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−644.31±0.57 2.94 ±0.13 – 0.40 ±0.07 1
Note. —aExtrapolated to r2500using the best-ﬁt relation between YSZD2
A(350kpc) and YSZD2
A(r2500) for eight clusters in common
between B08 and M09.
Note. — Redshift zand velocity dispersion σpare computed for galaxies deﬁned as members using the causti cs. Masses M100,vand
M100,care evaluated using the virial mass proﬁle and caustic mass p roﬁle respectively.
Note. — REFERENCES: SZE data are from (1) Bonamente et al. 2008 and (2) Marrone et al. 2009.
Our shallow slopes may also arise in part from the fact
that our sample, which has been assembled from the lit-
erature and whose selection function is diﬃcult to deter-
mine, is likely to be biased against clusters with small
mass and low SZE signal. Larger samples should deter-
mine whether unknown observational biases or issues in
the physical understanding of the relation account for
this discrepancy.
4.DISCUSSION
Thestrongcorrelationbetweenmassesfromgalaxydy-
namics and SZE signals indicates that the SZE is a rea-
sonableproxyforcluster mass. B08compareSZEsignals
toX-rayobservables,inparticularthetemperature TXof
the intracluster medium and YX=MgasTX, whereMgas
isthemassoftheICM(seealsoPlagge et al.2010). Both
of these quantities are measured within r500, a signiﬁ-
cantly smaller radius than r100where we measure virial
mass. M09 compare SZE signals to masses estimated
from gravitational lensing measurements. The lensing
masses are measured within a radius of 350 kpc. For the
clusters studied here, this radius is smaller than r2500
and much smaller than r100. Numerical simulations indi-
cate that the scatter in masses measured within an over-
densityδdecreases as δdecreases (White 2002), largely
because variations in cluster cores are averaged out at
larger radii. Thus, the dynamical measurement reaching
to larger radius may provide a more robust indication
of the relationship between the SZE measurements and
cluster mass.
TheYSZD2
A−Mlensdata presented in M09 show a
weakercorrelationthanouropticaldynamicalproperties.
A Spearman test rejects the hypothesis of uncorrelated
data for the M09 data at only the 94.8% conﬁdence level,
compared to the 98.4-99.8% conﬁdence levels for our op-
tical dynamical properties. One possibility is that Mlensis more strongly aﬀected by substructure in cluster cores
and by line-of-sight structures than are the virial masses
and velocity dispersions we derive.
Few measurements of SZE at large radii ( > r500) are
currently available. Hopefully, future SZ data will allow
a comparisonbetween virialmass and YSZwithin similar
apertures.
5.CONCLUSIONS
Our ﬁrst direct comparison of virial masses, velocity
dispersions, and SZ measurements for a sizable clus-
ter sample demonstrates a strong correlation between
these observables (98.4-99.8% conﬁdence). The SZE sig-
nal increases with cluster mass. However, the slopes of
both the YSZ−σrelation ( YSZ∝σ2.94±0.74
p) and the
YSZ−M100relation ( YSZ∝M1.11±0.16
100) are signiﬁcantly
shallower(giventheformaluncertainties)thantheslopes
predictedbynumericalsimulations(4.76and1.60respec-
tively).
This result may be partly explained by a bias against
less massive clusters that could artiﬁcially ﬂatten our
measured slopes. Unfortunately, the selection function
of our sample is unknown and we are unable to quan-
tify the size of this eﬀect. More importantly, our sample
indicates that the relation between SZE and virial mass
estimates (or velocity dispersion) has a non-negligible in-
trinsicscatter. Acomplete, representativeclustersample
is required to robustly determine the size of this scatter,
its origin, and its possible eﬀect on the SZE as a mass
proxy.
Curiously, YSZis more strongly correlated with both
σpandM100than with Mlens(M09). Comparison of
lensingmassesandclustervelocitydispersions(andvirial
masses)forlarger,complete, objectivelyselected samples
of clusters may resolve these diﬀerences.
Thefull HeCS sampleof53clusterswill providealargeHectospec Virial Masses and SZE 5
Fig. 2.— Integrated S-Z Compton parameter YSZD2
Aversus dynamical properties for 15 clusters from HeCS. Left panels: SZE data
versus virial mass M100estimated from the virial mass proﬁle (top) and the caustic m ass proﬁle (bottom). Solid and open points indicate
SZ measurements from B08 and M09 respectively. The dashed li ne shows the slope of the scaling predicted from numerical si mulations:
YSZ∝M1.6(Motl et al. 2005), while the solid line shows the ordinary le ast-squares bisector. Arrows show the aperture correction s to
the SZE measurements (see text). Right panels: SZE data versus projected velocity dispersions measured fo r galaxies inside the caustics
and (top) inside r100,cestimated from the caustic mass proﬁle and (bottom) inside t he Abell radius 2.14 Mpc. The dashed line shows the
scaling predicted from simulations: YSZ∝M1.6(Motl et al. 2005) and σ∝M0.33(Evrard et al. 2008). The solid line shows the ordinary
least-squares bisector. Data points and arrows are deﬁned a s in the left panels.
sample of clusters with robustly measured velocity dis-
persions and virial masses as a partial foundation for
these comparisons.
We thank Stefano Andreon for fruitful discussions
about ﬁtting scaling relations with measurement errorsand intrinsic scatter in both quantities. AD gratefully
acknowledges partial support from INFN grant PD51.
We thank Susan Tokarz for reducing the spectroscopic
data and Perry Berlind and Mike Calkins for assisting
with the observations.
Facilities: MMT (Hectospec)
REFERENCES

−644.31±0.57 2.94 ±0.13 – 0.40 ±0.07 1

Note. —aExtrapolated to r2500using the best-ﬁt relation between YSZD2

A(350kpc) and YSZD2

A(r2500) for eight clusters in common

between B08 and M09.

Note. — Redshift zand velocity dispersion σpare computed for galaxies deﬁned as members using the causti cs. Masses M100,vand

M100,care evaluated using the virial mass proﬁle and caustic mass p roﬁle respectively.

Note. — REFERENCES: SZE data are from (1) Bonamente et al. 2008 and (2) Marrone et al. 2009.

Our shallow slopes may also arise in part from the fact

that our sample, which has been assembled from the lit-

erature and whose selection function is diﬃcult to deter-

mine, is likely to be biased against clusters with small

mass and low SZE signal. Larger samples should deter-

mine whether unknown observational biases or issues in

the physical understanding of the relation account for

this discrepancy.

4.DISCUSSION

Thestrongcorrelationbetweenmassesfromgalaxydy-

namics and SZE signals indicates that the SZE is a rea-

sonableproxyforcluster mass. B08compareSZEsignals

toX-rayobservables,inparticularthetemperature TXof

the intracluster medium and YX=MgasTX, whereMgas

isthemassoftheICM(seealsoPlagge et al.2010). Both

of these quantities are measured within r500, a signiﬁ-

cantly smaller radius than r100where we measure virial

mass. M09 compare SZE signals to masses estimated

from gravitational lensing measurements. The lensing

masses are measured within a radius of 350 kpc. For the

clusters studied here, this radius is smaller than r2500

and much smaller than r100. Numerical simulations indi-

cate that the scatter in masses measured within an over-

densityδdecreases as δdecreases (White 2002), largely

because variations in cluster cores are averaged out at

larger radii. Thus, the dynamical measurement reaching

to larger radius may provide a more robust indication

of the relationship between the SZE measurements and

cluster mass.

TheYSZD2

A−Mlensdata presented in M09 show a

weakercorrelationthanouropticaldynamicalproperties.

A Spearman test rejects the hypothesis of uncorrelated

data for the M09 data at only the 94.8% conﬁdence level,

compared to the 98.4-99.8% conﬁdence levels for our op-

tical dynamical properties. One possibility is that Mlensis more strongly aﬀected by substructure in cluster cores

and by line-of-sight structures than are the virial masses

and velocity dispersions we derive.

Few measurements of SZE at large radii ( > r500) are

currently available. Hopefully, future SZ data will allow

a comparisonbetween virialmass and YSZwithin similar

apertures.

5.CONCLUSIONS

Our ﬁrst direct comparison of virial masses, velocity

dispersions, and SZ measurements for a sizable clus-

ter sample demonstrates a strong correlation between

these observables (98.4-99.8% conﬁdence). The SZE sig-

nal increases with cluster mass. However, the slopes of

both the YSZ−σrelation ( YSZ∝σ2.94±0.74

p) and the

YSZ−M100relation ( YSZ∝M1.11±0.16

100) are signiﬁcantly

shallower(giventheformaluncertainties)thantheslopes

predictedbynumericalsimulations(4.76and1.60respec-

tively).

This result may be partly explained by a bias against

less massive clusters that could artiﬁcially ﬂatten our

measured slopes. Unfortunately, the selection function

of our sample is unknown and we are unable to quan-

tify the size of this eﬀect. More importantly, our sample

indicates that the relation between SZE and virial mass

estimates (or velocity dispersion) has a non-negligible in-

trinsicscatter. Acomplete, representativeclustersample

is required to robustly determine the size of this scatter,

its origin, and its possible eﬀect on the SZE as a mass

proxy.

Curiously, YSZis more strongly correlated with both

σpandM100than with Mlens(M09). Comparison of

lensingmassesandclustervelocitydispersions(andvirial

masses)forlarger,complete, objectivelyselected samples

of clusters may resolve these diﬀerences.

Thefull HeCS sampleof53clusterswill providealargeHectospec Virial Masses and SZE 5

Fig. 2.— Integrated S-Z Compton parameter YSZD2

Aversus dynamical properties for 15 clusters from HeCS. Left panels: SZE data

versus virial mass M100estimated from the virial mass proﬁle (top) and the caustic m ass proﬁle (bottom). Solid and open points indicate

SZ measurements from B08 and M09 respectively. The dashed li ne shows the slope of the scaling predicted from numerical si mulations:

YSZ∝M1.6(Motl et al. 2005), while the solid line shows the ordinary le ast-squares bisector. Arrows show the aperture correction s to

the SZE measurements (see text). Right panels: SZE data versus projected velocity dispersions measured fo r galaxies inside the caustics

and (top) inside r100,cestimated from the caustic mass proﬁle and (bottom) inside t he Abell radius 2.14 Mpc. The dashed line shows the

scaling predicted from simulations: YSZ∝M1.6(Motl et al. 2005) and σ∝M0.33(Evrard et al. 2008). The solid line shows the ordinary

least-squares bisector. Data points and arrows are deﬁned a s in the left panels.

sample of clusters with robustly measured velocity dis-

persions and virial masses as a partial foundation for

these comparisons.

We thank Stefano Andreon for fruitful discussions

about ﬁtting scaling relations with measurement errorsand intrinsic scatter in both quantities. AD gratefully

acknowledges partial support from INFN grant PD51.

We thank Susan Tokarz for reducing the spectroscopic

data and Perry Berlind and Mike Calkins for assisting

with the observations.

Facilities: MMT (Hectospec)

REFERENCES