Mon. Not. R. Astron. Soc. 350, 893–913 (2004)

doi:10.1111/j.1365-2966.2004.07691.x

Searching for massive clusters in weak lensing surveys
Takashi Hamana,1,2  Masahiro Takada3 and Naoki Yoshida2,4
1 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, F 75014 Paris, France
2 National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan
3 Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd Street, Philadelphia, PA 19104, USA
4 Harvard-Smithsonian Centre for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

Accepted 2004 January 30. Received 2004 January 30; in original form 2003 October 24

ABSTRACT

We explore the ability of weak lensing surveys to locate massive clusters. We use both analytic
models of dark matter haloes and mock weak lensing surveys generated from a large cosmological N-body simulation. The analytic models describe the average properties of weak lensing
haloes and predict the number counts, enabling us to compute an effective survey selection
function. We argue that the detectability of massive haloes depends not only on the halo mass
but also strongly on the redshift where the halo is located. We test the model prediction for the
peak number counts in weak lensing mass maps against mock numerical data, and find thatthe noise resulting from intrinsic galaxy ellipticities causes a systematic effect which increases
the peak counts. We develop a correction scheme for the systematic effect in an empiricalmanner, and show that, after correction, the model prediction agrees well with the mock data.
The mock data is also used to examine the completeness and efficiency of the weak lensing
halo search by fully taking into account the noise and the projection effect by large-scale structures. We show that the detection threshold of S/N = 4 ∼ 5 gives an optimal balance between
completeness and efficiency. Our results suggest that, for a weak lensing survey with a galaxy
number density of n g = 30 arcmin−2 with a mean redshift of z = 1, the mean number of haloes
which are expected to cause lensing signals above S/N = 4 is N halo (S/N > 4) = 37 per 10 deg2 ,whereas 23 of the haloes are actually detected with S/N > 4, giving the effective completenessas good as 63 per cent. Alternatively, the mean number of peaks in the same area is N peak =62 for a detection threshold of S/N = 4. Among the 62 peaks, 23 are caused by haloes withthe expected peak height S/N > 4, 13 result from haloes with 3 < S/N < 4 and the remaining
26 peaks are either the false peaks caused by the noise or haloes with a lower expected peakheight. Therefore the contamination rate is 44 per cent (this could be an overestimation). Weaklensing surveys thus provide a reasonably efficient way to search for massive clusters.
Key words: gravitational lensing – galaxies: haloes – cosmology: theory – dark matter –large-scale structure of Universe.

1 INTRODUCTION
Recently, it has become feasible to locate massive clusters directly as
density enhancements using weak gravitational lensing. Miyazaki
et al. (2002) indeed discovered many clusters in a weak lensing halo
survey over a 2.1-deg2 field, proving the ability of weak lensing to
identify massive clusters (see also Wittman et al. 2001). Unlike conventional optical or X-ray selected cluster catalogues, weak lensing
cluster catalogues are free from a bias towards luminous objects because this type of cluster finding is not based on flux enhancement
but on projected mass enhancement. This is one great advantage of E-mail: hamana@iap.fr

C 2004 RAS

gravitational lensing. However, like every other cluster survey technique (i.e. optical, X-ray and Sunyaev–Zel’dovich survey) lensing
does have disadvantages; for example, the projection effect of unrelated structures in the same line of sight (Reblinsky & Bartelmann
1999; Metzler, White & Loken 2001). As galaxy clusters are someof the most important cosmological probes, it is of fundamental
importance to construct large unbiased samples. Given the advantages and disadvantages involved in every survey technique, it isdesirable to combine, in a complementary manner, several techniques to construct an unbiased catalogue. Such catalogues provide
a valuable data set to investigate, for example, the physical stateof the intracluster medium. To do this, it is important to correctly
understand the selection function and completeness of each survey
technique.

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T. Hamana, M. Takada and N. Yoshida

It has often been argued that weak lensing provides a truly massselected sample of clusters. This is not strictly true; weak lensing
measures the two-dimensional tidal field which is a line of sightprojection of the three-dimensional tidal field weighted by the distance ratio, D l D ls /D s (see Section 2 for definitions and details; see(Mellier 1999; Bartelmann & Schneider 2001) for reviews of weaklensing). In simpler terms, the amplitude of the lensing signal froma cluster is not solely determined by its mass, but also depends on
the redshift and the shape of the gravitational potential caused by
the cluster. In addition, both foreground and background large-scale
structures contribute to the lensing signal, and this projection effectusually adds a noise to the lensing signal from a halo. Further, since
the weak lensing signal is measured from tiny coherent distortions
in galaxy shapes, intrinsic ellipticities of galaxies introduce an irremovable noise. Thus, the detectability of the weak lensing halo
depends on all these factors.
The primary purpose of the present paper is to explore the abilityof weak lensing surveys to locate massive clusters. We especiallyaddress the following three points:(i) to examine the selection function of weak lensing cluster surveys;(ii) to develop a theoretical model that describes the weak lensing
halo counts; and(iii) to examine how the detailed structure of clusters, projection effect and noise affect weak lensing cluster surveys under atypical observational condition from a ground-based 4–10 m classtelescope.
The last point is especially important to understand a bias thatweak lensing halo catalogues may have. To address these things,
we use both simple analytic descriptions of dark matter haloes andmock numerical weak lensing survey data. The former offers a useful way to compute expected lensing properties of massive haloes.
We use them to compute the selection function. We also develop atheoretical model for the number counts of the weak lensing haloes
based on the analytic descriptions. Alternatively, the latter allow us
to examine all the factors listed above in a direct manner. Mocknumerical catalogues are generated using weak lensing ray-tracing
simulations. We extensively use the mock data and the halo catalogues directly produced from the simulation outputs to examine thecompleteness and efficiency of the weak lensing halo survey. The
mock data are also used to test the model prediction of the weaklensing halo counts.
White, Van Waerbeke & Mackey (2002) and Padmanabhan,
Seljak & Pen (2003) addressed the ability of weak lensing surveys focusing on the projection effect. In those studies, the completeness of weak lensing surveys is defined as the fraction of detected haloes with mass above a certain value relative to haloes
that lie in a given volume. This definition compromises two completely different elements: (i) the selection effect; and (ii) the effects from individuality of the halo mass distribution, the projection
and noise. For our purpose, these elements needs to be examined
separately. We first compute the selection function using the analytical descriptions of dark matter haloes which enables us to define ‘potentially detectable haloes’. Here, ‘potentially’ refers to an
ideal case in the absence of noise. Then, we shall adopt the different definition of the completeness that is the fraction of actuallydetected weak lensing haloes relative to the potentially detectablehaloes.
Our results have important implications to real observations.
When making an observational strategy, one would like to compute the expected number of detectable clusters using, for exam-

ple, a simplified model, and also needs to estimate the expectedcompleteness and false detection rate. Our theoretical model of the
weak lensing halo counts may be useful for the former case, andour results from the mock data are directly applicable to derive thelatter estimates. In addition, from an observational point of view, we
adopt a directly observable quantity, a peak height of weak lensing
mass map divided by the noise rms, that is the signal-to-noise ratio,
S/N, of the detection, as the fundamental estimate of the lensing
signal.
The rest of this paper is organized as follows. In Section 2, wesummarize lensing properties of massive haloes assuming the universal density profile proposed by (Navarro, Frenk & White 1996,1997). Using the expected properties, we compute the selection
function of the weak lensing cluster survey. Then, we combine thedetectability of weak lensing haloes with the halo mass function
based on the Press–Schechter approach (Press & Schechter 1974;
we adopt the modified fitting function by Sheth & Tormen 1999) tocompute the weak lensing halo counts and the selection functions
with respect to the halo mass and redshift. This simple analytical approach should offer a theoretical basis to explore properties of weaklensing haloes. In Section 3, we describe our numerical experimentof taking weak lensing surveys and our weak lensing ray-tracing
technique. Statistical properties of halo catalogues are obtained andcompared with the theoretical predictions derived in Section 2. Using the mock weak lensing survey data, in Section 4 we examine thenumber counts of peaks in the weak lensing convergence map, andthe theoretical prediction developed in Section 2 is tested. In Section 5, we investigate the relationships between haloes in the halo
catalogue and peaks identified in the lensing convergence map. In
particular, we examine the completeness and efficiency of the weaklensing halo search paying special attention to points (1)–(3) above.
A summary and discussion are given in Section 6. In Appendix A,
we present some lensing convergence maps in which a missing halo
or a false peak exists for illustrative examples of irregular systems.
In Appendix B, we develop a correction scheme for the theoreticalprediction of weak lensing peak counts. In Appendix C, the relationships between the halo shape (and orientation) and the strengthof the lensing signal are examined.
Throughout this paper, we work with the standard flat -colddark matter (CDM) cosmological model with the density parameter m = 0.3, the cosmological constant   = 0.7, and the Hubble constant H 0 = 100 h km s−1 Mpc−1 with h = 0.7. We adopt the fitting
function of the CDM power spectrum by Bardeen et al. (1986) with
the normalization of σ 8 = 0.9. We take the observational parametersof a typical weak lensing survey from a present-day ground-based
4–10 m telescope. Specifically, we consider an imaging observation
with a limiting magnitude fainter than R = 25.5 mag in a subarcsecond seeing condition; this provides the galaxy number density
n g  30 arcmin−2 with the mean redshift z  1.

2 P R E D I C T I O N S F RO M S TA N DA R D M O D E L S
2.1 Lensing properties of an NFW halo
The universal density profile of dark matter haloes proposed by(Navarro et al. 1996, 1997 NFW hereafter) is given by
ρNFW (x) =ρs,x(1 + x)2

x=

r
,
rs

(1)

where r s is the scaleradius. It is convenient to introduce theconcentration parameter c NFW = r vir /r s where r vir is the virial
C 2004 RAS, MNRAS 350, 893–913

Searching for massive clusters in weak lensing surveys
radius. Bullock et al. (2001) found from N-body simulations that theconcentration parameter is related to the halo mass by
cNFW (M, z) =

c∗
1+z



M
1014 h −1 M

−0.13,(2)where we set c ∗ = 8 for the CDM model. It is important to noticethat there is a relatively large scatter in this relationship (Jing 2000).
The mass enclosed within a sphere of radius r vir gives the virial
mass by definition, i.e.



34πρs rvir
cNFW
log(1 + cNFW ) −.
3
1 + cNFW
cNFW

Mvir =(3)

The virial mass is also defined by the spherical top-hat collapse
model as4π
3δvir (z)ρ̄0 rvir
Mvir =,(4)
3
where δ vir is the threshold overdensity for spherical collapse (seeNakamura & Suto 1997; Henry 2000 for useful fitting functions).
From equations (3) and (4) one can express ρ s in terms of δ vir (z)and c NFW . Introducing δs = ρs /ρ̄0 − 1, one finds
δs =

3
cNFW
δvir
.
3 log(1 + cNFW ) − cNFW /(1 + cNFW )(5)

Although the NFW profile formally extends to infinity, an actual
halo has a finite extent. We consider a truncated NFW halo, thedensity profile of which is truncated at the virial radius. The surface mass density of the truncated NFW halo is given by the lineof-sight integration of the density profile (Takada & Jain 2003),
i.e.


(y) =





−

with

f (y) =

2
cNFW
−y 2

dzρNFW (y, z) = 2ρs rs f (y),

2
cNFW
−y 2

r
y= ,
rs

(6)

 
2
cNFW
−y 2

y 2 +cNFW+ (1−y12 )3/2 arccosh y(1+c, − (1−y 2 )(1+c

NFW )
NFW )


 fory < 1






 c2 −1
NFW

3(1+cNFW )

1 + 1+c1NFW ,for y = 1,



2
cNFW
−y 2

y 2 +cNFW
1− (1−y 2 )(1+c− (y 2 −1),
3/2 arccos y(1+c

NFW )
NFW )




for1 < y  c,



0,

(7)

(y)

= κs f (y),(8)

with the critical surface mass density
Dsc2
4πG Dl Dls

andκs = 3m δs rs



H0
c

(9)

2

χl (χs − χl ),a(z l )χs

(10)

where a is the scalefactor normalized at the present time, D denotes
the angular diameter distance, χ denotes the comoving radial disC 2004 RAS, MNRAS 350, 893–913

WG (x) =1x2
exp − 2
2
πθG
θG


.

(11)

The Gaussian smoothing is indeed adopted in actual weak lensing surveys (e.g. Miyazaki et al. 2002). As shown in the studyof Van Waerbeke (2000), if the intrinsic ellipticities are uncorrelated between different source galaxies, the statistical propertiesof the resulting noise field can be described by Gaussian random
field theory (Bardeen et al. 1986; Bond & Efstathiou 1987), on
scales where the discreteness effect of the source galaxies can beignored. The Gaussian field is specified by the variance of thenoise field that is specified by the number of galaxies contained
within a smoothing aperture (Kaiser & Squires 1993; Van Waerbeke
2000)
2
σnoise=σ21,2 2πθG2 n g(12)where σ  is the rms amplitude of the intrinsic ellipticity distribution and n g is the number density of the source galaxies. Throughout this paper we adopt σ  = 0.4 and n g = 30 arcmin−2 , which
are typical values for a survey with a limiting magnitude that isfainter than R = 25.5 mag in a subarcsecond seeing condition. With
this choice, the noise rms is σ noise = 0.02 × (1 arcmin/θ G ). It is
worth noting that σ noise is a direct observable in The sense that it
can be estimated from a reconstructed weak lensing mass map obtained after randomizing the orientation of the images of the source
galaxy.
We also need to take into account the smoothing effect on thelensing signal from a halo. The peak amplitude of the convergenceprofile of haloes is modified by smoothing into



cr

cr =

tance (which is equal to the comoving angular diameter distancefor a flat universe), and the subscripts l and s stand for lens andsource, respectively. It is worth noting that κ scales with c NFW approximately as κ ∝ c NFW for 1  c NFW  5 and κ ∝ c1.5
NFW for 5 
c NFW  30. This clearly proves that the convergence amplitude issensitive to the halo concentration.
With the number density of source galaxies and the intrinsic ellipticity rms that we are interested in, a raw (unsmoothed) convergence
map is very noisy. To get a lensing mass map with a reasonably small
noise amplitude, we smooth the convergence map with a Gaussian
window function, i.e.

κG (θG ) =fory > c.

Note that the above expression differs from that by Bartelmann(1996) and Wright & Brainerd (2000) which is for a non-truncated
NFW halo and gives a larger surface mass density than ours. The
lensing convergence profile is given by
κ(y) =

895

d2 φWG (φ; θG ) κ(φ),(13)

where the centre of the smoothing kernel is set to the halo centre.
Hereafter we refer to this peak amplitude as ‘the lensing signal
resulting from a halo’. Fig. 1 plots the lensing signal as a function of
the smoothing scale. The dotted line shows a scaling of the noise rms
with the smoothing scale (σ noise ∝ θ −1
G ). The comparison between
the solid and dotted curves thus shows how S/N varies with thesmoothing scale. Clearly, an optimal S/N can be attained for θ G =(1 − 2) × θ s . Fig. 2 shows the contour plot of θ s in the halo mass
and redshift plane, where the mean mass–concentration relationshipgiven by equation (2) is employed. Note that throughout this paper,
the source redshift is fixed at z s = 1, which is a typical value for themean redshift of source galaxies for an actual survey with limiting
magnitude R  25.5. By combining Figs 1 and 2, it can be seenthat θ G ∼ 1 arcmin would be the optimal choice for an efficient
survey of massive haloes with M halo  1014 h −1 M at intermediateredshifts.

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Figure 3. The grey scale with contour lines shows the S/N value for weaklensing halo detection defined by equation (14). The S/N value is computed
assuming a universal NFW profile. We adopt, throughout this paper, σ  =0.4, n g = 30 arcmin−1 and θ G = 1 arcmin.
Figure 1. The solid line shows the lensing signal from an NFW halosmoothed with the Gaussian kernel (equation 13) against the smoothinglength. For the concentration parameter, c NFW = 5 is adopted here. The dotted line shows scaling of the noise rms with the smoothing scale (σ noise ∝θ −1
G , see equation 12).

equations 9 and 10). This makes the mass sensitivity of weak lensing surveys strongly redshift dependent. In other words, the lower
mass limit of the detectable haloes depends strongly on the redshift.
This figure also implies that the effective survey volume of a weaklensing halo search depends on the minimum halo mass that one islocating.
It should be noted that we adopt the mean mass–concentration
relationship to compute the model prediction for the S/N value,even though the relationship has large scatters (Bullock et al. 2001;
Jing 2000). As the lensing signal, κ G , depends on the concentration
parameter by κ G ∝ c1−1.5
NFW , the scatter likely translates to scatter inthe κ G value or equivalently in the S/N. It should also be noted thatthe lensing signal for a realistic halo is affected by substructures andasphericity of the mass distribution seen in N-body simulations. We
will carefully investigate these effects, using mock data of a weaklensing survey.
2.2 Model prediction of weak lensing halo counts

Figure 2. The grey scale with contour lines show the angular scaleradiusθ s (in arcmin), in the halo mass (M halo ) and redshift (z l ) plane. The source
redshift is fixed at z s = 1.

We define S/N for a weak lensing halo detection by1
κGν=.
σnoise(14)

Fig. 3 shows the contour plot of the S/N value computed assuming
the universal NFW profile in the halo mass and redshift plane. It
is evident from this figure that the lensing signal not only depends
on the mass but also on the redshift. This redshift dependence enters mainly through the lensing efficiency function D l D ls /D s (see1 With respect to notation throughout this paper, we denote the S/N of weaklensing signals by ν and distinguish its type with a subscript; ν NFW stands for
the S/N computed assuming a universal NFW profile, while ν lens and ν noisy
stands for the S/N measured concerning noise-free and noisy numerical
weak lensing mock data, respectively (see Sections 3 and 4). We will alsouse ν th and ν min to denote a threshold detection value and the minimum
value (to be set; see Sections 3 and 4), respectively.

Haloes that induce a sufficiently high-lensing signal are likely tobe identified as peaks in the convergence map, even if the map iscontaminated by noise caused by intrinsic ellipticities. Hence, it isconvenient to work with the number counts of peaks above a certain
threshold value: N (ν > ν th ;θ G ). To compute the number counts,
we assume that a high peak is generated by the lensing signal froma single massive halo and that the number of false peaks owing
to noise (i.e. contamination) is much smaller than that of the real
peaks, as will be shown in detail later. With our fiducial choice of thesmoothing scale θ G = 1 arcmin, this assumption is reasonably valid
for sufficiently high peaks (say ν > 4), and thus we focus on suchhigh peaks. In this case, the number density of peaks with heightgreater than the threshold ν th value per unit solid angle is given by(Kruse & Schneider 2000; Bartelmann, King & Schneider 2001)

N (ν > νth ; θG , z s ) =

dχ

dV
dχ



dM

dn halo [M, z(χ )]
dM

(15)

×H[ν(M, z) − νth ],where χ is the radial comoving distance, dV /dχ is the comoving
volume element per unit solid angle (dV /dχ = χ 2 for a flat cosmology), dn halo /dM is the halo mass function, for which we adopt
the fitting function by Sheth & Tormen (1999), and H(x) is the
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Searching for massive clusters in weak lensing surveys

897

Table 1. Summary of the weak lensing peak counts computed using equation (15) for various smoothing scales. The observational parameters adopted
are σ  = 0.4, n g = 30 arcmin−1 and a fixed source redshift of z s = 1.
The second and third columns are for the counts with ν > 5 and ν > 4,
respectively.
θ G (arcmin)

N (ν > 5) (deg−2 )N (ν > 4) (deg−2 )

1.47
2.08
1.76
1.30
0.97

3.38
4.05
3.15
2.27
1.67

0.5
1
2
3
4

Heaviside step function (H(x) = 1 for x > 0 and 0 otherwise). The
differential counts are obtained byn peak (ν) =dN (ν > νth ; θG , z s )
.
dν

(16)

Table 1 summarizes the predicted weak lensing peak counts forvarious smoothing scales. It is shown that θ G = 1 arcmin gives thelargest counts, for σ  = 0.4 and n g = 30 arcmin−2 . In the following
discussion, we will adopt this smoothing scale.
In Fig. 4, we plot the selection function of a weak lensing halosearch with respect to the halo mass and redshift. The grey scalewith the solid line contours shows the number density of haloes per
1 deg2 against a given redshift interval of dz and a given logarithmicmass interval of dlog M as follows:2d NhalodV dn halo≡.
dzd log M
dz d log M(17)

Note that this is the integrand of equation (15). It is clear that, for agiven halo mass, the number density rapidly increases with redshift
at low redshifts (z  0.2), and then gradually decreases towardhigher redshifts. This is because the volume element, dV ∝ χ 2 dχ,increases with redshift, whereas the halo number density decreases.
The dashed contours show the expected peak height, ν NFW , for theuniversal NFW halo at intervals of ν NFW = 1. The comparison
between the solid and dashed contours clarifies that, if a threshold

Figure 4. The grey scale with solid contour lines shows the number densityof haloes per 1-deg2 area, a given redshift interval dz and a given logarithmicmass interval d log M: d 2 N halo /dz/d log M ≡ dV /dz dn halo /d log M. Thisis the integrand of equation (15). The dashed contours show the expectedpeak height ν NFW from 3 to 9 at an interval of ν NFW = 1. Integrating thisnumber over the peak height above a threshold value ν th gives the expected
number of peaks per 1 deg2 as given by equation (15).

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Figure 5. The solid lines show the redshift distribution of haloes with apeak height above a threshold value, ν th , per 1 deg2 , dN (ν NFW > ν th )/dz/1
deg2 , computed using equation (15). The upper and lower solid lines are forν th = 4 and 5, respectively. Note that we employed θ G = 1 arcmin and z s =1. The broken lines show the same redshift distributions but for haloes with
masses above a given minimum mass, dN halo (M > M th )/dz/1 deg2 .

peak height of ν th =4–5 is employed, the weak lensing halo search
is most sensitive to haloes that are located at the redshift between
0.1  z  0.7 and have masses greater than 1014 h −1 M .
Let us look further into the redshift and mass dependence of the
weak lensing selected halo counts. The solid curves in Fig. 5 show
the redshift distribution of haloes selected by the weak lensing signal. These curves can be compared with the distribution of haloes
above a certain mass threshold (broken lines). A remarkable difference between these two is evidently seen at redshifts of z > 0.5;
the distribution of weak lensing-selected haloes drops very rapidly,
while the mass-selected ones decrease very gradually. This is again
because the lensing efficiency peaks around z = 0.4 for source galaxies of z = 1. Fig. 6 shows essentially the same information but froma different angle. The left-hand panel shows the mass function of
haloes with peak height exceeding a threshold value (the cases ofν th = 4 and 5 are displayed). The lower mass limit in the mass function is determined by the detection threshold, while the decline in
the high-mass side (which has an approximate exponential slope) iscaused by one property of the halo mass function (i.e. the exponential shape of its high-mass end, see Kruse & Schneider 2000). The
right-hand panels of Fig. 6 shows the mass distribution of haloes
for different redshift intervals as labelled in each plot. The solid(dashed) lines are for haloes with ν NFW > 5 (ν NFW > 4), while thedotted lines are for all the haloes in the corresponding volume. The
upper two plots clearly show that, up to z = 0.4, almost all thehaloes with M > 2 × 1014 h −1 M indeed produce high peaks with
ν NFW > 5. However, for higher redshifts, only very massive haloes
can produce high peaks.
To summarize the results shown in Figs 4–6 , we argue that weaklensing is not an efficient technique for finding high-redshift haloes(say z > 0.6), except for very massive haloes of M > 5 × 1014 h −1
M , if a mean source redshift is around z s = 1. It should be notedthat the capability of the weak lensing halo survey is dependent on
observational conditions. As a survey is deeper, a halo search enables
us to detect less massive haloes and haloes at higher redshift.

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T. Hamana, M. Takada and N. Yoshida
3.1 N-body simulation
We used N-body data from the Very Large N-body Simulation (VLS)
carried out by the Virgo Consortium (Jenkins et al. 2001, and see
also Yoshida, Sheth & Diaferio 2001 for simulation details). The
simulation was performed using a parallel P3 M code (Macfarland
et al. 1998) with a force softening length of l soft ∼ 30 h −1 kpc. The
simulation employed 5123 CDM particles in a cubic box of 479 h −1
Mpc on a side. It uses a flat cosmological model with  0 = 0.3,  = 0.7 and h = 0.7. The initial matter power spectrum was computed
using CMBFAST (Seljak & Zaldarriaga 1996) assuming a baryonicmatter density of  b = 0.04. Normalization of the power spectrum isgiven by σ 8 = 0.9. The particle mass (m part = 6.86 × 1010 h −1 M )of the simulation is sufficiently small so as to practically guarantee
no discreteness effect on dark matter clustering on scales down to
the softening length in the redshift range of interest here (Hamana,
Yoshida & Suto 2002).

Figure 6. Left-hand panel: the mass function of haloes with ν NFW > ν thcomputed using equation (15). The threshold values are ν th = 4 and 5 for thedashed and solid lines, respectively, The source redshift is taken to be z s = 1.
Right-hand panels: same as the left-hand panel but divided into four rangesof halo redshift as denoted in each plot. The dashed lines show the massfunction of haloes within the corresponding volume (which is equivalent tosetting ν th > 0, that is, no limit on the weak lensing signal is imposed). Notethat the scales of the y-axis are labelled on the right-hand side of the plots.
Table 2. Summary of the angular number density (per 1 deg2 ) of haloes with
masses more massive than a given minimum halo mass (the first column)
defined in equation (17) (see Section 2 for details). The second and thirdcolumns are for the maximum redshifts of z < 1 and z < 0.7, respectively.
Halo mass (h −1 M )z<1
M > 5 × 1013
M > 1 × 1014
M > 2 × 1014
M > 5 × 1014

55
17
4.1
0.36

N halo (deg−2 )z < 0.7
30
10
2.7
0.28

Table 2 gives the number density of haloes with mass above agiven minimum mass, which is obtained by integrating the halo massfunction up to a given maximum redshift value. Comparison with
Table 1 provides a rough estimate of how a weak lensing survey
misses mass-selected haloes in the given survey volume. A large
discrepancy between the halo counts and the weak lensing halo
counts is seen, which arises from the redshift dependence of thelensing selection function as shown in Figs 5 and 6.
3 N U M E R I C A L S I M U L AT I O N
A N D M O C K C ATA L O G U E S
The theoretical model prediction discussed in the previous section
provides us with a basic selection function of the weak lensing cluster survey. However, since our model is based on the simple averaged
descriptions of dark matter haloes, it does not take into account theindividuality of haloes which causes a substantial scatter in the lensing signal and may even cause a bias. Furthermore, there are two
additional sources of scatter; the projection effect and intrinsic ellipticity noise. In this section, we will examine these effects in detailusing mock numerical experiments.

3.2 Weak lensing ray-tracing simulation
The multiple-lens plane ray-tracing algorithm we used is detailed in
Hamana & Mellier (2001; see also Bartelmann & Schneider 1992;
Jain, Seljak & White 2000; Vale & White 2003 for the theoreticalbasics and technical issues); here we thus only describe the aspects
specific to the VLS N-body data. To generate a density field betweenz = 0 and z  1, we use a stack of six snapshot outputs from two
runs of the N-body simulation, which differ only in the realizationof the initial fluctuation field. Each cubic box is divided into four
subboxes of 4792 × 119.75h −3 Mpc3 with the shorter box side beingaligned with the line-of-sight direction. N-body particles in eachsubbox are projected on to the plane perpendicular to the shorter box
side and thus to the line-of-sight direction. In this way, the particle
distribution between the observer and z  1 is projected on to 19 lensplanes separated equally by 119.75h −1 Mpc. Note that to minimize
the difference in redshift between a lens plane and an output of Nbody data, only one half of the outputs (i.e. two subboxes) from thez = 0 N-body box was used.
The particle distribution on each plane is converted into the surface density field on a 40962 regular grid using the triangular-shaped
cloud (TSC) assignment scheme (Hockney & Eastwood 1988). A
40962 grid is chosen to maintain the resolution provided by the Nbody simulation as well as to remove at the same time the shot noisecaused by the discreteness in the N-body simulation. The computation follows the procedure described in Hamana & Mellier (2001)
and Jain et al. (2000).
Having produced surface density fields on all lens planes, we trace
5122 rays backwards from the observer’s point to the source plane
using the multiple-lens plane algorithm (e.g. Schneider, Ehlers &Falco 1992). The initial ray directions are set on 5122 grids witha grid spacing of 0.24 arcmin, and thus the total area covered by
the rays is 2.0482 deg2 . We made 50 realizations of the underlying
density field by randomly shifting the simulation boxes in the direction perpendicular to the line of sight using the periodic boundary
conditions of the simulation.2 Note that the lens planes coming fromthe same box are shifted in the same way to maintain the clusteringof matter in one box.
In the following analyses, we use the lensing convergence mapof the source plane closest to z = 1. This is, we call, the noise-freelensing map.
2 The weak lensing ray-tracing simulation data used in this paper are avail-able from T. Hamana on request, hamanatk@cc.nao.ac.jp.
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Searching for massive clusters in weak lensing surveys
3.3 Adding noise
To simulate a more realistic lensing survey, we add noises caused
by intrinsic ellipticities of source galaxies to the simulated lensing
map. The actual observations are further affected by other systematics such as imperfections in the telescope and/or the electric devicesof the detector. Examining these effects is beyond the scope of thepresent paper, and thus we do not consider such observational effects.
As extensively shown in Van Waerbeke (2000), the noise field
caused by intrinsic ellipticities is well approximated by the Gaussian random field on the angular scales of interest, if the ellipticities
are uncorrelated between different sources (see also Jain & Van
Waerbeke 2000). The Gaussian noise added to our κ map has variance as follows:
2
σpix
=

σ2 1
,
2
2 n g θpix

(18)

where θ pix = 0.24 arcmin is the pixel size of our κ maps. The noiserms on the pixels is σ pix = 0.22 which is much larger than the rms
of the lensing convergence field.

3.4 Halo catalogues on the light cone
We identify dark matter haloes in five low-redshift outputs of the
N-body simulation using the standard friends-of-friends algorithmwith a linking parameter of b = 0.164 (in units of mean particle
separation). Haloes with mass greater than 1 × 1013 h −1 M , whichcorresponds to the mass of 146 simulation particles, are used in thefollowing analyses. We define the halo centre to be the centre-ofmass position of the member particles.
To evaluate the halo shape, we compute the inertia tensor of themass distribution, i.e.


Ii j =

dx 3 (xi − x̄i )(x j − x̄ j )ρ(x)

= mp

Np


(19)
(xi,k − x̄i )(x j,k − x̄ j ),k=1

where i, j = 1–3, x̄i denotes the centre-of-mass position and N p is
the number of member particles of the halo. From the inertia tensor
we compute the eigenvectors and convert them to the axial ratios
a/c and a/b (a  b  c). We also compute the trace of the inertia
tensor,Q≡

1(I11 + I22 + I33 ).
3(20)This is compared with the expected value for the NFW halo, i.e.
Q NFW =

4π
3

 rvirdrr 4 ρNFW (r )02
= Mvir rvir
q(cNFW ),

(21)

with
q(cNFW ) =

3
2
cNFW
− 3cNFW− 6cNFW + 6(1 + cNFW ) log(1 + cNFW )
.
2
6cNFW
[(1 + cNFW ) log(1 + cNFW ) − cNFW ](22)
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899

Note that q(c NFW ) can be approximated by q(c NFW )  0.22c−0.26
NFW
for 3  c NFW  30. The ratio Q/Q NFW gives an estimate for theconcentration of the mass distribution. We also define the following
quantity as an estimate of the line-of-sight elongation of the halo
mass distribution:
R=

2I33
,
I11 + I22

(23)

where x 3 is taken by the direction parallel to the line of sight. Although this is rather a crude estimation, R > 1 or R < 1 generally
implies haloes elongated along or perpendicular to the line of sight.
Halo catalogues on the light cone are generated by stacking thesimulation outputs in the same manner as in the ray-tracing experiments. We use the 50 realizations that have the same underlying
matter distribution as used in the ray-tracing data. This combined
data set allows us to directly relate the distribution of dark matter
haloes to the weak lensing convergence map.
In summary, each halo in the mock catalogue has data on the mass,estimates of the halo shape Q and R defined by equations (20) and(23), respectively, the redshift computed from the radial distance toa halo, and the angular position converted from the spatial position.
In addition, for each halo, we compute the concentration parameterand the virial radius using equations (2) and (4), from the halo massand the redshift of the N-body data output time (i.e. z = 0, 0.1, 0.3,
0.5 and 0.8). These values allow us to evaluate the expected lensing
signal using equation (13) with equation (8), which is correct only
if the halo has the NFW profile. Similarly, we also compute Q NFWusing equation (21). Moreover, we introduce a ‘flag’ parameter foreach halo, to quantify the following possible problem that arises
from our use of the multiple-lens plane ray-tracing algorithm. As aresult of the projection of N-body particles into four lens planes, themember particles of one halo located close to the boundary could be
separated into different planes. Such a split halo would generate an
artificially deformed (or even separated) lensing signal in a κ map.
We assign flag = 1 for such haloes whose virial radius is larger thanthe distance between the halo centre and the closest plane, flag = 2
for haloes whose scaleradius is larger than the distance to the closest
plane, and flag = 0 otherwise. The fraction of flagged haloes is verysmall; 0.88 per cent for flag = 1 and 0.19 per cent for flag = 2, thus it
does not have significant influence on our analyses, but we carefully
check if the flagged haloes cause any artificial eccentric signals in
the lensing convergence map.
The expected lensing signal of each halo is converted into thepeak height in the same way as in equation (14), which we denoteas ν NFW . The total numbers of haloes with ν NFW > 5 and ν NFW > 4 inour 50 realizations are 459 and 871, respectively; this is sufficiently
large to allow for the statistical analyses of weak lensing haloes.
The redshift and mass distributions of haloes in the halo catalogue
agree well with the predictions given using the model in Section
2.2. Fig. 7 plots the accumulate number counts of haloes against
the expected peak height, ν NFW . The solid curve is the theoreticalprediction of equation (15) and agrees well with the simulation
result. This agreement is not surprising, because ingredients of thetheoretical model (the universal profile and halo mass function) arecalibrated by N-body simulations.
The counts of haloes with ν NFW > 5 in 2.0482 deg2 are found to
be N (ν NFW > 5) = 9.18 ± 3.63, where the error represents the rms
among 50 realizations. This is in good agreement with the theoreticalprediction of 8.71. However, the rms is somewhat large and slightly
larger than the Poisson fluctuation. Fig. 8 shows the distributionof numbers of haloes with ν NFW > 5 in each realization, quite awide spread is evidently seen. This is mainly caused by very strong

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T. Hamana, M. Takada and N. Yoshida

Figure 7. The number counts of the lensing peaks above a given thresholdν NFW per 4-deg2 area. The solid curve shows the model prediction computedfrom equation (15), while the filled circles with error bars denote the mean
and rms among 50 realizations of mock simulation data. Note that the peakheight of a halo is estimated from the halo mass and redshift in the manner
described in Section 3.4.

Figure 9. Upper panel: the scatter plot showing the relationship between
Q/Q NFW and the axial ratio a/c for haloes with M > 3 × 1013 h −1 M in
the z = 0.1 output. Here Q NFW is defined by equation (21) and is computed
using equation (4). Lower panel: the PDF of Q/Q NFW of haloes with M> 3 × 1013 h −1 M , the solid and dashed lines for z = 0.1 and z = 0.5
outputs, respectively.

extended mass distribution is likely to be associated with eccentricmass distributions such as haloes having substructures or an ongoing merger. We note that the PDF of the axial ratio peaks at a/c ∼0.5 and thus most of the haloes significantly deviate from sphericalsymmetry. This broad distribution leads to significant scatter in therelationship between the expected and observed lensing signals as
seen below.
Fig. 10 shows the PDF of the shape parameter, R, defined by equation (23) for massive haloes with M > 3 × 1013 h −1 M . A broad
distribution is evidently seen and we checked that the distribution

Figure 8. The frequency distribution of numbers of haloes with expectedpeak height (see Section 3.4 for a definition) ν NFW > 5 from 50 realizationsof 4 deg2 .

clustering of massive haloes. Accordingly, given a limited survey
area, the halo counts may vary significantly from survey to survey.
The lower panel of Fig. 9 shows the probability distributionfunctions (PDFs) of Q/Q NFW computed from massive haloes with
M > 3 × 1013 h −1 M (Q NFW is computed using the mean
mass–concentration relationship in equation 2). The PDF peaks at
Q/Q NFW ∼ 1 as expected, but has a rather broad distribution. No
significant difference in the PDFs between the five different redshift boxes was found. The upper panel shows the scatter plot in the
Q/Q NFW − a/c plane for haloes with M > 3 × 1013 h −1 M froman output of z = 0.1. It is clear that the mass concentration definedby Q/Q NFW correlates with the axial ratio, which implies that an

Figure 10. The PDF of the R defined by equation (23) for haloes with M >3 × 1013 h −1 M at z = 0.1 (solid line) and z = 0.5 (dashed line) outputs.
Note that the median of the distributions is ∼0.92, and 55 per cent of all thesamples has R < 1.
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Searching for massive clusters in weak lensing surveys

901

Figure 11. Upper left: the celestial distribution of haloes identified from one realization of mock light-cone data generated stacking N-body outputs. The large
filled circles denote haloes with M halo [h −1 M ] > 3 × 1014 , the small filled circles are for 3 × 1014 > M halo [h −1 M ] > 8 × 1013 and the dots are for 8 ×
1013 > M halo [h −1 M ] > 1 × 1013 . Only the haloes within the redshift interval of between 0.1 and 0.7 are displayed. Upper right: the grey map shows thelensing convergence map on 5122 pixels with a pixel size of 0.24 arcmin obtained from a ray-tracing experiment through the same mass distribution as usedin the upper-left panel. The circles show the positions of haloes with ν NFW > 4, the centre of which is set at the position of each halo and the radius of whichshows the projected virial radius (see Section 3.4 for details). Bottom left: same as the top-left panel but the convergence map is smoothed by the Gaussian
window function with θ G = 1 arcmin. The edge regions of the field with a 5-arcmin width are masked owing to incomplete smoothing caused by the lack of κdata on the outer region. The grey scale of this plot is the same as that of the bottom-right panel. Bottom right: same as the bottom-left panel but the Gaussian
noise is added before smoothing is applied. From comparison with the other plots, it is clear that high peaks resulting from massive haloes are still identifiable.

does not largely change over the redshift range under consideration(z < 0.8). It is important to note that the median of the distributionis less than unity (∼0.92), thus more than half of the massive haloes(specifically 55 per cent for haloes with M > 3 × 1013 h −1 M irrespective of the redshift) have R < 1. This is a natural consequenceof the random orientation of the haloes’ major axis; among the spatial three directions, it is more likely that the major axis of a halois elongated along either of two directions perpendicular to the lineof sight rather than along it. The skewed distribution leads to systematic bias in the relationship between the expected and observed
lensing signals (see Section 5.2 and Appendix C for details).
3.5 Visual impressions
Fig. 11 shows the lensing convergence maps and the celestial distribution of the haloes, which are taken from one of the 50 realizations.
In the top-right-hand panel, the grey scale shows the κ value with
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panel shows the same convergence map as in the top-right-handpanel but after being smoothed by the Gaussian filter function ofθ G = 1 arcmin. Adding the Gaussian noise yields the noisy map
shown in the bottom-right-hand panel. While the noise indeed alters the appearance of the convergence map, the very high peaks
remain identifiable.
The top-left-hand panel shows the distribution of all the haloes
identified by the friends-of-friends (FOF) scheme in the original
N-body simulations from which the convergence maps in the otherplots are generated. Note that the point size scales with the halo
masses. A good correspondence with respect to position between
massive haloes and high lensing signals on the κ map is seen. This
correspondence is more clearly seen if haloes are selected by meansof the expected ν NFW value: in the convergence maps (the top-righthand and two bottom panels) the positions of haloes with ν NFW >4 are marked by the circles with their centres being placed at thecentre-of-mass of the halo, and their radii show their angular virial
radius (see Section 3.4). The improvement in the correspondence

902

T. Hamana, M. Takada and N. Yoshidareflects the fact that the lensing signal is determined by combined
contributions from the halo mass, the redshift and the halo concentration, rather by the mass alone.
4 R E S U LT S O N T H E P E A K C O U N T S
In this section, we study the number counts of high peaks in thelensing convergence map. To do this, we use three kinds of κ maps:(i) the noise-free κ maps, (ii) the noisy κ maps where the Gaussian
noise was added in the manner described in Section 3; and (iii) the
pure-noise maps without the cosmological lensing signals. All thesemaps are smoothed with a Gaussian window function with θ G =1 arcmin (equation 11). We identify positive (negative) curvaturepeaks in the maps as pixels that have higher (lower) values of κ
than all the surrounding eight pixels (Jain & Van Waerbeke 2000,
and see also Miyazaki et al. 2002 for the first application to actual
observational data). The peak height is defined as the ratio of theconvergence value at the peak to the noise rms, as given by equation(14). For all the three cases, σ noise = 0.02 (σ  = 0.4 and n g =30 arcmin−2 ) is adopted.
4.1 The peak PDF
The PDF of peaks in the convergence map is studied in detail in Jain
& Van Waerbeke (2000). Here we briefly summarize the characteristics of the peak PDF that are relevant for the subsequent discussion.
Fig. 12 shows the peak PDFs in the noise-free κ (dashed histogram), the noisy κ map (solid histogram) and the pure-noise map(dotted histogram). The upper panel shows the sum of the positive and negative curvature peaks, while the lower panel shows thecontribution from only the positive peaks. In the absence of noise,the peak PDF reflects the characteristics of non-linear gravitational
clustering. Most of the negative peaks are a result of negative curvature peaks (troughs). There is a cut-off in the negative peak heightbecause the mass density contrast cannot be lower than −1. One canalso find a tail toward the high positive peak height which arises fromthe lensing signals caused by massive haloes (Kruse & Schneider2000). Another noticeable feature seen from the lower panel is thatsome of the positive curvature peaks have a negative peak height,
which occurs when the peak caused by a halo is superposed on avoid region in the same line of sight. If the noise field is Gaussian,the PDF from the pure-noise map can be derived from Gaussian
random field theory (Bardeen et al. 1986; Bond & Efstathiou 1987;
Van Waerbeke 2000). The symmetric bimodal feature of the peak
PDF is owing to the superposition of two contributions from peaks
with positive and negative curvatures. While adding the noise to thecosmological κ map alters the peak PDF drastically, there remainsome characteristic features that contain useful cosmological information. First, the asymmetric double peaks are caused by the skewed
feature of the cosmological κ PDF as stated above. Second, there
are excesses over the noise PDF on both the high positive (ν > 3)
and negative peaks (ν < −3). These features were indeed detectedin the recent weak lensing survey by Miyazaki et al. (2002).
It is worth pointing out that the pure-noise PDF can be directly
estimated from actual observed data. First, we can generate a purenoise map by applying mass reconstruction to a randomized catalogue where the orientation of the source galaxy images is randomized (to remove coherent image distortions from lensing signals).
Then we can measure the peak PDF from the pure-noise map, as
performed in Miyazaki et al. (2002). This allows one to estimate thecontamination rate (the false peak rate) directly from the data.
4.2 High-peak counts
In the following, we restrict our discussion to the number counts ofhigh peaks that are mainly a result of massive haloes. Fig. 13 shows
the number density of (positive curvature) peaks per 1 deg2 . The

Figure 12. The simulation results for the PDFs of the peaks in the noisefree κ map (dashed histogram), in the noisy κ map (solid histogram) andin the pure-noise map (dotted histogram). Shown is the cumulative resultsfrom the 50 realizations. The upper panel shows the sum of the positive andnegative curvature peaks, while the lower panel shows the contribution fromthe positive peaks only. In the total 50 × 2.0482 -deg2 ’ area, there are 59 908,
107 165 and 114 250 peaks for the noise-free κ map, the noisy κ map andthe pure-noise map, respectively. The corresponding numbers for positive
curvature peaks are 31 043, 53 123, and 57 141.

Figure 13. Counts of peaks per 1 deg2 . The histograms are from the numerical experiments; the noisy κ maps (solid), the noise-free κ maps (dashed)
and the pure-noise realizations (dotted). The long-dashed line shows the prediction using the model (equation 16 with equation 15), and the dot-dashedline shows the prediction using Gaussian random field theory (e.g. Bond &Efstathiou 1987; Van Waerbeke 2000).
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Searching for massive clusters in weak lensing surveys
histograms show the average values computed from the 50 realizations of the numerical experiments; the noisy κ maps (solid), thenoise-free κ maps (dashed) and the pure-noise maps (dotted). The
long-dashed line shows the predictions of the model from equation (16), while the dot-dashed line shows the prediction for thepure-noise counts (Van Waerbeke 2000). When comparing the results for the lensing maps with and without noise, it is clear thatthe noise significantly increases the peak counts over the range ofpeak heights under consideration. Even high peaks with ν > 5 aresimilarly boosted, while there is no such event in the pure-noise
counts, as shown by the dotted curve. There are two main reasonsfor this enhancement as follows (see Section 5.3 and Appendix B
for further details). First, the addition of noise leads to modifications in the peak height as well in the peak position. Second, thenoise generates spurious peaks in the noisy maps. In particular,
the latter effect is likely to show up in a biased way depending on
the peak height. As the convergence field around a given high peakhas greater amplitude than the average field, adding the noise to sucha region is more likely to induce spurious high peaks around the real
peak. We develop a correction scheme to the model peak counts inAppendix B. The corrected counts are in reasonable agreement withthe simulation result for high peaks, ν > 5.

5 R E S U LT S O N T H E C O R R E S P O N D E N C E
BETWEEN HALOES AND PEAKSIn this section, we examine the correspondence between massive
haloes and peaks in the weak lensing convergence maps, using theresults of our ray-tracing experiments and the halo catalogues created from the N-body simulation outputs. In doing so, we follow
a realistic procedure which is to be used in weak lensing surveys:
we focus on peaks above a certain threshold in the lensing map
as being strong candidates for haloes. The usefulness of the weaklensing halo search can be quantified in terms of completeness andefficiency. As will be shown later in detail, choosing a threshold ofν th  4–5 gives an optimal balance between efficiency and completeness. We will also carefully examine if there is a correlation between the shape of haloes and their detectability, because this could
produce a bias in halo catalogues constructed from the weak lensing
method.
To make realistic weak lensing survey catalogues, we impose
a criterion on the field selection of simulations so that no nearby
massive halo is included in the target fields. Specifically, we discard
a simulated field if there is a halo (or possibly more than one) with
M halo > 1014 h −1 M at z < 0.1. Similar criteria, such as no known
nearby X-ray luminous cluster of galaxies or no nearby optically
identified massive clusters, are usually imposed on the field selectionof an actual observation, because a nearby massive cluster would
occupy a large portion of a survey field and makes a distant halosearch difficult and inefficient. Of the simulated fields 31 out of
50 pass this criterion. In addition, we discard one field in which
there is a massive halo with ν NFW > 5 having flag = 2 (see Section
3.4 for a description of flag).3 We thus analyse the remaining 30
fields. We note that we do not use data within 5 arcmin of the edgeof each simulated field owing to the incomplete smoothing causedby the lack of data at the outer region. The total field size is then
106 deg2 .

3 Including this field in our analysis does not change our conclusions, but

we decided to discard it for having conclusive results.

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903

5.1 Matching haloes with peaks in lensing mass maps
We shall now examine the relationships between the peak heightin weak-lensing maps and the properties of haloes through the expected peak height for the NFW haloes (ν NFW , see Section 2.1).
We restrict our discussion to haloes with ν NFW > 3 only. The halo
catalogues contain 856 such haloes in total. Alternatively, the raytracing simulations directly enable us to find peaks in the κ maps.
We consider two kinds of the κ maps; those with and without thenoise caused by intrinsic ellipticities. The peaks in the two maps
are identified in the manner described in Section 4. The noise-freemaps contain 669 peaks with ν lens > 3, while the noisy maps contain2417 peaks with ν noisy > 3. Thus, the noise significantly enhances
the peak counts. It is also worth noting that the 669 peaks in thenoise-free maps are significantly smaller than the expected numberof peaks from the halo catalogues. We will carefully investigate thisdiscrepancy in the subsequent sections.
We consider haloes and peaks with only ν > 3 to avoid mismatches between peaks and physically unrelated haloes. With thelarge number density of low-mass haloes it should be noted that
including such haloes rapidly increases the chance of mismatching.
This is especially the case when one attempts to match peaks inthe noisy κ maps with haloes because false peaks caused by noise
can be accidentally matched with physically unrelated haloes. The
criterion ν > 3 is chosen based on the fact that most such (high)peaks are expected to be caused by real haloes, even when the lensing map is contaminated by noise (see Fig. 13). In addition, thecriterion value is reasonably lower than the observational thresholdvalue (generally defined by ν  4). This guarantees most haloes ofinterest to us are included in the final halo catalogue.
We carry out the peak–halo matching in two directions: whether
a peak has a corresponding halo in the halo catalogue; and whether
a halo has a counter-peak in the simulated κ map. Note that thepeak position does not necessarily agree with the centre-of-massof the halo exactly and this is more likely the case for haloes thathave substructures or are in the merger process. For this reason,
we search for a matched pair candidate within a radius of 12 pixel(2.88 arcmin) from the peak position or from the halo centre. Thismaximum angular separation of 2.88 arcmin is chosen so that it islarger than the smoothing radius of θ G = 1 arcmin, but still smaller
than the angular virial radius of a massive halo at 0.1  z  0.7. If
there are more than one pair candidate within the radius, we takethe closest one as the primary candidate and consider the others as
the secondary candidate.
We classify the peak–halo matching into the following fiveclasses.
(i) Class (i) is that both directions agree; both a halo and a peak
are the primary pair candidate.
(ii) Class (ii) is defined as haloes that have no paired peak withina radius of 12 pixels from the halo centre. We will call such haloes
as ‘missing haloes’.
(iii) Class (iii) is defined as peaks with no paired halo within aradius of 12 pixels from the peak position.We will often refer tosuch peaks as ‘false peaks’.
(iv) Class (iv) denotes a halo that has a matched peak candidate,but is not the primary matched halo of that peak.
(v) Class (v), which is the opposite to the class (iv), is that a peakhas a matched halo candidate but is not the primary matched peakof that halo.
The results for the noise-free and noisy maps are summarized inTable 3.

904

T. Hamana, M. Takada and N. Yoshida

Table 3. Classification of matching relationships between haloes and peaks.
The double-lined arrows denote the primary match, while the single-lined
arrows denote the secondary match. The last two columns summarize thenumber of haloes/peaks in each class, the third (fourth) column is for matching between the halo catalogue and the peak list from the noise-free (noisy)κ map. Note that haloes with ν NFW > 3 (856 haloes in total) and peaks withν lens > 3 (669 peaks) and ν noisy > 3 (2417 peaks) are considered.
Class

Matching relationship

i
ii
iii
iv
v

halo=⇒⇐= peakhalo with no paired peak
peak with no paired halo
halo=⇒←- peak→ peakhalo-⇐=

Number of objects
ν NFW − ν noisy
ν NFW − ν lens
475
367
177
14
17

624
222
1637
10
156

The typical properties of haloes or peaks in classes (ii)–(v) are
explained as follows. Class (ii) haloes are those which have a peakheight lower than the criterion value (ν = 3) although the expectedpeak height, ν NFW , exceeds it. This is mainly caused by departuresfrom the NFW profile or/and the projection effect. In the class (iii)peaks there are two kinds of peaks. The first are peaks which havecorresponding haloes but have expected peak heights, ν NFW , lower
than the criterion value, even though the measured peak heights in a
κ map exceed it. This is again caused by departures from the NFW
profile or/and the projection effect. The other type are peaks which
do not have corresponding haloes but are generated by the noise.
The former accounts for most of the class (iii) peaks in noise-freeκ maps, while the latter accounts for a large fraction of those innoisy κ maps. Class (iv) haloes are caused by the projection effect;
if two or more haloes are located at different redshifts along almost
the same line of sight, after smoothing the κ map only one peak
appears, because the smoothing merges those into one structure.
Finally, in class (v) peaks there are two kinds of peaks. The first are
peaks caused by substructures within a halo. This accounts for mostof the class (v) peaks in noise-free κ maps. The other are false peaks
caused by the noise generated around the primary peak owing to themassive halo. Because the convergence field around the high peakis biased and thus larger than the average field, adding the noise intosuch a region is more likely to generate false peaks with a relatively
high peak height. This accounts for most of the class (v) peaks innoisy κ maps.
In the rest of this section, we consider only objects from classes(i)–(iii), and do not deal with objects from classes (iv) and (v), because objects from classes (iv) and (v) are secondary matched pairs
associated with nearby primary matched pairs. This roughly mimics
the actual observation procedure in which the secondary objects areinvestigated in detail in the course of a follow-up observation of the
primary matched object.
5.2 Haloes versus peaks in noise-free κ maps
Let us first look into the correspondence between high peaks identified in the noise-free κ maps (ν lens ) and properties of haloes throughthe expected peak height (ν NFW ). The main purpose of this subsection is to examine the amplitude of the irreducible scatter in therelationship induced by both the individuality of the mass distribution of a halo and the projection effect.
Fig. 14 shows the distribution of the objects of classes (i)–(iii) inthe ν NFW − ν lens plane. As class (ii) haloes do not have a matchedpeak with height ν lens > 3 in the κ map (although the haloes have

Figure 14. The correspondences between haloes and peaks identified in thenoise-free κ maps in the ν NFW − ν lens plane. Objects from classes (i)–(iii)
are shown. The class (ii) objects are placed within a narrow track aroundν lens ∼ 2, regardless of their true value, and the class (iii) objects are also
placed at ν NFW ∼ 2.

a high expected peak height of ν NFW > 3), such haloes are placedin a narrow locus around ν lens = 2 for illustrative purposes. For
similar reasons, the class (iii) peaks are placed around ν NFW = 2.
The distribution of the class (i) haloes shows large scatter aroundthe ideal relationship of ν lens = ν NFW (shown by the solid line in
the upper panel) over the range of ν under consideration. It is worth
noting that the distribution for a fixed ν NFW value is slightly biased
toward lower ν lens values. The mean and rms of the differences,ν lens − ν NFW , are −0.24 and 1.2, respectively. As no noise is added
to the κ map in this case, the scatter and bias must be ascribed totwo effects as follows. (i) The mass distribution within a halo is notspherical nor completely universal, causing deviations of the peakheights from the expected values computed in our simple model.
(ii) A chance projection of structures along the same line of sightalters the lensing signal caused by the halo. For the former, although
we have so far used the average relationship for the halo concentration, as given by equation (2), it was shown in Jing (2000) that thehalo concentrations among haloes with a given mass obey the lognormal distribution with variance σ (ln c NFW ) ≈ 0.2. This roughly
corresponds to a 20 per cent rms scatter in the c NFW relationship.
Given the approximate relationship κ ∝ c1−1.5
NFW for the peak heightcaused by a massive halo (see Section 2), the rms scatter in thec NFW distribution translates into 20 − 30 percent rms scatters in thelensing signal ν NFW and therefore equivalently in the ν lens − ν NFW
relationship. For example, for haloes with ν NFW = 5, a rms of σ ν ∼1 results. Second, lensing projection caused by physically unrelated
structures along the same line of sight of a halo also contributes tothe scatter in ν lens . Following Hoekstra (2001), this scatter can be
estimated from the cosmic weak lensing variance under the assumption that the halo and the large-scale structures are uncorrelated. For
the Gaussian smoothing scale θ G = 1 , the amplitude of the (linear)convergence variance is σ κ (1 ) = 0.01 for the CDM model, corresponding to the rms deviation of σ ν = 0.5 caused by the noise,
σ noise (1 ) = 0.02 (see equation 12). Hence, these two effects canbasically explain the amplitude of the scatter in the ν lens − ν NFW
relationship in Fig. 14.
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Searching for massive clusters in weak lensing surveys

Figure 15. The scatter plot for the distribution of the haloes in the z −Q/Q NFW plane (lower panel) and in the z − R plane (upper panel). The
filled circle and triangle symbols are for the class (ii) haloes with ν NFW >5. For comparison, class (i) haloes with 5 < ν NFW < 7.5 are also plotted by
the crosses. It is apparent from the lower panel that the class (ii) haloes canbe separated into two subclasses: those with Q/Q NFW > 2 are plotted by
the filled triangles, while the rest are plotted by the filled circles.

We also argue that the systematic bias in the ν lens − ν NFW relationship can be explained by the following two effects. The first is
the fact that more than half of the haloes have R < 1 as discussed in
Section 3.4 and Fig. 10. As R positively correlates with ν lens /ν NFW(see Fig. C2 in Appendix C), it causes the bias. The second is the
projection with a void region in the same line of sight which reduces the lensing signal (see Metzler et al. 2001; White et al. 2002;
Padmanabhan et al. 2003, for detailed study on the projection effect).
Let us turn to the class (ii) haloes in Fig. 14. Among the 367 class(ii) haloes, there are 17 haloes that are supposed to produce peaks
with ν > 5 in the κ maps if the haloes have the NFW profile (i.e. ν NFW> 5 in our notation). Why are these haloes missing? Fig. 15 explains
this in terms of deviation of the mass distribution within a halo fromthe spherically symmetric NFW profile. We again use the parameters Q and R to quantify the mass distribution within a halo (seeequations 20 and 23, respectively): Q/Q NFW < 1(>1) characterizes
a halo that has centrally more (less) concentrated mass distribution
than expected from the NFW profile, while R > 1(<1) describesthat the mass distribution is elongated along (perpendicular) the lineof sight (R = 1 for a spherically symmetric halo). Therefore, a halowith Q/Q NFW > 1 and/or R < 1 probably leads to a smaller lensing
signal than that expected from the NFW profile (see Appendix C for
more detailed analysis of the relationships between the halo shapeand the amplitude of the lensing signal). The upper panel of Fig. 15
shows the distribution of the class (ii) haloes in the z − Q/Q NFWplane, while the lower panel shows the distribution in the z − Rplane. For comparison, the figure also shows the distributions forthe class (i) haloes with 5 < ν NFW < 7.5 by the cross symbols.
From the lower panel, it is clear that class (ii) haloes can be divided
into two distinct subgroups; those which have Q/Q NFW > 2 (plottedby the filled triangles) or Q/Q NFW < 2 (plotted by the filled circles).
We indeed find that haloes with Q/Q NFW > 2 show clear signaturesof substructures or ongoing merging, resulting in an extended mass
distribution (the κ map of such haloes is shown in Fig. A1, for ex
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ample). The remaining objects with Q/Q NFW < 2 tend to be locatedat a low redshift (z < 0.25), where the lensing efficiency is relatively
small for the source redshift, z s = 1, under consideration, and havesmall R values, mostly R < 1. The examples are shown in Figs A2
and A3. These haloes typically have prominent substructures and thewhole mass distribution within the halo is indeed elongated along
the perpendicular direction to the line of sight. With the exceptionof such irregular systems, almost all massive haloes with ν NFW > 5
generate a high peak with ν lens > 3. It is important to note that the
distribution of the class (ii) haloes in Fig. 15 is not clearly distinguished from that of the class (i) haloes, reflecting the fact that thelensing signal is also affected by the projection effect.
We turn next to the class (iii) objects in Fig. 14 which are peaks
with ν lens  3 in the lensing map, but do not have corresponding
haloes in the halo catalogue (contains only haloes with ν NFW > 3).
Among the class (iii) objects there are eight events with ν lens > 5
which we call ‘false peaks’. We carefully looked into each case in theκ map and searched for a possibly related halo in the halo catalogue.
We found that the five events are associated with the substructuresof massive haloes or the secondary local peaks of merger systems(note that the peaks are separated from the primary peak of the halo
centre by more than our searching distance, 12pixels = 2.88 arcmin).
Fig. A4 shows the κ map of one such case. The remaining three peaks
are caused by a chance projection of two moderately massive haloes
at different redshifts (Fig. A5 shows one example). Therefore, withthe except of the three chance projection systems, the false peaks
with ν lens > 5 are all associated with a real massive halo as thesecondary peak.
To quantify the ability of a weak lensing survey to locate massive
haloes, we consider its completeness and efficiency. We define thecompleteness by the fraction of haloes, which are detected as highpeaks above a given threshold, ν th , in the κ map, relative to all thehaloes with the expected lensing signal, ν NFW , greater than a given
minimum value ν min :Ni (νlens > νth | νNFW > νmin )f c (νlens > νth | νNFW > νmin ) =, (24)
Ni+ii (νNFW > νmin )where N i+ii (ν NFW > ν min ) denotes the number of haloes with ν NFW> ν min which, under our classification, corresponds to the numberof such haloes in classes (i) and (ii), and N i (ν lens > ν th |ν NFW > ν min )denotes the haloes with ν NFW > ν min matched with the peaks withν lens > ν th (these are thus class (i) objects). Similarly, we define
the efficiency by the fraction of peaks, which are associated withhaloes with the expected peak height exceeding a given minimum
value ν min , relative to all the peaks with height greater than a given
threshold ν thNi (νNFW > νmin | νlens > νth )f e (νNFW > νmin | νlens > νth ) =, (25)
Ni+iii (νlens > νth )where N i+iii (ν lens > ν th ) denotes the number of peaks with ν lens > ν thwhich corresponds to the number of such peaks in classes (i) and (iii).
The results are plotted in the left-hand panels of Fig. 16 as afunction of ν th . The upper panels show the completeness ( f c ) andthe lower panels show the efficiency ( f e ). As expected, these two
estimates behave inversely, that is, when f e goes up, f c goes down.
Therefore the optimal choice of the ν th value should be determinedby a balance between these two estimates, which we discuss in thelatter section with taking into account the noise effects.
The completeness plot shows that only about half of the haloes
with ν NFW > 5 produce peaks with ν lens > 5 in the noise-free κmaps, and that 77 per cent of the haloes have ν lens > 4. As expected,
the completeness deteriorates if ν th  ν min . It is also shown that agood level of completeness (say f c  0.8) is attainable if one sets

906

T. Hamana, M. Takada and N. Yoshidaν th  ν min − 1. This can be understood from the fact that the rms
scatter caused by the individuality of haloes and the projection effectis σ ν ∼ 1.2. Note that some fraction of the haloes with ν lens values
that are much smaller than the expected peak heights ν NFW are irregular systems or unrelaxed systems, and therefore the completenessis likely to be improved for samples of regular, well-relaxed haloes.
It should be also noted that noise increases the completeness (seeSection 5.4 and the upper right-hand panel of Fig. 16).
Let us turn to the efficiency issue. As no noise is added to the κmaps, the efficiency given here should be regarded as an estimate
for an ideal lensing survey. Clearly, the efficiency is very high. If
one sets the detection threshold to ν th = 5, 73 per cent of all the
peaks correspond to real haloes with ν NFW > 5, and 95 per cent forν NFW > 3. Even for ν th = 4 the efficiency is still high, 90 per cent forν NFW > 3. Therefore, in the absence of noise, the contamination rate(the fraction of false peaks) is very small for a detection thresholdof ν th > 4.
Fig. 17 shows the scatter plot of the haloes of classes (i) and (ii)in the redshift–halo mass plane. The contours show tracks of the
expected peak height ν NFW from 3 to 9 at an interval of ν NFW =1. The class (i) haloes matched with peaks in the noise-free κ maps
are plotted in the upper-left-hand (ν lens > 5), upper-right-hand (5 >ν lens > 4) and lower-left-hand panels (4 > ν lens > 3), respectively.
The lower-right-hand panel shows the scatter plot for the class (ii)haloes, namely ‘missing haloes’. Importantly, haloes within a small
range of the peak height are spread over a wide range of masses as aresult of the projection effect and the individuality of the halo mass
distribution.

tively examine how the noise alters the peaks caused by haloes andgenerates false peaks in the lensing map.
Matching between the noise-free peaks and the noisy peaks wasdone in the same manner described in Section 5.1. In what follows,
the peaks classified in classes (i)–(iii) (their analogy to ν lens − ν noisy
correspondence) are considered. Again, we only deal with peaks
with ν > 3.
Fig. 18 shows the relationships between ν lens and ν noisy plottedin the same manner as in Fig. 14. We emphasize that the statisticalproperties of the ν lens − ν noisy relationship do not follow a Gaussian
distribution, even though Gaussian noise is added. The reason is
as follows. The noise not only alters the peak height from massive
haloes but also changes the peak position. For example, if a large
positive noise is accidentally added on a neighbouring pixel aroundthe original peak position, the pixel could be identified as a peak thatis even higher than the original peak. Therefore, the distribution ofthe ν noisy values is not simply expressed as the convolution of that ofν lens values with Gaussian noise, but the ν lens − ν noisy relationshipbecomes asymmetric and is biased toward large ν noisy values. This
explains the asymmetric shown in Fig. 18 ν lens − ν noisy relationshipclearly. The mean and rms of the differences, ν noisy − ν lens , among
the class (i) objects with ν lens > 4 (this restriction is imposed to
avoid an incomplete sampling of low ν noisy pairs) are 0.31 and 0.90,
respectively. Therefore the peak heights of massive haloes identified
in a noisy κ map are, on average, boosted by the noise. This bias
partly accounts for the excess in the peak number counts over thesimple theoretical prediction (equation 16) as discussed in Section
4.2.
The class (ii) objects here correspond to the missing peaks, thepeaks that have ν lens > 3 but are absent in the corresponding noisy κ
map or have ν noisy < 3. With the exception of one peak with ν lens =5.1, the missing peaks sit in the range of ν lens  4. These peaks
are likely to be erased or their heights decreased by the addition ofnegative noise. The one exception (the peak with ν lens = 5.1 at themissing peak locus) is a local peak associated with a substructureof a massive halo which disappears in the noisy κ map. We may,therefore, safely conclude that more than 80 per cent of the peaks
with ν lens  3 in the noise-free map remain in the noisy κ map.
Next, we consider the false peaks, the peaks that are identified in
the noisy κ map with ν noisy > 3 but does not correspond to peaksin the noise-free κ map or have ν lens < 3, which are placed aroundν lens ∼ 2 in Fig. 18. A large fraction of the false peaks, especially forν noisy < 4, are the false peaks generated by the noise (see Section 4for details), and the small fraction (especially those with relatively
high ν noisy ) is associated with real peaks on which a large noise isaccidentally added. There are actually 14 objects with ν noisy > 5 in
the false peaks locus. All of those have a corresponding peak in thenoise-free κ map with low peak height (1.5 < ν lens < 2.9) on whicha relatively large noise of 2 ∼ 3.5σ is accidentally added. Therefore,
all peaks with ν noisy > 5 identified in the noisy κ map are associatedwith real peaks in the noise-free κ map. Alternatively, most of theclass (iii) peaks with lower heights are false peaks generated by
the noise. In fact, one can find the rapid increasing number of thefalse peaks toward low peak heights, especially for ν noisy < 4, asshown in Section 4. It immediately follows from this that to reducethe contamination rate, a high threshold value, ν th  4 at least, isrequired.

5.3 Effects of the noise on the lensing peaks

5.4 Haloes versus peaks in noisy κ maps

Before examining any correspondences between peaks identified in
the noisy κ maps and haloes in the halo catalogue, we quantita-

In this subsection, we examine correspondences between haloes in
the halo catalogue and peaks in the κ map including the noise caused

Figure 16. Upper panels: the completeness, f c (ν > ν th |ν NFW > ν min ),defined by equation (24), as a function of the threshold peak height ν th . Threecases for the minimum expected peak value, ν min = 3, 4 and 5 are shownby the filled squares, triangles and circles, respectively. Lower panels: theefficiency, f e (ν NFW > ν min |ν > ν th ), defined by equation (25), as a functionof the threshold peak height ν th . The left panels are for the noise-free case(ν = ν lens ), while the right panels are for the noisy case (ν = ν noisy ).

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Figure 17. The scatter plot for the haloes in the redshift–halo mass plane. The contours show the expected peak height ν NFW from 3 to 9 at an interval ofν NFW = 1. The class (i) haloes matched with peaks identified in noise-free κ maps are plotted in the upper-left (ν lens > 5), upper-right (5 > ν lens > 4) andlower-left panels (4 > ν lens > 3). The lower-right panel shows the scatter plot for the class (ii) ‘missing haloes’.

Figure 18. The correspondences between the peak heights identified in thenoise-free and noisy κ maps. The peaks, which have ν lens > 3 but are missedin the noisy κ map or have ν noisy < 3, are placed within a narrow locus atν noisy ∼ 2, regardless of their true ν noisy value. Similarly, the peaks, whichhave ν noisy > 3 but no paired peak with ν lens > 3, are placed at ν lens ∼ 2.

by intrinsic ellipticities. Fig. 19 shows the distribution of the class(i–iii) objects in the ν NFW − ν noisy plane plotted in the same manneras in Fig. 14. The comparison with Fig. 14 reveals that the noise
not only increases the scatter in the ν noisy − ν NFW relationship but
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Figure 19. Same as for Fig. 14 but for the correspondence between haloes
and peaks identified in the noisy κ maps.

also causes systematic bias in that the peak heights in the noisy
map are statistically boosted, as seen in the previous subsection.
Interestingly, this bias mean the completeness of the weak-lensing
halo search can be improved. The upper-right-hand and lower-righthand panels in Fig. 16 show the completeness and efficiency, definedin the same manner as equations (24) and (25), for the noisy maps.

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T. Hamana, M. Takada and N. Yoshida

About 81 (63) per cent of massive haloes with ν NFW > 5 (>4) are
identified as peaks with ν noisy  4 in the noisy maps. This is oneof the most important results of this paper, indicating that a highcompleteness of the weak lensing halo search could be attained ina realistic data.
There are 13 missing haloes that have ν NFW > 5 but ν noisy < 3 orhave no corresponding peak in the noisy κ map. We look into eachcase in the κ map and find that 11 haloes out of 13 are irregularsystems such as unrelaxed systems with substructures or ongoingmerger. Therefore, the completeness of the weak lensing halo search
for regular, well-relaxed systems would be higher than the estimate
plotted in Fig. 16. The remaining two missing haloes, which have
ν NFW = 5.3 and 5.4, are haloes which accidentally meet a large
negative noise so that their κ values are below ν noisy = 3. The fractionof such haloes over all haloes with ν NFW > 5 is 0.8 per cent (two out
of 250 haloes), which is very small indeed. Hence, we can safely
conclude that almost all haloes with ν NFW > 5 generate peaks withν noisy > 3 even in the presence of noise.
Let us turn to the class (iii) objects. As the noise generates false
peaks in the κ map, the efficiency deteriorates and becomes worse
than in the noise-free case (see Fig. 16). The fraction of contamination by false peaks is larger with a lower peak height because thenoise generates relatively lower peaks, mostly ν noisy < 4. Therefore,
if ν th = 3 is employed as the detection threshold, only 28 per cent of
all peaks are associated with haloes with ν NFW > 3. Note that as weare only considering haloes with ν NFW > 3, some fraction of missing
peaks could be associated with haloes with ν NFW < 3. Alternatively,
the efficiencies for higher detection thresholds are reasonably high:for ν th = 5 (ν th = 4), 85 per cent (56 per cent) of all peaks withν noisy > ν th are associated with haloes with ν NFW > 3.

From Fig. 16, it is clear that the improved efficiency achievedby setting a higher detection threshold is a ‘trade-off’ for poorer
completeness. We also note that the peak counts vary sensitively withthe threshold. For example, the counts with N (ν > 4) are about twiceas large as N (ν > 5) (see Section 4). Hence, the detection thresholdshould be determined so that the efficiency and completeness are
optimized for specific survey strategies. The results we have so far
shown indicate that ν noisy  4–5 is reasonably optimal for current
typical weak lensing surveys. To be more specific, the results fromthe simulated κ maps in the CDM model indicate that the mean
number of haloes with ν NFW > 4 is N halo (ν NFW > 4) = 37 per
10 deg2 , and on average 63 per cent of such haloes (23 out of 37) aredetected with ν noisy > 4. The completeness is better for higher ν NFW
haloes: for haloes with ν NFW > 5 the mean number is N halo (ν NFW >5) = 19 per 10 deg2 , where 81 per cent of the haloes (15 out of 19)
are detected with ν noisy > 4, and 12 haloes have a high-lensing signalof ν noisy > 5. With respects to efficiency, if one sets the detection
threshold at ν th = 4, the mean number of peaks is N peak (ν noisy >4) = 62 per 10 deg2 . Among the 62 peaks, 23 are accountable to real
haloes with ν NFW > 4, 13 peaks are haloes with 3 < ν NFW < 4 andthe remaining 26 peaks are likely to be either haloes with ν NFW < 3or false peaks from the noise. Alternatively, if one adopts a higher
detection threshold of ν th = 5, the mean number of peaks is reduced
to N peak (ν noisy > 5) = 25 per 10 deg2 , However, completeness is
improved significantly: in fact it is found that 19 (3) peaks out of 25
are signals from real haloes with ν NFW > 4 (3 < ν NFW < 4).
Fig. 20 shows the scatter plot for the class (i) and class (ii) haloes
in the redshift–halo mass plane as in Fig. 17. From comparison with
Fig. 17, one finds that more haloes pass a certain detection threshold
than in the noise-free map, owing to the systematic bias caused by

Figure 20. Same as for Fig. 17 but for haloes matched (not matched, for lower-right-hand panel) with peaks identified in noisy κ maps.
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Figure 21. Same as for Fig. 19 but the halo mass is used instead of the
expected peak height of ν NFW . Note that only haloes with ν NFW > 3 areconsidered.

the noise. Another important point revealed in this figure is that thepeak height in the noisy κ map is a poor estimator of the halo mass.
For example, masses of the haloes detected with 4 < ν noisy < 5 spanover an order of magnitude. Even in a narrow redshift range such as
0.3 < z < 0.4, there still exits a significant scatter in the halo mass.
Note that we do not plot haloes with ν NFW < 3, as this could cause alarger scatter toward small mass ranges. It should also be noted thatin the distribution of the missing haloes (lower-right-hand panel) nostrong tendency with redshift is seen, indicating that completenesswould not vary significantly with redshift. This is largely a result ofour definition of completeness, that is, we define completeness sothat the effect of the selection function, which very strongly depends
on redshift, is decoupled.
So far we have examined the relationship between haloes andpeaks in a κ map in terms of peak height as opposed to halo mass
because the observable lensing is determined by a combination of
halo mass, redshift and noise level, rather than by halo mass alone.
Nevertheless, it is interesting to see what mass range of haloes produce the peaks in the noisy κ maps, which is shown in Fig. 21 as inFig. 19. Again only haloes with ν NFW > 3 are considered. As can beseen, the peaks of a given height are caused by haloes with a wide
range of masses. We define the number fraction in a similar way
to how completeness is defined by equation (24) but replacing theminimum peak height with the minimum halo mass as follows:f m (νnoisy > νth | Mhalo > Mmin ) =Ni (νnoisy > νth | Mhalo > Mmin ),Ni+ii (Mhalo > Mmin )(26)

where N i+ii (M halo > M min ) is the number of [classes (i) and (ii)]haloes with mass greater than M min , and N i (ν noisy > ν th |M halo >M min ) is the number of haloes with mass greater than M min having peak height greater than ν th . Fig. 22 shows this estimate asa function of ν th . It is apparent that f m declines toward a higher
detection threshold and for lower minimum mass. For a detection
threshold of ν th = 4, more than 70 per cent of haloes with M halo >2 × 1014 h −1 M are identified, while about half of haloes with
M halo > 5 × 1013 h −1 M are identified
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909

Figure 22. The number fraction of haloes identified by peaks with heights
higher than ν th relative to all the haloes with mass greater than a given
minimum mass, M min , f m (ν noisy > ν th |M halo > M min ) (see equation 26).
Three cases of M min = 5 × 1013 , 1 × 1014 and 2 × 1014 h −1 M areplotted. Note that only haloes with ν NFW > 3 are considered.

6 S U M M A RY A N D D I S C U S S I O N
We have investigated various aspects of weak lensing cluster surveys employing both analytic descriptions of dark matter haloes andthe mock data of weak lensing surveys generated from numericalsimulations. For the latter, we combined weak lensing ray tracing
through mass distribution and dark matter halo catalogues. Our major findings are summarized as follows.
(1) In Section 2, we examined the expected properties of weaklensing haloes using the analytic descriptions of the dark matter
haloes, including the universal density profile (Navarro et al. 1996;
Navarro et al. 1997) and the Press–Schechter halo mass function(Press & Schechter 1974; Sheth & Tormen 1999). We found that
Gaussian smoothing with θ G  1 arcmin gives the largest expected
weak lensing halo counts (which may, however, depend on thesource redshift and noise properties). We computed the selection
function of the weak lensing cluster survey and examined, in detail,the mass and redshift distribution of the weak lensing haloes. It was
shown that the detectability of haloes depends not only on the mass
but also strongly on the redshift.
(2) In Section 3.4, we compared the model prediction of the weaklensing halo counts developed by Kruse & Schneider (2000) and
Bartelmann et al. (2001) with the halo counts in the mock catalogue,
and found a good agreement. We also found a large scatter in thenumbers of weak lensing haloes within a field of 4 deg2 , which islarger than the Poisson fluctuation. This can be explained by thestrong clustering of massive haloes.
(3) In Section 4, we tested the predictions of the model of thepeak counts against the mock weak lensing data. It was found thata systematic bias is induced by the noise caused by intrinsic galaxy
ellipticities. This bias increases the peak counts, and hence the prediction using the model underestimates the counts. We developeda correction scheme in an empirical manner in Appendix B, andshowed that the improved model reproduces the peak counts reasonably well.

910

T. Hamana, M. Takada and N. Yoshida(4) In Section 5.2, using the mock weak lensing data combinedwith the halo catalogue, we examined the matching between haloes
and peaks identified in the noise-free κ map. This was done toclarify the influence of halo individuality and the projection effecton a weak lensing halo search. We showed that these effects cause
not only a large scatter in the ν lens − ν NFW relationship but alsoa systematic bias. The mean and rms of the differences ν lens −ν NFW are −0.24 and 1.2, respectively. The level of scatter causedby the effects is comparable to that from the noise. We argue thatthe negative mean values are caused by a population of unrelaxed,
less centrally concentrated haloes and the fact that more than halfof the haloes are elongated in the direction perpendicular to the lineof sight. In addition, the projection of large under-dense regions
may partly account for the scatter. It was also shown that the chance
projection of massive haloes in the same line of sight is very rare.
(5) In Section 5.3, we used the noise-free κ map and the noisy
κ map to clarify how the peak distribution is affected by noise andhow false peaks are generated. We showed that the ν lens − ν noisy
distribution is biased toward large ν noisy values. The mean and rmsof the differences, ν noisy − ν lens , among the matched peaks with ν lens
> 4 and ν noisy > 4 are 0.31 and 0.90, respectively. Thus the noise
not only generates the scatter but also systematically ‘boosts’ thepeak heights. We found that almost all the peaks identified in thenoise-free κ map with ν lens > 4 are identified in the noisy κ map,
thus the noise rarely erases the high peaks. All the peaks with ν noisy> 5 identified in the noisy κ map are associated with real peaksin the noise-free κ map, whereas most of the false peaks (resulting
from the noise) have a relatively lower peak height, ν noisy < 4.
(6) In Section 5.4, we examined the correspondence between the
haloes and peaks identified in the noisy κ map. In particular, westudied the efficiency and completeness of the weak lensing halo
survey. We found that the detection threshold of ν th  4–5 gives anoptimal balance between efficiency and completeness. It was shownthat about 81 (62) per cent of massive haloes with ν NFW > 5 (>4)
are identified as high peaks with ν th = 4. This suggests that thecompleteness of the weak lensing cluster survey is reasonably high.
Concerning the efficiency, we found that for ν th = 4 (5), 56 per cent(82 per cent) of all the peaks are real signals from haloes with ν NFW> 3, while 37 per cent (63 per cent) are signals from haloes with ahigher expected peak height of ν NFW > 4. Therefore, it is possible
to attain relatively high levels of efficiency and completeness byselecting a moderately high detection threshold.
(7) In Section 5.4, we conclude that, for a detection threshold ofν th = 4, about half of the haloes with M halo > 5 × 1013 h −1 M
and more than 70 per cent of haloes with M halo > 2 × 1014 h −1 M
indeed produce the lensing signals with ν noisy  4.
The weak lensing cluster search technique explored in this paper
can be directly applied to data obtained by ongoing/future wide fieldsurveys. Extending the pilot 2.1-deg2 survey (Miyazaki et al. 2002),
a wide field weak lensing survey (Suprime33, PI: S. Miyazaki) is
being carried out using the wide field prime focus camera on the
Subaru telescope, Suprime-Cam (Miyazaki et al., private communication). Suprime33 will cover 33 deg2 with a limiting magnitudeof R = 25.5 (which provides n g  30 and z s   1). For the survey
exposure time of 1800 s and the field-of-view of the Suprime-Cam,
0.25 deg2 , the effective survey cost is 2.5 h per 1 deg2 including
overheads. The expected number of clusters that should be locatedby Suprime33 is 70 (120) at a threshold of ν th = 5 (ν th = 4). It
can thus be seen that weak lensing surveys exploiting a wide fieldcamera on a large telescope offer a reasonably efficient way of locat-

ing massive clusters. The CFHT Legacy Survey4 will significantly
enlarge the survey area. Its ‘wide’ survey will observe 170 deg2
in total with a limiting magnitude of i  = 25.5 (providing n g ∼30 and z s   1). The expected number of clusters is 360 (600)for ν th = 5 (ν th = 4), and thus it will provide invaluable data forstudies on large-scale structures of the Universe. Although herein
we have primarily considered weak lensing surveys that are feasible with current ground-based telescopes, future wide field surveys
using a space telescope will enable to us accurately measure theshape of distant, very small galaxy images, which will significantly
improve the capabilities of weak lensing cluster surveys and allow
the detection of lower mass and/or higher redshift clusters.
Finally, we note that it is, in principle, possible to enhance thedetection S/N by choosing a suitable smoothing function (Whiteet al. 2002; Padmanabhan et al. 2003). Padmanabhan et al. (2003)
proposed the function of W (θ) = (1 + θ/θ c )−2 with θ c ∼ 1 arcmin,which is motivated by the asymptotic behaviour of the outer part ofthe projected NFW profile. Such optimized filters may improve theS/N if the matter distribution is indeed close to the NFW profile.
Alternatively, its applicability to clusters whose mass distribution
deviates from the NFW profile remains unclear. It needs to be explored how much improvement can be obtained both in efficiency
and completeness. Optimizing the smoothing function clearly warrants further studies.
AC K N OW L E D G M E N T S
We are grateful to Bhuvnesh Jain for many valuable discussions.
We would like to thank Satoshi Miyazaki for useful discussions.
We thank the anonymous referee for detailed and constructive comments on the earlier manuscript which improved various aspectsof the paper. TH and NY thank University of Pennsylvania for
the warm hospitality during their visit where this work was initiated. TH and NY acknowledge support from the Japan Society for
Promotion of Science (JSPS) Research Fellowships. The N-bodysimulations used in this work were carried out by the Virgo Consortium at the computer centre at the Max-Planck-Institut, Garching(http://www.mpa-garching.mpg.de/NumCos). The numerical computations presented in this paper were partly carried out at ADAC
(the Astronomical Data Analysis Centre) of the National Astronomical Observatory, Japan.
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911

Figure A1. Missing halo, an example of a halo with an extended matter
distribution (possibly in the process of merging). The plus symbol denotes
the centre-of-mass position of the halo, and the circle shows the virial radius.
The map apparently shows a very eccentric density distribution. The halo
has M halo = 3.4 × 1014 h −1 M at z = 0.37, and the shape parameters
are Q/Q NFW = 2.1, R = 1.4 and c/a = 0.23. The expected peak value isν NFW = 7.0, but the ν value measured in the lensing map is less than 3.

A P P E N D I X A : T H E V I S UA L A P P E A R A N C E O FSYSTEMS WITH IRREGULAR SIGNALS
Here we present some κ maps, in which missing haloes (class ii) or
false peaks (class iii) with high signals (ν > 5) exists, for examplesof irregular systems. Figs A1–A5 show the (noise-free) convergencemaps smoothed with the Gaussian window function with θ G = 1
arcmin. The plus symbols denote the centre-of-mass position of thehaloes concerned and the circles show their virial radius. The crosses
denote the positions of the peaks.
Figs A1–A3 show three examples of missing haloes with the
expected peak value of ν NFW > 5. Such high ν NFW value missing
haloes mostly have an unrelaxed appearance and are probably ongoing merger systems. Figs A4 and A5 show two examples of false
peaks with ν lens > 5. Note that false peaks are rare in noise-freeκ maps. Most false peaks with ν lens > 5 are associated with substructures of massive haloes. The occurrence of chance projections
is very rare; we found three cases with ν lens > 5 in our 30 maps.

APPENDIX B: A SIMPLE CORRECTIONSCHEME TO THE PEAK COUNTSAs shown in Section 4 the theoretical model of the peak counts
given by equation (16) underestimates the counts obtained fromthe realistic noisy κ map by about 20 per cent. We here discuss thereasons for this change, and develop a correction scheme to the peakcounts in an empirical manner. Here we focus on the high peaks of
ν > 5.
The sources that modify the peak counts are as follows.
(1) The biased relationship between ν NFW and ν lens . In Section
5.2, it is found that the mean of the differences ν NFW − ν lens overclass (i) objects is −0.24. This means that the peak height computed
assuming the NFW profile tends to overestimate the real peak height.
The reason for this is discussed in Section 5.2.
C 2004 RAS, MNRAS 350, 893–913

Figure A2. Missing halo, an example of low-z unrelaxed system. The halo
(M halo = 2.0 × 1014 h −1 M , z = 0.13 and the shape parameters are
Q/Q NFW = 1.2, R = 0.36 and c/a = 0.41) does not seem to have reached arelaxed stage, but is probably an ongoing merger system. The expected peakvalue is ν NFW = 6.6 but the actual ν value is less than 3.

(2) Spurious peaks generated by the noise. It is found in Section 5that spurious peaks yield about a 10 per cent fraction of the total
number of high peaks (ν > 5).
(3) The scatter in the relationship between ν noisy and ν lens resulting from the noise. As will be shown, this scatter increases the peakcounts.
(4) The biased relationship between ν noisy and ν lens . In Section 5.3, it is found that the noise produces not only the scatterbut also the systematic bias, and that the mean of the differencesν noisy − ν lens over class (i) objects is 0.31. The peak heights in the
noisy κ map are thus statistically boosted by the noise. The reason
for this is discussed in Section 5.3.

912

T. Hamana, M. Takada and N. Yoshida

Figure A3. Missing halo, an example of an irregular system. The map
apparently shows a very eccentric density distribution. It is likely that thehalo (M halo = 1.9 × 1014 h −1 M , z = 0.20 and the shape parameters are
Q/Q NFW = 1.3, R = 0.25 and c/a = 0.36) is an ongoing merger system.
The expected peak value is ν NFW = 6.1 but the actual ν value is less than 3.

Figure A5. A very rare example of a chance projection of two moderately
massive haloes. The cross denotes the position of the peak with ν lens = 5.1.
This peak is not generated by a single halo but is generated by the chance
projection of two moderately massive haloes (M halo = 7.5 × 1013 h −1 M ,z = 0.35 and M halo = 5.4 × 1013 h −1 M , z = 0.68). Neither of the two
haloes on its own can generate a lensing signal that is larger than ν > 3.

ponential form n peak (ν) = n ∗ exp( pν), with a constant exponential
index of p. As can be seen in Fig. B1, this would be a reasonableapproximation over the range of 3  ν  8. This approximation
allows us to compute the integration in equation (B1) analytically,i.e.

n noisy (ν) 



ν 2
1
dν √ exp −
2
2π




= exp

Figure A4. An example of a local peak associated with the substructureof a massive halo. The cross denotes the position of the peak with ν lens =5.3. This is the local peak associated with the substructure of the massive
halo (M halo = 3.5 × 1014 h −1 M , z = 0.20, ν NFW = 8.8 and the shapeparameters are Q/Q NFW = 0.94, R = 0.61 and c/a = 0.66). The plus
symbol denotes the centre-of-mass position of the halo and the circle shows
the virial radius.

First, we develop the correction scheme for point (3). To do this,we use a simplified method, employing the assumption that very
high peaks are neither removed nor generated by the noise but theirpeak heights are altered by its addition. In other words, we ignore,
for the moment, any spurious peaks resulting from the noise andother sources. Under this above assumption, the peak counts in the
noisy κ map are given by combining the noise-free PDF (equation16) with the Gaussian noise PDF as follows:n noisy (ν) =



1
ν 2
dν  √ exp −22π

n peak (ν − ν  ).

(B1)

We further assume that within a small range of ν values the peakcounts in the original lensing map can be approximated by the ex-

p2
2





n ∗ exp p(ν − ν  )(B2)n peak (ν).

Thus the scatter resulting from the noise increases the counts by afactor of exp( p 2 /2).
Turn now to points (1) and (4) in the above list. From the biases
found in the mock data in the ν NFW − ν lens and ν lens − ν noisy relationships, we derive an approximate mean relationship of ν noisy− ν NFW = 0.07. We take this bias by simply shifting the counts as
n noisy (ν) → n noisy (ν + 0.07).
Finally, point (2) is corrected by simply increasing the peak countsby 10 per cent.
In Fig. B1, the corrected model prediction is plotted and compared with the results from numerical experiments. To compute thecorrected model prediction, n noisy (ν), we first computed the uncorrected counts, n peak (ν), at a small interval of δν using equation (16).
The local exponential slope of the counts is then evaluated by a finite differential scheme, p(ν i ) = [log n(ν i+1 ) − log n(ν i )]/δν. We
compute n noisy (ν) using equation (B2) with p(ν i ). Then, the correction n noisy (ν) → n noisy (ν + 0.07) was made for p(ν i ) at every
ν i to correct points (1) and (4). Finally the counts are increasedby 10 per cent to correct point (2). The corrected prediction agrees
with the noisy counts reasonably well for at least ν > 5, where theabove approximations are valid. On lower peak heights, i.e. ν < 4,
the prediction is systematically smaller than the measured values,
where false peaks resulting from noise make a larger contribution.
Note that the amplitude of effects (2) and (4) may depend on thenoise properties and on the choice of the smoothing scheme.
It is clearly seen in Fig. B1 that the corrected peak counts hasa slope that is very similar to that of the uncorrected counts. One
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Searching for massive clusters in weak lensing surveys

Figure B1. The number count of the lensing peaks per 1 deg2 . The histograms are from the numerical experiments from the noisy κ maps (solid)
and the noise-free κ maps (dashed). The long-dashed line shows the uncorrected model prediction computed from equation (16), while the solid line
shows the corrected model prediction (see Appendix B and Section 2.2 fordetails).

913

Figure C1. Shown is the scatter plot of the distribution of the haloes in the
Q/Q NFW − ν lens /ν NFW plane. Just class (i) haloes with ν NFW > 4 and ν lens
> 4 are plotted. The filled circles show the average value over 35 haloes
within bins whose ranges are denoted by the x-error bars, while the y-error
bars denote rms among the 35 haloes.

may thus make a simple correction to the model prediction obtained
from equation (16) by multiplying by a factor of 1.2, which gives
an alternative approximate way of computing the corrected peakcounts with a reasonable level of accuracy.
A P P E N D I X C : R E L AT I O N S H I P B E T W E E N T H E
HALO SHAPE AND κ AMPLITUDE
Here we examine the effect of individuality in the halo mass distribution on the amplitude of the κ peak. Figs C1 and C2 show how theratio ν lens /ν NFW varies with the halo shape parameters Q and R (seeequations 20 and 23 for definitions), where we only consider class(i) haloes with ν NFW > 4 and ν lens > 4. The filled circles show theaverage over haloes sitting in the range denoted by the x-direction
bar, where each interval is defined so that it contains 35 haloes, andthe error bar in the y-direction denotes the rms among them.
The correlations between the shape parameters and amplitude ofthe lensing signal are observed in both the figures, although the statistical significance is not very high. The anti-correlation in ν lens − Q
is because a higher concentration of mass distribution (Q < 1) compared to the average NFW profile gives a greater lensing signal. The
lensing signal is also amplified if the mass distribution within a halois elongated along the line of sight (R > 1). However, we also draw
attention to the rather large scatters in these relationships that may
arise from the following two sources. First, the shape parameters
we employ characterize a rather global mass distribution within aC 2004 RAS, MNRAS 350, 893–913

Figure C2. Same as for Fig. C1 but for the R − ν lens /ν NFW plane.

halo and cannot describe detailed structures. Second (and probably
more important) is the projection effect resulting from different
structures along the same line of sight of the halo.
This paper has been typeset from a TEX/LATEX file prepared by the author.