%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % BAORadio : LAL/UPS, Irfu/SPP % 21cm LSS P(k) sensitivity and foreground substraction % R. Ansari, C. Magneville, J. Rich, C. Yeche et al % 2010 - 2011 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % aa.dem % AA vers. 7.0, LaTeX class for Astronomy & Astrophysics % demonstration file % (c) Springer-Verlag HD % revised by EDP Sciences %----------------------------------------------------------------------- % % \documentclass[referee]{aa} % for a referee version %\documentclass[onecolumn]{aa} % for a paper on 1 column %\documentclass[longauth]{aa} % for the long lists of affiliations %\documentclass[rnote]{aa} % for the research notes %\documentclass[letter]{aa} % for the letters % \documentclass[structabstract]{aa} %\documentclass[traditabstract]{aa} % for the abstract without structuration % (traditional abstract) % \usepackage{amsmath} \usepackage{amssymb} \usepackage{graphicx} \usepackage{color} %% Commande pour les references \newcommand{\citep}[1]{(\cite{#1})} %% \newcommand{\citep}[1]{ { (\tt{#1}) } } %% Definitions diverses \newcommand{\HI}{$\mathrm{H_I}$ } \newcommand{\kb}{k_B} % Constante de Boltzmann \newcommand{\Tsys}{T_{sys}} % instrument noise (system) temperature \newcommand{\TTnu}{ T_{21}(\vec{\Theta} ,\nu) } \newcommand{\TTnuz}{ T_{21}(\vec{\Theta} ,\nu(z)) } \newcommand{\TTlam}{ T_{21}(\vec{\Theta} ,\lambda) } \newcommand{\TTlamz}{ T_{21}(\vec{\Theta} ,\lambda(z)) } \newcommand{\dlum}{d_L} \newcommand{\dang}{d_A} \newcommand{\hub}{ h_{70} } \newcommand{\hubb}{ h_{100} } % h_100 \newcommand{\etaHI}{ n_{\tiny HI} } \newcommand{\fHI}{ f_{H_I}(z)} \newcommand{\gHI}{ f_{H_I}} \newcommand{\gHIz}{ f_{H_I}(z)} \newcommand{\vis}{{\cal V}_{12} } \newcommand{\LCDM}{$\Lambda \mathrm{CDM}$ } \newcommand{\lgd}{\mathrm{log_{10}}} %% Definition fonction de transfer \newcommand{\TrF}{\mathbf{T}} %% Definition (u,v) , ... \def\uv{\mathrm{u,v}} \def\uvu{\mathrm{u}} \def\uvv{\mathrm{v}} \def\dudv{\mathrm{d u d v}} % Commande pour marquer les changements du papiers pour le referee \def\changemark{\bf } %%% Definition pour la section sur les param DE par C.Y \def\Mpc{\mathrm{Mpc}} \def\hMpcm{\,h \,\Mpc^{-1}} \def\hmMpc{\,h^{-1}\Mpc} \def\kperp{k_\perp} \def\kpar{k_\parallel} \def\koperp{k_{BAO\perp }} \def\kopar{k_{BAO\parallel}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{txfonts} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{document} % \title{21 cm observation of LSS at z $\sim$ 1 } \subtitle{Instrument sensitivity and foreground subtraction} \author{ R. Ansari \inst{1} \inst{2} \and J.E. Campagne \inst{3} \and P.Colom \inst{5} \and J.M. Le Goff \inst{4} \and C. Magneville \inst{4} \and J.M. Martin \inst{5} \and M. Moniez \inst{3} \and J.Rich \inst{4} \and C.Y\`eche \inst{4} } \institute{ Universit\'e Paris-Sud, LAL, UMR 8607, F-91898 Orsay Cedex, France \and CNRS/IN2P3, F-91405 Orsay, France \\ \email{ansari@lal.in2p3.fr} \and Laboratoire de lÍAcc\'el\'erateur Lin\'eaire, CNRS-IN2P3, Universit\'e Paris-Sud, B.P. 34, 91898 Orsay Cedex, France % \thanks{The university of heaven temporarily does not % accept e-mails} \and CEA, DSM/IRFU, Centre d'Etudes de Saclay, F-91191 Gif-sur-Yvette, France \and GEPI, UMR 8111, Observatoire de Paris, 61 Ave de l'Observatoire, 75014 Paris, France } \date{Received August 5, 2011; accepted xxxx, 2011} % \abstract{}{}{}{}{} % 5 {} token are mandatory \abstract % context heading (optional) % {} leave it empty if necessary { Large Scale Structures (LSS) in the universe can be traced using the neutral atomic hydrogen \HI through its 21 cm emission. Such a 3D matter distribution map can be used to test the Cosmological model and to constrain the Dark Energy properties or its equation of state. A novel approach, called intensity mapping can be used to map the \HI distribution, using radio interferometers with large instantaneous field of view and waveband.} % aims heading (mandatory) { In this paper, we study the sensitivity of different radio interferometer configurations, or multi-beam instruments for the observation of large scale structures and BAO oscillations in 21 cm and we discuss the problem of foreground removal. } % methods heading (mandatory) { For each configuration, we determine instrument response by computing the (u,v) or Fourier angular frequency plane coverage using visibilities. The (u,v) plane response is the noise power spectrum, hence the instrument sensitivity for LSS P(k) measurement. We describe also a simple foreground subtraction method to separate LSS 21 cm signal from the foreground due to the galactic synchrotron and radio sources emission. } % results heading (mandatory) { We have computed the noise power spectrum for different instrument configuration as well as the extracted LSS power spectrum, after separation of 21cm-LSS signal from the foregrounds. We have also obtained the uncertainties on the Dark Energy parameters for an optimized 21 cm BAO survey.} % conclusions heading (optional), leave it empty if necessary { We show that a radio instrument with few hundred simultaneous beams and a collecting area of $\sim 10000 \mathrm{m^2}$ will be able to detect BAO signal at redshift z $\sim 1$ and will be competitive with optical surveys. } \keywords{ large-scale structure of Universe -- dark energy -- Instrumentation: interferometers -- Radio lines; galaxies -- Radio continuum: general } \maketitle % %________________________________________________________________ % {\color{red} \large \bf A discuter : liste des auteurs, plans du papier et repartition des taches % Toutes les figures sont provisoires } \section{Introduction} % {\color{red} \large \it Jim ( + M. Moniez ) } \\[1mm] The study of the statistical properties of Large Scale Structure (LSS) in the Universe and their evolution with redshift is one the major tools in observational cosmology. These structures are usually mapped through optical observation of galaxies which are used as a tracer of the underlying matter distribution. An alternative and elegant approach for mapping the matter distribution, using neutral atomic hydrogen (\HI) as a tracer with intensity mapping, has been proposed in recent years \citep{peterson.06} \citep{chang.08}. Mapping the matter distribution using HI 21 cm emission as a tracer has been extensively discussed in literature \citep{furlanetto.06} \citep{tegmark.09} and is being used in projects such as LOFAR \citep{rottgering.06} or MWA \citep{bowman.07} to observe reionisation at redshifts z $\sim$ 10. Evidence in favor of the acceleration of the expansion of the universe have been accumulated over the last twelve years, thanks to the observation of distant supernovae, CMB anisotropies and detailed analysis of the LSS. A cosmological Constant ($\Lambda$) or new cosmological energy density called {\em Dark Energy} has been advocated as the origin of this acceleration. Dark Energy is considered as one of the most intriguing puzzles in Physics and Cosmology. % Constraining the properties of this new cosmic fluid, more precisely % its equation of state is central to current cosmological researches. Several cosmological probes can be used to constrain the properties of this new cosmic fluid, more precisely its equation of state: The Hubble Diagram, or luminosity distance as a function of redshift of supernovae as standard candles, galaxy clusters, weak shear observations and Baryon Acoustic Oscillations (BAO). BAO are features imprinted in the distribution of galaxies, due to the frozen sound waves which were present in the photon-baryon plasma prior to recombination at z $\sim$ 1100. This scale can be considered as a standard ruler with a comoving length of $\sim 150 \mathrm{Mpc}$. These features have been first observed in the CMB anisotropies and are usually referred to as {\em acoustic peaks} (\cite{mauskopf.00}, \cite{larson.11}). The BAO modulation has been subsequently observed in the distribution of galaxies at low redshift ( $z < 1$) in the galaxy-galaxy correlation function by the SDSS \citep{eisenstein.05} \citep{percival.07} \citep{percival.10}, 2dGFRS \citep{cole.05} as well as WiggleZ \citep{blake.11} optical galaxy surveys. Ongoing \citep{eisenstein.11} or future surveys \citep{lsst.science} plan to measure precisely the BAO scale in the redshift range $0 \lesssim z \lesssim 3$, using either optical observation of galaxies or through 3D mapping Lyman $\alpha$ absorption lines toward distant quasars \citep{baolya},\citep{baolya2}. Radio observation of the 21 cm emission of neutral hydrogen appears as a very promising technique to map matter distribution up to redshift $z \sim 3$, complementary to optical surveys, especially in the optical redshift desert range $1 \lesssim z \lesssim 2$, and possibly up to the reionization redshift \citep{wyithe.08}. In section 2, we discuss the intensity mapping and its potential for measurement of the \HI mass distribution power spectrum. The method used in this paper to characterize a radio instrument response and sensitivity for $P_{\mathrm{H_I}}(k)$ is presented in section 3. We show also the results for the 3D noise power spectrum for several instrument configurations. The contribution of foreground emissions due to the galactic synchrotron and radio sources is described in section 4, as well as a simple component separation method. The performance of this method using two different sky models is also presented in section 4. The constraints which can be obtained on the Dark Energy parameters and DETF figure of merit for typical 21 cm intensity mapping survey are discussed in section 5. %__________________________________________________________________ \section{Intensity mapping and \HI power spectrum} % {\color{red} \large \it Reza (+ P. Colom ?) } \\[1mm] \subsection{21 cm intensity mapping} %%% Most of the cosmological information in the LSS is located at large scales ($ \gtrsim 1 \mathrm{deg}$), while the interpretation at smallest scales might suffer from the uncertainties on the non linear clustering effects. The BAO features in particular are at the degree angular scale on the sky and thus can be resolved easily with a rather modest size radio instrument (diameter $D \lesssim 100 \, \mathrm{m}$). The specific BAO clustering scale ($k_{\mathrm{BAO}}$) can be measured both in the transverse plane (angular correlation function, ($k_{\mathrm{BAO}}^\perp$) or along the longitudinal (line of sight or redshift ($k_{\mathrm{BAO}}^\parallel$) direction. A direct measurement of the Hubble parameter $H(z)$ can be obtained by comparing the longitudinal and transverse BAO scales. A reasonably good redshift resolution $\delta z \lesssim 0.01$ is needed to resolve longitudinal BAO clustering, which is a challenge for photometric optical surveys. In order to obtain a measurement of the LSS power spectrum with small enough statistical uncertainties (sample or cosmic variance), a large volume of the universe should be observed, typically few $\mathrm{Gpc^3}$. Moreover, stringent constraint on DE parameters can only be obtained when comparing the distance or Hubble parameter measurements with DE models as a function of redshift, which requires a significant survey depth $\Delta z \gtrsim 1$. Radio instruments intended for BAO surveys must thus have large instantaneous field of view (FOV $\gtrsim 10 \, \mathrm{deg^2}$) and large bandwidth ($\Delta \nu \gtrsim 100 \, \mathrm{MHz}$) to explore large redshift domains. Although the application of 21 cm radio survey to cosmology, in particular LSS mapping has been discussed in length in the framework of large future instruments, such as the SKA (e.g \cite{ska.science}, \cite{abdalla.05}), the method envisaged has been mostly through the detection of galaxies as \HI compact sources. However, extremely large radio telescopes are required to detected \HI sources at cosmological distances. The sensitivity (or detection threshold) limit $S_{lim}$ for the total power from the two polarisations of a radio instrument characterized by an effective collecting area $A$, and system temperature $\Tsys$ can be written as \begin{equation} S_{lim} = \frac{ \sqrt{2} \, \kb \, \Tsys }{ A \, \sqrt{t_{int} \delta \nu} } \end{equation} where $t_{int}$ is the total integration time and $\delta \nu$ is the detection frequency band. In table \ref{slims21} (left) we have computed the sensitivity for 6 different sets of instrument effective area and system temperature, with a total integration time of 86400 seconds (1 day) over a frequency band of 1 MHz. The width of this frequency band is well adapted to detection of \HI source with an intrinsic velocity dispersion of few 100 km/s. These detection limits should be compared with the expected 21 cm brightness $S_{21}$ of compact sources which can be computed using the expression below (e.g.\cite{binney.98}) : \begin{equation} S_{21} \simeq 0.021 \mathrm{\mu Jy} \, \frac{M_{H_I} }{M_\odot} \times \left( \frac{ 1\, \mathrm{Mpc}}{\dlum(z)} \right)^2 \times \frac{200 \, \mathrm{km/s}}{\sigma_v} (1+z) \end{equation} where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ is the luminosity distance and $\sigma_v$ is the source velocity dispersion. {\changemark The 1 MHz bandwidth mentioned above is only used for computing the galaxy detection thresholds and does not determine the total survey bandwidth or frequency resolution of an intensity mapping survey.} % {\color{red} Faut-il developper le calcul en annexe ? } In table \ref{slims21} (right), we show the 21 cm brightness for compact objects with a total \HI \, mass of $10^{10} M_\odot$ and an intrinsic velocity dispersion of $200 \, \mathrm{km/s}$. The luminosity distance is computed for the standard WMAP \LCDM universe \citep{komatsu.11}. $10^9 - 10^{10} M_\odot$ of neutral gas mass is typical for large galaxies \citep{lah.09}. It is clear that detection of \HI sources at cosmological distances would require collecting area in the range of $10^6 \mathrm{m^2}$. Intensity mapping has been suggested as an alternative and economic method to map the 3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} \citep{seo.10}. In this approach, sky brightness map with angular resolution $\sim 10-30 \, \mathrm{arc.min}$ is made for a wide range of frequencies. Each 3D pixel (2 angles $\vec{\Theta}$, frequency $\nu$ or wavelength $\lambda$) would correspond to a cell with a volume of $\sim 10^3 \mathrm{Mpc^3}$, containing ten to hundred galaxies and a total \HI mass $ \sim 10^{12} M_\odot$. If we neglect local velocities relative to the Hubble flow, the observed frequency $\nu$ would be translated to the emission redshift $z$ through the well known relation: \begin{eqnarray} z(\nu) & = & \frac{\nu_{21} -\nu}{\nu} \, ; \, \nu(z) = \frac{\nu_{21}}{(1+z)} \hspace{1mm} \mathrm{with} \hspace{1mm} \nu_{21} = 1420.4 \, \mathrm{MHz} \\ z(\lambda) & = & \frac{\lambda - \lambda_{21}}{\lambda_{21}} \, ; \, \lambda(z) = \lambda_{21} \times (1+z) \hspace{1mm} \mathrm{with} \hspace{1mm} \lambda_{21} = 0.211 \, \mathrm{m} \end{eqnarray} The large scale distribution of the neutral hydrogen, down to angular scales of $\sim 10 \mathrm{arc.min}$ can then be observed without the detection of individual compact \HI sources, using the set of sky brightness map as a function of frequency (3D-brightness map) $B_{21}(\vec{\Theta},\lambda)$. The sky brightness $B_{21}$ (radiation power/unit solid angle/unit surface/unit frequency) can be converted to brightness temperature using the well known black body Rayleigh-Jeans approximation: $$ B(T,\lambda) = \frac{ 2 \kb T }{\lambda^2} $$ %%%%%%%% \begin{table} \begin{center} \begin{tabular}{|c|c|c|} \hline $A (\mathrm{m^2})$ & $ T_{sys} (K) $ & $ S_{lim} \, \mathrm{\mu Jy} $ \\ \hline 5000 & 50 & 66 \\ 5000 & 25 & 33 \\ 100 000 & 50 & 3.3 \\ 100 000 & 25 & 1.66 \\ 500 000 & 50 & 0.66 \\ 500 000 & 25 & 0.33 \\ \hline \end{tabular} %% \hspace{3mm} %% \begin{tabular}{|c|c|c|} \hline $z$ & $\dlum \mathrm{(Mpc)}$ & $S_{21} \mathrm{( \mu Jy)} $ \\ \hline % dernier chiffre : sans le facteur (1+z) 0.25 & 1235 & 175 \\ % 140 0.50 & 2800 & 40 \\ % 27 1.0 & 6600 & 9.6 \\ % 4.8 1.5 & 10980 & 3.5 \\ % 1.74 2.0 & 15710 & 2.5 \\ % 0.85 2.5 & 20690 & 1.7 \\ % 0.49 \hline \end{tabular} \caption{Sensitivity or source detection limit for 1 day integration time (86400 s) and 1 MHz frequency band (left). Source 21 cm brightness for $10^{10} M_\odot$ \HI for different redshifts (right) } \label{slims21} \end{center} \end{table} \subsection{ \HI power spectrum and BAO} In the absence of any foreground or background radiation {\changemark and assuming high spin temperature $\kb T_{spin} \gg h \nu_{21}$}, the brightness temperature for a given direction and wavelength $\TTlam$ would be proportional to the local \HI number density $\etaHI(\vec{\Theta},z)$ through the relation: \begin{equation} \TTlamz = \frac{3}{32 \pi} \, \frac{h}{\kb} \, A_{21} \, \lambda_{21}^2 \times \frac{c}{H(z)} \, (1+z)^2 \times \etaHI (\vec{\Theta}, z) \end{equation} where $A_{21}=2.85 \, 10^{-15} \mathrm{s^{-1}}$ \citep{astroformul} is the spontaneous 21 cm emission coefficient, $h$ is the Planck constant, $c$ the speed of light, $\kb$ the Boltzmann constant and $H(z)$ is the Hubble parameter at the emission redshift. For a \LCDM universe and neglecting radiation energy density, the Hubble parameter can be expressed as: \begin{equation} H(z) \simeq \hubb \, \left[ \Omega_m (1+z)^3 + \Omega_\Lambda \right]^{\frac{1}{2}} \times 100 \, \, \mathrm{km/s/Mpc} \label{eq:expHz} \end{equation} Introducing the \HI mass fraction relative to the total baryon mass $\gHI$, the neutral hydrogen number density and the corresponding 21 cm emission temperature can be written as a function of \HI relative density fluctuations: \begin{eqnarray} \etaHI (\vec{\Theta}, z(\lambda) ) & = & \gHIz \times \Omega_B \frac{\rho_{crit}}{m_{H}} \times \left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) + 1 \right) \\ \TTlamz & = & \bar{T}_{21}(z) \times \left( \frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}} (\vec{\Theta},z) + 1 \right) \end{eqnarray} where $\Omega_B, \rho_{crit}$ are respectively the present day mean baryon cosmological and critical densities, $m_{H}$ is the hydrogen atom mass, and $\frac{\delta \rho_{H_I}}{\bar{\rho}_{H_I}}$ is the \HI density fluctuations. The present day neutral hydrogen fraction $\gHI(0)$ present in local galaxies has been measured to be $\sim 1\%$ of the baryon density \citep{zwann.05}: $$ \Omega_{H_I} \simeq 3.5 \, 10^{-4} \sim 0.008 \times \Omega_B $$ The neutral hydrogen fraction is expected to increase with redshift, as gas is used in star formation during galaxy formation and evolution. Study of Lyman-$\alpha$ absorption indicate a factor 3 increase in the neutral hydrogen fraction at $z=1.5$ in the intergalactic medium \citep{wolf.05}, compared to its present day value $\gHI(z=1.5) \sim 0.025$. The 21 cm brightness temperature and the corresponding power spectrum can be written as (\cite{barkana.07} and \cite{madau.97}) : \begin{eqnarray} P_{T_{21}}(k) & = & \left( \bar{T}_{21}(z) \right)^2 \, P(k) \label{eq:pk21z} \\ \bar{T}_{21}(z) & \simeq & 0.084 \, \mathrm{mK} \frac{ (1+z)^2 \, \hubb }{\sqrt{ \Omega_m (1+z)^3 + \Omega_\Lambda } } \dfrac{\Omega_B}{0.044} \, \frac{\gHIz}{0.01} \label{eq:tbar21z} \end{eqnarray} The table \ref{tabcct21} shows the mean 21 cm brightness temperature for the standard \LCDM cosmology and either a constant \HI mass fraction $\gHI = 0.01$, or linearly increasing $\gHI \simeq 0.008 \times (1+z) $. Figure \ref{figpk21} shows the 21 cm emission power spectrum at several redshifts, with a constant neutral fraction at 2\% ($\gHI=0.02$). The matter power spectrum has been computed using the \cite{eisenhu.98} parametrisation. The correspondence with the angular scales is also shown for the standard WMAP \LCDM cosmology, according to the relation: \begin{equation} \theta_k = \frac{2 \pi}{k \, \dang(z) \, (1+z) } \hspace{3mm} k = \frac{2 \pi}{ \theta_k \, \dang(z) \, (1+z) } \end{equation} where $k$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance. { \changemark The matter power spectrum $P(k)$ has been measured using galaxy surveys, for example by SDSS and 2dF at low redshift $z \lesssim 0.3$ (\cite{cole.05}, \cite{tegmark.04}). The 21 cm brightness power spectra $P_{T_{21}}(k)$ shown here are comparable to one from the galaxy surveys, once the mean 21 cm temperature conversion factor $\left( \bar{T}_{21}(z) \right)^2$ and redshift evolution have been accounted for. } % It should be noted that the maximum transverse $k^{comov} $ sensitivity range % for an instrument corresponds approximately to half of its angular resolution. % {\color{red} Faut-il developper completement le calcul en annexe ? } \begin{table} \begin{center} \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline \hline z & 0.25 & 0.5 & 1. & 1.5 & 2. & 2.5 & 3. \\ \hline (a) $\bar{T}_{21}$ & 0.085 & 0.107 & 0.145 & 0.174 & 0.195 & 0.216 & 0.234 \\ \hline (b) $\bar{T}_{21}$ & 0.085 & 0.128 & 0.232 & 0.348 & 0.468 & 0.605 & 0.749 \\ \hline \hline \end{tabular} \caption{Mean 21 cm brightness temperature in mK, as a function of redshift, for the standard \LCDM cosmology with constant \HI mass fraction at $\gHIz$=0.01 (a) or linearly increasing mass fraction (b) $\gHIz=0.008(1+z)$ } \label{tabcct21} \end{center} \end{table} \begin{figure} \vspace*{-11mm} \hspace{-5mm} \includegraphics[width=0.57\textwidth]{Figs/pk21cmz12.pdf} \vspace*{-10mm} \caption{\HI 21 cm emission power spectrum at redshifts z=1 (blue) and z=2 (red), with neutral gas fraction $\gHI=2\%$} \label{figpk21} \end{figure} \section{interferometric observations and P(k) measurement sensitivity } \label{pkmessens} \subsection{Instrument response} \label{instrumresp} We introduce briefly here the principles of interferometric observations and the definition of quantities useful for our calculations. Interested reader may refer to \citep{radastron} for a detailed and complete presentation of observation methods and signal processing in radio astronomy. In astronomy we are usually interested in measuring the sky emission intensity, $I(\vec{\Theta},\lambda)$ in a given wave band, as a function of the sky direction. In radio astronomy and interferometry in particular, receivers are sensitive to the sky emission complex amplitudes. However, for most sources, the phases vary randomly with a spatial correlation length significantly smaller than the instrument resolution. \begin{eqnarray} & & I(\vec{\Theta},\lambda) = | A(\vec{\Theta},\lambda) |^2 \hspace{2mm} , \hspace{1mm} I \in \mathbb{R}, A \in \mathbb{C} \\ & & < A(\vec{\Theta},\lambda) A^*(\vec{\Theta '},\lambda) >_{time} = 0 \hspace{2mm} \mathrm{for} \hspace{1mm} \vec{\Theta} \ne \vec{\Theta '} \end{eqnarray} A single receiver can be characterized by its angular complex amplitude response $B(\vec{\Theta},\nu)$ and its position $\vec{r}$ in a reference frame. the waveform complex amplitude $s$ measured by the receiver, for each frequency can be written as a function of the electromagnetic wave vector $\vec{k}_{EM}(\vec{\Theta}, \lambda) $ : \begin{equation} s(\lambda) = \iint d \vec{\Theta} \, \, \, A(\vec{\Theta},\lambda) B(\vec{\Theta},\lambda) e^{i ( \vec{k}_{EM} . \vec{r} )} \\ \end{equation} We have set the electromagnetic (EM) phase origin at the center of the coordinate frame and the EM wave vector is related to the wavelength $\lambda$ through the usual equation $ | \vec{k}_{EM} | = 2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta},\lambda)$ corresponds to the receiver intensity response: \begin{equation} L(\vec{\Theta}, \lambda) = B(\vec{\Theta},\lambda) \, B^*(\vec{\Theta},\lambda) \end{equation} The visibility signal of two receivers corresponds to the time averaged correlation between signals from two receivers. If we assume a sky signal with random uncorrelated phase, the visibility $\vis$ signal from two identical receivers, located at the position $\vec{r_1}$ and $\vec{r_2}$ can simply be written as a function of their position difference $\vec{\Delta r} = \vec{r_1}-\vec{r_2}$ \begin{equation} \vis(\lambda) = < s_1(\lambda) s_2(\lambda)^* > = \iint d \vec{\Theta} \, \, I(\vec{\Theta},\lambda) L(\vec{\Theta},\lambda) e^{i ( \vec{k}_{EM} . \vec{\Delta r} ) } \end{equation} This expression can be simplified if we consider receivers with narrow field of view ($ L(\vec{\Theta},\lambda) \simeq 0$ for $| \vec{\Theta} | \gtrsim 10 \, \mathrm{deg.} $ ), and coplanar in respect to their common axis. If we introduce two {\em Cartesian} like angular coordinates $(\alpha,\beta)$ centered at the common receivers axis, the visibilty would be written as the 2D Fourier transform of the product of the sky intensity and the receiver beam, for the angular frequency \mbox{$(\uv)_{12} = ( \frac{\Delta x}{\lambda} , \frac{\Delta y}{\lambda} )$}: \begin{equation} \vis(\lambda) \simeq \iint d\alpha d\beta \, \, I(\alpha, \beta) \, L(\alpha, \beta) \exp \left[ i 2 \pi \left( \alpha \frac{\Delta x}{\lambda} + \beta \frac{\Delta y}{\lambda} \right) \right] \end{equation} where $(\Delta x, \Delta y)$ are the two receiver distances on a plane perpendicular to the receiver axis. The $x$ and $y$ axis in the receiver plane are taken parallel to the two $(\alpha, \beta)$ angular planes. Furthermore, we introduce the conjugate Fourier variables $(\uv)$ and the Fourier transforms of the sky intensity and the receiver beam: \begin{center} \begin{tabular}{ccc} $(\alpha, \beta)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & $(\uv)$ \\ $I(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal I}(\uv, \lambda)$ \\ $L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(\uv, \lambda)$ \\ \end{tabular} \end{center} The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular wave number domain located around $(\uv)_{12}=2 \pi( \frac{\Delta x}{\lambda} , \frac{\Delta y}{\lambda} )$. The weight function is given by the receiver beam Fourier transform. \begin{equation} \vis(\lambda) \simeq \iint \dudv \, \, {\cal I}(\uv, \lambda) \, {\cal L}(\uvu - \frac{\Delta x}{\lambda} , \uvv - \frac{\Delta y}{\lambda} , \lambda) \end{equation} A single receiver instrument would measure the total power integrated in a spot centered around the origin in the $(\uv)$ or the angular wave mode plane. The shape of the spot depends on the receiver beam pattern, but its extent would be $\sim 2 \pi D / \lambda$, where $D$ is the receiver physical size. The correlation signal from a pair of receivers would measure the integrated signal on a similar spot, located around the central angular wave mode $(\uv)_{12}$ determined by the relative position of the two receivers (see figure \ref{figuvplane}). In an interferometer with multiple receivers, the area covered by different receiver pairs in the $(\uv)$ plane might overlap and some pairs might measure the same area (same base lines). Several beams can be formed using different combination of the correlations from a set of antenna pairs. An instrument can thus be characterized by its $(\uv)$ plane coverage or response ${\cal R}(\uv,\lambda)$. For a single dish with a single receiver in the focal plane, the instrument response is simply the Fourier transform of the beam. For a single dish with multiple receivers, either as a Focal Plane Array (FPA) or a multi-horn system, each beam (b) will have its own response ${\cal R}_b(\uv,\lambda)$. For an interferometer, we can compute a raw instrument response ${\cal R}_{raw}(\uv,\lambda)$ which corresponds to $(u,v)$ plane coverage by all receiver pairs with uniform weighting. Obviously, different weighting schemes can be used, changing the effective beam shape and thus the response ${\cal R}_{w}(\uv,\lambda)$ and the noise behaviour. If the same Fourier angular frequency mode is measured by several receiver pairs, the raw instrument response might then be larger that unity. This non normalized instrument response is used to compute the projected noise power spectrum in the following section (\ref{instrumnoise}). We can also define a normalized instrument response, ${\cal R}_{norm}(\uv,\lambda) \lesssim 1$ as: \begin{equation} {\cal R}_{norm}(\uv,\lambda) = {\cal R}(\uv,\lambda) / \mathrm{Max_{(\uv)}} \left[ {\cal R}(\uv,\lambda) \right] \end{equation} This normalized instrument response can be used to compute the effective instrument beam, in particular in section \ref{recsec}. {\changemark Detection of the reionisation at 21 cm band has been an active field in the last decade and several groups (\cite{rottgering.06}, \cite{bowman.07}, \cite{lonsdale.09}, \cite{parsons.09}) have built instruments to detect reionisation signal around 100 MHz. Several authors have studied the instrumental noise and statistical uncertainties when measuring the reionisation signal power spectrum; the methods presented here to compute the instrument response and sensitivities are similar to the ones developed in these publications (\cite{morales.04}, \cite{bowman.06}, \cite{mcquinn.06}). } \begin{figure} % \vspace*{-2mm} \centering \mbox{ \includegraphics[width=0.5\textwidth]{Figs/uvplane.pdf} } \vspace*{-15mm} \caption{Schematic view of the $(\uv)$ plane coverage by interferometric measurement.} \label{figuvplane} \end{figure} \subsection{Noise power spectrum computation} \label{instrumnoise} Let's consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency bandwidth $\delta \nu$ centered on $\nu_0$, with an integration time $t_{int}$, characterized by a system temperature $\Tsys$. The uncertainty or fluctuations of this measurement due to the receiver noise can be written as $\sigma_{noise}^2 = \frac{2 \Tsys^2}{t_{int} \, \delta \nu}$. This term corresponds also to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated noise. If the receiver has an effective area $A \simeq \pi D^2/4$ or $A \simeq D_x D_y$, the measurement corresponds to the integration of power over a spot in the angular frequency plane with an area $\sim A/\lambda^2$. The noise spectral density, in the angular frequencies plane (per unit area of angular frequencies $\delta \uvu \times \uvv$), corresponding to a visibility measurement from a pair of receivers can be written as: \begin{eqnarray} P_{noise}^{\mathrm{pair}} & = & \frac{\sigma_{noise}^2}{ A / \lambda^2 } \\ P_{noise}^{\mathrm{pair}} & \simeq & \frac{2 \, \Tsys^2 }{t_{int} \, \delta \nu} \, \frac{ \lambda^2 }{ D^2 } \hspace{5mm} \mathrm{units:} \, \mathrm{K^2 \times rad^2} \label{eq:pnoisepairD} \end{eqnarray} The sky temperature measurement can thus be characterized by the noise spectral power density in the angular frequencies plane $P_{noise}^{(\uv)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$ per unit area of angular frequencies $\delta \uvu \times \delta \uvv$: We can characterize the sky temperature measurement with a radio instrument by the noise spectral power density in the angular frequencies plane $P_{noise}(\uv)$ in units of $\mathrm{Kelvin^2}$ per unit area of angular frequencies $\delta \uvu \times \delta \uvv$. For an interferometer made of identical receiver elements, several ($n$) receiver pairs might have the same baseline. The noise power density in the corresponding $(\uv)$ plane area is then reduced by a factor $1/n$. More generally, we can write the instrument noise spectral power density using the instrument response defined in section \ref{instrumresp} : \begin{equation} P_{noise}(\uv) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(\uv,\lambda) } \label{eq:pnoiseuv} \end{equation} When the intensity maps are projected in a three dimensional box in the universe and the 3D power spectrum $P(k)$ is computed, angles are translated into comoving transverse distances, and frequencies or wavelengths into comoving radial distance, using the following relations: { \changemark \begin{eqnarray} \alpha , \beta & \rightarrow & \ell_\perp = l_x, l_y = (1+z) \, \dang(z) \, \alpha,\beta \\ \uv & \rightarrow & k_\perp = k_x, k_y = 2 \pi \frac{ \uvu , \uvv }{ (1+z) \, \dang(z) } \label{eq:uvkxky} \\ \delta \nu & \rightarrow & \delta \ell_\parallel = (1+z) \frac{c}{H(z)} \frac{\delta \nu}{\nu} = (1+z) \frac{\lambda}{H(z)} \delta \nu \\ % \delta \uvu , \delta \uvv & \rightarrow & \delta k_\perp = 2 \pi \frac{ \delta \uvu \, , \, \delta \uvv }{ (1+z) \, \dang(z) } \\ \frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = 2 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu} = \frac{H(z)}{c} \frac{1}{(1+z)^2} \, \frac{\nu_{21}}{\delta \nu} \end{eqnarray} } { \changemark A brightness measurement at a point $(\uv,\lambda)$, covering the 3D spot $(\delta \uvu, \delta \uvv, \delta \nu)$, would correspond to cosmological power spectrum measurement at a transverse wave mode $(k_x,k_y)$ defined by the equation \ref{eq:uvkxky}, measured at a redshift given by the observation frequency. The measurement noise spectral density given by the equation \ref{eq:pnoisepairD} can then be translated into a 3D noise power spectrum, per unit of spatial frequencies $ \frac{\delta k_x \times \delta k_y \times \delta k_z}{8 \pi^3} $ (units: $ \mathrm{K^2 \times Mpc^3}$) : \begin{eqnarray} (\uv , \lambda) & \rightarrow & k_x(\uvu),k_y(\uvv), z(\lambda) \\ P_{noise}(k_x,k_y, z) & = & P_{noise}(\uv) \frac{ 8 \pi^3 \delta \uvu \times \delta \uvv }{\delta k_x \times \delta k_y \times \delta k_z} \\ P_{noise}(k_x,k_y, z) & = & \left( 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2} \right) \, \frac{1}{{\cal R}_{raw}} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \label{eq:pnoisekxkz} \end{eqnarray} It is worthwhile to notice that the cosmological 3D noise power spectrum does not depend anymore on the individual measurement bandwidth. In the following paragraph, we will first consider an ideal instrument with uniform $(\uv)$ coverage in order to establish the general noise power spectrum behaviour for cosmological 21 cm surveys. The numerical method used to compute the 3D noise power spectrum is then presented in section \ref{pnoisemeth}. } \subsubsection{Uniform $(\uv)$ coverage} If we consider a uniform noise spectral density in the $(\uv)$ plane corresponding to the equation \ref{eq:pnoisepairD} above, the three dimensional projected noise spectral density can then be written as: \begin{equation} P_{noise}(k) = 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \label{ctepnoisek} \end{equation} $P_{noise}(k)$ would be in units of $\mathrm{mK^2 \, Mpc^3}$ with $\Tsys$ expressed in $\mathrm{mK}$, $t_{int}$ is the integration time expressed in second, $\nu_{21}$ in $\mathrm{Hz}$, $c$ in $\mathrm{km/s}$, $\dang$ in $\mathrm{Mpc}$ and $H(z)$ in $\mathrm{km/s/Mpc}$. The matter or \HI distribution power spectrum determination statistical errors vary as the number of observed Fourier modes, which is inversely proportional to volume of the universe which is observed (sample variance). As the observed volume is proportional to the surveyed solid angle, we consider the survey of a fixed fraction of the sky, defined by total solid angle $\Omega_{tot}$, performed during a determined total observation time $t_{obs}$. A single dish instrument with diameter $D$ would have an instantaneous field of view $\Omega_{FOV} \sim \left( \frac{\lambda}{D} \right)^2$, and would require a number of pointings $N_{point} = \frac{\Omega_{tot}}{\Omega_{FOV}}$ to cover the survey area. Each sky direction or pixel of size $\Omega_{FOV}$ will be observed during an integration time $t_{int} = t_{obs}/N_{point} $. Using equation \ref{ctepnoisek} and the previous expression for the integration time, we can compute a simple expression for the noise spectral power density by a single dish instrument of diameter $D$: \begin{equation} P_{noise}^{survey}(k) = 2 \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \end{equation} It is important to note that any real instrument do not have a flat response in the $(u,v)$ plane, and the observations provide no information above a certain maximum angular frequency $u_{max},v_{max}$. One has to take into account either a damping of the observed sky power spectrum or an increase of the noise spectral power if the observed power spectrum is corrected for damping. The white noise expressions given below should thus be considered as a lower limit or floor of the instrument noise spectral density. For a single dish instrument of diameter $D$ equipped with a multi-feed or phase array receiver system, with $N$ independent beams on sky, the noise spectral density decreases by a factor $N$, thanks to the increase of per pointing integration time: \begin{equation} P_{noise}^{survey}(k) = \frac{2}{N} \, \frac{\Tsys^2 \, \Omega_{tot} }{t_{obs} \, \nu_{21} } \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \label{eq:pnoiseNbeam} \end{equation} This expression (eq. \ref{eq:pnoiseNbeam}) can also be used for a filled interferometric array of $N$ identical receivers with a total collection area $\sim D^2$. Such an array could be made for example of $N=q \times q$ {\it small dishes}, each with diameter $D/q$, arranged as $q \times q$ square. For a single dish of diameter $D$, or an interferometric instrument with maximal extent $D$, observations provide information up to $\uvu_{max},\uvv_{max} \lesssim D / \lambda $. This value of $\uvu_{max},\uvv_{max}$ would be mapped to a maximum transverse cosmological wave number $k_{\perp}^{max}$: \begin{equation} k_{\perp}^{max} \lesssim \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}} \label{kperpmax} \end{equation} Figure \ref{pnkmaxfz} shows the evolution of the noise spectral density $P_{noise}^{survey}(k)$ as a function of redshift, for a radio survey of the sky, using an instrument with $N=100$ beams and a system noise temperature $\Tsys = 50 \mathrm{K}$. The survey is supposed to cover a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$, in one year. The maximum comoving wave number $k^{max}$ is also shown as a function of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. In order to take into account the radial component of $\vec{k}$ and the increase of the instrument noise level with $k_{\perp}$, we have taken the effective $k_{ max} $ as half of the maximum transverse $k_{\perp} ^{max}$ of \mbox{eq. \ref{kperpmax}}: \begin{equation} k_{max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}} \end{equation} \begin{figure} \vspace*{-25mm} \centering \mbox{ \hspace*{-10mm} \includegraphics[width=0.65\textwidth]{Figs/pnkmaxfz.pdf} } \vspace*{-40mm} \caption{Minimal noise level for a 100 beams instrument with \mbox{$\Tsys=50 \mathrm{K}$} as a function of redshift (top). Maximum $k$ value for a 100 meter diameter primary antenna (bottom) } \label{pnkmaxfz} \end{figure} \subsubsection{3D noise power spectrum computation} \label{pnoisemeth} { \changemark We describe here the numerical method used to compute the 3D noise power spectrum, for a given instrument response, as presented in section \ref{instrumnoise}. The noise power spectrum is a good indicator to compare sensitivities for different instrument configurations, albeit the noise realization for a real instrument would not be isotropic. \begin{itemize} \item We start by a 3D Fourier coefficient grid, with the two first coordinates being the transverse angular wave modes, and the third being the frequency $(k_x,k_y,\nu)$. The grid is positioned at the mean redshift $z_0$ for which we want to compute $P_{noise}(k)$. For the results at redshift \mbox{$z_0=1$} discussed in section \ref{instrumnoise}, the grid cell size and dimensions have been chosen to represent a box in the universe \mbox{$\sim 1500 \times 1500 \times 750 \mathrm{Mpc^3}$}, with $3\times3\times3 \mathrm{Mpc^3}$ cells. This correspond to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given the small angular extent, we have neglected the curvature of redshift shells. \item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. $(k_x,k_y)$ is converted to the $\uv$ coordinates using the equation \ref{eq:uvkxky}, and the angular diameter distance $\dang(z)$ for \LCDM model with WMAP parameters. The noise variance is taken proportional to $P_{noise}$ : \begin{equation} \sigma_{re}^2=\sigma_{im}^2 \propto \frac{1}{{\cal R}_{raw}(\uv,\lambda)} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 \end{equation} \item an FFT is then performed in the frequency or redshift direction to obtain the noise Fourier complex coefficients $n(k_x,k_y,k_z)$ and the power spectrum is estimated as : \begin{equation} \tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \hspace{2mm} \mathrm{for} \hspace{2mm} \sqrt{k_x^2+k_y^2+k_z^2} = k \end{equation} Noise samples corresponding to small instrument response, typically less than 1\% of the maximum instrument response are ignored when calculating $\tilde{P}_{noise}(k)$. However, we require to have a significant fraction, typically 20\% to 50\% of the possible modes $(k_x,k_y,k_z)$ measured for a given $k$ value. \item the above steps are repeated a number of time to decrease the statistical fluctuations due to the random generations. The averaged computed noise power spectrum is normalized using equation \ref{eq:pnoisekxkz} and the instrument and survey parameters ($\Tsys \ldots$). \end{itemize} It should be noted that it is possible to obtain a good approximation of noise power spectrum shape, neglecting the redshift or frequency dependence of the instrument response function and $\dang(z)$ for a small redshift interval around $z_0$, using a fixed instrument response ${\cal R}(u,v,\lambda(z_0))$ and a constant the radial distance $\dang(z_0)*(1+z_0)$. \begin{equation} \tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \simeq < P_{noise}(u,v) , k_z > \end{equation} The approximate power spectrum obtain through this simplified and much faster method is shown as dashed curves on figure \ref{figpnoisea2g} for few instrument configurations. } \subsection{Instrument configurations and noise power spectrum} \label{instrumnoise} We have numerically computed the instrument response ${\cal R}(\uv,\lambda)$ with uniform weights in the $(\uv)$ plane for several instrument configurations: \begin{itemize} \item[{\bf a} :] A packed array of $n=121 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in a square $11 \times 11$ configuration ($q=11$). This array covers an area of $55 \times 55 \, \mathrm{m^2}$ \item [{\bf b} :] An array of $n=128 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in 8 rows, each with 16 dishes. These 128 dishes are spread over an area $80 \times 80 \, \mathrm{m^2}$. The array layout for this configuration is shown in figure \ref{figconfbc}. \item [{\bf c} :] An array of $n=129 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged over an area $80 \times 80 \, \mathrm{m^2}$. This configuration has in particular 4 sub-arrays of packed 16 dishes ($4\times4$), located in the four array corners. This array layout is also shown figure \ref{figconfbc}. \item [{\bf d} :] A single dish instrument, with diameter $D=75 \, \mathrm{m}$, equipped with a 100 beam focal plane receiver array. \item[{\bf e} :] A packed array of $n=400 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in a square $20 \times 20$ configuration ($q=20$). This array covers an area of $100 \times 100 \, \mathrm{m^2}$ \item[{\bf f} :] A packed array of 4 cylindrical reflectors, each 85 meter long and 12 meter wide. The focal line of each cylinder is equipped with 100 receivers, each $2 \lambda$ long, corresponding to $\sim 0.85 \, \mathrm{m}$ at $z=1$. This array covers an area of $48 \times 85 \, \mathrm{m^2}$, and have a total of $400$ receivers per polarisation, as in the (e) configuration. We have computed the noise power spectrum for {\em perfect} cylinders, where all receiver pair correlations are used (fp), or for a non perfect instrument, where only correlations between receivers from different cylinders are used. \item[{\bf g} :] A packed array of 8 cylindrical reflectors, each 102 meter long and 12 meter wide. The focal line of each cylinder is equipped with 120 receivers, each $2 \lambda$ long, corresponding to $\sim 0.85 \, \mathrm{m}$ at $z=1$. This array covers an area of $96 \times 102 \, \mathrm{m^2}$ and has a total of 960 receivers per polarisation. As for the (f) configuration, we have computed the noise power spectrum for {\em perfect} cylinders, where all receiver pair correlations are used (gp), or for a non perfect instrument, where only correlations between receivers from different cylinders are used. \end{itemize} \begin{figure} \centering \vspace*{-15mm} \mbox{ \hspace*{-10mm} \includegraphics[width=0.5\textwidth]{Figs/configab.pdf} } \vspace*{-15mm} \caption{ Array layout for configurations (b) and (c) with 128 and 129 D=5 meter diameter dishes. } \label{figconfbc} \end{figure} We have used simple triangular shaped dish response in the $(\uv)$ plane. However, we have introduced a filling factor or illumination efficiency $\eta$, relating the effective dish diameter $D_{ill}$ to the mechanical dish size $D^{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales as $\eta^2$ or $\eta_x \eta_y$. \begin{eqnarray} {\cal L}_\circ (\uv,\lambda) & = & \bigwedge_{[\pm D^{ill}/ \lambda]}(\sqrt{u^2+v^2}) \\ L_\circ (\alpha,\beta,\lambda) & = & \left[ \frac{ \sin (\pi (D^{ill}/\lambda) \sin \theta ) }{\pi (D^{ill}/\lambda) \sin \theta} \right]^2 \hspace{4mm} \theta=\sqrt{\alpha^2+\beta^2} \end{eqnarray} For the multi-dish configuration studied here, we have taken the illumination efficiency factor {\bf $\eta = 0.9$}. For the receivers along the focal line of cylinders, we have assumed that the individual receiver response in the $(u,v)$ plane corresponds to one from a rectangular shaped antenna. The illumination efficiency factor has been taken equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$ along the cylinder length. {\changemark We have used double triangular shaped response function in the $(\uv)$ for each of the receiver elements along the cylinder: \begin{equation} {\cal L}_\Box(\uv,\lambda) = \bigwedge_{[\pm \eta_x D_x / \lambda]} (\uvu ) \times \bigwedge_{[\pm \eta_y D_y / \lambda ]} (\uvv ) \end{equation} } It should be noted that the small angle approximation used here for the expression of visibilities is not valid for the receivers along the cylinder axis. However, some preliminary numerical checks indicate that the results obtained here for the noise spectral power density would not change significantly. The instrument responses shown here correspond to fixed pointing toward the zenith, which is the case for a transit type telescope. Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}(u,v,\lambda)$ for the four configurations (a,b,c,d) with $\sim 100$ receivers per polarisation. {\changemark Using the numerical method sketched in section \ref{pnoisemeth}, we have computed the 3D noise power spectrum for each of the eight instrument configurations discussed here, with a system noise temperature $\Tsys = 50 \mathrm{K}$, for a one year survey of a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$ at a mean redshift $z_0=1, \nu_0=710 \mathrm{MHz}$.} The resulting noise spectral power densities are shown in figure \ref{figpnoisea2g}. The increase of $P_{noise}(k)$ at low $k^{comov} \lesssim 0.02$ is due to the fact that we have ignored all auto-correlation measurements. It can be seen that an instrument with $100-200$ beams and $\Tsys = 50 \mathrm{K}$ should have enough sensitivity to map LSS in 21 cm at redshift z=1. \begin{figure*} \centering \mbox{ % \hspace*{-10mm} \includegraphics[width=\textwidth]{Figs/uvcovabcd.pdf} } \caption{$(\uv)$ plane coverage (raw instrument response ${\cal R}(\uv,\lambda)$ for four configurations. (a) 121 $D_{dish}=5$ meter diameter dishes arranged in a compact, square array of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each (fig. \ref{figconfbc}), (c) 129 dishes arranged as shown in figure \ref{figconfbc} , (d) single D=75 meter diameter, with 100 beams. The common color scale (1 \ldots 80) is shown on the right. } \label{figuvcovabcd} \end{figure*} \begin{figure*} \vspace*{-10mm} \centering \mbox{ % \hspace*{-5mm} \includegraphics[width=\textwidth]{Figs/pkna2h.pdf} } \vspace*{-20mm} \caption{P(k) LSS power and noise power spectrum for several interferometer configurations ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers.} \label{figpnoisea2g} \end{figure*} \section{ Foregrounds and Component separation } \label{foregroundcompsep} Reaching the required sensitivities is not the only difficulty of observing the large scale structures in 21 cm. Indeed, the synchrotron emission of the Milky Way and the extra galactic radio sources are a thousand times brighter than the emission of the neutral hydrogen distributed in the universe. Extracting the LSS signal using Intensity Mapping, without identifying the \HI point sources is the main challenge for this novel observation method. Although this task might seem impossible at first, it has been suggested that the smooth frequency dependence of the synchrotron emissions can be used to separate the faint LSS signal from the Galactic and radio source emissions. {\changemark Discussion of contribution of different sources to foregrounds for measurement of reionization through redshifted 21 cm, as well foreground subtraction using their smooth frequency dependence can be found in (\cite{shaver.99}, \cite{matteo.02},\cite{oh.03}) } However, any real radio instrument has a beam shape which changes with frequency: this instrumental effect significantly increases the difficulty and complexity of this component separation technique. The effect of frequency dependent beam shape is some time referred to as {\em mode mixing}. {\changemark Effect of frequency dependent beam shape for foreground subtraction and its application to MWA has been discussed in \citep{morales.06} \citep{bowman.09}.} In this section, we present a short description of the foreground emissions and the simple models we have used for computing the sky radio emissions in the GHz frequency range. We present also a simple component separation method to extract the LSS signal and its performance. We show in particular the effect of the instrument response on the recovered power spectrum. The results presented in this section concern the total sky emission and the LSS 21 cm signal extraction in the $z \sim 0.6$ redshift range, corresponding to the central frequency $\nu \sim 884$ MHz. \subsection{ Synchrotron and radio sources } We have modeled the radio sky in the form of three dimensional maps (data cubes) of sky temperature brightness $T(\alpha, \delta, \nu)$ as a function of two equatorial angular coordinates $(\alpha, \delta)$ and the frequency $\nu$. Unless otherwise specified, the results presented here are based on simulations of $90 \times 30 \simeq 2500 \, \mathrm{deg^2}$ of the sky, centered on $\alpha= 10\mathrm{h}00\mathrm{m} , \delta=+10 \, \mathrm{deg.}$, and covering 128 MHz in frequency. We have selected this particular area of the sky in order to minimize the Galactic synchrotron foreground. The sky cube characteristics (coordinate range, size, resolution) used in the simulations are given in the table \ref{skycubechars}. \begin{table} \begin{center} \begin{tabular}{|c|c|c|} \hline & range & center \\ \hline Right ascension & 105 $ < \alpha < $ 195 deg. & 150 deg.\\ Declination & -5 $ < \delta < $ 25 deg. & +10 deg. \\ Frequency & 820 $ < \nu < $ 948 MHz & 884 MHz \\ Wavelength & 36.6 $ < \lambda < $ 31.6 cm & 33.9 cm \\ Redshift & 0.73 $ < z < $ 0.5 & 0.61 \\ \hline \hline & resolution & N-cells \\ \hline Right ascension & 3 arcmin & 1800 \\ Declination & 3 arcmin & 600 \\ Frequency & 500 kHz ($d z \sim 10^{-3}$) & 256 \\ \hline \end{tabular} \\[1mm] \end{center} \caption{ Sky cube characteristics for the simulation performed in this paper. Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$ $ 1800 \times 600 \times 256 \simeq 123 \, 10^6$ cells } \label{skycubechars} \end{table} %%%% \par Two different methods have been used to compute the sky temperature data cubes. We have used the Global Sky Model (GSM) \citep{gsm.08} tools to generate full sky maps of the emission temperature at different frequencies, from which we have extracted the brightness temperature cube for the region defined above (Model-I/GSM $T_{gsm}(\alpha, \delta, \nu)$). As the GSM maps have an intrinsic resolution of $\sim$ 0.5 degree, it is difficult to have reliable results for the effect of point sources on the reconstructed LSS power spectrum. We have thus made also a simple sky model using the Haslam Galactic synchrotron map at 408 MHz \citep{haslam.82} and the NRAO VLA Sky Survey (NVSS) 1.4 GHz radio source catalog \citep{nvss.98}. The sky temperature cube in this model (Model-II/Haslam+NVSS) has been computed through the following steps: \begin{enumerate} \item The Galactic synchrotron emission is modeled as a power law with spatially varying spectral index. We assign a power law index $\beta = -2.8 \pm 0.15$ to each sky direction. $\beta$ has a gaussian distribution centered at -2.8 and with standard deviation $\sigma_\beta = 0.15$. {\changemark The diffuse radio background spectral index has been measured for example by \citep{platania.98} or \cite{rogers.08} } The synchrotron contribution to the sky temperature for each cell is then obtained through the formula: $$ T_{sync}(\alpha, \delta, \nu) = T_{haslam} \times \left(\frac{\nu}{408 \, \mathrm{MHz}}\right)^\beta $$ %% \item A two dimensional $T_{nvss}(\alpha,\delta)$ sky brightness temperature at 1.4 GHz is computed by projecting the radio sources in the NVSS catalog to a grid with the same angular resolution as the sky cubes. The source brightness in Jansky is converted to temperature taking the pixel angular size into account ($ \sim 21 \mathrm{mK / mJansky}$ at 1.4 GHz and $3' \times 3'$ pixels). A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source map; we have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the contribution of the radiosources to the sky temperature is computed as follows: $$ T_{radsrc}(\alpha, \delta, \nu) = T_{nvss} \times \left(\frac{\nu}{1420 \, \mathrm{MHz}}\right)^{\beta_{src}} $$ %% \item The sky brightness temperature data cube is obtained through the sum of the two contributions, Galactic synchrotron and resolved radio sources: $$ T_{fgnd}(\alpha, \delta, \nu) = T_{sync}(\alpha, \delta, \nu) + T_{radsrc}(\alpha, \delta, \nu) $$ \end{enumerate} The 21 cm temperature fluctuations due to neutral hydrogen in large scale structures $T_{lss}(\alpha, \delta, \nu)$ have been computed using the SimLSS \footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} } software package: % complex normal Gaussian fields were first generated in Fourier space. The amplitude of each mode was then multiplied by the square root of the power spectrum $P(k)$ at $z=0$ computed according to the parametrization of \citep{eisenhu.98}. We have used the standard cosmological parameters, $H_0=71 \, \mathrm{km/s/Mpc}$, $\Omega_m=0.264$, $\Omega_b=0.045$, $\Omega_\lambda=0.73$ and $w=-1$ \citep{komatsu.11}. An inverse FFT was then performed to compute the matter density fluctuations $\delta \rho / \rho$ in the linear regime, in a box of $3420 \times 1140 \times 716 \, \mathrm{Mpc^3}$ and evolved to redshift $z=0.6$. The size of the box is about 2500 $\mathrm{deg^2}$ in the transverse direction and $\Delta z \simeq 0.23$ in the longitudinal direction. The size of the cells is $1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the sky cube angular and frequency resolution defined above. The mass fluctuations has been converted into temperature through a factor $0.13 \, \mathrm{mK}$, corresponding to a hydrogen fraction $0.008 \times (1+0.6)$, using equation \ref{eq:tbar21z}. The total sky brightness temperature is then computed as the sum of foregrounds and the LSS 21 cm emission: $$ T_{sky} = T_{sync}+T_{radsrc}+T_{lss} \hspace{5mm} OR \hspace{5mm} T_{sky} = T_{gsm}+T_{lss} $$ Table \ref{sigtsky} summarizes the mean and standard deviation of the sky brightness temperature $T(\alpha, \delta, \nu)$ for the different components computed in this study. It should be noted that the standard deviation depends on the map resolution and the values given in table \ref{sigtsky} correspond to sky cubes computed here, with $\sim 3$ arc minute angular and 500 kHz frequency resolutions (see table \ref{skycubechars}). Figure \ref{compgsmmap} shows the comparison of the GSM temperature map at 884 MHz with Haslam+NVSS map, smoothed with a 35 arcmin gaussian beam. Figure \ref{compgsmhtemp} shows the comparison of the sky cube temperature distribution for Model-I/GSM and Model-II. There is good agreement between the two models, although the mean temperature for Model-II is slightly higher ($\sim 10\%$) than Model-I. \begin{table} \centering \begin{tabular}{|c|c|c|} \hline & mean (K) & std.dev (K) \\ \hline Haslam & 2.17 & 0.6 \\ NVSS & 0.13 & 7.73 \\ Haslam+NVSS & 2.3 & 7.75 \\ (Haslam+NVSS)*Lobe(35') & 2.3 & 0.72 \\ GSM & 2.1 & 0.8 \\ \hline \end{tabular} \caption{ Mean temperature and standard deviation for the different sky brightness data cubes computed for this study (see table \ref{skycubechars} for sky cube resolution and size).} \label{sigtsky} \end{table} we have computed the power spectrum for the 21cm-LSS sky temperature cube, as well as for the radio foreground temperature cubes obtained from the two models. We have also computed the power spectrum on sky brightness temperature cubes, as measured by a perfect instrument having a 25 arcmin (FWHM) gaussian beam. The resulting computed power spectra are shown on figure \ref{pkgsmlss}. The GSM model has more large scale power compared to our simple Haslam+NVSS model, while it lacks power at higher spatial frequencies. The mode mixing due to frequency dependent response will thus be stronger in Model-II (Haslam+NVSS) case. It can also be seen that the radio foreground power spectrum is more than $\sim 10^6$ times higher than the 21 cm signal from large scale structures. This corresponds to the factor $\sim 10^3$ of the sky brightness temperature fluctuations ($\sim$ K), compared to the mK LSS signal. { \changemark Contrary to most similar studies, where it is assumed that bright sources can be nearly perfectly subtracted, our aim was to compute also their effect in the foreground subtraction process. The GSM model lacks the angular resolution needed to compute correctly the effect of bright compact sources for 21 cm LSS observations and the mode mixing due to frequency dependent instrument, while Model-II provides a reasonable description of these compact sources. Our simulated sky cubes have an angular resolution $3'\times3'$, well below the typical $15'$ resolution of the instrument configuration considered here. However, Model-II might lack spatial structures at large scales, above a degree, compared to Model-I/GSM, and the frequency variations as a simple power law might not be realistic enough. The differences for the two sky models can be seen in their power spectra shown in figure \ref{pkgsmlss}. We hope that by using these two models, we have explored some of the systematic uncertainties related to foreground subtraction.} It should also be noted that in section 3, we presented the different instrument configuration noise levels after {\em correcting or deconvolving} the instrument response. The LSS power spectrum is recovered unaffected in this case, while the noise power spectrum increases at high k values (small scales). In practice, clean deconvolution is difficult to implement for real data and the power spectra presented in this section are NOT corrected for the instrumental response. The observed structures have thus a scale dependent damping according to the instrument response, while the instrument noise is flat (white noise or scale independent). \begin{figure} \centering \vspace*{-10mm} \mbox{ \hspace*{-20mm} \includegraphics[width=0.6\textwidth]{Figs/comptempgsm.pdf} } \vspace*{-10mm} \caption{Comparison of GSM (black) Model-II (red) sky cube temperature distribution. The Model-II (Haslam+NVSS), has been smoothed with a 35 arcmin gaussian beam. } \label{compgsmhtemp} \end{figure} \begin{figure*} \centering \mbox{ % \hspace*{-10mm} \includegraphics[width=0.9\textwidth]{Figs/compmapgsm.pdf} } \caption{Comparison of GSM map (top) and Model-II sky map at 884 MHz (bottom). The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin (FWHM) gaussian beam.} \label{compgsmmap} \end{figure*} \begin{figure} \centering \vspace*{-25mm} \mbox{ \hspace*{-15mm} \includegraphics[width=0.65\textwidth]{Figs/pk_gsm_lss.pdf} } \vspace*{-40mm} \caption{Comparison of the 21cm LSS power spectrum (red, orange) with the radio foreground power spectrum. The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple model based on Haslam+NVSS (Model-II, black). The curves with circle markers show the power spectrum as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam.} \label{pkgsmlss} \end{figure} \subsection{ Instrument response and LSS signal extraction } \label{recsec} The {\it observed} data cube is obtained from the sky brightness temperature 3D map $T_{sky}(\alpha, \delta, \nu)$ by applying the frequency or wavelength dependent instrument response ${\cal R}(u,v,\lambda)$. We have considered the simple case where the instrument response is constant throughout the survey area, or independent of the sky direction. For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$ : \begin{enumerate} \item Apply a 2D Fourier transform to compute sky angular Fourier amplitudes $$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(u, v, \lambda_k)$$ \item Apply instrument response in the angular wave mode plane. We use here the normalized instrument response $ {\cal R}(u,v,\lambda_k) \lesssim 1$. $$ {\cal T}_{sky}(u, v, \lambda_k) \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda_k) $$ \item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map, without instrumental (electronic/$\Tsys$) white noise: $$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda) \rightarrow \mathrm{Inv-2D-FFT} \rightarrow T_{mes1}(\alpha, \delta, \lambda_k) $$ \item Add white noise (gaussian fluctuations) to the pixel map temperatures to obtain the measured sky brightness temperature $T_{mes}(\alpha, \delta, \nu_k)$. We have also considered that the system temperature and thus the additive white noise level was independent of the frequency or wavelength. \end{enumerate} The LSS signal extraction depends indeed on the white noise level. The results shown here correspond to the (a) instrument configuration, a packed array of $11 \times 11 = 121$ dishes (5 meter diameter), with a white noise level corresponding to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500$ kHz cell. A brief description of the simple component separation procedure that we have applied is given here: \begin{enumerate} \item The measured sky brightness temperature is first {\em corrected} for the frequency dependent beam effects through a convolution by a fiducial frequency independent beam. This {\em correction} corresponds to a smearing or degradation of the angular resolution. We assume that we have a perfect knowledge of the intrinsic instrument response, up to a threshold numerical level of about $ \gtrsim 1 \%$ for ${\cal R}(u,v,\lambda)$. We recall that this is the normalized instrument response, ${\cal R}(u,v,\lambda) \lesssim 1$. $$ T_{mes}(\alpha, \delta, \nu) \longrightarrow T_{mes}^{bcor}(\alpha,\delta,\nu) $$ The virtual target instrument has a beam width larger than the worst real instrument beam, i.e at the lowest observed frequency. \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$ is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in the $\lgd ( T ) , \lgd (\nu)$ plane and we show here the results for a pure power law (P1) or modified power law (P2): \begin{eqnarray*} P1 & : & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) \\ P2 & : & \lgd ( T_{mes}^{bcor}(\nu) ) = a + b \, \lgd ( \nu / \nu_0 ) + c \, \lgd ( \nu/\nu_0 ) ^2 \end{eqnarray*} where $b$ is the power law index and $T_0 = 10^a$ is the brightness temperature at the reference frequency $\nu_0$. {\changemark Interferometers have poor response at small $(\uv)$ corresponding to baselines smaller than interferometer element size. The $(0,0)$ mode, corresponding the mean temperature can not be measured with an interferometer. We have used a simple trick to make the power law fitting procedure to work: we have set the mean value of the temperature for each frequency plane to a power law with an index close to the synchrotron index and we have checked that results are not too sensitive to the arbitrarily fixed mean temperature power law parameters. } \item The difference between the beam-corrected sky temperature and the fitted power law $(T_0(\alpha, \delta), b(\alpha, \delta))$ is our extracted 21 cm LSS signal. \end{enumerate} Figure \ref{extlsspk} shows the performance of this procedure at a redshift $\sim 0.6$, for the two radio sky models used here: GSM/Model-I and Haslam+NVSS/Model-II. The 21 cm LSS power spectrum, as seen by a perfect instrument with a 25 arcmin (FWHM) gaussian frequency independent beam is shown in orange (solid line), and the extracted power spectrum, after beam {\em correction} and foreground separation with second order polynomial fit (P2) is shown in red (circle markers). We have also represented the obtained power spectrum without applying the beam correction (step 1 above), or with the first order polynomial fit (P1). Figure \ref{extlssmap} shows a comparison of the original 21 cm brightness temperature map at 884 MHz with the recovered 21 cm map, after subtraction of the radio continuum component. It can be seen that structures present in the original map have been correctly recovered, although the amplitude of the temperature fluctuations on the recovered map is significantly smaller (factor $\sim 5$) than in the original map. This is mostly due to the damping of the large scale ($k \lesssim 0.04 h \mathrm{Mpc^{-1}} $) due the poor interferometer response at large angle ($\theta \gtrsim 4^\circ $). We have shown that it should be possible to measure the red shifted 21 cm emission fluctuations in the presence of the strong radio continuum signal, provided that this latter has a smooth frequency dependence. However, a rather precise knowledge of the instrument beam and the beam {\em correction} or smearing procedure described here are key ingredient for recovering the 21 cm LSS power spectrum. It is also important to note that while it is enough to correct the beam to the lowest resolution instrument beam ($\sim 30'$ or $D \sim 50$ meter @ 820 MHz) for the GSM sky model, a stronger beam correction has to be applied (($\sim 36'$ or $D \sim 40$ meter @ 820 MHz) for the Model-II to reduce significantly the ripples from bright radio sources. We have also applied the same procedure to simulate observations and LSS signal extraction for an instrument with a frequency dependent gaussian beam shape. The mode mixing effect is greatly reduced for such a smooth beam, compared to the more complex instrument response ${\cal R}(u,v,\lambda)$ used for the results shown in figure \ref{extlsspk}. \begin{figure*} \centering \vspace*{-25mm} \mbox{ \hspace*{-20mm} \includegraphics[width=1.15\textwidth]{Figs/extlsspk.pdf} } \vspace*{-35mm} \caption{Recovered power spectrum of the 21cm LSS temperature fluctuations, separated from the continuum radio emissions at $z \sim 0.6$, for the instrument configuration (a), $11\times11$ packed array interferometer. Left: GSM/Model-I , right: Haslam+NVSS/Model-II. black curve shows the residual after foreground subtraction, corresponding to the 21 cm signal, WITHOUT applying the beam correction. Red curve shows the recovered 21 cm signal power spectrum, for P2 type fit of the frequency dependence of the radio continuum, and violet curve is the P1 fit (see text). The orange/yellow curve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency independent gaussian beam. } \label{extlsspk} \end{figure*} \begin{figure*} \centering \vspace*{-20mm} \mbox{ \hspace*{-25mm} \includegraphics[width=1.20\textwidth]{Figs/extlssmap.pdf} } \vspace*{-25mm} \caption{Comparison of the original 21 cm LSS temperature map @ 884 MHz ($z \sim 0.6$), smoothed with 25 arc.min (FWHM) beam (top), and the recovered LSS map, after foreground subtraction for Model-I (GSM) (bottom), for the instrument configuration (a), $11\times11$ packed array interferometer. Notice the difference between the temperature color scales (mK) for the top and bottom maps. } \label{extlssmap} \end{figure*} \subsection{$P_{21}(k)$ measurement transfer function} \label{tfpkdef} The recovered red shifted 21 cm emission power spectrum $P_{21}^{rec}(k)$ suffers a number of distortions, mostly damping, compared to the original $P_{21}(k)$ due to the instrument response and the component separation procedure. We expect damping at small scales, or larges $k$, due to the finite instrument size, but also at large scales, small $k$, if total power measurements (auto-correlations) are not used in the case of interferometers. The sky reconstruction and the component separation introduce additional filtering and distortions. Ideally, one has to define a power spectrum measurement response or {\it transfer function} in the radial direction, ($\lambda$ or redshift, $\TrF(k_\parallel)$) and in the transverse plane ( $\TrF(k_\perp)$ ). The real transverse plane transfer function might even be anisotropic. However, in the scope of the present study, we define an overall transfer function $\TrF(k)$ as the ratio of the recovered 3D power spectrum $P_{21}^{rec}(k)$ to the original $P_{21}(k)$: \begin{equation} \TrF(k) = P_{21}^{rec}(k) / P_{21}(k) \end{equation} Figure \ref{extlssratio} shows this overall transfer function for the simulations and component separation performed here, around $z \sim 0.6$, for the instrumental setup (a), a filled array of 121 $D_{dish}=5$ m dishes. The orange/yellow curve shows the ratio $P_{21}^{smoothed}(k)/P_{21}(k)$ of the computed to the original power spectrum, if the original LSS temperature cube is smoothed by the frequency independent target beam FWHM=30' for the GSM simulations (left), 36' for Model-II (right). This orange/yellow curve shows the damping effect due to the finite instrument size at small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}, \theta \lesssim 1^\circ$). The recovered power spectrum suffers also significant damping at large scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}, $ due to poor interferometer response at large angles ($ \theta \gtrsim 4^\circ-5^\circ$), as well as to the filtering of radial or longitudinal Fourier modes along the frequency or redshift direction ($k_\parallel$) by the component separation algorithm. The red curve shows the ratio of $P(k)$ computed on the recovered or extracted 21 cm LSS signal, to the original LSS temperature cube $P_{21}^{rec}(k)/P_{21}(k)$ and corresponds to the transfer function $\TrF(k)$ defined above, for $z=0.6$ and instrument setup (a). The black (thin line) curve shows the ratio of recovered to the smoothed power spectrum $P_{21}^{rec}(k)/P_{21}^{smoothed}(k)$. This latter ratio (black curve) exceeds one for $k \gtrsim 0.2$, which is due to the noise or system temperature. It should be stressed that the simulations presented in this section were focused on the study of the radio foreground effects and have been carried intently with a very low instrumental noise level of $0.25$ mK per pixel, corresponding to several years of continuous observations ($\sim 10$ hours per $3' \times 3'$ pixel). This transfer function is well represented by the analytical form: \begin{equation} \TrF(k) = \sqrt{ \frac{ k-k_A}{ k_B} } \times \exp \left( - \frac{k}{k_C} \right) \label{eq:tfanalytique} \end{equation} We have performed simulation of observations and radio foreground subtraction using the procedure described here for different redshifts and instrument configurations, in particular for the (e) configuration with 400 five-meter dishes. As the synchrotron and radio source strength increases quickly with decreasing frequency, we have seen that recovering the 21 cm LSS signal becomes difficult for larger redshifts, in particular for $z \gtrsim 2$. We have determined the transfer function parameters of eq. \ref{eq:tfanalytique} $k_A, k_B, k_C$ for setup (e) for three redshifts, $z=0.5, 1 , 1.5$, and then extrapolated the value of the parameters for redshift $z=2, 2.5$. The value of the parameters are grouped in table \ref{tab:paramtfk} and the smoothed transfer functions are shown on figure \ref{tfpkz0525}. \begin{table}[hbt] \begin{center} \begin{tabular}{|c|ccccc|} \hline \hspace{2mm} z \hspace{2mm} & \hspace{2mm} 0.5 \hspace{2mm} & \hspace{2mm} 1.0 \hspace{2mm} & \hspace{2mm} 1.5 \hspace{2mm} & \hspace{2mm} 2.0 \hspace{2mm} & \hspace{2mm} 2.5 \hspace{2mm} \\ \hline $k_A$ & 0.006 & 0.005 & 0.004 & 0.0035 & 0.003 \\ $k_B$ & 0.038 & 0.019 & 0.012 & 0.0093 & 0.008 \\ $k_C$ & 0.16 & 0.08 & 0.05 & 0.038 & 0.032 \\ \hline \end{tabular} \end{center} \caption{Value of the parameters for the transfer function (eq. \ref{eq:tfanalytique}) at different redshift for instrumental setup (e), $20\times20$ packed array interferometer. } \label{tab:paramtfk} \end{table} \begin{figure*} \centering \vspace*{-30mm} \mbox{ \hspace*{-20mm} \includegraphics[width=1.15\textwidth]{Figs/extlssratio.pdf} } \vspace*{-35mm} \caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $, at $z \sim 0.6$, for the instrument configuration (a), $11\times11$ packed array interferometer. Left: GSM/Model-I , right: Haslam+NVSS/Model-II. } \label{extlssratio} \end{figure*} \begin{figure} \centering \vspace*{-25mm} \mbox{ \hspace*{-10mm} \includegraphics[width=0.55\textwidth]{Figs/tfpkz0525.pdf} } \vspace*{-30mm} \caption{Fitted/smoothed transfer function $\TrF(k)$ obtained for the recovered 21 cm power spectrum at different redshifts, $z=0.5 , 1.0 , 1.5 , 2.0 , 2.5$ for the instrument configuration (e), $20\times20$ packed array interferometer. } \label{tfpkz0525} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% \section{ BAO scale determination and constrain on dark energy parameters} % {\color{red} \large \it CY ( + JR ) } \\[1mm] %% We compute reconstructed LSS-P(k) (after component separation) at different z's %% and determine BAO scale as a function of redshifts. %% Method: %% \begin{itemize} %% \item Compute/guess the overall transfer function for several redshifts (0.5 , 1.0 1.5 2.0 2.5 ) \\ %% \item Compute / guess the instrument noise level for the same redshit values %% \item Compute the observed P(k) and extract $k_{BAO}$ , and the corresponding error %% \item Compute the DETF ellipse with different priors %% \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% Figures et texte fournis par C. Yeche - 10 Juin 2011 %%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Sensitivity to cosmological parameters} \label{cosmosec} The impact of the various telescope configurations on the sensitivity for 21 cm power spectrum measurement has been discussed in section \ref{pkmessens}. Fig. \ref{figpnoisea2g} shows the noise power spectra, and allows us to rank visually the configurations in terms of instrument noise contribution to P(k) measurement. The differences in $P_{noise}$ will translate into differing precisions in the reconstruction of the BAO peak positions and in the estimation of cosmological parameters. In addition, we have seen (sec. \ref{recsec}) that subtraction of continuum radio emissions, Galactic synchrotron and radio sources, has also an effect on the measured 21 cm power spectrum. In this paragraph, we present our method and the results for the precisions on the estimation of Dark Energy parameters, through a radio survey of the redshifted 21 cm emission of LSS, with an instrumental setup similar to the (e) configuration (sec. \ref{instrumnoise}), 400 five-meter diameter dishes, arranged into a filled $20 \times 20$ array. \subsection{BAO peak precision} In order to estimate the precision with which BAO peak positions can be measured, we used a method similar to the one established in \citep{blake.03} and \citep{glazebrook.05}. To this end, we generated reconstructed power spectra $P^{rec}(k)$ for slices of Universe with a quarter-sky coverage and a redshift depth, $\Delta z=0.5$ for $0.25