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Changeset 4031 in Sophya
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- Oct 27, 2011, 7:46:04 PM (13 years ago)
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trunk/Cosmo/RadioBeam/sensfgnd21cm.tex (modified) (24 diffs)
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trunk/Cosmo/RadioBeam/sensfgnd21cm.tex
r4030 r4031 274 274 is the source velocity dispersion. 275 275 {\changemark The 1 MHz bandwidth mentioned above is only used for computing the 276 galaxy detection thresholds and does not determine the total surveybandwidth or frequency resolution276 galaxy detection thresholds and does not determine the total bandwidth or frequency resolution 277 277 of an intensity mapping survey.} 278 278 % {\color{red} Faut-il developper le calcul en annexe ? } … … 287 287 Intensity mapping has been suggested as an alternative and economic method to map the 288 288 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 GBT 290 \citep{chang.10} and at GMRT \citep{ghosh.11}}. 289 291 In this approach, sky brightness map with angular resolution $\sim 10-30 \, \mathrm{arc.min}$ is made for a 290 292 wide range of frequencies. Each 3D pixel (2 angles $\vec{\Theta}$, frequency $\nu$ or wavelength $\lambda$) … … 409 411 galaxy surveys, for example by SDSS and 2dF at low redshift $z \lesssim 0.3$ 410 412 (\cite{cole.05}, \cite{tegmark.04}). The 21 cm brightness power spectra $P_{T_{21}}(k)$ 411 shown here are comparable to one from the galaxy surveys, once the mean 21 cm412 temperature conversion factor $\left( \bar{T}_{21}(z) \right)^2$ and redshift evolution 413 have been accounted for. }413 shown here are comparable to the power spectrum measured from the galaxy surveys, 414 once the mean 21 cm temperature conversion factor $\left( \bar{T}_{21}(z) \right)^2$, 415 redshift evolution and different bias factors have been accounted for. } 414 416 % It should be noted that the maximum transverse $k^{comov} $ sensitivity range 415 417 % for an instrument corresponds approximately to half of its angular resolution. … … 555 557 556 558 {\changemark Detection of the reionisation at 21 cm band has been an active field 557 in the last decade and several groups558 (\cite{rottgering.06}, \cite{bowman.07}, \cite{lonsdale.09}, \cite{parsons.09}) have built 559 instruments to detect reionisation signal around 100 MHz.559 in the last decade and different groups have built 560 instruments to detect reionisation signal around 100 MHz: LOFAR 561 \citep{rottgering.06}, MWA (\cite{bowman.07}, \cite{lonsdale.09}) and PAPER \citep{parsons.09} . 560 562 Several authors have studied the instrumental noise 561 563 and statistical uncertainties when measuring the reionisation signal power spectrum; … … 733 735 } 734 736 \vspace*{-40mm} 735 \caption{Minimal noise level for a 100 beams instrument with \mbox{$\Tsys=50 \mathrm{K}$} 736 as a function of redshift (top). Maximum $k$ value for a 100 meter diameter primary antenna (bottom) } 737 \caption{Top: minimal noise level for a 100 beams instrument with \mbox{$\Tsys=50 \mathrm{K}$} 738 as a function of redshift (top), for a one year survey of a quarter of the sky. Bottom: 739 maximum $k$ value for 21 cm LSS power spectrum measurement by a 100 meter diameter 740 primary antenna (bottom) } 737 741 \label{pnkmaxfz} 738 742 \end{figure} … … 755 759 the small angular extent, we have neglected the curvature of redshift shells. 756 760 \item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. $(k_x,k_y)$ is 757 converted to the $ \uv$coordinates using the equation \ref{eq:uvkxky}, and the758 angular diameter distance $\dang(z)$ for \LCDM model with WMAP parameters.761 converted to the $(\uv)$ angular frequency coordinates using the equation \ref{eq:uvkxky}, and the 762 angular diameter distance $\dang(z)$ for \LCDM model with standard WMAP parameters \citep{komatsu.11}. 759 763 The noise variance is taken proportional to $P_{noise}$ : 760 764 \begin{equation} … … 771 775 However, we require to have a significant fraction, typically 20\% to 50\% of the possible modes 772 776 $(k_x,k_y,k_z)$ measured for a given $k$ value. 773 \item the above steps are repeated a number of timeto decrease the statistical fluctuations774 due to therandom generations. The averaged computed noise power spectrum is normalized using777 \item the above steps are repeated $\sim$ 50 times to decrease the statistical fluctuations 778 from random generations. The averaged computed noise power spectrum is normalized using 775 779 equation \ref{eq:pnoisekxkz} and the instrument and survey parameters ($\Tsys \ldots$). 776 780 \end{itemize} 777 781 778 It should be noted that it is possible to obtain a good approximation of noise782 It should be noted that it is possible to obtain a good approximation of the noise 779 783 power spectrum shape, neglecting the redshift or frequency dependence of the 780 784 instrument response function and $\dang(z)$ for a small redshift interval around $z_0$, … … 782 786 a constant the radial distance $\dang(z_0)*(1+z_0)$. 783 787 \begin{equation} 784 \tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \simeq < P_{noise}(u,v ) , k_z>788 \tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \simeq < P_{noise}(u,v, k_z) > 785 789 \end{equation} 786 790 The approximate power spectrum obtain through this simplified and much faster … … 861 865 equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$ 862 866 along the cylinder length. {\changemark We have used double triangular shaped 863 response function in the $(\uv)$ for each of the receiver elements along the cylinder:867 response function in the $(\uv)$ plane for each of the receiver elements along the cylinder: 864 868 \begin{equation} 865 869 {\cal L}_\Box(\uv,\lambda) = … … 880 884 881 885 {\changemark Using the numerical method sketched in section \ref{pnoisemeth}, we have 882 computed the 3D noise power spectrum for each of the eight instrument configurations discussed886 computed the 3D noise power spectrum for each of the eight instrument configurations presented 883 887 here, with a system noise temperature $\Tsys = 50 \mathrm{K}$, for a one year survey 884 888 of a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$ at a mean redshift $z_0=1, \nu_0=710 \mathrm{MHz}$.} … … 912 916 } 913 917 \vspace*{-20mm} 914 \caption{P(k) LSS power and noise power spectrum for several interferometer 915 configurations ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers.} 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 computed for all configurations assuming a survey of a quarter of the sky over one year, 921 with a system temperature $\Tsys = 50 \mathrm{K}$. } 916 922 \label{figpnoisea2g} 917 923 \end{figure*} … … 929 935 emissions can be used to separate the faint LSS signal from the Galactic and radio source 930 936 emissions. {\changemark Discussion of contribution of different sources 931 to foregrounds for measurement of reionization through redshifted 21 cm,937 of radio foregrounds for measurement of reionization through redshifted 21 cm, 932 938 as well foreground subtraction using their smooth frequency dependence can 933 be found in (\cite{shaver.99}, \cite{matteo.02},\cite{oh.03}) 939 be found in (\cite{shaver.99}, \cite{matteo.02},\cite{oh.03}).} 934 940 However, any real radio instrument has a beam shape which changes with 935 941 frequency: this instrumental effect significantly increases the difficulty and complexity of this component separation 936 942 technique. The effect of frequency dependent beam shape is some time referred to as {\em 937 mode mixing}. {\changemark Effect of frequency dependent beam shape for foreground subtraction and938 its application to MWA has been discussed in \citep{morales.06} \citep{bowman.09}.} 943 mode mixing}. {\changemark Effect of the frequency dependent beam shape on foreground subtraction 944 has been discussed for example in \cite{morales.06}.} 939 945 940 946 In this section, we present a short description of the foreground emissions and 941 947 the simple models we have used for computing the sky radio emissions in the GHz frequency 942 948 range. We present also a simple component separation method to extract the LSS signal and 943 its performance. We show in particular the effect of the instrument response on the recovered 949 its performance. {\changemark The analysis presented here follow a similar path to 950 a detailed foreground subtraction study carried for MWA at $\sim$ 150 MHz by \cite{bowman.09}. } 951 We compute in particular the effect of the instrument response on the recovered 944 952 power spectrum. The results presented in this section concern the 945 953 total sky emission and the LSS 21 cm signal extraction in the $z \sim 0.6$ redshift range, … … 1003 1011 deviation $\sigma_\beta = 0.15$. {\changemark The 1004 1012 diffuse radio background spectral index has been measured for example by 1005 \citep{platania.98} or \cite{rogers.08} 1013 \citep{platania.98} or \cite{rogers.08}.} 1006 1014 The synchrotron contribution to the sky temperature for each cell is then 1007 1015 obtained through the formula: … … 1103 1111 compared to Model-I/GSM, and the frequency variations as a simple power law 1104 1112 might not be realistic enough. The differences for the two sky models can be seen 1105 in their power spectra shown in figure \ref{pkgsmlss}. We hope that by using 1113 in their power spectra shown in figure \ref{pkgsmlss}. The smoothing or convolution with 1114 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 using 1106 1116 these two models, we have explored some of the systematic uncertainties 1107 1117 related to foreground subtraction.} … … 1161 1171 The {\it observed} data cube is obtained from the sky brightness temperature 3D map 1162 1172 $T_{sky}(\alpha, \delta, \nu)$ by applying the frequency or wavelength dependent instrument response 1163 ${\cal R}( u,v,\lambda)$.1173 ${\cal R}(\uv,\lambda)$. 1164 1174 We have considered the simple case where the instrument response is constant throughout the survey area, or independent 1165 1175 of the sky direction. … … 1167 1177 \begin{enumerate} 1168 1178 \item Apply a 2D Fourier transform to compute sky angular Fourier amplitudes 1169 $$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}( u,v, \lambda_k)$$1179 $$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(\uv, \lambda_k)$$ 1170 1180 \item Apply instrument response in the angular wave mode plane. We use here the normalized instrument response 1171 $ {\cal R}( u,v,\lambda_k) \lesssim 1$.1172 $$ {\cal T}_{sky}( u,v, \lambda_k) \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda_k) $$1181 $ {\cal R}(\uv,\lambda_k) \lesssim 1$. 1182 $$ {\cal T}_{sky}(\uv, \lambda_k) \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda_k) $$ 1173 1183 \item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map, 1174 1184 without instrumental (electronic/$\Tsys$) white noise: 1175 $$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}( u,v,\lambda)1185 $$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(\uv,\lambda) 1176 1186 \rightarrow \mathrm{Inv-2D-FFT} \rightarrow T_{mes1}(\alpha, \delta, \lambda_k) $$ 1177 1187 \item Add white noise (gaussian fluctuations) to the pixel map temperatures to obtain 1178 1188 the measured sky brightness temperature $T_{mes}(\alpha, \delta, \nu_k)$. 1189 {\changemark The white noise hypothesis is reasonable at this level, as $(\uv)$ 1190 dependent instrument response has already been applied.} 1179 1191 We have also considered that the system temperature and thus the 1180 1192 additive white noise level was independent of the frequency or wavelength. … … 1189 1201 \begin{enumerate} 1190 1202 \item The measured sky brightness temperature is first {\em corrected} for the frequency dependent 1191 beam effects through a convolution by a fiducial frequency independent beam. This {\em correction} 1192 corresponds to a smearing or degradation of the angular resolution. We assume 1193 that we have a perfect knowledge of the intrinsic instrument response, up to a threshold numerical level 1194 of about $ \gtrsim 1 \%$ for ${\cal R}(u,v,\lambda)$. We recall that this is the normalized instrument response, 1195 ${\cal R}(u,v,\lambda) \lesssim 1$. 1196 $$ T_{mes}(\alpha, \delta, \nu) \longrightarrow T_{mes}^{bcor}(\alpha,\delta,\nu) $$ 1197 The virtual target instrument has a beam width larger than the worst real instrument beam, 1198 i.e at the lowest observed frequency. 1203 beam effects through a convolution by a fiducial frequency independent beam ${\cal R}_f(\uv)$ This {\em correction} 1204 corresponds to a smearing or degradation of the angular resolution. 1205 \begin{eqnarray*} 1206 {\cal T}_{mes}(u, v, \lambda_k) & \rightarrow & {\cal T}_{mes}^{bcor}(u, v, \lambda_k) \\ 1207 {\cal T}_{mes}^{bcor}(u, v, \lambda_k) & = & 1208 {\cal T}_{mes}(u, v, \lambda_k) \times \sqrt{ \frac{{\cal R}_f(\uv)}{{\cal R}(\uv,\lambda)} } \\ 1209 {\cal T}_{mes}^{bcor}(u, v, \lambda_k) & \rightarrow & \mathrm{2D-FFT} \rightarrow T_{mes}^{bcor}(\alpha,\delta,\lambda) 1210 \end{eqnarray*} 1211 {\changemark 1212 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. } 1199 1217 \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$ 1200 1218 is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in … … 1209 1227 1210 1228 {\changemark Interferometers have poor response at small $(\uv)$ corresponding to baselines 1211 smaller than interferometer element size. The $(0,0)$ mode, corresponding the mean temperature 1212 can not be measured with an interferometer. We have used a simple trick to make the power law 1213 fitting procedure to work: we have set the mean value of the temperature for 1214 each frequency plane to a power law with an index close to the synchrotron index 1215 and we have checked that results are not too sensitive to the arbitrarily fixed mean temperature 1216 power law parameters. } 1229 smaller than interferometer element size. The zero spacing baseline, the $(\uv)=(0,0)$ mode, represents 1230 the mean temperature for a given frequency plane and can not be measured with interferometers. 1231 We have used a simple trick to make the power law fitting procedure applicable: 1232 we have set the mean value of the temperature for 1233 each frequency plane according to a power law with an index close to the synchrotron index 1234 ($\beta\sim-2.8$) and we have checked that results are not too sensitive to the 1235 arbitrarily fixed mean temperature power law parameters. } 1217 1236 1218 1237 \item The difference between the beam-corrected sky temperature and the fitted power law … … 1232 1251 with the recovered 21 cm map, after subtraction of the radio continuum component. It can be seen that structures 1233 1252 present in the original map have been correctly recovered, although the amplitude of the temperature 1234 fluctuations on the recovered map is significantly smaller (factor $\sim 5$) than in the original map. This is mostly1235 due to the damping of the large scale ($k \lesssim 0.04 h \mathrm{Mpc^{-1}} $) due the poor interferometer1236 response at large angle ($\theta \gtrsim 4^\circ $). 1253 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}} $) 1255 due to the foreground subtraction procedure (see figure \ref{extlssratio}).} 1237 1256 1238 1257 We have shown that it should be possible to measure the red shifted 21 cm emission fluctuations in the … … 1299 1318 1300 1319 Figure \ref{extlssratio} shows this overall transfer function for the simulations and component 1301 separation performed here, around $z \sim 0.6$, for the instrumental setup (a), a filled array of 121 $D_{dish}=5$ m dishes. 1302 The orange/yellow curve shows the ratio $P_{21}^{smoothed}(k)/P_{21}(k)$ of the computed to the original 1303 power spectrum, if the original LSS temperature cube is smoothed by the frequency independent target beam 1304 FWHM=30' for the GSM simulations (left), 36' for Model-II (right). This orange/yellow 1305 curve shows the damping effect due to the finite instrument size at small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}, \theta \lesssim 1^\circ$). 1306 The recovered power spectrum suffers also significant damping at large scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}, $ due to poor interferometer 1307 response at large angles ($ \theta \gtrsim 4^\circ-5^\circ$), as well as to the filtering of radial or longitudinal Fourier modes along 1308 the frequency or redshift direction ($k_\parallel$) by the component separation algorithm. 1309 The red curve shows the ratio of $P(k)$ computed on the recovered or extracted 21 cm LSS signal, to the original 1310 LSS temperature cube $P_{21}^{rec}(k)/P_{21}(k)$ and corresponds to the transfer function $\TrF(k)$ defined above, 1311 for $z=0.6$ and instrument setup (a). 1312 The black (thin line) curve shows the ratio of recovered to the smoothed 1313 power spectrum $P_{21}^{rec}(k)/P_{21}^{smoothed}(k)$. This latter ratio (black curve) exceeds one for $k \gtrsim 0.2$, which is 1314 due to the noise or system temperature. It should be stressed that the simulations presented in this section were 1320 separation performed here, around $z \sim 0.6$, for the instrumental setup (a), a filled array of 121 $D_{dish}=5$ m dishes. {\changemark This transfer function has been obtained after averaging the reconstructed 1321 $ P_{21}^{rec}(k)$ for several realizations (50) of the LSS temperature field. 1322 The black curve shows the ratio $\TrF(k)=P_{21}^{beam}(k)/P_{21}(k)$ of the computed to the original 1323 power spectrum, if the original LSS temperature cube is smoothed by the frequency independent 1324 target beam FWHM=30'. This black curve shows the damping effect due to the finite instrument size at 1325 small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}, \theta \lesssim 1^\circ$). 1326 The red curve shows the transfer function for the GSM foreground model (Model-I) and the $11\times11$ filled array 1327 interferometer (setup (a)), while the dashed red curve represents the transfer function for a D=55 meter 1328 diameter dish. The transfer function for the Model-II/Haslam+NVSS and the setup (a) filled interferometer 1329 array is also shown (orange curve). The recovered power spectrum suffers also significant damping at large 1330 scales $k \lesssim 0.05 \, h \, \mathrm{Mpc^{-1}}, $, mostly due to the filtering of radial or 1331 longitudinal Fourier modes along the frequency or redshift direction ($k_\parallel$) 1332 by the component separation algorithm. We have been able to remove the ripples on the reconstructed 1333 power spectrum due to bright sources in the Model-II by applying a stronger beam correction, $\sim$37' 1334 target beam resolution, compared to $\sim$30' for the GSM model. This explains the lower transfer function 1335 obtained for Model-II at small scales ($k \gtrsim 0.1 \, h \, \mathrm{Mpc^{-1}}$). } 1336 1337 It should be stressed that the simulations presented in this section were 1315 1338 focused on the study of the radio foreground effects and have been carried intently with a very low instrumental noise level of 1316 1339 $0.25$ mK per pixel, corresponding to several years of continuous observations ($\sim 10$ hours per $3' \times 3'$ pixel). … … 1351 1374 \end{table} 1352 1375 1353 \begin{figure*}1354 \centering1355 \vspace*{-30mm}1356 \mbox{1357 \hspace*{-20mm}1358 \includegraphics[width=1.15\textwidth]{Figs/extlssratio.pdf}1359 }1360 \vspace*{-35mm}1361 \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.1362 Left: GSM/Model-I , right: Haslam+NVSS/Model-II. }1363 \label{extlssratio}1364 \end{figure*}1365 1366 1367 1376 \begin{figure} 1368 1377 \centering … … 1370 1379 \mbox{ 1371 1380 \hspace*{-10mm} 1372 \includegraphics[width=0.55\textwidth]{Figs/tfpkz0525.pdf} 1381 \includegraphics[width=0.6\textwidth]{Figs/extlssratio.pdf} 1382 } 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. 1385 The transfer function $\TrF(k)$ for the instrument configuration (a), $11\times11$ packed array interferometer, 1386 for the GSM/Model-I is shown in red, and in orange for Haslam+NVSS/Model-II. The transfer function 1387 for a D=55 meter diameter dish for the GSM model is also shown as the dashed red curve. } 1388 \label{extlssratio} 1389 \end{figure} 1390 1391 1392 \begin{figure} 1393 \centering 1394 \vspace*{-25mm} 1395 \mbox{ 1396 \hspace*{-10mm} 1397 \includegraphics[width=0.6\textwidth]{Figs/tfpkz0525.pdf} 1373 1398 } 1374 1399 \vspace*{-30mm} … … 1785 1810 1786 1811 % Intensity mapping/HSHS 1787 \bibitem[Chang et al. (2008)]{chang.08} Chang, T., Pen, U.-L., Peterson, J.B. \& McDonald, P. 2008, \prl, 100, 091303 1812 \bibitem[Chang et al. (2008)]{chang.08} Chang, T., Pen, U.-L., Peterson, J.B. \& McDonald, P., 2008, \prl, 100, 091303 1813 1814 % Mesure 21 cm avec le GBT (papier Nature ) 1815 \bibitem[Chang et al. (2010)]{chang.10} Chang T-C, Pen U-L, Bandura K., Peterson J.B., 2010, \nat, 466, 463-465 1788 1816 1789 1817 % 2dFRS BAO observation … … 1808 1836 % 21 cm emission for mapping matter distribution 1809 1837 \bibitem[Furlanetto et al. (2006)]{furlanetto.06} Furlanetto, S., Peng Oh, S. \& Briggs, F. 2006, \physrep, 433, 181-301 1838 1839 % Mesure 21 cm a 610 MHz par GMRT 1840 \bibitem[Ghosh et al. (2011)]{ghosh.11} Ghosh A., Bharadwaj S., Ali Sk. S., Chengalur J. N., 2011, \mnras, 411, 2426-2438 1841 1810 1842 1811 1843 % Haslam 400 MHz synchrotron map
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