Cool Core Bias in Sunyaev-Zel'dovich Galaxy Cluster Surveys

1.4 MB·English

by  Henry W. Lin

Draft version January 21, 2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
COOL CORE BIAS IN SUNYAEV-ZEL’DOVICH GALAXY CLUSTER SURVEYS
Henry W. Lin1, Michael McDonald2, Bradford Benson3,4,5, and Eric Miller2
Draft version January 21, 2015
ABSTRACT
Sunyaev-Zeldovich(SZ)surveysfindmassiveclustersofgalaxiesbymeasuringtheinverseCompton
scatteringofcosmicmicrowavebackgroundoffofintra-clustergas. Theclusterselectionfunctionfrom
such surveys is expected to be nearly independent of redshift and cluster astrophysics. In this work,
5 we estimate the effect on the observed SZ signal of centrally-peaked gas density profiles (cool cores)
1 andradioemissionfromthebrightestclustergalaxy(BCG)bycreatingmockobservationsofasample
0 of clusters that span the observed range of classical cooling rates and radio luminosities. For each
2 cluster, we make simulated SZ observations by the South Pole Telescope and characterize the cluster
selection function, but note that our results are broadly applicable to other SZ surveys. We find that
n
the inclusion of a cool core can cause a change in the measured SPT significance of a cluster between
a
0.01%–10% at z > 0.3, increasing with cuspiness of the cool core and angular size on the sky of the
J
cluster (i.e., decreasing redshift, increasing mass). We provide quantitative estimates of the bias in
9
theSZsignalasafunctionofagasdensitycuspinessparameter,redshift,mass,andthe1.4GHzradio
1
luminosity of the central AGN. Based on this work, we estimate that, for the Phoenix cluster (one

of the strongest cool cores known), the presence of a cool core is biasing the SZ significance high by
]
O ∼6%. The ubiquity of radio galaxies at the centers of cool core clusters will offset the cool core bias
to varying degrees.
C
Subject headings: cosmology: observations – galaxies: cooling flows – galaxies: clusters
.
h
p
- 1. INTRODUCTION Many potential systematics have been considered in
o SZ-selectedclustersurveys. Simulationshaveshownthat
Galaxyclustersarepotentiallypowerfultoolstostudy
r SZsurveysareexpectedtoberelativelyinsensitivetothe
t dark energy and cosmology (see review by Allen et al.
s effectsofnon-gravitationalphysics(Nagai2006), projec-
a 2011); whereas standard candles test cosmology on ho-
tioneffects(Shawetal.2008),andthedynamicalstateof
[ mogeneous scales, clusters trace the growth of inhomo-
geneity via N(M,z), the cluster number density at a the cluster (Krause et al. 2012). The presence of a “cool
1 core” — a central region of over-dense gas accompanied
given mass and redshift. Accurate cluster cosmology re-
v by a drop in temperature — could also conceivably bias
quires not only reliable estimators of clusters properties
0 mass estimates. Finally, contamination from radio-loud
such as total mass (i.e., M ), but also an accurate es-
5 500 active galactic nuclei (AGN), generally located in the
timation of the survey selection function.
6 brightest cluster galaxy, could introduce bias by “filling
Anewtechniqueofassemblinganearlyunbiasedsam-
4 in” the SZ decrement (Sayers et al. 2012).
ple of galaxy clusters uses the Sunyaev-Zel’dovich (SZ)
0 Cool core bias is also of particular astrophysical inter-
effect(Sunyaev&Zeldovich1972): theinverseCompton
1. scattering of photons as they pass through the hot intr- est,asestimatesofthecoolcorefractionasafunctionof
0 acluster medium, leading to distorted-spectrum patches redshift might lead to valuable insights into the cooling
5 inthecosmicmicrowavebackground(CMB).Oneattrac- flowproblem(seereviewbyFabian1994). Recentstudies
1 have found evidence for a lack of dense, cool cores in the
tive characteristic of SZ surveys is that the SZ signal is
v: virtuallyredshiftindependent,asopposedtothe∼1/d2 centers of high-redshift galaxy clusters (e.g., Vikhlinin
L et al. 2007; Santos et al. 2010; McDonald 2011; McDon-
i fluxdimmingininfrared,optical,andX-raysurveys. Re-
X cently,largesamples((cid:38)100)ofgalaxyclustershavebeen ald et al. 2013). These results may be evidence for an
evolution in the cooling/feedback balance, or may be a
r assembledusingSZselectionthatincludeclustersoutto
a result of biases in X-ray and optical surveys (e.g., Sem-
z ∼ 1.5 (e.g., Ade et al. 2011; Hasselfield et al. 2013;
ler et al. 2012). Thus, in order to fully understand the
Vanderlinde et al. 2010; Reichardt et al. 2013) including
evolution of cooling and heating processes in the cores
many of the most massive known galaxy clusters (e.g.,
of galaxy clusters, we must understand how our sample
Menanteauetal.2012;Foleyetal.2011;McDonaldetal.
selection affects the observed cool core fraction.
2012).
Throughout this paper, we take the fractional mat-
ter density Ω = 0.27 and the fractional vacuum den-
m
[email protected]
sity Ω = 0.73, and neglect radiation Ω and curvature
1HarvardUniversity,Cambridge,MA02138,USA Λ r
2Kavli Institute for Astrophysics and Space Research, MIT, Ωk. We introduce two dimensionless forms of the Hub-
Cambridge,MA02139,USA ble parameter, h ≡ H /(100kms−1Mpc−1) = 0.7 and
0
3Fermi National Accelerator Laboratory, Batavia, IL 60510- E(z)≡H(z)/H . We define the cluster radius r such
0 500
0500
that the average matter density interior to r is 500
4Kavli Institute for Cosmological Physics, University of 500
Chicago,5640SouthEllisAvenue,Chicago,IL60637 times the critical density ρcr of the universe as well as
5Department of Astronomy and Astrophysics, University of the cluster mass M = 4πr3 ×500ρ .
Chicago,5640SouthEllisAvenue,Chicago,IL60637 500 3 500 cr
2
2. METHODS positive nuclei density n . The density normalization
p
n was chosen to enforce an average gas fraction f =
In order to measure the influence of cool cores and 0 gas
0.125 within r .
radio-loud AGN on the SZ signal, we generate one- 500
For the unperturbed, non-cool core clusters, we adopt
dimensional (spherically-symmetric) mock galaxy clus-
the simple temperature profile
terswithpropertiesdrawnfromexistingsamplesofhigh-
mass, relaxed clusters, and add a perturbation either to
the gas density (cool core) or mm flux (radio source). T(r) (cid:34) (cid:18) r (cid:19)2(cid:35)−0.45
Simulated observations of these mock clusters with the =1.35 1+ , (3)
T 0.6r
South Pole Telescope are generated, including realistic 0 500
noise and background, yielding a realistic SZ signal-to-
whichisthe“universal”temperatureprofileofVikhlinin
noise measurement.
et al. (2006) with the term accounting for the central
temperature drop suppressed.
2.1. Constructing Mock Galaxy Clusters
Foreachnon-coolcorecluster,wegenerate14progres-
Since we are mostly interested in the SZ signal of our sively cuspier cool-core clusters. We start by duplicat-
mockclusters,thekeyingredientinourclustermodelsis ing each non-cool core cluster and shifting ρ → ρ+δρ.
the pressure profile P(r). However, we must also model To avoid adding more free parameters (a cooling radius,
theclusterdensityprofilesρ(r)sincewewishtocalculate etc.) which may be correlated with the other model pa-
thechangeinthepressureprofileδP duetoadensityper- rameters, we simply increment α to steepen the density
turbationδρ,whichingeneralisnotδP ∝δρasthetem- profile towards the center of the cluster. Each successive
perature profile also shifts T → T −δT. Consequently, clusterhasα =α +0.25. Thedensitynormalization
n n−1
we need an additional constraint on our clusters to cal- n wasalsoreducedbyanappropriatefactortokeepthe
0
culate δP, which we take to be hydrostatic equilibrium: gas mass within r constant.
200
dP = −ρdΦ, where Φ is gravitational potential. Since Finally,theassumptionofhydrostaticequilibriumwas
dr dr
thereisnoevidencethattheunderlyingdarkmatterdis- used to recalculate the pressure profile of the cool core
tribution is affected by the baryonic processes driving cluster P →P +δP:
cool cores (Blanchard et al. 2013), δΦ is straightforward
tocalculate,andusingP(r )asaboundaryvalue,itis (cid:90) (cid:20)ρ+δρdP (ρ+δρ)GδM(cid:21)
500 P +δP = dr˜ − . (4)
possible to calculate P +δP. ρ dr˜ r˜2
The details of our density and pressure parameteriza-
(cid:82)
tions are now presented. For each non-cool core cluster, We vary P(r200) until the “internal energy” ∝ PdV of
wedrawaclustermassM500 andaredshiftfromuniform the cool core cluster out to r200 agrees with its non-cool
distributions(seeTable1),withrangesmotivatedbythe core counterpart. Note that in general, this means that
observed ranges in the SPT 2500 deg2 survey (Bleem δP(r200)(cid:54)=0, though at r200, δP (cid:28)P.
et al. 2014). The gas density profiles were then modeled The above methods were used to generate 200 × 15
by a slightly modified version of the functional form of mock clusters which ranged from pure non-cool core
Vikhlinin et al. (2006): α = 0 to strong cool core α = 3.5. We chose the maxi-
mum value α=3.5 as even strong cool core clusters like
n n = n20(r/rc+δ)−α 1 , (1) Phoenix have α<3.5.
p e [1+(r/r )2]3β−α/2[1+(r/r )3](cid:15)/3
c s
2.2. Model Validation
where the various length scales and slopes were sampled
Additionalquantitieswerederivedtodemonstratehow
from realistic ranges for massive galaxy clusters by a
the wide range of observed cluster properties are cap-
Monte Carlo process, which we detail in Table 1. For
tured by our methodology.
this work, we have “regulated” the parameterization of
Vikhlininetal.(2006)byinsertingasmallfactorδ =0.1
1. Temperature. Ourderivedtemperatureprofilesare
to obtain a finite density at r = 0. This is necessary
displayed in Figure 1(b). The cool core profiles ex-
as we will eventually extract an SZ signal from the en-
hibit a central temperature drop similar to the av-
tirecluster,whereasthebehavioroftheprofileverynear
erage cool core profiles observed in X-ray selected
r =0 is not of interest to observational works due to the
samples (Vikhlinin et al. 2006). The characteristic
limited spatial resolution of any survey.
temperatures of our clusters range from ∼1 keV
For a pure non-cool core, α = 0 and equation (1) re-
to ∼30 keV, which is in agreement with observa-
duces to the simpler form
tions of massive clusters. Furthermore, T /T ,
min 0
n2 1 which corresponds to the ratio of the central cool
n n = 0 . (2)
p e [1+(r/r )2]3β [1+(r/r )3](cid:15)/3 core core temperature to the α = 0 central tem-
c s perature, range from 0.1 to 0.85 in the (Vikhlinin
where α, β, and (cid:15) are slope parameters for the r (cid:28) r , et al. 2006) clusters, which is completely covered
c
r (cid:46)r (cid:46)r and r (cid:29)r radial regimes. We convert n n by our simulated clusters. Similarly, our simula-
c s s p e
to the gas mass density via the relation ρ = m n A/Z tions cover the range of temperature profiles ob-
p e
where Z = 1.199 is the average nuclear charge, which is served in (McDonald et al. 2014), which find that
greater than unity because of ionized elements heavier the fractional temperature in the inner 0.04R is
500
than hydrogen, A = 1.397 is the average nuclear mass, T /T = 0.740.09 for high redshift, cool core
core 500 −0.04
which is greater than Z due to the presence of neutrons, clusters and higher for low redshift cool core clus-
and n = Zn is the electron density in terms of the ters.
e p
3
TABLE 1
Parameters. All radii are in units of kpc. We report 1σ uncertainties.
Parameter Range Distribution Notes
M /M [1×1014,2.1×1015] Log-Uniform SPT 2500 deg2; Bleem et al. (2014)
500 (cid:12)
z [0,2] Uniform SPT 2500 deg2; Bleem et al. (2014)
β (0.28,0.88,0.71) Trianglea McDonald et al. (2013),Vikhlinin et al. (2006)
(cid:15) 7.1−6βb — Andersson et al. (2011)
r /r (0.067,0.26,0.47) Trianglea McDonald et al. (2013)
c 500
logr /r 1.21 — McDonald et al. (2013)
s 500
aWefittriangledistributionstotheempiricaldistributionsderivedfromMcDonaldetal.(2013)toavoidthe
longGaussiantailsthatleadtounphysicalclusters. Wereportthe(min,max,mode)ofthedistributioninthe
rangecolumn.
bThemockSZsignalisvirtually(cid:15)-independent. Wechoose(cid:15)toletdlogP/dlogr=−0.90atlargeradii,as
givenbytheuniversaltemperatureprofileofArnaudetal.(2010).
Fig. 1.— Clockwise from the upper left: density, temperature (Tx = (cid:82) n2TdV/(cid:82) n2dV), cooling time, and entropy as a function of
cluster radius for a typical mock cluster. All of the displayed profiles were obtained by cloning a single non-cool core core cluster and
addingdensityperturbations. Theseplotsrepresent1/200thofthetotaldataset. Intheupperrightplot,thethickdashedlinerepresent
the universal temperature profile of Vikhlinin et al. (2006). In the lower left plot, the thick dashed line represents the predicted entropy
profileslopefromnon-radiativesimulationsofVoitetal.(2005).
2. Entropy Parameter. The entropy profiles of our slope of our entropy profiles outside of the core
mockclustersarederivedinanefforttoshowtheir is also in reasonable agreement with the accretion
resemblancetoobservedclusters. FollowingtheX- shock model of Voit et al. (2005), which represents
raysurveyconventions,anentropyparameterK ≡ the“baseline”entropyprofileofclustersifradiative
k Tn−2/3 is introduced. We plot this parameter and other non-gravitational processes are ignored.
B e
as a function of radius in Figure 1(c). Consistent
with Cavagnolo et al. (2009), Hudson et al. (2010) 3. Cooling Time. The cooling time t has been
cool
and McDonald et al. (2013), the cool core clusters shown by Hudson et al. (2010) to be a quantity
have a central entropy K0 (cid:46)100keVcm2, whereas that can be used to segregate cool cores from non-
for non-cool core clusters K0 (cid:38) 100keVcm2. The cool cores. In Figure 1(d), tcool = 3kT/(neΛ) is
4
plotted as a function of radius. Here, Λ(T,Z) is where L is the 1.4 GHz luminosity in WHz−1. All
1.4
thecoolingfunctiongivenbySutherland&Dopita other parameters are set to their median values. The
(1993). Most non-cool cores have central cooling rangeinlogL coversthemostluminousbrightestclus-
1.4
times t (cid:38)1/H whereas the cool core clusters ter galaxy (BCG) in the 152 X-ray BCG sample of Sun
cool,0 0
have cooling times t ∼ 1Gyr, consistent with (2009), allowing for unusually shallow spectrum systems
cool,0
the findings of Hudson et al. (2010). that would generate a large bias.
We convert these luminosities to the SPT observing
2.3. Sunyaev Zel’dovich Maps frequencies by assuming a spectral slope of α = 0.89,
s
Foreachmockcluster,weconstructedanSZmapwith where flux scales with ναs. This slope was the median
spectrum between 1.4 and 30 GHz for radio galaxies in
an angular resolution of ∆θ = 0.125arcmin, which was
a sample of 45 massive clusters observed by Sayers et al.
converted to spatial resolution ∆r = d (z)∆θ where
2D A (2012), however it is always possible to reinterpret the
d (z) is the angular diameter distance to the cluster.
A resultsfordifferentchoicesslopes. Wediscussdeviations
The SZ line of sight integral was then calculated at each
fromthisassumptioninSection3.1. Asacontrol,apoint
pixel:
sourceisnotaddedtoonecool-core/non-coolcorecluster
(cid:90) (cid:26) σ n k T(cid:27) at each mass and redshift.
∆T/T = dl f (ν,T) t e B (5)
CMB θ(cid:126) sz mec2 3. RESULTS
(cid:18) eX +1 (cid:19) In Figure 2, we show the fractional change in the
fsz(ν,T)= XeX −1 −4 [1+δsz(X,T)] (6) SPT detection significance as a function of the cool
core cuspiness (Vikhlinin et al. 2007), angular size
whereX ≡hν/k T ,σ istheThomsoncrosssection, (Θ ≡ r /d (z)), and redshift. At Θ < 3(cid:48), the
B CMB t 500 500 A 500
m isthemassoftheelectron,cisthespeedoflight,and SPT detection significance changes by ∆ζ/ζ = (ζ −
e CC
δ is the relativistic correction as given by Itoh et al. ζ )/ζ (cid:46) 0.1 for any type of cool core at any red-
sz NCC NCC
(1998). Forourpurposes,itsufficedtoevaluateδ (T)at shift. The cool core bias is most pronounced for low-
sz
T(r ). We made simulated SZ maps at 97.6 and 152.9 redshift, high mass systems, where the angular size of
500
GHz,correspondingtotheeffectivefrequencyoftheSPT the SPT beam is much smaller than the angular size of
observingbandsusedforcluster-finding(Carlstrometal. theclustersothattheinnerpressureprofilebecomesim-
2011). OurmockSZmapsweregeneratedataresolution portant. In principle, the resulting redshift-dependence
(cid:28)1(cid:48) andthenlaterconvolvedwiththeSPTbeamwhich in the bias might lead to a ∼2% bias towards cool core
has an effective FWHM of ∼1.2(cid:48). clusters from z ∼ 1.5 to z ∼ 0.3 in the observed cool
ThemockSZmapswerethenspatiallyfilteredinaway core fraction CCF ≡ N /N evolution, but
coolcore clusters
identical to the SPT cluster-finding algorithm used in these effects should be negligible when compared to the
Reichardt et al. (2013). A multi-frequency matched spa- ∼ 30% evolution in CCF reported in e.g. McDonald
tial filter (Haehnelt & Tegmark 1996; Melin et al. 2006) et al. (2013).
was applied to the SZ maps in Fourier space, which ac- In real observational studies it may be difficult to de-
counted for the cluster gas profile, and the other sources terminethedensityshapeparameterα,asthisrequiresa
of astrophysical signal and instrumental noise expected multi-parameter fit to the n n profile and assumptions
p e
in the SPT maps. The spatially filtered signal at the on the shape of the cool core/non-cool core profiles. To
true cluster position was then re-normalized so that it amelioratethesituation,wecalculatethe“cuspiness”pa-
was equivalent to the unbiased SPT significance, ζ, used rameter γ ≡ −dlogρ/dlogr| (Vikhlinin et al.
r=0.04r500
inVanderlindeetal.(2010), whichwouldeffectivelycor- 2007) of all simulated clusters. Since this quantity
respond to the signal-to-noise of cluster in a SPT map. does not refer to the details of our parameterizations,
These SZ maps were run through the SPT cluster- it serves as a model-independent indicator of cool core
finding pipeline, which adds noise in the form of point strength;correlationsbetweenthesequantitiesand∆ζ/ζ
sourcesandCMBanisotropymaps,beforeattemptingto are shown in Figure 2. The bias is well-captured by the
detect clusters. For each cluster, the SPT pipeline then following relation:
reported ζ, the “unbiased significance” (see Vanderlinde
etal.2010;Bensonetal.2011)thatroughlycorresponds (cid:104)δζ/ζ(cid:105)=10−2.08±0.01(∆γ)0.85±0.01(cid:18)Θ500(z)(cid:19)1.52±0.02
to the cluster signal-to-noise ratio. arcmin
(7)
2.4. Active Galactic Nuclei where ∆γ = γ−γ| ≈ γ−0.1. This expression gives
α=0
Inordertotesttheeffectsofincludingradio-loudAGN, a practical method for estimating the SZ bias of a given
we created mock clusters with the following grid of pa- cool core cluster, assuming that z >0.1.
rameters:
3.1. Radio bias
• M /2×1014M =100,101/2,101
500 (cid:12) Assuming a typical spectral slope of α = 0.89 (Say-
s
ers et al. 2012), only sources with logL >26 result in
• z =0.3,0.4,0.5,0.7,1.1,1.4,1.7 1.4
|∆ζ/ζ| (cid:29) 10% for z > 0.3 and M > 3×1014. Conse-
and we include a radio-loud AGN at the cluster center, quently, none of the 152 radio-loud AGN listed in Sun
with a range of luminosities: (2009) would produce a bias in excess of 10% if located
in massive clusters. More generally, the radio-loud point
• logL =23 – 29, ∆logL =1.0 source bias can be readily estimated by the following fit
1.4 1.4
5
1 1 1
0.1 0.1
0.1
0.01 0.01
0.01
Out[364]=(cid:68)Ζ(cid:144)Ζ Out[362]=(cid:68)Ζ(cid:144)Ζ Out[442]=(cid:68)Ζ(cid:144)Ζ
0.001 0.001
0.001 0.1(cid:60)z(cid:60)0.3 (cid:81)500(cid:60)1.75' 0.1(cid:60)z(cid:60)0.3
0.3(cid:60)z(cid:60)0.6 1.75'(cid:60)(cid:81)500(cid:60)3' 0.3(cid:60)z(cid:60)0.6
0.6(cid:60)z(cid:60)1.2 10(cid:45)4 3'(cid:60)(cid:81)500(cid:60)5' 10(cid:45)4 0.6(cid:60)z(cid:60)1.2
1.2(cid:60)z 5'(cid:60)(cid:81)500 1.2(cid:60)z
10(cid:45)4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 10(cid:45)50.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 10(cid:45)5(cid:45)3.5 (cid:45)3.0 (cid:45)2.5 (cid:45)2.0 (cid:45)1.5 (cid:45)1.0
Γ Γ log 10(cid:45)2.08Γ0.85Θ1.52
Fig. 2.—DependenceoftheSZbiasonthecuspiness(γ ≡−ddllooggρr), redshift(z), andangularsize(Θ500 ≡r500/dA(z))for3000mock
clusters. Intheleftpanel,thepointcolorcorrespondstoredshiftofthemockcluster. Inthecentralpanel,wecolorbytheangularsizeof
thecluster. Theclearseparationintocoloredbandsinthisplotsuggeststhatthebiasismorefundamentallytiedtotheangularsizethan
it is to the redshift. In the right-most panel, we show the edge-on projection of the best-fit three-dimensional plane (Equation 7). The
∼0.5dexscatterisprimarilyduetovariationsintheclustergasdensityprofile,andtheimperfectclassificationofcoolcorestrengthbased
ontheγ parameter.
2.0 "6 2.0 "8 "7 "5 2.0 "7 "5
"7 "4 "3
M#1015M , z#0.3 "2 M#1015M , z#0.6 "8 M#1015M , z#1
" " "
1.5 1.5 1.5 "3
! ! !
! !! ! !! ! !!
! ! !
Α 1.0! Α 1.0! Α 1.0!
s ! s "6 ! "1 s !
"5! ! 0 ! ! ! !
0.5 ! "3 !! 0.5 ! "4 !"!2 0.5"6 ! "4 !! "1
"1
"2
0.0 1 0.0 0 0.0 0
22 23 24 25 26 27 22 23 24 25 26 27 22 23 24 25 26 27
logL W Hz logL W Hz logL W Hz
1.4 1.4 1.4
2.0 "5 "3 2.0 "6 2.0 "8 "7 "5 "3
"6 M#3$10!14M"", z##0.3 "7 M#3$101!4M""",4z##0.6 "2 M#3$1014!M"", z##1
1.5 1.5 1.5
! ! !
! !! ! !! ! !!
! ! !
Α 1.0! 0 Α 1.0! Α 1.0!
s ! s ! s "6 ! "1
! ! "1 ! ! ! !
"5 0
"4 ! !! ! !! ! "4 !!
0.5 0.5 0.5
"2 "2
"3
"1
0.0 1 2 0.0 1 0.0 0
22 23 24 25 26 27 22 23 24 25 26 27 22 23 24 25 26 27
logL W Hz logL W Hz logL W Hz
1.4 1.4 1.4
Fig. 3.—Dependencyof∆ζ/ζonM500,redshift,radioluminosity,andspectralslope. Thecontoursrepresentlinesofconstantlog(∆ζ/ζ).
If an AGN exists in the shaded region, it will overpower the SZ signal. The large black dot represents a Phoenix-like cluster at multiple
redshifts. Evenatz=0.3,aPhoenix-likeclusterwillexhibitonlya1%bias. Themostextremesystemsexhibita∼10%biasatz∼0.3.
Red dots represent a sam!ple"of X#-ray selected massive elliptical galaxi!es fr"om#Dunn et al. (2010) and blue dots repr!esen"t ex#treme AGN
hostedinmassiveclustersHlavacek-Larrondoetal.(2012).
6
ple, low-redshift galaxy clusters that harbor a powerful,
0.15 radio galaxy (logL (cid:38) 25) but lack a cool core will be
1.4
Strong CC under-represented in SZ surveys. However in practice,
0.10 such systems appearto be uncommonin nature (seee.g.
Radio-loud
Sun (2009) and Dunn et al. (2010)). The AGN bias is
morestronglyredshiftdependentthanthecoolcorebias
0.05
due to the ∼ 1/d2 flux dimming where d is the lumi-
L L
nosity distance, whereas the evolution in the simulated
Dz 0.00 cool core bias is a result of SPT beam effects and weak
Out[106]= dependencies of our parameterizations on ρ (z).
z cr
-0.05
4. ROBUSTNESS OF SIMULATIONS
In this section we estimate how relaxing several of our
-0.10
Phoenix simplifyingassumptionscouldchangeourbiasestimates.
Though a thorough treatment of each of these issues is
-0.15 beyondthescopeofthispaper(andshouldbeaddressed
by, e.g. N-bodysimulations), anestimateontherelative
0.0 0.5 1.0 1.5 2.0 importance of these effects can be obtained by recasting
these effects into an M dependence.
500
z
4.1. Deviations from hydrostatic equilibrium and
Fig. 4.— Strong cool core (α=2.75) bias and radio-loud AGN non-thermal pressure
bias. The black, dotted lines represent the massive cool core
In the absence of complicated astrophysics, a
“Phoenix” cluster, with radio spectral slope αs =1.3 (McDonald
etal.2014)andX-raypropertiesgiveninMcDonaldetal.(2013), galaxy cluster will equilibrate on dynamical timescales
forarangeofredshifts. Thepinkshadedregionrepresentsaseries t ∼R /v ∼(Gρ)−1/2 ∼109 yr. For hydrostatic
ofradio-loudAGNwithspectralslopeαs≥0,whichdemonstrates edqyunialmibircium 5t0o0 generically hold, the cooling timescale
therangeinpossibleAGNbias. Thedarkbluelinesarethemedian
biasineachbinandthelightblueregionsrepresent2σ confidence should be much longer than the dynamical timescale
intervals. In general, the cool core bias and the radio-loud AGN t (cid:29) t so that hydrostatic equilibrium will
cool dynamic
biasworktoroughlycanceleachother. be restored efficiently when gas cools. If the cooling
timescaleisshorterthantheequilibriumtimescaleinthe
to our mock clusters (see §2.4): core of the cluster, we would expect pressure to be lower
than what is required to maintain hydrostatic equilib-
(cid:104)δζ/ζ(cid:105)=−0.03(cid:16) νSZ (cid:17)−αs(cid:18)S1.4(cid:19)(cid:18) M500 (cid:19)−1 rwiuoumld, sminecaengatshamtutshtebbeiasisnkshinogultdowbaerdgsenthereiccaolrlye.loTwheisr
1.4GHz mJy 1014M
(cid:12) thanwhatwehavederived, sincetheobservedchangein
(8)
pressure should be smaller than that required to main-
In Figure 3 we show a contour plot of ∆ζ/ζ on
tain hydrostatic equilibrium δP ≤δP .
Θ (M ,z),α,L parameter space. We display flux obs eq
500 500 1.4 Throughout our discussion, we have assume that pres-
and luminosity data from a sample of X-ray selected
sureispurelythermal,e.g. P =nkT. Analyticwork(Shi
massive elliptical galaxies from Dunn et al. (2010)
& Komatsu 2014), numerical simulations (Dolag et al.
and extreme AGN hosted in massive clusters Hlavacek-
2005;Sembolinietal.2013;Nelsonetal.2014;Battaglia
Larrondo et al. (2012). As can be seen by Figure 3, the
etal.2014a),andmulti-wavelengthobservations(Planck
fractionalchangeinSPTsignificancewillbestronglyde-
Collaboration et al. 2013; von der Linden et al. 2014;
pendent on the assumed value of α , which has a fairly
s Donahue et al. 2014; Sereno et al. 2014), however, have
large scatter. For example, Coble et al. (2007) measures
shown that non-thermal pressure can contribute signifi-
a median value of α =0.72 between 1.4 and 28.5 GHz.
s cantly(∼20%;increasingwithclusterradius),whichcan
Lin et al. (2009) explicitly measure the radio emission
leadtoanunderestimateofM byabout∼10%(Shi&
at frequencies ranging from 1.4 to 43 GHz in an X-ray 500
Komatsu2014). Sourcesofnon-thermalpressureinclude
selected sample of clusters and find mostly steep spectra
turbulencefromintraclustershocks,magneticfields,and
(α > 0.5), but also find a substantial number of flat
s cosmic rays. If turbulence dominates the non-thermal
or even inverted spectra. Neither of these studies – or
pressure,non-thermalpressurewillpersistforonroughly
any published studies of radio galaxies in clusters, for
the dynamical timescale (Shi & Komatsu 2014).
that matter – extend to ∼100GHz, so there is still sub-
From equation (8) we see that bias scales as 1/M, so
stantial uncertainty in the value of α that we should be
s to first order, we expect that adding non-thermal pres-
using.
sure will rescale the bias by (1+P /P )−1,
In Figure 4 we show the strong cool core (α = 2.75) non-therm therm
if we assume that the processes responsible for forming
bias and the radio-loud AGN bias as a function of red-
cool cores and providing non-thermal pressure are un-
shift. Interestingly, while SZ surveys are biased towards
correlated. As clusters evolve, this term could introduce
cool cores, they are biased against radio-loud AGN, and
an additional redshift dependence on the cool core bias,
this bias is of the same order of magnitude as the strong
if the processes generating non-thermal pressure evolve
cool core bias, defined as the average ∆ζ/ζ over cool
with redshift.
cores with α = 2.75. Since powerful radio galaxies tend
to live in the center of strongly-cooling galaxy clusters,
4.2. Varying f
thenetbiasispartiallycanceled(seefigure4). Inprinci- gas
7
Throughout this work, we have assumed that f = ples of radio-mode AGN feedback known. At z = 0.6,
gas
0.125 with no scatter. However, the numerical simula- thesetwoclusterswouldbebiasedlowinSZsignificance
tionsofBattagliaetal.(2014b)whichincludedradiative by1.8%and110%, respectively. Atz =1.0, thesebiases
physics and AGN feedback estimate a scatter ∼ 10% in are further reduced to 0.5% and 31%. Thus, while SZ
f at r < r . Eckert et al. (2013) suggest that cool surveys may miss the most extreme radio-loud clusters,
gas 500
coreclustersandnon-coolcoreclustersdifferinf bya this bias is rapidly reduced with increasing redshift. In
gas
fewpercent,withcoolcoreclustersmorereliablytracing nearby X-ray-selected clusters, (cid:28)1% of clusters have ra-
the cosmic baryon fraction. dio luminosities as high as Cygnus A (Hogan et al. in
For a fixed number of baryons, a lower f will re- prep), so we expect this bias to be small overall.
gas
sult in an increase in the pressure normalization, in or-
der to overcome the steeper potential due to additional 6. MASS BIAS IN SZ SURVEYS
dark matter. For a cluster in hydrostatic equilibrium,
To translate SZ observations to cosmological con-
kT ∝ M/r giving P ∝ M2/3 ∼ (1/fgas)2/3Mb2a/r3yon, straints, it is necessary to estimate mass distribution of
which means that the effects of changing 1/f will be an ensemble of clusters. To this end, Vanderlinde et al.
gas
roughly equivalent to changing M . Equations (7) and (2010) and Benson et al. (2011) have adopted ζ as a
500
(8) can therefore be rescaled by a factor (f /0.125). proxy for M by assuming a scaling relation ζ ∝ MB.
gas 500
Adding intrinsic scatter in f would then simply add As a consequence, the true M of cool core clusters
gas 500
scatter in the SZ bias. If f varied with redshift, this will be underestimated by an amount on the order of
gas
(cid:82)
could lead to a further evolutionary factor in Equation P(Θ ,γ)(cid:104)δζ/ζ(cid:105)dΘ dγ onaverage(forB ∼1)ifthe
500 500
(8). However, Battaglia et al. (2014b) found little evi- calibration was performed using only a sample of non-
dence for such a redshift dependence, as has long been cool core clusters. In principle, this redshift-dependent
assumed (White et al. 1993). bias should translate into a distorted N(M,z); however
the log-normal intrinsic scatter in ζ given mass has been
4.3. Effects and Evolution of AGN measured to be 0.21±0.10, calibrated using X-ray ob-
We purposefully exclude any evolution in AGN prop- servations (Benson et al. 2013). Given that the Phoenix
erties as a function of cluster mass and redshift, despite cluster has the most-cuspy X-ray surface brightness pro-
evidencethatsuchlinksexist(e.g.,Maetal.2013). How- file of the SPT clusters with Chandra X-ray follow-up
ever,bycastingEquation(8)intermsoftheclustermass, (McDonaldetal. 2013), whichincludes∼90clusters, we
redshift, and radio luminosity, we allow the direct incor- expect the bias in ζ to represent an extreme example.
poration of such trends into the bias estimate. By fully Therefore, the scatter in ζ given mass due to cool cores
samplingagridofparameters(see§2.4),wemakecertain should be a factor of several below the overall scatter
that, regardless of how AGN evolve, we understand how in ζ and its current measurement uncertainty. The un-
they influence the SZ signal at all redshifts and cluster certainty in scatter has a negligible effect on the cosmo-
masses. logical constraints from current SZ cluster surveys (e.g.,
Radio-mode AGN feedback can modify the gas distri- Benson et al. (2013)), so the additional scatter from the
bution of galaxy clusters as gas is heated and expelled effect of cool cores will be even less significant.
from the core, leading to deviations from hydrostatic Although we have focused on the SPT SZ survey, our
equilibriumintheinnerregion. Suchprocesseswilltypi- results should be applicable to other SZ surveys, e.g.
callyresultinlesspressurethanpredictedbyhydrostatic Planck, ACT, etc. Broadlyspeaking, we expect less bias
equilibrium,sincethegasissupportedbyfeedbackinad- inasurveywithlowerangularresolutionandgreaterbias
dition to thermal pressure. inahigher-resolutionsurvey,sinceconvolvingawidesur-
vey beam function with a cuspy SZ signal will smooth it
5. APPLICATION TO WELL-KNOWN SYSTEMS to a less cuspy signal.
Using Equations 7 and 8, we can now calculate how
biased SZ surveys are (or are not) for some well-studied 7. CONCLUSION
extreme systems. First, we consider two of the strongest Using extensive Monte Carlo simulations of galaxy
known cool cores: Abell 1835 (γ = 0.85, M500 ∼ 1015 clusters and mock SPT observations, we have estimated
M(cid:12); McNamara et al. 2006) and the Phoenix cluster the SZ bias due to cool cores in relaxed, massive sys-
(γ =1.29,M ∼1.3×1015 M ;McDonaldetal.2012). tems. By doing so, we have constrained the cosmolog-
500 (cid:12)
Assuming both of these clusters are at z = 0.6, we find ical and astrophysical bias due to the presence of cool
∆ζ/ζ =3.6% and 5.9%, for Abell 1835 and Phoenix, re- cores and radio-loud AGN in SZ surveys. We find that
spectively. At z = 1.0, these biases are reduced to 2.8% the bias from cool cores is no larger than ∼ 10% for
and4.2%(seealsoFigure4). Giventhatthesearetwoof z > 0.1 systems, and for typical high redshift objects
the most extreme cool cores known, we expect the typi- (z (cid:38) 1,r (cid:46) 800kpc,∆γ (cid:46) 1) the bias is at the per-
500
cal bias towards selecting cool cores in SZ surveys to be cent level. Further, the presence of radio-loud sources in
(cid:28)5% at z >0.5. cool cores should reduce the overall bias, though at low
Figure 3 quantifies the radio bias for a variety of redshifts z (cid:46)0.3, the bias from radio-loud point sources
nearby clusters from Dunn et al. (2010) and Hlavacek- should dominate any cool core bias.
Larrondo et al. (2012), but here we specifically look at Our results support the long-asserted claim that an
two well-known nearby radio-loud central galaxies: Hy- SZ-selected sample of galaxy clusters is a robust cosmo-
dra A (S = 40.8 Jy, z = 0.055; Bˆırzan et al. 2004) logicalprobe: thoughweobserveasmallbiasinthemass
1.4
and Cygnus A (S = 1600 Jy; z = 0.056; Bˆırzan et al. estimator,themagnitudeofthebiasismuchsmallerthan
1.4
2004), the latter being one of the most powerful exam- the typical scatter in the mass relationship.
8
We provide estimates of the SZ bias as a function of SZ-selecteddatasetareonlysubjecttoasystematicbias
redshift,mass,andcuspiness,parametersthataremodel- of order one percent, a significant reduction over X-ray
independent, as well as the radio bias. One can estimate selection. Since there is a stringent upper limit on the
thebiasofagivensystemeasilybyplugginginvaluesto redshift evolution of the cool core fraction bias, we can
ourfunction∆ζ/ζ =f(z,r ,γ)+g(L ,z,M )where now confidently say that almost all observed evolution
500 1.4 500
f(z,r ,γ) is given by equation (7) and g(L ,z,M ) in the cool core fraction reflects genuine cool core evolu-
500 1.4 500
is given by equation (8). By quantifying the cool core tion. With the arrival of results from Planck, SPT, and
bias, astrophysical constraints on cool core properties the Atacama Cosmology Telescope and others, SZ sur-
fromSZsurveyscannowbemorereliablyinterpreted. In veys might be ideal for studying the mysterious balance
particular, constraints on the cool core fraction from an between heating and cooling.
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