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stat_sec.tex
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The expected statistical errors on the stacked cluster weak lensing
measurement for DES are dominated by the density of sources, and the
number of clusters present in each mass and redshift bin. The statistical error we model
combines the scatter due to shape noise in background galaxies, and the
scatter due to cluster halos not being spherical mass distributions as
described in \citep{obscos, mbecker}. To estimate the number of background galaxies per arcmin$^2$ ($\overline{ngal}$)
and their mean redshift for a survey like DES we use a redshift
distribution of galaxies
\begin{equation}
f(z) = z^m exp(-( z/z_* )^{\beta})
\end{equation}
where $m=2.0 $, $z_*=0.5$ and $\beta = 2.0 $ as given in
\citep{obscos}. To determine the number of background sources at a
given redshift the background fraction of galaxies is
\begin{equation}
F_{bg} = \frac{\int_z^{\infty} dz' f(z')}{\int_0^{\infty} dz' f(z')}
\end{equation}
which is then used to determine the number of galaxies in sq. arcmin.
\begin{equation}
\overline{ngal} = 10 F_{bg}(z)
\end{equation}
for the DES survey \citep{edr}. The average redshift of sources
behind the lens for each bin is shown in \ref{tab:state}
\indent
The statistical error or mass uncertainty $\Delta ln M $ as
described in \citet{obscos} for stacked weak lensing is :
\begin{equation}
\Delta ln M = \sqrt{ (\Delta ln M_{shape})^2 +
(\sigma_{wl} )^2 }
\end{equation}
which combines the error due to the shape noise and the error due to
scatter based on clusters not being spherical. From \citep{mbecker}
we take
\begin{equation}
\sigma_{wl} = \frac{0.3}{\sqrt{N}}
\end{equation}
where N is the number of clusters in a given bin. To calculate the
shape noise we use the equation
\begin{equation}
\Delta ln M_{shape} \approx 6.0*10^3
(\frac{N}{4000})^{1/2}(\frac{\sigma_e}{0.3})(\frac{M}{2*10^{14}
M_{\odot}})^{-2/3} (\frac{\overline{ngal}}{30
arcmin^{-2}})^{-1/2}(\frac{D_{ls}/D_{s}}{0.5})^{-1},
\end{equation}
from \citep{obscos} with $\sigma_e = 0.4$, $\overline{ngal} = 5.72$
and $D_{ls}/D_{s} = 0.27$ which yields the statistical limits shown in
Table \ref{table:Eduardoerror}. To model the concentration we expect
for clusters at this redshift we assign an initial concentration c
\begin{equation}
c= A*( m_{200}/(2.0*1.e12))^{B}*(1 + z_{\rm{cluster}})^{C}
\end{equation}
Where $ A = 7.85 $ , $ B = -0.081 $ , $ C= -0.71 $ and $m_{200}$ is
the mass within $r_{200}$ from \citep{oguri}.
\begin{table*}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
N & Z lens & Z source & Dls/Ds & Mvir & C & M200m & C & M200c & C & $\Delta ln M$ \\
449 & 0.16 & 0.58 & 0.69 & 1.08 $10^{14}$ & 5.12 & 1.21 $10^{14}$ & 5.98 & 0.898 $10^{14}$ & 4.0 & 0.044 \\
\hline
228 & 0.16 & 0.58 & 0.69 & 1.86 $10^{14}$ & 4.91 & 2.07 $10^{14}$ & 5.73 & 1.53 $10^{14}$ & 3.82 & 0.046 \\
\hline
54 & 0.16 & 0.58 & 0.69 & 3.60 $10^{14}$ & 4.66 & 4.05 $10^{14}$ & 5.46 & 2.96 $10^{14}$ & 3.62 & 0.068 \\
\hline
5 & 0.17 & 0.58 & 0.69 & 7.51 $10^{14}$ & 4.35 & 8.45 $10^{14}$ & 5.08 & 6.15 $10^{14}$ & 3.39 & 0.173 \\
\hline
1079 & 0.26 & 0.62 & 0.54 & 1.06 $10^{14}$ & 4.85 & 1.16 $10^{14}$ & 5.5 & 0.898 $10^{14}$ & 3.89 & 0.037 \\
\hline
528 & 0.25 & 0.62 & 0.54 & 1.87 $10^{14}$ & 4.63 & 2.06 $10^{14}$ & 5.26 & 1.57 $10^{14}$ & 3.71 & 0.038 \\
\hline
109 & 0.26 & 0.62 & 0.54 & 3.60 $10^{14}$ & 4.39 & 3.95 $10^{14}$ & 4.97 & 3.0 $10^{14}$ & 3.51 & 0.058 \\
\hline
10 & 0.26 & 0.62 & 0.54 & 6.83 $10^{14}$ & 4.17 & 7.54 $10^{14}$ & 4.73 & 5.7 $10^{14}$ & 3.33 & 0.144 \\
\hline
1744 & 0.35 & 0.66 & 0.42 & 1.05 $10^{14}$ & 4.60 & 1.13 $10^{14}$ & 5.09 & 0.902 $10^{14}$ & 3.77 & 0.040 \\
\hline
775 & 0.35 & 0.66 & 0.42 & 1.83 $10^{14}$ & 4.40 & 1.98 $10^{14}$ & 4.86 & 1.57 $10^{14}$ & 3.6 & 0.042 \\
\hline
120 & 0.35 & 0.66 & 0.42 & 3.64 $10^{14}$ & 4.17 & 3.94 $10^{14}$ & 4.62 & 3.1 $10^{14}$ & 3.41 & 0.071 \\
\hline
13 & 0.36 & 0.66 & 0.42 & 7.30 $10^{14}$ & 3.92 & 7.92 $10^{14}$ & 4.34 & 6.19 $10^{14}$ & 3.2 & 0.147 \\
\hline
2395 & 0.45 & 0.73 & 0.34 & 1.04 $10^{14}$ & 4.38 & 1.11 $10^{14}$ & 4.74 & 0.906 $10^{14}$ & 3.67 & 0.046 \\
\hline
990 & 0.45 & 0.73 & 0.34 & 1.85 $10^{14}$ & 4.18 & 1.97 $10^{14}$ & 4.52 & 1.61 $10^{14}$ & 3.5 & 0.049 \\
\hline
150 & 0.45 & 0.73 & 0.34 & 3.53 $10^{14}$ & 3.97 & 3.77 $10^{14}$ & 4.3 & 3.05 $10^{14}$ & 3.32 & 0.084 \\
\hline
16 & 0.46 & 0.73 & 0.34 & 6.81 $10^{14}$ & 3.75 & 7.26 $10^{14}$ & 4.06 & 5.84 $10^{14}$ & 3.13 & 0.173 \\
\hline
2753 & 0.55 & 0.79 & 0.26 & 1.02 $10^{14}$ & 4.19 & 1.071 $10^{14}$ & 4.44 & 0.899 $10^{14}$ & 3.56 & 0.065 \\
\hline
985 & 0.55 & 0.79 & 0.26 & 1.84 $10^{14}$ & 4.00 & 1.93 $10^{14}$ & 4.25 & 1.62 $10^{14}$ & 3.4 & 0.073 \\
\hline
138 & 0.55 & 0.79 & 0.26 & 3.52 $10^{14}$ & 3.80 & 3.71 $10^{14}$ & 4.04 & 3.08 $10^{14}$ & 3.23 & 0.128 \\
\hline
13 & 0.54 & 0.79 & 0.26 & 6.73 $10^{14}$ & 3.61 & 7.09 $10^{14}$ & 3.85 & 5.84 $10^{14}$ & 3.06 & 0.278 \\
\hline
3053 & 0.65 & 0.87 & 0.21 & 1.01 $10^{14}$ & 4.01 & 1.05 $10^{14}$ & 4.19 & 0.89 $10^{14}$ & 3.46 & 0.091 \\
\hline
915 & 0.65 & 0.87 & 0.21 & 1.82 $10^{14}$ & 3.82 & 1.89 $10^{14}$ & 4.0 & 1.61 $10^{14}$ & 3.3 & 0.112 \\
\hline
121 & 0.65 & 0.87 & 0.21 & 3.52 $10^{14}$ & 3.63 & 3.66 $10^{14}$ & 3.8 & 3.11 $10^{14}$ & 3.13 & 0.200 \\
\hline
8 & 0.65 & 0.87 & 0.21 & 6.97 $10^{14}$ & 3.43 & 7.25 $10^{14}$ & 3.6 & 6.13 $10^{14}$ & 2.96 & 0.499 \\
\hline
2971 & 0.75 & 0.95 & 0.17 & 1.00 $10^{14}$ & 3.85 & 1.03 $10^{14}$ & 3.98 & 0.894 $10^{14}$ & 3.37 & 0.142 \\
\hline
793 & 0.75 & 0.95 & 0.17 & 1.82 $10^{14}$ & 3.67 & 1.87 $10^{14}$ & 3.79 & 1.63 $10^{14}$ & 3.2 & 0.184 \\
\hline
99 & 0.75 & 0.95 & 0.17 & 3.49 $10^{14}$ & 3.48 & 3.59 $10^{14}$ & 3.59 & 3.11 $10^{14}$ & 3.04 & 0.338 \\
\hline
5 & 0.75 & 0.95 & 0.17 & 7.13 $10^{14}$ & 3.28 & 7.34 $10^{14}$ & 3.39 & 6.32 $10^{14}$ & 2.85 & 0.942 \\
\hline
\end{tabular}
\caption{ The statistical error or mass
uncertainty $\Delta ln(M)$ as described in Weinberg et
al. (2012), for each stacked weak lensing bin modeled for the DES survey.}
\label{table:state}
\end{table*}