SZ power spectrum and cluster numbers from an extended merger-tree model

0.24 MB·English

by  Irina Dvorkin

Mon.Not.R.Astron.Soc.000,1–8(2011) Printed9January2012 (MNLATEXstylefilev2.2)
SZ power spectrum and cluster numbers from an extended
merger-tree model
Irina Dvorkin1⋆, Yoel Rephaeli1,2, Meir Shimon1
2 1School of Physics and Astronomy, TelAviv University,TelAviv, 69978, Israel
1 2Centerfor Astrophysics and Space Sciences, Universityof California, San Diego, La Jolla, CA 92093-0424
0
2

n
a 9January2012
J

5
ABSTRACT

] We haverecently developedanextendedmerger-treemodelthat efficiently followshi-
O
erarchicalevolution ofgalaxy clusters and providesa quantitative descriptionof both
C their dark matter and gas properties. We employed this diagnostic tool to calculate
. the thermal SZ power spectrum and cluster number counts, accounting explicitly for
h
uncertaintiesintherelevantstatisticalandintrinsicclusterproperties,suchasthehalo
p
mass function and the gas equation of state. Results of these calculations are com-
-
o paredwiththoseobtainedfroma directanalytictreatmentandfromhydrodynamical
r simulations. We show that under certain assumptions on the gas mass fraction our
t
s results are consistent with the latest SPT measurement. Our approachcan be partic-
a
ularlyuseful inpredicting clusternumber counts andtheir dependence onclusterand
[
cosmologicalparameters.

1
Key words: galaxies: clusters: general - large-scale structure of Universe - cosmic
v
4 backgroundradiation
0
3
1
.
1 1 INTRODUCTION trumissignificantlylargerthanthestatisticalerrors.Several
0 theoreticalstudies(Shaw et al.2010;Trac, Bode & Ostriker
2 The importance of the Sunyaev-Zel’dovich (SZ) effect
2011;Efstathiou & Migliaccio2011)havealsodemonstrated
1 as a valuable cosmological probe and a usefull tool for
the sensitivity of the SZ effect to simplified modeling of IC
: discovery of high-redshift clusters is now firmly estab-
v gas, particularly its equation of state.
lished. In view of the sensitivity of the effect to the
i
X halo mass function (MF), in particular to the normal- TherearecurrentlytwoapproachestomodelingtheSZ
powerspectrum:Ananalyticalapproach(Komatsu & Seljak
r ization of the matter power spectrum, the SZ effect can
a be used to constrain various cosmological parameters of 2002) assumes universal profiles of the dark matter (DM)
density and the IC gas density (and pressure), whose pa-
the(standard)ΛCDMmodel(e.g.,Holder, Haiman & Mohr
rameters follow simple scaling laws with mass and theclus-
2001;Allen, Evrard & Mantz2011,andreferences therein),
ter redshift (of observation). The relative contribution of
including the equation of state of dark energy (e.g.,
clustersatdifferentredshiftsisweightedaccordingtoanap-
Wang et al. 2004) and the total neutrino mass (e.g.,
propriate MF. The computational efficiency of this method
Shimon, Sadeh & Rephaeli2011),aswellasalternativecos-
makes it feasible to explore a large portion of the parame-
mological models (e.g., Sadeh,Rephaeli & Silk 2007).
ter space, but it relies heavily on average scaling relations
However,theSZeffectisalsosensitivetotheintraclus-
between the cluster observables, such as the concentration
ter(IC)gasphysics,whichispoorlyknownathighredshifts.
The latest observational results from the Atacama Cosmol- parameter, gas temperature, and the cluster mass. In addi-
ogyTelescope (ACT; Fowler et al.2010)andtheSouthPole tion, this description does not include the intrinsic scatter
Telescope (SPT; Luekeret al. 2010; Shirokoff et al. 2011; in these scaling relations which may depend on the cluster
mass and redshift.
Reichardt et al.2011)underlinetheneedforrealisticmodel-
ingofICgaspropertiesasaprerequisiteforuseofclustersas The second approach is based on numerical simula-
probestodeterminecosmological parametersfrom SZmea- tions,eitherdynamical(e.g., Shaw et al.2009;Sehgal et al.
surements. Indeed, Reichardt et al. (2011) concluded that 2010) or hydrodynamical (e.g., Seljak, Burwell & Pen
thetheoreticaluncertaintyinpredictingtheSZpowerspec- 2001; Springel, White & Hernquist 2001; Bond et al. 2005;
Sch¨afer et al. 2006; Roncarelli et al. 2007; Battaglia et al.
2010), from which large catalogues of clusters are created.
⋆ E-mail:[email protected] This method reproduces thegreat variety in theobservable
(cid:13)c 2011RAS
2 I. Dvorkin, Y. Rephaeli, M. Shimon
propertiesofclusters,butthelargecomputationalcostslead somewhatfromouroriginaltreatment(Dvorkin & Rephaeli
to two shortcomings: the volume of the simulation is lim- 2011) in which we assumed that the concentration parame-
ited,whichleads toinsufficientstatistical samplingof high- ter, c, is constant duringslow accretion.
mass (∼ 1015M⊙) clusters, and it is highly inefficient in Having built the merger tree, we start at the highest
testing various cosmological and IC gas models. A varia- redshift with the smallest masses and calculate the density
tion on the numerical approach uses the Lagrangian per- profile of each halo in the tree. To conform with common
turbation theory (of which the Zel’dovich approximation is practice, and for direct comparison with previous work, we
the first term) to generate DM halo catalogues with much assume a Navarro-Frenk-White (Navarro, Frenk & White
less computational effort (Monaco, Theuns& Taffoni 2002; 1995) profile for all halos at all times:
Holder, McCarthy & Babul 2007). However, this technique
too is limited bythe size of thesimulation box. ρd(x)= x(14+ρsx)2 (1)
Inthisworkweuseour(Dvorkin& Rephaeli2011)ex-
tended merger-tree model of cluster evolution to calculate wherex=r/rs,withrsthecharacteristicscaleradiusofthe
thethermalSZpowerspectrumandclusternumbercounts. profile, and ρs = ρ(1). Starting with an initial distribution
Thismethodenablesquantitativepredictionsoftheoutcome of concentration parameters c(M,z) for the earliest halos,
of numerical simulations by taking into account the intrin- wecalculate theconcentrationparameterofeachsuccessive
sicscatterinclusterpropertiesthatresultfrom theirdiffer- halo in thetree from considerations of energy conservation.
ent formation histories. Our model is an improvement over Theoutcomeofthiscalculationistheconcentrationparam-
standard analytical methods in that it accounts for the full eter of the halo we started with at the specified redshift of
merger history and formation redshift of each cluster, and observation.Generatingalargenumberoftreesgivesanes-
does not depend on pre-calibrated scaling relations. On the timateoftheprobabilitydensityfunction(PDF)foragiven
otherhand,ourapproachcanbeusedinstudiesofcosmolog- massandredshift.Furtherdetailsonthemerger-treemodel
ical parameter estimation in a much more computationally can befound in Dvorkin & Rephaeli (2011).
efficient way than numerical simulations. The next step is to model IC gas, which is assumed
This paper is organized as follows. In Section 2 we re- to constitute a small fraction of the cluster mass, and to
viewourmerger-treemodelofclusterevolutionanddescribe not significantly affect the evolution of thecluster. The gas
how it is integrated into the calculations of the SZ power equation of state is assumed to be polytropic, namely that
spectrumandnumbercounts.InSection3wepresentourre- thepressure and density are related by
sfoullltoswainngddcoisscmuosslotghiceairlipmaprlaimcaettioernss:iΩnmSe=cti0o.n254,.ΩWΛe=use0.t7h5e, P =P0(ρ/ρ0)Γ (2)
Ωb =0.045, H0 =73 km/s/Mpc, σ8 =0.8. with Γ=1.2.
Thesolution oftheequationofhydrostaticequilibrium
for a polytropic gas inside a potential well of a DM halo is
(Ostriker,Bode & Babul 2005):
2 EXTENDED MERGER-TREE MODEL
B ln(1+x) n
We build merger trees of DM halos using the modified ρ(x)=ρ0 1− 1− , (3)
(cid:20) 1+n(cid:18) x (cid:19)(cid:21)
galform algorithm (Parkinson, Cole & Helly 2008) which
is based on the excursion set formalism (Lacey & Cole where n=(Γ−1)−1, B is given by:
1993). The conditional MF used in this algorithm is cal-
ibrated to the outcome of the Millennium Simulation B= 4πGρsrs2µmp, (4)
(Springel et al. 2005) and is consistent with the Sheth- kBT0
Tormen MF (Sheth & Tormen 1999). Consequently, we use and µmp is the mean molecular weight. The temperature
thisMFinallofourcalculations.ThemoreupdatedMFpre- profile is given by:
sentedinTinkeret al.(2008)differsfromtheSheth-Tormen
B ln(1+x)
MF by up to ∼ 20% at z = 0, and by a higher fraction at T(x)=T0 1− 1− . (5)
(cid:20) 1+n(cid:18) x (cid:19)(cid:21)
larger redshifts. The adoption of a specific (especially an
analytic, theoretically-based) MF, introduces modeling un- As a boundary condition we assume that the gas pres-
certainties; these are discussed in Section 3. sureatthevirialradiusobeysPg =fgPd,wherefg isthegas
Foraclusterwithagivenmassandatagivenobserva- mass fraction and Pd = ρdσ2. We obtain σ2, the DM (3D)
tionredshiftwebuildamergertreewhichrepresentsitspos- velocity dispersion, by solving the Jeans equation (in the
sibleevolutionarytrack.Wedefinethemajormergerevents fieldofNFW-distributedDM).Forthegasmassfractionwe
as those with a mass ratio M>/M< < q for some q whose adopt a (nominal) mean value of fg = 0.1, estimated from
value is to be determined, and treat all other processes of thedatain Bonamente et al. (2008),although wenotethat
mass growth as continuous (relatively) slow accretion. We these measurements refer to the mass inside R2500, within
assume that during the violent major merger events DM is which the mean density is 2500 the background density (at
redistributed in the cluster potential well, while the minor the cluster redshift). While this may be considered a rela-
mergers and accretion processes affect mainly the outskirts tively small inner (and perhaps unrepresentative) region of
oftheclusterbutnotitscentralsphericalregion.Weassume thecluster,asimilarmeanvaluecanbededucedfromX-ray
thatthescaleradiusrs ofthehaloremainsconstant during measurementswithinR500 byVikhlinin et al.(2009).Below
slowaccretion ofmatterontothecluster,sothattheradius we explore otherassumptions about fg.
ofthehaloRgrows duringaccretion buttheinteriorregion We also consider an alternative model for the IC gas,
remains essentially unchanged. Note that we deviate here theβ−model with β =2/3, in which case thedensity is
(cid:13)c 2011RAS,MNRAS000,1–8
SZ power spectrum from a merger-tree model 3
ρ(x)= 1+ρ0x2 (6) ∆Iν = 2((khBcT)20)3 ·y·g(χ). (12)
andthetemperatureisgivenbythesolutionoftheequation
Thespectraldependenceisg(χ)=χ4eχ(eχ−1)−2s(χ),
of hydrostatic equilibrium with the (approximate) bound-
ary condition T → 0 at large radii. Note that the non- andT0istheCMBtemperature.Thentheexpectednumber
dimensionalradialcoordinateisgenerallydifferentfromthat of clusters with flux greater than some threshold ∆F¯ν is:
in eq.(1).
TheSZpowerspectrumiscomputedusingthehaloap- N(∆F¯ν)= dV dz B(M,z)dNdM. (13)
proximation (Komatsu & Seljak 2002): Z dz Z∆F¯ν dM
zmax dV(z) Mmax dn The selection function is B(M,z) = 1 if the flux of a
C =s(χ)2 dz dM |y (M,z)|2
ℓ Z dz Z dM ℓ givenclusterisabovethethreshold;otherwise,B(M,z)=0.
0 Mmin
(7) ThecalculationswerecarriedoutwithPlanck HFI143GHz
where s(χ) is the spectral dependence of the SZ signal channel, with a beam size of σB = 7.1′, ∆ν/ν = 0.33, and
given by: flux sensitivity threshold of ∆F¯ν = 12.6 mJy for 14 month
observation.
eχ+1
s(χ)=χ −4, (8) The computation time obviously depends on the num-
eχ−1 ber of trees we grow for each mass and redshift, and the
whereχ=hν/kBT0,V(z)isthecomovingvolumeperstera- number of mass and redshift bins used to approximate the
dian,dn/dM istheMF,andy isthe2DFouriertransform integralinequation(7).Calculatingthepowerspectrumfor
ℓ
of the projected Comptonization parameter, different numbers of trees, and with various mass and red-
shift resolutions, we find that the calculation converges for
yℓ = 4πℓ2srs Z0cdxx2sinℓ(xℓ/xℓ/sℓs)ζ(x) (9) gtNaivtkreeenses∼ℓ.=3B5eml,oiNwnu∆wtMeespt=ore1csae0nl0ctuarlnaedstueNltths∆efzopr=o1w81e0rv0as,pluienecstwrouhfmiℓchbCecℓtawfsoeereinat
where c=Rv/rs is the concentration parameter (Rv is the ℓ=100 and ℓ=9000.
virial radius), ℓs =dA(z)/rs, dA(z) is the angular diameter
distance to the cluster, and ζ(x) is the gas (normalized)
pressure
3 RESULTS
ζ(x)= kmBeσcT2ne(x)Te(x). (10) oTuhremSeZrgpeorwtreeresmpeocdterlu,missahtoνwn=in15F3igGurHez1,.cFoomrpcuotmedpaurissionng
Typical parameters are zmax =2, Mmin =1013h−1M⊙ and we show results of a standard calculation where we use the
Mmax =1016h−1M⊙. scalingrelationfortheconcentrationparameterdeducedby
Webuildonetreeforeachmassandredshiftinequation Duffyet al.(2008)fromasampleofrelaxedhalos.Alsoplot-
(7), and sum over all themass and redshift range to obtain ted is the result of the standard calculation where we used
the power spectrum. When we repeat this calculation, the theaverage c(M,z) relation from the merger-tree model:
concentration parameter of each halo is slightly different,
M
since it is effectively drawn from a probability distribution, logc(M,z)=2.417−0.1081·log
resulting in a different power spectrum. However, for a suf- (cid:18)M⊙(cid:19)
ficiently large number of mass and redshift bins that are M
−1.57·log(1+z)+0.1039·log log(1+z)
usedtocalculatetheintegralinequation(7),thecalculation (cid:18)M⊙(cid:19)
converges.BelowwepresenttheresultsofNtrees =5subse- −0.2486·log2(1+z).
quentcalculationsforeachℓ.Itisremarkablethatthecalcu-
(14)
lation converges for such a small Ntrees in comparison with
the number of trees that were needed to estimate the con- The normalization of the power spectrum is known to
centration parameter PDF in Dvorkin& Rephaeli (2011). be very sensitive to σ8; we find the scaling ASZ ∝ σ87.4, in
This is due to the fact that the concentration parameter accord with previous studies. The grey bands in Figure 1
PDF changes slowly as a function of mass and redshift, so correspond to σ8 =0.8±0.03, a level of uncertainty deter-
for a dense grid of M and z in the evaluation of equation mined from a joint analysis of WMAP7, BAO and SN data
(7) we adequately sample each PDF. (Komatsu et al. 2011).
We have also calculated the expected cluster number The fiducial model presented in Figure 1 results in an
counts, following the method of Sadeh & Rephaeli (2004). overallnormalization wellabovethelevelobtainedfromthe
The SZ fluxof a cluster is given by: latest SPT measurement (Reichardt et al. 2011): ASZ(ℓ =
3000) = 3.65±0.69 µK2. Below we will show that within
the uncertainties in the gas model, the gas mass fraction
∆Fν =Z Rs(Ωˆ,σB)∆Iν(Ωˆ)dΩ, (11) andthevalueof σ8, ourcalculation isconsistent with these
measurements.
whereΩˆ isthedirectiononthesky,Rsdescribesthedetector To compare the above results with those from simu-
beamwithbeamsizeσB and∆Iν isthe(spectral)intensity lations, we show in Figure 1 the power spectrum deter-
change, mined from an adiabatic hydrodynamical simulation by
(cid:13)c 2011RAS,MNRAS000,1–8
4 I. Dvorkin, Y. Rephaeli, M. Shimon
of relaxed halos A = 9.23+0.17, B = −0.09 ± 0.009 and
−0.16
14 C = −0.69±0.05 (1σ CL). Thus, the merger-tree model
trees
predicts a slightly higher normalization of the c(M,z) re-
12 Duffy et al.
uncertainty due to σ lation, which is partly responsible for the higher normal-
8
2] 10 Battaglia et al. ization of the SZ power spectrum, and a significantly lower
K redshift dependence, which also contributes to the increase
µ c(M,z) from trees
π [ 8 in the power spectrum normalization and shifts the peak
+1)/2 6 mtoohreighcoenrcmenutlrtaipteodlesan(sdinhcaevceluastlearrsgeart choignhtreirburetdiosnhifttos tahree
⋅C l(ll 4 SZ power). Indeed, when we calculate the SZ power spec-
trumwith thescaling relation from Duffyet al. (2008),but
with the redshift dependence parameter C = −0.34 taken
2
fromourmerger-treemodel,thepeakshiftstoℓ∼5000and
0 the normalization increases by ∼ 14%. An additional dif-
102 103 104 ference between thecalculation performed with thec(M,z)
l
relation from equation (14) and the full merger-tree calcu-
Figure1.SZpowerspectrumatν=153GHz.Resultsshownare lation,appearsbecausetheconcentration parameterhasan
fromthestandardcalculation(thinblueline),merger-treecalcu- approximately log-normal distribution function: there are
lation (thick pink line), a standard calculation with c(M,z) fit morehaloswithhigher-than-averagecthanwithlower-than-
tothemerger-treemodel(dot-dashed redline),adiabatichydro- average.
dynamicalsimulationsfromBattagliaetal.(2010)(dashedblack To what extent are these predictions for the concen-
line).Thegreyareashowstheuncertaintyofthemerger-treecal- tration parameter reliable? There is some observational ev-
culationduetovariationsofσ8. idence that the concentration parameter is in fact higher
thanpredictedbyN-bodysimulations.Forexample,thecon-
Battaglia et al. (2010) 1, rescaled to ν = 153 GHz. It is centrationparametersmeasuredbySchmidt & Allen(2007)
from X-ray observations of 34 dynamically relaxed clus-
very interesting that both the normalization and the shape
ters in the redshift range z = 0.06−0.7 are significantly
ofthepowerspectrumobtainedinourtreatmentagreewell
greater than those predicted by Duffyet al. (2008) for the
withthefullnumericalcalculation,inspiteoftheverydiffer-
same masses and redshifts. In a large dataset compiled by
entapproachusedinthesimulations.Battaglia et al.(2010)
Comerford & Natarajan (2007), which consists of 62 clus-
also calculated the power spectrum in the case when ra-
ters at redshifts up to z = 0.89 and masses up to at least
diative cooling, star formation, supernova and AGN feed-
back were included, and found a reduction of power at ∼ 2×1015M⊙, the normalization of the concentration pa-
rameterisfoundtobehigherbyatleast20%thantheresults
small scales due to expansion of the gas to larger radii.
ofnumericalsimulations.Asimilarconclusion isreachedby
Similar results were obtained by Shaw et al. (2009) and
Wojtak &L okas(2010),whoanalyzedkinematicdataof41
Sehgal et al. (2010), with different normalizations depend-
clusters at redshifts z < 0.1. A recent X-ray analysis of 44
ing on the amount of energy feedback. While we have not
clustersintheredshiftrange0.1−0.3byEttori et al.(2010)
attempted to model feedback mechanisms in this work, the
found a normalization of c consistent with numerical simu-
agreement of our model with numerical calculations in the
lations, although note that their constraint on the cosmo-
simpleadiabaticcaseshowsthatourdescriptionofthehier-
archical growth of clusters and the intrinsic scatter in their logical parameters produces a rather high σ8 =1.0±0.2.
propertiesisreasonablyrealistic.Westressthateventhough Interestingly, several strong lensing studies of massive
we make several simplifying assumptions, including that of clusters (e.g., Zitrin et al. 2011) found anomalously large
hydrostaticequilibrium,westillgetabetteragreementwith Einsteinradiiandconcentrationparameterswhencompared
simulationsthaninthestandardapproach.Themodelingof with predictions from numerical simulations. The results of
IC gas can be furtherrefined, as we discuss below. themerger-treemodelsuggestthatperhapstheseconcentra-
There are two major differences between the merger- tion parameters, although higher than the average values,
tree calculation and the standard treatment with the scal- are not incompatible with theΛCDM model.
ing relation from Duffy et al. (2008): the normalization in It is important to note that the simulations of
theformercaseishigher,andthepeakshiftstohighermul- Duffyet al. (2008) span the mass range of 1011 −
tipoles (ℓ ∼ 6000 vs. ℓ ∼ 4000 in the latter case). These 1015h−1M⊙, similar to other numerical studies, while the
featurescanbeunderstoodintermsofthemodifiedc(M,z) masses that are important in the context of the SZ effect
relation that is deduced from the merger-tree algorithm. are1013−1016h−1M⊙.Low-massandlow-redshifthalosout-
When we fit the average c(M,z) relation from the merger- numberhigh-masshalosbyseveralordersofmagnitude;the
treemodeltoageneralformc=A(M/Mpivot)B(1+z)C that c(M,z) relations arethusheavily weighted by thestatistics
is used by Duffy et al. (2008), we obtain A = 11.99+0.42, of thelow mass halos.
−0.43
B = −0.079±0.005 and C = −0.34±0.03 for Mpivot = Forexample,thesampleofDuffyet al.(2008)contains
2 × 1012h−1M⊙, with 2σ confidence level (CL), but this 1269 sufficiently well-resolved halos at z =0, of which typ-
function provides a poorer fit to our results than equation ically less than ∼ 0.1% are above 1014h−1M⊙ if their rel-
(14), whereas Duffyet al. (2008) obtain for their sample ative abundance approximately follows the Sheth-Tormen
MF. At z =0.5 only ∼0.03% of all the halos are expected
to have masses above 1014h−1M⊙. On the other hand,
1 http://www.astro.utoronto.ca/∼battaglia/ the parameters B and C are expected to vary with both
(cid:13)c 2011RAS,MNRAS000,1–8
SZ power spectrum from a merger-tree model 5
10 12
NFW
10
8 α=3
2K] 2K] 8
µ 6 µ
π [ π [
1)/2 1)/2 6
⋅C l(l+l 4 ⋅C l(l+l 4
2
2
0 0
102 103 104 102 103 104
l l
Figure2.SZpowerspectrumfordifferentdensityprofilesofDM Figure4.SZpowerspectrumwithmerger-treemodelparameters
halos. Shown are results for anNFW profile (solidline)and the in the range q = 7−13, κ = 4−6 (grey area) and the fiducial
profileinequation(15)withα=3(dashedline). modelq=10,κ=5(thickline).
solution for the new DM potential). Figure 2 shows that
14
this modification with α=3 has a minor effect on the am-
12 Γ=1.2 plitudeof thepower spectrum,as it affectsonly weakly the
β−model integrated gas pressure in theouter regions of clusters.
2] 10 Γ=1.3 All the calculations in this work were performed with
µ K Γ=1.1 the simple IC gas model described in Section 2. As
π2 [ 8 δrel=0.1 was recently shown in several studies (Shawet al. 2010;
⋅C l(l+1)/l 46 TtIChreagcsa,tsrBepnohdgyteshi&cosf,OtfohsrtereiSxkZearme2ffp0el1ce1ta;isEmvfosedtrayetshsteiaonmusiot&iuvneMttioogfltinhaocencdi-oteht2ae0irl1ms1oa)fl,
pressurecansignificantlylowerthepowerspectrum.Herewe
2 explore the possibility of non-thermal pressure support, as
well as slight variations in the parameters of the polytropic
0 model. Figure 3 compares the results for different values of
102 103 104
the adiabatic index Γ, a β−model and a polytropic model
l
with a non-thermal pressure component. For simplicity we
Figure3.SZpowerspectrumfordifferentgasmodels:polytropic assume that thenon-thermalcomponent is a constant frac-
withdifferentvaluesoftheadiabaticindex,non-thermalpressure tion of thetotal pressure:
component, andβ-modelwithβ=2/3.
Ptot=Pthermal+Prel =(1+δrel)Pthermal, (16)
mass and redshift (Gao et al. 2008; Mun˜oz-Cuartas et al.
2011; Prada et al. 2011). Indeed, when we fit the average with δrel =0.1 as a representative value.
merger-tree results only in the range z = 0 − 1, M = It can beseen that theSZ powerspectrum is very sen-
1013−1014h−1M⊙ weobtainB=−0.09,C =−0.46,closer sitive to slight changes in the value of the adiabatic index
to the values in Duffyet al. (2008). This example demon- away from Γ=1.2, an average value deduced from numeri-
strates the inherent difficulty of numerical simulations to calsimulations.Inaccordwithpreviousstudies,wefindthat
produce large samples of halos due to computational lim- asmall constant fraction of non-thermalpressureobviously
itations. It is possible that because of the relatively small reduces the normalization of the power spectrum but does
sample size numerical simulations overestimate theredshift not affect its shape.
evolution of the concentration parameter which biases the Figure 4 shows the uncertainty in our calculation due
predicted SZ power spectrum. to the merger-tree model parameters q, which is the max-
Ourmerger-treeapproachenablesustotestalternative imal allowed ratio for major merger events, and κ, which
DMhalodensityprofiles,whichisnotpossibleinanalytical describes the initial distance between two clusters that are
calculations that are based on pre-calibrated scaling rela- about to merge. As expected, this uncertainty is smaller
tions.Asanexample,weexplorethefollowing modification than ∼7%.
of the NFW profile: Our fiducial model assumes fg = 0.1, a constant gas
mass fraction across the whole mass and redshift range.
ρd(r)= r/rs(12α+ρrs/rs)α (15) Hafrnoodmwervevedarsr,hiotihuftesfogrfaacloatbicostneicrvoafptrhiooocnte.gssaTesshienincmclluaussdtseinrdsgevpsaetranierdsefnwocrietmhaamtriioasnesss,
for α > 2 (equations (3-5) are replaced by the appropriate ram pressure stripping of gas from galaxies, and galactic
(cid:13)c 2011RAS,MNRAS000,1–8
6 I. Dvorkin, Y. Rephaeli, M. Shimon
10 10000
trees, f~M0.2
g
8 uncertainty due to σ8 8000
SPT
2K]
µ 6 6000
π [ z)
2 >
1)/ N(
⋅C l(l+l 4 4000
2 2000
100 2 103 104 100−2 10−1 100 101
l z
Figure5.SZpowerspectrumcalculatedwiththegasmassfrac- Figure 6. SZ cluster number counts above a specified redshift.
tiontakenfromequation(17)andδrel=0.1.Bluedotwitherror Shownareresultsofastandardcalculationwithac(M,z)relation
barsrepresents the SPT measurement(Reichardtetal.2011)at fromDuffyetal.(2008)(reddot-dashedline),withc(M,z)froma
ℓ = 3000. The grey area shows the uncertainty due to variation fittothemerger-treemodel(bluesolidline),andthefullmerger-
inσ8. treecalculation(blacklines).
winds. While the full redshift evolution of the gas mass merger-tree model and the standard calculation, a level of
fraction is not known, it clearly reflects aspects of cluster variancethatcouldhaveappreciableramificationsonprecise
formation and evolution, which include also the effects of cosmological parameterextraction based onclusternumber
internal processes that re-distribute cluster baryons, those counts(Holder, Haiman & Mohr 2001; Wang et al. 2004).
same processes that also imprint the (related) mass depen- Amajor sourceofuncertaintyincalculations oftheSZ
dence.ThelatterwasdeducedinseveralX-raystudiesoflow effect is the halo mass function. It is evident that the ac-
redshift clusters(Vikhlinin et al. 2009;Giodini et al. 2009). curacy with which the MF can be determined in a given
Giodini et al.(2009)derivedthefollowingdependencebased numerical simulation is reduced at higher redshifts, espe-
on a combined sample of 41 clusters at z ≤ 0.2 observed cially at the high mass end, due to the limited volume of
by Vikhlinin et al. (2006), Arnaud,Pointecouteau & Pratt thesimulation.Toillustratethispoint,wecanestimate the
(2007) and Sun et al. (2009): expectednumberofhalosatdifferentredshiftsaccordingto
theTinkerMFinthesimulationsthatwereusedtocalibrate
it. For example, for z = 1.25 the number of halos per unit
fg =(9.3±0.2)×10−2(cid:18)2×M1500104M⊙(cid:19)0.21±0.03(cid:18)0h.7(cid:19)−1.5. wlnhMileiastezxp=ec2t.e5dthtoerbeeislelesssstthhaann∼on1e0h0afloorwMith>m3a·s1s0a1r4oMun⊙d,
Wenote that this fit is not valid beyond M ≃1015M⊙.(17) 2·101I4tMca⊙npbeeraurngiuteldnMthaitnltahrgeecommabssinesedarseimraurlaetaiotnhvigohlurmede-.
Not knowing the explicit scaling of the gas mass frac-
shifts and their contribution to the SZ power spectrum, al-
tion with redshift, we can only account for its mass depen-
though uncertain, is not important. Figure 7 shows therel-
dence by adopting the relation fg = Mgas,500/Mtot,500 = ative contribution of different masses and redshifts to the
M /M . To avoid having a baryon fraction
gas,virial tot,virial integral in equation (7) in the following way: we calculate
greater than the cosmic mean, which arises in high-mass
thisexpressionforℓ=4000withdifferentzmax (x-axis)and
clusters if the scaling relations above are strictly followed,
Mmax(y-axis)andplottheresultasafractionofthefiducial
we set an upperlimit of fcosmic =Ωb/Ωm. valueforwhichwetookzmax =2andMmax =1016h−1M⊙.
Thepowerspectrum,calculatedwiththegasmassfrac- Thecontoursareforconstantfractionofthefiducialvalue2.
tion from equation (17) and assuming δ = 0.1 is shown
rel TwoimportantfeaturescanbeseeninFigure7:(a)wehave
inFigure5.Itcanbeseenthatourresultisconsistentwith
tointegrateatleastuptoz ∼1.4tohaveareliableestimate
the SPT measurement if we allow for an uncertainty in σ8 of the effect, and (b) masses above 2·1014M⊙ contribute
(Komatsu et al. 2011):σ8 =0.8±0.03. roughly∼50%oftheeffect(overthewholeredshiftrange).
While we did not account for thepossible redshift evo-
It is clear that a large error in the MF even at z ∼ 1.25
lution of the gas mass fraction, we note that precise mea-
(whichonlyworsensat higherredshifts) isverydetrimental
surements of the SZ power spectrum may actually be used
to thereliability of thecalculation.
to deduce, or significantly constrain the redshift evolution
Finally,it shouldbeemphasized thatthereisan inher-
of fg, especially so when σ8 is more precisely determined. ent limitation in the accuracy of cluster MF deduced from
The expected cluster number counts are presented in
dynamical cosmological simulations. This stems from the
Figure6.Weshowtheresultsofastandardcalculationwith
c(M,z) taken from Duffy et al. (2008) and from equation
(14),aswellasthemerger-treecalculationrepeated10times 2 Asimilarplotwaspresented inTrac,Bode&Ostriker(2011),
soastoprobethedistributionfunctionsoftheclusterprop- exceptthatherethevariableisthemaximummass,notthemin-
erties. There is ∼15% difference between the results of the imummass.
(cid:13)c 2011RAS,MNRAS000,1–8
SZ power spectrum from a merger-tree model 7
(2008). We argue that the higher normalization of the con-
centrationparameterismorecompatiblewithobservations,
1155..46 0.1 2.0 4.0 0.5 0.6 8.0 9.0 while the faster redshift evolution of Duffy et al. (2008) re-
sults from the fact that their sample is heavily weighted by
15.2
low-mass low-redshift halos, where the dependence on red-
un 15 shift is expected to be stronger. The excellent agreement
s
M/M10max111444...468 0.1 0.20.3 0.40.50.6 0.7 0.80.5 0.60.90.7 buclleuatstwiiooennenss.boyurBraetstualgtsliaanedt athl.e(a2d01ia0b)aftuicrthhyedrrsoudpypnoarmtiocuarlcsiomn--
g 0.4
lo 14.2 0.3 As anticipated, we have found the SZ power spectrum
0.2 0.3
14 0.1 0.2 is quite sensitive to the assumed IC gas model. Among the
13.8 0.1 parametersthatcanstronglyinfluencetheSZpowerarethe
gas mass fraction and the amount of non-thermal pressure
13.6
andtheirvariationwithclustermassandredshift.Ourfidu-
0.2 0.4 0.6 0.8 1 1.2 1.4
z
max cial model, which assumes a constant fg = 0.1 produces a
leveloftheSZpowerspectrum significantly higherthanre-
Figure 7. The value of the integral in equation (7) for various cent measurements by the SPT. When we adopt a scaling
zzmmaaxx (=x-a2xiasn)danMdmMaxma=x (1y0-1a6xhi−s)1Mrel⊙at,ivfoertoℓ t=he40fi0d0u.ciCalonvtaoluuerss Gofiotdhiengiaest mal.as(s20fr0a9c)t,iothnew(ictuhrrmenats)sSfPgT∝mMea0.s2u,rienmfeernrtedfabllys
showconstantfractionsofthefiducialvalue;thecalculationswere
performedusingthefittingformulainequation(14). withintherangeofuncertaintyinσ8whichextendstolower
powerlevelsduetothemass-dependentgasfraction.Clearly,
more detailed modeling of the gas mass fraction is needed
appreciable impact of baryonic processes (both stellar and formeaningfulcomparison withobservationsofthethermal
gaseous)onclusterevolution.Asanexamplewebrieflymen- SZ effect. Note also that we adopted a class of polytropic
tion here the conclusion of Stanek, Rudd& Evrard (2009) gas models; different models can be readily explored in the
that number densities of clusters in their hydrodynamical context of our merger-tree approach.
simulations deviate by 10%−60% from those predicted by
Another basic source of uncertainty is the halo MF.
Tinker et al. (2008).
We have shown that the high-mass, high-redshift halos, for
which the MF calibration is least certain, have a relatively
large contribution to the SZ power spectrum. In this work
4 DISCUSSION
we haveused the Sheth-TormenMF, but our approach can
be extended to include other mass functions. This can be
We have presented a new analytical method of calculating
achieved by either calibrating the conditional MF that we
thethermalSZpowerspectrumandclusternumbercounts.
use to build the merger trees to an up-to-date numerical
This merger-tree method takes into account the intrinsic
simulation, in the spirit of Parkinson, Cole & Helly (2008),
scatter in thehalo parameters and theirdependenceon the
or constructing a theoretically motivated conditional MF
redshiftsofformation andobservation thatresultsfrom the
that provides a betterfit to simulations.
hierarchical evolution of clusters. In particular, our model
providesagood statistical description of thehigh-massand Inthecalculationspresentedhereweassumedspherical
high-redshift halos. symmetryforallhalos.However,itispossibletoextendour
We have demonstrated that our approach, which does approach to account for halo triaxiality by modeling each
notrelyonpre-calibratedscalingrelations,allowstoexplore halo in the tree as an ellipsoid either with constant values
differentdensityprofilesofDMhalos. Inaddition,sincethe of the axis ratios, or with values drawn randomly from a
cosmological parameters can be changed in each run of the probability distribution.
merger-tree code and no scaling relations are inserted by
In this work we calculated the power spectrum of the
hand,ourapproachismoresuitableforcosmologicalparam-
SZthermalcomponent.Usingourapproachitispossibleto
eter estimation studies than thestandard analytic method.
calculatealsothepowerduetotheSZkinematiccomponent
Westressthattheapproachpresentedherediffersfrom
duetorandomclustermotionsbyassigningapeculiarveloc-
adding scatter in the various observable parameters to the
itytoeachhalo.Thiscontributionisexpectedtobeasmall
standard analytic calculation, since in this case the scaling
fraction of that from the thermal component. However, we
relationsandthescatteritselfareinsertedbyhand,whereas
cannotuseourmethodtocalculatethefullkinematicpower
here theyare intrinsic to themodel and its predictions.
whichalsohascontributionsfrommatteroutsideofclusters.
Our calculations predict a higher power spectrum nor-
Therefore, our approach is better suited for calculations of
malization than the standard calculation which uses the
cluster number counts and cosmological parameter estima-
c(M,z) relation from Duffyet al. (2008) (for the same IC
tion which relies on cluster counts, since they depend only
gas model), and a shift of thepeak to highermultipoles. In
on the signal that originates from matter within clusters.
addition,themerger-treemodelpredictshigherclusternum-
bercountsthanthestandardapproach(again,forthesame The approach presented in this work provides a com-
IC gas model). These differences can be explained by the putationally efficient way to explore the uncertainties dis-
modified c(M,z) relation that is predicted by the merger- cussedabove,andtogainanimprovedunderstandingofthe
treemodel.Inparticular,ithasahighernormalizationanda influenceoftheevolutionofgalaxy clustersontheirvarious
slowerevolutionwithredshiftthantheresultsofDuffyet al. observable properties.
(cid:13)c 2011RAS,MNRAS000,1–8
8 I. Dvorkin, Y. Rephaeli, M. Shimon
ACKNOWLEDGMENTS Shaw L. D., Zahn O., Holder G. P., Dor´e O., 2009, ApJ,
702, 368
The authors wish to thank the galform team for making
Sheth R.K., Tormen G., 1999, MNRAS,308, 119
the code publicly available. Work was partly supported by
Shimon M., Sadeh S., Rephaeli Y., 2011, MNRAS, 412,
theUS-ILBinationalScienceFoundationgrant2008452,and
1895
by a grant from theJames B. Ax Family Foundation.
Shirokoff E. et al., 2011, ApJ,736, 61
Springel V., WhiteM., Hernquist L., 2001, ApJ, 549, 681
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