Changeset 4043 in Sophya


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Timestamp:
Dec 12, 2011, 10:24:25 PM (12 years ago)
Author:
ansari
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Modifs texte papier pour version V3 modifee en reponse au 2eme rapport du referee, Reza 12/12/2011

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  • trunk/Cosmo/RadioBeam/sensfgnd21cm.tex

    r4032 r4043  
    6666
    6767% Commande pour marquer les changements du papiers pour le referee
    68 \def\changemark{\bf }
    69 % \def\changemark{ }
     68% \def\changemark{\bf }
     69\def\changemark{}
     70\def\changemarkb{\bf }
     71
    7072
    7173%%% Definition pour la section sur les param DE par C.Y
    … …  
    9294          \inst{1} \inst{2}
    9395          \and
    94           J.E. Campagne \inst{3}
     96          J.E. Campagne \inst{2} 
    9597         \and
    96           P.Colom  \inst{5}
     98          P.Colom  \inst{3}
    9799          \and
    98100          J.M. Le Goff \inst{4}
    … …  
    102104          J.M. Martin  \inst{5}
    103105          \and
    104           M. Moniez \inst{3}
     106          M. Moniez \inst{2} 
    105107        \and
    106108          J.Rich \inst{4}
    … …  
    110112
    111113   \institute{
    112      Universit\'e Paris-Sud, LAL, UMR 8607, F-91898 Orsay Cedex, France
    113    \and
    114     CNRS/IN2P3,  F-91405 Orsay, France \\
     114     Universit\'e Paris-Sud, LAL, UMR 8607, CNRS/IN2P3,  F-91405 Orsay, France
    115115     \email{ansari@lal.in2p3.fr}
    116116    \and
    117     Laboratoire de lÍAcc\'el\'erateur Lin\'eaire, CNRS-IN2P3, Universit\'e Paris-Sud,
     117    CNRS/IN2P3, Laboratoire de l'Acc\'el\'erateur Lin\'eaire (LAL)
    118118    B.P. 34, 91898 Orsay Cedex, France
     119    \and
     120    LESIA, UMR 8109, Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon Cedex, France
    119121    % \thanks{The university of heaven temporarily does not
    120122    %                 accept e-mails}
    … …  
    123125    \and
    124126    GEPI, UMR 8111, Observatoire de Paris, 61 Ave de l'Observatoire, 75014 Paris, France
    125    }
     127  }
    126128
    127129   \date{Received August 5, 2011; accepted xxxx, 2011}
    … …  
    142144  % methods heading (mandatory)
    143145 { For each configuration, we determine instrument response by computing the $(\uv)$ or Fourier angular frequency
    144 plane  coverage using visibilities. The $(\uv)$ plane response is the noise power spectrum,
     146plane  coverage using visibilities. The $(\uv)$ plane response determines the noise power spectrum,
    145147hence the instrument sensitivity for LSS P(k) measurement. We describe also   a simple foreground subtraction method to
    146148separate LSS 21 cm signal from the foreground due to the galactic synchrotron and radio sources emission. }
    147149  % results heading (mandatory)
    148    { We have computed the noise power spectrum for different instrument configuration as well as the extracted
     150   { We have computed the noise power spectrum for different instrument configurations as well as the extracted
    149151   LSS power spectrum, after separation of 21cm-LSS signal from the foregrounds. We have also obtained
    150152  the uncertainties on the Dark Energy parameters for an optimized 21 cm BAO survey.}
    151153  % conclusions heading (optional), leave it empty if necessary
    152154   { We show that a radio instrument with few hundred simultaneous beams and a collecting area of
    153   $\sim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ and will be
     155  \mbox{$\sim 10000 \, \mathrm{m^2}$} will be able to  detect BAO signal at redshift z $\sim 1$ and will be
    154156  competitive with optical surveys. }
    155157
    … …  
    171173optical observation of galaxies which are used as a tracer of the underlying matter distribution.
    172174An alternative and elegant approach for mapping the matter distribution, using neutral atomic hydrogen
    173 (\HI) as a tracer with intensity mapping, has been proposed in recent years \citep{peterson.06} \citep{chang.08}.
    174 Mapping the matter distribution using HI 21 cm emission as a tracer has been extensively discussed in literature
     175(\HI) as a tracer with intensity mapping has been proposed in recent years (\cite{peterson.06} , \cite{chang.08}).
     176Mapping the matter distribution using \HI 21 cm emission as a tracer has been extensively discussed in literature
    175177\citep{furlanetto.06} \citep{tegmark.09} and is being used in projects such as LOFAR \citep{rottgering.06} or
    176178MWA \citep{bowman.07} to observe reionisation  at redshifts z $\sim$ 10.
    … …  
    191193BAO are features imprinted  in the distribution of galaxies, due to the frozen
    192194sound waves which were present in the photon-baryon plasma prior to recombination
    193 at z $\sim$ 1100.
     195at \mbox{$z \sim 1100$}.
    194196This scale can be considered as a standard ruler with a comoving
    195 length  of $\sim 150 \mathrm{Mpc}$.
     197length  of \mbox{$\sim 150 \mathrm{Mpc}$}.
    196198These features have been first observed in the CMB anisotropies
    197199and are usually referred to as {\em acoustic peaks} (\cite{mauskopf.00}, \cite{larson.11}).
    … …  
    201203WiggleZ \citep{blake.11} optical galaxy surveys.
    202204
    203 Ongoing \citep{eisenstein.11}   or future surveys \citep{lsst.science} 
     205Ongoing {\changemarkb surveys such as BOSS} \citep{eisenstein.11}  or future surveys
     206{\changemarkb such as LSST} \citep{lsst.science}
    204207plan to measure precisely the BAO scale in the redshift range
    205208$0 \lesssim z \lesssim 3$, using either optical observation of galaxies 
    206 or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars
     209or through 3D mapping of Lyman $\alpha$ absorption lines toward distant quasars
    207210\citep{baolya},\citep{baolya2}.
    208211Radio observation of the 21 cm emission of neutral hydrogen appears as
    … …  
    236239and thus can be resolved easily with a rather modest size radio instrument
    237240(diameter $D \lesssim 100 \, \mathrm{m}$). The specific BAO clustering scale ($k_{\mathrm{BAO}}$)
    238 can be measured both in the transverse plane (angular correlation function, ($k_{\mathrm{BAO}}^\perp$)
    239 or along the longitudinal (line of sight or redshift ($k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of
     241can be measured both in the transverse plane (angular correlation function, $k_{\mathrm{BAO}}^\perp$)
     242or along the longitudinal (line of sight or redshift $k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of
    240243the Hubble parameter $H(z)$ can be obtained by comparing   the longitudinal and transverse
    241244BAO scales. A reasonably good redshift resolution $\delta z \lesssim 0.01$ is needed to resolve
    … …  
    284287WMAP \LCDM universe \citep{komatsu.11}. $10^9 - 10^{10} M_\odot$ of neutral gas mass
    285288is typical for large galaxies \citep{lah.09}. It is clear that detection of \HI sources at cosmological distances
    286 would require collecting area in the range of $10^6 \mathrm{m^2}$.
     289would require collecting area in the range of \mbox{$10^6 \, \mathrm{m^2}$}.
    287290
    288291Intensity mapping has been suggested as an alternative and economic method to map the
    289 3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} \citep{seo.10}.
    290 {\changemark There have been attempts to detect the 21 cm LSS signal at GBT
     2923D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} and \citep{seo.10}.
     293{\changemark There have also been attempts to detect the 21 cm LSS signal at GBT
    291294\citep{chang.10} and at GMRT \citep{ghosh.11}}.
    292 In this approach, sky brightness map with angular resolution $\sim 10-30 \, \mathrm{arc.min}$ is made for a
     295In this approach, sky brightness map with angular resolution \mbox{$\sim 10-30 \, \mathrm{arc.min}$} is made for a
    293296wide range of frequencies. Each 3D pixel  (2 angles $\vec{\Theta}$, frequency $\nu$ or wavelength $\lambda$) 
    294297would correspond to a cell with a volume of $\sim 10^3 \mathrm{Mpc^3}$, containing ten to hundred galaxies
    … …  
    304307\hspace{1mm} \mathrm{with}   \hspace{1mm}  \lambda_{21} = 0.211 \, \mathrm{m}
    305308\end{eqnarray}
    306 The large scale distribution of the neutral hydrogen, down to angular scales of $\sim 10 \mathrm{arc.min}$
     309The large scale distribution of the neutral hydrogen, down to angular scale of \mbox{$\sim 10 \, \mathrm{arc.min}$}
    307310can then be observed without the detection of individual compact \HI sources, using the set of sky brightness
    308311map as a function of frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$
    309312(radiation power/unit solid angle/unit surface/unit frequency)
    310 can be converted to brightness temperature using the well known black body Rayleigh-Jeans approximation:
     313can be converted to brightness temperature using the Rayleigh-Jeans approximation of  black body radiation law:
    311314$$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} $$
    312315 
    313316%%%%%%%%
    314317\begin{table}
     318\caption{Sensitivity or source detection limit for 1 day integration time (86400 s) and 1 MHz
     319frequency band (left). 21 cm brightness for $10^{10} M_\odot$ \HI for different redshifts (right)  }
     320\label{slims21}
    315321\begin{center}
    316322\begin{tabular}{|c|c|c|}
    … …  
    341347\hline
    342348\end{tabular}
    343 \caption{Sensitivity or source detection limit for 1 day integration time (86400 s) and 1 MHz
    344 frequency band (left). Source 21 cm brightness for $10^{10} M_\odot$ \HI for different redshifts (right)  }
    345 \label{slims21}
    346349\end{center}
    347350\end{table}
    … …  
    358361where $A_{21}=2.85 \, 10^{-15} \mathrm{s^{-1}}$ \citep{astroformul} is the spontaneous 21 cm emission
    359362coefficient, $h$ is the Planck constant, $c$ the speed of light, $\kb$ the Boltzmann
    360 constant and $H(z)$ is the Hubble parameter at the emission redshift.
     363constant and $H(z)$ is the Hubble parameter at the emission
     364redshift {\changemarkb (\cite{field.59} , \cite{zaldarriaga.04})}.
    361365For a \LCDM universe and neglecting radiation energy density, the Hubble parameter
    362366can be expressed as:
    … …  
    387391compared to its present day value $\gHI(z=1.5) \sim 0.025$.
    388392The 21 cm brightness temperature and the corresponding power spectrum can be written as
    389 (\cite{barkana.07} and \cite{madau.97}) :
     393(\cite{madau.97}, \cite{zaldarriaga.04}), \cite{barkana.07}) :
    390394\begin{eqnarray}
    391395 P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z)  \right)^2 \, P(k)    \label{eq:pk21z} \\
    … …  
    420424
    421425\begin{table}
    422 \begin{center}
     426\caption{Mean 21 cm brightness temperature in mK, as a function of redshift, for the
     427standard \LCDM cosmology with constant \HI mass fraction at $\gHIz$=0.01  (a) or linearly
     428increasing mass fraction (b)  $\gHIz=0.008(1+z)$ }
     429\label{tabcct21}
     430% \begin{center}
    423431\begin{tabular}{|l|c|c|c|c|c|c|c|}
    424432\hline
    … …  
    432440\hline 
    433441\end{tabular}
    434 \caption{Mean 21 cm brightness temperature in mK, as a function of redshift, for the
    435 standard \LCDM cosmology with constant \HI mass fraction at $\gHIz$=0.01  (a) or linearly
    436 increasing mass fraction (b)  $\gHIz=0.008(1+z)$ }
    437 \label{tabcct21}
    438 \end{center}
     442%\end{center}
    439443\end{table}
    440444
    441445\begin{figure}
    442 \vspace*{-11mm}
     446\vspace*{-4mm}
    443447\hspace{-5mm}
    444448\includegraphics[width=0.57\textwidth]{Figs/pk21cmz12.pdf}
    … …  
    516520The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular
    517521wave number domain located around
    518 $(\uv)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is
     522$(\uv)_{12}=( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is
    519523given by the receiver beam Fourier transform.
    520524\begin{equation}
    … …  
    586590corresponds also to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated
    587591noise. If the receiver has an effective area $A \simeq \pi D^2/4$ or $A \simeq D_x D_y$, the measurement
    588 corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. The noise spectral density, in the angular frequencies plane (per unit area of angular frequencies  $\delta \uvu \times \uvv$), corresponding to a visibility
     592corresponds 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 
     593\mbox{$\delta \uvu \times \delta \uvv$}), corresponding to a visibility
    589594measurement from a pair of receivers can be written as:
    590595\begin{eqnarray}
    … …  
    595600\end{eqnarray}
    596601
    597 The sky temperature measurement can thus be characterized by the noise spectral power density in
    598 the angular frequencies plane $P_{noise}^{(\uv)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$ 
    599 per unit area of angular frequencies  $\delta \uvu \times \delta \uvv$:
    600602We can characterize the sky temperature measurement with a radio instrument by the noise
    601603spectral power density in the angular frequencies plane $P_{noise}(\uv)$ in units of $\mathrm{Kelvin^2}$ 
    … …  
    612614When the intensity maps are projected in a three dimensional box in the universe and the 3D power spectrum
    613615$P(k)$ is computed, angles are translated into comoving transverse distances,
    614 and frequencies or wavelengths into comoving radial distance, using the following relations:
     616and frequencies or wavelengths into comoving radial distance, using the following relations
     617{\changemarkb (e.g. \cite{cosmo.peebles} chap. 13, \cite{cosmo.rich})} :
    615618{ \changemark
    616619\begin{eqnarray}
    … …  
    620623  = (1+z) \frac{\lambda}{H(z)} \delta \nu \\
    621624% \delta \uvu , \delta \uvv & \rightarrow & \delta k_\perp = 2 \pi \frac{ \delta \uvu \, , \, \delta \uvv }{  (1+z) \, \dang(z)  } \\
    622 \frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = 2 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
     625\frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = \delta k_z =
     6262 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
    623627 =  \frac{H(z)}{c} \frac{1}{(1+z)^2} \, \frac{\nu_{21}}{\delta \nu}
    624628\end{eqnarray}
    … …  
    631635The measurement noise spectral density given by the equation \ref{eq:pnoisepairD} can then be
    632636translated into a 3D noise power spectrum, per unit of spatial frequencies
    633 $ \frac{\delta k_x \times \delta k_y \times \delta k_z}{8 \pi^3} $ (units: $ \mathrm{K^2 \times Mpc^3}$) : 
     637$ \delta k_x \times \delta k_y \times \delta k_z / 8 \pi^3 $ (units: $ \mathrm{K^2 \times Mpc^3}$) : 
    634638
    635639\begin{eqnarray}
    … …  
    652656 
    653657\subsubsection{Uniform $(\uv)$ coverage}
    654 
    655 If we consider a uniform noise spectral density in the $(\uv)$ plane corresponding to the
    656 equation \ref{eq:pnoisepairD} above,  the three dimensional projected noise spectral density
    657 can then be written as:
    658 \begin{equation}
    659 P_{noise}(k) = 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 
     658{ \changemarkb We consider here an instrument with uniform $(\uv)$ plane coverage  (${\cal R}(\uv)=1$),
     659and measurements at regularly spaced frequencies centered on a central frequency $\nu_0$ or redshift $z(\nu_0)$.
     660The noise spectral power  density from equation (\ref{eq:pnoisekxkz}) would then be 
     661constant, independent of $(k_x, k_y, \ell_\parallel(\nu)$. Such a noise power spectrum corresponds thus
     662to a 3D white noise, with a uniform noise spectral density:}
     663\begin{equation}
     664P_{noise}(k_\perp, l_\parallel(\nu) ) = P_{noise} = 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 
    660665\label{ctepnoisek}
    661666\end{equation}
    662667
    663 $P_{noise}(k)$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$,
     668$P_{noise}$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$,
    664669$t_{int}$ is the integration time expressed in second,
    665670$\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and
    666671 $H(z)$ in $\mathrm{km/s/Mpc}$.
    667672
    668 The matter or \HI distribution power spectrum determination statistical errors vary as the number of
    669 observed Fourier modes, which is inversely proportional to volume of the universe
     673The statistical uncertainties of matter or \HI distribution power spectrum estimate decreases
     674with the number of observed Fourier modes; this number is proportional to the volume of the universe
    670675which is observed (sample variance).  As the observed volume is proportional to the
    671676surveyed solid angle, we  consider the survey of a fixed
    672 fraction of the sky, defined by  total solid angle $\Omega_{tot}$, performed during a determined
     677fraction of the sky, defined by  total solid angle $\Omega_{tot}$, performed during a given
    673678total observation time $t_{obs}$.
    674679A single dish instrument with diameter $D$ would have an instantaneous field of view
    675680$\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and would require
    676681a number of pointings  $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area.
    677 Each sky direction or pixel of size $\Omega_{FOV}$ will be observed during an integration
     682Each sky direction or patch of size $\Omega_{FOV}$ will be observed during an integration
    678683time $t_{int} = t_{obs}/N_{point} $. Using equation \ref{ctepnoisek} and the previous expression
    679684for the integration time, we can compute a simple expression
    … …  
    685690It is important to note that any real instrument do not have a flat
    686691response in the $(\uv)$ plane, and the observations provide no information above
    687 a certain maximum angular frequency $u_{max},v_{max}$.
     692a certain maximum angular frequency $\uvu_{max},\uvv_{max}$.
    688693One has to take into account either a damping of the observed sky power
    689 spectrum or an increase of the noise spectral power if
     694spectrum or an increase of the noise spectral density if
    690695the observed power spectrum is corrected for damping. The white noise
    691696expressions given below should thus be considered as a lower limit or floor of the
    … …  
    755760we want to compute $P_{noise}(k)$. For the results at redshift \mbox{$z_0=1$} discussed in section  \ref{instrumnoise},
    756761the grid cell size and dimensions have been chosen to represent a box in the universe 
    757 \mbox{$\sim 1500 \times 1500 \times 750 \mathrm{Mpc^3}$},
    758 with $3\times3\times3 \mathrm{Mpc^3}$ cells.
     762\mbox{$\sim 1500 \times 1500 \times 750 \, \mathrm{Mpc^3}$},
     763with \mbox{$3\times3\times3 \, \mathrm{Mpc^3}$} cells.
    759764This correspond to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given
    760765the small angular extent, we have neglected the curvature of redshift shells.
    … …  
    779784\item the above steps are repeated $\sim$ 50 times to decrease the statistical fluctuations
    780785from random generations. The averaged computed noise power spectrum is normalized using
    781 equation \ref{eq:pnoisekxkz}  and the instrument and survey parameters ($\Tsys \ldots$).
     786equation \ref{eq:pnoisekxkz}  and the instrument and survey parameters:
     787{\changemarkb system temperature $\Tsys= 50 \mathrm{K}$,
     788individual receiver size $D^2$ or $D_x D_y$ and the integration time $t_{int}$.
     789This last parameter is obtained through the relation
     790$t_{int} = t_{obs} \Omega_{FOV}  / \Omega_{tot}$ using the total survey duration
     791$t_{obs}=1 \mathrm{year}$ and the instantaneous field of view
     792$\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, for a total survey sky coverage
     793of $\pi$ srad. }
    782794\end{itemize}
    783795
    … …  
    786798instrument response function and $\dang(z)$ for a small redshift interval around $z_0$,
    787799using a fixed  instrument response ${\cal R}(\uv,\lambda(z_0))$ and
    788 a constant the radial distance $\dang(z_0)*(1+z_0)$.
     800a constant radial distance $\dang(z_0)*(1+z_0)$.
    789801\begin{equation}
    790802\tilde{P}_{noise}(k) = < |  {\cal F}_n (k_x,k_y,k_z) |^2 > \simeq < P_{noise}(\uv, k_z) >
    … …  
    863875
    864876For the receivers along the focal line of cylinders, we have assumed that the
    865 individual receiver response in the $(\uv)$ plane corresponds to one from a
     877individual receiver response in the $(\uv)$ plane corresponds to a
    866878rectangular shaped antenna. The illumination efficiency factor has been taken
    867879equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$
    … …  
    901913\includegraphics[width=\textwidth]{Figs/uvcovabcd.pdf}
    902914}
    903 \caption{Raw instrument response ${\cal R}(\uv,\lambda)$ or the $(\uv)$ plane coverage
     915\caption{Raw instrument response ${\cal R}_{raw}(\uv,\lambda)$ or the $(\uv)$ plane coverage
    904916at 710 MHz (redshift $z=1$) for four configurations.
    905917(a) 121 $D_{dish}=5$ meter diameter dishes arranged in a compact, square array
    906 of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each (fig. \ref{figconfbc}),
     918of $11 \times 11$, (b) 128 dishes arranged in 8 rows of 16 dishes each (fig. \ref{figconfbc}),
    907919(c) 129 dishes arranged as shown in figure \ref{figconfbc} , (d) single D=75 meter diameter, with 100 beams.
    908920The common color scale (1 \ldots 80) is shown on the right. }
    … …  
    965977used in the simulations are given in the table \ref{skycubechars}.
    966978\begin{table}
     979\caption{
     980Sky cube characteristics for the simulation performed in this paper.
     981Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$ ;
     982$1800 \times 600 \times 256 \simeq 123 \times 10^6$ cells
     983}
     984\label{skycubechars}
    967985\begin{center}
    968986\begin{tabular}{|c|c|c|}
    … …  
    9851003\end{tabular} \\[1mm]
    9861004\end{center}
    987 \caption{
    988 Sky cube characteristics for the simulation performed in this paper.
    989 Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$   
    990 $ 1800 \times 600 \times 256 \simeq 123 \, 10^6$ cells
    991 }
    992 \label{skycubechars}
    9931005\end{table}
    9941006%%%%
    … …  
    10171029The synchrotron contribution to the sky temperature for each cell is then
    10181030obtained  through the formula:
    1019 $$ T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 \, \mathrm{MHz}}\right)^\beta $$
     1031\begin{equation}
     1032 T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 \, \mathrm{MHz}}\right)^\beta
     1033\end{equation}
    10201034%%
    10211035\item A two dimensional $T_{nvss}(\alpha,\delta)$ sky brightness temperature at 1.4 GHz is computed
    10221036by projecting the radio sources in the NVSS catalog to a grid with the same angular resolution as
    10231037the sky cubes. The source brightness in Jansky is converted to temperature taking the
    1024 pixel angular size into account ($ \sim 21 \mathrm{mK / mJansky}$ at 1.4 GHz and $3' \times 3'$ pixels). 
     1038pixel angular size into account ($ \sim 21 \mathrm{mK/mJy}$ at 1.4 GHz and $3' \times 3'$ pixels). 
    10251039A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source
    10261040map; we have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the
    10271041contribution of the radiosources to the sky temperature is computed as follows:
    1028 $$ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}} $$
     1042\begin{equation}
     1043 T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}}
     1044\end{equation}
    10291045%%
    10301046\item The sky brightness temperature data cube is obtained through the sum of
    10311047the two contributions, Galactic synchrotron and resolved radio sources:
    1032 $$ T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu) $$
     1048\begin{equation}
     1049 T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu)
     1050\end{equation}
    10331051\end{enumerate}
    10341052
    … …  
    10511069The size of the cells is  $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the
    10521070sky cube angular and frequency resolution defined above. 
    1053 
    1054 The mass fluctuations has been
    1055 converted into temperature through a factor $0.13 \, \mathrm{mK}$, corresponding to a hydrogen
    1056 fraction $0.008 \times (1+0.6)$, using equation \ref{eq:tbar21z}. 
    1057 The total sky brightness temperature is then computed as the sum
     1071{\changemarkb
     1072We haven't taken into account the curvature of redshift shells when
     1073converting SimLSS euclidean coordinates to angles and frequency coordinates
     1074of the sky cubes analyzed here, which introduces distortions visible at large angles $\gtrsim 10^\circ$.
     1075These angular scales, corresponding to small wave modes $k \lesssim 0.02 \mathrm{h \, Mpc^{-1}}$
     1076 are excluded for results presented in this paper.
     1077} 
     1078
     1079The mass fluctuations have been converted into temperature using equation \ref{eq:tbar21z},
     1080and a neutral  hydrogen fraction \mbox{$0.008 \times (1+0.6)$}, leading to a mean temperature of
     1081$0.13 \, \mathrm{mK}$.
     1082The total sky brightness temperature is computed as the sum
    10581083of foregrounds and the LSS 21 cm emission:
    1059 $$  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm}
    1060 T_{sky} = T_{gsm}+T_{lss} $$
     1084\begin{equation}
     1085  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm}
     1086T_{sky} = T_{gsm}+T_{lss}
     1087\end{equation}
    10611088
    10621089Table \ref{sigtsky} summarizes the mean and standard deviation of the sky brightness
    … …  
    10721099
    10731100\begin{table}
     1101\caption{ Mean temperature and standard deviation for the different sky brightness
     1102data cubes computed for this study (see table \ref{skycubechars} for sky cube resolution and size).}
     1103\label{sigtsky}
    10741104\centering
    10751105\begin{tabular}{|c|c|c|}
    … …  
    10841114\hline
    10851115\end{tabular}
    1086 \caption{ Mean temperature and standard deviation for the different sky brightness
    1087 data cubes computed for this study (see table \ref{skycubechars} for sky cube resolution and size).}
    1088 \label{sigtsky}
    10891116\end{table}
    10901117
    … …  
    11611188}
    11621189\vspace*{-5mm}
    1163 \caption{Comparison of the 21cm LSS power spectrum at $z=0.6$ with $\gHI=1\%$ (red, orange)
     1190\caption{Comparison of the 21cm LSS power spectrum at $z=0.6$ with \mbox{$\gHI\simeq1.3\%$} (red, orange)
    11641191with the radio foreground power spectrum.
    11651192The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple
    … …  
    11981225additive white noise level was independent of the frequency or wavelength.   
    11991226\end{enumerate}
    1200 The LSS signal extraction depends indeed on the white noise level.
     1227The LSS signal extraction performance depends obviously on the white noise level.
    12011228The results shown here correspond to the (a) instrument configuration, a packed array of
    12021229$11 \times 11 = 121$ dishes  (5 meter diameter), with a white noise level corresponding
    12031230to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500$ kHz
    1204 cell.
    1205 
    1206 A brief description of the simple component separation procedure that we have applied is given here:
     1231cell. \\[1mm]
     1232
     1233The different steps of the simple component separation procedure that has been applied are
     1234briefly described here.
    12071235\begin{enumerate}
    12081236\item The measured sky brightness temperature is first {\em corrected} for the frequency dependent
    … …  
    12231251${\cal R}(\uv,\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$  has also a numerical upper bound $\sim 100$. }
    12241252\item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
    1225  is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in
    1226 the $\lgd ( T ) , \lgd (\nu)$ plane and we show here the results for a pure power law (P1)
    1227 or modified power law (P2):
     1253 is fitted to the beam-corrected brightness temperature. The parameters have been obtained
     1254using a linear $\chi^2$ fit in the $\lgd ( T ) , \lgd (\nu)$ plane.
     1255We show here the results for a pure power law (P1), as well as a modified power law (P2):
    12281256\begin{eqnarray*}
    12291257P1 & :  & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) \\
    … …  
    12391267we have set the mean value of the temperature for
    12401268each frequency plane according to a power law with an index close to the synchrotron index
    1241 ($\beta\sim-2.8$) and we have checked that results are not too sensitive to the
     1269($\beta\sim-2.8$) and we have checked that the results are not too sensitive to the
    12421270arbitrarily fixed mean temperature power law parameters. }
    12431271
    … …  
    12841312% \vspace*{-10mm}
    12851313\caption{Recovered power spectrum of the 21cm LSS temperature fluctuations, separated from the
    1286 continuum radio emissions at $z \sim 0.6, \gHI=1\%$, for the instrument configuration (a), $11\times11$
     1314continuum radio emissions at $z \sim 0.6$, \mbox{$\gHI\simeq1.3\%$}, for the instrument configuration (a), $11\times11$
    12871315packed array interferometer.
    12881316Left: GSM/Model-I , right: Haslam+NVSS/Model-II. The black curve shows the residual after foreground subtraction,
    … …  
    13111339The recovered red shifted 21 cm emission power spectrum $P_{21}^{rec}(k)$ suffers a number of distortions, mostly damping,
    13121340 compared to the original $P_{21}(k)$ due to  the instrument response and the component separation procedure.
     1341{\changemarkb
     1342We remind that we have neglected the curvature of redshift or frequency shells
     1343in this numerical study, which affect our result at large angles $\gtrsim 10^\circ$.
     1344The results presented here and our conclusions are thus restricted to wave mode range
     1345$k \gtrsim 0.02 \mathrm{h \, Mpc^{-1}}$.
     1346}
    13131347We expect damping at small scales, or larges $k$, due to the finite instrument size, but also at large scales, small $k$,
    13141348if total power measurements (auto-correlations) are not used in the case of interferometers.
    13151349The sky reconstruction and the component separation introduce additional filtering and distortions.
    1316 Ideally, one has to define a power spectrum measurement response or {\it transfer function} in the
    1317 radial direction,  ($\lambda$ or redshift, $\TrF(k_\parallel)$) and in the transverse plane ( $\TrF(k_\perp)$ ).
    13181350The real transverse plane transfer function might even be anisotropic.
    13191351
    13201352However, in the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the
    1321 recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$:
     1353recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$
     1354{\changemarkb , similar to the one defined by \cite{bowman.09} , equation (23):}
    13221355\begin{equation}
    13231356\TrF(k) = P_{21}^{rec}(k) / P_{21}(k)
    … …  
    13381371longitudinal Fourier modes along the frequency or redshift direction ($k_\parallel$)
    13391372by the component separation algorithm. We have been able to remove the ripples on the reconstructed
    1340 power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$37'
     1373power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$36'
    13411374target beam resolution, compared to $\sim$30' for the GSM model. This explains the lower transfer function
    13421375obtained for Model-II at small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}$). }
    … …  
    13611394for setup (e) for three redshifts, $z=0.5, 1 , 1.5$, and then extrapolated the value of the parameters
    13621395for redshift $z=2, 2.5$. The value of the parameters are grouped in table \ref{tab:paramtfk}
    1363 and the smoothed transfer functions are shown on  figure \ref{tfpkz0525}.
     1396and the corresponding transfer functions are shown on  figure \ref{tfpkz0525}.
    13641397
    13651398\begin{table}[hbt]
     1399\caption{Value of the parameters for the transfer function (eq. \ref{eq:tfanalytique}) at different redshift
     1400for instrumental setup (e), $20\times20$ packed array interferometer.  }
     1401\label{tab:paramtfk}
    13661402\begin{center}
    13671403\begin{tabular}{|c|ccccc|}
    … …  
    13761412\end{tabular}
    13771413\end{center}
    1378 \caption{Value of the parameters for the transfer function (eq. \ref{eq:tfanalytique}) at different redshift
    1379 for instrumental setup (e), $20\times20$ packed array interferometer.  }
    1380 \label{tab:paramtfk}
    13811414\end{table}
    13821415
    … …  
    15901623spectrum are directly related to the number of modes in the surveyed volume $V$ corresponding to
    15911624 $\Delta z=0.5$ slice with the solid angle $\Omega_{tot}$ = 1 $\pi$ sr.
    1592 The number of mode $N_{\delta k}$ in the wave number interval $\delta k$ can be written as:
     1625The number of modes $N_{\delta k}$ in the wave number interval $\delta k$ can be written as:
    15931626\begin{equation}
    15941627V  =  \frac{c}{H(z)} \Delta z  \times (1+z)^2 \dang^2  \Omega_{tot} \hspace{10mm}
    … …  
    15981631\ref {eq:pnoiseNbeam}. Table \ref{tab:pnoiselevel} gives the white noise level for
    15991632$\Tsys = 50 \mathrm{K}$ and one year total observation time to survey $\Omega_{tot}$ = 1 $\pi$ sr.
    1600 \item {\it Noise with transfer function}: we take into account of the interferometer and radio foreground
     1633\item {\it Noise with transfer function}: we take into account the interferometer response and radio foreground
    16011634subtraction represented as the measured P(k) transfer function $T(k)$ (section \ref{tfpkdef}), as
    1602 well as instrument noise $P_{noise}$.
     1635well as the instrument noise $P_{noise}$.
    16031636\end{itemize}
    16041637
    16051638\begin{table}
     1639\caption{Instrument or electronic noise spectral power $P_{noise}$ for a $N=400$ dish interferometer with $\Tsys=50$ K and $t_{obs} =$ 1 year to survey $\Omega_{tot} = \pi$ sr }
     1640\label{tab:pnoiselevel}
    16061641\begin{tabular}{|l|ccccc|}
    16071642\hline
    … …  
    16121647\hline
    16131648\end{tabular}
    1614 \caption{Instrument or electronic noise spectral power $P_{noise}$ for a $N=400$ dish interferometer with $\Tsys=50$ K and $t_{obs} =$ 1 year to survey $\Omega_{tot} = \pi$ sr }
    1615 \label{tab:pnoiselevel}
    16161649\end{table}
    16171650
    … …  
    16201653
    16211654\begin{table*}[ht]
     1655\caption{Sensitivity on the measurement of $\koperp$ and $\kopar$ as a
     1656function of the redshift $z$ for various simulation configuration.
     1657$1^{\rm st}$ row: simulations without noise with pure cosmic variance;
     1658$2^{\rm nd}$ row: simulations with electronics noise for a telescope with dishes;
     1659$3^{\rm rd}$ row: simulations with the same electronics noise and with the transfer function ;
     1660$4^{\rm th}$ row: optimized survey with a total observation time of 3 years (3 months, 3 months, 6 months, 1 year and 1 year respectively for redshift 0.5, 1.0, 1.5, 2.0 and 2.5 ).}
     1661\label{tab:ErrorOnK}
    16221662\begin{center}
    16231663\begin{tabular}{lc|c c c c c }
    … …  
    16381678\end{tabular}
    16391679\end{center}
    1640 \caption{Sensitivity on the measurement of $\koperp$ and $\kopar$ as a
    1641 function of the redshift $z$ for various simulation configuration.
    1642 $1^{\rm st}$ row: simulations without noise with pure cosmic variance;
    1643 $2^{\rm nd}$
    1644 row: simulations with electronics noise for a telescope with dishes;
    1645 $3^{\rm th}$ row: simulations
    1646 with same electronics noise and with correction with the transfer function ;
    1647 $4^{\rm th}$ row: optimized survey with a total observation time of 3 years (3 months, 3 months, 6 months, 1 year and 1 year respectively for redshift 0.5, 1.0, 1.5, 2.0 and 2.5 ).}
    1648 \label{tab:ErrorOnK}
    16491680\end{table*}%
    16501681
    … …  
    17901821
    17911822%%%
     1823%%%% LSST Science book
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     1825{\it LSST Science book}, LSST Science Collaborations, Abell, P.A. {\it et al.} 2009, arXiv:0912.0201 
     1826
    17921827%% reference SKA - BAO / DE en radio avec les sources
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    18211856
     1857%%%  SKA-Science
     1858\bibitem[Carilli et al. (2004)]{ska.science}
     1859{\it Science with the Square Kilometre Array}, eds: C. Carilli, S. Rawlings,
     1860New Astronomy Reviews, Vol.48, Elsevier, December 2004 \\
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     1862
    18221863%  Intensity mapping/HSHS
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     1888% Papier de Field sur la profondeur optique HI en 1959
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    18771920%% Description MWA
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    1879 % LSST Science book
    1880 \bibitem[LSST.Science]{lsst.science}
    1881 {\it LSST Science book}, LSST Science Collaborations, 2009, arXiv:0912.0201 
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     1958% Livre Cosmo de Peebles
     1959\bibitem[Peebles (1993)]{cosmo.peebles} Peebles, P.J.E., {\it Principles of Physical Cosmology},
     1960Princeton University Press (1993)
     1961
     1962% Original CRT HSHS paper (Moriond Cosmo 2006 Proceedings)
     1963\bibitem[Peterson et al. (2006)]{peterson.06} Peterson, J.B.,  Bandura, K., \& Pen, U.-L. 2006, arXiv:0606104 
     1964
    19181965% Synchrotron index =-2.8 in the freq range 1.4-7.5 GHz
    19191966\bibitem[Platania et al. (1998)]{platania.98}   Platania P., Bensadoun M., Bersanelli M. {\it al.} 1998,  \apj 505, 473-483
    19201967
    1921 % Original CRT HSHS paper (Moriond Cosmo 2006 Proceedings)
    1922 \bibitem[Peterson et al. (2006)]{peterson.06} Peterson, J.B.,  Bandura, K., \& Pen, U.-L. 2006, arXiv:0606104 
    1923 
    19241968% SDSS BAO 2007
    19251969\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
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    19271971% SDSS BAO 2010  - arXiv:0907.1660
    19281972\bibitem[Percival et al. (2010)]{percival.10}   Percival, W.J., Reid, B.A., Eisenstein, D.J. {\it et al.},  2010, \mnras, 401, 2148-2168   
     1973
     1974% Livre Cosmo de Jim Rich
     1975\bibitem[Rich (2001)]{cosmo.rich} James Rich,  {\it Fundamentals of Cosmology}, Springer (2001)
    19291976
    19301977% Radio spectral index between 100-200 MHz
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    19431990%  Frank H. Briggs, Matthew Colless, Roberto De Propris, Shaun Ferris, Brian P. Schmidt, Bradley E. Tucker
    19441991
    1945 \bibitem[SKA.Science]{ska.science}
    1946 {\it Science with the Square Kilometre Array}, eds: C. Carilli, S. Rawlings,
    1947 New Astronomy Reviews, Vol.48, Elsevier, December 2004 \\
    1948 { \tt http://www.skatelescope.org/pages/page\_sciencegen.htm }
    1949 
    19501992% Papier 21cm-BAO Fermilab  ( arXiv:0910.5007)
    19511993\bibitem[Seo et al (2010)]{seo.10} Seo, H.J. Dodelson, S., Marriner, J. et al,  2010, \apj, 721, 164-173
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    19662008%  BAO à 21 cm et reionisation
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     2011%% Papier fluctuations 21 cm par Zaldarriaga et al
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    19682014
    19692015%% Today HI cosmological density
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