Changeset 4030 in Sophya for trunk


Ignore:
Timestamp:
Oct 26, 2011, 7:42:21 PM (14 years ago)
Author:
ansari
Message:

Corrections papiers avec les commentaires du referee (1) - Reza 26/10/2011

Location:
trunk/Cosmo/RadioBeam
Files:
6 edited

Legend:

Unmodified
Added
Removed

  • trunk/Cosmo/RadioBeam/mdish.cc

    r3991 r4030  
    4848    }
    4949}
    50 // Return a vector representing the power spectrum (for checking)
     50// Return a 2 D histrogram as the response(kx,ky)
    5151Histo2D Four2DResponse::GetResponse(int nx, int ny)
    5252{
    … …  
    6161      h2.BinCenter(i,j,xbc,ybc);
    6262      h2(i,j) = Value(xbc,ybc);
     63    }
     64  return h2;   
     65}
     66
     67// Return a 2 D histrogram as the response(u=kx/2 pi, v=ky/2 pi)
     68Histo2D Four2DResponse::GetResponseUV(int nx, int ny)
     69{
     70  double kxmx = 1.2*dx_;
     71  double kymx = 1.2*dy_;
     72  if (typ_<3) kymx=kxmx;
     73  Histo2D h2(-kxmx,kxmx,nx,-kymx,kymx,ny);
     74
     75  double xbc,ybc;
     76  for(int_4 j=0; j<h2.NBinY(); j++)
     77    for(int_4 i=0; i<h2.NBinX(); i++) {
     78      h2.BinCenter(i,j,xbc,ybc);
     79      h2(i,j) = Value(xbc*DeuxPI,ybc*DeuxPI);
    6380    }
    6481  return h2;   

  • trunk/Cosmo/RadioBeam/mdish.h

    r3991 r4030  
    5050  inline  double operator()(double kx, double ky)
    5151  { return Value(kx, ky); }
     52  // Retourne l'histo de reponse en fonction de k_x,k_y
    5253  virtual Histo2D GetResponse(int nx=255, int ny=255); 
     54  // Retourne l'histo de reponse en fonction de u=k_x/2 pi, v=k_y/2 pi
     55  virtual Histo2D GetResponseUV(int nx=255, int ny=255); 
     56
    5357  // Retourne le niveau moyen du bruit projete 1D en fonction (sqrt(u^2+v^2)
    5458  HProf GetProjNoiseLevel(int nbin=128, bool fgnorm1=true);

  • trunk/Cosmo/RadioBeam/pknoise.cc

    r4028 r4030  
    5050       << "    -cszmpc sx,sy,sz : 3D real space box cell size in Mpc (default=3x3x1.5) \n"   
    5151       << "    -z redshift : redshift  (default=1.0) \n"
     52       << "    -pnf : Apply frequency dependent noise factor=ComovDa^2*(1+z)^2/Hz (default= NON) \n"
    5253       << "    -scut SCutValue (default= -100.) \n"
    5354       << "       if SCutValue<0. ==> SCut=MinNoisePower*(-SCutValue) \n"
    … …  
    9192  int box3dsz[3]={512,512,256};
    9293  double cellsz[3]={3.,3.,3.};
     94  bool fgnoisefreq=false;
    9395  int NBINPK=256;
    9496  int prtlev=0;
    … …  
    122124    else if (strcmp(arg[ka],"-nbin")==0) {
    123125      NBINPK=atoi(arg[ka+1]);    ka+=2;
     126    }
     127    else if (strcmp(arg[ka],"-pnf")==0) {
     128      fgnoisefreq=true;   ka++;
    124129    }
    125130    else if (strcmp(arg[ka],"-prt")==0) {
    … …  
    191196              << " DoL=" << DIAMETRE/lambda << " ) " << endl;
    192197    Histo2D h2drep = arep_p->GetResponse();
     198    Histo2D h2drepuv = arep_p->GetResponseUV();
    193199    double repmax= h2drep.VMax();
    194200    if (fgautoscut) {
    … …  
    210216    double angscale=1.;
    211217    Vector angscales(box3dsz[2]);
     218    double pnoise0=comov_dA_z*comov_dA_z*(1.+z_Redshift)*(1.+z_Redshift)/H_z;
     219    Vector pnoisevec(box3dsz[2]);
     220    pnoisevec=1.;
    212221    angscales=angscale;
    213222    if (NMAX<0)  { box3dsz[0]=256; box3dsz[1]=256;  box3dsz[2]=128; }
    … …  
    222231    cout << " pknoise[2.c]: Freq0=" << freq0 << " dFreq=" << dfreq << " freq(z=" << z_Redshift << ")="
    223232         << conv.getFrequency() << " zmin=" << 1420.4/freq0-1. << " zmax=" << 1420.4/(freq0+box3dsz[2]*dfreq)-1. << endl;
    224     cout << " pknoise[2.d]: Four3DPk m3d(rg," << box3dsz[0]/2 << "," << box3dsz[1] << ","
     233    if(fgnoisefreq) {
     234      for(int kz=0; kz<box3dsz[2]; kz++) {
     235        double zreds=1420.4/(freq0+(double)kz*dfreq)-1.;
     236        double comda=redshift2da(zreds);
     237        pnoisevec(kz)=comda*comda*(1.+zreds)*(1.+zreds)/redshift2hz(zreds)/pnoise0;
     238      }
     239      cout << " pknoise[2.d]: PnoiseFactor:" << pnoisevec(0) << " ... " << pnoisevec(pnoisevec.Size()-1) << endl;
     240    }
     241    cout << " pknoise[2.e]: Four3DPk m3d(rg," << box3dsz[0]/2 << "," << box3dsz[1] << ","
    225242         << box3dsz[2] << ")" << endl;
    226243    Four3DPk m3d(rg,box3dsz[0]/2,box3dsz[1],box3dsz[2]);
    227     cout << " pknoise[2.d]: m3d.SetCellSize(" << dkxmpc << "," << dkympc << "," << dkzmpc
     244    cout << " pknoise[2.f]: m3d.SetCellSize(" << dkxmpc << "," << dkympc << "," << dkzmpc
    228245         << ") cell size (Mpc) : " << cellsz[0] << "x" << cellsz[1] << "x" << cellsz[2] << endl;
    229246    m3d.SetCellSize(dkxmpc, dkympc, dkzmpc);
    … …  
    235252      pkn.SetFreqRange(freq0, dfreq);
    236253      pkn.SetAngScaleConversion(angscales);
     254      pkn.SetPNoiseFactor(pnoisevec);
    237255      pkn.SetPrtLevel(prtlev,prtmod);
    238256      HProf hpn = pkn.Compute(NBINPK);
    … …  
    255273    po << PPFNameTag("h1drep") << h1drep;
    256274    po << PPFNameTag("h2drep") << h2drep;
    257  
     275    po << PPFNameTag("h2drepuv") << h2drepuv;
     276
    258277    rc = 0;
    259278  }  // End of try bloc

  • trunk/Cosmo/RadioBeam/sensfgnd21cm.tex

    r4022 r4030  
    5959%% Definition fonction de transfer
    6060\newcommand{\TrF}{\mathbf{T}}
    61 
     61%% Definition (u,v) , ...
     62\def\uv{\mathrm{u,v}}
     63\def\uvu{\mathrm{u}}
     64\def\uvv{\mathrm{v}}
     65\def\dudv{\mathrm{d u d v}}
     66
     67% Commande pour marquer les changements du papiers pour le referee
     68\def\changemark{\bf }
    6269
    6370%%% Definition pour la section sur les param DE par C.Y
    … …  
    257264temperature, with a total integration time of 86400 seconds (1 day) over a frequency band of 1 MHz.
    258265The width of this frequency band is well adapted  to detection of \HI source with an intrinsic velocity
    259 dispersion of few 100 km/s. These detection limits should be compared with the expected 21 cm brightness
     266dispersion of few 100 km/s.
     267These detection limits should be compared with the expected 21 cm brightness
    260268$S_{21}$ of compact sources which can be computed using the expression below (e.g.\cite{binney.98}) :
    261269\begin{equation}
    … …  
    264272\end{equation}
    265273 where $ M_{H_I} $ is the neutral hydrogen mass, $\dlum(z)$ is the luminosity distance and $\sigma_v$
    266 is the source velocity dispersion.
     274is the source velocity dispersion. 
     275{\changemark The 1 MHz bandwidth mentioned above is only used for computing the
     276galaxy detection thresholds and does not determine the total survey bandwidth or frequency resolution
     277of an intensity mapping survey.}
    267278% {\color{red} Faut-il developper le calcul en annexe ? }
    268279 
    … …  
    270281compact objects with a total \HI \, mass of $10^{10} M_\odot$ and an intrinsic velocity dispersion of
    271282$200 \, \mathrm{km/s}$. The luminosity distance is computed for the standard
    272 WMAP \LCDM universe. $10^9 - 10^{10} M_\odot$ of neutral gas mass
     283WMAP \LCDM universe \citep{komatsu.11}. $10^9 - 10^{10} M_\odot$ of neutral gas mass
    273284is typical for large galaxies \citep{lah.09}. It is clear that detection of \HI sources at cosmological distances
    274285would require collecting area in the range of $10^6 \mathrm{m^2}$.
    … …  
    334345
    335346\subsection{ \HI power spectrum and BAO}
    336 In the absence of any foreground or background radiation, the brightness temperature
    337 for a given direction and wavelength $\TTlam$ would be proportional to
     347In the absence of any foreground or background radiation
     348{\changemark and assuming high spin temperature $\kb T_{spin} \gg h \nu_{21}$},
     349the brightness temperature for a given direction and wavelength $\TTlam$ would be proportional to
    338350the local \HI number density $\etaHI(\vec{\Theta},z)$ through the relation:
    339351\begin{equation}
    … …  
    394406\end{equation}
    395407where $k$ is the comoving wave vector and $ \dang(z) $ is the angular diameter distance.
     408{ \changemark The matter power spectrum $P(k)$ has been measured using
     409galaxy surveys, for example by SDSS and 2dF at low redshift $z \lesssim 0.3$
     410(\cite{cole.05}, \cite{tegmark.04}). The 21 cm brightness power spectra $P_{T_{21}}(k)$
     411shown here are comparable to one from the galaxy surveys, once the mean 21 cm
     412temperature conversion factor $\left( \bar{T}_{21}(z)  \right)^2$ and redshift evolution
     413have been accounted for. }
    396414% It should be noted that the maximum transverse $k^{comov} $ sensitivity range
    397415% for an instrument corresponds approximately to half of its angular resolution.
    … …  
    474492the common receivers axis, the visibilty would be written as the 2D Fourier transform
    475493of the product of the sky intensity and the receiver beam, for the angular frequency
    476 \mbox{$(u,v)_{12} = 2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$}:
     494\mbox{$(\uv)_{12} = ( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$}:
    477495\begin{equation}
    478496\vis(\lambda) \simeq  \iint d\alpha d\beta \, \, I(\alpha, \beta) \,  L(\alpha, \beta)
    … …  
    483501two $(\alpha, \beta)$ angular planes.
    484502
    485 Furthermore, we introduce the conjugate Fourier variables $(u,v)$ and the Fourier transforms
     503Furthermore, we introduce the conjugate Fourier variables $(\uv)$ and the Fourier transforms
    486504of the sky intensity and the receiver beam:
    487505\begin{center}
    488506\begin{tabular}{ccc}
    489 $(\alpha, \beta)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & $(u,v)$ \\
    490 $I(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal I}(u,v, \lambda)$ \\
    491 $L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(u,v, \lambda)$ \\
     507$(\alpha, \beta)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & $(\uv)$ \\
     508$I(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal I}(\uv, \lambda)$ \\
     509$L(\alpha, \beta, \lambda)$ & \hspace{2mm} $\longrightarrow $ \hspace{2mm} & ${\cal L}(\uv, \lambda)$ \\
    492510\end{tabular}
    493511\end{center}
    … …  
    495513The visibility can then be interpreted as the weighted sum of the sky intensity, in an angular
    496514wave number domain located around
    497 $(u, v)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is
     515$(\uv)_{12}=2 \pi( \frac{\Delta x}{\lambda} ,  \frac{\Delta y}{\lambda} )$. The weight function is
    498516given by the receiver beam Fourier transform.
    499517\begin{equation}
    500 \vis(\lambda) \simeq  \iint d u d v \, \, {\cal I}(u,v, \lambda) \, {\cal L}(u - 2 \pi \frac{\Delta x}{\lambda} , v - 2 \pi \frac{\Delta y}{\lambda} , \lambda)
     518\vis(\lambda) \simeq  \iint \dudv \, \, {\cal I}(\uv, \lambda) \, {\cal L}(\uvu - \frac{\Delta x}{\lambda} , \uvv - \frac{\Delta y}{\lambda} , \lambda)
    501519\end{equation}
    502520
    503521A single receiver instrument would measure the total power integrated in a spot centered around the
    504 origin in the $(u,v)$ or the angular wave mode plane. The shape of the spot depends on the receiver
     522origin in the $(\uv)$ or the angular wave mode plane. The shape of the spot depends on the receiver
    505523beam pattern, but its extent would be $\sim 2 \pi D / \lambda$, where $D$ is the receiver physical
    506524size.
    507525
    508526The correlation signal from a pair of receivers would measure the integrated signal on a similar
    509 spot, located around the central angular wave mode  $(u, v)_{12}$ determined by the relative
     527spot, located around the central angular wave mode  $(\uv)_{12}$ determined by the relative
    510528position of the two receivers (see figure \ref{figuvplane}).
    511529In an interferometer with multiple receivers, the area covered by different receiver pairs in the
    512 $(u,v)$ plane might overlap and some pairs might measure the same area (same base lines).
     530$(\uv)$ plane might overlap and some pairs might measure the same area (same base lines).
    513531Several beams can be formed using different combination of the correlations from a set of 
    514532antenna pairs. 
    515533
    516 An instrument can thus be characterized by its $(u,v)$ plane coverage or response
    517 ${\cal R}(u,v,\lambda)$. For a single dish with a single receiver in the focal plane,
     534An instrument can thus be characterized by its $(\uv)$ plane coverage or response
     535${\cal R}(\uv,\lambda)$. For a single dish with a single receiver in the focal plane,
    518536the instrument response is simply the Fourier transform of the beam.
    519537For a single dish with multiple receivers, either as a Focal Plane Array (FPA) or
    520538a multi-horn system, each beam (b) will have its own response
    521 ${\cal R}_b(u,v,\lambda)$.
     539${\cal R}_b(\uv,\lambda)$.
    522540For an interferometer, we can compute a raw instrument response
    523 ${\cal R}_{raw}(u,v,\lambda)$ which corresponds to $(u,v)$ plane coverage by all
     541${\cal R}_{raw}(\uv,\lambda)$ which corresponds to $(u,v)$ plane coverage by all
    524542receiver pairs  with uniform weighting.
    525543Obviously, different weighting schemes can be used, changing
    526 the effective beam shape and thus the response ${\cal R}_{w}(u,v,\lambda)$
     544the effective beam shape and thus the response ${\cal R}_{w}(\uv,\lambda)$
    527545and the noise behaviour. If the same Fourier angular frequency mode is measured
    528546by several receiver pairs, the raw instrument response might then be larger
    529547that unity. This non normalized instrument response is used to compute the projected
    530548noise power spectrum in the following section (\ref{instrumnoise}).
    531 We can also define a  normalized instrument response, ${\cal R}_{norm}(u,v,\lambda) \lesssim 1$ as:
    532 \begin{equation}
    533 {\cal R}_{norm}(u,v,\lambda) = {\cal R}(u,v,\lambda) / \mathrm{Max_{(u,v)}} \left[ {\cal R}(u,v,\lambda) \right]
     549We can also define a  normalized instrument response, ${\cal R}_{norm}(\uv,\lambda) \lesssim 1$ as:
     550\begin{equation}
     551{\cal R}_{norm}(\uv,\lambda) = {\cal R}(\uv,\lambda) / \mathrm{Max_{(\uv)}} \left[ {\cal R}(\uv,\lambda) \right]
    534552\end{equation}
    535553This normalized  instrument response can be used to compute the effective instrument beam,
    536554in particular in section \ref{recsec}. 
     555
     556{\changemark Detection of the reionisation at 21 cm band has been an active field
     557in the last decade and several groups
     558(\cite{rottgering.06}, \cite{bowman.07}, \cite{lonsdale.09}, \cite{parsons.09})  have built
     559instruments to detect reionisation signal around 100 MHz.
     560Several authors have studied the instrumental noise
     561and statistical uncertainties when measuring the reionisation signal power spectrum;
     562the methods presented here to compute the instrument response
     563and sensitivities are similar to the ones developed in these publications
     564(\cite{morales.04}, \cite{bowman.06}, \cite{mcquinn.06}). }
    537565
    538566\begin{figure}
    … …  
    543571}
    544572\vspace*{-15mm}
    545 \caption{Schematic view of the $(u,v)$ plane coverage by interferometric measurement.}
     573\caption{Schematic view of the $(\uv)$ plane coverage by interferometric measurement.}
    546574\label{figuvplane}
    547575\end{figure}
    548576
    549 \subsection{Noise power spectrum}
     577\subsection{Noise power spectrum computation}
    550578\label{instrumnoise}
    551579Let's consider a total power measurement using a receiver at wavelength $\lambda$, over a frequency
    … …  
    555583corresponds also to the noise for the visibility $\vis$ measured from two identical receivers, with uncorrelated
    556584noise. If the receiver has an effective area $A \simeq \pi D^2/4$ or $A \simeq D_x D_y$, the measurement
    557 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  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$), corresponding to a visibility
     585corresponds 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
    558586measurement from a pair of receivers can be written as:
    559587\begin{eqnarray}
    … …  
    565593
    566594The sky temperature measurement can thus be characterized by the noise spectral power density in
    567 the angular frequencies plane $P_{noise}^{(u,v)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$ 
    568 per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$:
     595the angular frequencies plane $P_{noise}^{(\uv)} \simeq \frac{\sigma_{noise}^2}{A / \lambda^2}$, in $\mathrm{Kelvin^2}$ 
     596per unit area of angular frequencies  $\delta \uvu \times \delta \uvv$:
    569597We can characterize the sky temperature measurement with a radio instrument by the noise
    570 spectral power density in the angular frequencies plane $P_{noise}(u,v)$ in units of $\mathrm{Kelvin^2}$ 
    571 per unit area of angular frequencies  $\frac{\delta u}{ 2 \pi} \times \frac{\delta v}{2 \pi}$.
     598spectral power density in the angular frequencies plane $P_{noise}(\uv)$ in units of $\mathrm{Kelvin^2}$ 
     599per unit area of angular frequencies  $\delta \uvu \times \delta \uvv$.
    572600For an interferometer made of identical receiver elements, several ($n$) receiver pairs
    573 might have the same baseline. The noise power density in the corresponding $(u,v)$ plane area
     601might have the same baseline. The noise power density in the corresponding $(\uv)$ plane area
    574602is then reduced by a factor $1/n$. More generally, we can write the instrument  noise
    575603spectral power density using the instrument response defined in section \ref{instrumresp} :
    576604\begin{equation}
    577 P_{noise}(u,v) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(u,v,\lambda) }
     605P_{noise}(\uv) = \frac{ P_{noise}^{\mathrm{pair}} } { {\cal R}_{raw}(\uv,\lambda) }
     606\label{eq:pnoiseuv}
    578607\end{equation}
    579608
    … …  
    581610$P(k)$ is computed, angles are translated into comoving transverse distances,
    582611and frequencies or wavelengths into comoving radial distance, using the following relations:
     612{ \changemark
    583613\begin{eqnarray}
    584 \delta \alpha , \beta & \rightarrow & \delta \ell_\perp = (1+z) \, \dang(z) \, \delta \alpha,\beta  \\
     614\alpha , \beta & \rightarrow &  \ell_\perp = l_x, l_y = (1+z) \, \dang(z) \,  \alpha,\beta  \\
     615\uv & \rightarrow & k_\perp = k_x, k_y = 2 \pi \frac{ \uvu , \uvv }{  (1+z) \, \dang(z)  }   \label{eq:uvkxky} \\
    585616\delta \nu & \rightarrow & \delta \ell_\parallel = (1+z) \frac{c}{H(z)} \frac{\delta \nu}{\nu}
    586617  = (1+z) \frac{\lambda}{H(z)} \delta \nu \\
    587 \delta u , \delta v & \rightarrow & \delta k_\perp = \frac{ \delta u \, , \, \delta v }{  (1+z) \, \dang(z)  } \\
    588 \frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
     618% \delta \uvu , \delta \uvv & \rightarrow & \delta k_\perp = 2 \pi \frac{ \delta \uvu \, , \, \delta \uvv }{  (1+z) \, \dang(z)  } \\
     619\frac{1}{\delta \nu} & \rightarrow & \delta k_\parallel = 2 \pi \, \frac{H(z)}{c} \frac{1}{(1+z)} \, \frac{\nu}{\delta \nu}
    589620 =  \frac{H(z)}{c} \frac{1}{(1+z)^2} \, \frac{\nu_{21}}{\delta \nu}
    590621\end{eqnarray}
    591 
    592 If we consider a uniform noise spectral density in the $(u,v)$ plane corresponding to the
     622}
     623{ \changemark
     624A brightness measurement at a  point $(\uv,\lambda)$, covering
     625the 3D spot  $(\delta \uvu, \delta \uvv, \delta \nu)$, would correspond
     626to cosmological power spectrum measurement at a transverse wave mode $(k_x,k_y)$
     627defined by the equation \ref{eq:uvkxky}, measured at a redshift given by the observation frequency.
     628The measurement noise spectral density given by the equation \ref{eq:pnoisepairD} can then be
     629translated into a 3D noise power spectrum, per unit of spatial frequencies
     630$ \frac{\delta k_x \times \delta k_y \times \delta k_z}{8 \pi^3} $ (units: $ \mathrm{K^2 \times Mpc^3}$) : 
     631
     632\begin{eqnarray}
     633(\uv , \lambda) & \rightarrow & k_x(\uvu),k_y(\uvv), z(\lambda) \\
     634P_{noise}(k_x,k_y, z)  & = & P_{noise}(\uv)
     635 \frac{ 8 \pi^3 \delta \uvu \times \delta \uvv }{\delta k_x \times \delta k_y \times \delta k_z} \\
     636P_{noise}(k_x,k_y, z) & = &   \left( 2 \, \frac{\Tsys^2}{t_{int} \, \nu_{21} } \, \frac{\lambda^2}{D^2}  \right)
     637   \, \frac{1}{{\cal R}_{raw}} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4 
     638\label{eq:pnoisekxkz}
     639\end{eqnarray}
     640
     641It is worthwhile to notice that the cosmological 3D noise power spectrum does not depend anymore on the
     642individual measurement bandwidth.
     643In the following paragraph, we will first consider an ideal instrument
     644with uniform $(\uv)$ coverage
     645in order to establish the general noise power spectrum behaviour for cosmological 21 cm surveys.
     646The numerical method used to compute the 3D noise power spectrum is then presented in section
     647\ref{pnoisemeth}.
     648}
     649 
     650\subsubsection{Uniform $(\uv)$ coverage}
     651
     652If we consider a uniform noise spectral density in the $(\uv)$ plane corresponding to the
    593653equation \ref{eq:pnoisepairD} above,  the three dimensional projected noise spectral density
    594654can then be written as:
    … …  
    644704
    645705For a single dish of diameter $D$, or an interferometric instrument with maximal extent $D$,
    646 observations provide information up to $u_{max},v_{max} \lesssim 2 \pi D / \lambda $. This value of
    647 $u_{max},v_{max}$ would be mapped to a maximum transverse cosmological wave number
    648 $k^{\perp}_{max}$:
    649 \begin{equation}
    650 k^{\perp}  =  \frac{(u,v)}{(1+z) \dang}  \hspace{8mm}
    651 k^{\perp}_{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}}
     706observations provide information up to $\uvu_{max},\uvv_{max} \lesssim D / \lambda $. This value of
     707$\uvu_{max},\uvv_{max}$ would be mapped to a maximum transverse cosmological wave number
     708$k_{\perp}^{max}$:
     709\begin{equation}
     710k_{\perp}^{max}  \lesssim  \frac{2 \pi}{\dang \, (1+z)^2} \frac{D}{\lambda_{21}}
    652711\label{kperpmax}
    653712\end{equation}   
    … …  
    657716beams and a system noise temperature $\Tsys = 50 \mathrm{K}$.
    658717The survey is supposed to cover a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$, in one
    659 year. The maximum comoving wave number $k_{max}$  is also shown as a function
     718year. The maximum comoving wave number $k^{max}$  is also shown as a function
    660719of redshift, for an instrument with $D=100 \, \mathrm{m}$ maximum extent. In order
    661720to take into account the radial component of $\vec{k}$ and the increase of
    662 the instrument noise level with $k^{\perp}$, we have taken the effective $k_{ max} $
    663 as half of the maximum transverse $k^{\perp} _{max}$ of \mbox{eq. \ref{kperpmax}}:
     721the instrument noise level with $k_{\perp}$, we have taken the effective $k_{ max} $
     722as half of the maximum transverse $k_{\perp} ^{max}$ of \mbox{eq. \ref{kperpmax}}:
    664723\begin{equation}
    665724k_{max} (z) = \frac{\pi}{\dang \, (1+z)^2} \frac{D=100 \mathrm{m}}{\lambda_{21}}
    … …  
    678737\label{pnkmaxfz}
    679738\end{figure}
    680 
     739 
     740\subsubsection{3D noise power spectrum computation} 
     741\label{pnoisemeth}
     742{ \changemark
     743We describe here the numerical method used to compute the 3D noise power spectrum, for a given instrument
     744response, as presented in section \ref{instrumnoise}. The noise power spectrum is a good indicator to compare
     745sensitivities for different instrument configurations, albeit the noise realization for a real instrument would not be
     746isotropic.
     747\begin{itemize}
     748\item We start by a 3D Fourier coefficient grid, with the two first coordinates being the transverse angular wave modes,
     749and the third being the frequency $(k_x,k_y,\nu)$. The grid is positioned at the mean redshift $z_0$ for which
     750we want to compute $P_{noise}(k)$. For the results at redshift \mbox{$z_0=1$} discussed in section  \ref{instrumnoise},
     751the grid cell size and dimensions have been chosen to represent a box in the universe 
     752\mbox{$\sim 1500 \times 1500 \times 750 \mathrm{Mpc^3}$},
     753with $3\times3\times3 \mathrm{Mpc^3}$ cells.
     754This correspond to an angular wedge $\sim 25^\circ \times 25^\circ \times (\Delta z \simeq 0.3)$. Given
     755the small angular extent, we have neglected the curvature of redshift shells.
     756\item For each redshift shell $z(\nu)$, we compute a Gaussian noise realization. $(k_x,k_y)$ is
     757converted to the $\uv$ coordinates using the equation \ref{eq:uvkxky}, and the
     758angular diameter distance $\dang(z)$ for \LCDM model with WMAP parameters.
     759The noise variance is taken proportional to $P_{noise}$ :
     760\begin{equation}
     761\sigma_{re}^2=\sigma_{im}^2 \propto \frac{1}{{\cal R}_{raw}(\uv,\lambda)} \, \dang^2(z) \frac{c}{H(z)} \, (1+z)^4
     762\end{equation}
     763\item an FFT is then performed in the frequency or redshift direction to obtain the noise Fourier
     764complex coefficients $n(k_x,k_y,k_z)$ and the power spectrum is estimated as :
     765\begin{equation}
     766\tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 >  \hspace{2mm}  \mathrm{for} \hspace{2mm}
     767  \sqrt{k_x^2+k_y^2+k_z^2} = k
     768\end{equation}
     769Noise samples corresponding to small instrument response, typically less than 1\% of the
     770maximum instrument response are ignored when calculating  $\tilde{P}_{noise}(k)$.
     771However, we require to have a significant fraction, typically 20\% to 50\% of the possible modes
     772$(k_x,k_y,k_z)$ measured for a given $k$ value.
     773\item the above steps are repeated a number of time to decrease the statistical fluctuations
     774due to the random generations. The averaged computed noise power spectrum is normalized using
     775equation \ref{eq:pnoisekxkz}  and the instrument and survey parameters ($\Tsys \ldots$).
     776\end{itemize}
     777
     778It should be noted that it is possible to obtain a  good approximation of noise
     779power spectrum shape, neglecting the redshift or frequency dependence of the
     780instrument response function and $\dang(z)$ for a small redshift interval around $z_0$,
     781using a fixed  instrument response ${\cal R}(u,v,\lambda(z_0))$ and
     782a constant the radial distance $\dang(z_0)*(1+z_0)$.
     783\begin{equation}
     784\tilde{P}_{noise}(k) = < | n(k_x,k_y,k_z) |^2 > \simeq < P_{noise}(u,v) , k_z >
     785\end{equation}
     786The approximate power spectrum obtain through this simplified and much faster
     787method is shown as dashed curves on figure \ref{figpnoisea2g} for few instrument
     788configurations.
     789}
    681790
    682791\subsection{Instrument configurations and noise power spectrum}
    683792\label{instrumnoise}
    684 We have numerically computed the instrument response ${\cal R}(u,v,\lambda)$
    685 with uniform weights in the $(u,v)$ plane for several instrument configurations:
     793We have numerically computed the instrument response ${\cal R}(\uv,\lambda)$
     794with uniform weights in the $(\uv)$ plane for several instrument configurations:
    686795\begin{itemize}
    687796\item[{\bf a} :] A packed array of $n=121 \, D_{dish}=5 \, \mathrm{m}$ dishes, arranged in
    … …  
    734843\end{figure}
    735844
    736 We have used simple triangular shaped dish response in the $(u,v)$ plane.
     845We have used simple triangular shaped dish response in the $(\uv)$ plane.
    737846However, we have introduced a filling factor or illumination efficiency
    738847$\eta$, relating the effective dish diameter $D_{ill}$ to the
    … …  
    740849as $\eta^2$ or $\eta_x \eta_y$.
    741850\begin{eqnarray}
    742 {\cal L}_\circ (u,v,\lambda) & = & \bigwedge_{[\pm 2 \pi D^{ill}/ \lambda]}(\sqrt{u^2+v^2})  \\
     851{\cal L}_\circ (\uv,\lambda) & = & \bigwedge_{[\pm D^{ill}/ \lambda]}(\sqrt{u^2+v^2})  \\
    743852 L_\circ (\alpha,\beta,\lambda) & = & \left[ \frac{ \sin (\pi (D^{ill}/\lambda) \sin \theta ) }{\pi (D^{ill}/\lambda) \sin \theta} \right]^2
    744853\hspace{4mm} \theta=\sqrt{\alpha^2+\beta^2}
    … …  
    751860rectangular shaped antenna. The illumination efficiency factor has been taken
    752861equal to $\eta_x = 0.9$ in the direction of the cylinder width, and $\eta_y = 0.8$
    753 along the cylinder length. It should be noted that the small angle approximation
     862along the cylinder length. {\changemark  We have used double triangular shaped
     863response function in the $(\uv)$ for each of the receiver elements along the cylinder:
     864\begin{equation}
     865 {\cal L}_\Box(\uv,\lambda)  =
     866\bigwedge_{[\pm \eta_x D_x / \lambda]} (\uvu ) \times
     867\bigwedge_{[\pm \eta_y D_y / \lambda ]} (\uvv )
     868\end{equation}
     869}
     870It should be noted that the small angle approximation
    754871used here for the expression of visibilities is not valid for the receivers along
    755872the cylinder axis. However, some preliminary numerical checks indicate that
    … …  
    758875is the case for a transit type telescope.
    759876
    760 \begin{equation}
    761  {\cal L}_\Box(u,v,\lambda)  =
    762 \bigwedge_{[\pm 2 \pi D^{ill}_x / \lambda]} (u ) \times
    763 \bigwedge_{[\pm 2 \pi D^{ill}_y / \lambda ]} (v )
    764 \end{equation}
    765877Figure \ref{figuvcovabcd} shows the instrument response ${\cal R}(u,v,\lambda)$
    766878for the four configurations (a,b,c,d) with $\sim 100$ receivers per
    767 polarisation. The resulting projected noise spectral power density is shown in figure
     879polarisation.
     880
     881{\changemark Using the numerical method sketched in section \ref{pnoisemeth}, we have
     882computed the 3D noise power spectrum for each of the eight instrument configurations discussed
     883here, with a system noise temperature $\Tsys = 50 \mathrm{K}$, for a one year survey
     884of a quarter of sky $\Omega_{tot} = \pi \, \mathrm{srad}$ at a mean redshift $z_0=1, \nu_0=710 \mathrm{MHz}$.}
     885The resulting noise spectral power densities are shown in figure
    768886\ref{figpnoisea2g}. The increase of $P_{noise}(k)$ at low $k^{comov} \lesssim 0.02$
    769887is due to the fact that we have ignored all auto-correlation measurements. 
    … …  
    774892\centering
    775893\mbox{
    776 \hspace*{-10mm}
    777 \includegraphics[width=0.90\textwidth]{Figs/uvcovabcd.pdf}
     894% \hspace*{-10mm}
     895\includegraphics[width=\textwidth]{Figs/uvcovabcd.pdf}
    778896}
    779 \caption{(u,v) plane coverage (raw instrument response ${\cal R}(u,v,\lambda)$
     897\caption{$(\uv)$ plane coverage (raw instrument response ${\cal R}(\uv,\lambda)$
    780898for four configurations.
    781899(a) 121 $D_{dish}=5$ meter diameter dishes arranged in a compact, square array
    782900of $11 \times 11$, (b) 128 dishes arranged in 8 row of 16 dishes each (fig. \ref{figconfbc}),
    783901(c) 129 dishes arranged as shown in figure \ref{figconfbc} , (d) single D=75 meter diameter, with 100 beams.
    784 (color scale : black $<1$, blue, green, yellow, red $\gtrsim 80$) }
     902The common color scale (1 \ldots 80) is shown on the right. }
    785903\label{figuvcovabcd}
    786904\end{figure*}
    787905 
    788906\begin{figure*}
    789 \vspace*{-25mm}
     907\vspace*{-10mm}
    790908\centering
    791909\mbox{
    792 \hspace*{-20mm}
    793 \includegraphics[width=1.15\textwidth]{Figs/pkna2h.pdf}
     910% \hspace*{-5mm}
     911\includegraphics[width=\textwidth]{Figs/pkna2h.pdf}
    794912}
    795 \vspace*{-40mm}
     913\vspace*{-20mm}
    796914\caption{P(k) LSS power  and noise power spectrum for several interferometer
    797915configurations ((a),(b),(c),(d),(e),(f),(g)) with 121, 128, 129, 400 and 960 receivers.}
    … …  
    810928it has been suggested that the smooth frequency dependence of the synchrotron
    811929emissions can be used to separate the faint LSS signal from the Galactic and radio source
    812 emissions.
     930emissions. {\changemark Discussion of contribution of different sources
     931to foregrounds for measurement of reionization through redshifted 21 cm,
     932as well foreground subtraction using their smooth frequency dependence can
     933be found in (\cite{shaver.99}, \cite{matteo.02},\cite{oh.03}) }
    813934However, any real radio instrument has a beam shape which changes with
    814935frequency: this instrumental effect significantly increases the difficulty and complexity of this component separation
    815936technique. The effect of frequency dependent beam shape is some time referred to as {\em
    816 mode mixing}. See for example \citep{morales.06}, \citep{bowman.07}.
     937mode mixing}.  {\changemark Effect of frequency dependent beam shape for foreground subtraction and
     938its application to MWA has been discussed in \citep{morales.06}  \citep{bowman.09}.}
    817939
    818940In this section, we present a short description of the foreground emissions and
    … …  
    8791001varying spectral index. We assign a power law index $\beta = -2.8  \pm 0.15$ to each sky direction.
    8801002$\beta$ has a gaussian distribution centered at -2.8 and with standard
    881 deviation $\sigma_\beta = 0.15$.
     1003deviation $\sigma_\beta = 0.15$. {\changemark The
     1004diffuse radio background spectral index has been measured  for example by
     1005\citep{platania.98} or \cite{rogers.08} }
    8821006The synchrotron contribution to the sky temperature for each cell is then
    8831007obtained  through the formula:
    … …  
    9061030of the power spectrum $P(k)$ at $z=0$ computed according to the parametrization of
    9071031\citep{eisenhu.98}. We have used the standard cosmological parameters,
    908  $H_0=71 \, \mathrm{km/s/Mpc}$, $\Omega_m=0.27$, $\Omega_b=0.044$,
    909 $\Omega_\lambda=0.73$ and $w=-1$.
     1032 $H_0=71 \, \mathrm{km/s/Mpc}$, $\Omega_m=0.264$, $\Omega_b=0.045$,
     1033$\Omega_\lambda=0.73$ and $w=-1$ \citep{komatsu.11}.
    9101034An inverse FFT was then performed to compute the matter density fluctuations $\delta \rho / \rho$
    9111035in the linear regime,
    … …  
    9661090to the factor $\sim 10^3$ of the sky brightness temperature fluctuations ($\sim$ K),
    9671091compared to the mK LSS signal. 
     1092
     1093{ \changemark Contrary to most similar studies, where it is assumed that bright sources
     1094can be nearly perfectly subtracted, our aim was to compute also their
     1095effect in the foreground subtraction process.
     1096The GSM model lacks the angular resolution needed to compute
     1097correctly the effect of bright compact sources for 21 cm LSS observations and
     1098the mode mixing due to frequency dependent instrument, while Model-II
     1099provides a reasonable description of these compact sources. Our simulated
     1100sky cubes have an angular resolution $3'\times3'$, well below the typical
     1101$15'$ resolution of the instrument configuration considered here.
     1102However, Model-II might lack spatial structures at large scales, above a degree,
     1103compared to Model-I/GSM, and the frequency variations as a simple power law
     1104might not be realistic enough. The differences for the two sky models can be seen
     1105in their power spectra shown in figure  \ref{pkgsmlss}. We hope that by using
     1106these two models, we have explored some of the systematic uncertainties
     1107related to foreground subtraction.}
    9681108
    9691109It should also be noted that in section 3, we presented the different instrument
    … …  
    10571197The virtual target instrument has a beam width larger than the worst real instrument beam,
    10581198i.e at the lowest observed frequency. 
    1059  \item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
     1199\item For each sky direction $(\alpha, \delta)$, a power law $T = T_0 \left( \frac{\nu}{\nu_0} \right)^b$
    10601200 is fitted to the beam-corrected brightness temperature. The fit is done through a linear $\chi^2$ fit in
    10611201the $\lgd ( T ) , \lgd (\nu)$ plane and we show here the results for a pure power law (P1)
    … …  
    10661206\end{eqnarray*}
    10671207where $b$ is the power law index and  $T_0 = 10^a$ is the brightness temperature at the
    1068 reference frequency $\nu_0$:
     1208reference frequency $\nu_0$.
     1209
     1210{\changemark Interferometers have poor response at small $(\uv)$ corresponding to baselines
     1211smaller than interferometer element size. The $(0,0)$ mode, corresponding the mean temperature
     1212can not be measured with an interferometer. We have used a simple trick to make the power law
     1213fitting procedure to work: we have set the mean value of the temperature for
     1214each frequency plane to a power law with an index close to the synchrotron index
     1215and we have checked that results are not too sensitive to the arbitrarily fixed mean temperature
     1216power law parameters. }
     1217
    10691218\item The difference between the beam-corrected sky temperature and the fitted power law
    10701219$(T_0(\alpha, \delta), b(\alpha, \delta))$ is our extracted 21 cm LSS signal.
    … …  
    16081757\bibitem[Abdalla \& Rawlings (2005)]{abdalla.05} Abdalla, F.B. \& Rawlings, S.  2005,  \mnras, 360,  27     
    16091758
     1759% reference DETF - DE eq.state parameter figure of merit
    16101760\bibitem[Albrecht et al. (2006)]{DETF}  Albrecht, A., Bernstein, G., Cahn, R. {\it et al.} (Dark Energy Task Force) 2006, arXiv:astro-ph/0609591
    16111761
     1762% Papier sensibilite/reconstruction CRT (cylindres) ansari et al 2008
    16121763\bibitem[Ansari et al. (2008)]{ansari.08} Ansari R., J.-M. Le Goff, C. Magneville, M. Moniez, N. Palanque-Delabrouille, J. Rich,
    16131764    V. Ruhlmann-Kleider, \& C. Y\`eche , 2008 , arXiv:0807.3614
    … …  
    16251776% Galactic astronomy, emission HI d'une galaxie
    16261777\bibitem[Binney \& Merrifield (1998)]{binney.98} Binney J. \& Merrifield M. , 1998 {\it Galactic Astronomy} Princeton University Press
     1778% 21cm reionisation P(k) estimation and sensitivities
     1779\bibitem[Bowman et al. (2006)]{bowman.06} Bowman, J.D.,  Morales, M.F., Hewitt, J.N. 2006, \apj, 638, 20-26
    16271780% MWA description
    16281781\bibitem[Bowman et al. (2007)]{bowman.07} Bowman, J. D., Barnes, D.G., Briggs, F.H. et al 2007, \aj, 133, 1505-1518 
    16291782
    16301783%% Soustraction avant plans ds MWA
    1631 \bibitem[Bowman et al. (2009)]{bowman.07} Bowman, J. D., Morales, M., Hewitt, J.N., 2009, \apj, 695, 183-199 
     1784\bibitem[Bowman et al. (2009)]{bowman.09} Bowman, J. D., Morales, M., Hewitt, J.N., 2009, \apj, 695, 183-199 
    16321785
    16331786%  Intensity mapping/HSHS
    … …  
    16411794Taylor, G. B., \& Broderick, J. J. 1998, AJ, 115, 1693
    16421795
    1643 %  Parametrisation P(k)
    1644 \bibitem[Eisenstein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, ApJ 496:605-614 (astro-ph/9709112)
     1796% Effet des radio-sources sur le signal 21 cm reionisation
     1797\bibitem[Di Matteo  et al. (2002)]{matteo.02}  Di Matteo, T., Perna R., Abel T., Rees M.J. 2002, \apj, 564, 576-580
     1798
     1799%  Parametrisation P(k)   - (astro-ph/9709112)
     1800\bibitem[Eisenstein \& Hu  (1998)]{eisenhu.98}  Eisenstein D. \& Hu W. 1998, \apj 496, 605-614
    16451801
    16461802% SDSS first BAO observation
    … …  
    16611817\bibitem[Jackson (2004)]{jackson.04} Jackson, C.A. 2004, \na, 48, 1187 
    16621818
     1819% WMAP 7 years cosmological parameters
     1820\bibitem[Komatsu et al. (2011)]{komatsu.11}   E. Komatsu, K. M. Smith, J. Dunkley {\it et al.}  2011, \apjs, 192, p. 18 \\
     1821\mbox{\tt http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm\_sz\_lens\_wmap7.cfm}
     1822
    16631823% HI mass in galaxies
    1664 \bibitem[Lah et al. (2009)]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur et al.  2009,  \mnras, 399, 1447
     1824\bibitem[Lah et al. (2009)]{lah.09}  Philip Lah, Michael B. Pracy, Jayaram N. Chengalur {\it et al.}  2009,  \mnras, 399, 1447
    16651825% ( astro-ph/0907.1416)
    16661826
    … …  
    16721832\bibitem[Larson et al. (2011)]{larson.11}  Larson, D.,  {\it et al.}  (WMAP) 2011, \apjs, 192, 16
    16731833
     1834%% Description MWA
     1835\bibitem[Lonsdale et al. (2009)]{lonsdale.09}  Lonsdale C.J., Cappallo R.J., Morales M.F.  {\it et al.}  2009, arXiv:0903.1828
    16741836% LSST Science book
    16751837\bibitem[LSST.Science]{lsst.science}
    … …  
    16881850\bibitem[Mauskopf et al. (2000)]{mauskopf.00} Mauskopf, P. D., Ade, P. A. R., de Bernardis, P. {\it et al.}  2000, \apjl, 536,59
    16891851
     1852%% PNoise and cosmological parameters with reionization
     1853\bibitem[McQuinn et al. (2006)]{mcquinn.06}  McQuinn M., Zahn O., Zaldarriaga M., Hernquist L.,  Furlanetto S.R.
     18542006, \apj 653, 815-834
     1855
     1856%  Papier sur la mesure de sensibilite P(k)_reionisation
     1857\bibitem[Morales \& Hewitt (2004)]{morales.04}  Morales M. \& Hewitt J., 2004, \apj, 615, 7-18 
     1858
    16901859%  Papier sur le traitement des observations radio / mode mixing
    16911860\bibitem[Morales et al. (2006)]{morales.06} Morales, M., Bowman, J.D., Hewitt, J.N., 2006, \apj, 648, 767-773
     1861
     1862%% Foreground removal using smooth frequency dependence
     1863\bibitem[Oh \& Mack (2003)]{oh.03} Oh S.P. \& Mack K.J., 2003, \mnras, 346, 871-877
    16921864
    16931865%  Global Sky Model Paper
    … …  
    16951867\mnras, 388, 247-260
    16961868
     1869%%  Description+ resultats PAPER
     1870\bibitem[Parsons et al. (2009)]{parsons.09}  Parsons A.R.,Backer D.C.,Bradley  R.F. {\it et al.}  2009, arXiv:0904.2334
     1871
     1872% Synchrotron index =-2.8 in the freq range 1.4-7.5 GHz
     1873\bibitem[Platania et al. (1998)]{platania.98}   Platania P., Bensadoun M., Bersanelli M. {\it al.} 1998,  \apj 505, 473-483
     1874
    16971875% Original CRT HSHS paper (Moriond Cosmo 2006 Proceedings)
    16981876\bibitem[Peterson et al. (2006)]{peterson.06} Peterson, J.B.,  Bandura, K., \& Pen, U.-L. 2006, arXiv:0606104 
    … …  
    17031881% SDSS BAO 2010  - arXiv:0907.1660
    17041882\bibitem[Percival et al. (2010)]{percival.10}   Percival, W.J., Reid, B.A., Eisenstein, D.J. {\it et al.},  2010, \mnras, 401, 2148-2168   
     1883
     1884% Radio spectral index between 100-200 MHz
     1885\bibitem[Rogers \& Bowman (2008)]{rogers.08} Rogers, A.E.E. \& Bowman, J. D. 2008, \aj 136, 641-648
    17051886
    17061887%% LOFAR description
    … …  
    17101891%% SDSS-3
    17111892\bibitem[SDSS-III(2008)]{sdss3} SDSS-III 2008, http://www.sdss3.org/collaboration/description.pdf
     1893
     1894% Reionisation: Can the reionization epoch be detected as a global signature in the cosmic background?
     1895\bibitem[Shaver  et al. (1999))]{shaver.99} Shaver P.A., Windhorst R. A., Madau P., de Bruyn A.G. \aap, 345, 380-390
    17121896
    17131897%  Frank H. Briggs, Matthew Colless, Roberto De Propris, Shaun Ferris, Brian P. Schmidt, Bradley E. Tucker
    … …  
    17201904% Papier 21cm-BAO Fermilab  ( arXiv:0910.5007)
    17211905\bibitem[Seo et al (2010)]{seo.10} Seo, H.J. Dodelson, S., Marriner, J. et al,  2010, \apj, 721, 164-173
     1906
     1907% Mesure P(k) par SDSS
     1908\bibitem[Tegmark et al.  (2004)]{tegmark.04}  Tegmark M., Blanton  M.R, Strauss M.A. et al. 2004, \apj, 606, 702-740
    17221909
    17231910% FFT telescope

  • trunk/Cosmo/RadioBeam/specpk.cc

    r4028 r4030  
    209209
    210210// Generate mass field Fourier Coefficient
    211 void Four3DPk::ComputeNoiseFourierAmp(Four2DResponse& resp, double f0, double df, Vector& angscales)
     211void Four3DPk::ComputeNoiseFourierAmp(Four2DResponse& resp, double f0, double df, Vector& angscales, Vector& noisevec)
    212212// angscale is a multiplicative factor converting transverse k (wave number) values to angular wave numbers
    213213// typically = ComovRadialDistance
    214214{
    215215  uint_8 nmodeok=0;
    216   if (angscales.Size() != fourAmp.SizeZ())
    217     throw SzMismatchError("ComputeNoiseFourierAmp(): angscales.Size()!=fourAmp.SizeZ()");
     216  if ((angscales.Size() != fourAmp.SizeZ())||(noisevec.Size() != fourAmp.SizeZ()))
     217    throw SzMismatchError("ComputeNoiseFourierAmp(): angscales/noisevec.Size()!=fourAmp.SizeZ()");
    218218  H21Conversions conv;
    219219  // fourAmp represent 3-D fourier transform of a real input array.
    … …  
    225225    resp.setLambda(conv.getLambda());
    226226    double angsc=angscales(kz);
     227    double noisepow=noisevec(kz);
    227228    if (prtlev_>2) 
    228       cout << " Four3DPk::ComputeNoiseFourierAmp(...) - freq=" << f0+kz*df << " -> AngSc=" << angsc << endl;
     229      cout << " Four3DPk::ComputeNoiseFourierAmp(...) - freq=" << f0+kz*df << " -> AngSc=" << angsc
     230           << " NoisePow=" << noisepow << endl;
    229231    for(sa_size_t ky=0; ky<fourAmp.SizeY(); ky++) {
    230232      kyy =  (ky>fourAmp.SizeY()/2) ? -(double)(fourAmp.SizeY()-ky)*dky_ : (double)ky*dky_;
    … …  
    234236        if (rep<1.e-19)  rep=1.e-19;
    235237        else  nmodeok++;
    236         amp = 1./sqrt(rep)/sqrt(2.);
     238        amp = sqrt(noisepow/rep/2.);
    237239        fourAmp(kx, ky, kz) = complex<TF>(rg_.Gaussian(amp), rg_.Gaussian(amp));   
    238240      }
    … …  
    445447PkNoiseCalculator::PkNoiseCalculator(Four3DPk& pk3, Four2DResponse& rep, double s2cut, int ngen,
    446448                                     const char* tit)
    447   : pkn3d(pk3), frep(rep), S2CUT(s2cut), NGEN(ngen), title(tit), angscales_(pk3.SizeZ())
     449  : pkn3d(pk3), frep(rep), S2CUT(s2cut), NGEN(ngen), title(tit), angscales_(pk3.SizeZ()), pnoisefac_(pk3.SizeZ())
    448450{
    449451  SetFreqRange();
    450452  SetAngScaleConversion();
     453  SetPNoiseFactor();
    451454  SetPrtLevel();
    452455}
    … …  
    461464       << " ... " << angscales_(angscales_.Size()-1) << endl;
    462465  for(int igen=0; igen<NGEN; igen++) {
    463     pkn3d.ComputeNoiseFourierAmp(frep, freq0_, dfreq_, angscales_);
     466    pkn3d.ComputeNoiseFourierAmp(frep, freq0_, dfreq_, angscales_, pnoisefac_);
    464467    if (igen==0) hnd = pkn3d.ComputePk(S2CUT,nbin,kmin,kmax,true);
    465468    else pkn3d.ComputePkCumul();

  • trunk/Cosmo/RadioBeam/specpk.h

    r4027 r4030  
    5656// typically = ComovRadialDistance
    5757  void ComputeNoiseFourierAmp(Four2DResponse& resp, double angscale=1., bool crmask=false);
    58   void ComputeNoiseFourierAmp(Four2DResponse& resp, double f0, double df, Vector& angscales);
     58  void ComputeNoiseFourierAmp(Four2DResponse& resp, double f0, double df, Vector& angscales, Vector& noisp);
    5959
    6060// Return the array size
    … …  
    112112  inline void SetAngScaleConversion(Vector& angscs)
    113113  { angscales_=angscs; }
     114  inline void SetPNoiseFactor(double pnoisef=1.)
     115  { pnoisefac_=pnoisef; }
     116  inline void SetPNoiseFactor(Vector& pnoisefac)
     117  { pnoisefac_=pnoisefac; }
    114118  inline void SetS2Cut(double s2cut=100.)
    115119  {  S2CUT=s2cut; }
    … …  
    124128  double freq0_,dfreq_;
    125129  Vector angscales_;
     130  Vector pnoisefac_;
    126131  double S2CUT;
    127132  int NGEN;
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