of the instrument response.
 We recall that this is the normalized instrument response,
-${\cal R}(\uv,\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$  has also a numerical upper bound $\sim 100$. }
+${\cal R}(\uv,\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$ 
+also has a numerical upper bound $\sim 100$. }
 \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
- is fitted to the beam-corrected brightness temperature. The parameters have been obtained 
+ is fitted to the beam-corrected brightness temperature. The parameters were obtained 
 using a linear $\chi^2$ fit in the $\lgd ( T ) , \lgd (\nu)$ plane. 
 We show here the results for a pure power law (P1), as well as a modified power law (P2):
@@ -1264 +1266 @@
 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 
+where $b$ is the power law index and  $T_0 = 10^a$ the brightness temperature at the 
 reference frequency $\nu_0$.
 
-{\changemark Interferometers have poor response at small $(\uv)$ corresponding to baselines 
-smaller than interferometer element size. The zero spacing baseline, the $(\uv)=(0,0)$ mode,  represents 
-the mean temperature for a given frequency plane and can not be measured with interferometers. 
-We have used a simple trick to make the power law fitting procedure applicable: 
-we have set the mean value of the temperature for 
+{\changemark Interferometers have a poor response at small $(\uv)$ corresponding to baselines 
+smaller than interferometer element size. The zero-spacing baseline, the $(\uv)=(0,0)$ mode,  represents 
+the mean temperature for a given frequency plane and cannot be measured with interferometers. 
+We used a simple trick to make the power-law fitting procedure applicable,
+by setting the mean value of the temperature for 
 each frequency plane according to a power law with an index close to the synchrotron index 
-($\beta\sim-2.8$) and we have checked that the results are not too sensitive to the 
+($\beta\sim-2.8$). And we checked that the results are not too sensitive to the 
 arbitrarily fixed mean temperature power law parameters. } 
 
@@ -1283 +1285 @@
 for the two radio sky models used here: GSM/Model-I and Haslam+NVSS/Model-II. The 
 21 cm LSS power spectrum, as seen by a perfect instrument with a 25 arcmin (FWHM) 
-gaussian frequency independent beam is shown 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).
+Gaussian frequency independent beam is shown, as well as  
+the extracted power spectrum, after beam {\em correction} 
+and foreground separation with second order polynomial fit (P2).
 We have also represented the obtained power spectrum without applying the beam correction (step 1 above),
-or with the first order polynomial fit (P1). 
+or with the first-order polynomial fit (P1). 
 
 Figure \ref{extlssmap} shows a comparison of  the original 21 cm brightness temperature map at 884 MHz 
-with the recovered 21 cm map, after subtraction of the radio continuum component. It can be seen that structures 
+with the recovered 21 cm map, after subtracting the radio continuum component. It can be seen that structures 
 present in the original map have been correctly recovered, although the amplitude of the temperature 
 fluctuations on the recovered map is significantly smaller (factor $\sim 5$) than in the original map. 
-{\changemark This is mostly due to the damping of the large scale power ($k \lesssim 0.1 h \mathrm{Mpc^{-1}} $) 
+{\changemark This is mostly due to the damping of the large-scale power ($k \lesssim 0.1 h \mathrm{Mpc^{-1}} $) 
 due to the foreground subtraction procedure (see figure \ref{extlssratio}).}
 
-We have shown that it should be possible to measure the red shifted 21 cm emission fluctuations in the 
-presence of the strong radio continuum signal, provided that this latter has a smooth frequency dependence.
+We have shown that it should be possible to measure the red-shifted 21 cm emission fluctuations in the 
+presence of the strong radio continuum signal, provided that the latter has a smooth frequency dependence.
 However, a rather  precise knowledge of the instrument beam and the beam {\em correction} 
-or smearing procedure described here  are key 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 
+or smearing procedure described here  are key ingredients for recovering the 21 cm LSS power spectrum. 
+It is also important to note that, while it is enough to correct the beam to the lowest resolution instrument beam 
 ($\sim 30'$ or $D \sim 50$ meter @ 820 MHz) for the GSM sky model, a stronger beam correction 
-has to be applied (($\sim 36'$ or $D \sim 40$ meter @ 820 MHz) for the Model-II to reduce 
+has to be applied ($\sim 36'$ or $D \sim 40$ meter @ 820 MHz) for Model-II to reduce 
 significantly the ripples from bright radio sources. 
 We have also applied the same procedure to simulate observations and LSS signal extraction for an instrument 
-with a frequency dependent gaussian beam shape. The mode mixing effect is greatly reduced for
+with a frequency-dependent Gaussian beam shape. The mode mixing effect is greatly reduced for
 such a smooth beam, compared to the more complex instrument response 
 ${\cal R}(\uv,\lambda)$ used for the results shown in figure \ref{extlsspk}.
@@ -1322 +1324 @@
 Left: GSM/Model-I , right: Haslam+NVSS/Model-II. The black curve shows the residual after foreground subtraction,
 corresponding to the 21 cm signal, WITHOUT applying the beam correction. The red curve shows the recovered 21 cm 
-signal power spectrum, for P2 type fit of the frequency dependence of the radio continuum, and violet curve is the P1 fit (see text). The orange curve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency independent gaussian beam. }
+signal power spectrum, for P2 type fit of the frequency dependence of the radio continuum, and violet curve is the P1 fit (see text). The orange curve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency-independent Gaussian beam. }
 \label{extlsspk}
 \end{figure*}
@@ -1336 +1338 @@
 \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. }
+with 25 arc.min (FWHM) beam (top), and the recovered LSS map, after foreground subtraction for Model-I (GSM) (bottom),  for the instrument configuration (a), $11\times11$ packed array interferometer. }
 \label{extlssmap}
 \end{figure*}
@@ -1346 +1347 @@
  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
+We recall that we have neglected the curvature of redshift or frequency shells
 in this numerical study, which affect our result at large angles $\gtrsim 10^\circ$. 
-The results presented here and our conclusions are thus restricted to wave mode range 
+The results presented here and our conclusions are thus restricted to the wave-mode range 
 $k \gtrsim 0.02 \mathrm{h \, Mpc^{-1}}$. 
 } 
-We expect damping at small scales, or larges $k$, due to the finite instrument size, but also at large scales, small $k$,
+We expect damping on small scales, or large $k$, due to the finite instrument size, but also on large scales, small $k$,
 if total power measurements (auto-correlations) are not used in the case of interferometers. 
 The sky reconstruction and the component separation introduce additional filtering and distortions.
 The real transverse plane transfer function might even be anisotropic. 
 
-However, in the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the 
+However, within the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the 
 recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$
-{\changemarkb , similar to the one defined by \cite{bowman.09} , equation (23):}
-\begin{equation}
-\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) 
+{\changemarkb , similar to the one defined by \cite{bowman.09}, equation (23):}
+\begin{equation}
+\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) \hspace{3mm} .
 \end{equation}
 
 Figure \ref{extlssratio} shows this overall transfer function for the simulations and component 
-separation performed here, around $z \sim 0.6$, for the instrumental setup (a), a filled array of 121 $D_{dish}=5$ m dishes. {\changemark This transfer function has been obtained after averaging the reconstructed  
+separation performed here, around $z \sim 0.6$, for the instrumental setup (a), 
+a filled array of 121 $D_{dish}=5$ m dishes. {\changemark This transfer function has been obtained after averaging the reconstructed  
 $ P_{21}^{rec}(k)$ for several realizations (50) of the LSS temperature field.
 The black curve shows the ratio $\TrF(k)=P_{21}^{beam}(k)/P_{21}(k)$ of the computed to the original 
@@ -1370 +1372 @@
 target beam FWHM=30'. This black curve shows the damping effect due to the finite instrument size at 
 small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}, \theta \lesssim 1^\circ$).  
-The red curve shows the transfer function for the GSM foreground model (Model-I) and  the $11\times11$ filled array 
-interferometer (setup (a)), while the dashed red curve represents the transfer function for a D=55 meter 
+The transfer function for the GSM foreground model (Model-I) and  the $11\times11$ filled array 
+interferometer (setup (a)) is represented, as well as  the transfer function for a D=55 meter 
 diameter dish. The transfer function for the Model-II/Haslam+NVSS and the setup (a) filled interferometer 
-array is also shown (orange curve). The recovered power spectrum suffers also significant damping at large 
-scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}, $, mostly due to the filtering of radial or 
+array is also shown. The recovered power spectrum also suffers significant damping on large 
+scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}$, mostly due to the filtering of radial or 
 longitudinal Fourier modes along the frequency or redshift direction ($k_\parallel$) 
-by the component separation algorithm. We have been able to remove the ripples on the reconstructed 
+by the component separation algorithm. We were able to remove the ripples on the reconstructed 
 power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$36' 
 target beam resolution, compared to $\sim$30' for the GSM model. This explains the lower transfer function
-obtained for Model-II at small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}$). }
+obtained for Model-II on small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}$). }
 
  It should be stressed that the simulations presented in this section were 
-focused on the study of the radio foreground effects and have been carried 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).
-
+focused on the study of the radio foreground effects and have been carried 
+intentionally with a very low instrumental noise level of 
+$0.25$ mK per pixel, corresponding to several years of continuous 
+observations ($\sim 10$ hours per $3' \times 3'$ pixel).
+%
 This transfer function is well represented by the analytical form:
 \begin{equation}
-\TrF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right) 
+\TrF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right)  \hspace{1mm} .
 \label{eq:tfanalytique}
 \end{equation} 
 
-We have performed  simulation of observations and radio foreground subtraction using 
+We simulated observations and radio foreground subtraction using 
 the procedure described here for different redshifts and instrument configurations, in particular 
 for the (e) configuration with 400 five-meter dishes. As the synchrotron and radio source strength 
 increases quickly with decreasing frequency, we have seen that recovering the 21 cm LSS signal 
-becomes difficult for larger redshifts, in particular for $z \gtrsim 2$. 
+becomes difficult for higher redshifts, in particular for $z \gtrsim 2$. 
 
 We have determined the transfer function parameters of equation (\ref{eq:tfanalytique}) $k_A, k_B, k_C$ 
 for setup (e) for three redshifts, $z=0.5, 1 , 1.5$, and then extrapolated the value of the parameters 
-for redshift $z=2, 2.5$. The value of the parameters are grouped in table \ref{tab:paramtfk} 
-and the corresponding transfer functions are shown on  figure \ref{tfpkz0525}. 
+for redshift $z=2, 2.5$. The value of the parameters are grouped in Table \ref{tab:paramtfk}, 
+and the corresponding transfer functions are shown in Fig. \ref{tfpkz0525}. 
 
 \begin{table}[hbt]
@@ -1417 +1421 @@
 \end{tabular}
 \end{center}
-\tablefoot{ The transfer function parameters, $(k_A,k_B,k_C)$  (eq. \ref{eq:tfanalytique}) 
+\tablefoot{ The transfer function parameters, $(k_A,k_B,k_C)$  (Eq. \ref{eq:tfanalytique}) 
 at different redshifts and for instrumental setup (e), $20\times20$ packed array interferometer,
 are given in $\mathrm{Mpc^{-1}}$ unit, and not in $\mathrm{h \, Mpc^{-1}}$. } 
@@ -1430 +1434 @@
 }
 % \vspace*{-30mm}
-\caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $ (transfer function), at $z \sim 0.6$.  for the instrument configuration (a), $11\times11$ packed array interferometer. The effect of a frequency independent 
-gaussian beam of $\sim 30'$ is shown in black.
+\caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 
+21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $ (transfer function), at $z \sim 0.6$ 
+for the instrument configuration (a), $11\times11$ packed array interferometer. 
+The effect of a frequency-independent Gaussian beam of $\sim 30'$ is shown in black.
 The transfer function $\TrF(k)$  for the instrument configuration (a), $11\times11$ packed array interferometer,
 for the GSM/Model-I is shown in red, and in orange for Haslam+NVSS/Model-II. The transfer function 
@@ -1475 +1481 @@
 
 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{figpnoisea2g} shows the noise power spectra, and allows us to rank visually the configurations 
-in terms of instrument noise contribution to P(k) measurement. 
+power spectrum measurement has been discussed in Sec. \ref{pkmessens}. 
+Figure \ref{figpnoisea2g} shows the noise power spectra and allows us to visually rank 
+the configurations in terms of instrument noise contribution to P(k) measurement. 
 The differences in $P_{noise}$ will translate into differing precisions
 in the reconstruction of the BAO peak positions and in
-the estimation of cosmological parameters. In addition, we have seen (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. 
+the estimation of cosmological parameters. In addition, we have seen (Sect. \ref{recsec})
+that subtraction of continuum radio emissions, Galactic synchrotron, and radio sources
+also has an effect on the measured 21 cm power spectrum. 
 In this paragraph, we present our method and the results for the precisions on the estimation 
-of Dark Energy parameters, through a radio survey of the redshifted 21 cm emission of LSS, 
+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. 
@@ -1491 +1497 @@
 \subsection{BAO peak precision}
 
-In order to estimate the precision with which BAO peak positions can be
+To estimate the precision with which BAO peak positions can be
 measured, we  used a method similar to the one established in 
 \citep{blake.03} and \citep{glazebrook.05}.
-
-
-
+%
 To this end, we  generated reconstructed power spectra $P^{rec}(k)$ for
- slices of Universe with a quarter-sky coverage and a redshift depth,
+ slices of the  Universe with a quarter-sky coverage and a redshift depth,
  $\Delta z=0.5$ for  $0.25<z<2.75$. 
 The peaks in the generated spectra were then determined by a 
@@ -1504 +1508 @@
 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 $\TrF(k)$ (Eq. \ref{eq:tfanalytique} , table \ref{tab:paramtfk})
-and a white noise component $P_{noise}$ calculated according to the equation \ref{eq:pnoiseNbeam}, 
+the sum of the expected \HI signal term, corresponding to Eqs. \ref{eq:pk21z} and \ref{eq:tbar21z}, 
+damped by the transfer function $\TrF(k)$ (Eq. \ref{eq:tfanalytique} , Table \ref{tab:paramtfk})
+and a white noise component $P_{noise}$ calculated according to the Eq. \ref{eq:pnoiseNbeam}, 
 established in section \ref{instrumnoise} with $N=400$:
 \begin{equation}
@@ -1512 +1516 @@
 \end{equation}
 where the different terms ($P_{21}(k) , \TrF(k), P_{noise}$) depend on the slice redshift.  
-The expected 21 cm power spectrum $P_{21}(k)$ has been generated according to the formula:
+The expected 21 cm power spectrum $P_{21}(k)$ has been generated according to the formula
 %\begin{equation}
 \begin{eqnarray}
@@ -1526 +1530 @@
 \end{eqnarray}
 %\end{equation}
-where $k=\sqrt{\kperp^2 + \kpar^2}$, the parameters $A$, $\alpha$ and $\tau$ 
+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. 
+\citep{eisenhu.98}, and $P_{ref}(\kperp,\kpar)$ is the 
+envelope curve of the HI power spectrum without baryonic oscillations. 
 The parameters $\koperp$ and $\kopar$ 
 are the inverses of the oscillation periods in k-space. 
-The following values have been used for these 
+The following values were used for these 
 parameters for the results presented here: $A=1.0$, $\tau=0.1 \, \hMpcm$, 
-$\alpha=1.4$ and $\koperp=\kopar=0.060 \, \hMpcm$.
-
-Each simulation is performed for a given set of parameters 
-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 
+$\alpha=1.4$, and $\koperp=\kopar=0.060 \, \hMpcm$.
+
+Each simulation is performed for a given set of parameters:
+the system temperature $\Tsys$, an observation time 
+$t_{obs}$, an average redshift, and a redshift depth $\Delta z=0.5$. 
+Then,  each simulated  power spectrum  is fitted with a 2D
+normalized function $P_{tot}(\kperp,\kpar)/P_{ref}(\kperp,\kpar)$, which is 
 the sum of the signal power spectrum damped by the transfer function and the 
 noise power spectrum  multiplied by a 
 linear term,  $a_0+a_1k$. The upper limit $k_{max}$ in $k$ of the fit 
-corresponds to the approximate position of the linear/non-linear transition. 
+corresponds to the approximate position of the linear/nonlinear transition. 
 This limit is established on the basis of the criterion discussed in  
 \citep{blake.03}. 
-In practice, we used for the redshifts 
-$z=0.5,\,\, 1.0$ and  $1.5$ respectively $k_{max}= 0.145 \hMpcm,\,\, 0.18\hMpcm$ 
-and $0.23 \hMpcm$. 
+In practice, we used $k_{max}= 0.145 \hMpcm,\,\, 0.18\hMpcm$, 
+and $0.23 \hMpcm$ for the redshifts $z=0.5,\,\, 1.0$, and  $1.5$, respectively. 
  
-Figure \ref{fig:fitOscill} shows the result of the fit for 
-one of these simulations. 
-Figure \ref{fig:McV2} histograms the recovered values of  $\koperp$ and $\kopar$
+Figure \ref{fig:fitOscill} shows the result of the fit for one of these simulations. 
+Figure \ref{fig:McV2} histogram show the recovered values of  $\koperp$ and $\kopar$
 for 100 simulations.
 The widths of the two distributions give an estimate 
@@ -1559 +1561 @@
 
 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 
+signal $A$, $\tau$, $\alpha$, and the parameter correcting the noise power 
+spectrum $(a_0,a_1)$ are floated to take the possible 
 ignorance  of the signal shape and the uncertainties in the 
-computation of the noise power spectrum. 
-In this way, we can correct possible imperfections and the 
+computation of the noise power spectrum into account.
+In this way, we can correct possible imperfections, and the 
 systematic uncertainties are directly propagated to statistical errors 
 on the relevant parameters  $\koperp$ and $\kopar$. By subtracting the 
 fitted noise contribution to each simulation, the baryonic oscillations 
-are clearly observed, for instance, on Fig.~\ref{fig:AverPk}.
+are clearly observed, for instance, in Fig.~\ref{fig:AverPk}.
 
  
@@ -1591 +1593 @@
 \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
+wavelength  $\koperp$ and $\kopar$ perpendicular and parallel, 
+respectively, to the line of sight
 for simulations as in Fig. \ref{fig:fitOscill}. 
 The fit by a Gaussian of the distribution (solid line) gives the 
-width of the distribution  which represents the statistical error 
+width of the distribution,  which represents the statistical error 
 expected on these parameters.}
 \label{fig:McV2}
@@ -1608 +1610 @@
 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, 
+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, $\Delta z=0.5$. 
+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 
+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 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}
@@ -1630 +1632 @@
 \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 modes $N_{\delta k}$ in the wave number interval $\delta k$ can be written as:
+the $\Delta z=0.5$ slice with the solid angle $\Omega_{tot}$ = 1 $\pi$ sr. 
+The number of modes $N_{\delta k}$ in the wave number interval $\delta k$ can be written as
 \begin{equation}
 V  =  \frac{c}{H(z)} \Delta z  \times (1+z)^2 \dang^2  \Omega_{tot} \hspace{10mm}
-N_{\delta k}  =  \frac{ V }{4 \pi^2} k^2 \delta k 
+N_{\delta k}  =  \frac{ V }{4 \pi^2} k^2 \delta k \hspace{3mm} .
 \end{equation}   
 \item {\it Noise}: we add the instrument noise as a constant term $P_{noise}$ as described in Eq. 
-\ref {eq:pnoiseNbeam}. Table \ref{tab:pnoiselevel} gives the white noise level for a $N=400$ dish interferometer 
+\ref {eq:pnoiseNbeam}. Table \ref{tab:pnoiselevel} gives the white noise level for an $N=400$ dish interferometer 
 with $\Tsys = 50 \mathrm{K}$ and one year total observation time to survey $\Omega_{tot}$ = 1 $\pi$ sr.
-\item {\it Noise with transfer function}: we take into account the interferometer response and radio foreground
+\item {\it Noise with transfer function}: we consider the interferometer response and radio foreground
 subtraction represented as the measured P(k) transfer function $T(k)$ (section \ref{tfpkdef}), as 
 well as the instrument noise $P_{noise}$.
@@ -1658 +1660 @@
 
 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.
+decrease as a function of redshift for simulations without electronic noise because the volume 
+of the universe probed is larger. Once we apply the electronics noise, each slice in redshift gives
+comparable results.  Finally, after applying the full reconstruction of the interferometer, the best 
+accuracy is obtained for the first slices in redshift around 0.5 and 1.0 for an identical time of 
+observation. We can optimize the survey by using a different observation time for each 
+slice in redshift. Finally, for a 3-year survey we can split in five observation periods 
+with durations that are three months, three months, six months, one year and one year 
+for redshift 0.5, 1.0, 1.5, 2.0, and 2.5, respectively (Table  \ref{tab:ErrorOnK}, 4$^{\rm th}$ row). 
 
 \begin{table*}[ht]
@@ -1667 +1676 @@
 \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 (a) & $\sigma(\koperp)/\koperp$  (\%) & 1.8 & 0.8 & 0.6 & 0.5 &0.5\\
+\bf No noise, pure cosmic variance & $\sigma(\koperp)/\koperp$  (\%) & 1.8 & 0.8 & 0.6 & 0.5 &0.5\\
  & $\sigma(\kopar)/\kopar$  (\%) & 3.0 & 1.3 & 0.9 &  0.8 & 0.8\\
  \hline
- \bf  Noise without Transfer Function (b)  & $\sigma(\koperp)/\koperp$  (\%) & 2.3 & 1.8 & 2.2 & 2.4 & 2.8\\
+ \bf  Noise without transfer function (a)  & $\sigma(\koperp)/\koperp$  (\%) & 2.3 & 1.8 & 2.2 & 2.4 & 2.8\\
  (3-months/redshift bin)& $\sigma(\kopar)/\kopar$  (\%) & 4.1 & 3.1  & 3.6 & 4.3 & 4.4\\
  \hline
- \bf   Noise with Transfer Function  (c) & $\sigma(\koperp)/\koperp$  (\%) & 3.0 & 2.5 & 3.5 & 5.2 & 6.5 \\
+ \bf   Noise with transfer function (a)  & $\sigma(\koperp)/\koperp$  (\%) & 3.0 & 2.5 & 3.5 & 5.2 & 6.5 \\
  (3-months/redshift bin)& $\sigma(\kopar)/\kopar$  (\%) & 4.8 & 4.0 & 6.2 & 9.3 & 10.3\\
  \hline
- \bf  Optimized survey (d) & $\sigma(\koperp)/\koperp$  (\%)   & 3.0 & 2.5 & 2.3 &  2.0 &  2.7\\
+ \bf  Optimized survey (b) & $\sigma(\koperp)/\koperp$  (\%)   & 3.0 & 2.5 & 2.3 &  2.0 &  2.7\\
  (Observation time :  3 years)& $\sigma(\kopar)/\kopar$  (\%) & 4.8 & 4.0 & 4.1 &  3.6  & 4.3 \\
  \hline
@@ -1683 +1692 @@
 \tablefoot{Relative errors on $\koperp$ and $\kopar$ measurements are given 
 as a function of the redshift $z$ for various simulation configurations: \\
-\tablefoottext{a}{$1^{\rm st}$ row: simulations without noise with pure cosmic variance; } \\
-\tablefoottext{b}{$2^{\rm nd}$ row: simulations with electronics noise for a telescope with dishes;  } \\
-\tablefoottext{c}{$3^{\rm rd}$ row: simulations with the same electronics noise and with the transfer function; } \\
-\tablefoottext{d}{$4^{\rm th}$ row: optimized survey with a total observation time of 3 years: 3 months, 3 months, 
-6 months, 1 year and 1 year respectively for \\ redshifts 0.5, 1.0, 1.5, 2.0 and 2.5.} 
+\tablefoottext{a}{simulations with electronics noise, without ($2^{\rm nd}$ row) and with ($3^{\rm rd}$ row) the transfer function;  } \\
+\tablefoottext{b}{optimized survey, simulations with electronic noise and the transfer function} 
 }
 \end{table*}%
@@ -1726 +1732 @@
 sonic horizon.
 The peaks in the angular spectrum are proportional to 
-$d_T(z)/a_s$ and those in the redshift spectrum to $d_H(z)/a_s$.
+$d_T(z)/a_s$ and those in the redshift spectrum to $d_H(z)/a_s$, where 
 $a_s \sim 105  h^{-1} \mathrm{Mpc}$ is the acoustic horizon comoving size at recombination, 
-$d_T(z) = (1+z) \dang$ is the comoving angular distance and $d_H=c/H(z)$ is the Hubble distance
+$d_T(z) = (1+z) \dang$ is the comoving angular distance and $d_H=c/H(z)$ the Hubble distance
 (see Eq. \ref{eq:expHz}):
 \begin{equation}
@@ -1735 +1741 @@
 \label{eq:dTdH}
 \end{equation}
-The quantities $d_T$, $d_H$ and $a_s$ all depend on
+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
@@ -1750 +1756 @@
 
 To estimate the sensitivity 
-to parameters describing dark energy equation of 
+to parameters describing the dark energy equation of 
 state, we follow the procedure explained in 
 \citep{blake.03}. We can introduce the equation of 
-state of dark energy, $w(z)=w_0 + w_a\cdot z/(1+z)$ by 
+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:
+(Eq. \ref{eq:dTdH}) by
 \begin{equation}
 \Omega_\Lambda \rightarrow \Omega_{\Lambda} \exp \left[ 3  \int_0^z   
@@ -1763 +1769 @@
 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  
+Table \ref{tab:ErrorOnK}, we can compute the Fisher matrix for  
 five cosmological parameter: $(\Omega_m, \Omega_b, h, w_0, w_a)$. 
 Then, the combination of this BAO Fisher 
-matrix with the Fisher matrix obtained for Planck mission, allows us to 
+matrix with the Fisher matrix obtained for Planck mission allows us to 
 compute the errors on dark energy parameters.
-{\changemark We have used the Planck Fisher matrix, computed for the 
+{\changemark We used the Planck Fisher matrix, computed for the 
 Euclid proposal \citep{laureijs.09}, for the 8 parameters: 
 $\Omega_m$, $\Omega_b$, $h$, $w_0$, $w_a$,
@@ -1776 +1782 @@
 
 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 
+observation time of three years, the packed 400-dish interferometer array has a 
 precision of  12\% on $w_0$ and 48\% on $w_a$. 
-The  Figure of Merit, the inverse of the area in the 95\% confidence level 
-contours  is 38.
+The  figure of merit (FOM), the inverse of the area in the 95\% confidence level 
+contours, is 38.
 Finally, Fig.~\ref{fig:Compw0wa} 
 shows a comparison of different BAO projects, with a set of priors on 
 $(\Omega_m, \Omega_b, h)$ corresponding to the expected precision on 
-these parameters in early 2010's. {\changemark The confidence contour 
-level in the plane $(w_0,w_a)$  have been obtained by marginalizing 
+these parameters in early 2010s. {\changemark The confidence contour 
+level in the plane $(w_0,w_a)$  were obtained by marginalizing 
 over all the other parameters.} This BAO project based on \HI intensity 
 mapping is clearly competitive with the current generation of optical 
-surveys such as SDSS-III \citep{sdss3}. 
+surveys such as SDSS-III \citep{eisenstein.11}. 
 
 
@@ -1802 +1808 @@
 
 \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 
+The 3D mapping of redshifted 21 cm emission though {\it intensity mapping} is a novel and complementary 
+approach to optical surveys for studying the statistical properties of the LSS in the universe
+up to redshifts $z \lesssim 3$. A radio instrument with a large instantaneous field of view 
 (10-100 deg$^2$) and large bandwidth ($\gtrsim 100$ MHz) with $\sim 10$ arcmin resolution is needed 
 to perform a cosmological neutral hydrogen survey over a significant fraction of the sky. We have shown that 
-a nearly packed interferometer array with few hundred receiver elements spread over an hectare or a hundred beam 
+a nearly packed interferometer array with a few hundred receiver elements spread over an hectare or a hundred beam 
 focal plane array with a $\sim \hspace{-1.5mm} 100  \, \mathrm{meter}$ primary reflector will have the required sensitivity to measure 
-the 21 cm power spectrum. A method 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 
+the 21 cm power spectrum. A method of computing the instrument response for interferometers  
+was developed, and we  computed the noise power spectrum for various telescope configurations.
+The Galactic synchrotron and radio sources are a thousand times brighter than the redshifted 21 cm signal, 
+making the measurement of the latter signal a major scientific and technical challenge. 
+We also studied  the performance of a simple foreground subtraction method through realistic models of the sky 
 emissions in the GHz domain and simulation of interferometric observations. 
-We 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 us to define and 
+We were able to show that the cosmological 21 cm signal from the LSS should be observable, but 
+requires a very good knowledge of the instrument response. Our method allowed us to define and 
 compute the overall  {\it transfer function} or {\it response function} for the measurement of the 21 cm 
 power spectrum. 
-Finally, we have used the computed noise power spectrum and  $P(k)$ 
+Finally, we used the computed noise power spectrum and  $P(k)$ 
 measurement response function to estimate 
-the precision on the determination of Dark Energy parameters, for a 21 cm BAO survey. Such a radio survey 
+the precision on the determination of dark energy parameters, for a 21 cm BAO survey. This radio survey 
 could be carried using the current technology and would be competitive with the ongoing or planned 
 optical surveys for dark energy,  with a fraction of their cost.
@@ -1833 +1840 @@
 %%%
 %%%% 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  
+\bibitem[Abell et al. 2009]{lsst.science} 
+Abell, P.A. {\it et al.} {\it LSST Science book}, LSST Science Collaborations, {\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     
+\bibitem[Abdalla \& Rawlings 2005]{abdalla.05} Abdalla, F.B. \& Rawlings, S.  2005,  \mnras, 360,  27     
 
 % reference DETF - DE eq.state parameter figure of merit 
-\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[Albrecht et al. 2006]{DETF}  Albrecht, A., Bernstein, G., Cahn, R. {\it et al.} (Dark Energy Task Force), 2006, arXiv:astro-ph/0609591 
 
 % Papier sensibilite/reconstruction CRT (cylindres) ansari et al 2008 
-\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
+\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
 
 %%   Temperature HI 21 cm (Valeur pour la reionisation) 
-\bibitem[Barkana \& Loeb (2007)]{barkana.07} Barkana, R., and Loeb, A. 2007, Rep. Prog. Phys, 70,  627 
+\bibitem[Barkana \& Loeb 2007]{barkana.07} Barkana, R., and Loeb, A. 2007, Rep. Prog. Phys, 70,  627 
 
 %% Methode de generation/fit k_bao  (Section 5 - C. Yeche) 
-\bibitem[Blake and Glazebrook (2003)]{blake.03} Blake, C. \& Glazebrook, K. 2003, \apj, 594, 665
-\bibitem[Glazebrook and Blake (2005)]{glazebrook.05} 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,  (accepted, arXiv/1105.2862)
+\bibitem[Blake \& Glazebrook 2003]{blake.03} Blake, C. \& Glazebrook, K. 2003, \apj, 594, 665
+\bibitem[Glazebrook \& Blake 2005]{glazebrook.05} Glazebrook, K. \&  Blake, C. 2005 \apj, 631, 1
+
+% WiggleZ BAO observation ( arXiv/1105.2862 )
+\bibitem[Blake et al. 2011]{blake.11} Blake, Davis, T., Poole, G.B.  {\it et al.}  2011, \mnras, 415, 2892-2909 
 
 % Galactic astronomy, emission HI d'une galaxie
-\bibitem[Binney \& Merrifield (1998)]{binney.98} Binney J. \& Merrifield M. , 1998 {\it Galactic Astronomy} Princeton University Press 
+\bibitem[Binney \& Merrifield 1998]{binney.98} Binney J. \& Merrifield M. , 1998 {\it Galactic Astronomy} Princeton University Press 
 % 21cm reionisation P(k) estimation and sensitivities 
-\bibitem[Bowman et al. (2006)]{bowman.06} Bowman, J.D.,  Morales, M.F., Hewitt, J.N. 2006, \apj, 638, 20-26
+\bibitem[Bowman et al. 2006]{bowman.06} Bowman, J.D.,  Morales, M.F., Hewitt, J.N. 2006, \apj, 638, 20-26
 % 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  
+\bibitem[Bowman et al. 2007]{bowman.07} Bowman, J. D., Barnes, D.G., Briggs, F.H. {\it et al.} 2007, \aj, 133, 1505-1518  
 
 %% Soustraction avant plans ds MWA 
 \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 } 
+%%%  SKA-Science Elsevier, December 2004   http://www.skatelescope.org/pages/page\_sciencegen.htm 
+\bibitem[Carilli et al. 2004]{ska.science} 
+{\it Science with the Square Kilometre Array}, eds: C. Carilli, S. Rawlings, 2004, New Astronomy Reviews, 48  
 
 %  Intensity mapping/HSHS 
-\bibitem[Chang et al. (2008)]{chang.08}  Chang, T.,  Pen, U.-L., Peterson, J.B. \&  McDonald, P., 2008, \prl, 100, 091303
+\bibitem[Chang et al. 2008]{chang.08}  Chang, T.,  Pen, U.-L., Peterson, J.B. \&  McDonald, P., 2008, \prl, 100, 091303
 
 % Mesure 21 cm avec le GBT (papier Nature )
-\bibitem[Chang et al. (2010)]{chang.10}   Chang T-C, Pen U-L, Bandura K., Peterson  J.B., 2010, \nat, 466, 463-465
+\bibitem[Chang et al. 2010]{chang.10}   Chang T-C, Pen U-L, Bandura K., Peterson  J.B., 2010, \nat, 466, 463-465
 
 % 2dFRS BAO observation 
-\bibitem[Cole et al. (2005)]{cole.05} Cole, S. Percival, W.J., Peacock, J.A.  {\it et al.} (the 2dFGRS Team) 2005,  \mnras, 362, 505
+\bibitem[Cole et al. 2005]{cole.05} Cole, S. Percival, W.J., Peacock, J.A.  {\it et al.} 2005,  \mnras, 362, 505
 
 % NVSS radio source catalog : NRAO VLA Sky Survey (NVSS) is a 1.4 GHz
-\bibitem[Condon et al. (1998)]{nvss.98} Condon J. J., Cotton W. D., Greisen E. W., Yin Q. F., Perley R. A., 
+\bibitem[Condon et al. 1998]{nvss.98} Condon J. J., Cotton W. D., Greisen E. W., Yin Q. F., Perley R. A., 
 Taylor, G. B., \& Broderick, J. J. 1998, AJ, 115, 1693
 
 % Effet des radio-sources sur le signal 21 cm reionisation 
-\bibitem[Di Matteo  et al. (2002)]{matteo.02}  Di Matteo, T., Perna R., Abel T., Rees M.J. 2002, \apj, 564, 576-580
+\bibitem[Di Matteo  et al. 2002]{matteo.02}  Di Matteo, T., Perna R., Abel T., Rees M.J. 2002, \apj, 564, 576-580
 
 %  Parametrisation P(k)   - (astro-ph/9709112) 
-\bibitem[Eisenstein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, \apj 496, 605-614 
+\bibitem[Eisenstein \& Hu  1998]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, \apj, 496, 605-614 
 
 % SDSS first BAO observation 
-\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
+\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[Eisenstein 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 \\
+{ \tt http://www.sdss3.org/ }
 
 % Papier de Field sur la profondeur optique HI en 1959 
-\bibitem[Field (1959)]{field.59} Field G.B., 1959, \apj, 129, 155 
+\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
+\bibitem[Furlanetto et al. 2006]{furlanetto.06} Furlanetto, S., Peng Oh, S. \&  Briggs, F. 2006, \physrep, 433, 181-301
 
 % Mesure 21 cm a 610 MHz par GMRT 
-\bibitem[Ghosh et al. (2011)]{ghosh.11}  Ghosh A., Bharadwaj S., Ali Sk. S., Chengalur  J. N., 2011, \mnras, 411, 2426-2438
+\bibitem[Ghosh et al. 2011]{ghosh.11}  Ghosh A., Bharadwaj S., Ali Sk. S., Chengalur  J. N., 2011, \mnras, 411, 2426-2438
 
 
 % Haslam 400 MHz synchrotron map 
-\bibitem[Haslam et al. (1982)]{haslam.82} Haslam  C. G. T.,  Salter C. J., Stoffel H., Wilson W. E., 1982, 
-Astron. \& Astrophys.  Supp.  Vol 47,  \\ {\tt (http://lambda.gsfc.nasa.gov/product/foreground/)}
+\bibitem[Haslam et al. 1982]{haslam.82} Haslam  C. G. T.,  Salter C. J., Stoffel H., Wilson W. E., 1982, 
+Astron. \& Astrophys.  Supp.  Vol 47 %%  {\tt (http://lambda.gsfc.nasa.gov/product/foreground/)}
 
 
 % Distribution des radio sources
-\bibitem[Jackson (2004)]{jackson.04} Jackson, C.A. 2004, \na, 48, 1187  
+\bibitem[Jackson 2004]{jackson.04} Jackson, C.A. 2004, \na, 48, 1187  
 
 % WMAP 7 years cosmological parameters 
-\bibitem[Komatsu et al. (2011)]{komatsu.11}   E. Komatsu, K. M. Smith, J. Dunkley {\it et al.}  2011, \apjs, 192, p. 18 \\
-\mbox{\tt http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm\_sz\_lens\_wmap7.cfm} 
+\bibitem[Komatsu et al. 2011]{komatsu.11}   E. Komatsu, K. M. Smith, J. Dunkley {\it et al.}  2011, \apjs, 192, p. 18 
+% \mbox{\tt http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm\_sz\_lens\_wmap7.cfm} 
 
 % HI mass in galaxies 
-\bibitem[Lah et al. (2009)]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur {\it et al.}  2009,  \mnras, 399, 1447 
+\bibitem[Lah et al. 2009]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur {\it et al.}  2009,  \mnras, 399, 1447 
 % ( astro-ph/0907.1416) 
 
 % Livre Astrophysical Formulae de Lang
-\bibitem[Lang (1999)]{astroformul} Lang, K.R. {\it Astrophysical Formulae}, Springer, 3rd Edition 1999 
+\bibitem[Lang 1999]{astroformul} Lang, K.R. {\it Astrophysical Formulae}, Springer, 3rd Edition 1999 
 
 %  WMAP CMB 7 years power spectrum 2011 
 % \bibitem[Hinshaw et al. (2008)]{hinshaw.08}  Hinshaw, G., Weiland, J.L., Hill, R.S.  {\it et al.} 2008, arXiv:0803.0732) 
-\bibitem[Larson et al. (2011)]{larson.11}  Larson, D.,  {\it et al.}  (WMAP) 2011, \apjs, 192, 16 
+\bibitem[Larson et al. 2011]{larson.11}  Larson, D.,  {\it et al.}  (WMAP) 2011, \apjs, 192, 16 
 
 %% 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
+\bibitem[Lonsdale et al. 2009]{lonsdale.09}  Lonsdale C.J., Cappallo  R.J., Morales M.F.  {\it et al.}, 2009,
+IEEE Proceeding, 97, 1497-1506 (arXiv:0903.1828)
 
 % Planck Fischer matrix, computed for EUCLID 
-\bibitem[Laureijs (2009)]{laureijs.09} Laureijs, R. 2009, ArXiv:0912.0914
+\bibitem[Laureijs 2009]{laureijs.09} Laureijs, R. 2009, ArXiv:0912.0914
 
 % Temperature du 21 cm 
-\bibitem[Madau et al. (1997)]{madau.97} Madau, P., Meiksin, A. and Rees, M.J., 1997, \apj 475, 429 
+\bibitem[Madau et al. 1997]{madau.97} Madau, P., Meiksin, A. and Rees, M.J., 1997, \apj 475, 429 
  
 % Foret Ly alpha - 1 
-\bibitem[McDonald et al. (2006)]{baolya} McDonald P., Seljak, U. and Burles, S.  {\it et al.}  2006, \apjs, 163, 80
+\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
+\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
+\bibitem[Mauskopf et al. 2000]{mauskopf.00} Mauskopf, P. D., Ade, P. A. R., de Bernardis, P. {\it et al.}  2000, \apjl, 536,59
 
 %% PNoise and cosmological parameters with reionization
-\bibitem[McQuinn et al. (2006)]{mcquinn.06}  McQuinn M., Zahn O., Zaldarriaga M., Hernquist L.,  Furlanetto S.R.
-2006, \apj 653, 815-834 
+\bibitem[McQuinn et al. 2006]{mcquinn.06}  McQuinn M., Zahn O., Zaldarriaga M., Hernquist L.,  Furlanetto S.R.
+2006, \apj, 653, 815-834 
 
 %  Papier sur la mesure de sensibilite P(k)_reionisation 
-\bibitem[Morales \& Hewitt (2004)]{morales.04}  Morales M. \& Hewitt J., 2004, \apj, 615, 7-18  
+\bibitem[Morales \& Hewitt 2004]{morales.04}  Morales M. \& Hewitt J., 2004, \apj, 615, 7-18  
 
 %  Papier sur le traitement des observations radio / mode mixing 
@@ -1958 +1965 @@
 
 %% Foreground removal using smooth frequency dependence 
-\bibitem[Oh \& Mack (2003)]{oh.03} Oh S.P. \& Mack K.J., 2003, \mnras, 346, 871-877
+\bibitem[Oh \& Mack 2003]{oh.03} Oh S.P. \& Mack K.J., 2003, \mnras, 346, 871-877
 
 %  Global Sky Model Paper 
-\bibitem[Oliveira-Costa et al. (2008)]{gsm.08} de Oliveira-Costa, A., Tegmark, M., Gaensler, B.~M.  {\it et al.} 2008, 
+\bibitem[Oliveira-Costa et al. 2008]{gsm.08} de Oliveira-Costa, A., Tegmark, M., Gaensler, B.~M.  {\it et al.} 2008, 
 \mnras, 388, 247-260 
 
-%%  Description+ resultats PAPER 
-\bibitem[Parsons et al. (2009)]{parsons.09}  Parsons A.R.,Backer D.C.,Bradley  R.F. {\it et al.}  2009, arXiv:0904.2334
+%%  Description+ resultats PAPER - arXiv:0904.2334
+\bibitem[Parsons et al. 2010]{parsons.10}  Parsons A.R.,Backer D.C.,Bradley  R.F. {\it et al.}  2010, 
+\aj, 2010,  139,  1468-1480
 
 % Livre Cosmo de Peebles 
-\bibitem[Peebles (1993)]{cosmo.peebles} Peebles, P.J.E., {\it Principles of Physical Cosmology}, 
-Princeton University Press (1993)
+\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  
+\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
+\bibitem[Platania et al. 1998]{platania.98}   Platania P., Bensadoun M., Bersanelli M. {\it al.} 1998,  \apj, 505, 473-483
 
 % 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
+\bibitem[Percival et al. 2007]{percival.07}   Percival, W.J., Nichol, R.C., Eisenstein, D.J. {\it et al.}, 2007, \apj, 657, 645
 
 % SDSS BAO 2010  - arXiv:0907.1660 
-\bibitem[Percival et al. (2010)]{percival.10}   Percival, W.J., Reid, B.A., Eisenstein, D.J. {\it et al.},  2010, \mnras, 401, 2148-2168   
+\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)
+\bibitem[Rich 2001]{cosmo.rich} James Rich,  {\it Fundamentals of Cosmology}, Springer, 2001
 
 % Radio spectral index between 100-200 MHz 
-\bibitem[Rogers \& Bowman (2008)]{rogers.08} Rogers, A.E.E. \& Bowman, J. D. 2008, \aj 136, 641-648
+\bibitem[Rogers \& Bowman 2008]{rogers.08} Rogers, A.E.E. \& Bowman, J. D. 2008, \aj 136, 641-648
 
 %% LOFAR description 
-\bibitem[Rottering et al. (2006)]{rottgering.06} Rottgering H.J.A., Braun, r., Barthel, P.D. {\it et al.}  2006, arXiv:astro-ph/0610596
+\bibitem[Rottering et al. 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 
+%  \bibitem[SDSS-III 2008]{sdss3} SDSS-III 2008, http://www.sdss3.org/collaboration/description.pdf 
 
 % Reionisation: Can the reionization epoch be detected as a global signature in the cosmic background?
-\bibitem[Shaver  et al. (1999))]{shaver.99} Shaver P.A., Windhorst R. A., Madau P., de Bruyn A.G. \aap, 345, 380-390
+\bibitem[Shaver  et al. 1999]{shaver.99} Shaver P.A., Windhorst R. A., Madau P., de Bruyn A.G. \aap, 345, 380-390
 
 %  Frank H. Briggs, Matthew Colless, Roberto De Propris, Shaun Ferris, Brian P. Schmidt, Bradley E. Tucker
 
 % Papier 21cm-BAO Fermilab  ( arXiv:0910.5007)
-\bibitem[Seo et al (2010)]{seo.10} Seo, H.J. Dodelson, S., Marriner, J. et al,  2010, \apj, 721, 164-173 
+\bibitem[Seo et al 2010]{seo.10} Seo, H.J. Dodelson, S., Marriner, J. {\it et al.}  2010, \apj, 721, 164-173 
 
 % Mesure P(k) par SDSS 
-\bibitem[Tegmark et al.  (2004)]{tegmark.04}  Tegmark M., Blanton  M.R, Strauss M.A. et al. 2004, \apj, 606, 702-740
+\bibitem[Tegmark et al.  2004]{tegmark.04}  Tegmark M., Blanton  M.R, Strauss M.A. {\it et al.} 2004, \apj, 606, 702-740
 
 % FFT telescope 
-\bibitem[Tegmark \& Zaldarriaga (2009)]{tegmark.09} Tegmark, M. \& Zaldarriaga, M., 2009, \prd, 79, 8, p. 083530 % arXiv:0802.1710 
+\bibitem[Tegmark \& Zaldarriaga 2009]{tegmark.09} Tegmark, M. \& Zaldarriaga, M., 2009, \prd, 79, 8, p. 083530 % arXiv:0802.1710 
 
 %  Thomson-Morane livre interferometry 
@@ -2015 +2023 @@
 
 % Lyman-alpha, HI fraction 
-\bibitem[Wolf et al.(2005)]{wolf.05} Wolfe, A. M., Gawiser, E. \& Prochaska, J.X. 2005  \araa, 43, 861
+\bibitem[Wolf et al. 2005]{wolf.05} Wolfe, A. M., Gawiser, E. \& Prochaska, J.X. 2005  \araa, 43, 861
 
 %  BAO à 21 cm et reionisation 
-\bibitem[Wyithe et al.(2008)]{wyithe.08} Wyithe, S., Loeb, A. \& Geil, P. 2008, \mnras, 383, 1195 %  http://fr.arxiv.org/abs/0709.2955, 
+\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, 
+\bibitem[Zaldarriaga et al. 2004]{zaldarriaga.04}  Zaldarriaga, M., Furlanetto, S.R., Hernquist, L., 2004, 
 \apj, 608, 622-635
 
 %% Today HI cosmological density 
-\bibitem[Zwaan et al.(2005)]{zwann.05} Zwaan, M.A., Meyer, M.J., Staveley-Smith, L., Webster, R.L. 2005,  \mnras, 359, L30
+\bibitem[Zwaan et al. 2005]{zwann.05} Zwaan, M.A., Meyer, M.J., Staveley-Smith, L., Webster, R.L. 2005, \mnras, 359, L30
 
 \end{thebibliography}