Changeset 3977 in Sophya


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Timestamp:
May 2, 2011, 4:19:09 PM (14 years ago)
Author:
ansari
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Papier sensibilite, presque complet jusqu'a la fin section 4 (radio-sources), Reza 02/05/2011

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  • trunk/Cosmo/RadioBeam/sensfgnd21cm.tex

    r3976 r3977  
    100100   }
    101101
    102    \date{Received December 15, 2010; accepted xxxx, 2011}
     102   \date{Received June 15, 2011; accepted xxxx, 2011}
    103103
    104104% \abstract{}{}{}{}{}
    … …  
    111111cm emission. Such a 3D matter distribution map can be used to test the Cosmological model and to constrain the Dark Energy
    112112properties or its equation of state. A novel approach, called intensity mapping can be used to map the \HI distribution,
    113 using radio interferometers with large instanteneous field of view and waveband}
     113using radio interferometers with large instanteneous field of view and waveband.}
    114114 % aims heading (mandatory)
    115   { In this paper, we study the sensitivity of different radio interferometer configuration for the observation of large scale structures
    116   and BAO oscillations in 21 cm and we discuss the problem of foreground removal. }
     115  { In this paper, we study the sensitivity of different radio interferometer configurations, or multi-beam
     116instruments for the observation of large scale structures and BAO oscillations in 21 cm and we discuss the problem of foreground removal. }
    117117  % methods heading (mandatory)
    118118 { For each configuration, we determine instrument response by computing the (u,v) plane (Fourier angular frequency plane)
    … …  
    124124   LSS power spectrum, after separation of 21cm-LSS signal from the foregrounds. }
    125125  % conclusions heading (optional), leave it empty if necessary
    126    { We show that an interferometer with few hundred elements and a surface coverage of
     126   { We show that a radio instrument with few hundred simultaneous beamns and a surface coverage of
    127127  $\lesssim 10000 \mathrm{m^2}$ will be able to  detect BAO signal at redshift z $\sim 1$ }
    128128
    … …  
    422422We have set the electromagnetic (EM) phase origin at the center of the coordinate frame and
    423423the EM wave vector is related to the wavelength $\lambda$ through the usual
    424 $ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta})$
     424$ | \vec{k}_{EM} |  =  2 \pi / \lambda $. The receiver beam or antenna lobe $L(\vec{\Theta},\lambda)$
    425425corresponds to the receiver intensity response:
    426426\begin{equation}
    427 L(\vec{\Theta}) = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda)
     427L(\vec{\Theta}), \lambda = B(\vec{\Theta},\lambda)  \,  B^*(\vec{\Theta},\lambda)
    428428\end{equation}
    429429The visibility signal between two receivers corresponds to the time averaged correlation between
    … …  
    441441the common receivers axis, the visibilty would be written as the 2D Fourier transform
    442442of the product of the sky intensity and the receiver beam, for the angular frequency
    443 $2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$:
     443\mbox{$(u,v)_{12} = 2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta x}{\lambda} )$}:
    444444\begin{equation}
    445445\vis(\lambda) \simeq  \iint d\alpha d\beta \, \, I(\alpha, \beta) \,  L(\alpha, \beta)
    … …  
    478478Several beam can be formed using different combination of the correlation from different
    479479antenna pairs. 
     480
     481An instrument can thus be characterized by its $(u,v)$ plane coverage or response
     482${\cal R}(u,v,\lambda)$. For a single dish with a single receiver in the focal plane,
     483the instrument response is simply the Fourier transform of the beam.
     484For a single dish with multiple receivers, either as a Focal Plane Array (FPA) or
     485a multi horn system, each beam (b) will have its own response
     486${\cal R}_b(u,v,\lambda)$.
     487For an interferometer, we can compute a raw instrument response
     488${\cal R}_{raw}(u,v,\lambda)$ which corresponds to $(u,v)$ plane coverage by all
     489receiver pairs  with uniform weighting.
     490Obviously, different weighting schemes can be used, changing
     491the effective beam shape and thus the response ${\cal R}_{w}(u,v,\lambda)$
     492and the noise behaviour.
    480493
    481494\begin{figure}
    … …  
    587600
    588601\begin{figure}
    589 \vspace*{-10mm}
     602\vspace*{-25mm}
    590603\centering
    591604\mbox{
    592605\hspace*{-10mm}
    593 \includegraphics[width=0.6\textwidth]{Figs/pnkmaxfz.pdf}
     606\includegraphics[width=0.65\textwidth]{Figs/pnkmaxfz.pdf}
    594607}
    595 \vspace*{-35mm}
     608\vspace*{-40mm}
    596609\caption{Minimal noise level for a 100 beam instrument as a function of redshift (top).
    597610 Maximum $k$ value for  a 100 meter diameter primary antenna (bottom) }
    … …  
    602615\subsection{Instrument configurations and noise power spectrum}
    603616
    604 We have numerically computed the instrument response in the (u,v) plane for several
    605 instrument configurations, at redshift $z=1$.
     617We have numerically computed the instrument response ${\cal R}(u,v,\lambda)$
     618with uniform weights in the $(u,v)$ plane for several instrument configurations:
    606619\begin{itemize}
    607620\item[{\bf a} :] A packed array of $n=121 \, D_{dish}=5 \mathrm{m}$ dishes, arranged in
    … …  
    678691\bigwedge_{[\pm 2 \pi D^{ill}_y / \lambda ]} (v )
    679692\end{equation}
    680 Figure \ref{figuvcovabcd} for the four configurations with $\sim 100$ receivers per
     693Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}(u,v,\lambda)$
     694for the four configurations (a,b,c,d) with $\sim 100$ receivers per
    681695polarisation. The resulting projected noise spectral power density is shown in figure
    682696\ref{figpnoisea2g}. The increase of $P_{noise}(k)$ at low $k^{comov} \lesssim 0.02$
    683697is due to the fact that we have ignored all auto-correlation measurements. 
    684 It can be seen that an instrument with $100$ beams and $\Tsys = 50 \mathrm{K}$
     698It can be seen that an instrument with $100-200$ beams and $\Tsys = 50 \mathrm{K}$
    685699should have enough sensitivity to map LSS in 21 cm at redshift z=1.
    686700
    … …  
    716730Reaching the required sensitivities is not the only difficulty of observing the large
    717731scale structures in 21 cm. Indeed, the synchrotron emission of the
    718 Milky Way and the extra galactic radio sources is a thousand time brighter than the
     732Milky Way and the extra galactic radio sources are a thousand time brighter than the
    719733emission of the neutral hydrogen distributed in the universe. Extracting the LSS signal
    720734using Intensity Mapping, without identifying the \HI point sources is the main challenge
    … …  
    723737emissions can be used to separate the faint LSS signal from the Galactic and radio source
    724738emissions. However, any real radio instrument has a beam shape which changes with
    725 frequency which significantly increases the difficulty and complexity of this component separation
     739frequency: this instrumental effect significantly increases the difficulty and complexity of this component separation
    726740technique. The effect of frequency dependent beam shape is often referred to as {\em
    727741mode mixing} \citep{morales.09}.
    … …  
    730744the simple models we have used for computing the sky radio emissions in the GHz frequency
    731745range. We present also a simple component separation method to extract the LSS signal and
    732 its performance. We show in particular the effect of the instrument response and possible
    733 way of getting around this difficulty. The results presented in this section concern the
     746its performance. We show in particular the effect of the instrument response on the recovered
     747power spectrum, and possible way of getting around this difficulty. The results presented in this section concern the
    734748total sky emission and the LSS 21 cm signal extraction in the $z \sim 0.6$ redshift range,
    735749corresponding to the central frequency $\nu \sim 884$ MHz. 
    736750 
    737751\subsection{ Synchrotron and radio sources }
    738 We have modeled the radio in the form of three dimensional maps (data cubes) of sky temperature
     752We have modeled the radio sky in the form of three dimensional maps (data cubes) of sky temperature
    739753brightness $T(\alpha, \delta, \nu)$ as a function of two equatorial angular coordinates $(\alpha, \delta)$
    740754and the frequency $\nu$. Unless otherwise specified, the results presented here are based on simulations of
    … …  
    760774Frequency & 500 kHz ($d z \sim 10^{-3}$) & 256 \\
    761775\hline
    762 \end{tabular} \\
     776\end{tabular} \\[1mm]
    763777Cube size : $ 90 \, \mathrm{deg.} \times 30 \, \mathrm{deg.} \times 128 \, \mathrm{MHz}$   \\
    764778$ 1800 \times 600 \times 256 \simeq 123 \, 10^6$ cells
    … …  
    775789
    776790We have thus also created a simple sky model using the Haslam Galactic synchrotron map
    777 at 408 Mhz \citep{haslam.08} and the NRAO VLA Sky Survey (NVSS) 1.4 GHz radio source
     791at 408 Mhz \citep{haslam.82} and the NRAO VLA Sky Survey (NVSS) 1.4 GHz radio source
    778792catalog \cite{nvss.98}. The sky temperature cube in this model (Model-II/Haslam+NVSS)
    779793has been computed through the following steps:
    780794
    781795\begin{enumerate}
    782 \item The Galactic synchrotron emission is modeled as a sum of two power law.
    783 We assign a power law index $\beta = -2.8  \pm 0.15$ to each sky direction.
     796\item The Galactic synchrotron emission is modeled power law with spatially
     797varying spectral index. We assign a power law index $\beta = -2.8  \pm 0.15$ to each sky direction.
    784798$\beta$ has a gaussian distribution centered at -2.8 and with standard
    785799deviation $\sigma_\beta = 0.15$.
    … …  
    790804\item A two dimensional $T_{nvss}(\alpha,\delta)$sky brightness temperature at 1.4 GHz is computed
    791805by projecting the radio sources in the NVSS catalog to a grid with the same angular resolution as
    792 the sky cubes is computed. The source brightness in Jansky is converted to temperature taking the
    793 pixel angular size into account ($ \sim 21 \mathrm{mK / mJansky}$ at 1.4 Ghz and $3' \times 3'$ pixels). 
    794 A sepctral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source
     806the sky cubes. The source brightness in Jansky is converted to temperature taking the
     807pixel angular size into account ($ \sim 21 \mathrm{mK / mJansky}$ at 1.4 GHz and $3' \times 3'$ pixels). 
     808A spectral index $\beta_{src} \in [-1.5,-2]$ is also assigned to each sky direction for the radio source
    795809map; we have taken $\beta_{src}$ as a flat random number in the range $[-1.5,-2]$, and the
    796810contribution of the radiosources to the sky temperature is computed as follow:
    … …  
    806820\footnote{SimLSS : {\tt http://www.sophya.org/SimLSS} }. 
    807821{\color{red}: CMV, please add few line description of SimLSS}.
    808 We have generated the mass fluctuations $\delta \rho/rho$ at $z=0.6$, in cells of size
     822We have generated the mass fluctuations $\delta \rho/\rho$ at $z=0.6$, in cells of size
    809823$1.9 \times 1.9 \times 2.8 \, \mathrm{Mpc^3}$, which correspond approximately to the
    810824sky cube angular and frequency resolution defined above.  The mass fluctuations has been
    811 converted into temperature through a factor $0.13 mK$, corresponding to a hydrogen
    812 fraction $0.008x(1+0.6)$.  The total sky brightness temperature is then computed as the sum
     825converted into temperature through a factor $0.13 \mathrm{mK}$, corresponding to a hydrogen
     826fraction $0.008 \times (1+0.6)$.  The total sky brightness temperature is then computed as the sum
    813827of foregrounds and the LSS 21 cm emission:
    814828$$  T_{sky} = T_{sync}+T_{radsrc}+T_{lss}   \hspace{5mm} OR \hspace{5mm}
    … …  
    840854\end{table}
    841855
    842 we have computed the power spectrum on the 21cm-LSS sky temperature cube, as well
    843 as on the radio foreground temperature cubes computed using our two foreground
     856we have computed the power spectrum for the 21cm-LSS sky temperature cube, as well
     857as for the radio foreground temperature cubes computed using our two foreground
    844858models. We have also computed the power spectrum on sky brightness temperature
    845859cubes, as measured by a perfect instrument having a 25 arcmin gaussian beam.
    846 The resulting computed power spectrum are shown on figure \ref{pkgsmlss}.
     860The resulting computed power spectra are shown on figure \ref{pkgsmlss}.
    847861The GSM model has more large scale power compared to our simple model, while
    848862it lacks power at higher spatial frequencies. The mode mixing due to
    … …  
    853867compared to the mK LSS signal. 
    854868
     869It should also be noted that in section 3, we presented the different instrument
     870configuration noise level after {\em correcting or deconvolving} the instrument response. The LSS
     871power spectrum is recovered unaffected in this case, while the noise power spectrum
     872increases at high k values (small scales). In practice, clean deconvolution is difficult to
     873implement for real data and the power spectra presented in this section are NOT corrected
     874for the instrumental response.
     875
    855876\begin{figure}
    856877\centering
     878\vspace*{-10mm}
    857879\mbox{
    858 \hspace*{-10mm}
    859 \includegraphics[width=0.5\textwidth]{Figs/comptempgsm.pdf}
     880\hspace*{-20mm}
     881\includegraphics[width=0.6\textwidth]{Figs/comptempgsm.pdf}
    860882}
     883\vspace*{-10mm}
    861884\caption{Comparison of GSM (black) Model-II (red) sky cube temperature distribution.
    862885The Model-II (Haslam+NVSS),
    … …  
    869892\mbox{
    870893\hspace*{-10mm}
    871 \includegraphics[width=\textwidth]{Figs/compmapgsm.pdf}
     894\includegraphics[width=0.9\textwidth]{Figs/compmapgsm.pdf}
    872895}
    873896\caption{Comparison of GSM map (top) and Model-II sky map at 884 MHz (bottom).
    … …  
    883906\includegraphics[width=0.7\textwidth]{Figs/pk_gsm_lss.pdf}
    884907}
    885 \vspace*{-30mm}
     908\vspace*{-40mm}
    886909\caption{Comparison of the 21cm LSS power spectrum (red, orange) with the radio foreground power spectrum.
    887910The radio sky power spectrum is shown for the GSM (Model-I) sky model (dark blue), as well as for our simple
    … …  
    893916
    894917
    895 \subsection{ LSS signal extraction }
    896 % {\color{red} \large \it  CMV + Reza  +  J.M. Martin  } \\[1mm]
    897 Description of the component separation method and the results
    898 \begin{itemize}
    899 \item Component separation method, based on instrument response correction and frequency
    900 smoothness / power law
    901 \item Foreground power spectrum
    902 \item Performance of component separation : comparison of frequency slices of recovered LSS
    903 and foreground maps, source catalogs
    904 \item Performance in statistical sense (power spectrum) : comparison of recovered P(k)-LSS
    905 and true P(k), residual noise/systematic effect power spectrum
    906 \end{itemize}
    907 
     918\subsection{ Instrument response and LSS signal extraction }
     919
     920The observed data cube is obtained from the sky brightness temperature 3D map
     921$T_{sky}(\alpha, \delta, \nu)$ by applying the frequency dependent instrument response
     922${\cal R}(u,v,\lambda)$.
     923As a simplification, we have considered that the instrument response is independent
     924of the sky direction.
     925For each frequency $\nu_k$ or wavelength $\lambda_k=c/\nu_k$ :
     926\begin{enumerate}
     927\item Apply a 2D Fourier transform to compute sky angular Fourier amplitudes
     928$$ T_{sky}(\alpha, \delta, \lambda_k) \rightarrow \mathrm{2D-FFT} \rightarrow {\cal T}_{sky}(u, v, \lambda_k)$$
     929\item Apply instrument response in the angular wave mode plane
     930$$  {\cal T}_{sky}(u, v, \lambda_k)  \longrightarrow {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda) $$
     931\item Apply inverse 2D Fourier transform to compute the measured sky brightness temperature map,
     932without instrumental (electronic/$\Tsys$) white noise:
     933$$ {\cal T}_{sky}(u, v, \lambda_k) \times {\cal R}(u,v,\lambda)   
     934\rightarrow \mathrm{Inv-2D-FFT} \rightarrow T_{mes1}(\alpha, \delta, \lambda_k) $$
     935\item Add white noise (gaussian fluctuations) to obtain the measured sky brightness temperature
     936$T_{mes}(\alpha, \delta, \nu_k)$. We have also considered that the system temperature and thus the
     937additive white noise level was independent of the frequency or wavelength.   
     938\end{enumerate}
     939The LSS signal extraction depends indeed on the white noise level.
     940The results shown here correspond to the (a) instrument configuration, a packed array of
     941$11 \times 11 = 121$ 5 meter diameter dishes, with a white noise level corresponding
     942to $\sigma_{noise} = 0.25 \mathrm{mK}$ per $3 \times 3 \mathrm{arcmin^2} \times 500 kHz$
     943cell.
     944
     945Our simple component separation procedure is described below:
     946\begin{enumerate}
     947\item The measured sky brightness temperature is first corrected for the frequency dependent
     948beam effects through a convolution by a virtual, frequency independent beam. We assume
     949that we have a perfect knowledge of the intrinsic instrument response.
     950$$  T_{mes}(\alpha, \delta, \nu) \longrightarrow T_{mes}^{bcor}(\alpha,\delta,\nu) $$
     951The virtual target instrument has a beam width larger to the worst real instrument beam,
     952i.e at the lowest observed frequency. 
     953 \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
     954 is fitted to the beam-corrected brightness temperature. $b$ is the power law index and  $10^a$
     955is the brightness temperature at the reference frequency $\nu_0$:
     956\begin{eqnarray*}
     957P1 & :  & \log10 ( T_{mes}^{bcor}(\nu) ) = a + b \log10 ( \nu / \nu_0 ) \\
     958P2 & :  & \log10 ( T_{mes}^{bcor}(\nu) ) = a + b \log10 ( \nu / \nu_0 ) + c \log10 ( \nu/\nu_0 ) ^2
     959\end{eqnarray*}
     960\item The difference between the beam-corrected sky temperature and the fitted power law
     961$(T_0(\alpha, \delta), b(\alpha, \delta))$ is our extracted 21 cm LSS signal.
     962\end{enumerate}
     963
     964Figure \ref{extlsspk} shows the performance of this procedure at a redshift $\sim 0.6$,
     965for the two radio sky models used here: GSM/Model-I and Haslam+NVSS/Model-II. The
     96621 cm LSS power spectrum, as seen by a perfect instrument with a gaussian frequency independent
     967beam is shown in orange (solid line), and the extracted power spectrum, after beam correction
     968and foreground separation with second order polynomial fit (P2) is shown in red (circle markers).
     969We have also represented the obtained power spectrum without applying the beam correction (step 1 above),
     970or with the first order polynomial fit (P1).
     971
     972It can be seen that a precise knowledge of the instrument beam and the beam correction
     973is a key ingredient for recovering the 21 cm LSS power spectrum. It is also worthwhile to
     974note that while it is enough to correct the beam to the lowest resolution instrument beam
     975($\sim 30'$ or $D \sim 50$ meter @ 820 MHz) for the GSM model, a stronger beam correction
     976has to be applied (($\sim 36'$ or $D \sim 40$ meter @ 820 MHz) for the Model-II to reduce
     977significantly the ripples from bright radio sources. The effect of mode mixing is reduced for
     978an instrument with smooth (gaussian) beam, compared to the instrument response
     979${\cal R}(u,v,\lambda)$ used here.
     980
     981Figure \ref{extlssratio} shows the overall {\em transfer function} for 21 cm LSS power
     982spectrum measurement. We have shown (solid line, orange) the ratio of measured LSS power spectrum
     983by a perfect instrument $P_{perf-obs}(k)$, with a gaussian beam of $\sim$ 36 arcmin, respectively $\sim$ 30 arcmin,
     984in the absence of any foregrounds or instrument noise, to the original 21 cm power spectrum $P_{21cm}(k)$.
     985The ratio of the recovered LSS power spectrum $P_{ext}(k)$ to $P_{perf-obs}(k)$ is shown in red, and the
     986ratio of the recovered spectrum to  $P_{21cm}(k)$  is shown in black (thin line).
     987
     988\begin{figure*}
     989\centering
     990\vspace*{-20mm}
     991\mbox{
     992\hspace*{-20mm}
     993\includegraphics[width=1.1\textwidth]{Figs/extlsspk.pdf}
     994}
     995\vspace*{-30mm}
     996\caption{Power spectrum of the 21cm LSS temperature fluctuations, separated from the
     997continuum radio emissions at $z \sim 0.6$.
     998Left: GSM/Model-I , right: Haslam+NVSS/Model-II.  }
     999\label{extlsspk}
     1000\end{figure*}
     1001
     1002
     1003\begin{figure*}
     1004\centering
     1005\vspace*{-20mm}
     1006\mbox{
     1007\hspace*{-20mm}
     1008\includegraphics[width=1.1\textwidth]{Figs/extlssratio.pdf}
     1009}
     1010\vspace*{-30mm}
     1011\caption{Power spectrum of the 21cm LSS temperature fluctuations, separated from the
     1012continuum radio emissions at $z \sim 0.6$.
     1013Left: GSM/Model-I , right: Haslam+NVSS/Model-II.  }
     1014\label{extlssratio}
     1015\end{figure*}
    9081016
    9091017\section{ BAO scale determination and constrain on dark energy parameters}
    … …  
    9111019We compute  reconstructed LSS-P(k) (after component separation) at different z's
    9121020and determine BAO scale as a function of redshifts.
    913 We can this a large number of time ( ~ 100 \ldots 1000 ) to have the reconstructed P(k)
    914 with {\it realistic } errors. We can then determine the error on the reconstructed DE
    915 parameters
     1021Method:
     1022\begin{itemize}
     1023\item Compute/guess the overall transfer function for several redshifts (0.5 , 1.0 1.5 2.0 2.5 ) \\
     1024\item Compute / guess the instrument noise level for the same redshit values
     1025\item Compute the observed P(k) and extract $k_{BAO}$ , and the corresponding error
     1026\item Compute the DETF ellipse with different priors
     1027\end{itemize}
     1028
    9161029
    9171030\section{Conclusions}
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