Scatter and bias incluster mass estimates
Masamune Oguri(University of Tokyo)

2015/3/25 Astroparticle View of Galaxy Clusters @ Hiroshima

Scatter and bias inlensing cluster mass estimates
Masamune Oguri(University of Tokyo)

2015/3/25 Astroparticle View of Galaxy Clusters @ Hiroshima

Cluster mass

• one of the most fundamental parametersthat characterize clusters• not easy to measure because it is dominatedby the mass of dark matter• critically important for cluster cosmology

Planck 2015

Planck Collaboration: Cosmology from SZ cluster• cosmology with Planck
SZ cluster counts

different
cosmo.
constraints• different mass estimatesyield quite different
cosmology results

Ωm−σ8

Planck Collaboration: Cosmology from SZ cluster counts• uncertainty in cluster
mass estimates is themost outstanding issue
in cluster cosmology!

Fig. 7: Comparison of constraints from the CMB to those fromthe cluster counts in the (⌦m , 8 )-plane. The green, blue andviolet contours give the cluster constraints (two-dimensional
likelihood) at 1 and 2 for the WtG, CCCP, and CMB lensing mass calibrations, respectively, as listed in Table 2. These
constraints are obtained from the MMF3 catalogue with the
SZ+BAO+BBN data set and ↵ free. Constraints from the Planck
TT, TE, EE+lowP CMB likelihood (hereafter, Planck primary
CMB) are shown as the dashed contours enclosing 1 and 2 confidence regions (Planck Collaboration XIII 2015), while the grey
shaded region also include BAO. The red contours give resultsfrom a joint analysis of the cluster counts, primary CMB and
the Planck lensing power spectrum (Planck Collaboration XV2015), leaving the mass bias parameter free and ↵ constrainedby the X-ray prior.

Fig. 7: Comparison of constraints from the CMB to those fromthe cluster counts in the (⌦ , )-plane. The green, blue andFig. 8: Compariso
the base ⇤CDM
1 b. The solid blquired to reconcil
is found as the po
the Planck cluster
mass bias free. Th
distributions on thdifferent
massestimates

Planck collaboration, arXiv:1502.01597

Fig. 8: Comparison of cluster and primary CMB constraints in
6.3. Constraints on ⌦ and : comparison to primary CMB

sion with the prim
extensions to the bThomson optical dof-state, and the n"CMB" in the foll
fined in Planck Co
confidence and all

Mass estimates: scatter and bias
“accuracy” of mass estimates?

• scatter

p(M)

bias

important for analysis ofindividual clusters• biasimportant for statisticalanalysis even for the casescatter ≫ bias

scatter

Mest Mtrue

MCluster mass estimates• X-ray hydrostatic equilibrium
small scatter, large bias

• weak gravitational lensing
large scatter, small bias

X-ray hydrostatic equilibrium• X-ray mass derived w/

TESTING X-RAY CLUSTER MEASU

hydrostatic equilibrium
is known to be biased
low by ~10−40%• need independent massestimates to quantify
the X-ray mass bias

Nagai et al. (2007)

FIG. 5.— Same as the upper-panels in Fig. 4, except that the estimated mass
(Mest ) is evaluated within the estimated virial radii (rest ); hence, Mest andMtrue are measured in diﬀerent physical regions.

the inner region of relaxed clusters, while the deviation fromthe hydrostatic equilibrium become more prominent in cluster outskirts and/or unrelaxed systems, as expected. Similar
results are obtained for clusters at high redshift.
The middle panels in Figure 4 illustrate that the X-ray gas
mass determinations are remarkably accurate and robust. The
Mgas measurements are accurate to better than a few percent
for both relaxed and unrelaxed clusters and independent ofredshift. For the relaxed cluster samples, the accuracy of thetrueMgas measurements is as good as 1% at r500c
with 3% 1 σ

Weak lensing• purely gravitational effect• direct measurements of
total mass, including
dark matter!

S. Colombi

Scatter and bias in lensing mass• scatterstatistical error − shot noise, LSS
halo triaxiality• biasprofile mismatch
substructure? (not in this talk)
photo-z, dilution, ... (not in this talk)Halo triaxiality• ΛCDM model predicts
highly non-sphericalhalo shape• typical major-to-minor
axis ratio 2:1

http://www.mpa-garching.mpg.de/galform/millennium/

Observational evidence• direct measurements ofhalo shapes w/ WL

Combined strong

Oguri et al. (2012)• non-sphericity detectedat ≳ 5σ〈e2D〉= 0.46±0.04(Oguri et al. 2010 w/ LoCuSS)〈e2D〉= 0.47±0.06(Oguri et al. 2012 w/ SGAS)• e2D quite consistent with
ΛCDM predictions!

Figure 11. The two-dimensional weak lensing shear maps obtained fr
directions and strengths of weak lensing shear distortion. Colour contour

Oguri, Takada, Umetsu, Broadhurst ApJ 632(2005)841

Projection effect2

Ogu

triaxialclusterradialprofile• projected mass profile depends sensitively

F IG . 1.— Convergence profiles of the triaxial halo with the virial mass
Mvir = 1015 h−1 M⊙ , the concentration parameter ce = 1.15, and the triaxial
axis ratios of a/c = 0.4 and b/c = 0.7. The halo is placed at zl = 0.3, and weassume the source redshift of zs = 1.0. We consider the projection along eachof the three principal axes: from upper to lower the dashed lines show profiles
projected along z, y, and x (see eq. [1]), respectively. The convergence profileof the corresponding spherical NFW halo is also plotted by the solid line
for comparison (see text for details). The vertical arrow indicates the virial
radius.

on the projection direction

Oguri, Takada, Umetsu, Broadhurst ApJ 632(2005)841

Projection effectOguri et al.

(mass)

triaxialcluster

1.— Convergence profiles of the triaxial halo with the virial mass
015 h−1 M⊙ , the concentration parameter ce = 1.15, and the triaxial
os of a/c = 0.4 and b/c = 0.7. The halo is placed at zl = 0.3, and wethe source redshift of zs = 1.0. We consider the projection along eachree principal axes: from upper to lower the dashed lines show profiles
d along z, y, and x (see eq. [1]), respectively. The convergence profile
orresponding spherical NFW halo is also plotted by the solid line
parison (see text for details). The vertical arrow indicates the virial

projection effect(concentration)• lensing-derived mass and concentration are

F IG . 2.— Constraint contours in the virial mass and halo concentration
parameter space, obtained by fitting the mock data of triaxial halos to thespherical NFW halo model. The contours show 68%, 95%, 99.7% confidence
limits (corresponding to ∆χ2 = 2.3, 6.17 and 11.8, respectively). From left toright, the constraint contours from the convergence profiles of the triaxial halo
projected along the principal axes x, y, and z (as in Figure 1), respectively,
are shown. For comparison, the square symbol shows the best-fitting modelfor the convergence profile obtained by projecting the a priori sphericallyaveraged mass profile of the input triaxial halo.

significantly affected by the cluster orientation→ ~20-30% scatter in lensing mass

Bias from radial mass profile• data have often been analyzed assuming asimple NFW profile• any mismatch between assumed and true
mass profiles can cause bias

12

M. Oguri et al.

CLASH (Umetsu et al. 2014)

Umetsu et al.

SGAS (Oguri et al. 2012)by the c–M degeneracy discussed in Section 5.4.1. On

the basis of this comparison, we estimate the systematic
uncertainty in the overall mass calibration to be of theorder of ±8%.
6. CLASH STACKED LENSING ANALYSIS

Stacking an ensemble of clusters helps average out the
projection e↵ects of cluster asphericity and substructure,
as well as the cosmic noise from projected uncorrelated
LSS, inherent in lensing measurements. The statisticalprecision can be greatly improved by stacking together
a large number of clusters, especially on small angularscales (Okabe et al. 2010), allowing a tighter comparisonof the averaged lensing profile with theoretical models.
Here our stacked lensing analysis will focus on the
CLASH X-ray-selected subsample of Postman et al.
(2012), which comprises a population of high-mass X-ray
regular clusters. The four high-magnification clusters arethus excluded from this part of the analysis.
In Section 6.1 we present a stacked tangentialdistortion (shear-only) analysis of the 16 X-ray regularclusters, and examine the form of their underlying halo
FigureThe stacked
shear profile obtained
mass6.profile
usingtangential
the ensemble-averaged
hh by⌃+comii probining the 25 clusters. The average diﬀerential surface density
file. In Section 6.2 we derive the ensemble-averaged total⟨∆Σ+ (r)⟩ (see equation 27) is plotted as a function of the physmass profile hh⌃ii from our cluster mass-profile dataset
ical radius r. Grey points indicate stacked tangential shear mea(Figure 3), for comparison with theoretical predictions
surements from weak lensing that are not used for fitting. The
taking into account both one- and two-halo term contriupper left point with a horizontal error-bar is the constraint frombutions.
the average Einstein radius. The solid line with shading is the

LoCuSS (Okabe et al. 2013)

best-fit NFW model with 1σ error range. The lower panel plots
6.1. CLASH Stacked Shear-only Analysis
the stacked profile of the 45◦ rotated component, ⟨∆Σ× (r)⟩.

6.1.1. Stacking the Weak Shear Signal
The azimuthally-averaged tangential distortion is abeen
successful
for radially-modulated
constraining mean dark
mattermeasureof thesurfacemassdistridensity
butionsofclustersamples(e.g.,
Mandelbaum
etal.
2006b;and is insensitive to sheet-like mass overdensities, which
Johnston
et al.
Leauthaud
et al.
2010;
Okabe
et al. ofresemblethe 2007;
projected
two-halotermwithina couple2010).
conduct
stacking2011).
analysisof thetanvirialHere
radiiwe(Oguri& HamanaHence,
the stacked
gentialshear profile forsignal
our around
lensing asample
for studytangential-distortion
large sample
of clusingtersthe ismass-concentration
another(one-haloviewa sensitive proberelationof the fromcluster-only
term)
mass
distribution.
point.
Notethat
the oﬀ-centreing eﬀect, which has been
In Figure
we showstackedtangential-shear
proknown
to be 5oneof thethemost
significant
systematic
file hhin ⌃for oursamplewhere
individual
cluserrors
stackedlensinganalysis(e.g.,
Johnston
et al.
+ ii derived
tersMandelbaum,
and backgroundgalaxies
are2008;weighted
shear2007;
Seljak,& Hirata
Oguriby& the
Takadasensitivity
kernel(trW+forinour
Section
3.4.1).
Theofindivid2011),
should be
negligible
analysis,
becausethe17

Table 5. Summary of stacked tangential shear analysis
Sample

N

⟨z⟩

⟨θE ⟩
(arcsec)⟨Mvir ⟩
14
(10 h−1 M⟨cvir ⟩
⊙)

all

25

0.469

14.4+10.6−7.0

4.57+0.33−0.31

5.75+0.70
−0.57

θE -1
θE -2
θE -3

4
5
7

0.379
0.416
0.471

22.8+4.3−2.8
13.0+2.4−2.8
8.3+2.6−2.4

6.03+0.74−0.72
3.13+0.50−0.50
3.51+0.52−0.52

7.94+1.28−1.02
10.23+2.65−1.82
7.08+1.53−1.12

Mvir -1
Mvir -2
Mvir -3

5
10
10

0.480
0.472
0.460

17.1+12.9−6.4
14.7+10.2−5.7
12.6+9.3−7.8

9.55+1.17−1.04
5.62+0.61−0.55
1.97+0.32−0.29

4.90+0.79−0.73
5.37+0.87−0.75
10.59+4.54−2.83

linear dependence on the mass profile. In fact, the reduced
shear is defined by g+ ≡ γ+ /(1 − κ), where γ+ and κ aretangential shear and convergence. Thus, the quantity definedby equation (27) still depends slightly on the source5redshift
via the factor 1/(1 − κ), particularly near the halo centre.
Thus,
in comparison
with theNFW predictions,Fig.
5.— The averagetangential-shear
profile hhwe⌃assume+ ii (upper
thepanel,sourceblack
redshiftof zsobtained= 1.1, whichis the typicaleﬀectivesquares)from stacking
the X-ray-selectedsubsample
of 16
shown
in unitsof projectedmass density.
sourceredshiftforclusters,
our weaklensinganalysis(see Table
3).
The thick-solid (red), dashed (blue), dashed-dotted (magenta),
Also
the non-linear
makeslines
it somewhat
dotted(green), anddependence
thin-solid (orange)
corresponddiﬃcult
to the bestto fit
interpretthe average profile,
stackedtan-SIS,NFW, truncated-NFW(Baltzandet al.hence
2009,our
BMO),
Einasto,andcored
isothermal
sphereprofiles,respectively.
Thegray-shaded
gential profile measurement near the centre should be takenarea shows the composite halo mass-profile prediction (1 ) derived
withcaution.
froma weighted average of individual NFW profiles based on the
It isshear+magnification
known that the matter
fluctuations
thelinejoint
analysis(Figure 3), alongin goodagreementwiththestackedshear-only
constraints(Figure
6).
The
lower
panel
of-sight contributes to the total error budget (e.g., Hoekstra
shows the 45 -rotated ⇥ component hh ⌃⇥ ii, which is consistent
2003;
et al.
2004;
Gruen theet al.
withHoekstra
a null signal
well 2011;
within Dodelson
2 at all radii,
indicating
reliabilityWhileof ourwedistortion
analysis.thisTherightforverticalaxes represent2011).
have ignoredeﬀect
the analysisof1
the corresponding
shear components,
hh⌃cweiihh
ii andindividual
clusters presented
in Section +4, =heretake⌃+
into1hh⌃c error
iihh ⌃fromto themean
depth of in
weak-lensing⇥ = the⇥ ii, scaled
account
the large
scale
structure
fitting
observations.
the stacked tangential shear profile by including the full coa totalbetween
S/N of diﬀerent' 2.8, indicating
variance
radial binsthat(seeresidual
Oguri &systematic
Takadaerrors
are
at
least
an
order
of
magnitude
smallerthan
2011; Umetsu et al. 2011b, for the calculation of the covarithe measured lensing signal.
ance matrix). We, however, comment that the error of thelarge scale6.1.2.
structure
is subdominant
in Weak
our analysis,
becauseModeling
the Stacked
Shear Signalof the relatively small number density of background galaxHere we quantify and characterize the ensembleies after the colour cut (see also Oguri et al. 2010).
averaged mass distribution of our cluster sample using
In addition to weak lensing, we stack strong lensing

all data from Subaru S-cam

observed density profiles
are consistent with
the NFW profile!

(←see talk by N. Okabe)Possible origins of bias

• concentration c=rvir/rs

ρ(r)central concentrationof radial profile−1r• steepening−3r

cutoff of radial density
profile beyond ~rvir• 2-halo term
rs

rvir

r

correlated matter
fluctuations aroundthe cluster

Halo concentration• in some work, specific values ρ(r)or forms of concentrations
are assumed to get mass

high c• the mass would be biased if
the assumed concentration
is wrong• concentrations in real clusters
consistent with ΛCDM predictions?
(cf. anomalously high concentrations claimed by Broadhurst et al.)

low c

r

CLASH: The c-M relation

17

11

Combined strong and weak lensing analysis of 28 clusters

CLASH (Merten et al. 2014)

SGAS (Oguri et al. 2012)

5
5.1

STACKING ANALYSIS
Stacked tangential shear profile

We can study the average properties of a given sample bystacking lensing signals. This stacked lensing analysis has
been successful for constraining mean dark matter distributions of cluster samples (e.g., Mandelbaum et al. 2006b;
Johnston et al. 2007; Leauthaud et al. 2010; Okabe et al.
2010). Here we conduct stacking analysis of the tangential
shear profile for our lensing sample for studying the massconcentration relation from another viewpoint. Note thatthe oﬀ-centreing eﬀect, which has been known to be oneof the most significant systematic errors in stacked lensing
analysis (e.g., Johnston et al. 2007; Mandelbaum, Seljak, &5Hirata 2008; Oguri & Takada 2011), should be negligible forour analysis, because of the detection of weak lensing signals
Figure 5. The mass-concentration relation obtained from comfor individual clusters and the presence of giant arcs whichbined strong and weak lensing analysis. Filled triangles show ourassure that selected centres (positions of the brightest galaxresults presented in this paper, whereas filled squares show rein theregion)indeedcorrespond
Figure 12. A c-M ies
comparison
in 2D. strong
This figure islensing
identical to Fig.
11, but shows
the comparison
between di↵erenttoc-M that
relations, basedondi↵erent
halo
subsets
from
M14
in
2D.
In
addition,
weoverlay
again
the
c-M
likelihood
contours
fromUmetsuetal.
(2014).
sults from literature; A1689, A370, CL0024, RXJ1347 (Umetsuof the mass distribution.
We configure X-MAS to reproduce the X-ray observaet al. 2011b), and A383 (Zitrin et al. 2011b). The black shaded
Tablescale.
9
We Allenperformstacking
Speciftions (Maughan et al. 2008;
et al. 2008;
Ebeling in the physical length
Goodness-of-fit: Meneghetti et al. 2014
region indicates the predicted concentration parameters as a et
funcal. 2007; Cavagnolo et al. 2008; Mantz et al. 2010)
ically,we computethe diﬀerential surface density ∆Σ+ (r)according to which
the CLASH
clusters were selected.
tion of the halo mass with the lensing bias taken into accountSamplehc /c iQ
Q
Q
p-value
Using this set of simulated X-ray images we apply thewhichisdefine
by
3D
full
1.00
±
0.18
1.03
0.86
1.15
9.5
0.85
very same selection criteria which were used to select the(see Appendix A for details). The solid line is the best-fit mass-

LoCuSS (Okabe et al. 2013)

concentration values
also consistent with
ΛCDM prediction!
sim

2

1

3

0.80 ± 0.16
1.03 ± 0.19
0.86 ± 0.16
0.91 ± 0.19
0.94 ± 0.20

0.84
1.06
0.88
0.93
0.96

0.68
0.89
0.73
0.78
0.80

0.93
1.09
0.98
1.03
1.06

obs

CLASH X-ray selected clusters. For a more detailed de-3D relaxed

2D fullconcentration relation from fitting of our cluster sample (i.e., scription
filled of these∆Σcriteria
and≡theΣ
selection
processsee ol ), 2D relaxed
= r/D
+ (r)
cr g+ (θM14.
2D SL
triangles), with the 1σ range indicated by dotted lines.
2D X-ray

2

29.4
9.2
32.1
18.0
16.1

(27)

0.01
0.87
0.01
0.26
0.38

This CLASH-like, X-ray selected sample in 2D is thewheretheto criticalsurface mass density for lensone simulated sample
whichΣcomesthe real
cr isclosest
CLASH clusters, both with respect to the selection criNote. — The column explanations are identical to Table 8.
ing.method.
We stack∆Σ+between(r) for diﬀerent
clusters to obtain theteria and the analysis
The comparison
from
Sec.
5.1
and the X-ray selected c-M relation andthe c-M relation of this sample and the observed CLASH
averagediﬀerential
surfacedensity.
We
doagreenotat theinclude
find
that
the
two
relations
90% confidence
clusters shows indeed significant improvement over thelevel.
limiting case of the
fully relaxed sample in theandlast secSDSSJ1226+2149
SDSSJ1226+2152 in our stacking
tion. The qualitative agreement between the data points
6.3. Individual CLASH Clusters in Our Simulated
and the X-ray selectedc-M relationin Fig. these
12 is quite
analysis,
becausefields clearly
have complicatedmassSampleobvious. The median concentration ratio shows that the
As the finalcores
analysisseparated
in this work weselect closewithstrong lensing
bynow
only
observed CLASHdistributions
concentrations are only
4% twolower thanmatches
toindividual
CLASH
clustersoutof our 2D
the ones from the X-ray
selected
simulation
sampleandAllen 2007; Buote et al. 2007; Ettori et al. 2010) analysis.
∼ 3′ . Furthermore,
we exclude
SDSSJ1110+6459
as wellbeset ofsimulated halos. We do thisin orderto gather
the p-value 0.38 indicatesno strong tension between
theadditional
confirmationthat
ourspecificwayof selecttwo
samples(compareTable
9).
Finally,
wecalculate
Our result suggests that the observed mass-concentration
cause the two-dimensional weaklensing
mapa numericalsuggeststhe2
ing CLASH
clusters fromsimulation
is sufthe
value for the fits of the CLASH c-M relation
ficiently
accuratetocharacterize
the
CLASH
selection
relation is in reasonable agreement with the simulation represence of a very complicated mass distribution around the(←see talk by N. Okabe)Outer density profile of clusters
ρ(r)• has not attracted muchattention until recently−1r• turned out to be quite

−3
r

rs

rvir

important for accuratemass measurementsrOguri & Hamana MNRAS 414(2011)1851

Outer lensing profiles
1-halo2-halo• detailed lensing profiles from ray-tracing in
N-body simulations• truncated NFW profile + 2-halo term can fit
the profile in the simulation wellng simulations are made in Section 3.
del calibrated by the ray-tracing, we
in various lens studies in Section 4.
ts in Section 5. In Appendices A ande which should be useful for various
moothly truncated NFW profile.

ρs parameters (Mvir , cvir ) for the NFWThus there are 2 fitting
ρTJand(r) TJ= profiles, and 3 fitting parametersΘ(rvir(M−virr),, cvir , τv ) fo
2
(r/r
)(1
+
r/r
)
s
s
Oguri
& Hamana MNRAS 414(2011)1851
the BMO profile.
It has been known that very central density profiles
withdarkΘ(x)being the Heaviside step functio
haloes in N -body simulations are not reliable becau
propertiesof this eﬀects
profilecanalso
be computof several numericalsuch
as thetwo-body
relaxatio
the finite&time
step2003a,b).
size. Based on the detailed analy(seeand
TakadaJain
sis of Fukushige & Makino (2001), we estimate that densit
One arepotential
problemofvirtheTJNprofileisprofilesreliable downto ∼ 0.08r
for our-body simu
4
lations
in
which
N
∼
10
particles
includedin each ha
and convergence profiles
arearenot
diﬀerentiab
analysed in the present paper. Indeed, our analysis resul
cationradius,thatwhichcausesthe divergencealso indicate
convergenceand tangential
shear profilnear thevery centre(θ/θvir ∼&0.01)
tend to(2009)
be significant
profile.
Baltz,
Marshall,Oguri
prosmaller than our analytic model calculations (see below
ent
forminofallthe
truncation(hereafter the BM(Baltz,& OguriThus, Marshallmass andredshift 2009)bins fitting is performed%nfittin$sure thatthe radial bins θ/θvir ! 0.079 to makeour
2rtresults are not aﬀectedρbys numerical artifacts.
ρBMO (r) =,2
2
2(r/rs )(1 + r/rs )r + rt3.3 Result

Truncation radius

ALO MASS PROFILES

the radial profiles of the convergenceround clusters. At small scales theby the signals from the dark haloes
ters (the so-called 1-halo term). The• profiles in simulations are
fitted with a smoothly
truncated NFW profile• truncation radius weakly

igure 1. Bets-fit values of the dimensionless truncation radius(eq. [8]) in the BMO profile, for diﬀerent redshift (upper) andass (lower) bins. Filled squares plot the case when the concenation parameter is computed from equation (5), whereas open
rcles are the case when the concentration parameter is fitted
multaneously. Dotted lines connect median values of τv in difrent redshift and mass bins. The median values of all bins are= 3.0 for the fixed concentration parameter, and τv = 2.6 forhe fitted concentration parameter.

where
rt ischeckthethetruncation
radius. In the pa
First webest-fit values of the truncation radiu[8]) in the BMO
profile. Figure truncation
1 show the best-fi
use τthedimensionlessrav (eq.following

values for diﬀerent
redshiftand massbins. We find the bes
dependsonhalo
masses,fit values to be ∼ 2−3, showing no strong dependence on thcthe RAS,⃝
MN
mass and massive
redshift. We consider
two cases,
case that
th
more
halos
have
concentration parameter is fixed to the value computed
equation (5) andthe case that the concentration
parametsmallertruncation
radiusis also fitted to the data. We find that the results of bot

cases agree reasonably well with each other, which implithat the best-fit values of τv is not sensitive to how th
concentration parameter is treated. The median values aτv = 3.0 when the concentration parameter is fixed, anτv = 2.6 when the concentration parameter is fitted to thAdhikari, Dalal & Chamberlain (2014)

Physical origin of steepening?

• steepening can be explainedby “turn-around” of infallingmaterial(velocity)

to determine the enclosed overdensity inside the splashback radius, ∆s . Our results do not strongly depend on
our assumed mass profile inside the halo. For example,
using an isothermal profile instead of NFW gives resultsthat are consistent at the ∼ 10% level. Figure 2 shows thepredicted values of the enclosed overdensity. Throughout
this paper, we define overdensities relative to the meanmatter density, not the critical density. In our model,∆s depends only on the halo’s accretion rate s, alongwith the values of the background cosmological parameters ΩM and ΩΛ at the time the halo is observed. The
behavior we find is unsurprising. As the accretion rate isincreased (larger s), the potential deepens more quickly
in time, resulting in splashback occuring at a smaller radius, or equivalently, at a larger enclosed overdensity ∆s .
Similarly, at low redshift when ΩM diminishes and ΩΛ
increases, the mean background density of the universeρ̄m decreases more during the time between turnaroundand splashback, again resulting in a larger ∆s .
Finally, although the model presented here is extremely simple to evaluate, we also provide a very roughfitting function to approximate the location of splashback:3• this suggests that the cutoff
radius is partly determinedby the accretion rate

10

(3)

with fitted parameters A = 38, b = 0.57, c = 0.02, d =0.2, e = 0.52. This fitting function has ∼ 5% accuracy
over the range 0.5 < s < 4, 0.1 < ΩM < 1. The results
shown in this paper do not use this fitting function, since
it is equally simple to evaluate the spherical toy model.

(radius)

r−1
r−3

ρ

−b−c s dΩM +e s3/4
∆s ≈ A ΩM
e
,

ρ(r)

e=0.1
e=0.05
e=0

3

102

1

cutoffr

Oguri & Hamana MNRAS 414(2011)1851

Mass bias and outer density profile• fitting shear profiles
assuming traditional
NFW without 2-haloOguri and T. Hamana

virial radius• best-fit mass can bebiased low up to~10%• bias can be reduced
by restricting fitting
region to ≲ rvir

(or including steepening
and 2-halo in profile fitting)thick: rt=2.6rvir
thin: rt=2.0rvir

Scatter and bias in lensing mass• scatterstatistical error − shot noise, LSS
halo triaxiality• biasprofile mismatch
substructure? (not in this talk)getting more and more understood by
ray-tracing simulations and observations

Statistical lensing mass estimate• weak lensing mass estimates have large scatter• one way to beat down scatter is to combine
many measurements → stacked weak lensing• very powerful approach in the era of wide-fieldsurveysPower of stacking
4• stacking significantly
enhances S/N• one can get average

properties very well

• particularly powerfulwhen applied to wide-field survey data

Okabe et al. (2013)

Fig. 2.— The projected mass distribution reconstructed from our weak-lensing catalogs, from one typical cluster (N = 1; A
upper left) to the full sample (N = 50; bottom right). Contours start at S/N = 3, and are spaced at ∆S/N = 2. A Gaussian
scale of FWHM = 2 arcmin is used in all panels (hatched region at lower right).

of the best fit density profile models directly, obtainingβ(r = 0.01r200 ) = −d log ρ/d log r = 1.1 for the gNFW
and Einasto models, in good agreement with ⟨β⟩ ≃ 1.1

in Section3.1.

Shear calibration – We confirmed the reliabi

Oguri MNRAS 444(2014)147

Scaling relation

Cluster finding algorithm

13• stacking of CFHTLenS
data for SDSS clusters• calibrate richness-mass
relation well from~120 deg2 lensing data,
14
down to ~10 MSunre 16. Stacked surface mass density profiles from the
IRA SDSS DR8 clusters and CFHTLenS shear catalogue.
we plot the low cluster redshift bin defined by 0.1 < zcl <Diﬀerent symbols show results for diﬀerent richness bins;
N̂cor < 25 (filled triangles), 25 < N̂cor < 35 (filled squares),N̂cor < 50 (crosses), and 50 < N̂cor < 90 (open circles),are shifted vertically by −0.5, 0, 0.5, and 1 dex, respecfor illustrative purpose. Solid lines show best-fitting NFW
es including the miscentring eﬀect (see Section 5.2 for more

Figure 18. Scaling relations between the mean richness ⟨N̂cor ⟩and the mean halo mass ⟨Mvir ⟩ inferred from the CFHTLenS
stacked weak lensing analysis. Filled triangles show the relation
for the low-redshift (0.1 < zcl < 0.3) cluster sample, whereas open
squares show the relation for the high-redshift (0.4 < zcl < 0.6)
cluster sample (see also Table 1). Solid lines with shading are
power-law fits (equation 40) and 1σ error of the scaling relations.

Bias in stacking analysis• you need to understand your sample very wellfor proper understanding of stacking results• if you take the interpretation of the sample

wrong, stacking analysis results will be biased• selection function is critically important!

Possible cluster selection effects• concentration/formation history• merger• orientation bias

6

Effects of merger• merger can have large impact on observables• change X-ray properties drastically• strong lensing cross section also enhanced
6

Clowe et al. (2006)

Simulation (Nelson et al. 2011)

Ota, Oguri et al. ApJ 758(2012)26
Oguri et al. MNRAS 429(2013)482

An extreme example?

HST

SDSS J1029+2623•(“theHidden Fortress”)• mass distribution
very concentrated• mass discrepancy
MX/Mlens~2−3 (!)• interpretation:line-of-sight merger

8

M. Oguri et al.

for c
mass
is qu
appe

diﬀer
most
is an
due t
this
mass
davi
the d
tion
decre
sitive
eﬀect
dyna
be co

Cluster selection in optical surveys

optical (galaxy)

weak lensing(see talk by S. Miyazaki)

Orientation bias

• triaxiality affects some cluster selection methods• then resulting cluster sample has orientation bias,

i.e., cluster orientations w.r.t line-of-sight direction
is not random• since lensing properties are sensitive to cluster

orientations, the orientation bias can have large
impact on stacked lensing analysis
θEstimated orientation bias
optical (galaxy) Cluster Orientationweak
lensing
Bias and its
Impact on Weak Lensing
Dietrich et al. (2014)

Scatter and bias in weak lensing selected clusters

Hamana, Oguri et al. (2012)7

the threshold adopted by the developers for inclusion of objects
the cluster catalogue. It is also important to note that RED M A PP E
haloescluster detection and l richness estimation are strongly intertwin
1.8
redMaPPerand tuned to each other. Thus the two lowest richness points of t
C4
GM BCG and WA Z P cluster are only presented for completeness a
1.6
gmBCG
should not be interpreted as having reliable richness measuremen
and they should not be understood as an ability of these clust
WAzP
1.4
finders to find lower richness clusters than RED M A PP ER. Given t
complex interplay between cluster selection and richness estimatio
1.2
we make no attempt at an interpretation of these points.
Binning by richness as in Fig. 5 does not reproduce the treof decreasing prolateness with increasing mass and thus increasi
1.0
richness seen in Fig. 4. We observe that the scatter in optical richne
101
102
leads to substantial mixing of cluster masses between richness biRichness l
We must, however, caution that there are indications that the intrins
scatter in the simulations exceeds that of real data and thus artificia
Figure 5. Measured axis ratio q as a function of estimated halo and cluster
enhances
this mixing.
Figure 6.
Probability
distribution functions of the halo orienrichness l . Again horizontal offsets are applied to the data points for greatertation with respect
to thetheline-of-sightdirection,| cos(θ)|,pushfor cluste
Furthermore,
net effect oforientationbiasztoclarity.
weak lensing
selectedclusters.
We
adopt thethe
peakheightofthreshinto higher
richnessbins
counteracts
decrease
axis ratios w
old valueincreasingof ν = mass.
4. TheThedashedis mass
for theNFW-which a
reasonhistogramis that lower
clusters,
corresponding
peaks, i.e.,peak heights
by thespherical
ure 5. Relationship between ν1000 and νκ as a function of
more subject
to orientationbiasevaluatedin the cluster
finding
step and th
NFW model via spherical overdensity masses, the long-dashed
a/c assume(bottomthat
left)
and
| cos(θ(rightpanels).
z )| anappear more elliptical, are preferentially measured to have highH (top
We left),
therefore
the
mocks
have
intrinsic
scatterinthehistogramfor the peak heights measured from the noise-free K
13−1
haloeswith M1000relation> 3 × that
10 exceeds
h M⊙ theat 0.1< zin<the0.4realareUniverse.
richness as compared to the seemingly rounder high mass cluste
mass-richness
scattermap, and the solid histogram for the peak heights measured fromdered. The dots show the relationship for each halo and filled
Thus low mass clusters are preferentially pushed into higher richne(los elongation) Axis ratio q

2.0(major axis-los
angle)• optical and weak lensing selected clusters are

preferentially aligned with line-of-sight direction!

Summary• origins of scatter and bias in cluster (lensing) massestimates are getting more and more understood− triaxiality, outer profile, ...

• stacking analysis is powerful in reducing scatter,but understanding of the selection function is
critically important (e.g., orientation bias)