SIMULATION OF A COMBINED SZE AND WEAK LENSING CLUSTER SURVEY FOR AMIBA EXPERIMENT
\footnotesize KEIICHI UMETSU %\footnote{ %Typeset names in 8-pt Times Roman, uppercase. Use the footnote to %indicate the present or permanent address of the author.}
%We explore the expected number counts of galaxy clusters %We have simulated the forth-coming %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %We have made a quantitative study % studied %of halo number counts % the detectability of galaxy clusters %in a combined interferometiric Sunyaev-Zel'dovich %effect (SZE) and optical weak lensing survey for the forth-coming %AMiBA experiment. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %We explore the ponetntial of combining AMiBA cluster survey with %weak lensing %We have simulated the forth-coming interferometric %Sunyaev-Zel'dovich effect (SZE) observations with AMiBA %and the We present simulations of interferometric Sunyaev-Zel'dovich effect (SZE) and optical weak lenisng observations for the forthcoming AMiBA experiment, aiming at searching for high-redshift clusters of galaxies. On the basis of simulated sky maps, we have derived theoretical halo number counts and redshift distributions of selected halo samples for an AMiBA SZE survey and a weak lensing follow-up survey. By utilizing the conditional number counts of weak lensing halos with the faint SZE detection, we show that a combined SZE and weak lensing survey can gain an additional fainter halo sample at a given false positive rate, which cannot be obtained from either survey alone. %from %AMiBA SZE and follow-up weak lensing surveys. %We have made a quantitative study of the halo detection %in simulated AMiBA and weak lensing maps %Using hydrodynamical simulations in a $\Lambda$CDM universe, %We investigate the cluster detections in mock AMiBA and weak lensing maps %to derive the number counts of SZE and weak lensing halos %to derive the %cumulative distribution functions of SZE and %weak lensing halos as a function of peak flux. %and the false positive rate as a function of peak. %We use hydrodynamical simulations in a $\Lambda$CDM universe %to generate mock AMiBA SZE and weak lensing maps. %We have made a quantitative study of the halo number counts %in mock SZE and weak lensing sky maps %We study quantitatively the number counts of halos in %We explore the expected cluster number counts in % %We have simulated sky maps of %Sunyaev-Zel'dovich effect (SZE) and %weak gravitational lensing in a $\Lambda$CDM uninverse %to make predictions %for the forth-coming AMiBA experiment aiming at searching for %high-redshift galaxy clusters. \keywords{Cosmology; cosmic microwave background; gravitational lensing; clusters.}
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\markboth{Umetsu et al.}
{Simulation of a combined SZE and weak lensing cluster survey for AMiBA experiment}

%%%%%%%%%%%%%%%%%%%%% Publisher's Area please ignore %%%%%%%%%%%%%%%
%
\catchline{}{}{}{}{}
%
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\title{SIMULATION OF A COMBINED SZE AND WEAK LENSING CLUSTER SURVEY
 FOR AMIBA EXPERIMENT
}

\author{\footnotesize KEIICHI UMETSU
%\footnote{
%Typeset names in 8-pt Times Roman, uppercase. Use the footnote to 
%indicate the present or permanent address of the author.}
}

\address{Institute of Astronomy and Astrophysics, Academia Sinica,
P.O. Box 23-141, Taipei 106, Taiwan, Republic of China\\
%University Department, University Name, Address\\
%Taipei
%City, State ZIP/Zone,
%Country
%\footnote{State completely without abbreviations, the 
%affiliation and mailing address, including country and e-mail address. 
%Typeset in 8-pt Times Italic.}\\
keiichi@asiaa.sinica.edu.tw}

\author{TZIHONG CHIUEH, KAI-YANG LIN, JUN-MEIN WU}
\address{Department of Physics, National Taiwan University, 
Taipei 106, Taiwan, Republic of China
}

\author{YAO-HUAN TSENG}
\address{Institute of Astronomy and Astrophysics, Academia Sinica,
P.O. Box 23-141, Taipei 106, Taiwan, Republic of China\\
}


\maketitle

\pub{Received (Day Month Year)}{Revised (Day Month Year)}

\begin{abstract}
%We explore the expected number counts of galaxy clusters
%We have simulated the forth-coming 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%We have made a quantitative study %    studied 
%of halo number counts %    the detectability of galaxy clusters 
%in a combined interferometiric Sunyaev-Zel'dovich
%effect (SZE) and optical weak lensing survey for the forth-coming
%AMiBA experiment. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%We explore the ponetntial of combining AMiBA cluster survey with
%weak lensing 
%We have simulated the forth-coming  interferometric
%Sunyaev-Zel'dovich effect (SZE) observations with AMiBA
%and the 

We present simulations of 
interferometric 
Sunyaev-Zel'dovich effect (SZE) and
optical weak lenisng observations
for the forthcoming AMiBA experiment,
aiming at searching for high-redshift clusters of galaxies.
On the basis of simulated sky maps,
we have derived 
theoretical halo number counts and redshift distributions of
selected halo samples 
for an AMiBA SZE survey and a 
weak lensing follow-up  survey.
By utilizing the conditional number counts of weak lensing halos
with the faint SZE detection, we show that a combined SZE and
weak lensing survey can 
gain an additional fainter halo sample 
at a given false positive rate,
which cannot be obtained from either survey alone.




%from
%AMiBA SZE and follow-up weak lensing surveys.

%We have made a quantitative study of the halo detection
%in simulated AMiBA and weak lensing maps 
%Using hydrodynamical simulations in a $\Lambda$CDM universe,
%We investigate the cluster detections in mock AMiBA and weak lensing maps
%to derive the number counts of SZE and weak lensing halos
%to derive the 
%cumulative distribution functions of SZE and
%weak lensing halos as a function of peak flux.
%and the false positive rate as a function of peak.


%We use hydrodynamical simulations in a $\Lambda$CDM universe
%to generate mock AMiBA SZE and weak lensing maps.
%We have made a quantitative study of the halo number counts
%in mock SZE and weak lensing sky maps 
%We study quantitatively the number counts of halos in 
%We explore the expected cluster number counts in 
%
%We have simulated sky maps of
%Sunyaev-Zel'dovich effect (SZE) and 
%weak gravitational lensing in a $\Lambda$CDM uninverse %to make predictions 
%for the forth-coming AMiBA experiment aiming at searching for
%high-redshift galaxy clusters.


\keywords{Cosmology; cosmic microwave background; 
gravitational lensing; clusters.}
\end{abstract}

\ccode{PACS Nos.: include PACS Nos.}

\section{Introduction}	
The thermal Sunyaev-Zel'dovich effect (SZE\cite{Birkinshaw}) is a spectral
distortion
of the Cosmic Microwave Background (CMB) radiation due to 
the inverse-Compton scattering of CMB photons by high energy 
electrons in the intracluster medium (ICM). 
The most remarkable properties of the SZE is that its surface brightness is 
redshift independent, which make it as an ideal probe of the
high-redshift universe. Further, since the SZE is proportional to
the thermal energy content of the ICM,
%or the
%temperature-weighted thermal electron density, 
SZE imaging surveys allow us to 
select clusters over a wide range of the redshift
with physically meaningful selection criteria. Weak gravitational
lensing, on the other hand, probes the total mass projected along
the line-of-sight, and hence 
provides complementary information 
on the mass of galaxy clusters.\cite{Bartelmann} 
%(See, e.g., Ref.~\refcite{Bartelmann}). 
Array for Microwave Background Anisotropy (AMiBA\cite{AMiBA})
is a 19-element interferometric array with full polarization
capabilities operating at $95$GHz, specifically designed for the CMB
observations. One of the main scientific goals of AMiBA is to conduct
blind SZE surveys 
%with $2$arcmin resolution
to search for high-redshift clusters.
AMiBA will also conduct 
follow-up optical imaging observations 
with wide-field camera, MegaCam, at {\it Canada France Hawaii Telescope}
(CFHT). 
In this paper, we simulate
the forthcoming AMiBA SZE experiment combined with the planned follow-up
weak lensing observations to examine the expected cluster number counts
%
%and quantify false positive detections due to experimental noise.
for individual surveys, and explore the potential of a combined 
AMiBA SZE and weak lensing cluser survey.
%The remarkable properties of the SZE are that its surface brightness is 
%redshift independent and
%that it is proportinal to the thermal energy content of the ICM.


%The remarkable propertiy of the SZE that it  is essentially a redshift independent  measure of galaxy clusters makes it an ideal probe of the distant universe. The lensing and SZE are also complementary in that they both measure the projection of the mass of galaxy clusters; the SZE measure the cluster's baryonic mass component, while the lensing measures the total mass. A very large scale SZE survey covering thousands of square degrees should, in principle, allow a tight constraint on the equation of state of the Dark Energy.
%Clusters of galaxies are the largest gravitationally-bound structures in
%the Universe. They are of great interest in their own right, and also
%are a powerful cosmological probe. 
%    The Sunyaev-Zel'dovich (SZ) Effect is a distortion in the spectrum
%    of the Cosmic Microwave Background (CMB) radiation caused by the
%    scattering of CMB photons off hot gas between us and the surface of
%    last scattering. The CMB is a relic radiation field emitted 14
%    billion years ago, about 300,000 years after the Big Bang, at a time
%    when the universe was much hotter and denser. The CMB is incredibly
%    uniform and isotropic (meaning it looks the same in all directions)
%    which makes it a sensitive probe of everything between us and the
%    so-called "surface of last scattering" 14 billion light years
%    away. CMB photons which inverse Compton scatter off intervening hot
%    gas will on average gain energy causing an overall distortion in the
%   spectrum of the CMB. This distortion is called the SZ effect.
%
%The SZ effect is most often used to study clusters of galaxies. Most of
%the hot gas in the universe is contained in the potential wells of
%galaxy clusters which can be detected through X-ray emission. By
%measuring the SZ effect in the direction of a galaxy cluster (see Figure
%1) one can infer physical properties of the cluster (e.g.- the overall
%mass, the electron temperature, the peculiar velocity,...) and other
%cosmological interesting parameters (e.g.- the recessional velocity of
%the cluster, a.k.a - the Hubble constant, the local CMB temperature,
%...) The SZ effect is a particularly useful cosmological probe because
%its intensity is redshift independent, meaning a cluster 100 light years
%away would have the same intensity if it were 10 billion light years
%away.
%The SZ effect can be broken into two components, the thermal and
%kinematic SZ effect. The thermal SZ effect is due to the random thermal
%motions of the scattering electrons. The kinematic SZ effect is due to
%the bulk velocity of the cluster relative to the rest frame of the CMB,
%this

%The Sunyaev-Zel'dovich (SZ) effect is a structure in the microwave
%background radiation caused by the scattering of photons of the
%radiation by electrons in a hot intervening medium. Although those
%electrons are generally thermal (i.e., present in a hot gas, such as the
%X-ray emitting atmosphere of a cluster of galaxies), they may also be
%non-thermal (e.g., present in a relativistic plasma, such as the plasma
%in radio galaxy lobes).
%In the low-frequency part of the spectrum of the microwave background radiation (at wavelengths > 2 mm) the scattering process causes the brightness of the background radiation to decrease at a fixed frequency. Thus the SZ effect is often referred to as a decrement, and appears as a patch of sky that is cooler than the surrounding regions of sky. At wavelengths < 2 mm the effect becomes positive. 

%\section{Method}
\section{Simulation Data and Mock Observations}
%
%\subsection{Simulation data and sky map making}
%
%To make SZE and weak lensing maps, 
%we use results from cosmoligical simulations in a $\Lambda$ cold
%dark matter (CDM) cosmology
%($\Omega_m=0.34$,$\Omega_{\Lambda}=0.66$, $\Omega_b=0.044$,$h=0.7$,
%$\sigma_8=0.94$) generated with a N-body/hydrodynamics code
%GADGET\cite{GADGET}.  To simulate realistic cluster SZE signals,
%we adopt an instantaneous heating scheme at $z=2$ 

To make sky maps with realistic SZE and weak lensing signals,
we use results from preheating cosmological simulations
of a $\Lambda$CDM model 
%$with
%$(\Omega_m, \Omega_{\Lambda}, \Omega_b, h, \sigma_8) 
%=(0.34, 0.66, 0.044, 0.7, 0.94)$
$(\Omega_m=0.34,\Omega_{\Lambda}=0.66,\Omega_b=0.044,h=0.7,\sigma_8=0.94)$
in a $100h^{-1}$Mpc co-moving box 
%generated with a N-body/hydrodynamics code
%GADGET\cite{GADGET},
which reproduce the observed cluster $M_X$-$T_X$
and $L_X$-$T_X$ relations at $z=0$.\cite{Kyle}
%To simulate realistic cluster SZE signals,
%we adopt an instantaneous heating scheme at $z=2$ 
%normalized to reproduce the observed ...
%We then proceed to 
We construct 36 SZE sky maps
each with $1$deg$^2$ on a $1024^2$ grid, by projecting the electron
pressure through the randomly displaced and oriented simulation boxes,
separated by $100h^{-1}$ Mpc, along a viewing cone out  to the redshift
of $z=2$. Similarly, weak lensing convergence ($\kappa$)
maps are constructed by projecting
the distance-weighted mass over-density $\delta\rho$
out to a source plane at $z=z_s$. 
%A total of $36$ sky maps are generated for each SZE
%and weak lensing convergence. 

%\subsection{Mock observations}%Observation Strategies
As an AMiBA specification,
we adopt a close-packed hexagonal configuration of $19\times 1.2$m dishes
%with $1.2$m diameter mounted
on a single platform.
%where the gap width between adjacent dishes is $5$ cm.
%where the $1.2$m dishes are separated by $1.25$m.
% ($5$cm gap).
This array configuration yields a synthesized beam of ${\rm
FWHM}\simeq 2'$ ($0.6h^{-1}$Mpc
at $z=0.8$),
which is  optimized to detect high-redshift clusters. The field-of-view
of the primary beam is about $10'$.
%We assume the following telescope system: bandwidth $B=20$GHz,
%system temperature
%$T_{\rm sys}=70$K, system efficiency $\eta=0.6$, and dual polarizations
%$n_{\rm p}=2$.  
To generate the mock AMiBA visibility data, 
we perform a mosaic survey of  $9\times 9$ pointings with 
%exposure time of $t_{\rm exp}=1.5$ hours per pointing 
each exposure of $t_{\rm exp}=1.5$ hours 
covering 
%a contiguous field of 
$1$deg$^2$ with spacing of  $4\farcm 5$.
%Here the adopted medium deep AMiBA survey will cover about $17$ deg$^2$ 
%(3 weeks per ${\rm deg}^2$) in a 
%year assuming 8 hours observation per day. 
%
%mosaic observation strategy 
%As an AMiBA SZE observation strategy, we adopt
%a medium-deep survey 
The AMiBA sensitivity is then given as $\sigma_{\rm noise}^S=
0.78(c_{\rm mos}/0.83)(\eta/0.6)^{-1}(T_{\rm sys}/70{\rm K})
(n_{\rm pol}/2)^{-1/2}(B/20{\rm GHz})^{-1/2}(t_{\rm exp}/1.5{\rm
hours})^{-1/2}$ mJy/beam,
where $c_{\rm mos}$ depends on the mosaicking strategy.
% and
%the primary beam pattern. 
The non-linear maximum entropy method is
applied to reconstruct an AMiBA SZE map 
%$S(\vec{\theta})=-\Omega_{\rm beam}\Delta I_{\rm SZE}(\vec{\theta})$
from the mock visibility data.
%
%
For a weak lensing survey,
we assume moderately deep optical observations with 
a mean source number density of $\bar{n}=30{\rm arcmin}^{-2}$ and 
$z_s=1$.
We take into account noise in observable image ellipticities
due to the random-phase intrinsic source ellipticities.
We choose the intrinsic dispersion of $\sigma_{\rm int}=0.4$.
We then apply a pixelization on the mock weak lensing data using
a Gaussian filter with ${\rm FWHM}=2'$, and perform a linear mass
inversion to reconstruct the $\kappa$-map.
The sensitivity in reconstructed $\kappa$-maps is given as
$\sigma_{\rm noise}^{\kappa}=1.75\times 10^{-2}(\sigma_{\rm int}/0.4)
(\bar{n}/30{\rm arcmin}^{-2})^{-1/2}({\rm FWHM}/2')^{-1}$.
We carry out the statistical analysis on reconstructed sky maps
by using the inner $41' \times 41'{\rm arcmin}^2$ subfield 
in order to avoid the noisier boundaries. Thus the effective survey
area used for our statistical analysis is $36\times 0.47{\rm deg}^2
\sim 17{\rm deg^2}$.

\section{Results}

Figure 1 shows the theoretical halo number counts $N(>\nu)$
and false positive rate
as a function of peak threshold $\nu(=S/\sigma_{\rm noise}^S,
\kappa/\sigma_{\rm noise}^{\kappa})$ 
obtained from mock AMiBA SZE and weak
lensing surveys. To quantify the spurious peak detections
 due to experimental noise, 
we followed the prescription given by Ref.~\refcite{Pen}.
The limiting halo masses $M_{\rm lim}(z)$
for both surveys are shown in Figure 2.
For both surveys, halo samples are 
defined at a false positive rate of $10\%$, and the 
redshift distributions of the halo samples are derived (see Figure 2).
%
Further, we utilize the conditional number counts of weak lensing
halos with faint SZE detection, $N(\kappa>\kappa_{\rm lim}
|S>S_{\rm lim})$,
to explore the potential of  a combined AMiBA SZE and weak lensing
survey. Here we choose the SZE peak threshold of
$S_{\lim}=2.5\sigma_{\rm noise}^S=2{\rm mJy/beam}$ (see Figure 1).
Table 1 summarizes the main properties of halos samples obtained from
mock AMiBA SZE and weak lensing observations. 





\section{Conclusions}
We have examined the expected cluster number counts 
and redshift distributions of cluster samples
for the forthcoming
AMiBA SZE/weak lensing cluster survey on the basis of 
$\Lambda$CDM cosmological simulations.
%from simulated sky maps.
%The number count statistics of SZE and weak lensing
%halos are measured from the simulated sky maps. 
%We derive the redshift distribution
%functions for the SZE and weak lensing halo samples
%defined at the false positive rate of $10\%$.
By utilizing the conditional halo number counts 
$N(\kappa>\kappa_{\rm lim}
|S>S_{\rm lim})$,
we have demonstrated that a combined SZE and
weak lensing survey can %find 
gain an additional fainter halo sample 
at a given false positive rate,
%,
which cannot be obtained from either survey alone.


\section*{Acknowledgments}
We would like to thank Mark Birkinshaw for fruitful discussions.
AMiBA is funded by the Ministry of Education and National Science
Council in Taiwan.

\section*{References}

\vspace*{6pt}

\begin{thebibliography}{0}

%\bibitem{SZ} R. A. Sunyaev and Y. B. Zel'dovich, {\it
%	Comm. Astrophys. Sp. Phys.}, {\bf 4}, 173 (1972).

\bibitem{Birkinshaw} M. Birkinshaw, {\it Phys. Rep.}, {\bf 310}, 97 (1999).

\bibitem{Bartelmann} M. Bartelmann and P. Schneider, {\it
	Astron. Astrophys.}, {\bf 345}, 17

\bibitem{AMiBA} 
%K. Y. Lo, T. Chiueh, and R. N. Martin,
%{\it AAS}, {\it 32}, 1497 (2000)
%American Astronomical Society, 197th AAS Meeting, #56.03; 
%Bulletin of the American Astronomical Society, Vol. 32, p.1497
%2000AAS...197.5603L
 K. Y. Lo, T. Chiueh, H. Liang, C.-P. Ma, R. N. Martin, K.-W. Ng,
 U.-L. Pen, and R. Subramanyan, {\it in IAU Symp. 201, New
	Cosmological Data and the Values of the Fundamental Parameters,
	ed. A. Lasenby and A. Wilkinson (San Francisco: ASP)}, 31 (2000).

\bibitem{Kyle}
K.-Y. Lin, L. Lin, T.-P. Woo, Y.-H. Tseng, and T. Chiueh,
{\it astro-ph}/0210323

%\bibitem{KS93}
%N. Kaiser and G. Squires, {\it Astrophys. J.}, {\bf 404}, 441

\bibitem{Pen}
P. Zhang, U.-L. Pen, and B. Wang, {\it Astrophys. J.}, {\bf 577}, 555


%\bibitem{Bohr} A. Bohr and B. R. Mottelson, {\it Nuclear Structure} 
%(Benjamin, 1969), Vol.~1, pp.~100--102.
%\bibitem{Webb} R. C. Webb, Ph.D. thesis, Princeton University, 1972.
%\bibitem{Toimela} T. Toimela, Helsinki Research Institute for 
%Theoretical Physics, Report No. HU-TFT-82-37, 1982 (unpublished).
\end{thebibliography}

\clearpage




\begin{figure}[th]
\centerline{\psfig{file=Fig1.eps,width=6in}}
\vspace*{8pt}
\caption{Left: Cumulative distribution functions $N(>\nu)$ 
of halo number counts
%clusters 
in mock AMiBA (solid)
%; $\nu=S_{\rm lim}/\sigma_{\rm noise}^{S}) 
and weak lensing (dashed)
%; $\nu=K_{\rm lim}/\sigma_{\rm noise}^{K}$) 
observations as a function of peak threshold $\nu
(=S/\sigma_{\rm noise}^S, \kappa/\sigma_{\rm noise}^{\kappa})$
derived from 36 simulated sky maps ($17{\rm deg}^2$). 
The long-dashed curve represents
the  conditional number counts of weak lensing halos 
%$N(K>K_{\rm lim}| S>S_{\rm lim})$ 
with faint SZE detection of $S>2.5\sigma_{\rm noise}^{S}=2$mJy/beam. The shaded regions indicate
false positive detections due to noise. 
%For each curve, the detection
%threshold at which the false positive rate becomes $10\%$ is indicated.
Right: fraction of false detections  as a function of peak threshold.
}
\end{figure}

\clearpage

\begin{figure}[th]
\centerline{\psfig{file=Fig2.eps,width=6in}}
\vspace*{8pt}
\caption{
Top: the limiting mass $M_{\rm lim}$ of halos as a function
of redshift $z$ for (a) AMiBA SZE and (c) weak lensing surveys
(see Section 2). $M_{\rm lim}(z)$ is calculated for three different
peak thresholds. The solid curves correspond to a false positive
rate of $10\%$. Bottom: redshift distributions of halo samples
selected from mock $17{\rm deg}^2$ 
(b) AMiBA SZE and (d) weak lensing surveys.
}
\end{figure}


\clearpage

\begin{table}[h]
\tbl{Halo samples selected from mock AMiBA SZE and weak lensing surveys}
{\begin{tabular}{@{}lccc@{}} \toprule
Halo sample & limiting peak-flux $S$ & limiting peak $\kappa$-value & $N$ (deg$^{-2}$)\\
       & $\nu = S/\sigma_{\rm noise}^S$ & $\nu=\kappa/\sigma_{\rm
  noise}^{\kappa}$& \\ \colrule
AMiBA SZE    & $\nu >4.2$ &  & 4.2 \\
weak lensing &            & $\nu>4.5$ & 5.2 \\
combined AMiBA/lensing & $\nu>2.5$           & $\nu>3.5$ & 4.9 \\
%1.0\hphantom{00} & \hphantom{0}281.0 & \hphantom{0}280.81 & 0.07 \\
%0.1\hphantom{00} & \hphantom{0}876.0 & \hphantom{0}875.74 & 0.03 \\
%0.01\hphantom{0} & 2441.0 & 2441.0\hphantom{0} & 0.0\hphantom{0} \\
%0.001 & 4130.0 & 4129.3\hphantom{0} & 0.16\\ \botrule
faint & $2.5 < \nu <4.2$  & $3.5<\nu<4.5$  & 1.4 \\ \botrule
\end{tabular}}
\end{table}


\end{document}



astro-ph0402009.extracted_bib
\begin{thebibliography}{0}

%\bibitem{SZ} R. A. Sunyaev and Y. B. Zel'dovich, {\it
%	Comm. Astrophys. Sp. Phys.}, {\bf 4}, 173 (1972).

\bibitem{Birkinshaw} M. Birkinshaw, {\it Phys. Rep.}, {\bf 310}, 97 (1999).

\bibitem{Bartelmann} M. Bartelmann and P. Schneider, {\it
	Astron. Astrophys.}, {\bf 345}, 17

\bibitem{AMiBA} 
%K. Y. Lo, T. Chiueh, and R. N. Martin,
%{\it AAS}, {\it 32}, 1497 (2000)
%American Astronomical Society, 197th AAS Meeting, #56.03; 
%Bulletin of the American Astronomical Society, Vol. 32, p.1497
%2000AAS...197.5603L
 K. Y. Lo, T. Chiueh, H. Liang, C.-P. Ma, R. N. Martin, K.-W. Ng,
 U.-L. Pen, and R. Subramanyan, {\it in IAU Symp. 201, New
	Cosmological Data and the Values of the Fundamental Parameters,
	ed. A. Lasenby and A. Wilkinson (San Francisco: ASP)}, 31 (2000).

\bibitem{Kyle}
K.-Y. Lin, L. Lin, T.-P. Woo, Y.-H. Tseng, and T. Chiueh,
{\it astro-ph}/0210323

%\bibitem{KS93}
%N. Kaiser and G. Squires, {\it Astrophys. J.}, {\bf 404}, 441

\bibitem{Pen}
P. Zhang, U.-L. Pen, and B. Wang, {\it Astrophys. J.}, {\bf 577}, 555


%\bibitem{Bohr} A. Bohr and B. R. Mottelson, {\it Nuclear Structure} 
%(Benjamin, 1969), Vol.~1, pp.~100--102.
%\bibitem{Webb} R. C. Webb, Ph.D. thesis, Princeton University, 1972.
%\bibitem{Toimela} T. Toimela, Helsinki Research Institute for 
%Theoretical Physics, Report No. HU-TFT-82-37, 1982 (unpublished).
\end{thebibliography}