Cluster Baryon Fraction and Structure from the Convergence/SZ Effect Diagram
Keiichi Umetsu\altaffilmark{1}
Jun-Mein Wu\altaffilmark{2}
Tzihong Chiueh\altaffilmark{2}
Mark Birkinshaw\altaffilmark{3}
The cross-correlation of Sunyaev-Zel'dovich effect (SZ) and weak-lensing imaging surveys can be used to test how well hot baryons trace dark matter in clusters of galaxies. We examine this concept using mock %We use mock SZ and weak-lensing surveys based on the forthcoming AMiBA experiment and generated from a pre-heated cosmological $N$-body/hydrodynamic simulation. A cross-correlation diagram between matched lensing convergence and Compton $y$ maps exhibits butterfly-wing-like structures, corresponding to individual clusters, that %structures corresponding to individual clusters that encode rich information about the distributions of hot gas and dark matter. When the cluster redshift and temperature are available the slope of a wing reveals the cluster gas fraction and the width of the wing indicates how badly the hot gas traces dark matter. On the basis of simulated data we discuss systematic errors in the projected gas fraction estimates that would be obtained from such survey comparisons.
ms.tex
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\newcommand{\kaco}[1]{\left\langle{#1}\right\rangle}
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\newcommand{\ereion}{\eta_{\rm reion}}
\newcommand{\solam}{M_\odot}
\newcommand{\tildel}{\tilde{\kappa}}
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\newcommand{\btheta}{\mbox{\boldmath $\theta$}}

 
  
\lefthead{Umetsu et al.}
\righthead{Cluster Baryon Fraction and Structure}

%\received{2005 Jan 31}

\begin{document}

\title{Cluster Baryon Fraction and Structure from the 
Convergence/SZ Effect Diagram}
 
\author{Keiichi Umetsu\altaffilmark{1}, 
Jun-Mein Wu\altaffilmark{2},
Tzihong Chiueh\altaffilmark{2},
Mark Birkinshaw\altaffilmark{3}}
\altaffiltext{1}{Institute of Astronomy and Astrophysics, Academia
Sinica,  P.~O. Box 23-141, Taipei 106,  Taiwan, Republic of China}
\altaffiltext{2}{Department of Physics, National Taiwan University, Wenshan Chiu, Taipei 116, Taiwan}
\altaffiltext{3}{Department of Physics, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK.}


\begin{abstract}
The cross-correlation of Sunyaev-Zel'dovich effect (SZ)
and weak-lensing imaging surveys can be used to test how 
well hot baryons trace dark matter in clusters of galaxies.
We examine this concept using mock
%We use mock 
SZ and weak-lensing surveys based on the forthcoming 
AMiBA experiment and
generated from a pre-heated cosmological $N$-body/hydrodynamic
simulation. A cross-correlation diagram between matched lensing
convergence and Compton $y$ maps exhibits butterfly-wing-like
structures, corresponding to individual clusters, that
%structures corresponding to individual clusters that 
encode rich
information about the distributions of hot gas and dark matter. When
the cluster redshift and temperature are available the slope of a wing
reveals the cluster gas fraction and the width of the wing indicates
how badly the hot gas traces dark matter. On the basis of simulated data we
discuss systematic errors in the projected gas fraction estimates that
would be obtained from such survey comparisons. 
\end{abstract}



\keywords{cosmic microwave background --- cosmology: theory --- dark matter --- galaxies: clusters: general --- gravitational lensing}
%\keywords{gravitational lensing, galaxies: clusters, cosmology: theory, dark matter}


\section{Introduction}

Clusters of galaxies, the largest virialized systems known,
   are key tracers of the matter distribution in the Universe.
   Clusters are composed of the hot, diffuse intracluster medium (ICM),
   luminous galaxies (stars), and a dominant mass component in the
   form of the unseen dark matter. The mass ratio of hot gas to dark
   matter in clusters (the gas fraction) is believed to be a fair
   estimate of the universal baryon fraction (White et
   al. 1993). However, non-gravitational processes associated with cluster
   formation, such as radiative gas cooling or pre-heating, would
   break the self-similarities in cluster properties and hence cause 
   the gas fraction to acquire some mass dependence (Kravtsov et
   al. 2005; Bialek et al. 2001).
%
Weak gravitational lensing on background galaxies 
probes directly the projected cluster mass distribution 
(Bartelmann \& Schneider 2001).
The thermal 
Sunyaev-Zel'dovich (SZ) effect on the cosmic microwave background 
spectrum, on the other hand, 
measures the projected
thermal electron pressure (Birkinshaw 1999).
Therefore, 
%in combination of weak-lensing and SZ measurements, one can 
a combination of weak-lensing and SZ measurements will allow us to
measure directly the cluster gas fractions 
%provided that 
when the ICM
temperature is available
% from X-ray observations 
(Myers et al. 1997; Holder, Carlstrom, \& Evrard 2000).
%%
%%

Both blind SZ surveys, such as that to be done with AMiBA (Ho et
   al. 2004), and gravitational weak lensing surveys, such as the
   CFHT Legacy Survey\footnote{http://www.cfht.hawaii.edu/Science/CFHTLS}, aim to detect massive clusters of
   $\simgt 10^{14} M_{\odot}$ to study the formation of structure
   through the abundance and properties of the detected clusters.
   A comparison of the two types of survey should reveal much 
   about the different structures of hot gas and dark matter in
   clusters of different mass. 
Here we investigate this possibility using $N$-body simulations.




\section{Cluster gas fractions from weak-lensing and SZ observations}

%\subsection{Lensing convergence and Compton y}

The lensing convergence $\kappa$ is essentially the surface mass
density projected on the sky,
\begin{equation}
\label{eq:kappa}
\kappa
= \Sigma_m/\Sigma_{\rm crit}(z_d,z_s),
\end{equation}
where 
$\Sigma_m$ is the surface mass density of a halo acting as
gravitational lens of redshift $z_d$, and 
$\Sigma_{\rm crit}$
is  the lensing critical surface mass density 
as a function of 
lens redshift 
$z_d$ and source redshift $z_s$
specified for a given background cosmology
(e.g., Bartelmann \& Schneider 2001).
%(e.g., Schneider, Ehlers, \& Falco 1992).
%\begin{equation}
%\Sigma_{\rm crit} = \frac{c^2}{4\pi G}\frac{D_s}{D_dD_{ds}}
%\end{equation}
%with the angular diameter distances $D_d,D_s,D_{ds}$
%between observer and lens, observer and source and lens and source,
%respectively (see, e.g., Bartelmann \& Schneider 2001). 

The SZ effect is described by the 
Compton $y$ parameter defined as
a  line-of-sight integral of the temperature-weighted 
thermal electron  density (Birkinshaw 1999):
% \begin{equation}
% y = \frac{\sigma_T}{m_e c^2}\int\!dl\, n_e k_{\rm B}T_g
% \end{equation}
% where
% $n_e$ is the electron number density, $T_g$ the gas
% temperature, $m_e$ the electron mass, $\sigma_{\rm T}$ the Thomson cross
% section,
%Assuming the cluster is isothermal, we have
\begin{equation}
\label{eq:y}
y=
 \frac{\sigma_T}{m_e c^2}\int\!dl\, n_e k_{\rm B}T_g \approx
 \frac{1+X}{2}
\frac{\sigma_T}{m_p} \frac{k_{\rm B}T_g}{m_e c^2}\Sigma_g.
\end{equation}
where  $n_e$ is the electron number density, $T_g$ the gas
temperature, $m_e$ the electron mass, $\sigma_{\rm T}$ the Thomson cross
section, $\Sigma_g$ the surface gas mass density,
and $X=0.76$ the hydrogen abundance.
Note that we have assumed isothermality in the second
equality in Eq.~(\ref{eq:y}).
%{\bf
The presence of temperature gradients in the ICM
%intra-cluster medium (ICM)
still remains controversial  (see Piffaretti et al. 2005 and references
therein).
In the paper, 
we assume the ICM is isothermal as a zeroth-order approximation; 
we found 
a temperature variation within the virial radius 
by a factor two at most
from our sample of massive halos (see \S 4).
%from our simulation data. 
%(c.f., Ettori et al. 2004).
%}
%temperature gradients in the intra-cluster medium (ICM) ... remains a
%controversial issue (see Piffaretti et al. 2005 and references therein).
%{\bf Description about the temperature gradients from X-ray observations
%(Allen, Schmidt, Fabian 2001)
%and numerical simulations (??). And comment that the cluster isothermality is
%still a zero-th order approximation to the ICM....}
%We will turn back to this issue in \S??.

%  remains a controversial issue

% From the cosmological hydrodynamic simulations in a $\Lambda$CDM
% universe (see \S\ref{sect:data}), 
% we found the isothermality is a good approximation to describe
% the intra-cluster hot gas.

%Note that the dimensionless lensing convergence $\kappa$ is
%independent of the Hubble parameter $h$ while the Compton y parameter
%$y$ depends on $h$.
 

%\subsection{2D-based gas fraction estimators}


In general,
the lensing $\kappa$ and the Compton
$y$ reconstructed from noisy observations are smoothed by a 
certain window function
in the map making process.
In interferometric observations, the synthesized beam (or PSF) is well
approximated by a Gaussian window. In weak lensing surveys, Gaussian
smoothing is often adopted to find clusters as local maxima in the
reconstructed convergence map (e.g., Miyazaki et al. 2002). 
We hence express the convergence field 
smoothed with a Gaussian window $W_{\rm G}(\theta)= \exp(-\theta^2/\theta_{\rm G}^2)/\pi\theta^2_{\rm
G}$ as $\kappa_{\rm
G}(\btheta)$;
%\begin{equation}
%\kappa_{\rm G}(\btheta)=\int\!d^2\theta'\, W_{\rm G}(\btheta-\btheta')\,\kappa(\btheta')
%\end{equation}
%with
%%$W_{\rm G}(\theta)= \exp(-\theta^2/\theta_{\rm G}^2)/\pi\theta^2_{\rm
%G}$; 
$\theta_{\rm G}$ is related to FWHM of the Gaussian window
$\theta_{\rm fwhm}$ by
$\theta_{\rm G}=\theta_{\rm fwhm}/\sqrt{4\ln(2)}$.
Similarly, we define
the Gaussian smoothed Compton $y$ parameter, $y_{\rm G}(\btheta)$.
%We may interested in comparing the 
We then compare the reconstructed $\kappa_{\rm G}$ and
$y_{\rm G}$ values at a certain point $\btheta$.%, say, at a halo center.
Taking the ratio of smoothed $\kappa_{\rm G}$ and $y_{\rm G}$,
we have
\begin{eqnarray} 
\label{eq:y2k}
\eta_g \equiv 
\frac{y_{\rm G}}{\kappa_{\rm G}}
&=&
8.9h\times 10^{-4}
\left( \frac{k_{\rm B}T_g}{10{\rm keV}} \right)
\left( \frac{\Sigma_{\rm crit}}{1h\, {\rm g/cm^2}} \right)
\left(\frac{f_g}{0.1}\right)
\end{eqnarray}
where $f_g(\btheta)  = \Sigma_{g,{\rm G}}(\btheta)
/\Sigma_{m,{\rm G}}(\btheta) $ is the {\it local} gas fraction
defined with the Gaussian smoothed surface mass densities.
%% Note that the gas temperature $T_g$ also depends 
%% on the halo mass. The observed X-ray emission weighted 
%% $M$-$T_{\rm ew}$ relation 
%% is given as 
%% $T_{\rm ew}= 2.63\times 
%% ( M_{500}/1.4\times 10^{14}h^{-1} M_{\odot})^{0.54} \,{\rm keV}$
%% (Finoguenov et al. 2001).
%Assuming an isothermal gas, the SZ signal is hence proportional to
%$M^{1.54}$.
%
In order to extract information on the cluster gas fraction from
weak lensing and SZ observations, we thus need additional 
information on the ICM temperature 
$T_g$, the cluster redshift $z_d$,
and the redshift distribution of background galaxies.
Such information can be
in principle
available from
X-ray observations and photometric redshift measurements of background
galaxies.

We can define a {\it global} estimator for the cluster gas fraction 
based on weak-lensing and SZ observation as
\begin{equation}
\label{eq:eta}
\lan \eta_g \ran \equiv\frac
  {\sum_i w_i \eta_{g,i}}
  {\sum_i w_i}\propto \Sigma_{\rm crit}(z_d,z_s)\, \lan f_g T_g \ran
\end{equation}
where $\eta_{g,i}=\eta_g(\btheta_i)$ and $w_i=w(\btheta_i)$
is the weight for the $i$th pixel $\btheta_i$.
%in practice, we take the weight $w_i$ to be zero if a pixel at $\btheta_i$
%is located outsidde a certain aperture of interest.
Accordingly, the global gas fraction of a halo can be estimated
from $\kappa$ and $y$ maps
as $\lan f_g \ran \propto \lan \eta_g \ran T_g^{-1}\Sigma_{\rm crit}^{-1}$
if one
assumes 
the isothermality of the ICM.
The case with $w_i=\kappa_{\rm G}(\btheta_i)$ corresponds to a commonly used definition
of the gas fraction, $\eta_g^{(1)}\equiv
\lan y\ran /\lan \kappa\ran 
\propto
\lan \Sigma_g\ran /\lan \Sigma_m\ran $, as the ratio of the mean
surface mass densities. However, the mean convergence $\lan \kappa\ran $
is ill constrained from weak-lensing observations:
Weak lensing $\kappa$ reconstructions
based solely on image distortions
suffer from the {mass sheet} degeneracy 
%$\kappa \to \kappa+{\rm const.}$
(Schneider \& Seitz 1995)
unless additional information
such as the magnification bias is taken into account
(Broadhurst, Taylor \& Peacock 1995; Broadhurst et al. 2005).
%%
On the other hand, interferometric observations are insensitive to
the DC signal, and hence do not constrain the total flux in the
field-of-view (FoV). Bolometers also suffer from
a high level of contamination by
atmospheric and other environmental signals, and hence rely on the
differencing scheme to extract the sky signals.
%%
We therefore consider an alternative
 weight  $w_i=\kappa_{\rm G}^2(\btheta_i)$
that weights high density regions more strongly than $w_i=\kappa_{\rm G}(\btheta_i)$.
Then we have the gas fraction estimator of the form:
\begin{equation}
\label{eq:cor}
\eta_g^{(2)}  \equiv \frac
  {\sum_i \kappa_{\rm G}(\btheta_i)\,y_{\rm G}(\btheta_i)}
  {\sum_i \kappa_{\rm G}(\btheta_i)\,\kappa_{\rm G}(\btheta_i)} = 
\frac{\lan \kappa y\ran }{\lan \kappa^2\ran}.
\end{equation}
%
%Let us consider a case where $\kappa$ and $y$ satisfy a linear bias
%relation $y(\btheta)=a\,\kappa(\btheta)$, and introduce a global
%transformation such that  (i) $y\to y+\epsilon$ ($y \gg |\epsilon|$)
%or (ii) $\kappa\to\kappa+\epsilon$ ($\kappa \gg |\epsilon|)$.
%For the case (i), we have $\Delta
% \eta_g^{(1)}=\epsilon/\lan \kappa\ran $ and 
% $\Delta
% \eta_g^{(2)}=\epsilon\lan \kappa \ran /\lan \kappa^2\ran $,
%and hence
%$|\Delta \eta^{(1)} |-|\Delta \eta^{(2)} |=
%\left(|\epsilon| / \lan\kappa \ran \right)
%\left({\rm Var}[\kappa]/ \lan \kappa^2\ran \right)\ge 0$;
%the equality is satisfied only if the variance in $\kappa$ vanishes, i.e.,
%${\rm Var}[\kappa]=0$, in which case we are not interested.
%Similarly, for the case (ii), we have 
%$|\Delta \eta^{(1)}|-|\Delta \eta^{(2)}  |
%\approx 
%\left(a|\epsilon| / \lan\kappa \ran \right)
%\left({\rm Var}[\kappa]/ \lan \kappa^2\ran \right)\ge 0$.
%Therefore, we see qualitatively
% the cross-correlation based estimator $\eta_g^{(2)}$
%is less sensitive than $\eta_g^{(1)}$ 
%against the error for
%the normalization in $\kappa$ and/or $y$ maps. 
Suppose that $\kappa$ and $y$ satisfy a linear bias relation
   $y(\btheta)=a\,\kappa(\btheta)$. Then errors in the Compton
   parameter of the form $y\to y+\epsilon$ (where $y \gg |\epsilon|$)
   lead to errors in the two estimators of $\eta_g$ of
   $\Delta \eta_g^{(1)}=\epsilon/\lan \kappa\ran $ and 
   $\Delta \eta_g^{(2)}=\epsilon\lan \kappa \ran /\lan \kappa^2\ran $.
   The difference between these errors
   $|\Delta \eta_g^{(1)} |-|\Delta \eta_g^{(2)} |=
   \left(|\epsilon| / \lan\kappa \ran \right)
   \left({\rm Var}[\kappa]/ \lan \kappa^2\ran \right)\ge 0$, with
   equality only if variance in $\kappa$ vanishes. Thus the 
   $\eta_g^{(2)}$ estimator is always better than the 
   $\eta_g^{(1)}$ estimator. Similarly, an error on the convergence of
   the form $\kappa \to \kappa+\epsilon$ (where $\kappa \gg |\epsilon|)$
   leads to 
   $|\Delta \eta^{(1)}|-|\Delta \eta^{(2)}  |
   \approx 
   \left(a|\epsilon| / \lan\kappa \ran \right)
   \left({\rm Var}[\kappa]/ \lan \kappa^2\ran \right)\ge 0$, again
   showing that the cross-correlation based estimator $\eta_g^{(2)}$
   is more robust than $\eta_g^{(1)}$.


%In the presence of noise, 
%noise contributions to the estimator (\ref{eq:eta})
%need to be properly taken into account. 
%Since noise properties between $\kappa$ and $y$ maps are not
%correlated, we only have to subtract off the noise variance from the
%denominator of Eq.~(\ref{eq:cor}) as $\lan \kappa^2 \ran \to 
%\lan \kappa^2\ran -\sigma_{\kappa}^2$ with $\sigma_{\kappa}^2$
%being the noise variance in the smoothed convergence field, $\kappa_{\rm
%G}$ (\S 3.2).


In practical observations, noise contributions to the estimator
   (\ref{eq:eta}) must be taken into account. Since the noise
   properties between the $\kappa$ and $y$ maps will not be
   correlated, the required correction is in the
   denominator of Eq.~(\ref{eq:cor}), where we replace
   $\lan \kappa^2 \ran$ by $\lan \kappa^2\ran -\sigma_{\kappa}^2$ with 
   $\sigma_{\kappa}^2$ being the noise variance in the smoothed
   convergence field, $\kappa_{\rm G}$ (\S 3.3).

\section{Mock observations}

We use cosmological simulation data by Lin et al. (2004)
to demonstrate what useful information 
we can extract from 
the cross correlation between weak-lensing and SZ surveys.
Specifically,
we simulate the forthcoming AMiBA experiment as an illustration
of such combined surveys.  


\subsection{$N$-body/hydrodynamic simulations}\label{sect:data}
%To make SZ 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 an N-body/hydrodynamics code
%GADGET\cite{GADGET}.  To simulate realistic cluster SZ signals,
%we adopt an instantaneous heating scheme at $z=2$ 

To make sky maps with realistic SZ and weak lensing signals,
we use results from preheating cosmological simulations
of a $\Lambda$CDM model 
$(\Omega_m=0.34,\Omega_{\Lambda}=0.66,\Omega_b=0.044,h=0.66,\sigma_8=0.94)$
in a $100h^{-1}$Mpc co-moving box 
generated with an N-body/hydrodynamics code
GADGET (Springel et al. 2001),
which reproduce the observed cluster $M_X$-$T_X$
and $L_X$-$T_X$ relations at $z=0$ (Lin et al. 2004).
We construct a total of $29$ SZ 
sky maps, each $1 {\rm deg}^2$ in size and containing $1024^2$ pixels,
%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 ($\kappa$)
29 $\kappa$-maps are constructed by projecting
the distance-weighted mass overdensity $\delta\rho$
out to a source plane 
at a fixed redshift, $z = z_s$.
%at a certain redshift of $z=z_s$. 

\subsection{AMiBA SZ cluster survey}

%As an AMiBA specification,
%we adopt 
As an AMiBA specification, we adopt
a close-packed hexagonal configuration of $19\times 1.2$m dishes
on a single platform.
AMiBA operates at $95$ GHz, and
this array configuration yields a synthesized beam of 
 $\theta_{\rm fwhm}\approx 2'$ ($0.6h^{-1}$Mpc at $z=0.8$),
which is optimized to detect high-$z$ clusters (Zhang et al. 2002).
%(Lo et al. 2000). 
The primary beam has an FoV of 
${\rm FWHM} \approx 11'$.
We assume the following telescope system: bandwidth $\Delta\nu=20$ GHz,
system temperature
$T_{\rm sys}=70$ K, system efficiency of $0.7$, and dual polarizations.

AMiBA will operate as a drift scan interferometer to remove
   ground contamination to high order (Pen et al.
   2002\footnote{http://www.cita.utoronto.ca/\~{}pen/download/Amiba};
   Zhang et al. 2002; Park et al. 2003).
   We approximate the resulting synthesized beam in the AMiBA map 
   by a Gaussian with $\theta_{\rm fwhm}=2'$, and we convolve 
   the simulated Compton $y$ maps with this Gaussian beam to generate a
   set of {\it noise-free} AMiBA $y_{\rm G}$-maps. The expected rms
   noise over a sky area of $\Omega_s$ with a total integration time
   of $t_{\rm int}$ is (Pearson et al. 2003),
   \begin{equation}
    \label{eq:ynoise}
    \sigma_{y_{\rm G}} =  
    2.0\times 10^{-6}
    \left(  \frac{T_{\rm sys}}{70{\rm K}} \right)\!
    \left(  \frac{\Delta\nu}{20{\rm GHz}} \right)^{-1/2}\!\!
    \left( \frac{\Omega_{\rm s}}{1{\rm deg}^2} \right)^{1/2}\!\!
    \left(   \frac{ t_{\rm int}}{ 240{\rm hrs}}\right)^{-1/2}.
   \end{equation}


%AMiBA will adopt a drift scan strategy to remove the
%ground contamination as an interferometry is insensitive to the ground
%emission that is a DC signal (Pen et
%al. 2002\footnote{http://www.cita.utoronto.ca/\~{}pen/download/Amiba}; Zhang et al. 2002; Park
%et al. 2003).
%In the AMiBA SZ-map making,
%we approximate the synthesized beam in drift scan images 
%by a Gaussian beam
%with $\theta_{\rm fwhm}=2'$, and we simply convolve 
%the Compton $y$ maps
%with the Gaussian beam to generate a set of
%{\it noise-free} $y_{\rm G}$-maps through AMiBA drift-scanning. 
%The rms noise in drift scan images over a sky area of $\Omega_s$
%with a total integration time of $t_{\rm int}$
%is given as (Pearson et al. 2003),
%\begin{equation}
%\label{eq:ynoise}
%\sigma_{y_{\rm G}} =  
%2.0\times 10^{-6}
%\left(  \frac{T_{\rm sys}}{70{\rm K}} \right)\!
%\left(  \frac{\Delta\nu}{20{\rm GHz}} \right)^{-1/2}\!\!
%\left( \frac{\Omega_{\rm s}}{1{\rm deg}^2} \right)^{1/2}\!\!
%\left(   \frac{ t_{\rm int}}{ 240{\rm hrs}}\right)^{-1/2}.
%\end{equation}
%which corresponds to $\approx 0.5$mJy/beam at $\nu=95$GHz.
%0.51mJy/beam




\subsection{Weak lensing cluster survey}

We assume a weak lensing survey with a mean galaxy number
density of $n_g=40$ arcmin$^{-2}$,
an rms amplitude of the intrinsic
ellipticity distribution of $\sigma_{\epsilon}=0.3$,
and a mean galaxy redshift of $z_s=1$,
which are close to typical values for a ground based optical imaging
survey with a magnitude limit of $R_{\rm lim}\simgt 25.5$ mag in a
sub-arcsecond seeing condition (e.g., Hamana, Takada, \& Yoshida 2004).
The rms noise in a Gaussian-smoothed convergence field, $\kappa_{\rm
G}$,
is given as
\begin{equation}
\label{eq:knoise}
\sigma_{\kappa_{\rm G}}=1.1\times 10^{-2}
\left(\frac{\sigma_{\epsilon}}{0.3}\right)
\left(\frac{n_g}{40{\rm arcmin}^{-2}}\right)^{-1/2}
\left(\frac{\theta_{\rm fwhm}}{2'}\right)^{-1}
\end{equation}
(e.g., van Waerbeke 2000).
We choose the same smoothing scale of  $\theta_{\rm fwhm}=2'$ as 
for the AMiBA SZ survey and
convolve the $\kappa$ maps from $N$-body simulations
(see \S 3.1) with the Gaussian beam
to produce a set of {\it noise-free} $\kappa_{\rm G}$ maps.
Note that the adopted value of $\theta_{\rm fwhm}$ is
close to the optimal smoothing scale of 
$\theta_{\rm fwhm} \approx 1\farcm 7$ (or $\theta_{\rm G}\approx 1'$)
for an efficient survey
of massive halos with $M\simgt 10^{14}h^{-1}M_{\odot}$ found 
by Hamana et al. (2004) based on cosmological $N$-body simulations.


\begin{figure}
\epsscale{1.0}
\plotone{f1.eps}
\label{fig:buttfly_mock}
\caption{
Cross correlation between 
$\kappa_{\rm G}$ and $y_{\rm G}$ maps as a 2D scatter plot histogram
%A 2D scatter plot histogram of
%lensing $\kappa$ vs.Compton $y$ 
%presented in units of rms dispersions $\kappa_*$ and $y_*$, respectively,
obtained from 
$29\times 1{\rm deg}^2$
{\it noise-free} sky maps of weak-lensing and SZ surveys for the AMiBA experiment.
The $\kappa_{\rm G}$ and $y_{\rm G}$ values are normalized in units
of rms dispersions  $\kappa_*$ and $y_*$, respectively.
Halo peaks with $\kappa \simgt 4.5\kappa_*$ and $y \simgt 6y_*$ are 
indicated with markers
according to their redshifts: 
$z<0.3$ ({\it circle}), $0.3\le z<0.5$ ({\it triangle}), $0.5\le z<1$
 ({\it square}).
%$z<0.3$ with {\it cross},   
%$0.3<z<0.5$ with {\it triangle}, and $z>0.5$ with {\it diamond}.
}
\label{fig:bfd1}
%\end{center}
\end{figure} 









\section{Cluster gas fractions from 
cross correlation between weak-lensing and SZ signals}

%\subsection{Cross correlation between  $\kappa$ and $y$ maps}

We carried out a statistical analysis on the simulated 
weak lensing $\kappa_{\rm G}$ and SZ Compton $y_{\rm G}$ maps of 
$29\times 1{\rm deg}^2$ for the AMiBA experiment.
Figure 1 shows the cross correlation between 
the noise-free  $\kappa_{\rm G}$ and $y_{\rm G}$ sky maps
of the entire $29{\rm deg}^2$
as a 2D scatter plot histogram. The scatter plot is presented in units
of rms dispersions $\kappa_*$ and $y_*$ of the smoothed noise-free
fields $\kappa_{\rm G}(\btheta)$ and $y_{\rm G}(\btheta)$, respectively;
$\kappa_*=1.4\times 10^{-2}$ and $y_*=2.2\times 10^{-6}$.
Halo peaks with
$\kappa_{\rm G} \simgt 4.5\kappa_*$ and $y_{\rm G}\simgt 6y_*$
($\approx 3.3{\rm mJy/beam}$ at $95$ GHz) 
are indicated in Fig.~1 
according to their redshifts.
%The here adopted peak thresholds effectively select massive halos with
%$M_{200}
%\simgt 2\times 10^{14}h^{-1}M_{\odot}$ (Fig.~2 of Umetsu et
%al. 2004); $M_{\Delta_c}\equiv (3/4\pi)\Delta_c\rho_{c}r_{\Delta_c}^3$ 
%is the mass enclosed by $r_{\Delta_c}$
%with the critical density of the universe $\rho_c$.
These thresholds effectively select massive halos
    with $M_{200} \simgt 2\times 10^{14}h^{-1}M_{\odot}$ (see Fig.~2 of
    Umetsu et al. 2004); $M_{\Delta_c}\equiv
    (4\pi/3)\Delta_c\rho_{c}r_{\Delta_c}^3$ is the mass enclosed by
    $r_{\Delta_c}$ with the critical density of the universe
    $\rho_c$.
Figure 1 clearly exhibits different capabilities of 
SZ and weak-lensing in cluster detections: weak lensing depends 
strongly on $z_d$ through $\Sigma_{\rm crit}$,
%$\Sigma_{\rm crit}(z_{\rm halo},z_s)$, 
while the surface brightness of
SZ effect is independent of $z_d$.
In particular, gravitational lensing is less efficient
for high-$z$ lensing halos due to pure geometrical effects. 
As a consequence, high-$z$ halos
tend to appear in low-$\kappa$ and high-$y$ regions.


Another important feature in Fig.~1 is a number of
   butterfly-wing-shaped structures extending from the origin to
   points corresponding to the halo peaks. Each blade of a butterfly
   wing corresponds to an individual cluster. 
%More detailed and quantitative discussions 
%are given in \S 4. 
   Hereafter, we call
   the $\kappa/y$ cross-correlation diagram the {\it butterfly
   diagram}. For illustrative purposes, we show in Fig.~2 a butterfly
   diagram from the noise-free result for the most massive halo, which
   has $M_{200}=7.0\times 10^{14}h^{-1}M_{\odot}$ at $z=0.142$.
%Another important feature observed from Fig.~1 is 
%a number of wing-shaped structures 
%in the diagram extending
%out to the points that correspond to the halo peaks: 
%each blade of the butterfly wing in the diagram 
%%each wing-like structure in the diagram 
%corresponds to an individual cluster.
% More detailed and quantitative discussions are given in 
%\S 4.
%Hereafter, we call such a cross-correlation diagram 
%between $\kappa$ and $y$
%as the {\it butterfly diagram}.
%%For an illustrative purpose, we show in Fig.~2 a {\it butterfly diagram} of 
%a particular halo from the noise-free result.
%This is the most massive halo 
%with 
%$M_{200}=7.0\times 10^{14}h^{-1}M_{\odot}$
%at $z=0.142$ 
%extracted from the present simulation run. 
%%
%%
The SZ signal for this halo
%As displayed in Fig.~2,
%the SZ signal 
is more extended than the weak lensing signal in this cluster
region. Furthermore, we can see a clear disparity between the 
shapes of the contours in the $\kappa$ and $y$ maps.
%lensing $\kappa$ and Compton $y$ maps.
%From the diagram, we can understand intuitively that the position angle
%of this wing provides us with information of the halo baryon fraction
%and that the width of the wing is related to the disparity between
%the spatial distributions of mass and hot gas in the cluster.





\begin{figure}
\plotone{f2.eps}
\epsscale{1.}
\caption{Same as Fig.~1 but for a $25'\times 25'$ region 
of a single massive halo at $z=0.142$
found in the simulation. 
The halo properties such as $M_{500}$ and $T_{500}$
are also indicated within the figure.
Also plotted are
the $y_{\rm G}$ and $\kappa_{\rm G}$ maps smoothed with $\theta_{\rm
 fwhm}=2'$ Gaussian in the left and the right subpanels, respectively.
A circle with radius of $\theta_{500}$ is shown in each subpanel.
}
\label{fig:bfd_halo}
\end{figure}

%\subsection{Cluster gas fractions from cross-correlation}


\begin{figure}
\epsscale{0.85}
\plotone{f3.eps}
\caption{Distributions of the ratio 
$f_g^{(2)}(<\theta_{500})/f_g(<r_{500})$
between the projected (Eq.~[5]) and true gas fractions against the halo mass
 $M_{200}$ for three redshift subsamples,
%three redshift bins of 
%(a) $0< z<0.3$, (b) $0.3\le z<0.5$, and (c) $0.5\le z<1$. 
obtained with the same halo selection as in Fig.~1
but from a total survey area of $\approx 20{\rm deg}^2$.
Each data point is measured from a pair of the Gaussian-smoothed 
{\it noise-free} $\kappa_{\rm G}$ and $y_{\rm G}$ maps,
 while each error-bar represents an observational uncertainty ($1\sigma$)
based on $50$ Monte-Carlo realizations of random noise in $\kappa_{\rm
 G}$ and $y_{\rm G}$.
%the smoothed 
% $\kappa_{\rm G}$ and $y_{\rm G}$ maps. 
%For each redshift-bin, the mean
The sample means
% value 
over the noise-free measurements are indicated as dashed lines.}
\label{fig:kap}
\end{figure} 
%since pixel noise can be reduced more significantly
%by averaging over a 



The butterfly diagram of a single cluster shows a wing
   that contains rich information on the relative distributions
   of dark matter and hot gas.
%The {\it butterfly} diagram of a single cluster yields 
%a butterfly wing
%that
%contains fairly 
%rich information on the relation between distributions of 
%dark-matter
%and hot-gas.
The tip of a butterfly wing shows the surface mass densities of both
dark matter and hot gas in the cluster core, and the bulk of the wing
reveals the degree to which the hot gas traces dark matter.
%%%%%%%%%%%%%%%%%%%%%%%%
The wing width serves as an indicator of the difference 
between projected distributions of dark matter and hot gas.
Most halos shown in Fig.~1 have significant wing widths. These
   widths arise because the baryon distribution is closer to 
   axisymmetric than the dark matter distribution due to collisional
   relaxation of the baryons.
%Most halos shown in Fig.~1 have 
%finite wing widths, which arises naturally from the well-known fact that the 
%baryon distribution is closer to having axisymmetry than the dark 
%matter distribution due to the collisional relaxation in baryons.  
%%%%%%%%%%%%%%%%%%%%%%
%To be more quantitative,
The local slope  $\eta_g(\btheta)$ of a wing
in conjunction with the cluster redshift and temperature
%is a function of $f_g$, $T_g$, and $\Sigma_{\rm crit}$
%(see Eq.~(\ref{eq:y2k})).  
%When the cluster redshift and temperature are provided, it can be hence
can be
an indicator of the local gas fraction $f_g(\btheta)$ (see Eq.~[\ref{eq:y2k}]).
Note that the estimator $\eta_{g}^{(2)}$ 
defined in Eq.~(\ref{eq:cor}) can be interpreted as  a global slope 
of a wing in the least-square sense. 
%Therefore,
%the $\left<\eta_{g}^{\rm C} \right>$ estimator with weight 
%$w_i=\kappa_{\rm G}^2(\btheta_i)$ is relevant to the {\it butterfly
%diagram}. 

We use the halo sample with $\kappa_{\rm G}\simgt 4.5\kappa_*$ and
 $y_{\rm G}\simgt 6y_*$ 
%derived from the 
%simulated $\kappa_{\rm G}$ and $y_{\rm G}$ 
derived from our noise-free sky maps of $29{\rm deg}^2$
%(see \S 4)
to compare 
%quantitatively
the halo-based cumulative gas fractions
$f_g(<r)=M_g(r)/M(r)$ %enclosed by $r$ 
with
their projected estimates
$f_g^{(2)}(<\theta)\propto \eta_g^{(2)}(<\theta) T_g^{-1}
\Sigma_{\rm crit}^{-1}(z_d,z_s)$ from cross correlation
between $\kappa$ and $y$;
$\theta=r/D_A(z_d)$ 
with $D_A(z)$ being the angular diameter distance 
to redshift $z$.
%We take this radius to be $r_{500}$ and calculate for each halo
%a gas fraction using Eqs.~(\ref{eq:eta}) and (\ref{eq:cor})
%within an aperture of projected radius $\theta_{500}=r_{500}/D_A(z_d)$.
%We assume the cluster isothermality and use the emission-weighted
%mean temperature $T_{500}$ inside a sphere of $r_{500}$ for each 
%halo as an observable X-ray temperature.
We make this comparison inside $r_{500}$, and calculate for each
   halo a gas fraction using Eqs.~(\ref{eq:eta}) and (\ref{eq:cor})
   within an aperture of projected radius
   $\theta_{500}=r_{500}/D_A(z_d)$. We assume cluster
   isothermality and take the emission-weighted mean temperature
   $T_{500}$ inside $r_{500}$ as the observable X-ray temperature.
%%
%%
The weight $w_i$ in Eq.~(\ref{eq:eta}) is set to zero if
$\kappa_i\le 0$ or $y_i\le 0$ to exclude low-signal regions
%where no meaningful information about cross correlation is available.
%{\bf
where cross correlation measurements are of low significance due to
projection effects and/or observational noise.
%}
%to minimize possible contributions
%from projection effects (???) and random noise 
%in low signal regions. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Further, we remove those halos 
whose centers are located outside the inner $50'\times 50'$ region 
of a $1{\rm deg}^2$ sky map, in order to avoid systematics in the gas
fraction estimates due to 
%the incomplete aperture coverage near 
the boundaries. The effective survey area is thus about
   $20\ {\rm deg}^2$
%; the effective survey area is thus $\approx 20{\rm deg}^2$. 

Figure~3 shows the distribution of the ratio $ f_g^{(2)}
   (<\theta_{500})/ f_g (<r_{500})$ obtained from the noise-free maps
   as a function of $M_{200}$ for three redshift subsamples:
   (a) $z\le 0.3$ (where we find 8 halos); (b) $0.3\le z<0.5$ (16
   halos); and (c) $0.5\le z<1$ (13 halos).
%Figure 3 shows the distribution of 
%the ratio $ f_g^{(2)} (<\theta_{500})/ f_g (<r_{500})$ 
%obtained from the {\it noise-free} maps
%against $M_{200}$
%separately for three redshift subsamples:
%(a) $z\le 0.3$;
%(b) $0.3\le z<0.5$; (c)  $0.5\le z<1$.
%We found (a) 8, (b) 16, and (C) 13 halos in respective redshift bins.
The sample means and their standard errors
of $f_g^{(2)}(<\theta_{500})/f_g(<r_{500})$ 
are (a) $0.96\pm 0.03$, (b) $1.10\pm 0.04$, and (c) $0.95\pm 0.05$.
Note that the quoted errors represent
the intrinsic scatter % in  $f_g^{(2)}(<\theta_{500})/f_g(<r_{500})$ 
of the estimator $f_g^{(2)}$
from the noise-free signal maps.
%%%%%%
%%%%%%
%%%%%%
On the whole, the estimator $f_g^{(2)}(<\theta_{500})$ for the gas
   fraction is close to unbiased, but there is a slight
   ($\sim 2\sigma$) over-estimation bias of $f_g$ in 
   $ 0.3 \le z < 0.5$. 
We found that most halos with overestimated values of $f_g$ are
   associated with mergers. The shock heating of gas in mergers boosts
   the temperature in the region between the merging halos. However,
   we ascribe the lower temperature $T_{500}$ to this gas, based on the
   emission-weighted temperature of the entire halo, and so
   overestimate the value of $f_g$ for the system as a whole.
   This overestimate of $f_g$ appears to be smaller at higher $z$
   because such halos have smaller angular sizes and so suffer larger
   beam smearing effects. If the 5 interacting/merger systems in
   subsample~(b) are rejected, then the sample mean
   $f_g^{(2)}(<\theta_{500})/f_g(<r_{500}) = 1.01\pm 0.03$, showing
   that the bias is eliminated. Such a debias should be possible based
   on X-ray observations of the survey field.
%We found that most halos with over-estimated
%   values of $f_g$ are associated with mergers. The shock heating of
%   gas in mergers boosts the temperature in the region between the
%   merging halos (Ishizaka \& Mineshige 1996).
%   {\bf However, the emission weighted temperature is biased towards 
%   the un-shocked core region, while the compton $y$ will have a 
%   significant contribution from the shocked region.
%   This fact, in conjunction with the isothermal assumption, 
%  leads to an under-estimate of the ICM temperature in the shock-heated 
%  region, and hence an over-estimate of the gas density there.}
%
%
%   and this, in
%   conjunction with the isothermal assumption, leads to an
%   over-estimate of the ICM temperature in the shock-heated 
%   region, and hence an under-estimate of the gas density there.
   %%
%   %%
%   This over-estimate of $f_g$ appears to be smaller at higher $z$
%   because such halos have smaller angular sizes and so suffer larger
%   beam smearing effects. If interacting/merger systems in
%   subsample~(b) are rejected, then the sample mean
%   $f_g^{(2)}(<\theta_{500})/f_g(<r_{500}) = 1.01\pm 0.03$, showing
%   that the bias is eliminated. Such a debias should be possible based
%   on X-ray observations of the survey field.

   On the other hand, the projection effect in $\kappa$ due to
   intervening matter (Metzler et al. 2001) also affects cluster 
   $f_g$ estimates made using weak lensing data. This is more severe
   for higher-$z$ clusters due to their lower geometrical lensing
   efficiency and smaller angular sizes. If an intervening mass
   concentration is projected onto a cluster, then the gas fraction 
   of that cluster could be under-estimated. Furthermore, the
   gradients in 3D cluster gas-fraction and temperature profiles
   (Ettori et al. 2004) can also induce a systematic bias 
   in projected $f_g^{(2)}$ estimates, especially for
   spatially-resolved low-$z$ clusters. Intrinsic gradients in $T_g$
   and $f_g$ are manifest as a curvature of the wing in a butterfly
   diagram, and they cannot be distinguished unless detailed
   information is available on the temperature distribution in the ICM.
%On the whole, the estimator $f_g^{(2)}(<\theta_{500})$ 
%for 
%$f_g(<r_{500})$
%is close to unbiased. 
%However, there is a slightly higher bias ($\simgt 2\sigma$) towards
%over-estimation of the gas fraction in the intermediate-$z$ subsample (b).
%We found most halos with such over-estimations are associated with
%ongoing mergers; namely, shock-heating of gas
%due to mergers boosts the temperature 
%of the region between
%merging halos 
%%(Burns et al. 1994; 
%(Ishizaka \& Mineshige 1996);
%this, in conjunction with the isothermal assumption, 
%would lead to an over-estimation of the ICM temperature in the shock-heated
%region, which results in an under-estimation of the gas density in that region.
%%and under estimation of the density
%%and temperature of the ICM, respectively, 
%%in the shock-heated region;
%%\footnote{
%%{\bf
%This
%over-estimation of $f_g^{(2)}$ is apparently less significant 
%for higher $z$ halos
%with smaller angular sizes because of the beam smearing effect.
%The rejection of interacting/merger systems
%%apparent mergers 
%in the subsample (b) 
%%gives the mean in
%eliminates the bias in the sample mean of 
%$f_g^{(2)}(<\theta_{500})/f_g(<r_{500})$ as $1.01\pm 0.03$. 
%%%%
%It is therefore important 
%to remove mergers in the sample in order to reduce such systematics in
%the $f_g$-statistic; 
%such a debias can practically be possible based on
%X-ray observations.
%On the other hand, the projection effect in $\kappa$ due to intervening matter
%(Metzler et
%al. 2001)
%%; White et al. 2002)
%also affects 
%cluster 
%%gas fraction 
%$f_g$ estimates using weak lensing data.
%%White, van Waerbeke, Mackey 2002).
%This is more severe for higher-$z$ clusters due to their
%less geometrical
%lensing efficiency
%and smaller angular sizes. If an intervening mass concentration 
%is projected
%onto the aperture of a cluster of interest, the gas fraction 
%could be under-estimated. 
%Furthermore, the gradients in 3D cluster
%gas-fraction and temperature profiles (Ettori et al. 2004)
%can also induce a systematic bias 
%in projected $f_g^{(2)}$ estimates especially for spatially resolved
%low-$z$ clusters; 
%both the intrinsic gradients in $T_g$ and $f_g$ can be observed
%as a curvature of the wing in a {\it butterfly diagram},
%and they cannot be distinguished unless detailed information on
%the temperature distribution in the ICM is available.


%}
%%%%%%%%}
%the former can be observed as a curvature of the
%wing in a {\it butterfly diagram}; however, the latter requires detailed
%information of the temperature distribution in the ICM to correct for
%the systematics.

 
The error bars in Fig.~(3) are based on 50 realizations of 
   Monte-Carlo simulated noise maps which are linearly added to
   the Gaussian smoothed signal maps of $\kappa_{\rm G}$ and 
   $y_{\rm G}$. Following van Waerbeke (2000), a correlated noise map
   of $\kappa_{\rm G}$ is generated for each realization
   using the noise auto-correlation function specified 
   by the smoothing scale $\theta_{\rm fwhm}$ and the observational
   noise rms $\sigma_{\kappa}$ (Eq.~[\ref{eq:knoise}]). A similar
   correlated noise map for $y_{\rm G}$ uses the noise rms
   $\sigma_{y}$ (Eq.~[\ref{eq:ynoise}]). For a fixed halo mass, 
   the random error in $f_g^{(2)}$ is smaller for lower-$z$ halos with
   larger angular scales, $\theta_{500}$, over which pixel noise is
   averaged, but systematic errors will be dominant for these clusters.

%%
%The error-bars in Fig.~(3) are based on 50 realizations of 
%a pair of Monte-Carlo simulated noise maps which are linearly added to
%the Gaussian smoothed signal maps of $\kappa_{\rm G}$ and $y_{\rm
%G}$;
%following van Waerbeke (2000),
%a correlated noise map of $\kappa_{\rm G}$
% is generated for each realization
%using the noise
%auto-correlation function that is fully specified 
%by the smoothing scale $\theta_{\rm fwhm}$ and the noise rms
%$\sigma_{\kappa}$
%given by Eq.~(\ref{eq:knoise}). Similarly, a correlated noise map is 
%produced for %the Gaussian smoothed 
%$y_{\rm G}$ for each realization.
%For a fixed halo mass, 
%the error in $f_g^{(2)}$ is smaller for lower-$z$ halos
%with larger angular scales, $\theta_{500}$, over which pixel noise is
%averaged.
%In the $f_g$ estimation from weak-lensing and SZ imaging observations,
%systematic errors become more important than observational errors for
%low-$z$ clusters.


\section{Conclusions}\label{sect:conclusion}

By simulating the AMiBA experiment, we have shown that the
   butterfly diagram that emerges from a comparison of the SZ and
   weak-lensing signals contains rich information on the
   distribution of the cluster gas fraction within an individual
   cluster and between different clusters (\S 4). 
   The cross-correlation based estimator $f_g^{(2)}(<\theta)$ should
   be used for a robust estimate of the cluster gas fraction. 
   Overall, this estimator is close to unbiased, but some halos show
   intrinsic scatter that exceeds the observational errors (\S 3)
   due to various sources of systematics (\S 4). The bias for the 
   the intermediate-$z$ subsample of halos can be eliminated by
   removing interacting/merger systems. Careful selection of clusters
   should minimize systematic uncertainties in $f_g$. Detailed
   information regarding the temperature distribution in clusters, 
   derived from X-ray observations, improves the calibration
   of the gas fraction estimator, and yields valuable information on
   the thermodynamic history of the ICM.

%We report a succinct means for examining how well hot baryons trace 
%dark matter in clusters when both SZ and weak-lensing 
%surveys are available for cross-correlation.
%By simulating the AMiBA experiment,
%we have shown that the {\it butterfly diagram} emerged from 
%a cross correlation between SZ and weak-lensing signals
%contains %fairly 
%rich information on the
%distribution of the cluster gas fraction within an individual cluster as
%well as that among different clusters (\S 4). 
%We proposed to use the cross-correlation based estimator
%$f_g^{(2)}(<\theta)$ for cluster gas fraction estimates based on
%weak-lensing and SZ imaging observations.
%Our results show that, overall, the 
%estimator $f_g^{(2)}(<\theta)$ 
%is close to unbiased; however, some halos show intrinsic scatter
%that is greater than observational errors (Fig.~3)
%due to various sources of systematics (\S 4).
%In particular, 
%the positive bias found in the intermediate-$z$ subsample can be 
%eliminated by removing interacting/merger systems from the subsample.
%It is therefore important to make 
%a secure sample selection in order to minimize systematic uncertainties
% in the
%cluster
%$f_g$ statistic. Detailed
%information regarding temperature distribution from X-ray observations
%will provide a better calibration of the gas fraction estimator
%as well as valuable information on the thermodynamic history of the ICM.


\acknowledgments 
We acknowledge Kai-Yang Lin and Masahiro Takada for useful discussions.


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astro-ph0506065.extracted_bib
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Broadhurst, T., Takada, M., Umetsu, K., Kong, X., Arimoto, N., Chiba,
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%Broadhurst, T. et al. 2005,  ApJ, 619, L143


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