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Changeset 4032 in Sophya for trunk/Cosmo
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- Nov 3, 2011, 6:50:50 PM (14 years ago)
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trunk/Cosmo/RadioBeam/sensfgnd21cm.tex (modified) (37 diffs)
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trunk/Cosmo/RadioBeam/sensfgnd21cm.tex
r4031 r4032 67 67 % Commande pour marquer les changements du papiers pour le referee 68 68 \def\changemark{\bf } 69 % \def\changemark{ } 69 70 70 71 %%% Definition pour la section sur les param DE par C.Y … … 140 141 instruments for the observation of large scale structures and BAO oscillations in 21 cm and we discuss the problem of foreground removal. } 141 142 % methods heading (mandatory) 142 { For each configuration, we determine instrument response by computing the (u,v)or Fourier angular frequency143 plane coverage using visibilities. The (u,v)plane response is the noise power spectrum,143 { 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, 144 145 hence the instrument sensitivity for LSS P(k) measurement. We describe also a simple foreground subtraction method to 145 146 separate LSS 21 cm signal from the foreground due to the galactic synchrotron and radio sources emission. } … … 287 288 Intensity mapping has been suggested as an alternative and economic method to map the 288 289 3D distribution of neutral hydrogen by \citep{chang.08} and further studied by \citep{ansari.08} \citep{seo.10}. 289 {\changemark There have even tentatives to detect the 21 cm LSS signal at GBT290 {\changemark There have been attempts to detect the 21 cm LSS signal at GBT 290 291 \citep{chang.10} and at GMRT \citep{ghosh.11}}. 291 292 In this approach, sky brightness map with angular resolution $\sim 10-30 \, \mathrm{arc.min}$ is made for a … … 348 349 \subsection{ \HI power spectrum and BAO} 349 350 In the absence of any foreground or background radiation 350 {\changemark and assuming high spin temperature $\kb T_{spin} \gg h \nu_{21}$},351 {\changemark and assuming high spin temperature, $\kb T_{spin} \gg h \nu_{21}$}, 351 352 the brightness temperature for a given direction and wavelength $\TTlam$ would be proportional to 352 353 the local \HI number density $\etaHI(\vec{\Theta},z)$ through the relation: … … 541 542 ${\cal R}_b(\uv,\lambda)$. 542 543 For an interferometer, we can compute a raw instrument response 543 ${\cal R}_{raw}(\uv,\lambda)$ which corresponds to $( u,v)$ plane coverage by all544 ${\cal R}_{raw}(\uv,\lambda)$ which corresponds to $(\uv)$ plane coverage by all 544 545 receiver pairs with uniform weighting. 545 546 Obviously, different weighting schemes can be used, changing … … 556 557 in particular in section \ref{recsec}. 557 558 558 {\changemark Detection of the reionisation at 21 cm bandhas been an active field559 {\changemark Detection of the reionisation at 21 cm has been an active field 559 560 in the last decade and different groups have built 560 561 instruments to detect reionisation signal around 100 MHz: LOFAR … … 636 637 P_{noise}(k_x,k_y, z) & = & P_{noise}(\uv) 637 638 \frac{ 8 \pi^3 \delta \uvu \times \delta \uvv }{\delta k_x \times \delta k_y \times \delta k_z} \\ 638 P_{noise}(k_x,k_y, z)& = & \left( 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2} \right)639 & = & \left( 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2} \right) 639 640 \, \frac{1}{{\cal R}_{raw}} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 640 641 \label{eq:pnoisekxkz} 641 642 \end{eqnarray} 642 643 643 It is worthwhile to not ice that the cosmological 3D noise power spectrum does not depend anymore on the644 individual measurement bandwidth.644 It is worthwhile to note that the ``cosmological'' 3D noise power spectrum does not depend 645 anymore on the individual measurement bandwidth. 645 646 In the following paragraph, we will first consider an ideal instrument 646 647 with uniform $(\uv)$ coverage … … 683 684 684 685 It is important to note that any real instrument do not have a flat 685 response in the $( u,v)$ plane, and the observations provide no information above686 response in the $(\uv)$ plane, and the observations provide no information above 686 687 a certain maximum angular frequency $u_{max},v_{max}$. 687 688 One has to take into account either a damping of the observed sky power … … 738 739 as a function of redshift (top), for a one year survey of a quarter of the sky. Bottom: 739 740 maximum $k$ value for 21 cm LSS power spectrum measurement by a 100 meter diameter 740 primary antenna (bottom)}741 primary antenna. } 741 742 \label{pnkmaxfz} 742 743 \end{figure} … … 758 759 This correspond to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given 759 760 the small angular extent, we have neglected the curvature of redshift shells. 760 \item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. $(k_x,k_y)$ is 761 converted to the $(\uv)$ angular frequency coordinates using the equation \ref{eq:uvkxky}, and the 761 \item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. 762 The coordinates $(k_x,k_y)$ are converted to the $(\uv)$ angular frequency coordinates 763 using equation (\ref{eq:uvkxky}), and the 762 764 angular diameter distance $\dang(z)$ for \LCDM model with standard WMAP parameters \citep{komatsu.11}. 763 765 The noise variance is taken proportional to $P_{noise}$ : … … 766 768 \end{equation} 767 769 \item an FFT is then performed in the frequency or redshift direction to obtain the noise Fourier 768 complex coefficients $ n(k_x,k_y,k_z)$ and the power spectrum is estimated as :769 \begin{equation} 770 \tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \hspace{2mm} \mathrm{for} \hspace{2mm}770 complex coefficients ${\cal F}_n(k_x,k_y,k_z)$ and the power spectrum is estimated as : 771 \begin{equation} 772 \tilde{P}_{noise}(k) = < | {\cal F}_n(k_x,k_y,k_z) |^2 > \hspace{2mm} \mathrm{for} \hspace{2mm} 771 773 \sqrt{k_x^2+k_y^2+k_z^2} = k 772 774 \end{equation} 773 775 Noise samples corresponding to small instrument response, typically less than 1\% of the 774 776 maximum instrument response are ignored when calculating $\tilde{P}_{noise}(k)$. 775 However, we require to have a significant fraction, typically 20\% to 50\% of thepossible modes777 However, we require to have a significant fraction, typically 20\% to 50\% of all possible modes 776 778 $(k_x,k_y,k_z)$ measured for a given $k$ value. 777 779 \item the above steps are repeated $\sim$ 50 times to decrease the statistical fluctuations … … 783 785 power spectrum shape, neglecting the redshift or frequency dependence of the 784 786 instrument response function and $\dang(z)$ for a small redshift interval around $z_0$, 785 using a fixed instrument response ${\cal R}( u,v,\lambda(z_0))$ and787 using a fixed instrument response ${\cal R}(\uv,\lambda(z_0))$ and 786 788 a constant the radial distance $\dang(z_0)*(1+z_0)$. 787 789 \begin{equation} 788 \tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \simeq < P_{noise}(u,v, k_z) >790 \tilde{P}_{noise}(k) = < | {\cal F}_n (k_x,k_y,k_z) |^2 > \simeq < P_{noise}(\uv, k_z) > 789 791 \end{equation} 790 The approximate power spectrum obtain through this simplified and much faster792 The approximate power spectrum obtained through this simplified and much faster 791 793 method is shown as dashed curves on figure \ref{figpnoisea2g} for few instrument 792 794 configurations. … … 850 852 However, we have introduced a filling factor or illumination efficiency 851 853 $\eta$, relating the effective dish diameter $D_{ill}$ to the 852 mechanical dish size $D ^{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales854 mechanical dish size $D_{ill} = \eta \, D_{dish}$. The effective area $A_e \propto \eta^2$ scales 853 855 as $\eta^2$ or $\eta_x \eta_y$. 854 856 \begin{eqnarray} 855 {\cal L}_\circ (\uv,\lambda) & = & \bigwedge_{[\pm D^{ill}/ \lambda]}(\sqrt{u^2+v^2}) \\857 {\cal L}_\circ (\uv,\lambda) & = & \bigwedge_{[\pm \eta D_{dish}/ \lambda]}(\sqrt{u^2+v^2}) \\ 856 858 L_\circ (\alpha,\beta,\lambda) & = & \left[ \frac{ \sin (\pi (D^{ill}/\lambda) \sin \theta ) }{\pi (D^{ill}/\lambda) \sin \theta} \right]^2 857 859 \hspace{4mm} \theta=\sqrt{\alpha^2+\beta^2} … … 861 863 862 864 For the receivers along the focal line of cylinders, we have assumed that the 863 individual receiver response in the $( u,v)$ plane corresponds to one from a865 individual receiver response in the $(\uv)$ plane corresponds to one from a 864 866 rectangular shaped antenna. The illumination efficiency factor has been taken 865 867 equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$ … … 879 881 is the case for a transit type telescope. 880 882 881 Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}( u,v,\lambda)$883 Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}(\uv,\lambda)$ 882 884 for the four configurations (a,b,c,d) with $\sim 100$ receivers per 883 885 polarisation. … … 899 901 \includegraphics[width=\textwidth]{Figs/uvcovabcd.pdf} 900 902 } 901 \caption{ $(\uv)$ plane coverage (raw instrument response ${\cal R}(\uv,\lambda)$902 for four configurations.903 \caption{Raw instrument response ${\cal R}(\uv,\lambda)$ or the $(\uv)$ plane coverage 904 at 710 MHz (redshift $z=1$) for four configurations. 903 905 (a) 121 $D_{dish}=5$ meter diameter dishes arranged in a compact, square array 904 906 of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each (fig. \ref{figconfbc}), … … 916 918 } 917 919 \vspace*{-20mm} 918 \caption{P(k) 21 cm LSS power spectrum at redshift $z=1$ and noise power spectrum for several interferometer 919 configurations ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers. The noise power spectrum has been 920 \caption{P(k) 21 cm LSS power spectrum at redshift $z=1$ with $\gHI=2\%$ 921 and the noise power spectrum for several interferometer configurations 922 ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers. The noise power spectrum has been 920 923 computed for all configurations assuming a survey of a quarter of the sky over one year, 921 924 with a system temperature $\Tsys = 50 \mathrm{K}$. } … … 941 944 frequency: this instrumental effect significantly increases the difficulty and complexity of this component separation 942 945 technique. The effect of frequency dependent beam shape is some time referred to as {\em 943 mode mixing}. {\changemark Effect of the frequency dependent beam shape on foreground subtraction946 mode mixing}. {\changemark The effect of the frequency dependent beam shape on foreground subtraction 944 947 has been discussed for example in \cite{morales.06}.} 945 948 … … 947 950 the simple models we have used for computing the sky radio emissions in the GHz frequency 948 951 range. We present also a simple component separation method to extract the LSS signal and 949 its performance. {\changemark The analysis presented here follow a similar path to952 its performance. {\changemark The analysis presented here follows a similar path to 950 953 a detailed foreground subtraction study carried for MWA at $\sim$ 150 MHz by \cite{bowman.09}. } 951 954 We compute in particular the effect of the instrument response on the recovered … … 1011 1014 deviation $\sigma_\beta = 0.15$. {\changemark The 1012 1015 diffuse radio background spectral index has been measured for example by 1013 \cite p{platania.98} or \cite{rogers.08}.}1016 \cite{platania.98} or \cite{rogers.08}.} 1014 1017 The synchrotron contribution to the sky temperature for each cell is then 1015 1018 obtained through the formula: … … 1104 1107 The GSM model lacks the angular resolution needed to compute 1105 1108 correctly the effect of bright compact sources for 21 cm LSS observations and 1106 the mode mixing due to frequency dependent instrument, while Model-II1107 provides a reasonable description of these compact sources. Our simulated1109 the mode mixing due to the frequency dependence of the instrumental response, 1110 while Model-II provides a reasonable description of these compact sources. Our simulated 1108 1111 sky cubes have an angular resolution $3'\times3'$, well below the typical 1109 1112 $15'$ resolution of the instrument configuration considered here. … … 1113 1116 in their power spectra shown in figure \ref{pkgsmlss}. The smoothing or convolution with 1114 1117 a 25' beam has negligible effect of the GSM power spectrum, as it originally lacks 1115 structures below 0.5 degree. We hope that by using1118 structures below 0.5 degree. By using 1116 1119 these two models, we have explored some of the systematic uncertainties 1117 1120 related to foreground subtraction.} … … 1133 1136 } 1134 1137 \vspace*{-10mm} 1135 \caption{Comparison of GSM (black) Model-II (red) sky cube temperature distribution.1138 \caption{Comparison of GSM (black) and Model-II (red) sky cube temperature distribution. 1136 1139 The Model-II (Haslam+NVSS), 1137 1140 has been smoothed with a 35 arcmin gaussian beam. } … … 1145 1148 \includegraphics[width=0.9\textwidth]{Figs/compmapgsm.pdf} 1146 1149 } 1147 \caption{Comparison of GSM map (top) and Model-II sky map at 884 MHz (bottom).1150 \caption{Comparison of GSM (top) and Model-II (bottom) sky maps at 884 MHz. 1148 1151 The Model-II (Haslam+NVSS) has been smoothed with a 35 arcmin (FWHM) gaussian beam.} 1149 1152 \label{compgsmmap} … … 1152 1155 \begin{figure} 1153 1156 \centering 1154 \vspace*{-25mm}1157 % \vspace*{-25mm} 1155 1158 \mbox{ 1156 \hspace*{- 15mm}1157 \includegraphics[width=0. 65\textwidth]{Figs/pk_gsm_lss.pdf}1159 \hspace*{-6mm} 1160 \includegraphics[width=0.52\textwidth]{Figs/pk_gsm_lss.pdf} 1158 1161 } 1159 \vspace*{-40mm} 1160 \caption{Comparison of the 21cm LSS power spectrum (red, orange) with the radio foreground power spectrum. 1162 \vspace*{-5mm} 1163 \caption{Comparison of the 21cm LSS power spectrum at $z=0.6$ with $\gHI=1\%$ (red, orange) 1164 with the radio foreground power spectrum. 1161 1165 The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple 1162 1166 model based on Haslam+NVSS (Model-II, black). The curves with circle markers show the power spectrum 1163 as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam.} 1167 as observed by a perfect instrument with a 25 arcmin (FWHM) gaussian beam. This beam has 1168 negligible effect on the GSM/Model-I power spectrum, as GSM has no structures below $\sim 0.5^\circ$. 1169 } 1164 1170 \label{pkgsmlss} 1165 1171 \end{figure} … … 1180 1186 \item Apply instrument response in the angular wave mode plane. We use here the normalized instrument response 1181 1187 $ {\cal R}(\uv,\lambda_k) \lesssim 1$. 1182 $$ {\cal T}_{sky}(\uv, \lambda_k) \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}( u,v,\lambda_k) $$1188 $$ {\cal T}_{sky}(\uv, \lambda_k) \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda_k) $$ 1183 1189 \item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map, 1184 1190 without instrumental (electronic/$\Tsys$) white noise: … … 1187 1193 \item Add white noise (gaussian fluctuations) to the pixel map temperatures to obtain 1188 1194 the measured sky brightness temperature $T_{mes}(\alpha, \delta, \nu_k)$. 1189 {\changemark The white noise hypothesis is reasonable at this level, as$(\uv)$1190 dependent instrument response has already been applied.}1195 {\changemark The white noise hypothesis is reasonable at this level, since $(\uv)$ 1196 dependent instrumental response has already been applied.} 1191 1197 We have also considered that the system temperature and thus the 1192 1198 additive white noise level was independent of the frequency or wavelength. … … 1211 1217 {\changemark 1212 1218 The virtual target beam ${\cal R}_f(\uv)$ has a lower resolution than the worst real instrument beam, 1213 i.e at the lowest observed frequency. We assume that the intrinsic instrument response is known up to a threshold 1214 numerical level of about $ \gtrsim 1 \%$ for ${\cal R}(u,v,\lambda)$. We recall that this is the normalized instrument response, 1215 ${\cal R}(\uv\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$ has also a numerical upper 1216 bound around $\sim$100. } 1219 i.e at the lowest observed frequency. 1220 No correction has been applied for modes with ${\cal R}(\uv,\lambda) \lesssim 1\%$, as a first 1221 attempt to represent imperfect knowledge of the instrument response. 1222 We recall that this is the normalized instrument response, 1223 ${\cal R}(\uv,\lambda) \lesssim 1$. The correction factor ${\cal R}_f(\uv) / {\cal R}(\uv,\lambda)$ has also a numerical upper bound $\sim 100$. } 1217 1224 \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$ 1218 1225 is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in … … 1252 1259 present in the original map have been correctly recovered, although the amplitude of the temperature 1253 1260 fluctuations on the recovered map is significantly smaller (factor $\sim 5$) than in the original map. 1254 {\changemark This is mostly due to the damping of the large scale ($k \lesssim 0.1 h \mathrm{Mpc^{-1}} $)1261 {\changemark This is mostly due to the damping of the large scale power ($k \lesssim 0.1 h \mathrm{Mpc^{-1}} $) 1255 1262 due to the foreground subtraction procedure (see figure \ref{extlssratio}).} 1256 1263 … … 1266 1273 with a frequency dependent gaussian beam shape. The mode mixing effect is greatly reduced for 1267 1274 such a smooth beam, compared to the more complex instrument response 1268 ${\cal R}( u,v,\lambda)$ used for the results shown in figure \ref{extlsspk}.1275 ${\cal R}(\uv,\lambda)$ used for the results shown in figure \ref{extlsspk}. 1269 1276 1270 1277 \begin{figure*} 1271 1278 \centering 1272 \vspace*{-25mm}1279 % \vspace*{-25mm} 1273 1280 \mbox{ 1274 \hspace*{-20mm}1275 \includegraphics[width= 1.15\textwidth]{Figs/extlsspk.pdf}1281 % \hspace*{-20mm} 1282 \includegraphics[width=\textwidth]{Figs/extlsspk.pdf} 1276 1283 } 1277 \vspace*{-35mm}1284 % \vspace*{-10mm} 1278 1285 \caption{Recovered power spectrum of the 21cm LSS temperature fluctuations, separated from the 1279 continuum radio emissions at $z \sim 0.6 $, for the instrument configuration (a), $11\times11$1286 continuum radio emissions at $z \sim 0.6, \gHI=1\%$, for the instrument configuration (a), $11\times11$ 1280 1287 packed array interferometer. 1281 Left: GSM/Model-I , right: Haslam+NVSS/Model-II. black curve shows the residual after foreground subtraction,1282 corresponding to the 21 cm signal, WITHOUT applying the beam correction. Red curve shows the recovered 21 cm1283 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 /yellowcurve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency independent gaussian beam. }1288 Left: GSM/Model-I , right: Haslam+NVSS/Model-II. The black curve shows the residual after foreground subtraction, 1289 corresponding to the 21 cm signal, WITHOUT applying the beam correction. The red curve shows the recovered 21 cm 1290 signal power spectrum, for P2 type fit of the frequency dependence of the radio continuum, and violet curve is the P1 fit (see text). The orange curve shows the original 21 cm signal power spectrum, smoothed with a perfect, frequency independent gaussian beam. } 1284 1291 \label{extlsspk} 1285 1292 \end{figure*} … … 1351 1358 becomes difficult for larger redshifts, in particular for $z \gtrsim 2$. 1352 1359 1353 We have determined the transfer function parameters of eq . \ref{eq:tfanalytique}$k_A, k_B, k_C$1360 We have determined the transfer function parameters of equation (\ref{eq:tfanalytique}) $k_A, k_B, k_C$ 1354 1361 for setup (e) for three redshifts, $z=0.5, 1 , 1.5$, and then extrapolated the value of the parameters 1355 1362 for redshift $z=2, 2.5$. The value of the parameters are grouped in table \ref{tab:paramtfk} … … 1376 1383 \begin{figure} 1377 1384 \centering 1378 \vspace*{-25mm}1385 % \vspace*{-25mm} 1379 1386 \mbox{ 1380 \hspace*{-10mm}1381 \includegraphics[width=0. 6\textwidth]{Figs/extlssratio.pdf}1387 % \hspace*{-10mm} 1388 \includegraphics[width=0.5\textwidth]{Figs/extlssratio.pdf} 1382 1389 } 1383 \vspace*{-30mm} 1384 \caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $ (transfer function), at $z \sim 0.6$. for the instrument configuration (a), $11\times11$ packed array interferometer. The effect of perfect gaussian beam of $\sim 30'$ is shown in black. 1390 % \vspace*{-30mm} 1391 \caption{Ratio of the reconstructed or extracted 21cm power spectrum, after foreground removal, to the initial 21 cm power spectrum, $\TrF(k) = P_{21}^{rec}(k) / P_{21}(k) $ (transfer function), at $z \sim 0.6$. for the instrument configuration (a), $11\times11$ packed array interferometer. The effect of a frequency independent 1392 gaussian beam of $\sim 30'$ is shown in black. 1385 1393 The transfer function $\TrF(k)$ for the instrument configuration (a), $11\times11$ packed array interferometer, 1386 1394 for the GSM/Model-I is shown in red, and in orange for Haslam+NVSS/Model-II. The transfer function … … 1392 1400 \begin{figure} 1393 1401 \centering 1394 \vspace*{-25mm}1402 % \vspace*{-25mm} 1395 1403 \mbox{ 1396 \hspace*{-10mm}1397 \includegraphics[width=0. 6\textwidth]{Figs/tfpkz0525.pdf}1404 % \hspace*{-10mm} 1405 \includegraphics[width=0.5\textwidth]{Figs/tfpkz0525.pdf} 1398 1406 } 1399 \vspace*{-30mm}1407 %\vspace*{-30mm} 1400 1408 \caption{Fitted/smoothed transfer function $\TrF(k)$ obtained for the recovered 21 cm power spectrum at different redshifts, 1401 1409 $z=0.5 , 1.0 , 1.5 , 2.0 , 2.5$ for the instrument configuration (e), $20\times20$ packed array interferometer. } … … 1718 1726 matrix with the Fisher matrix obtained for Planck mission, allows us to 1719 1727 compute the errors on dark energy parameters. 1720 The Planck Fisher matrix is 1721 obtained for the 8 parameters (assuming a flat universe):1728 {\changemark We have used the Planck Fisher matrix, computed for the 1729 Euclid proposal \citep{laureijs.09}, for the 8 parameters: 1722 1730 $\Omega_m$, $\Omega_b$, $h$, $w_0$, $w_a$, 1723 1731 $\sigma_8$, $n_s$ (spectral index of the primordial power spectrum) and 1724 $\tau$ (optical depth to the last-scatter surface) .1725 1732 $\tau$ (optical depth to the last-scatter surface), 1733 assuming a flat universe. } 1726 1734 1727 1735 For an optimized project over a redshift range, $0.25<z<2.75$, with a total … … 1730 1738 The Figure of Merit, the inverse of the area in the 95\% confidence level 1731 1739 contours is 38. 1732 1740 Finally, Fig.~\ref{fig:Compw0wa} 1733 1741 shows a comparison of different BAO projects, with a set of priors on 1734 1742 $(\Omega_m, \Omega_b, h)$ corresponding to the expected precision on 1735 these parameters in early 2010's. This BAO project based on \HI intensity 1743 these parameters in early 2010's. {\changemark The confidence contour 1744 level in the plane $(w_0,w_a)$ have been obtained by marginalizing 1745 over all the other parameters.} This BAO project based on \HI intensity 1736 1746 mapping is clearly competitive with the current generation of optical 1737 1747 surveys such as SDSS-III \citep{sdss3}. … … 1742 1752 \includegraphics[width=0.55\textwidth]{Figs/Ellipse21cm.pdf} 1743 1753 \caption{$1\sigma$ and $2\sigma$ confidence level contours in the 1744 parameter plane $(w_0,w_a)$ for two BAO projects: SDSS-III (LRG) project 1754 parameter plane $(w_0,w_a)$, marginalized over all the other parameters, 1755 for two BAO projects: SDSS-III (LRG) project 1745 1756 (blue dotted line), 21 cm project with HI intensity mapping (black solid line).} 1746 1757 \label{fig:Compw0wa} … … 1870 1881 {\it LSST Science book}, LSST Science Collaborations, 2009, arXiv:0912.0201 1871 1882 1883 % Planck Fischer matrix, computed for EUCLID 1884 \bibitem[Laureijs (2009)]{laureijs.09} Laureijs, R. 2009, ArXiv:0912.0914 1885 1872 1886 % Temperature du 21 cm 1873 1887 \bibitem[Madau et al. (1997)]{madau.97} Madau, P., Meiksin, A. and Rees, M.J., 1997, \apj 475, 429
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