Changeset 4013 in Sophya for trunk


Ignore:
Timestamp:
Aug 4, 2011, 6:38:33 PM (14 years ago)
Author:
ansari
Message:

Version definitine(?) du papier, pret pour la soumission, Reza 04/08/2011

Location:
trunk/Cosmo/RadioBeam
Files:
2 edited

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Removed

  • trunk/Cosmo/RadioBeam/radutil.cc

    r3947 r4013  
    4646  double cc=zz*zz/sqrt(OmegaMatter_*zz*zz*zz+OmegaLambda_);
    4747  cc *= ((h100_/0.7)*(OmegaBaryons_/0.044)*(fracHI_/0.01));
    48   return (cc*0.054);
     48  return (cc*0.059);
    4949}

  • trunk/Cosmo/RadioBeam/sensfgnd21cm.tex

    r4011 r4013  
    2828\usepackage{color}
    2929
     30%% Commande pour les references
     31\newcommand{\citep}[1]{ (\cite{#1}) }
     32%% \newcommand{\citep}[1]{ { (\tt{#1}) } }
     33
     34%% Definitions diverses
    3035\newcommand{\HI}{$\mathrm{H_I}$ }
    3136\newcommand{\kb}{k_B}         % Constante de Boltzmann
    … …  
    3944\newcommand{\dang}{d_A}
    4045\newcommand{\hub}{ h_{70} }
    41 \newcommand{\hubb}{ h }    % h_100
     46\newcommand{\hubb}{ h_{100} }    % h_100
    4247
    4348\newcommand{\etaHI}{ n_{\tiny HI} }
    4449\newcommand{\fHI}{ f_{H_I}(z)}
    45 \newcommand{\gHI}{ g_{H_I}}
    46 \newcommand{\gHIz}{ g_{H_I}(z)}
     50\newcommand{\gHI}{ f_{H_I}}
     51\newcommand{\gHIz}{ f_{H_I}(z)}
    4752
    4853\newcommand{\vis}{{\cal V}_{12} }
    … …  
    5055\newcommand{\LCDM}{$\Lambda \mathrm{CDM}$ }
    5156
    52 \newcommand{\citep}[1]{ (\cite{#1}) }
    53 %% \newcommand{\citep}[1]{ { (\tt{#1}) } }
     57\newcommand{\lgd}{\mathrm{log_{10}}}
     58
     59%% Definition fonction de transfer
     60\newcommand{\TrF}{\mathrm{Tr}}
     61
    5462
    5563%%% Definition pour la section sur les param DE par C.Y
    … …  
    126134  % methods heading (mandatory)
    127135 { For each configuration, we determine instrument response by computing the (u,v) or Fourier angular frequency
    128 plane  coverage using visibilities. The (u,v) plane response is then used to compute the three dimensional noise power spectrum,
     136plane  coverage using visibilities. The (u,v) plane response is the noise power spectrum,
    129137hence the instrument sensitivity for LSS P(k) measurement. We describe also   a simple foreground subtraction method to
    130138separate LSS 21 cm signal from the foreground due to the galactic synchrotron and radio sources emission. }
    … …  
    135143  % conclusions heading (optional), leave it empty if necessary
    136144   { We show that a radio instrument with few hundred simultaneous beams and a collecting area of
    137   $\lesssim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ and will be
     145  $\sim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ and will be
    138146  competitive with optical surveys. }
    139147
    … …  
    152160% {\color{red} \large \it Jim ( + M. Moniez ) }   \\[1mm]
    153161The study of the statistical properties of Large Scale Structure (LSS) in the Universe and their evolution
    154 with redshift is one the major tools in observational cosmology. Theses structures are usually mapped through
    155 optical observation of galaxies which are used as tracers of the underlying matter distribution.
     162with redshift is one the major tools in observational cosmology. These structures are usually mapped through
     163optical observation of galaxies which are used as a tracer of the underlying matter distribution.
    156164An alternative and elegant approach for mapping the matter distribution, using neutral atomic hydrogen
    157 (\HI) as tracer with Total Intensity Mapping, has been proposed in recent years \citep{peterson.06} \citep{chang.08}.
     165(\HI) as a tracer with intensity mapping, has been proposed in recent years \citep{peterson.06} \citep{chang.08}.
    158166Mapping the matter distribution using HI 21 cm emission as a tracer has been extensively discussed in literature
    159167\citep{furlanetto.06} \citep{tegmark.08} and is being used in projects such as LOFAR \citep{rottgering.06} or
    160168MWA \citep{bowman.07} to observe reionisation  at redshifts z $\sim$ 10.
    161169
    162 Evidences in favor of the acceleration of the expansion of the universe have been
    163 accumulated over the last twelve years, thank to the observation of distant supernovae,
     170Evidence in favor of the acceleration of the expansion of the universe have been
     171accumulated over the last twelve years, thanks to the observation of distant supernovae,
    164172CMB anisotropies and detailed analysis of the LSS. 
    165173A cosmological Constant ($\Lambda$) or new cosmological
    166174energy density called {\em Dark Energy} has been advocated as the origin of this acceleration.
    167 Dark Energy is considered as one the most intriguing puzzles in Physics and Cosmology.
     175Dark Energy is considered as one of the most intriguing puzzles in Physics and Cosmology.
    168176% Constraining the properties of this new cosmic fluid, more precisely
    169177% its equation of state is central to current cosmological researches.
    … …  
    174182
    175183BAO are features imprinted  in the distribution of galaxies, due to the frozen
    176 sound waves which were present in the photons baryons plasma prior to recombination
     184sound waves which were present in the photon-baryon plasma prior to recombination
    177185at z $\sim$ 1100.
    178 This scale, which can be considered as a standard ruler with a comoving
    179 length  of $\sim 150 Mpc$.
    180 Theses features have been first observed in the CMB anisotropies
    181 and are usually referred to as {\em acoustic pics} \citep{mauskopf.00} \citep{hinshaw.08}.
     186This scale can be considered as a standard ruler with a comoving
     187length  of $\sim 150 \mathrm{Mpc}$.
     188These features have been first observed in the CMB anisotropies
     189and are usually referred to as {\em acoustic peaks} \citep{mauskopf.00} \citep{hinshaw.08}.
    182190The BAO modulation has been subsequently observed in the distribution of galaxies
    183191at low redshift ( $z < 1$) in the galaxy-galaxy correlation function by the SDSS
    184 \citep{eisenstein.05}  \citep{percival.07}  \citep{percival.10}  and 2dGFRS  \citep{cole.05}  optical galaxy surveys.
     192\citep{eisenstein.05}  \citep{percival.07}  \citep{percival.10}, 2dGFRS  \citep{cole.05}  as well as
     193WiggleZ \citep{blake.11} optical galaxy surveys.
    185194
    186195Ongoing \citep{eisenstein.11}   or future surveys \citep{lsst.science} 
    187196plan to measure precisely the BAO scale in the redshift range
    188 $0 \lesssim z \lesssim 3$, using either optical observation of galaxies   %  CHECK/FIND baorss baolya references
    189 or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars \citep{baolya},\citep{baolya2}.
    190 Mapping matter distribution using 21 cm emission of neutral hydrogen appears as
     197$0 \lesssim z \lesssim 3$, using either optical observation of galaxies 
     198or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars
     199\citep{baolya},\citep{baolya2}.
     200Radio observation of the 21 cm emission of neutral hydrogen appears as
    191201a very promising technique to map matter distribution up to redshift $z \sim 3$,
    192202complementary to optical surveys, especially in the optical redshift desert range
    … …  
    217227The BAO features in particular are at the degree angular scale on the sky
    218228and thus can be resolved easily with a rather modest size radio instrument
    219 ($D \lesssim 100 \, \mathrm{m}$). The specific BAO clustering scale ($k_{\mathrm{BAO}}$)
     229(diameter $D \lesssim 100 \, \mathrm{m}$). The specific BAO clustering scale ($k_{\mathrm{BAO}}$)
    220230can be measured both in the transverse plane (angular correlation function, ($k_{\mathrm{BAO}}^\perp$)
    221231or along the longitudinal (line of sight or redshift ($k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of
    … …  
    226236In order to obtain a measurement of the LSS power spectrum with small enough statistical
    227237uncertainties (sample or cosmic variance),  a large volume of the universe should be observed,
    228 typically few $\mathrm{Gpc^3}$. Moreover, stringent constrain on DE parameters can be obtained when
    229 comparing the distance or Hubble parameter measurements as a function of redshift with
    230 DE models, which translates into a survey depth $\Delta z \gtrsim 1$.
     238typically few $\mathrm{Gpc^3}$. Moreover, stringent constraint on DE parameters can only be
     239obtained when comparing the distance or Hubble parameter measurements with
     240DE models as a function of redshift, which requires a significant survey depth $\Delta z \gtrsim 1$.
    231241
    232242Radio instruments intended for BAO surveys must thus have large instantaneous field
    233 of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$).
     243of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$)
     244to explore large redshift domains.
    234245
    235246Although the application of 21 cm radio survey to cosmology, in particular LSS mapping has been
    … …  
    237248the method envisaged has been mostly through the detection of galaxies as \HI compact sources.
    238249However, extremely large radio telescopes are required to detected \HI sources at cosmological distances.
    239 The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the of two polarisations
     250The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the two polarisations
    240251of a radio instrument characterized by an effective collecting area $A$, and system temperature $\Tsys$ can be written as
    241252\begin{equation}
    … …  
    243254\end{equation}
    244255where $t_{int}$ is the total integration time and $\delta \nu$ is the detection frequency band. In table
    245 \ref{slims21} (left)  we have computed the sensitivity for 6 different set of instrument effective area and system
     256\ref{slims21} (left)  we have computed the sensitivity for 6 different sets of instrument effective area and system
    246257temperature, with a total integration time of 86400 seconds (1 day) over a frequency band of 1 MHz.
    247258The width of this frequency band is well adapted  to detection of \HI source with an intrinsic velocity
    248 dispersion of few 100 km/s. Theses detection limits should be compared with the expected 21 cm brightness
     259dispersion of few 100 km/s. These detection limits should be compared with the expected 21 cm brightness
    249260$S_{21}$ of compact sources which can be computed using the expression below (e.g.\cite{binney.98}) :
    250261\begin{equation}
    251262 S_{21}  \simeq  0.021 \mathrm{\mu Jy} \, \frac{M_{H_I} }{M_\odot}   \times
    252 \left( \frac{ 1\, \mathrm{Mpc}}{\dlum} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v}
     263\left( \frac{ 1\, \mathrm{Mpc}}{\dlum(z)} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v}  (1+z)
    253264\end{equation}
    254  where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum$ is the luminosity distance and $\sigma_v$
     265 where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ is the luminosity distance and $\sigma_v$
    255266is the source velocity dispersion.
    256267% {\color{red} Faut-il developper le calcul en annexe ? }
    … …  
    264275
    265276Intensity mapping has been suggested as an alternative and economic method to map the
    266 3D distribution of neutral hydrogen \citep{chang.08} \citep{ansari.08} \citep{seo.10}.
     2773D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} \citep{seo.10}.
    267278In this approach, sky brightness map with angular resolution $\sim 10-30 \, \mathrm{arc.min}$ is made for a
    268279wide range of frequencies. Each 3D pixel  (2 angles $\vec{\Theta}$, frequency $\nu$ or wavelength $\lambda$) 
    269 would correspond to a cell with a volume of $\sim 10 \mathrm{Mpc^3}$, containing hundreds of galaxies and a total
    270 \HI mass $ \gtrsim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow,
     280would correspond to a cell with a volume of $\sim 10^3 \mathrm{Mpc^3}$, containing ten to hundred galaxies
     281and a total \HI mass $ \sim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow,
    271282the observed frequency $\nu$ would be translated to the emission redshift $z$ through
    272283the well known relation:
    … …  
    281292The large scale distribution of the neutral hydrogen, down to angular scales of $\sim 10 \mathrm{arc.min}$
    282293can then be observed without the detection of individual compact \HI sources, using the set of sky brightness
    283 map as a function frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$
     294map as a function of frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$
    284295(radiation power/unit solid angle/unit surface/unit frequency)
    285296can be converted to brightness temperature using the well known black body Rayleigh-Jeans approximation:
    … …  
    307318\hline
    308319$z$ &  $\dlum \mathrm{(Mpc)}$ & $S_{21}  \mathrm{( \mu Jy)} $ \\
    309 \hline
    310 0.25 &  1235   & 140 \\
    311 0.50 &  2800   & 27 \\
    312 1.0   &  6600   & 4.8 \\
    313 1.5   &  10980 & 1.74 \\
    314 2.0   &  15710  & 0.85 \\
    315 2.5  &  20690 & 0.49 \\
     320\hline      % dernier chiffre : sans le facteur (1+z)
     3210.25 &  1235   & 175 \\ % 140
     3220.50 &  2800   & 40   \\  % 27
     3231.0   &  6600   & 9.6  \\  % 4.8
     3241.5   &  10980 & 3.5  \\  % 1.74
     3252.0   &  15710  & 2.5  \\ % 0.85
     3262.5  &  20690 &  1.7  \\  % 0.49
    316327\hline
    317328\end{tabular}
    … …  
    330341  \frac{c}{H(z)} \, (1+z)^2 \times  \etaHI (\vec{\Theta}, z) 
    331342\end{equation}
    332 where $A_{21}=1.87 \, 10^{-15} \mathrm{s^{-1}}$ is the spontaneous 21 cm emission
     343where $A_{21}=2.85 \, 10^{-15} \mathrm{s^{-1}}$ \citep{lang.99} is the spontaneous 21 cm emission
    333344coefficient, $h$ is the Planck constant, $c$ the speed of light, $\kb$ the Boltzmann
    334345constant and $H(z)$ is the Hubble parameter at the emission redshift.
    … …  
    34435521 cm emission temperature can be written as:
    345356\begin{eqnarray}
    346 \frac{ \delta \etaHI}{\etaHI} (\vec{\Theta}, z(\lambda) ) & = & \gHIz \times \Omega_B  \frac{\rho_{crit}}{m_{H}}  \times
    347 \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) \\
    348  \TTlamz  &  = & \bar{T}_{21}(z) \times \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z)
     357\etaHI (\vec{\Theta}, z(\lambda) ) & = & \gHIz \times \Omega_B  \frac{\rho_{crit}}{m_{H}}  \times
     358\left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) + 1 \right) \\
     359 \TTlamz  &  = & \bar{T}_{21}(z) \times \left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z)  + 1 \right)
    349360\end{eqnarray}
    350361where $\Omega_B, \rho_{crit}$ are respectively the present day mean baryon cosmological
    … …  
    355366measured to be $\sim 1\%$ of the baryon density \citep{zwann.05}:
    356367$$ \Omega_{H_I} \simeq 3.5 \, 10^{-4} \sim 0.008 \times \Omega_B $$
    357 The neutral hydrogen fraction is expected to increase with redshift. Study
    358 of Lyman-$\alpha$ absorption indicate a factor 3 increase in the neutral hydrogen
     368The neutral hydrogen fraction is expected to increase with redshift, as gas is used
     369in star formation during galaxy formation and evolution. Study of Lyman-$\alpha$ absorption
     370indicate a factor 3 increase in the neutral hydrogen
    359371fraction at $z=1.5$ in the intergalactic medium \citep{wolf.05},
    360 compared to the its present day value $\gHI(z=1.5) \sim 0.025$.
     372compared to its present day value $\gHI(z=1.5) \sim 0.025$.
    361373The 21 cm brightness temperature and the corresponding power spectrum can be written as \citep{wyithe.07} :
    362374\begin{eqnarray}
    363375 P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
    364  \bar{T}_{21}(z)  & \simeq & 0.077  \, \mathrm{mK} 
     376 \bar{T}_{21}(z)  & \simeq & 0.084  \, \mathrm{mK} 
    365377\frac{ (1+z)^2 \, \hubb }{\sqrt{ \Omega_m (1+z)^3 + \Omega_\Lambda } }
    366378 \dfrac{\Omega_B}{0.044}  \,  \frac{\gHIz}{0.01}
    … …  
    368380\end{eqnarray}
    369381
    370 The table \ref{tabcct21} below shows the mean 21 cm brightness temperature for the
     382The table \ref{tabcct21} shows the mean 21 cm brightness temperature for the
    371383standard \LCDM cosmology and either a constant \HI mass fraction $\gHI = 0.01$, or
    372384linearly increasing  $\gHI \simeq 0.008 \times (1+z) $. Figure \ref{figpk21} shows the
    … …  
    376388shown for the standard WMAP \LCDM cosmology, according to the relation:
    377389\begin{equation}
    378 \theta_k = \frac{2 \pi}{k^{comov} \, \dang(z) \, (1+z) } 
     390\theta_k = \frac{2 \pi}{k \, \dang(z) \, (1+z) } 
    379391\hspace{3mm}
    380 k^{comov} = \frac{2 \pi}{ \theta_\mathrm{scale}  \, \dang(z) \, (1+z) } 
     392k = \frac{2 \pi}{ \theta_k  \, \dang(z) \, (1+z) } 
    381393\end{equation}
    382 where $k^{comov}$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance.
    383 It should be noted that the maximum transverse $k^{comov} $ sensitivity range
    384 for an instrument corresponds approximately to half of its angular resolution.
     394where $k$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance.
     395% It should be noted that the maximum transverse $k^{comov} $ sensitivity range
     396% for an instrument corresponds approximately to half of its angular resolution.
    385397% {\color{red} Faut-il developper completement le calcul en annexe ? }
    386398
    … …  
    390402\hline
    391403\hline
    392    & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\
     404 z  & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\
    393405\hline
    394 (a) $\bar{T}_{21}$ (mK) & 0.08 & 0.1 & 0.13 & 0.16 & 0.18 & 0.2 & 0.21 \\
     406(a) $\bar{T}_{21}$ & 0.085 & 0.107 & 0.145 & 0.174 & 0.195 & 0.216 & 0.234 \\
    395407\hline
    396 (b) $\bar{T}_{21}$ (mK)  & 0.08 & 0.12 & 0.21 & 0.32 & 0.43 & 0.56 & 0.68 \\
     408(b) $\bar{T}_{21}$  & 0.085 & 0.128 & 0.232 & 0.348 & 0.468 & 0.605 & 0.749 \\
    397409\hline
    398410\hline 
    … …  
    406418
    407419\begin{figure}
    408 \vspace*{-15mm}
     420\vspace*{-11mm}
    409421\hspace{-5mm}
    410422\includegraphics[width=0.57\textwidth]{Figs/pk21cmz12.pdf}
    … …  
    426438$I(\vec{\Theta},\lambda)$ in a given wave band, as a function of the sky direction. In radio astronomy
    427439and interferometry in particular, receivers are sensitive to the sky emission complex
    428 amplitudes. However, for most sources, the phases vary randomly and bear no information:
     440amplitudes. However, for most sources, the phases vary randomly with a spatial correlation
     441length significantly smaller than the instrument resolution.
    429442\begin{eqnarray}
    430443& &
    431444I(\vec{\Theta},\lambda)  =  | A(\vec{\Theta},\lambda) |^2  \hspace{2mm} , \hspace{1mm} I \in \mathbb{R}, A \in \mathbb{C} \\
    432 & & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  =  \delta(   \vec{\Theta} - \vec{\Theta '} )  I(\vec{\Theta},\lambda)
     445& & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time}  = 0 \hspace{2mm} \mathrm{for}   \hspace{1mm} \vec{\Theta} \ne \vec{\Theta '}   I(\vec{\Theta},\lambda)
    433446\end{eqnarray}
    434447A single receiver can be  characterized by its angular complex amplitude response $B(\vec{\Theta},\nu)$ and
    … …  
    440453\end{equation}
    441454We have set the electromagnetic (EM) phase origin at the center of the coordinate frame and
    442 the EM wave vector is related to the wavelength $\lambda$ through the usual
     455the EM wave vector is related to the wavelength $\lambda$ through the usual equation
    443456$ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta},\lambda)$
    444457corresponds to the receiver intensity response:
    445458\begin{equation}
    446 L(\vec{\Theta}), \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda)
     459L(\vec{\Theta}, \lambda) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda)
    447460\end{equation}
    448461The visibility signal of two receivers corresponds to the time averaged correlation between
    449462signals from two receivers. If we assume a sky signal with random uncorrelated phase, the
    450463visibility $\vis$ signal from two identical receivers, located at the position $\vec{r_1}$ and
    451 $\vec{r_2}$ can simply be written as a function their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
     464$\vec{r_2}$ can simply be written as a function of their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$
    452465\begin{equation}
    453466\vis(\lambda) = < s_1(\lambda) s_2(\lambda)^* > = \iint d \vec{\Theta} \, \, I(\vec{\Theta},\lambda) L(\vec{\Theta},\lambda)
    … …  
    455468\end{equation}
    456469This expression can be simplified if we consider receivers with narrow field of view
    457 ($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \mathrm{deg.} $ ),
     470($ L(\vec{\Theta},\lambda) \simeq  0$ for $| \vec{\Theta} | \gtrsim 10 \, \mathrm{deg.} $ ),
    458471and coplanar in respect to their common axis.
    459472If we introduce two {\em Cartesian} like angular coordinates $(\alpha,\beta)$ centered at
    460473the common receivers axis, the visibilty would be written as the 2D Fourier transform
    461474of the product of the sky intensity and the receiver beam, for the angular frequency
    462 \mbox{$(u,v)_{12} = 2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$}:
     475\mbox{$(u,v)_{12} = 2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$}:
    463476\begin{equation}
    464477\vis(\lambda) \simeq  \iint d\alpha d\beta \, \, I(\alpha, \beta) \,  L(\alpha, \beta)
    … …  
    481494The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular
    482495wave number domain located around
    483 $(u, v)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$. The weight function is
     496$(u, v)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is
    484497given by the receiver beam Fourier transform.
    485498\begin{equation}
    … …  
    504517the instrument response is simply the Fourier transform of the beam.
    505518For a single dish with multiple receivers, either as a Focal Plane Array (FPA) or
    506 a multi horn system, each beam (b) will have its own response
     519a multi-horn system, each beam (b) will have its own response
    507520${\cal R}_b(u,v,\lambda)$.
    508521For an interferometer, we can compute a raw instrument response
    … …  
    529542}
    530543\vspace*{-15mm}
    531 \caption{Schematic view of the $(u,v)$ plane coverage by interferometric measurement}
     544\caption{Schematic view of the $(u,v)$ plane coverage by interferometric measurement.}
    532545\label{figuvplane}
    533546\end{figure}
    … …  
    536549\label{instrumnoise}
    537550Let's consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency
    538 bandwidth $\delta \nu$, with an integration time $t_{int}$, characterized by a system temperature
     551bandwidth $\delta \nu$ centered on $\nu_0$, with an integration time $t_{int}$, characterized by a system temperature
    539552$\Tsys$. The uncertainty or fluctuations of this measurement due to the receiver noise can be written as
    540553$\sigma_{noise}^2 = \frac{2 \Tsys^2}{t_{int} \, \delta \nu}$. This term
    … …  
    553566the angular frequencies plane $P_{noise}^{(u,v)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$ 
    554567per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$:
    555 We can characterize the sky temperature measurement by a radio instrument by the noise
     568We can characterize the sky temperature measurement with a radio instrument by the noise
    556569spectral power density in the angular frequencies plane $P_{noise}(u,v)$ in units of $\mathrm{Kelvin^2}$ 
    557570per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$.
    558571For an interferometer made of identical receiver elements, several ($n$) receiver pairs
    559572might have the same baseline. The noise power density in the corresponding $(u,v)$ plane area
    560 is then reduced by a factor $1/n$. More generally, we cam write the instrument  noise
     573is then reduced by a factor $1/n$. More generally, we can write the instrument  noise
    561574spectral power density using the instrument response defined in section \ref{instrumresp} :
    562575\begin{equation}
    … …  
    585598
    586599$P_{noise}(k)$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$,
    587 $t_{int}$ in second, $\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and
     600$t_{int}$ is the integration time expressed in second,
     601$\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and
    588602 $H(z)$ in $\mathrm{km/s/Mpc}$.
    589603
    … …  
    596610A single dish instrument with diameter $D$ would have an instantaneous field of view
    597611$\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and would require
    598 a number of pointing $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area.
     612a number of pointings $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area.
    599613Each sky direction or pixel of size $\Omega_{FOV}$ will be observed during an integration
    600614time $t_{int} = t_{obs}/N_{point} $. Using equation \ref{ctepnoisek} and the previous expression
    … …  
    607621It is important to note that any real instrument do not have a flat
    608622response in the $(u,v)$ plane, and the observations provide no information above
    609 a maximum angular frequency $u_{max},v_{max}$.
     623a certain maximum angular frequency $u_{max},v_{max}$.
    610624One has to take into account either a damping of the observed sky power
    611625spectrum or an increase of the noise spectral power if
    … …  
    617631phase array receiver system, with $N$ independent beams on sky,
    618632the noise spectral density decreases by a factor $N$,
    619 thanks to the  increase of per pointing integration time.
     633thanks to the  increase of per pointing integration time:
    620634
    621635\begin{equation}
    … …  
    624638\end{equation}
    625639
    626 The expression above (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$
     640This expression (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$
    627641identical receivers with a  total collection area $\sim D^2$. Such an array could be made for example
    628642of $N=q \times q$ {\it small dishes}, each with diameter $D/q$, arranged as $q \times q$ square.   
    … …  
    631645observations provide information up to $u_{max},v_{max} \lesssim 2 \pi D / \lambda $. This value of
    632646$u_{max},v_{max}$ would be mapped to a maximum transverse cosmological wave number
    633 $k^{comov}_{\perp \, max}$:
    634 \begin{equation}
    635 k^{comov}_{\perp}  =  \frac{(u,v)}{(1+z) \dang}  \hspace{8mm}
    636 k^{comov}_{\perp \, max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}}
     647$k^{\perp}_{max}$:
     648\begin{equation}
     649k^{\perp}  =  \frac{(u,v)}{(1+z) \dang}  \hspace{8mm}
     650k^{\perp}_{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}}
    637651\label{kperpmax}
    638652\end{equation}   
    … …  
    642656beams and a system noise temperature $\Tsys = 50 \mathrm{K}$.
    643657The survey is supposed to cover a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$, in one
    644 year. The maximum comoving wave number $k^{comov}$  is also shown as a function
     658year. The maximum comoving wave number $k_{max}$  is also shown as a function
    645659of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. In order
    646 to take into account the radial component of $\vec{k^{comov}}$ and the increase of
    647 the instrument noise level with $k^{comov}_{\perp}$, we have taken the effective $k^{comov}_{ max}  $
    648 as half of the maximum transverse $k^{comov}_{\perp \, max} $ of \mbox{eq. \ref{kperpmax}}:
    649 \begin{equation}
    650 k^{comov}_{ max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}}
     660to take into account the radial component of $\vec{k}$ and the increase of
     661the instrument noise level with $k^{\perp}$, we have taken the effective $k_{ max}  $
     662as half of the maximum transverse $k^{\perp} _{max}$ of \mbox{eq. \ref{kperpmax}}:
     663\begin{equation}
     664k_{max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}}
    651665\end{equation}
    652666
    … …  
    676690in 8 rows, each with 16 dishes. These 128 dishes are spread over an area
    677691$80 \times 80  \, \mathrm{m^2}$. The array layout for this configuration is
    678 shown in figure \ref{figconfab}.
     692shown in figure \ref{figconfbc}.
    679693\item [{\bf c} :] An array of $n=129  \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged
    680694 over an area $80 \times 80  \, \mathrm{m^2}$. This configuration has in
    681695particular 4 sub-arrays of packed 16 dishes ($4\times4$), located in the
    682 four array corners. This array layout is also shown figure \ref{figconfab}.
     696four array corners. This array layout is also shown figure \ref{figconfbc}.
    683697\item [{\bf d} :] A single dish instrument, with diameter $D=75 \, \mathrm{m}$,
    684698equipped with a 100 beam focal plane receiver array.
    … …  
    716730\caption{ Array layout for configurations (b) and (c) with 128 and 129  D=5 meter
    717731diameter dishes. }
    718 \label{figconfab}
     732\label{figconfbc}
    719733\end{figure}
    720734
    … …  
    723737$\eta$, relating the effective dish diameter $D_{ill}$ to the
    724738mechanical dish size $D^{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales
    725 as $\eta^2$ or $eta_x \eta_y$.
     739as $\eta^2$ or $\eta_x \eta_y$.
    726740\begin{eqnarray}
    727741{\cal L}_\circ (u,v,\lambda) & = & \bigwedge_{[\pm 2 \pi D^{ill}/ \lambda]}(\sqrt{u^2+v^2})  \\
    … …  
    762776\includegraphics[width=0.90\textwidth]{Figs/uvcovabcd.pdf}
    763777}
    764 \caption{(u,v) plane coverage (non normalized instrument response ${\cal R}(u,v,\lambda)$
     778\caption{(u,v) plane coverage (raw instrument response ${\cal R}(u,v,\lambda)$
    765779for four configurations.
    766780(a) 121 $D_{dish}=5$ meter diameter dishes arranged in a compact, square array
    767 of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each,
    768 (c) 129 dishes arranged as above, single D=65 meter diameter, with 100 beams.
    769 color scale : black $<1$, blue, green, yellow, red $\gtrsim 80$ }
     781of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each (fig. \ref{figconfbc}),
     782(c) 129 dishes arranged as shown in figure \ref{figconfbc} , (d) single D=75 meter diameter, with 100 beams.
     783(color scale : black $<1$, blue, green, yellow, red $\gtrsim 80$) }
    770784\label{figuvcovabcd}
    771785\end{figure*}
    … …  
    789803Reaching the required sensitivities is not the only difficulty of observing the large
    790804scale structures in 21 cm. Indeed, the synchrotron emission of the
    791 Milky Way and the extra galactic radio sources are a thousand time brighter than the
     805Milky Way and the extra galactic radio sources are a thousand times brighter than the
    792806emission of the neutral hydrogen distributed in the universe. Extracting the LSS signal
    793807using Intensity Mapping, without identifying the \HI point sources is the main challenge
    … …  
    813827brightness $T(\alpha, \delta, \nu)$ as a function of two equatorial angular coordinates $(\alpha, \delta)$
    814828and the frequency $\nu$. Unless otherwise specified, the results presented here are based on simulations of
    815 $90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10:00 \, \mathrm{h} , \delta=+10 \, \mathrm{deg.}$,
    816 and  covering 128 MHz in frequency. The sky cube characteristics (coordinate range, size, resolution)
    817 used in the simulations is given in the table \ref{skycubechars}.
     829$90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10\mathrm{h}00\mathrm{m} , \delta=+10 \, \mathrm{deg.}$, and  covering 128 MHz in frequency. We have selected this particular area of the sky to in order to minimize
     830the Galactic synchrotron foreground. The sky cube characteristics (coordinate range, size, resolution)
     831used in the simulations are given in the table \ref{skycubechars}.
    818832\begin{table}
    819833\begin{center}
    … …  
    875889A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source
    876890map; we have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the
    877 contribution of the radiosources to the sky temperature is computed as follow:
     891contribution of the radiosources to the sky temperature is computed as follows:
    878892$$ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 MHz}\right)^{\beta_{src}} $$
    879893%%
    … …  
    884898
    885899 The 21 cm temperature fluctuations due to neutral hydrogen in large scale structures
    886 $T_{lss}(\alpha, \delta, \nu)$  has been computed using the SimLSS software package
    887 \footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }. 
    888 {\color{red}: CMV, please add few line description of SimLSS}.
    889 We have generated the mass fluctuations $\delta \rho/\rho$ at $z=0.6$, in cells of size
    890 $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the
    891 sky cube angular and frequency resolution defined above.  The mass fluctuations has been
     900$T_{lss}(\alpha, \delta, \nu)$  have been computed using the
     901SimLSS \footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }  software package:
     902%
     903complex normal Gaussian fields were first generated in Fourier space.
     904The amplitude of each mode was then multiplied by the square root
     905of the power spectrum $P(k)$ at $z=0$ computed according to the parametrization of
     906\citep{eisenhu.98}. We have used the standard cosmological parameters,
     907 $H_0=71 \mathrm{km/s/Mpc}$, $\Omega_m=0.27$, $\Omega_b=0.044$,
     908$\Omega_\lambda=0.73$ and $w=-1$.
     909An inverse FFT was then performed to compute the matter density fluctuations
     910in the linear regime,
     911$\delta \rho / \rho$ in a box of $3420 \times 1140 \times 716  \, \mathrm{Mpc^3}$ and evolved
     912to redshift $z=0.6$.
     913The size of the box is about 2500 $\mathrm{deg^2}$  in the transverse direction and
     914$\Delta z \simeq 0.23$ in the longitudinal direction. 
     915The size of the cells is  $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the
     916sky cube angular and frequency resolution defined above. 
     917
     918The mass fluctuations has been
    892919converted into temperature through a factor $0.13 \, \mathrm{mK}$, corresponding to a hydrogen
    893920fraction $0.008 \times (1+0.6)$, using equation \ref{eq:tbar21z}. 
    … …  
    940967
    941968It should also be noted that in section 3, we presented the different instrument
    942 configuration noise level after {\em correcting or deconvolving} the instrument response. The LSS
     969configuration noise levels after {\em correcting or deconvolving} the instrument response. The LSS
    943970power spectrum is recovered unaffected in this case, while the noise power spectrum
    944971increases at high k values (small scales). In practice, clean deconvolution is difficult to
    … …  
    9941021$T_{sky}(\alpha, \delta, \nu)$ by applying the frequency or wavelength dependent instrument response
    9951022${\cal R}(u,v,\lambda)$.
    996 we have considered the simple case where  the instrument response constant throughout the survey area, or independent
     1023We have considered the simple case where  the instrument response is constant throughout the survey area, or independent
    9971024of the sky direction.
    9981025For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$ :
    … …  
    10211048\begin{enumerate}
    10221049\item The measured sky brightness temperature is first {\em corrected} for the frequency dependent
    1023 beam effects through a convolution by a virtual, frequency independent beam. This {\em correction}
     1050beam effects through a convolution by a fiducial frequency independent beam. This {\em correction}
    10241051corresponds to a smearing or degradation of the angular resolution. We assume
    10251052that we have a perfect knowledge of the intrinsic instrument response, up to a threshold numerical level
    … …  
    10311058 \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
    10321059 is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in
    1033 the $\log10 ( T ) , \log10 (\nu)$ plane and we show here the results for a pure power law (P1)
     1060the $\lgd ( T ) , \lgd (\nu)$ plane and we show here the results for a pure power law (P1)
    10341061or modified power law (P2):
    10351062\begin{eqnarray*}
    1036 P1 & :  & \log10 ( T_{mes}^{bcor}(\nu) ) = a + b \log10 ( \nu / \nu_0 ) \\
    1037 P2 & :  & \log10 ( T_{mes}^{bcor}(\nu) ) = a + b \log10 ( \nu / \nu_0 ) + c \log10 ( \nu/\nu_0 ) ^2
     1063P1 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) \\
     1064P2 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) + c \, \lgd ( \nu/\nu_0 ) ^2
    10381065\end{eqnarray*}
    10391066where $b$ is the power law index and  $T_0 = 10^a$ is the brightness temperature at the
    … …  
    10551082with the recovered 21 cm map, after subtraction of the radio continuum component. It can be seen that structures
    10561083present in the original map have been correctly recovered, although the amplitude of the temperature
    1057 fluctuations on the recovered map is significantly smaller (factor $sim 5$) than in the original map. This is mostly
     1084fluctuations on the recovered map is significantly smaller (factor $\sim 5$) than in the original map. This is mostly
    10581085due to the damping of the large scale ($k \lesssim 0.04 h \mathrm{Mpc^{-1}} $) due the poor interferometer
    10591086response at large angle    ($\theta \gtrsim 4^\circ $).
    … …  
    11121139The sky reconstruction and the component separation introduce additional filtering and distortions.
    11131140Ideally, one has to define a power spectrum measurement response or {\it transfer function} in the
    1114 radial direction,  ($\lambda$ or redshift, $TF(k_\parallel)$) and in the transverse plane ( $TF(k_\perp)$ ).
     1141radial direction,  ($\lambda$ or redshift, $\TrF(k_\parallel)$) and in the transverse plane ( $\TrF(k_\perp)$ ).
    11151142The real transverse plane transfer function might even be anisotropic.
    11161143
    1117 However, in the scope of the present study, we define an overall transfer function $TF(k)$ as the ratio of the
     1144However, in the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the
    11181145recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$:
    11191146\begin{equation}
    1120 TF(k) = P_{21}^{rec}(k) / P_{21}(k)
     1147\TrF(k) = P_{21}^{rec}(k) / P_{21}(k)
    11211148\end{equation}
    11221149
    … …  
    11311158the frequency or redshift direction ($k_\parallel$) by the component separation algorithm.
    11321159The red curve shows the ratio of $P(k)$ computed on the recovered or extracted 21 cm LSS signal, to the original
    1133 LSS temperature cube $P_{21}^{rec}(k)/P_{21}(k)$ and corresponds to the transfer function $TF(k)$ defined above,
     1160LSS temperature cube $P_{21}^{rec}(k)/P_{21}(k)$ and corresponds to the transfer function $\TrF(k)$ defined above,
    11341161for $z=0.6$ and instrument setup (a).
    11351162The black (thin line) curve shows the ratio of recovered to the smoothed
    11361163power spectrum $P_{21}^{rec}(k)/P_{21}^{smoothed}(k)$. This latter ratio (black curve) exceeds one for $k \gtrsim 0.2$, which is
    1137 due to the noise or system temperature. It should stressed that the simulations presented in this section were
     1164due to the noise or system temperature. It should be stressed that the simulations presented in this section were
    11381165focused on the study of the radio foreground effects and have been carried intently with a very low instrumental noise level of
    11391166$0.25$ mK per pixel, corresponding to several years of continuous observations ($\sim 10$ hours per $3' \times 3'$ pixel).
    11401167
    1141 This transfer function is well represented a the analytical form:
    1142 \begin{equation}
    1143 TF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right)
     1168This transfer function is well represented by the analytical form:
     1169\begin{equation}
     1170\TrF(k) = \sqrt{ \frac{ k-k_A}{ k_B}  } \times \exp \left( - \frac{k}{k_C} \right)
    11441171\label{eq:tfanalytique}
    11451172\end{equation}
    … …  
    11571184
    11581185\begin{table}[hbt]
     1186\begin{center}
    11591187\begin{tabular}{|c|ccccc|}
    11601188\hline
    … …  
    11671195\hline
    11681196\end{tabular}
     1197\end{center}
    11691198\caption{Value of the parameters for the transfer function (eq. \ref{eq:tfanalytique}) at different redshift
    11701199for instrumental setup (e), $20\times20$ packed array interferometer.  }
    … …  
    11801209}
    11811210\vspace*{-35mm}
    1182 \caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $TF(k) = P_{21}^{rec}(k) / P_{21}(k) $, at $z \sim 0.6$,  for the instrument configuration (a), $11\times11$ packed array interferometer.
     1211\caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $, at $z \sim 0.6$,  for the instrument configuration (a), $11\times11$ packed array interferometer.
    11831212Left: GSM/Model-I , right: Haslam+NVSS/Model-II.  }
    11841213\label{extlssratio}
    … …  
    11941223}
    11951224\vspace*{-30mm}
    1196 \caption{Fitted/smoothed  transfer function obtained for the recovered 21 cm power spectrum at different redshifts,
     1225\caption{Fitted/smoothed  transfer function $\TrF(k)$ obtained for the recovered 21 cm power spectrum at different redshifts,
    11971226$z=0.5 , 1.0 , 1.5 , 2.0 , 2.5$ for the instrument configuration (e), $20\times20$ packed array interferometer. }
    11981227\label{tfpkz0525}
    … …  
    12211250\label{cosmosec}
    12221251
    1223 In section \ref{pkmessens},
    12241252The impact of the various telescope configurations on the sensitivity for 21 cm
    12251253power spectrum measurement has been discussed in section \ref{pkmessens}.
    1226 Fig.~\ref{powerfig} shows the noise power spectra, and allows us to rank visually the configurations
     1254Fig. \ref{figpnoisea2g} shows the noise power spectra, and allows us to rank visually the configurations
    12271255in terms of instrument noise contribution to P(k) measurement.
    12281256The differences in $P_{noise}$ will translate into differing precisions
    … …  
    12521280The reconstructed power spectrum used in the simulation is 
    12531281the sum of the expected \HI signal term, corresponding to equations \ref{eq:pk21z} and \ref{eq:tbar21z},
    1254 damped by the transfer function $TF(k)$ (Eq. \ref{eq:tfanalytique} , table \ref{tab:paramtfk})
     1282damped by the transfer function $\TrF(k)$ (Eq. \ref{eq:tfanalytique} , table \ref{tab:paramtfk})
    12551283and a white noise component $P_{noise}$ calculated according to the equation \ref{eq:pnoiseNbeam},
    12561284established in section \ref{instrumnoise} with $N=400$:
    12571285\begin{equation}
    1258  P^{rec}(k) = P_{21}(k) \times TF(k) + P_{noise}
     1286 P^{rec}(k) = P_{21}(k) \times \TrF(k) + P_{noise}
    12591287\end{equation}
    1260 where the different terms ($P_{21}(k) , TF(k), P_{noise}$depend on the slice redshift. 
     1288where the different terms ($P_{21}(k) , \TrF(k), P_{noise}$) depend on the slice redshift. 
    12611289The expected 21 cm power spectrum $P_{21}(k)$ has been generated according to the formula:
    12621290%\begin{equation}
    … …  
    12991327 
    13001328Figure \ref{fig:fitOscill} shows the result of the fit for
    1301 one of theses simulations.
     1329one of these simulations.
    13021330Figure \ref{fig:McV2} histograms the recovered values of  $\koperp$ and $\kopar$
    13031331for 100 simulations.
    13041332The widths of the two distributions give an estimate
    1305 the statistical errors.
     1333of the statistical errors.
    13061334
    13071335In addition, in the fitting procedure, both the parameters modeling the
    … …  
    15031531(Eq. \ref{eq:dTdH}) by:
    15041532\begin{equation}
    1505 \Omega_\Lambda = \Omega_{\Lambda}^0 \exp \left[ 3  \int_0^z   
     1533\Omega_\Lambda \rightarrow \Omega_{\Lambda} \exp \left[ 3  \int_0^z   
    15061534\frac{1+w(z^\prime)}{1+z^\prime } dz^\prime  \right]
    15071535\end{equation}
    … …  
    15511579to perform a cosmological neutral hydrogen survey over a significant fraction of the sky. We have shown that
    15521580a nearly packed interferometer array with few hundred receiver elements spread over an hectare or a hundred beam
    1553 focal plane array with a $\sim 100$ meter primary reflector will have the required sensitivity to measure
     1581focal plane array with a $\sim 100 \, \mathrm{meter}$ primary reflector will have the required sensitivity to measure
    15541582the 21 cm power spectrum. A method to compute the instrument response for interferometers 
    15551583has been developed and we have  computed the noise power spectrum for various telescope configurations.
    … …  
    15581586emissions in the GHz domain and simulation of interferometric observations.
    15591587We have been able to show that the cosmological 21 cm signal from the LSS should be observable, but
    1560 requires a very good knowledge of the instrument response. Our method has allowed to define and
     1588requires a very good knowledge of the instrument response. Our method has allowed us to define and
    15611589compute the overall  {\it transfer function} or {\it response function} for the measurement of the 21 cm
    15621590power spectrum.
    1563 Finally, we have used the computed noise power spectrum and  P(k)
     1591Finally, we have used the computed noise power spectrum and  $P(k)$
    15641592measurement response function to estimate
    15651593the precision on the determination of Dark Energy parameters, for a 21 cm BAO survey. Such a radio survey
    1566 could be carried using the current technology and would be comptetitive with the ongoing or planned
     1594could be carried using the current technology and would be competitive with the ongoing or planned
    15671595optical surveys for dark energy,  with a fraction of their cost.
    15681596 
    … …  
    15911619Glazebrook, K. \&  Blake, C. 2005 \apj, 631, 1
    15921620
     1621% WiggleZ BAO observation
     1622\bibitem[Blake et al. (2011)]{blake.11} Blake, Davis, T., Poole, G.B.  {\it et al.}  2011, \mnras  (arXiv/1105.2862)
     1623
    15931624% Galactic astronomy, emission HI d'une galaxie
    15941625\bibitem[Binney \& Merrifield (1998)]{binney.98} Binney J. \& Merrifield M. , 1998 {\it Galactic Astronomy} Princeton University Press
    … …  
    16101641
    16111642%  Parametrisation P(k)
    1612 \bibitem[Eisentein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, ApJ 496:605-614 (astro-ph/9709112)
     1643\bibitem[Eisenstein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, ApJ 496:605-614 (astro-ph/9709112)
    16131644
    16141645% SDSS first BAO observation
    1615 \bibitem[Eisentein et al. (2005)]{eisenstein.05}  Eisenstein D. J., Zehavi, I., Hogg, D.W. {\it et al.}, (the SDSS Collaboration) 2005,  \apj, 633, 560
     1646\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
    16161647
    16171648% SDSS-III description
    1618 \bibitem[Eisentein et al. (2011)]{eisenstein.11}  Eisenstein D. J., Weinberg, D.H., Agol, E. {\it et al.}, 2011, arXiv:1101.1529
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    16191650
    16201651%   21 cm emission for mapping matter distribution 
    … …  
    16341665\bibitem[Lah et al. (2009)]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur et al.  2009,  \mnras
    16351666( astro-ph/0907.1416)
     1667
     1668% Livre Astrophysical Formulae de Lang
     1669\bibitem[Lang (1999)]{radastron} Lang, K.R. {\it Astrophysical Formulae}, Springer, 3rd Edition 1999
    16361670
    16371671% LSST Science book
    … …  
    16851719
    16861720%  Thomson-Morane livre interferometry
    1687 \bibitem[Radio Astronomy (1998)]{radastron} Thompson, A.R., Moran, J.M., Swenson, G.W, {\it Interferometry and
     1721\bibitem[Thompson, Moran \& Swenson (2001)]{radastron} Thompson, A.R., Moran, J.M., Swenson, G.W, {\it Interferometry and
    16881722Synthesis in Radio Astronomy}, John Wiley \& sons, 2nd Edition 2001
    16891723
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