arXiv:1001.0006v2  [astro-ph.CO]  4 May 2010Draft version November 2, 2018
Preprint typeset using L ATEX style emulateapj v. 11/10/09
COMPARISON OF HECTOSPEC VIRIAL MASSES WITH SZE MEASUREMENT S
Kenneth Rines1,2, Margaret J. Geller2, and Antonaldo Diaferio3,4
Draft version November 2, 2018
ABSTRACT
We present the first comparison of virial masses of galaxy clusters with their Sunyaev-Zel’dovich
Effect (SZE) signals. We study 15 clusters from the Hectospec Clus ter Survey (HeCS) with
MMT/Hectospec spectroscopy and published SZE signals. We measu re virial masses of these clusters
from an average of 90 member redshifts inside the radius r100. The virial masses of the clusters are
strongly correlated with their SZE signals (at the 99% confidence lev el using a Spearman rank-sum
test). This correlation suggests that YSZcan be used as a measure of virial mass. Simulations predict
a powerlaw scaling of YSZ∝Mα
200withα≈1.6. Observationally, we find α=1.11±0.16, significantly
shallower (given the formal uncertainty) than the theoretical pr ediction. However, the selection func-
tion of our sample is unknown and a bias against less massive clusters c annot be excluded (such a
selection bias could artificially flatten the slope). Moreover, our sam ple indicates that the relation
between velocity dispersion (or virial mass estimate) and SZE signal has significant intrinsic scatter,
comparable to the range of our current sample. More detailed stud ies of scaling relations are therefore
needed to derive a robust determination of the relation between clu ster mass and SZE.
Subject headings: galaxies: clusters: individual — galaxies: kinematics and dynamics — co smology:
observations
1.INTRODUCTION
Clusters of galaxies are the most massive virialized
systems in the universe. The normalization and evo-
lution of the cluster mass function is therefore a sen-
sitive probe of the growth of structure and thus cos-
mology (e.g., Rines et al. 2007, 2008; Vikhlinin et al.
2009; Henry et al. 2009; Mantz et al. 2008; Rozo et al.
2008, and references therein). Many methods exist
to estimate cluster masses, including dynamical masses
from either galaxies (Zwicky 1937) or intracluster gas
(e.g., Fabricant et al. 1980), gravitational lensing (e.g.,
Smith et al.2005;Richard et al.2010), andthe Sunyaev-
Zel’dovich effect (SZE Sunyaev & Zeldovich 1972). In
practice, these estimates are often made using simple
observables, such as velocity dispersion for galaxy dy-
namics or X-ray temperature for the intracluster gas.
If one of these observable properties of clusters has a
well-defined relation to the cluster mass, a large survey
can yield tight constraints on cosmological parameters
(e.g., Majumdar & Mohr 2004). There is thus much
interest in identifying cluster observables that exhibit
tight scaling relations with mass (Kravtsov et al. 2006;
Rozo et al. 2008). Numerical simulations indicate that
X-ray gas observables (Nagai et al. 2007) and SZE sig-
nals (Motl et al. 2005) are both candidates for tight scal-
ing relations. Both methods are beginning to gain ob-
servational support (e.g., Henry et al. 2009; Lopes et al.
2009; Mantz et al. 2009; Locutus Huang et al. 2009).
Dynamical masses from galaxy velocities are unbiased
kenneth.rines@wwu.edu
1Department of Physics & Astronomy, Western Washington
University, Bellingham, WA 98225; kenneth.rines@wwu.edu
2Smithsonian Astrophysical Observatory, 60 Garden St, Cam-
bridge, MA 02138
3Universit` a degli Studi di Torino, Dipartimento di Fisica G en-
erale “Amedeo Avogadro”, Torino, Italy
4Istituto Nazionale di Fisica Nucleare (INFN), Sezione di
Torino, Torino, Italyin numerical simulations (Diaferio 1999; Evrard et al.
2008), and recent results from hydrodynamical simula-
tions indicate that virial masses may have scatter as
small as ∼5% (Lau et al. 2010).
Previous studies have compared SZE signals to hydro-
staticX-raymasses(Bonamente et al.2008;Plagge et al.
2010) and gravitational lensing masses (Marrone et al.
2009, hereafter M09). Here, we make the first compar-
ison between virial masses of galaxy clusters and their
SZE signals. We use SZE measurements from the lit-
erature and newly-measured virial masses of 15 clus-
ters from extensive MMT/Hectospec spectroscopy. This
comparison tests the robustness of the SZE as a proxy
for cluster mass and the physical relationship between
the SZE signal and cluster mass. Large SZ cluster sur-
veys are underway and are beginning to yield cosmologi-
calconstraints(Carlstrom et al.2010;Hincks et al.2010;
Staniszewski et al. 2009).
We assume a cosmology of Ω m=0.3, Ω Λ=0.7, and
H0=70 km s−1Mpc−1for all calculations.
2.OBSERVATIONS
2.1.Optical Photometry and Spectroscopy
We are completing the Hectospec Cluster Survey
(HeCS), a study of an X-ray flux-limited sample of 53
galaxy clusters at moderate redshift with extensive spec-
troscopy from MMT/Hectospec. HeCS includes all clus-
ters with ROSAT X-ray fluxes of fX>5×10−12erg
s−1at [0.5-2.0]keVfrom the Bright Cluster Survey (BCS
Ebeling et al.1998)orREFLEXsurvey(B¨ ohringer et al.
2004) with optical imaging in the Sixth Data Release
(DR6) of SDSS (Adelman-McCarthy et al. 2008). We
use DR6 photometry to select Hectospec targets. The
HeCS targets are all brighter than r=20.8 (SDSS cata-
logs are 95% complete for point sources to r≈22.2). Out
of the HeCS sample, 15 clusters have published SZ mea-
surements.2 Rines, Geller, & Diaferio
2.1.1.Spectroscopy: MMT/Hectospec and SDSS
HeCS is a spectroscopic survey of clusters in the red-
shift range 0.10 ≤z≤0.30. We measure spectra with
the Hectospec instrument (Fabricant et al. 2005) on the
MMT 6.5m telescope. Hectospec provides simultaneous
spectroscopy of up to 300 objects across a diameter of
1◦. This telescope and instrument combination is ideal
for studying the virial regions and outskirts of clusters
at these redshifts. We use the red sequence to preselect
likely cluster members as primary targets, and we fill
fibers with bluer targets (Rines et al. in prep. describes
the details of target selection). We eliminate all targets
withexistingSDSSspectroscopyfromourtargetlistsbut
include these in our final redshift catalogs.
Ofthe15clustersstudiedhere,onewasobservedwitha
single Hectospec pointing and the remaining 14 were ob-
served with two pointings. Using multiple pointings and
incorporatingSDSS redshifts of brighterobjectsmitigate
fiber collision issues. Because the galaxy targets are rel-
atively bright ( r≤20.8), the spectra were obtained with
relativelyshortexposuretimes of3x600sto 4x900sunder
a variety of observing conditions.
Figure 1 shows the redshifts of galaxies versus their
projected clustrocentric radii for the 15 clusters stud-
ied here. The infall patterns are clearly present in all
clusters. We use the caustic technique (Diaferio 1999)
to determine cluster membership. Briefly, the caustic
technique uses a redshift-radius diagram to isolate clus-
ter members in phase space by using an adaptive ker-
nel estimator to smooth out the galaxies in phase space,
and then determining the edges of this distribution (see
Diaferio 2009, for a recent review). This technique has
been successfully applied to optical studies of X-ray clus-
ters, and yields cluster mass estimates in agreement
with estimatesfromX-rayobservationsandgravitational
lensing (e.g., Rines et al. 2003; Biviano & Girardi 2003;
Diaferio et al. 2005; Rines & Diaferio 2006; Rines et al.
2007, and references therein).
We apply the prescription of Danese et al. (1980) to
determine the mean redshift cz⊙and projected velocity
dispersion σpof each cluster from all galaxies within the
caustics. We calculate σpusing only the cluster members
projected within r100estimated from the caustic mass
profile.
2.2.SZE Measurements
The SZE detections are primarily from
Bonamente et al. (2008, hereafter B08), supplemented
by three measurements from Marrone et al. (2009,
hereafter M09). Most of the SZ data were obtained with
the OVRO/BIMA arrays; the additional clusters from
M09 were observed with the Sunyaev-Zel’dovich Array
(SZA; e.g., Muchovej et al. 2007).
Numerical simulations indicate that the integrated
Compton y-parameter YSZhas smaller scatter than the
peak y-decrement ypeak(Motl et al. 2005), so B08 and
M09 report only YSZ. Although ypeakshould be nearly
independent of redshift, YSZdepends on the angular size
of the cluster. The quantity YSZD2
Aremoves this depen-
dence. Thus, we compare our dynamical mass estimates
to this quantity rather than ypeakorYSZ. Table 1 sum-
marizes the SZ data and optical spectroscopy.
It is also critical to determine the radius within whichYSZis determined. B08 use r2500, the radius that en-
closes an average density of 2500 times the critical den-
sity at the cluster’s redshift; r2500has physical values of
300-700kpc forthe massiveclustersstudied by B08(470-
670kpcforthesubsamplestudiedhere). M09useaphys-
ical radius of 350 kpc because this radius best matches
their lensing data.
To use both sets of data, we must estimate the con-
version between YSZ(r2500) measured within r2500and
YSZ(r= 350 kpc) measured within the smaller radius
r=350 kpc. There are 8 clusters analyzed in both B08
and M09 (5 of which are in HeCS). We perform a least-
squaresfit to YSZ(r2500)−YSZ(r= 350kpc) to determine
an approximate aperture correction for the M09 clusters.
We list both quantities in Table 1.
3.RESULTS
We examine two issues: (1) the strength of the corre-
lation between SZE signal and the dynamical mass and
(2) the slope of the relationship between them. Figure 2
shows the YSZ−σprelation. Here, we compute σpfor all
galaxies inside both the caustics and the radius r100,cde-
fined by the caustic mass profile [ rδis the radius within
which the enclosed density is δtimes the critical density
ρc(z)].
Because we make the first comparison of dynami-
cal properties and SZE signals, we first confirm that
these two variables are well correlated. A nonparametric
Spearman rank-sum test (one-tailed) rejects the hypoth-
esis of uncorrelated data at the 98.4% confidence level.
The strong correlation in the data suggests that both σp
andYSZD2
Aincrease with increasing cluster mass.
Hydrodynamic numerical simulations indicate that
YSZ(integrated to r500) scales with cluster mass as
YSZ∝Mα
500, whereα=1.60 with radiative cooling and
star formation, and 1.61 for simulations with radiative
cooling, star formation, and AGN feedback ( α=1.70 for
non-radiative simulations, Motl et al. 2005). Combin-
ing this result with the virial scaling relation of dark
matter particles, σp∝M0.336±0.003
200 (Evrard et al. 2008),
the expected scaling is YSZ∝σ4.76(we assume that
M100∝M500). The right panels of Figure 2 shows this
predicted slope (dashed lines).
The bisector of the least-squares fits to the data has
a slope of 2 .94±0.74, significantly shallower than the
predicted slope of 4.8.
We recompute the velocity dispersions σp,Afor all
galaxies within one Abell radius (2.14 Mpc) and in-
side the caustics. Surprisingly, the correlation is slightly
stronger (99.4% confidence level). This result supports
the idea that velocity dispersions computed within a
fixedphysicalradiusretainstrongcorrelationswith other
cluster observables, even though we measure the velocity
dispersion inside different fractions of the virial radius
for clusters of different masses. Because cluster veloc-
ity dispersions decline with radius (e.g. Rines et al. 2003;
Rines & Diaferio 2006), σp,Amay be smaller than σp,100
(measured within r100,c) for low-mass clusters, perhaps
exaggerating the difference in measured velocity disper-
sionsrelativeto the differences in virialmass(i.e., σp,Aof
a low-mass cluster may be measured within 2 r100while
σp,Aof a high-mass cluster may be measured within r100;
the ratio σp,Aof these clusters would be exaggerated rel-Hectospec Virial Masses and SZE 3
Fig. 1.— Redshift versus projected clustrocentric radius for the 15 HeCS clusters studied here. Clusters are ordered left-to-r ight and
top-to-bottom by decreasing values of YSZD2
A(r2500). The solid lines show the locations of the caustics, which w e use to identify cluster
members. The Hectospec data extend out to ∼8 Mpc; the figure shows only the inner 4 Mpc to focus on the viria l regions.
ative to the ratio σp,100). Future cluster surveys with
enough redshifts to estimate velocity dispersions but too
few to perform a caustic analysis should still be sufficient
for analyzing scaling relations.
Because of random errors in the mass estimation, the
virial mass and the caustic mass within a given radius
do not necessarily coincide. Therefore, the radius r100
depends on the mass estimator used. Figure 2 shows
the scaling relationsfor two estimated masses M100,cand
M100,v;M100,cis the mass estimated within r100,c(where
bothquantitiesaredefinedfromthecausticmassprofile),
andM100,vis the mass estimated within r100,v(both
quantities are estimated with the virial theorem, e.g.,
Rines & Diaferio 2006). including galaxies projected in-
sider100,v. Similar to σp, there is a clear correlation
between M100,vandYSZD2
A(99.0% confidence with a
Spearman test). The strong correlation of dynamical
mass with SZE also holds for M100,cestimated directlyfrom the caustic technique (99.8% confidence).
The bisector of the least-squares fits has a slope of
1.11±0.16, again significantly shallower than the pre-
dicted slope of 1.6. This discrepancy has two distinct
origins. By looking at the distribution of the SZE sig-
nals in Figure 2, we see that, at a given velocity disper-
sion or mass, the SZE signals have a scatter which is a
factor of ∼2. Alternatively, at fixed SZE signal, there
is a scatter of a factor of ∼2 in estimated virial mass.
Unless the observational uncertainties are significantly
underestimated, the data show substantial intrinsic scat-
ter. Moreover, this scatter is comparable to the range of
our sample and, therefore, the error on the slope derived
from our least-squares fit to the data is likely to be un-
derestimated (see Andreon & Hurn 2010, for a detailed
discussionofaBayesianapproachtofittingrelationswith
measurement uncertainties and intrinsic scatter in both
quantities).4 Rines, Geller, & Diaferio
TABLE 1
HeCS Dynamical Masses and SZE Signals
Cluster z σ p M100,vM100,c YSZD2
AYSZD2
ASZE
(350 kpc) ( r2500)
km s−11014M⊙1014M⊙10−5Mpc−210−4Mpc2Ref.
A267 0.2288 743+81
−616.86±0.82 4.26 ±0.14 3.08 ±0.34 0.42 ±0.06 1
A697 0.2812 784+77
−596.11±0.69 5.96 ±3.51 – 1.29 ±0.15 1
A773 0.2174 1066+77
−6318.4±1.7 16.3 ±0.7 5.40 ±0.57 0.90 ±0.10 1
Zw2701 0.2160 564+63
−473.47±0.42 2.69 ±0.30 1.46 ±0.016 0.17 ±0.02a2
Zw3146 0.2895 752+92
−676.87±0.89 4.96 ±0.91 – 0.71 ±0.09 1
A1413 0.1419 674+81
−606.60±0.85 3.49 ±0.15 3.47 ±0.24 0.81 ±0.12 1
A1689 0.1844 886+63
−5215.3±1.4 9.44 ±5.66 7.51 ±0.60 1.50 ±0.14 1
A1763 0.2315 1042+79
−6416.9±1.6 12.6 ±1.5 3.10 ±0.32 0.46 ±0.05a2
A1835 0.2507 1046+66
−5519.6±1.6 20.6 ±0.3 6.82 ±0.48 1.37 ±0.11 1
A1914 0.1659 698+46
−386.70±0.57 6.21 ±0.21 – 1.08 ±0.09 1
A2111 0.2290 661+57
−454.01±0.41 4.77 ±1.23 – 0.55 ±0.12 1
A2219 0.2256 915+53
−4512.8±1.0 12.0 ±4.7 6.27 ±0.26 1.19 ±0.05a2
A2259 0.1606 735+67
−535.59±0.60 4.90 ±1.69 – 0.27 ±0.10 1
A2261 0.2249 725+75
−577.13±0.83 5.10 ±2.07 – 0.71 ±0.09 1
RXJ2129 0.2338 684+88
−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)
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