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Low_Xray_Luminosity_.tex sectionIntroduction_.tex sectionObservational.tex sectionWeak_Lensing_.tex Weak Lensing Analysis:.tex sectionClusters_Dark.tex subsectioncluster_c2.tex

\section{Weak Lensing Measure in simulated catalogs:} \label{sec:method} We perform a pipeline based on python language to make the lensing analysis. Its detects and classifies the sources, determines the \textit{Point Spread Function} (henceforth PSF) at the position of every object, measures the shape of the galaxies accounting for the PSF deformation, selects the background galaxies and computes the shear profile. \\ In order to check the pipeline funcionality, we apply it to the \textit{DES Cluster Simulation} images, publically available (Gill et al. 2009).\\ The steps in the weak lensing analysis and the results of the aplication of it to the simulated data, are described in the next subsections. % \subsection{Object detection and classification} % We used SExtractor (Bertin \& Arnouts, 1996) for the detection and fotometry of the sources, in a two-pass mode. A first run is made to detect bright objects, with a detection level of 5$ \sigma $ above the background, in order to estimate the seeing and the saturation level of each image. The seeing is estimated using the average of the FWHM parameter of the point-like objects, selected from the FWHM/MAG\_BEST diagram since for these objects the FWHM is independent of the magnitude (\textbf{se entiende, o les parece mejor agregar el grafico?)}). Determining the seeing is important for SExtractor to perform the star-galaxy classification. The saturation level is estimated as 0.8 times the maximum value of the FLUX\_MAX parameter. These parameters, \textit{seeing} and \textit{saturation level}, are taking into account in the SExtractor configuration file for the second run, with a lower threshold detection limit of 1.5$ \sigma $. Second run is made in dual mode, detecting objects on the \textit{r'} image, while astrometric and photometric parameters are measured on all individual images.\\ % \begin{figure} \centering \includegraphics[width=.5\textwidth]{./../plots/new-ones/mu_mag.eps}~\\ \includegraphics[width=.5\textwidth]{./../plots/new-ones/mag_fwhm.eps} \caption{Objects detected in the galaxy cluster [VMF98]102 \textit{r'-}band image, stars (\textit{blue points}), galaxies (\textit{green points}), and artifacts (\textit{black points}). In the MU\_MAX/MAG\_BEST plane (\textit{left}), stars are in the region marked by the red solid line $\pm$ 0.4 magnitudes. To check the selection, we plot the sources in the MAG\_BEST/FWHM plane (\textit{left}), where point like object selected as stars have almost the same FWHM despite the magnitude.} \label{sources} \end{figure} % For the object classification in stars, galaxies and false detections, we considered a similar analysis as Bardeau\,et\,al.\,2005, taking into account the position of the source in the magnitude/central flux diagram, the FWHM respect to the seeing and the stellarity index, according to the CLASS\_STAR parameter. Objects that are more sharply peaked than the PSF (FWHM $<$ \textit{seeing} - 0.5 pixel) and with FLAG parameter $>$ 4, are considered as false detections. As the light distribution of a point source scales with magnitude, objects on the line magnitude/central flux, $\pm$ 0.4 magnitudes (Figure\,\ref{sources}), FWHM $<$ \textit{seeing} + 1 pixel and CLASS\_STAR $>$ 0.8 are considered as stars. The rest of the objects are considered as galaxies. \\ % \subsection{Shape measurements} % For the shape measurements we used IM2SHAPE (Bridle et al. 2002). Its computes the shapes parameters modeling the object as sums of Gaussians, convolved with a PSF which is also a sum of Gaussians. For simplicity, both the PSF and the object are modeled with a single elliptical Gaussian profile. \\ The PSF was determined measuring the shapes of the stars as they are intrinsically pointlike objects. Once we obtained the shape of the stars, we clean the catalogue removing those objects with ellipticity greater than 0.2, which mainly appear to be false detections or faint galaxies, and by looking at the 5 nearest stars at each star position and removing those that differ by more than 2$\sigma$ from the local average shape. \\ After cleaning the catalogue, we linearly interpolate the local PSF at each object position by averaging the shapes of the five closest stars. We checked that our Im2shape implementation can recover point-like objects by applying this PSF correction to each star. Figure\,\ref{PSF} shows the results of the shape parameters measurements for these stars, with and wihtout taking into account the PSF in the shape measurement: the size distribution is dominated by point sources, and the orientation is more uniformly distributed after the PSF correction.\\ \begin{figure} \centering \includegraphics[width=.25\textwidth]{./../plots/psf/ab.eps}~\hfill \includegraphics[width=.25\textwidth]{./../plots/psf/ab_correc.eps} \includegraphics[width=.25\textwidth]{./../plots/psf/map.eps}~\hfill \includegraphics[width=.25\textwidth]{./../plots/psf/map_correc.eps} \includegraphics[width=.25\textwidth]{./../plots/psf/e1.eps}~\hfill \includegraphics[width=.25\textwidth]{./../plots/psf/e1_correc.eps} \includegraphics[width=.25\textwidth]{./../plots/psf/e2.eps}~\hfill \includegraphics[width=.25\textwidth]{./../plots/psf/e2_correc.eps} \includegraphics[width=.25\textwidth]{./../plots/psf/Theta.eps}~\hfill \includegraphics[width=.25\textwidth]{./../plots/psf/Theta_correc.eps} \caption{PSF treatment applied to stars of one of the images of the DES simulations: Semiaxis ($a$ cos $\theta$, $a$ sin $\theta$) before (\textit{left}) and after (\textit{right}) the PSF deconvolution, which is more randomly distributed and considerably smaller (First row at top left corner indicates the scale).} \label{PSF} \end{figure} \subsection{Shear radial profiles} Gravitational lensing maps the unlensed image, specified by coordinates $(x,y)$, to the lensed image $(x',y')$ using a matrix transformation: \\ \begin{equation*} \begin{pmatrix} x\\ y \end{pmatrix} \begin{pmatrix} 1-g_{1} & -g_{2} \\ -g_{2} & 1+g_{1} \end{pmatrix} = \begin{pmatrix} x' \\ y' \\ \end{pmatrix} \end{equation*} \\ where $g_{1}$ and $g_{2}$ are the components of the reduced shear: \begin{equation*} g=\dfrac{\gamma}{1-\kappa} \end{equation*} If lensing is weak, the image of a circular source with ratio \textit{r}, appears elliptical, with axis given by \begin{equation*} a=\dfrac{r}{1-\kappa-\gamma} , b=\dfrac{r}{1-\kappa+\gamma} \end{equation*} Defining the ellipticity as \begin{equation*} e=\dfrac{a-b}{a+b}=\dfrac{\gamma}{1-\kappa}\approx\gamma \end{equation*} where \textit{g} becomes the normal shear, $\gamma$, since $\kappa \ll 1$, which generally holds in the weak lensing regime for clusters, and will be assumed henceforth here.\\ If the source has an intrinsic ellipticity $e_{s}$, the observed ellipticity in the weak lensing limit will be: \begin{equation*} e=e_{s}+\gamma \end{equation*} Assuming that unlensed galaxies are randomly oriented on the sky plane ($\langle e_{s} \rangle = 0$ ) and averaging over sufficiently many sources: \begin{equation*} \langle e \rangle=\langle \gamma \rangle \end{equation*} Hence, in the weaklensing approximation, we get an unbiased estimator of the reduced shear by averaging the shape of background galaxies in concentric annuli around the cluster center. Spherical symmetry also implies that the tangential component (E-mode) of the lensed galaxies traces the reduced shear, while the average of the component tilted at $\pi/4$ relative to the tangential component is the 'B-mode' and should be exactly zero for the case of perfect symmetry (see e.g. Sec. 4 of Bartelmann \& Schneider 2001 or Bernstein \& Jarvis 2002).\\ Because of the random orientation of the galaxies in the source plane, the error in the observed galaxy ellipticities and thus, on the estimated shear, will depend on the number of galaxies averaged together to measure the shear (Schneider et al. 2000). The error can be estimated as: \begin{equation*} \sigma_{\gamma}\approx\dfrac{\sigma_{\epsilon}}{\sqrt{N}} \end{equation*} where $\sigma_{\epsilon}$ is the dispersion of the intrinsic ellipticity distribution ($\sigma_{\epsilon} \approx 0.3$) and $N$ is the number of objects in the annular bin. \subsection{Background Galaxies selection and redshift distribution} % Background galaxies for the shear estimation, were selected as the galaxies with magnitude in the filter \textit{r'}, $m_{r}$, higher than $m_{L}$, and lower than $m_{max} + 1.0$, where $m_{L}$ is defined as the lowest magnitude $r'$ such that the probability that the galaxy is behind the cluster is higher than 0.7 and $m_{max}$ correspond to the peak of the magnitude distribution of the image in the filter \textit{r'} (both magnitudes for each cluster are listed in Table\,\ref{table:2}). The later cut in magnitude ensures that we are not taking into account galaxies that are too faint, given that they could have great errors in the shape measurements. Also, as we did for the simulated data, we discard galaxies with FWHM $<$ 5, with companions closer than 16 pixels and with $\sigma_{e} > 0.2$.\\ To compute $m_{L}$ we used the catalogue of photometric redshifts computed by Coupon et al. 2009, based on the public release Deep Field 1 of the Canada-France-Hawaii Telescope Legacy Survey. We estimated the fraction of galaxies with $z > z_{cluster}$ in magnitude bins for the \textit{r'} filter, and then we chose $m_{L}$ as the lowest magnitude for which the fraction of galaxies was greater than 0.7. Background galaxy density after the selection are listed for each cluster in Table 2.\\ Once we obtained a catalogue for the background galaxies, we average the two components of the ellipticities (E-mode and B-mode) in nonoverlapping logarithmic annuli, to conserve the S/N ratio in each bin. In order to take into account the contamination of foreground galaxies in the catalogue, we weighted the average value of the components of the galaxie's ellipticity with the probability that the galaxy was behind the cluster. We compute this probability using Coupon's catalogue, from the fraction of galaxies with $z > z_{cluster}$ for each bin in magnitude, \textit{r'}, and color (\textit{g' - r'} and \textit{r' - i'}) - Figure\,\ref{weigh}. Hence, given the magnitude and the color of each galaxy, we assigned to it a weigh, \textit{w}, as the fraction of galaxies with $z > z_{cluster}$ in that bin. \\ \begin{figure} \centering \includegraphics[scale=0.35]{./../plots/new-ones/distrib_z05.ps}\\ \caption{Fraction of galaxies with $z > 0.5$ ($n(z>0.5)/N_{tot}$), for different magnitudes in filter \textit{r'} and colors \textit{r'-i'}.} \label{weigh} \end{figure} The average E-mode components correspond to the shear value which depends on the geometrical factor $ \beta = D_{LS}/ D_{L} $, where $D_{LS}$ is the angular diameter distance from the lens to the background source galaxy, and $D_{L}$ is the distance from the observer to the background galaxy. A galaxy at the same radial distance from the center of the cluster but at a different background redshift is sheared differently. To remove this variation, we divide the estimated shear by $\beta$, which results in the infinite redshift limit (z $\rightarrow \infty$) shear value, that is only function of the radial distance from the cluster center. To estimate $\beta$ we used once more Coupon's catalogue, since we can consider that this catalogue is complete up to our magnitude limit. We applied the photometric selection criteria to the catalogue and then we computed $\beta$ for the whole distribution of field galaxies, taking into account the contamination by foreground galaxies given our selection criteria by setting $\beta(z_{phot} < z_{cluster}) = 0$. The averaged geometrical factors for each cluster are given in Table\,\ref{table:2}. \subsection{Fiting the profiles} We finally estimate the cluster mass fitting the shear data with the singular isothermal sphere (SIS) and the NFW profile (Navarro et al. 1997). We will explain briefly the lensing formulae for these two profiles: \subsubsection{SIS profile} The SIS mass model is the simplest one for describing a relaxed massive sphere with a constant and isotropic velocity dispersion, $\sigma_{V}$. This is mainly described by the density distribution: \begin{equation*} \rho(r) = \dfrac{\sigma_{V}^{2}}{2 \pi G r^{2}} \end{equation*} where $c$ is the speed of light and \textit{G} is the gravitational constant. From this ecuation, we can get the critical Einstein radius for source at $z \rightarrow \infty $ ($\beta =1$) as: \begin{equation*} \theta_{E} = \dfrac{4 \pi \sigma_{V}^{2}}{c^{2}} \end{equation*} In terms of wich one obtains: \begin{equation*} \kappa_{\theta} = \gamma_{\theta} = \dfrac{\theta_{E}}{2 \theta} \end{equation*} where $\theta$ is the distance to the cluster center. Hence, fitting the shear for different radius, we can estimate the Einstein radius, and from that, we can obtain an estimation of the mass within a radius $\theta$ as: \begin{equation*} M(\theta) = \dfrac{\theta_{E} c }{4G} \theta \end{equation*} \subsubsection{NFW profile} The NFW profile is derived from fitting the density profile of numerical simulations of cold dark matter halos (Navarro et al. 1995, 1997). This profiles depend on two parameters, the virial radius, $R_{200}$, and a dimensionless concentration parameter: \begin{equation*} \rho(r) = \dfrac{\rho_{c} \delta_{c}}{(r/r_{s})(1+r/r_{s})^{2}} \end{equation*} where $r_{s}$ is the scale radius, $r_{s} = R_{200}/c$ and $\delta_{c}$ is the characteristic overdensity of the halo, \begin{equation*} \delta_{c} = \frac{200}{3} \dfrac{c^{3}}{\ln(1+c)+c/(1+c)} \end{equation*} The properties of the NFW profile in the context of gravitational lensing have been discussed by many authors (Bartelmann 1996, Wright \& Brainerd 2000). The radial dependence of the shear as function of $x:=\theta / r_{s}$ is given by: \begin{equation*} \gamma(x) = \dfrac{2r_{s}\delta_{c}\rho_{c}}{\Sigma_{c}} j(x) \end{equation*} where $\Sigma_{c}$ is the \textit{critical surface mass density} and $j(x)$ is given by: \begin{eqnarray*} $j(x < 1) & = & \dfrac{4 \arctanh \sqrt{\frac{1-x}{1+x}}}{x^{2} \sqrt{1-x^{2}}} + \dfrac{2 \ln(\frac{x}{2})}{x^{2}} - \dfrac{1}{x^{2}-1} \\ & &+ \dfrac{2 \arctanh \sqrt{\frac{1-x}{1+x}}}{(x^{2}-1) \sqrt{1-x^{2}}} \\ $ \end{eqnarray*} \begin{eqnarray*} $j(x = 1) = 2 \ln \left(\dfrac{1}{2} \right) + \dfrac{5}{3}$\\ $j(x < 1) = \dfrac{4 \arctan \sqrt{\frac{1-x}{1+x}}}{x^{2} \sqrt{1-x^{2}}} + \dfrac{2 \ln(\frac{x}{2})}{x^{2}} - \dfrac{1}{x^{2}-1} + \dfrac{2 \arctan \sqrt{\frac{1-x}{1+x}}}{(x^{2}-1)^{3/2}}} $ \end{eqnarray*} If we fit the shear for different radius we can have an estimation of the parameters $c$ and $R_{200}$. Once we obtain these parameters we can compute the mass within a radius $\theta$ as: \begin{equation*} M(\theta) = \int^{\theta}_0 2\pi \theta \Sigma(\theta) d\theta = \int^{\theta}_0 4\pi \theta \rho_{c} \delta_{c} r_{s} F(x) d\theta \end{equation*} where, \begin{eqnarray*} $F(x < 1) & = & \dfrac{1}{x^{2}-1} (1 - \dfrac{1}{\sqrt{1-x^{2}}} \arcosh \dfrac{1}{x} )$ $F(x = 1) = \dfrac{1}{3}$ \\ $F(x > 1) & = & \dfrac{1}{x^{2}-1} (1 - \dfrac{1}{\sqrt{x^{2}-1}} \arccos \dfrac{1}{x} )$ \end{eqnarray*} There is a well-known degeneracy between the parameters $R_{200}$ and $c$ when fitting the shear profile in the weak lensing regime. This is due to the lack of information on the mass distribution near the cluster center and only a combination of strong and weak lensing can raise it and provide useful constraints on the concentration parameter. Since we do not have strong lensing modeling of the clusters in the sample, we decided to fix the concentration parameter, $c = 5$, and fit the mass profile with only one free parameter, $R_{200}$. \subsection{Testing the pipeline with simulated data} % To check the performance of our weak lensing analysis pipeline, we tested it on the DES cluster simulation images publically available \citep{2009arXiv0909.3856G}, with good results. This simulation consists of a sets of images, with different grades of difficulty, of sheared galaxies according to a SIS profile with a velocity dispersion of 1250\,km\,s$^{-1}$.\\ % \begin{figure}{h!} \centering \includegraphics[scale=0.4]{./../plots/des/DES4_shearprofile.eps}\\ \includegraphics[scale=0.4]{./../plots/des/DES2_shearprofile.eps}\\ \includegraphics[scale=0.4]{./../plots/des/DES3_shearprofile.eps}\\ \caption{Shear blabla des.} \label{DES} \end{figure} We applied our pipeline to three of the image files available, High Noise File, High Noise PSF Applied File and Low Noise PSF Applied File. For the PSF Applied files, we checked that our Im2shape implementation can recover point-like objects by applying the PSF correction to each star. The images contain only the sheared galaxies, hence all the galaxies detected were considered as background galaxies at $z=0.8$, which is the average of the redshifts of the galaxies. We cut the catalogue discarding the galaxies with FWHM $<$ 5, with companions closer than 16 pixels and with $\sigma_{e} > 0.2$, where $\sigma_{e}$ is defined as the quadratic sum of the errors $\sigma_{e1}$ and $\sigma_{e2}$ given by IM2SHAPE. These cuts are made in order to consider, for the shear measurements, insulated galaxies with good shape measurements. Shear profiles are shown in Figure\,\ref{DES}. For the most complex image that we treated, we obtained a deviation parameter of 1.6, defined as the number of $\sigma$ that the result is away from the input value of $\sigma_{V} = 1250 km/sec$, i.e. $\sigma = \dfrac{result - input}{error}$ . \begin{table*} \centering \caption{Summary of the weak lensing analysis}\label{tab:esp} \label{table:2} \begin{tabular}{@{}ccccccccc@{}} \hline \hline \rule{0pt}{1.05em}% [VMF\,98] & Density of background galaxies & $m_{L}$ & $m_{max}$ & $\langle\beta\rangle$ & $\sigma_{V}$ & M$^{(1)}$($r < 0.5 Mpc$) & $R_{200}$ & M$^{(2)}$($r < 0.5 Mpc$) \\ Id. & (galaxies/arcmin$^{2}$) & & & & (km/sec) & $10^{14} M_{\odot} h_{70}^{-1}$ & Mpc & $10^{14} M_{\odot} h_{70}^{-1}$ \\ \hline \rule{0pt}{1.05em}% 001 & 33 & 23.0 & 27.0 & 0.42 & - & - & - & - \\ 022 & 15 & 20.7 & 26.0 & 0.63 & 600 $\pm$ 30 & 1.3 & 1.7 $\pm$ 0.2 & 1.2 \\ 093 & 9 & 22.3 & 25.0 & 0.50 & - & - & - & - \\ 097 & 31 & 23.0 & 27.0 & 0.44 & 580 $\pm$ 40 & 1.2 & 1.2 $\pm$ 0.4 & 0.83 \\ 102 & 26 & 22.7 & 26.9 & 0.50 & 450 $\pm$ 40 & 0.75 & 1.1 $\pm$ 0.1 & 0.62 \\ 119 & 18 & 24.0 $^{(3)}$ & 26.4 & 0.30 & 860 $\pm$ 30 & 2.7 & 1.4 $\pm$ 0.1 & 1.4 \\ 124 & 26 & 19.5 & 26.7 & 0.72 & - & - & - & - \\ 148 & 19 & 24.0 $^{(3)}$ & 26.9 & 0.27 & 860 $\pm$ 140 & 2.7 & 1.4 $\pm$ 0.7 & 1.4 \\ \hline \end{tabular} \\ \begin{tablenotes}{\begin{footnotesize} \end{footnotesize} \begin{small} \begin{flushleft} \item [] (1) From the fit of the SIS profile \item [] (2) From the fit of the NFW profile \item [] (3) Here $m_{L}$ is defined as the lowest magnitude $r'$ such that the probability that the galaxy is behind the cluster is higher than 0.6} \end{flushleft} \end{small} \end{tablenote} \end{table*}