Shear Selected Cluster Cosmology I: Tomography and Optimal Filtering

3.4 MB·English

by  Joseph F. Hennawi

Draftversion February2,2008
PreprinttypesetusingLATEXstyleemulateapjv.4/12/04
SHEAR SELECTED CLUSTER COSMOLOGY: TOMOGRAPHY AND OPTIMAL FILTERING
Joseph F. Hennawi1,2,3, David N. Spergel2
Draft versionFebruary 2, 2008
ABSTRACT
We study the potential of weak lensing surveys to detect clusters of galaxies, using a fast Parti-
cle Mesh cosmological N-body simulation algorithm specifically tailored to investigate the statistics
of these shear selected clusters. In particular, we explore the degree to which the radial positions
of galaxy clusters can be determined tomographically, by using photometric redshifts of background
source galaxies. We quantify the errors in the tomographic redshifts, ∆z z z , and
5 tomography real
≡ −
study their dependence on mass, redshift, detection significance, and filtering scheme. For clusters
0
0 detectedwithsignaltonoiseratioS/N &4.5,thefractionofclusterswithtomographicredshifterrors
2 ∆z 0.2 is 45% and the root mean square deviation from the real redshifts is (∆z)2 1/2 = 0.41,
| | ≤ h i
with smaller errors for higher S/N ratios. A Tomographic Matched Filtering (TMF) scheme, which
n
combinestomographyandmatchedfiltering,isintroducedwhichoptimallydetectsclustersofgalaxies
a
J in weak lensing surveys. The TMF exploits the extra information provided by photometric redshifts
of background source galaxies, neglected in previous studies, to optimally weight the sources. The
1
efficacy and reliability of the TMF is investigated using a large ensemble of mock observations from
3
our simulations and detailed comparisons are made to other filters. Using photometric redshift in-

2 formation with the TMF enhances the number of clusters detected with S/N & 4.5 by as much as
v 76%, and it increases the dynamic range of weak lensing searches for clusters, detecting more high
9 redshift clusters and extending the mass sensitivity down to the scale of large groups. Furthermore,
4 we find that coarse redshift binning of source galaxies with as few as three bins is sufficient to realize
3 the gains of the TMF. Thus, the tomographic filtering techniques presented here can be applied to
4 currentgroundbasedweaklensingdatainasfewasthreebands,sincetwocolorsandmagnitudeprior
0 are sufficient to bin source galaxies this coarsely. Cosmological applications of shear selected cluster
4
samples are also discussed.
0
Subject headings: cosmology: theory – methods: numerical – clusters: general – large scale structure
/
h of the universe – gravitationallensing
p
-
o 1. INTRODUCTION Van Waerbeke & Mellier (2003) for a review). Recently,
r
t Mapping the distribution of dark matter in the uni- weak lensing by largescale structure, or “cosmic shear”,
s has been independently detected by several groups (see
a verse is now one of the primary goals of observational
Table1ofVan Waerbeke & Mellier(2003)foracompila-
: cosmology. The traditional approachhas been to survey
v tion of references). The primary focus of these observa-
the sky for a population of luminous objects, assuming
i tionshasbeento measurethe twopointstatisticalprop-
X a direct relationship between luminosity and dark mat-
erties of the shear field, which probes the matter power
r ter. While large galaxy surveys such as the 2dF Galaxy
a RedshiftSurvey(Colless et al. 2001)andtheSloanDig- spectrum in projection. However, as first suggested by
Kaiser(1995)andSchneider(1996),weaklensingsurveys
ital Sky Survey (York et al. 2000) are perhaps the first
can also be used to detect individual mass concentra-
examples that come to mind, future wide field X-ray
tions, allowing one to construct a shear selected sample
(Romer et al. 2001) and Sunyaev-Z’eldovich (SZ) sur-
of dark halos. Typically these will be galaxy cluster size
veys(Carlstrom, Holder, & Reese2002;Kosowsky2003)
will usher in an era where maps of the universe will be M & 1013.5 M⊙ objects, as the finite number of back-
groundsourcesandtheirintrinsicellipticityplacealower
availableatmultiple frequencies. Generically,allobjects
limitonthe sizeofahalothatcanbe detected. Todate,
thatemitradiationwillbe biasedtracersofthe underly-
there are only two such cases of spectroscopically con-
ing dark matter distribution. Hence, the interpretation
firmed cluster detections from ‘blank field’ weak lensing
of this data is limited by our ability to quantify the re-
observations(Wittman et al. 2001,2003). Although,re-
lationship between dark and luminous matter.
cently Schirmer et al. (2003, 2004) have used weak lens-
Weak gravitational lensing, or the coherent distor-
ing to confirm several color-selected optical cluster can-
tion of images of faint background galaxies by the fore-
didates.
ground matter distribution, provides a unique oppor-
A shear selected sample of galaxyclusters wouldbe of
tunity to map the dark matter directly without mak-
great astrophysical and cosmological interest. Probable
ing any assumptions about how baryons trace dark
biases in optical, X-ray, and SZ selected cluster samples
matter (see e.g., Bartelmann & Schneider (2001) or
with respect to richness, baryon fraction, morphology,
1 Department of Astronomy, University of California Berkeley, and dynamical state, would be revealed (Hughes et al.
Berkeley,CA94720 2004). Furthermore, if there does exist a population of
2 PrincetonUniversityObservatory,Princeton,NJ08544 high mass to light clusters that has hitherto gone un-
3 HubbleFellow
detected, this would cast serious doubt upon the “fair
2 HENNAWI & SPERGEL
sample” hypotheses (White et al. 1993), invoked to de- 1999;Levine, Schulz, & White2002;Majumdar & Mohr
termine the matter density parameter, Ω , from com- 2003; Hu 2003). Inclusion of uncertainties in the mass-
m
parisons of cluster baryon fraction to total cluster mass observablerelationssignificantlydegradesconstraintson
measurements (see e.g., Allen, Schmidt, & Bridle (2003) cosmologicalparametersobtainedfromsurveysthatrely
or Voevodkin & Vikhlinin (2004) for recent examples). on a baryonic proxy for mass (Levine, Schulz, & White
Perhaps a more contentious issue is the possible 2002; Majumdar & Mohr 2003). Although consistency
existence of a population of truly dark clusters. checks provide additional leverage (Hu 2003), expensive
Recently, several groups have purportedly detected follow up mass measurements are in general required to
dark mass concentrations from weak lensing (Fischer empirically calibrate the relationships between light and
1999; Erben et al. 2000; Umetsu & Futamase 2000; mass (Majumdar & Mohr 2003a;though see Lima & Hu
Miralles et al.2002; Dahle et al. 2003, thoughsee Erben 2004).
etal.2003). Iftheseobjectsareactually‘dark’virialized In this regard, the great advantage of a shear se-
mass concentrations as opposed to non-virialized ob- lected sample of galaxy clusters is that the selection
jects (Weinberg & Kamionkowski 2002) or projections function can be predicted ab initio, given a model
of large scale structure (Metzler, White, & Loken for structure formation. Because on the scales of
2001; White, van Waerbeke, & Mackey interest for weak lensing, only gravity is involved
2002; Padmanabhan, Seljak, & Pen 2003; and the mass is dominated by dark matter, pre-
Hamana, Takada, & Yoshida 2003), this would pose cision cosmological measurements will in principle
serious challenges for our current structure formation be limited by instrumental systematics rather than
paradigmandtheoriesofgalaxyformation. Theabsence unknown astrophysics. An accurate determination
of galaxies would require a complex form of feedback of the statistics of shear selected clusters from weak
that suppresses galaxy formation in clusters. If such a lensing requires N-body simulations of cosmological
dark cluster were X-ray or SZ faint, one would have to structure formation (though see Kruse & Schneider
invoke some exotic mechanism to segregate dark matter 1999; Bartelmann, Perrotta, & Baccigalupi 2002;
from baryons on Mpc scales. Weinberg & Kamionkowski 2003, for simple analytical
It is well known that a large sample of clus- treatments),which bringsus to the subjectof this work.
ters of galaxies out to moderate redshift z 0.5 In this paper we consider the detection of shear
∼
can be used to impose stringent constraints on cos- selected clusters using fast Particle Mesh (PM) N-
mological parameters. Several authors have re- body simulations of cosmological structure formation
cently suggested using the number count distribu- specifically tailored to investigate the statistics of
tion of galaxy clusters, from X-ray and SZ observa- weak lensing by clusters. Similar numerical stud-
tions (Haiman, Mohr, & Holder 2001; Weller & Battye ies carried out by other groups (Reblinsky et al.
2003), deep optical surveys (Newman et al. 2002), 1999; White, van Waerbeke, & Mackey
and weak lensing (Bartelmann, Perrotta,& Baccigalupi 2002; Padmanabhan, Seljak, & Pen 2003;
2002;Weinberg & Kamionkowski2003),toprobethena- Hamana, Takada, & Yoshida 2003), have focused
ture of the dark energy believed to be causing the accel- on the completeness and efficiency of weak lensing
erationoftheexpansionoftheuniverse. Besidesprobing cluster searches. Devising an optimal filtering scheme
the dark energy, these cluster samples would constrain to detect clusters has received little attention and
the matter density and amplitude of mass fluctuations furthermore, the additional information provided by
(see e.g., Bahcall & Fan 1998; Holder, Haiman, & Mohr photometric redshifts of background source galaxies has
2001; Pierpaoli, Borgani, Scott, & White 2003). In been neglected. In this paper we introduce the fast
addition, the clustering of such cluster samples PM simulation algorithm, deduce the optimal cluster
would provide another means to measure the mat- detection strategy, and investigate cluster tomography.
terpowerspectrum(Tadros, Efstathiou, & Dalton1998; Afundamentalobstacleforweaklensingstudiesofthe
Schuecker et al. 2001) which would provide complimen- matter distribution is the lack of radial information. All
tary parameter constraints (Majumdar & Mohr 2003) of the matter along the line-of-sight to a distant source
and might possibly allow for a measurement of the an- contributestothe lensing andsothe distortionreflectsa
gular diameter distance-redshift relation (Cooray et al. twodimensionalprojectionofthedarkmatter. Thislim-
2001) or baryon wiggles (Hu & Haiman 2003). itation can be overcome by using photometric redshifts
Yet, a limitation of the ‘baryon selection’ in optical, of the background source population, which will be dis-
X-ray, and SZ cluster samples, is that their selection tributed in redshift, thus allowing for a tomographic re-
function depends on observables (e.g., richness, veloc- constructionof the three dimensional foregroundmatter
ity dispersion, flux, temperature, SZ decrement) that distribution.
serve as a proxy for mass. Attempts to model these A variant of this technique has already been applied
mass-observable relationships depend on the uncertain to real weak lensing data. Wittman et al. (2001, 2003)
astrophysicsofgalaxyformationandthestateofbaryons tomographically determined the redshifts of two shear
in clusters. While these can be determined empirically, selected clusters at z =0.30 and z =0.70 respectively,
d d
scatter in the correlation between any two such observ- which were both confirmed by follow up spectroscopy to
ables is significant (see e.g., Kochanek et al. 2003); fur- be at z = 0.28 and z = 0.68. Recently, Taylor et al.
thermore, these mass-observable relations surely evolve (2004) conducted a weak lensing analysis of the previ-
with redshift. The mass function of galaxy clusters is ously knownAbell supercluster A901/2at z =0.16,and
steep and exponentially sensitive to changes in limiting tomographically confirmed its redshift from weak lens-
mass; moreover, errors in the mass-observable relations ing alone. They even constructed the first 3-D map of
mimic cosmological parameter changes (Viana & Liddle the dark matter potential of this cluster using the inver-
SHEAR SELECTED CLUSTERS 3
sion method introduced by Taylor (2001). Hu & Keeton The force on each particle is calculated by finite differ-
(2002) and Bacon & Taylor (2003) applied the Taylor encing the potential field, which is calculated from the
(2001)tomographicinversionmethodtoanalyticalmod- densityfieldinFourierspaceusingthekernel 1/k2 and
−
elsofclustersputin‘byhand’,withmixedsuccess. How- fastFouriertransforms(FFT). The griddeddensity field
ever, the reliability of these tomographic reconstruction is computed from the particles using the cloud-in-cell
techniques have yet to be characterized on realistic dis- (CIC)chargeassignmentscheme(Hockney & Eastwood
tributions of matter from N-body simulations; although 1981). The simulationsarestartedat1+z =50andare
it is clear that the fidelity of the reconstructions will be evolvedusingequalstepsintheexpansionfactorawitha
severely degraded by line of sight projections of large symplectic leapfrog integratordescribed in Quinn et al.
scale structure. (1997). The time step is taken to be less than the
In this work we introduce a complementary to- Courant condition.
mographic technique which is effectively two dimen- Beforeweproceedweshouldjustify ouruseofthe PM
sional. It is qualitatively similar to that used by algorithm for the N-body simulations. It is well known
Wittman et al. (2001, 2003) and Taylor et al. (2004) that the PM algorithm is ‘memory limited’ in the sense
and also to maximum likelihood methods developed that higher spatial resolution comes at the cost of stor-
to study cluster mass profiles (Geiger & Schneider ing an exceedingly large force mesh in random access
1998;Schneider, King, & Erben2000;King & Schneider memory. The main drawback of PM simulations is thus
2001). The efficacy and reliability of the method is limited dynamic range, whereas the primary advantage
investigated by applying it to our large ensemble of N- of the algorithm is speed. We compensate for the lack
bodysimulations. Additionally,aTomographicMatched of dynamic range by adopting the “tiling” algorithm in-
Filtering (TMF) scheme, which combines tomography troducedby White & Hu (2000), whereby the lightcone
and matched filtering, is introduced which takes ad- is tiled with a telescoping sequence of N-body simula-
vantage of the additional radial information provided tioncubes ofincreasingresolution. ThespeedofthePM
by photometric redshifts of source galaxies. The TMF algorithm is essential for characterizing the statistics of
is similar in spirit to the matched filtering algorithms shear selected clusters for the following reasons.
used to find clusters in optical surveys (Postman et al. First, massive clusters of galaxies are rare events.
1996; Kepner et al. 1999; White & Kochanek 2002; In order to quantify their statistical properties it is
Kochanek et al. 2003). Applying it to our ensemble essential to properly sample the primordial Gaus-
of simulations, we find that it is superior to fil- sian distribution of random phases of the large scale
tering techniques used previously (Reblinsky et al. structure along the light cone. Similar numerical
1999; White, van Waerbeke, & Mackey studies using N-body simulations (Reblinsky et al.
2002; Padmanabhan, Seljak, & Pen 2003; 1999; White, van Waerbeke, & Mackey
Hamana, Takada, & Yoshida 2003), detecting more 2002; Padmanabhan, Seljak, & Pen 2003;
clustersper squaredegreeandprobinglowermassesand Hamana, Takada, & Yoshida 2003) recycle the out-
higher redshifts. put of one simulation by reprojecting across the same
The outline of this paper is as follows. In 2 we de- simulation cube numerous times, hence avoiding the
scribeourimplementationofaPMcodetosimu§lateweak computationalchallengeofsimulatingthe entire Gpc3
lensing observations. The formalism behind our maxi- volume. However, the frequency of one very rar∼e event
mumlikelihoodtomographictechnique andthe adaptive could be grossly misrepresented by such a procedure,
matched filtering method is introduced in 3. In 4 we and there is a danger that the variance and tails of
§ §
apply the TMF and other filters to a large ensemble of the mass and redshift distributions of clusters will be
simulations and determine the ‘optimal’ filter for clus- incorrect. This misrepresentation is exacerbated by the
ter detection. The completeness of this optimal filter is fact that the primary leverage of cluster counts as a
discussed in 5. We assess the reliability of tomographic cosmological probe often comes from precisely those
§
redshifts in 6 and conclude in 7. objects on the exponential tail of the mass function. To
§ §
2. SIMULATINGWEAKLENSING avoid this problem we tile nearly the entire light cone
volume with unique simulations, clearly requiring a fast
2.1. The PM Code
algorithm.
Toevolvethedarkmatterdistributionintothenonlin- Second, a primary goal of this simulation program
ear regime, we use a particle-mesh (PM) N-body code. will be to explore how the distribution of shear se-
This code, written by Changbom Park, is described lected clusters depends on cosmological parameters
and tested in Park (1990) and Park & Gott (1991), (Hennawi & Spergel 2004), which requires simulating
and has been used in studies of peculiar velocities many cosmological models that span a large region of
(Berlind, Narayanan,& Weinberg 2000) and biased parameter space.
galaxy formation (Narayanan,Berlind, & Weinberg Finally, high resolution simulations are not re-
2000; Berlind, Narayanan,& Weinberg 2001). quired to accurately represent weak lensing because of
The simulation uses a staggered mesh to the small scale shot noise limit of the observations
compute forces on particles (Melott 1986; (White & Hu 2000). This can be understood heuris-
Centrella, Gallagher, Melott, & Bushouse 1988; Park tically from simple scaling arguments (Miralda-Escude
1990) , and those employed here use 2563 particles and 1991; Blandford, Saust, Brainerd, & Villumsen 1991).
a 5123 force mesh. For a galaxy cluster with a singular isothermal profile,
The initial conditions are generatedby displacing par- the effective lensing signal, γ√N, scales logarithmically
ticlesfromaregulargridusingtheZ’eldovichapproxima- with θ, the size of the smoothing aperture. This loga-
tion (see e.g., Efstathiou, Davis, White, & Frenk 1985).
4 HENNAWI & SPERGEL
rithmic scaling ensures that the dominant contribution in eqn. (1) and the rear of the current simulation cube.
to cluster weak lensing comes from scales of order the In addition, we begin calculating a new convergence in-
smoothing aperture, rather than much smaller scales tegral, κ(nˆ,z ) from z to the observer at each tile,
rear rear
which are not resolved. Since the noise in the obser- so that the final output of our simulation is a sequence
vationswesimulatewillrequireaveragingoverapertures of convergence planes densely spaced in redshift.
&1′,resolvingsmallscalepowerinthesimulationsisnot By evaluating the full convergence integral from the
essential. tiling sequence, the geometry of the rays and the evo-
All simulations described in this paper are carriedout lution of the potential are accurately represented. The
using the currentlyfavoredcolddarkmatter model with lines of sight originate on a square grid at z and con-
rear
a cosmological constant (ΛCDM). The parameters we vergeonanobserveratz =0. Theplanarlatticehasthe
simulated are a total (dark matter + baryons) matter same dimensionality as the force grid N = 512. For
grid
density parameter Ω = 0.295, a density of baryons staggered mesh PM algorithms, the effective resolution
m
Ω = 0.045, a cosmological constant Ω = 0.705, and of the simulation is set by the particle grid N =256;
b Λ part
a dimensionless Hubble constant h = 0.69, and normal- however, as we will be interested in the densest regions
ization parameter σ = 0.84, which are very close to occupied by clusters of galaxies, there will be no danger
8
the WMAP best fit model of Spergel et al. (2003). The of undersampling the particle distribution and the use
power spectrum is given by a scale invariant spectrum of the finer grid is justified. We work in the small an-
of adiabatic perturbations (n = 1), and for the transfer gle approximation,so that the sky can be treated as flat
function we use the fitting function of Eisenstein & Hu and hence all photons propagate along the z-axis of our
(1998). simulationcube foralllines ofsight. The convergenceof
the geodesics as the light cone narrows is implemented
2.2. Tiling the Line of Sight by multiplying by the appropriate geometric factor for
In whatfollows we describe our implementationof the each grid point, and by the tiling. Also as mentioned
White & Hu (2000) algorithm to tile the line of sight previously, the use of a unique simulation for each tile
with PM simulation cubes. (For a comprehensive dis- ensures the statistical independence of the fluctuations
cussion of numerical issues in weak lensing simulations, and accurately represents the statistics of rare events.
seee.g.,Jain, Seljak, & White2000;Vale & White2003; The tiling sequence is uniquely specified by an open-
White & Vale 2003). ing angle and a maximum and minimum redshift. The
The distortion of a source galaxy at redshift z in the redshiftzmax =4 is the highestsourceredshiftfor which
direction nˆ on the sky is determined by the shear field we evaluate the convergence. The source redshift distri-
γ(nˆ,z) of the matter between the observer and redshift butions we consider peak at z 1.0, thus there will be
∼
z. In the weak lensing approximation, the shear is com- a negligible number of source galaxies at redshift higher
pletelyspecifiedbytheconvergencebetweentheobserver than zmax. Below zmin = 0.2, we tile the remainder of
and comoving distance D D(z), given by the light cone with the same simulation, recycling the
z
≡ output of the last cube. This is necessary because we
3 Ω Dz D δ(Dnˆ,D) cannot shrink the box size beyond the nonlinear scale,
κ(nˆ,z)= m dDD 1 , (1)
2H0−2 Z0 (cid:18) − Dz(cid:19) a oretchtelrywbiseecatuhseePoMf thceodaebswenilclenooftmevoodlevecotuhpelidnegntsoitywacvoers-
and larger than the box. Furthermore, below z = 0.2,
dD 1 min
= , (2) it becomes computationally impractical to continue to
dz H(z) shrink the box and tile the light cone with successively
smaller unique simulations. The time step for the last
whichis valid for smallangles in the Limber approxima-
leg of the simulation with z < z is adjusted so as to
tion (Jain, Seljak, & White 2000; Vale & White 2003). min
resolve the next segment of the convergence integral in
Here δ is the density contrast field, a = 1/(1+z) is the
eqn. (1).
scale factor normalized to unity today, and H(z) is the
To maintain statistical independence of the fluctua-
redshiftdependentHubbleconstant. Theshearcanthen
tions during the short sequence where we recycle out-
be obtained via the Fourier relations
put, we project across a different axis of the simulation
l2 l2
γ = 1− 2κ, eachtimeandtheconvergencegridisdisplacedbyaran-
1 l2+l2 dom offset relative to the density grid. Rays that leave
1 2
2l l the box are remapped by periodicity. The sizes of the
γe2 = l2+1 2l2eκ, (3) simulationcubes in our tiling sequence are chosento ex-
1 2 actly reproduce the comoving volume of the light cone
where κ is the two-diemensional Feourier Transform (FT) for zmin = 0.2 < z < zmax = 4, ensuring that the num-
of the convergence field, and l = (l ,l ) is the Fourier ber of cluster size halos in the field of view is accurately
1 2
variable conjugate to position on the sky. represented. Accurately reproducing the volume of the
e
Following White & Hu (2000), we tile the line of sight light cone is not essential for studying two point statis-
with a sequence of telescoping simulation cubes of suc- tics, but is crucial for predicting cluster statistics and
cessively higher resolution. The integral in eqn. (1) is maybecomeincreasinglyrelevantforhigherorderstatis-
directly integrated across the tiles to the observer at ticswhichdependmoresensitivelyonrareevents. Below
z =0 using the overdensity field δ from the simulations. zmin =0.2, we misrepresentthe volume and geometry of
Specifically,foreachtile alongthe lightcone,the matter the light cone. The extra volume implies a slight excess
distribution is evolved from 1+z = 50 to the redshift ofclusterswithzmin 0.2andtheslabgeometryforthis
≤
z , corresponding to the next segment of the integral laststretchresultsinadeficitofsmallscalepowerinthe
rear
SHEAR SELECTED CLUSTERS 5
convergencefield,sincetheslabsprobelargerscalesthan
TABLE 1
the converging light cone. Both of these discrepancies Tiling Solution
have a negligible effect on the tomography and filtering
which we study in this paper. zrear Lbox Lgrid Mpart
Our simulation fields are 4◦ on a side and the angu- h−1 Mpc h−1 kpc 108 h−1 M⊙
lar pixel size of the convergence grids is 4◦/512= 0.47′. 4.00 (cid:0) 340.7 (cid:1) (cid:0) 665 (cid:1) (cid:0) 1933.1 (cid:1)
The simulations use 46 tiles of which 32 are unique PM 3.37 317.7 621 1567.6
2.88 296.3 579 1271.2
simulations (the last 14 tiles are recycled from the same
2.49 276.3 540 1030.9
simulation). Relevant parameters for the simulations in
2.18 257.6 503 836.0
our tiling scheme are listed in Table 1. The right panels
1.92 240.3 469 677.9
ofFigure1showthe dimensionless3-dmasspowerspec- 1.70 224.0 438 549.7
trum, ∆2mass(k,zrear), for four tiles, obtained by averag- 1.52 208.9 408 445.8
ing individual power spectra from 38 independent simu- 1.37 194.8 381 361.5
lations. Theyarecomparedtoanalyticalfittingformulae 1.23 181.7 355 293.2
for the nonlinear power spectrum using the prescription 1.11 169.4 331 237.7
of Smith et al. (2003). 1.01 158.0 309 192.8
0.92 147.3 288 156.3
From Table 1 is is apparent that we are forced to sim-
0.84 137.4 268 126.8
ulate small simulation cubes at low redshift, which re-
0.77 128.1 250 102.8
flects the need to achieve subarcminute angular resolu-
0.71 119.5 233 83.4
tionwhilereproducingthevolumeofthelightconeabove
0.65 111.4 218 67.6
zmin = 0.2. However, there is a danger that small cubes 0.60 103.9 203 54.8
misrepresent the evolution of structure if the fundamen- 0.55 96.9 189 44.5
tal mode of the simulation goes non-linear, because of 0.51 90.4 176 36.1
mode coupling to smaller scales that occurs during non- 0.47 84.3 165 29.2
linear evolution. However, note that the nonlinear scale 0.43 78.6 153 23.7
r =2π/k at z =0 defined by 0.40 73.3 143 19.2
nl nl
0.37 68.3 133 15.6
knl∆2 (k)dlnk=1, (4) 0.34 63.7 124 12.6
mass 0.32 59.4 116 10.3
isr =20.3 h−1Z0Mpcforthecosmologicalmodelusedin 0.29 55.4 108 8.3
nl 0.27 51.7 101 6.7
thispaper,where∆2 isthedimensionlesslinearpower
mass 0.25 48.2 94 5.5
spectrum. Accordingly, because even our smallest cubes 0.23 44.9 88 4.4
Lbox = 40.0 for z < zmin are larger than the nonlinear 0.22 41.9 82 3.6
scale, finite box effects should not be an issue. 0.20 40.0 78 3.1
0.19 40.0 78 3.1
2.3. Group Catalogs
0.17 40.0 78 3.1
Foreachtileusedtoevaluatetheconvergence,thepar- 0.16 40.0 78 3.1
ticle distributions aredumped and a halo catalog is pro- 0.14 40.0 78 3.1
0.13 40.0 78 3.1
duced by running a “friends–of–friends” (FOF) group
finder4 (see e.g., Davis, Efstathiou, Frenk, & White 0.12 40.0 78 3.1
0.10 40.0 78 3.1
1985) with the canonical linking length (in units of the
0.09 40.0 78 3.1
mean interparticle separation) b = 0.2. The FOF al-
0.07 40.0 78 3.1
gorithm groups the particles into equivalence classes by 0.06 40.0 78 3.1
linking together all pairs separated by less than b. We 0.05 40.0 78 3.1
imposeaminimumhalomassof1012.7h−1 M⊙,wherewe 0.03 40.0 78 3.1
followJenkins et al.(2001)anddefinethemassofahalo 0.02 40.0 78 3.1
as the mass of all the particles in the FOF group (with 0.01 40.0 78 3.1
b = 0.2). Note that this minimum mass corresponds to
NOTES.— Parameters for the 46 simulation cubes in our tiling
different numbers of particles for different tiles, as our
solution for the ΛCDM model. The sequence comprises output
larger tiles have poorer mass resolution. A cluster is de- from32uniquePMsimulations(the last14tilesrecyclethesame
finedasahalowithM >1013.5 h−1 M⊙. Itisclearfrom simulation). The column column zrear corresponds to the red-
Table 1 that although our mass resolution varies from shiftatwhichtheparticledistributionsofeach tileareoutput. Is
Mpart ∼ 3×108 −2×1011 h−1 M⊙ from the smallest κis(nˆal,szoretahr)e.soTuhrecesirzeedoshfitfhtethseimduelnasteiolynscpuabceedusceodnvfoerrgtehnactetpillean(eins
tiletothelargest,itisalwayssufficienttoresolvecluster comovingh−1 Mpc),thegridspacing(incomovingh−1 kpc),and
size halos. As we will see in §3, weak lensing selected theparticlemass(inh−1 M⊙)arealsogiven.
clusters span the redshift range 0.2 . z . 1.0, so that
we also spatially resolve the virial radii of cluster halos N2 calculation for each group, which is not com-
in this range. ∼ group
putationally feasible for a large ensemble of simulations.
Althoughthereexistseveralmethodstodefinethecen-
Insteadweoptforasimpler,fasterdefinitionofthegroup
ter of a FOF group, such as the location of the den-
centerwhichscalesasN . Asphereofradiusr is
sity (potential) maximum (minimum), these require an group sphere
placedatthe locationofa randomparticlein the group,
and the center of mass of the particles interior to the
4 We used the University of Washington NASA
sphere is calculated. Moving the sphere center to this
HPCC ESS group’s publicly available FOF code at
http://www-hpcc.astro.washington.edu new location, the procedure is iterated until the sphere
6 HENNAWI & SPERGEL
centerconvergestothecenterofmassoftheparticlesin-
side ofit. We set r =0.4h−1 Mpc oforderthe size
sphere 100
of a cluster virial radius, which gives halo centers that -3c ] 10-3
match the peaks in our lensing maps accurately. This Mp 10
moving sphere method is more robust than using the 3h
center of mass of all the particles in the FOF group be- M [ 10-4 k) 1
causethelatterquantityoftendoesnotcoincidewiththe g10 ∆2(δ
o
deTnshietylemftaxpiamnueml offorFdigisutruerb1edcoomrpealornesgattheed dstirffuecrteunrteias.l M)/dl 10-5 0.1
z,
mass function for several tiles, again averaged over 38 n( 0.01
d
simulations,with the universalmass function fitting for- 10-6
1014 1015 0.01 0.1 1
mula of Jenkins et al. (2001). The close agreement be- M [h-1 M ] k (h/Mpc)
sun
tween the mass function in the tiles and the fits indi- 100
cate that our simulations robustly reproduce the mass -3c ] 10-3
function down to 1013 h−1 M⊙. Figure 2 shows the cu- Mp 10
mulative number of halos in the light cone, dN(> M = 3h
1013 h−1 M⊙)/dz asafunctionofredshift,averagedover M [ 10-4 k) 1
the 38 16 deg2 fields, again compared to fitting formula. g10 ∆2(δ
o
Because we have been careful to accurately reproduce M)/dl 10-5 0.1
thevolumeofthelightcone,thetotalnumberofclusters z,
n(
as a function of redshift is reproduced for 0.2.z .2.5. d 0.01
10-6
Below zmin = 0.2, we overproduce clusters because of 1014 1015 0.1 1
theexcessvolumeofoursimulationswhereourtilingbe- M [h-1 Msun] k (h/Mpc)
comes inefficient. Above z 2.5, there is a tendency to
underproduce collapsed ob∼jects because the resolution -3pc ] 10-3 100
of our larger tiles approaches the size of the virial ra- M
lddeiunucsstiinrogvniriso≃fonc1ll.yu0sahtpe−rp1sreMactipazcbl.oeff0oc.lr2u0sa.t2nerd.s.uzAn.dsetr1h.p0e,reotffihdeucicoetvnioecrnyprffooorr- 3dlogM [ h 1010-4 ∆2(k)δ 101
z & 2.5 will not effect our conclusions on clusters from M)/ 10-5
weak lensing. z,
n( 0.1
d
10-6
2.4. Mock Observations 1014 1015 0.1 1 10
M [h-1 M ] k (h/Mpc)
sun
Theobservablesinweaklensingobservationsaretheel-
1000
ldiipsttiocrittieedsobfybtahcekgfrooruegnrdousonudrcmeagtatlearxideisstwrhibicuhtihonav.eEbveeenn -3pc ] 10-3
M 100
for the ideal case of no instrumental noise, observations 3h
athreelfiimniitteednubymtbheerionftrPinosiiscsoenllidpitsitcritibieustoedftshoeusrocuesrcoensatnhde gM [ 1010-4 ∆2(k)δ 10
o
sky. Here we simulate this ideal case by drawing a num- dl
ber density n of source galaxies from the source redshift M)/ 10-5 1
z,
distribution n(
d
pz(z)= 21z03z2e−z/z0, (5) 10-6 M 1 0[h14-1 Msun] 1015 0.10.1 k 1 (h/Mpc) 10
andplacingthematrandompositionsonour4◦ 4◦sim-
Fig. 1.— Mass functions and 3-d power spectra for four
×
ulated field. This redshift distribution peaks at 2z0, has tiles. From top to bottom the redshift of the tiles are zrear =
mean redshift z = 3z , and has been used in previous 1.52,1.01,0.60,0.20 (see Table 1 for simulation parameters). The
0
studies of cosmhicishear (Wittman et al. 2000). A frac- left panels show the mass function (histogram) averaged over 38
different simulations of the same tile compared to the fitting for-
tionf ofsourcegalaxieswillhavephotometricredshifts,
z mula(solid)ofJenkins etal.(2001). Therightpanelsshowthe3-d
wherethe errorsaredrawnfroma Gaussiandistribution power spectraof the density fieldforeach tile. Solidlines arethe
with dispersion σ = δ (1+z) and added to the source power spectra averaged over 38 simulations. Dashed curves show
z z
redshifts. We take n = 40 arcmin−2, z = 0.50 corre- thenonlinearpowerspectrafromtheanalyticalfittingformulaeof
0 Smithetal.(2003)and dotted curves arethe lineartheory power
sponding to a peak redshift of z = 1.0, and δz = 0.12. spectra.
Henceforth in this work we set f = 1 and assume all
z
source galaxies have photometric redshifts in order to
determine the efficacy of our tomographic and adaptive
matched filtering techniques for the best case. here is achievable if the optimal weighting discussed in
Theintrinsicellipticities,ǫ ,ofthesourcegalaxiesare (Bernstein & Jarvis 2002) is applied to the sources for
int
drawn from a Gaussian distribution with a single com- theasymptoticcaseofnoinstrumentalnoise(GaryBern-
ponent rms dispersion γ = 0.165 (Bernstein & Jarvis stein, private communication).
rms
2002). Note that there has been some confusion in the The simulations output convergence planes κ(nˆ,z )
rear
literature over the intrinsic ellipticity, and larger values for 0 z 4 densely spaced in redshift as listed
rear
have been employed in other studies. The value used in Tab≤le 1, and≤the shear fields γ(nˆ,z ) are obtained
rear
SHEAR SELECTED CLUSTERS 7
peak is identified if it is higher than all of its neighbor-
ing pixels. While more complex algorithms exist to find
peaks in pixelised data, because our maps have already
2g 10000 been smoothed, searching for local maxima is sufficient.
e Given a list of peaks from the smoothed map and the
d
0 1000 cluster catalogconstructedby applying FOF to the sim-
6. ulation tiles, we can correlate the peaks with the cluster
1
z/ catalog and locate all halos along the line of sight of a
d 100 given peak. A peak in a map is considered a cluster de-
)/
M tection if there is a cluster in the light cone within an
> aperture θ = 3′ centered on the peak. Although it
N( 10 might seemmamtchore appropriate to set this aperture size
d
to the field of view of some ‘follow up’ instrument, 3′ is
1
wellmatchedto the anglesubtended by the virialradius
0 1 2 3 4 of a typical cluster at z = 0.4, a typical redshift for a
z (redshift) shearselected cluster. Further our interesthere is in the
primaryobjectresponsibleforthe lensing signalandnot
Fig. 2.— Total number of clusters above M =1013.5 h−1 M⊙ objectsatlargerangularseparationthathappentobein
ina16deg−2patchofskyasafunctionofredshift. Thehistogram thevicinity ofthe peak. Inthe eventthatthere aremul-
shows the redshiftdistributionaveraged over 38simulations. The tiple clusters within the aperture centered on the peak,
solidlineistheredshiftdistributionobtainedfromthefittingfor- we take the most massive cluster to be the ‘match’ to
mulaofJenkinsetal.(2001).
that peak. Depending on the method of follow up, the
objectmostlikelytobedetectedwillbesomefunctionof
mass and redshift, though we neglect this subtlety here.
fromthe convergencevia FFT using eqns. (3). A source
Finally, to avoid the problem of multiple peaks in the
galaxyatredshiftz isshearedbythenearestshearplane
mapcorrespondingtothe samecluster,wekeeponlythe
with zrear <z, via the weak lensing relation highest peak in an aperture θ = 1′ centered on each
iso
ǫ=ǫ +γ, (6) peak,sothatlesssignificantpeaksnearhigherpeaksare
int
discarded.
which is valid for κ 1, γ 1. The shear is inter-
polated from the gri≪d onto| |th≪e source galaxy position 3. TOMOGRAPHYANDMATCHEDFILTERING
using bicubic spline interpolation. The end result is a Previous studies (Reblinsky et al.
mock catalog of source galaxy ellipticities with photo- 1999; White, van Waerbeke, & Mackey
metricredshifts,whichweusetoexplorethestatisticsof 2002; Padmanabhan, Seljak, & Pen 2003;
clusters below. Hamana, Takada, & Yoshida 2003), have neglected
In what follows we will want to compare mock obser- the extra information provided by photometric redshifts
vations with the noise properties described above with of source galaxies when searching for clusters in weak
noiseless weak lensing observations in order to assess lensing data. Without redshift information, the source
the intrinsic limitations oftomographyandweaklensing galaxy ellipticities provide a noisy measure of the mean
searchesfor clusters. We define noiseless observationsto shear
be an infinite number of source galaxies with zero in- γ¯(nˆ) dzp (z)γ(nˆ,z), (7)
z
trinsic ellipticity and photo-z error (i.e. γ = 0.0 and ≡
rms Z
δz = 0) drawn from the source redshift distribution in which is the shear out to a given redshift averaged over
eqn. (5), and placed on the simulation grid rather than the source redshift distribution p (z). Knowledge of the
z
at random positions. Placing the source galaxies on the source redshift enables one to measure the shear more
grid removes the Poisson clustering of the source galax- accurately,and in this section we present a tomographic
ies, so that the angular resolution is then limited by the matched filtering scheme which fully incorporates this
resolution of the simulation rather than shot noise. Be- extra information.
cause we cannot simulate an infinite number of sources, The tomographic matched filtering (TMF) technique
we draw 50 galaxies from eqn. (5) for each of the 5122 is similar in spirit to matched filtering algorithms
gridpoints, whichwe find is sufficient to convergeto the used to find clusters in optical surveys (Postman et al.
limitingcaseofnonoise. TheleftpanelofFigure3shows 1996; Kepner et al. 1999; White & Kochanek 2002;
a Kaiser-Squires reconstruction (Kaiser & Squires 1993) Kochanek et al. 2003) and is also qualitatively sim-
of the mean convergence field for noiseless data. ilar to the maximum likelihood techniques devel-
oped to study cluster mass profiles from weak lens-
2.5. Matching Peaks with Clusters
ing(Geiger & Schneider1998;Schneider, King, & Erben
Given a mock catalog of source galaxy ellipticities, we 2000; King & Schneider 2001). In what follows we
constructsmoothmapsbyconvolvingthedatawithsome present a well-defined procedure for identifying clusters
filters. Candidate mass selected clusters will correspond of galaxies in weak lensing surveys that makes optimal
to the peaks in these smoothed maps. We elaborate on use of both the shape and photometric redshift informa-
the filtering techniques in the next section, but focus tion of the background source galaxies. It can be ap-
here on the details of matching the peaks in smoothed plied to weak lensing data for which only the overall
maps with clusters from the simulation tiles. We locate source redshift distribution is known, for which photo-
peaks in the maps using a simple algorithm whereby a metric redshifts of the source galaxies are available, as
8 HENNAWI & SPERGEL
well as combinations of the two scenarios (i.e. a fraction and analogously for the tangential component of ǫ, and
of galaxies with photometric redshifts and the rest with hencethetangentialellipticityisrelatedtothetangential
only shapes). Furthermore, we will see in 4 that a few shear by ǫT = ǫT +γT. For our model cluster lens we
§ int
source redshift bins perform nearly as well as full pho- take the mass distribution to be spherically symmetric,
tometric redshift information, so simple color cuts and andhenceforth work primarily with the tangentialshear
apparent magnitude priors can be used to extract the and tangential ellipticity.
extra tomographic information. Foraclustercenteredatpositionθ~ withtheparameter
0
The matched filter identifies clusters in weak lensing vector (z ,θ ,A) the convergence can be written
d s
databyfindingthepeaksinaclusterlikelihoodmapgen-
Σ
eratedbyconvolvingthesourcegalaxyellipticitieswitha κ (~θ ,z ;z ,θ ,A)= model (14)
model j j d s
filterthatmodelsthedistortioncausedbytheforeground Σcrit
mass distribution. The peaks in the likelihood map will =Z(z ;z )A K(x ),
j d j
correspondtothelocationswherethematchbetweenthe
where x θ~ θ~ /θ , K(x) is the cluster convergence
cluster model and the data is maximized, hence giving j j 0 s
≡| − |
profile in units of θ , and the amplitude A is given by
the two dimensional location of the cluster. In addition, s
the algorithm produces a tomographic estimate of the Σ0Dd
A , (15)
cluster redshift using the lensing signal, similar to the ≡ c2/4πG
tomographic technique used by Wittman et al. (2001,
where Σ is the surface density normalization.
2003) and Taylor et al. (2004). 0
The tangential shear for this mass distribution is
3.1. Formalism γ (~θ ,z ;z ,θ ,A)=Z(z ;z )A G(x ) (16)
model j j d s j d j
Thegoalofthematchedfilteristomatchthedatatoa
where G is related to K by
modelthatdescribesthedistortionofbackgroundgalax-
ies by the foreground mass distribution, here a galaxy 2 x
G(x)= K(y)ydy K(x), (17)
cluster. For the sake of generality, we parameterize the x2 −
Z0
cluster lens by three quantities, a cluster redshift z , an
d from the relation γ = κ¯ κ. We omit the tangential
angular scale θ , and an amplitude A. −
s superscriptonγ andǫ fornotationalsimplicity.
Foraclusteratredshiftz theconvergenceforasource model model
d Theprobabilityofmeasuringasourcegalaxywithtan-
galaxy at position ~θj and redshift zj can be written gential ellipticity ǫT at position θ~ with source redshift
j j
κ(~θj,zj)=Z(zj;zd)κ∞(~θj) (8) zj, given the lens model in eqn. (16) is
where Dds P = 1 exp 1(ǫTj −γmodel)2 (18)
Z(z;zd)≡ Ds H(z−zd), (9) j 2πσj2 "−2 σj2 #
and
κ∞(~θj)= c2DΣ(/~θj4)πG. (10) inTphriencmipelaesuinrecqmluednetsecrornotrri(baustsiuomnsedfrtoombteheGainutsrsiinasnic) σelj2-
d lipticities of the source galaxies as well as an error term
Here Σ (z ,z ) = c2 Ds is the critical density for duetothephotometricredshifterrorofthesourcegalaxy
crit d s 4πGDdDds atz ,butinpracticetheformerwilldominatebecauseof
lensing with D , D , D the deflector, deflector-source, j
d ds s
thetheslowvariationofthefunctionZ(z;z )ineqn.(11)
and source distances, respectively. The Heaviside step d
with source redshift.
function accounts for the fact that sources in the fore-
Iftheonlyknowledgeofthesourceredshiftcomesfrom
groundofthedeflectorarenotlensed,andκ∞ isthecon-
theaggregatesourceredshiftdistribution,onemustwork
vergenceforahypotheticalsourceatinfinitedistance(see
with the first moment of Z(z) (Seitz & Schneider 1997;
e.g. Seitz & Schneider1997;King & Schneider2001,for
King & Schneider2001), Z = dz p (z)Z(z;z )where
similar treatments). The same relationshipholds for the z d
h i
p is the source redshift distribution in eqn. (5). In this
shear, γ(z) = Z(z;zd)γ∞. For a flat universe, eqn. (9) cazse, Z must be substituted foRr Z in eqn. (16).
simplifies to
h i
The likelihood of finding a cluster at position θ~ with
0
Z(z;z )=max 1 Dd,0 . (11) parameters(zd,θs,A), isthe productoverallthe dataof
d − D the individual probabilities
(cid:18) z (cid:19)
In the weak lensing regime κ 1, γ 1, the ob- (θ~,z ,θ ,A)= P (19)
0 d s j
≪ | | ≪ L
servedcomplexellipticity ofasourcegalaxyisrelatedto
j
Y
the shear field by eqn. (6). For a spherically symmet-
and the log likelihood is
ric mass distribution centered about the position θ~, the
shear will be entirely tangential 0 ln = 1 ln(2πσ2) 1 (ǫTj −γmodel)2. (20)
γ(θ)=−γT(~θ;θ~0)e2iφ (12) L −2Xj j − 2Xj σj2
whereφistheazimuthalangleaboutthepositionθ~. We Expanding the expression in eqn. (20) and dropping
0
employ the standard definition of the tangential shear terms that do not depend on the parameter vector gives
γT(~θ;θ~0)≡−[γ1cos(2φ)+γ2sin(2φ)] ln = 1 (2ǫTjγmodel−γm2odel). (21)
= [γ(~θ+θ~0)e−2iφ] (13) L 2 j σj2
−ℜ X
SHEAR SELECTED CLUSTERS 9
ThisexpressionhasalineardependenceonA,whichcan TABLE 2
be exploited to lower the dimensionality of the fit. Set- SourceRedshift Binning
ting the derivative of eqn.(21) with respect to A to zero
Type zlow – zhigh zcenter
gives
0 – 1.02 0.70
j σ12ǫTjZG Coarse 1.02 – 1.72 1.34
A(zd,θs)= 2P j σj12Z2G2 (22) 1.072 –– 0∞.77 02..5258
j
0.77 – 1.14 0.96
Substituting this back into eqPn. (21) finally gives
Fine 1.14 – 1.55 1.34
lnL(θ~0,zd,θs)= 12 ǫTjγσm2odel. (23) 12..5154 –– 2∞.14 12..8616
j j
X
NOTES.— Source redshift binnings used in this paper. The
Thus, the log likelihood is a convolution of the data
“Coarse” binning uses three redshift bins, while the “Fine” bin-
with a ‘matched filter’, with each source galaxy inverse
ning uses five. Bins are chosen to contain equal probability from
weightedby its respective error. The technique is ‘adap- thesource redshiftdistributionineqn. 5. Lower, upper, andcen-
tive’ in that it searches for parameters (zd,θs,A) that tralredshiftaredenotedbyzlow,zhigh,andzcenter,respectively.
maximizesthecontrastbetweentheclusterandtheback-
ground noise, making full use of the additional informa-
Here ǫT is the tangential ellipticity about θ~ at grid-
tion provided by photometric redshifts of source galax- ik 0
point k for source galaxies in the ith redshift bin. The
ies, which is reflected in the redshift ‘weights’ used in
number of grid points and source redshift bins are de-
eqn. (16).
noted by n and n respectively. The sum over i ex-
Ideally,onewouldperformtheconvolutionineqn.(23) g z
tends to n + 1 to indicate that the last bin will be
with the fast algorithmsintroducedand applied to weak z
those galaxies that do not have photometric redshifts,
lensing by Padmanabhan, Seljak, & Pen (2003), which
for which the mean value of Z (z ) must be used. For
do not require spatial binning. Here for simplicity, we d
h i
simplicitywehavetakenalltheellipticityerrorstobethe
opt to use FFT methods. Doing the convolutions with
same, σ σ2. This canbe easily generalizedto the gen-
FFT’srequiresbinningthesourcegalaxiesbothspatially ≡ j
andinredshift. Intheeventthatthescaleofthematched eralcase of different errorsfor eachsourcegalaxy,which
filter, θ , is kept fixed when maximizing the likelihood, would amount to modifying eqn. (24) to take the differ-
s
then the convolution in eqn. (23) can be done with a ent errors into account. Note that the sum overk in the
single FFT for each redshift bin, as we see below. numeratorofthislastexpressionforthelikelihoodisjust
For the spatial binning we follow Seitz & Schneider a convolution of the tangential shear with the ‘matched
(1996)andwritethe complexellipticity atanygridpoint filter’ G(xk) which can be done using an FFT.
k as If we define
ǫ(θ~k)= PjjWWjjǫj (24) Mi(θ~0)≡ PnknkggGǫTi2k(Gxk(x)k1)/2, (27)
with
P
θ~ θ~ 2 then the likelihood can fi(cid:2)nPally be writ(cid:3)ten
j k
W =exp | − | (25)
j − 2∆θ2 ! nz+1Z(z ;z )M (θ~) 2
ln (θ~,z ,θ )= 1 i i d i 0 , (28)
whInerech∆oθosiisnga stmheoostohuirncgelgenalgatxhy. redshift bins, we take L 0 d s 2σ2(cid:16)P niz+1Z2(zi;zd) (cid:17)
pz(z)into accountandrequire eachbin to haveanequal where M (θ~) is the convoPlution of the ellipticity of
i 0
fractionoftheprobability. BecausethefunctionZ(z;zd) the source galaxies in the ith redshift bin with the
in eqn. (11) varies slowly with redshift for z > zd, this normalized kernel G(x )/[ G2(x )]1/2. If there is no
binning can be relatively coarse. We consider two dif- k k k
photometric redshift information, n = 0, the redshift
ferent binnings: a coarse binnning with only 3 bins, as weight factors of Z 2(z ) Pin the numzerator and denom-
d
mightbe achievedby using aslittle astwo colorsandan h i
inator of eqn. (28) will cancel, and the log likelihood
apparentmagnitudeprior,andafinersetof5binswhich
just reduces to the square of a single map M(θ~),
would require multi color data. We don’t simulate the 0
which is a convolution of the tangential shear with the
effects of errors in assigning galaxies to their respective
normalized kernel. It is only this limiting case which
bins. The spacing of the bins is chosen to contain equal
has previously been considered by studies that neglect
probabilityasgivenbythesourceredshiftdistributionin
photometric redshift information (Reblinsky et al.
eqn. (5). The redshift ranges of the bins and the central
1999; White, van Waerbeke, & Mackey
redshiftarelistedinTable 2. We willhenceforthreferto
2002; Padmanabhan, Seljak, & Pen 2003;
the TMF with three coarse bins as the TMF3 and that
Hamana, Takada, & Yoshida 2003).
with five finer bins as the TMF5.
The expression for the likelihood in eqn. (28) can be
Combining eqns. (22) and (24) with the log likelihood
maximizedateachpointonthe sky, allowingone to cre-
in eqn. (23) gives
ate a cluster likelihood map. Clusters will be located at
2
ln (θ~,z ,θ )= 1 niz+1Z(zi;zd) nkgǫTikG(xk) wthiellpbeeatkhseitnomthoisgrmapahpicaensdtimthaetemfaoxritmhuemlenliskerelidhsohoidft.zd
L 0 d s 2σ2(cid:16)P inz+1Z2(zi;zdP) nkgG2(xk)(cid:17) An example of a likelihood map constructed from
(26) binned source galaxies is depicted in Figure 3. The
P P
10 HENNAWI & SPERGEL
left panel shows a Kaiser-Squires reconstruction of the ers that the lensing efficiency peaks at this redshift, and
mean convergence field (recall eqn. 7) for the case of large scale structure along the line of sight weighted by
no noise. The right panel shows the cluster likelihood thislensingkernelislikelytoyieldatomographicredshift
map constructed by applying the TMF to mock data at its peak.
which includes noise. Three coarse source redshift bins Although the tomographic redshift probability distri-
have been used to make this map (TMF3). For the bution radially resolved two projected clusters for peak
function G(x) we have used the ‘optimal filter’ deter- c, it led us to believe that peak d was a shear selected
mined in 4 below (eqn. (34) with a scale angle,θ = cluster just like any other, at z = 0.36, when in reality
s d
0.50′, and§ gaussian truncation radius, θ = 5.5′), it was a projection of large scale structure. A detailed
out
which maximizes the number of clusters detected. This investigation of the degree to which cluster tomography
filter approximates a projected Navarro-Frenk-White can distinguish projections from real clusters is beyond
Navarro,Frenk, & White (1997) (NFW) profile, but is thescopeofthiswork. Ifitwerepossibletomakeasuch
truncatedwithaGaussianto preventthe likelihoodsum adistinctionbasedonthetomography,onecouldimagine
frombeing diluted by distantuncorrelatedsourcegalax- flaggingcertainthesepeaks,sothatashearselectedclus-
ies. Peaks in the likelihood map are candidate shear ter sample could be cleaned of such projections, which
selected clusters and four of the more significant peaks will complicate attempts to determine cosmological pa-
in this map are circled and labeled a-d. rameters from the redshift distribution of shear selected
The probability distributions of the tomographic red- clusters. Ofcourse,theprojectionhypothesiscanalsobe
shiftforeachofthesefourpeaksisshowninFigure4. To tested by searching for overdensities of early type galax-
create the likelihood map in Figure 3 the source galax- ies (i.e. cluster galaxies associated with the lenses) as a
ies were binned spatially and in redshift as in eqn. (26); function of photometric redshift
however,tocompute theprobabilitydistributionsshown (Dahle et al. 2003; Schirmer et al. 2003, 2004;
in Figures 4, the convolution in eqn. (23) is performed Taylor et al. 2004).
exactly in real space with no binning. The mass and 4. THEOPTIMALFILTER
redshiftofthecluster(s)responsibleforthe peaksarela-
4.1. Discussion of Filter Functions
beled,andtheverticaldashedlinesindicatethetruelens
redshift(s). In this section we will consider several filter functions
Although we will consider the accuracy of the tomo- G(x) to convolve the tangential shear data with for the
graphic redshifts in detail in 5, the examples in Figure TMF introduced in 3. But first, a few general remarks
4 illustrate some qualitative fe§atures of the tomographic about searching for §clusters in weak lensing data are in
technique which we elaborate on them briefly. order.
Peak a corresponds to a fairly massive cluster at z = Like the ζ statistics
0.29 and the tomography performs well, giving a tomo- (Fahlman, Kaiser, Squires, & Woods 1994) and the
graphic redshift of z = 0.26. For the higher redshift aperture mass measures M (Schneider 1996), The
ap
cluster coincident with peak b, at z = 0.70, the tomog- TMF has the virtue that it operates directly on the
raphygiveszd =0.95,whichisincorrect,butthetomog- shear, which is the observable quantity in weak lensing
raphy succeeds to indicate that the deflector is at high observations. Some studies of shear selected clus-
redshift, which is still helpful as the source galaxies can ter detection search for clusters by first performing
be weighted for a high redshift deflector, increasing the a Kaiser-Squires density reconstruction, and then
contrast between the cluster and the noise. smoothing the reconstructed convergence field with
Projections of several halos at different redshifts are a filter and searching for peaks (Reblinsky et al.
quite common, which is illustrated by peak c which is 1999; White, van Waerbeke, & Mackey 2002;
a projection of a 2 1014 h−1 M⊙ cluster at z = 0.27 Hamana, Takada, & Yoshida 2003). However, working
and two large group×size halos of 5 1013 h−1 M⊙ at directly with the shear is crucial, when one recalls
×
z = 0.53 and z = 1.06, respectively. Notice in Figure that the surface mass density at any point depends
4, that the tomographygives a deflector probability dis- on the galaxy ellipticities at all distances (see e.g.,
tribution with two likelihood maxima at redshifts corre- Bartelmann & Schneider 2001). The kernel that the
sponding to the two redshifts of the lowerredshift halos. ellipticity data is convolved with in a density recon-
The fact that the peaks in the probability line up with struction is not local, decaying as θ−2. Because of edge
thedeflectorredshiftsissomewhatofacoincidence,since effects from the field boundaries and all the masked
wearefitting a modeltothe shearwithonly asinglede- bright stars, diffraction spikes, and bleed trails (see
flector. Figure 14 of Van Waerbeke & Mellier (2003) for a nice
Finally,aswewillelaborateonfurtherinthenextsec- example), the non-local Kaiser-Squires reconstruction
tion, mass selected cluster samples are plagued by pro- will propagate ringing from this ‘window function’ into
jectioneffectsduetolargescalestructureandsmallhalos other regions of the mass map, significantly amplifying
along the line of sight. Although the peak labeled d is noise. It should thus be avoided for the purposes of
the highest peak in the likelihood map, there is no cor- finding clusters.
responding halo in the light cone above 1013.5 h−1 M⊙ Besides avoiding the deleterious effect of the noisy
within an aperture of radius 3′ (which is our peak-halo window function or mask, convolving the shear with a
matching criteria). However, there is no indication that compact kernel seems sensible because clusters of galax-
this peak is a projection from the tomography, which ies have characteristic size on the sky. However, weak
gives a probability distribution similar to the others in- lensing statistics using small apertures are complicated
dicating the tomographic redshift is zd = 0.36. It is by the non-local relationship between shear and mass.
perhaps not surprising that zd 0.4, when one consid- Any filter convolved with the shear within some aper-
≈