DRAFTVERSIONAPRIL9,2010 PreprinttypesetusingLATEXstyleemulateapjv.11/10/09 THETHERMALSZTOMOGRAPHY JIAWEISHAO1,2,PENGJIEZHANG1,WEIPENGLIN1,YIPENGJING1 1KeyLaboratoryforResearchinGalaxiesandCosmology,ShanghaiAstronomicalObservatory,NandanRoad80,Shanghai,200030,China 2GraduateSchooloftheChineseAcademyofSciences,19A,YuquanRoad,Beijing,China Email:[email protected],[email protected] DraftversionApril9,2010 ABSTRACT 0 1 ThethermalSunyaev-Zel’dovich(tSZ)effectdirectlymeasuresthe thermalpressureof freeelectronsinte- 0 grated along the line of sight and thus contains valuable information on the thermal history of the universe. 2 However,theredshiftinformationisentangledintheprojectionalongthelineofsight. Thisprojectioneffect severelydegradesthepowerofthetSZeffecttoreconstructthethermalhistory. However,thisotherwiselost r p redshiftinformationcanberecoveredbytheSZtomographytechnique,namelybycrosscorrelatingthetSZef- A fectwithgalaxiesofknownredshifts,oralternativelywithdarkmatterdistributionreconstructedfromlensing tomography.Forthispurpose,weinvestigateindetailthe3Ddistributionofthegasthermalpressureanditsre- 8 lationwiththematterdistribution.Wecompareouradiabatichydrodynamicsimulationwiththeoneincluding radiativecoolingandstarformationwithsupernova(SN)feedback.Weconfirmthattheseadditionalprocesses ] ofgastrophysicssuppressthepressurepowerspectrumandthustheSZangularpowerspectrumsignificantly. O Ontheotherhand,theyonlyaffectthe3Dpressure-mattercorrelationcoefficientr(k)at∼1- 2%level. Since C r(k)isakeyinputintheSZtomography,thisfindingmakesthetSZtomographyrobustagainstuncertaintiesin . theunderstandingofcoolingandSNfeedback. h Subjectheadings:cosmicmicrowavebackground-large-scalestructureofuniverse-galaxies:clusters:general p - -methods:numerical o r t 1. INTRODUCTION independent. Photonsoriginatedfromredshiftzsufferafac- s a Free electrons in the universe reveal their existence in tor of1+z energyloss caused by the cosmic expansion. On [ theotherhand,CMBphotonsthatelectronsscatteratredshift 2 tChMe BCMphBotsoknys.thrTohueghintdhueciredinsveecrosendCaroymCptMonBscteamttepreirnagtuoref z are a factor of 1+z more energetic than CMB photons to- day. Thesetwoeffectscanceloutexactlyandenablesgalaxy v anisotropies, proportionalto the electron thermal energy in- clusterstobedetectedathighredshiftwithoutextraeffort,op- 7 tegrated along the line of sight, are the well known ther- positetotheX-rayclusterfinding.Anexcitingadvanceinthis 1 malSunyaev-Zel’dovicheffect(Sunyaev&Zeldovich1972, areaistherecentdiscoveryof3newgalaxyclustersthrough 3 1980). Since massive galaxyclusterscontainlarge reservoir thetSZeffectbytheSPTgroup(Staniszewskietal.2008). 5 ofhotelectrons,thegeneratedtSZeffectcanthusoverwhelm ThetSZeffectisalsoapowerfulprobetothethermalhis- . the primary CMB around cluster scales. For this reason, 3 tory of the universe, since it directly probes the thermal en- the thermal SZ effect of dozens of galaxy clusters has been 0 ergyofintergalacticmedium(IGM)andintraclustermedium measured by various experiments (refer to Carlstrometal. 9 (ICM). Numerical simulations show that, the amplitude of 0 2002; Reeseetal. 2002; Jonesetal. 2005; LaRoqueetal. the SZ signal is sensitive to the amount of radiative cool- : 2006;Bonamenteetal.2006forreviews).Ontheotherhand, v blindlydetectingthetSZeffectinrandomdirectionsofskyis ing and energyfeedback(Springeletal. 2001; daSilvaetal. i 2001; Whiteetal. 2002; Linetal. 2004). However, it is not X muchmoredifficult,sincetheexpectedsignalisoverwhelmed straightforwardto extractinformationonthese astrophysical by the primary CMB fluctuations. Currently there are only r processesfromthetSZ measurementsalone. Firstof all, re- a a few tentative detections through the observed small scale searchshowsthatthereexistgreatdegeneraciesbetweendif- CMB power excess (Dawsonetal. 2002, 2006; Masonetal. ferentcompetingprocesses. Evenworse,thetSZ effectonly 2003; Goldsteinetal. 2003; Runyanetal. 2003; Kuoetal. measurestheelectronthermalenergyprojectedalongtheline 2004;Readheadetal.2004;Bondetal.2005;Reichardtetal. 2008)1. However,thesituationwillbesignificantlyimproved ofsight. Theredshiftinformationoftheseastrophysicalpro- cessesisthusentangledintheprojection. inthenextfewyearsbyongoingandproposedgroundsurveys suchasSZA2,ACT3,APEX4,SPT5andthePlancksatellite. Zhang&Pen(2001)proposedtorecovertheredshiftinfor- mationbycrosscorrelatingthetSZeffectwithgalaxieswith Precisionmappingofthe SZ skyisof greatimportanceto at least photo-z information. The idea is that galaxies in a bothcosmologyandastrophysics. TheSZ effectisa power- givenredshiftbinshouldstronglycorrelatewiththetSZsignal fulfinderofgalaxyclustersathighredshifts. Theefficiency fromthesameredshiftbin. Akeylinkbetweenthemeasured offreeelectronstogeneratethethermalSZeffectisredshift crosscorrelationandthegaspressureauto-correlationthatwe want to extractis the cross correlationcoefficientr between 1However,refertotherecentSZAmeasurementfortheoppositeconclu- the thermal energy and the galaxy number density. Assum- sion(Sharpetal.2009) ingaconstantr,thetimeresolvedthermalenergydistribution 2http://astro.uchicago.edu/sza/ can be reconstructed self consistently. This SZ tomography 3http://www.physics.princeton.edu/act/ techniquewouldbeapplicableinreality,sinceSZsurveysof- 4http://bolo.berkeley.edu/apexsz/ tenhavefollow-upsofgalaxysurveys. Forexample,thedark 5http://pole.uchicago.edu/ 2 energy survey6 will cover the SPT sky and measure photo- 2.1. TheSZtomography metricredshiftsof∼108 galaxiesuptoz=1.3. Thelensing One of the most widely used statistical quantities of the tomographyalso helpstoreconstructthe3Dmatterdistribu- tSZ effect is the angular power spectrumCtSZ. Throughout tion, which can also be correlated with the SZ map to make l this paper, unless specified, we will focus on the Rayleigh- theSZtomography. Jeans limit ∆T/T = - 2y. Under the Limber’s approxima- In the current paper, we reformulate this SZ tomography tion(Limber1954),CtSZ isrelatedtothe3Dthermalpressure technique and explore the possibility to improve its robust- l powerspectrum∆2(k,z)bythefollowingrelation, ness. We no longer approximate r as a constant. Rather, P we rely on numerical simulations to quantify its scale and redshift dependence. We are able to show that, r is insensi- l2 CtSZ= χCMB∆2(k= l ,z)W2 (z)χdχ. (4) tivetogastrophysicssuchasradiativecoolingandsupernova 2π l Z P χ tSZ 0 (SN)feedback.Namely,rcalculatedfromadiabatichydrody- namicsimulationsshouldbe sufficientlyaccurate, evenwith Wehaveadoptedtheflatcosmologyintheaboveexpression. the presence of radiative cooling and SN feedback. We are Theweightingfunctionis then able to take this r as inputto performthe SZ tomogra- phy. hn k Ti The paper is organized as follows. In §2, we introduce WtSZ(z)=- 2σTa me Bc2e . (5) thetSZeffectandexplaintheSZtomographytechnique. We e thenanalyzeourhighprecisionhydrodynamicsimulationsto Thepressurepowerspectrum∆2 isthatofthefractionalther- quantifythedependenceofvariousSZ statisticsontheextra p gastrophysicsin§3. Althoughmostquantitiesaresensitiveto mal pressure fluctuations δP ≡nekBTe/hnekBTei- 1. WtSZ(z) thesegastrophysicalprocesses,wefindthatthecrosscorrela- tellsustheoverallthermalenergyoftheuniverseandthe∆2 p tion coefficientr onlyweakly dependson them(§3.3). This tellsustheclusteringofthethermalenergy. featureallowsustousercalculatedfromtheadiabaticsimula- OurSZtomographytechniqueaimstoreconstructthetime tionsastheinputoftheSZtomography,despitetheexistence resolved∆2(k,z)W2 (z).Thisquantitytellsustheoverallam- P tSZ ofcomplicatedgastrophysicsintherealuniverse.Wediscuss plitude of the thermal energy and the clustering strength at andmakeconclusionin§4. redshiftz. ThetSZeffectiscorrelatedwithtracersofthelarge scale structure. Giventhe redshiftinformationofthese trac- 2. THETHERMALSZEFFECTANDTHESZTOMOGRAPHY ers, such as galaxies, we can manage to recoverthe redshift The tSZ effectinducesa new sourceof CMB temperature informationofthetSZeffect.Theoriginaltomographyispre- fluctuationswiththeamplitude sentedinthevariationformalism(Zhang&Pen2001),while ∆T(θ) in the currentpaper, we reformulateit in a morestraightfor- =g(x)y(θ). (1) ward manner. The key idea of the SZ tomography is that, T CMB galaxiesdistributed in a certain redshiftrangecorrelatewith Here,θisthedirectiononthesky. g(x)describesthespectral the tSZ signals contributed by the IGM in the same redshift dependence.Inthenon-relativisticlimit, range. Given the photometric redshift information, we are ex+1 ableto split galaxiesintodifferentredshiftbins. The galaxy g(x)= x - 4 , (2) number distribution in the i-th redshift bin is n(z), which is (cid:18) ex- 1 (cid:19) relatedtothephoto-zdistributionnP(zp)by i i wherex≡hν/k T =ν/56.84GHzandν isthe observed B CMB frequencyof CMB photons. The Comptonizationparameter zi+∆zi n(z)= p(z|zP)nP(zP)dzP. (6) y(θ)is σ i Zzi- ∆zi/2 i y= T adχn k T, (3) mec2Z e B e Here, p(z|zP)isthephoto-zprobabilitydistributionfunction. wheren k T isthehotelectronpressure.χ,n ,k ,T andσ Duetonon-negligiblephoto-zerror,therealgalaxydistribu- e B e e B e T are the comoving diameter distance, number density of free tion is wider than ∆z. However,it can still be narrow, cen- i ∞ electrons, the Boltzmann constant, electron temperature and tered at z. We normalize n such that n(z)dz=1. The i i 0 i theThompsonscatteringcrosssectionrespectively. tSZ-galaxycrosspowerspectrumisthenR The spectral dependence of the tSZ effect is unique. The tGShZzeafnfedcatsshinocwresmasenCtMsaBtνte>mp2e1r8atGurhez.deTchreemspeencttsraaltfνu<nct2i1on8 2lπ2 CltSZ- g=Z ∞∆2Pg(k= χl ,z)WtSZ(z)ni(z)χdz (7) g(x)→- 2attheRayleigh-Jeansbandx≪1andg(x)→x- 4 0 at x ≫1. This unique spectral dependence allows a clean Notice that the integral in Eq. 7 is over the entire redshift separationof the tSZ effectfrom otherCMB componentsin range.Forasufficientlynarrowredshiftbin,thegalaxydistri- multi-bandCMBsurveys. butionn(z)peaksatz=z7withanarrowr.m.s.redshiftwidth The y parameter contains key information on the thermal i i ≃ σ2+(∆z)2/12,whereσ isthephoto-zerror.Fortypical historyof the universe. However,since it onlymeasuresthe z i z projectedelectronthermalenergyalongthelineofsight, the valpuesσz=0.05(1+z)and∆zi=0.2,theeffectiver.m.s.width is∼0.1-0.2. Overthisredshiftwidth, the functions∆2 ,W redshiftinformationissmearedbythisprojectioneffect. Our Pg SZ tomographytechniqueaimsto recoverthe otherwiselost redshiftinformationinthetSZeffect. 7 This relation assumes no catastrophic error in photo-z measurement. Withtheexistenceofcatastrophicerror,thepeakredshiftcanbeshiftedby 6http://www.darkenergysurvey.org/ outstandingoutliers. 3 andχvaryslowly. Wethushaveanapproximation This approach is self consistent and able to provide a quick realization of the SZ tomography. However, improvements l2 CtSZ- g≃∆2 (k= l ,z)W (z)χ ∞n(z)dz mustbe made to do precision SZ tomography. As shown in 2π l Pg χi i tSZ i iZ0 i §3.3, the assumption that r is a constant is only accurate at l thelevelof∼20%. Furthermore,toobtainthevalueofrun- =∆2Pg(k= χ ,zi)WtSZ(zi)χi. (8) biaslyfromtheSZpowerspectrum,thegalaxysurveysmust i wellcovertheSZ redshiftrange(z.2). Thisischallenging Here,χ ≡χ(z). Itisclearfromtheaboveequationthatthe forgalaxysurveys. i i cross correlation between the thermal SZ maps with galaxy Since r(k,z) is such a key quantity in the SZ tomography, distributionpicksupand,toagoodapproximation,onlypicks a natural question arises: Can one robustly predict r? This uprelevantinformationwithinthegivenredshiftbin. Thisis seems challenging, given the fact that various complicated a key step to recoverthe redshiftinformationof the thermal gastrophysical processes affect the tSZ effect. Surprisingly, SZeffect. robustpredictiononrislikelyfeasible. Thekeypointisthat, For similar argument, the galaxy angular power spectrum thesegastrophysicsaffects∆2 and∆2 inbasicallythesame P Pg is way, so that these effects roughly cancel out in r by defini- l2 ∞ l dz tion. Toquantifythedependenceofronthesegastrophysics, Cg = ∆2(k= ,z)n2(z)χ dz we comparetwo sets of hydrodynamicsimulationswith and 2π l Z0 g χ i dχ without radiative cooling, star formation and SN feedback. l ∞ dz These simulations confirm the above naive speculation and ≃∆2(k= ,z)χ n2(z) dz. (9) g χi i iZ0 i dχ fi1%ndltehvaetl,,tdheesspeitaedtdhietiofancatltghaasttrtohpeyhyaslitcesr cthaenSoZnlypoawffeercstpreca-t The next step is to recoverthe thermalSZ contributionin trumby∼40%. the given redshift bin with the correct weighting, namely, ∆2(k,z)W2 (z). This step requires a key input, namely the 3. SIMULATIONS P tSZ crosscorrelationcoefficientrbetweenfluctuationsinthegas The tSZ statistics has been studied by both semi- pressureandthegalaxydistribution,definedby analytical models (Cooray 2000; Zhang&Pen 2001; Komatsu&Seljak 2002; Zhang&Wu 2003; Zhangetal. ∆2Pg(k,z) 2004; Zhang&Sheth 2007) and numerical simulations r(k,z)= (10) ∆ (k,z)∆ (k,z) (Persietal.1995;Refregieretal.2000;Refregier&Teyssier P g 2002; daSilvaetal. 2000, 2001; Seljaketal. 2001; ∆2 (k,z)W (z) Pg tSZ Springeletal. 2001; Whiteetal. 2002; Linetal. 2004; = . [∆ (k,z)W (z)]∆ (k,z) Zhangetal. 2004; Roncarellietal. 2007; Hallmanetal. P tSZ g 2007, 2009). Beyond the gravitational heating mechanism, CombiningEq. 8andEq. 9,weobtain some works (daSilvaetal. 2001; Springeletal. 2001; [∆2 (k,z)W ]2 Whiteetal. 2002; Linetal. 2004; Roncarellietal. 2007; ∆2(k,z)W2 (z)=r- 2(k,z) Pg i tSZ (11) Scannapiecoetal. 2008) have incorporated additional gas- P i tSZ i i ∆2g(k,zi) trophysics such as radiative cooling, preheating, SN/AGN l2 CtSZ- g 2 ∞n2(z)dzdz ftheeedbsmacakl.l TschaelseeSstZudpieoswfeorunsdpetchtartumth.eseBpurotcnesosneessoufppthreemss ≃r- 2(k,zi) h2πlCg i R0 iχ dχ . addressedthedependenceofthecrosscorrelationcoefficient l i r on these processes, whose investigation is the key goal of In this relation, the angular power spectra CtSZ- g and Cg thispaper. l l Inthispaper,weanalyzeacontrolledsetofhydrodynamic are evaluatedat l =kχ. Both the angularpowerspectra can i simulations for the relevant SZ statistics. The cosmology be measured directly by combining the SZ surveys and the galaxysurveys. Thequantityn(z)isgivenbythegalaxysur- adoptedisaΛCDMcosmology:ΩΛ=0.732,Ω0=0.268,Ωb= i 0.04448,h = 0.71,σ = 0.85. The simulations are run by veys,andχcanbecalculatedgiventhecosmologymodel.As 8 GADGET2 code (Springel 2005), with 100 h- 1 Mpc box longasr(k,z)canbefiguredout,onecanapplythisequation todothethermalSZtomography. size, and 5123 particles for both dark matter and SPH par- WesummarizethethermalSZtomographyasfollows. ticles. The masses of the dark matter and SPH particles are 4.62×108 h- 1M⊙and9.20×107 h- 1M⊙respectively.One • Measure the cross powerspectrumCtSZ- g between the simulation only includes gravitational heating and the other l thermalSZeffectandthegalaxydistributioninagiven includes radiative cooling and star formation with SN feed- redshiftbin. back. We referthesetwoasadiabaticandnon-adiabaticrun, respectively. For the non-adiabatic run, 7.63% of gas parti- • MeasurethegalaxypowerspectrumCg inthisredshift cles turninto stars by z=0, resulting in stellar mass density l bin. Ω∗=0.0034.Bothsimulationsstartfromredshiftz=120.We output60snapshots, whichareequallyspacedinlna. Refer • ReconstructthethermalSZcontributionfromthisred- to Linetal. (2006); Jingetal. (2006) for more detailsof the shiftbinthroughEq.11. simulations. WederivethetemperatureT ofthei-thgasparticleasfol- A key input required in this SZ tomography is the quan- i lows tityr. Zhang&Pen(2001)adoptedasimplificationthatrisa k T =(γ- 1)µm u, (12) constant.Theyfurthershowedthat,thevalueofrcanbemea- B i p i suredcombiningEq.11andthemeasuredSZpowerspectrum. with µ=0.588 the mean molecule weight, γ =5/3 the spe- 4 FIG.1.—Theredshiftevolutionofthedensity-weightedtemperatureTρin FIG.2.—Thedistributionofinternalenergyatdifferenttemperatureu(T) theadiabatic(solidline)andthenon-adiabaticsimulation(dashedline)with whichisdefinedas u(T)dlnT=U,whereUisthetotalinternalenergy.The cooling,starformationandSNfeedback. additionalprocesseRsbringsthetotalinternalenergydownby12.0%,10.9%, 11.5%and16.6%atz=0,0,52,1.02and2.08respectively.Thesamearbitrary unitisadoptedforboththesimulations. Forboththeadiabatic(solidlines) and non-adiabatic (dashed lines) simulation, the most contributions come fromthe gas attemperature around 1- 2keV. Andthe lower the redshift, cificheatratioformonatomicgas,andu theinternalenergy i themorecontributionsfromhighertemperaturecomponents. perunitmassofthei-thparticle. Whenthetemperatureofa SPH particle is higher than 10000 K, we consider it as ion- Here Tρ ≡ hneTi/n¯e is the density weighted temperature, ized,otherwiseasneutral.WesumupalltheSPHparticlesto whichiscomputedbythefollowingequivalentexpression calculatethedensity-weightedtemperatureT atalltheavail- ρ Tm able snapshots. From these measurements, we are then able T = i i i . (14) ρ tocalculatetheyparameter§3.1. P imi We use the fast Fourier transform (FFT) to calculate the P The sum is over all gas particles, assuming gas with T ≥ powerspectraofdarkmatter,gaspressureandthecrosspower i 10000Kasionized,otherwiseneutral. spectrumbetweenthetwo. For darkmatterparticles, we as- Figure 1 shows the redshift evolution of the density signthemassesto5123gridsusingtheClouds-In-Cells(CIC) weighted temperature for the adiabatic (solid lines) and algorithm.Ontheotherhand,weassignthethermalenergyof the non-adiabatic (dashed lines) simulation. As shown, SPHparticlestothesamegridsusingtheSPHkernelwhileas- the additional gastrophysical processes suppress the mass suringtheenergyconservation. Thegaspressureonthegrid weighted temperature T , which confirms the earlier results ρ is converted from the gas thermal energy assigned. We re- byWhiteetal.(2002),thoughmoreevidentthantheirsatz=0. movetheshotnoisesfromthecorrespondingrawpowerspec- We furtherlookinto the effectof coolingand feedbackin tra. However, since we are mainly interested in scales well redistributing the energy budget into regions with different belowtheNyquistfrequency(e.g.k.16hMpc- 1),wedon’t temperatures. The logarithmiccontributionto the total ther- correctforthealiaseffect.8 Resultsareshownin§3.2&3.3. malenergyfromregionswithdifferenttemperaturesisshown inFig.2.Wecanseeclearlythat,forthe4presentedredshifts, 3.1. ThetSZmeanyparameterandthegasdensityweighted the mostcontributionsof the total energycome fromthe re- temperature gionswith T ∼1- 2keV at lower redshiftsexceptz=2.08, forbothadiabatic andnon-adiabaticsimulations. The figure ThemeanComptonizationyparameterisgivenbythefol- also shows that gas with temperature higher than ∼ 2 keV lowingequation isslightlyaffectedbytheseadditionalgastrophysics.Mostof σ thesegasliesinclusters,whichhavedeepgravitationalpoten- y¯= meTc2Z n¯ekBTρadχ. (13) triaatlewiselrleslatotiovefflsyeltothweifneecdlubsatcekrereffgeicotn.sA,rlseosu,tlhtiengstainrfloersmsfaeteiodn- 8 Thealias effect wouldaffect thepowerspectra toscales aslarge asa back intensity and also less depletion of hot gas by cooling. quarteroftheNyquistfrequency,namelyk∼4hMpc- 1inoursimulations Ontheotherhand,gascoolerthan∼1keVhasoppositefate. (Jing2005).ThemethodsproposedbyJing(2005)andCuietal.(2008)can MostofthesegasliesingalaxygroupsorinIGM,susceptible effectivelycorrectforthiseffect. However,tothefirstorderapproximation, tocoolingandfeedback.Thissuppressestheircontributionto itaffectsthecorrespondingpowerspectrainthetwosimulationsinthesame manner.Sincewe’remainlyconcernedaboutthedifferencesbetweenthetwo thetotalenergy.Thecombinedeffectisthat,thetotalthermal powerspectra,weneglectthiseffectinthispaper. energyissuppressedbyafactorof12.0%,10.9%,11.5%and 5 shFifItGs.z3=.0—,0.5T2o,p1.0p2anaenld: 2T.0h8e.poAwtesrmsaplelcstcraaleosf,gthaes pprreessssuurreep∆o2Pw(ekr)saptercetdra- FIG.4.—ThebiasbP(k)=q∆2P(k)/∆2m(k)ofgaspressurerelativetothe of non-adiabatic simulation (dashed lines) are greatly reduced by cooling darkmatteratthefourredshifts. ThesamelegendsareadoptedasinFigure andSNfeedback,evenmoresignificanttohigherredshifts,comparedtothe 1. adiabaticone(solidlines). However,thepowersatlargescalesareslightly boosted. Bottompanel: Thedifferencesbetweenthepowerspectradefined simulationoutput,wecancalculatethe3Dpowerspectrumof by∆(∆2)/∆2 =(∆2 (k)- ∆2 (k))/∆2 (k) the gasthermalpressure∆2. The2D SZ powerspectrumis P P,A P,NA P,A P,A P obtainedthroughEq. (4)andweinterpolatelinearlybetween 16.6%atz=0,0,52,1.02and2.08respectively. adjacentsnapshotstomodel∆2(k)atanyotherredshifts. p GivenTρ measuredatthe 60simulationsnapshots, we are ∆2 atz=0,0.52,1.02and2.08areshowninFigure3. At able to integrate it over the whole relevant redshift range P smallscales, thepressurepowerspectraofthenon-adiabatic to calculate y, according to Eq. 13. We adopt a simple simulation (dashed lines) fall below the corresponding adi- linear, while stable, interpolation to model T at redshift ρ abatic ones(solid lines). Most of tSZ signals atthese scales fallingbetweentwoadjacentoutputredshifts.Forbothcases, arecontributedbygasinlessmassivehalos.Thesehaloshave about a half of the y signal comes from z< 1. We obtain shallowerpotentialwellsandthusSNfeedbackiseasiertode- y = 2.07×10- 6 for the adiabatic run and y = 1.77×10- 6 pletegas.Thissuppressioneffectisevenlargertowardhigher for the non-adiabatic run. These values of y are lower than redshiftswherethegravitationalpotentialwellsareshallower, the results in e.g. (daSilvaetal. 2001; Whiteetal. 2002; leading to as high as 50% reduction at z=2.08. The same Zhangetal.2004),soastheresultofT . Ontheotherhand, ρ feedbackredistributesgasandmovestheclusteringpowerof it is higherthan some recentsimulation results, such as that the thermal energy from small scales to large scales. This of Roncarellietal. (2007) using GADGET2. Differencesin is likely the reason that we see enhancement at large scales cosmological parameters, such as difference in σ8 and Ωb, (Fig. 3). However, at these scales, massive halos contribute can only account for part of the discrepancy, while cosmic mostofthesignal. Sincetheyhavedeeperpotentialwellsto variance, differences in the input gastrophysicsand numeri- fightagainstfeedbackandconfinegas,thelargescalepressure cal codes may be responsible for the remains. We postpone powerspectrumislessaffected. furtherinvestigationofthisissueelsewhere.9 Thegaspressurepowerspectrumisoftenexpressedbythe gas pressure bias with respect to the matter distribution, de- 3.2. The3DgaspressurepowerspectrumandthetSZpower finedasb (k)= ∆2(k)/∆2(k). Themeasuredpressurebias spectrum P P m is shown in Figpure 4. We find in Figure 4 that the results Most of the simulated SZ power spectra in the literature areingoodagreementwiththesimulationresultsatz=0by are directly calculated from the simulated 2D SZ maps. In Refregier&Teyssier(2002). this paper we take an alternative approach, as adopted by TheSZpowerspectrumevaluatedfromtheLimberintegral Refregieretal.(2000);Refregier&Teyssier(2002).Foreach isshowninFigure5. Intheexistenceofthecoolingandself- regulated star formation, the tSZ angular power spectrum at 9 We thank Volker Springel for providing the result of an independent largescalesissuppressedby∼20%,whichismainlydueto GADGET2simulation,whichallowsustodoapreliminarycheck. Thecos- thereductionofdensityweightedtemperatureT . Atthesame mologicalparametersofthissimulationareidenticaltoours,exceptthedif- ρ ferentinitialconditionsandσ8=0.9. ThissimulationgivesTρ=0.283keV. time,thepowersarereducedbyaslargeasaround∼40%at TmaaktiengthathtethsicsasliinmgulraetliaotnionwoTuρld∝reσs83u.l0t5-in0.1T5ρΩ0=(0Z.h2a4n3gkeetVal.fo2r0σ084)=, w0.e85e,stiin- vbeyrythesmsuapllpsrceassleiosnaroofubnodthlt=he10d0e0n0si,tywhwiecihghistejdoitnetmlypecraautsuerde goodagreementwithours. and pressure power spectra compared to the adiabatic sim- 6 FIG.5.—TheSZangularpowerspectraCtSZforthetwosimulations. Ra- FIG.6.— Toppanel: Thecross correlation coefficient between the3-D diativecoolingandSNfeedbacksuppressthel SZpowerspectrumby∼20% gaspressureanddarkmatterinadiabaticsimulations(solidlines)andnon- at large scales and around ∼40% at small scales. The same legends are adiabaticsimulations(dashedlines).Forboththesimulations,r(k)isgreater adoptedasinFigure1. than0.9atk≃0.062 hMpc- 1forallthe4redshifts,asgaspressuretraces darkmatterontheverylargescales.Atintermediatescaleswhere0.4≤k≤3 ulation. These suppression effects confirm the previous re- hMpc- 1,there’sstillgreatcorrelationabout∼0.8atz=0,whiler(k)drops sultsbydaSilvaetal.(2001);Whiteetal.(2002).Andwhat’s by25%to∼0.6atz=2.08. Bottompanel: Thedifference ofrbetween more, the non-adiabatic power spectrum peaks on a slightly tfheeedtbwaocksimaruelainticolnusd:ed∆,rt(hke)r/er’sA(okn)l=y(∼rNA1(-k)2-%rAc(hka)n)/greAi(nk)r.(Ikf)caotomlinogstasncdalSeNs, larger scale, about l ∼8000, than the adiabatic one, which andonlyatthesmallestscalesdoesrchange∼5%. mayarisefromSNfeedbackthatwoulddriveawaythegasout to diffuse regions. Springeletal. (2001) also found that en- ergyinjectionwouldsuppressthepoweronsmallscalesand cient. pushthepowerstowardlargerscales. Roncarellietal.(2007) Theresultonr(k,z)isshowninFigure6. Wefindgaspres- consideredthe same non-gravitationalprocessesas our non- suretendstotracethematterdistributionfaithfullyonthevery adiabatic one in their simulation, and their power spectrum largescales. At the largestscale thatour simulation can ap- behavesconsistentlywithours,despiteoftheirlowy. proach,k≃2π/L=0.062 hMpc- 1,r(k)≥0.9forallofthe 3.3. Thecorrelationcoefficientsr(k,z) 4 redshifts. At 0.4≤k≤3 hMpc- 1, there’sstill greatcor- relation between pressure and matter, while r(k) changes by As addressed in §2, a key input of the SZ tomography is 25%from∼0.8atz=0to0.6atz=2.08. However,ateach thegaspressure-galaxynumberdensitycrosscorrelationco- redshift, the correlation strength nearly keeps unchanged in efficient r. State of art numerical simulations with purely thisrange. At smaller scales k≥3 hMpc- 1 , r(k) beginsto gravitationalheating(adiabaticsimulations)isabletomodel decreasefaster. r(k,z)robustly. Ifwecanfurtherquantifythedependenceof Wefurtherinvestigatetheinfluencescausedbycoolingand r(k,z) on additional gastrophysics such as radiative cooling starformationonr(k). Surprisingly,we findthattheseaddi- and feedback, we will be able to model r(k,z) robustly for tionalgastrophysicalprocesseshavelittle impactonr(k). In generalcasesandperformtheSZtomographyrobustly. mostoftherelevantkrange,thechangeis∼1%.Onlyatvery For this purpose, we analyze the behavior of r(k,z) in the twosimulations.Sincegalaxiesaretightlycorrelatedwiththe smallscales (correspondingto l&104), can the effectreach matterdistribution,andthestochasticityissmallforthemea- 5%level.Thisbehaviorishoweverexplainable,sincethegas- surementofr(k,z)betweenthegaspressureandgalaxynum- trophysicsinfluencesthe denominatorandnumeratorof r(k) berdensity,weareabletoapproximategalaxydistributionas inbasicallythesameway. A1%errorinr resultsintoa2% matterdistribution.Namely,whatweactuallymeasureisr , error in the reconstructed ∆2W2 , negligible to the ∼30% Pm P tSZ thecrosscorrelationcoefficientbetweenthegaspressureand changein ∆2W2 inducedby coolingand feedback. There- P tSZ thematterdensity.Tothefirstorderapproximation,thesetwo fore, this result enables us to extract robust predictions for shouldbeidentical. Forexample,fortheoftenadoptedcon- r from adiabatic simulations without considering additional stantgalaxybiasmodel,thetwoareequaltoeachother.Even complicatedgastrophysics,andthencarryoutthetSZtomog- for scale dependentgalaxy bias, as long as the stochasticity raphymethodbyapplyingrbackintotheobservations.Since isnegligible,thetwoshouldbeequaltoeachother. Infuture eventually we need a large sample of simulations with high works, we willpopulatedarkmatterhaloswithgalaxiesand resolutiontoimprovethecalculation,thisfeaturecansignifi- measuretherealgaspressure-galaxycrosscorrelationcoeffi- cantlysavethecomputationcost. Moreimportantly,itbreaks 7 theotherwisecircularproceduretodothetSZtomography.10 withmuchhigheraccuracy. We arethenabletocrosscorre- Thisfeatureassuresusthatwecanrelyonadiabaticsimula- latethethermalSZmeasurementwiththereconstructedmat- tionsaloneto provider. However,it doesnotmean thatthe ter distribution in a redshift bin to perform the thermal SZ simulationspresentedinthispaperhavealreadyrobustlypre- tomography.ThethermalSZtomographycanthenbecarried dictedr. Weexpectthefunctionr(k)tobesufficientlysmooth outasfollows. withrespecttok. However,sinceourboxsizeisonly100h- 1 Mpc, statistical fluctuations induced by cosmic variance are • Reconstructthe3Dmatterdistributionfromthelensing tomography. non-negligibleand the simulated r-k relation shows clear ir- regularities(Fig. 6). To beat down the cosmic variance, we • MeasurethecrosspowerspectrumCtSZ- m betweenthe need more simulations or larger box size. Nonetheless, we l thermalSZeffectandthematterdistributioninagiven onlyneedadiabaticsimulationsforthisimprovement. redshiftbin. 3.4. Thegalaxy-matterrelation • MeasurethematterangularpowerspectrumCm inthis Thermeasuredaboveisactuallybetweenthegaspressure l redshiftbin. and the matter distribution. We have shown its robustness againstvariousgastrophysicalprocesses.However,inthepro- • ReconstructthethermalSZcontributionfromthisred- posedSZtomography,whatwereallyneedistheonebetween shiftbinthroughthefollowingrelation the gas pressure and the galaxy distribution. At sufficiently largescale, thegalaxydistributionwithrespecttothematter ∆2 (k,z)W distributionisbelievedtobedeterministic(e.g.Bonoli&Pen ∆2P(k,zi)Wt2SZ(zi)=r- 2(k,zi) P∆m2(k,iz)tSZ (15) 2009; Baldaufetal. 2009). This is a genericbehaviorin the m i standard model of structure formation and is unlikely to be l2 CtSZ- m 2 ∞n2(z)dzdz changedbyrealistic gastrophysics. Thusat thesescales, our ≃r- 2(k,zi) (cid:2)2πlCm (cid:3) R0 iχ dχ . resultprovesthatrbetweengaspressureandgalaxiesisalso l i robust various gastrophysical processes. However, at small In this case the relevantcross correlationcoefficient r is the scales,stochasticitiesbetweengalaxyandmatterdistribution onebetweenthegaspressureandthematterdensity,exactly developandthe crosscorrelationcoefficientbetweengalaxy theonethatwestudyinthispaper. Wehavedemonstratedits and matter distribution r 6=1. In this case, we should also mg robustnessagainstvariousgastrophysicalprocessesandhence quantifytheimpactofvariousgastrophysicalprocessessuch therobustnessofthethermalSZtomography. asfeedbackandcoolingonr . mg Unfortunately, since even the state of art simulations are 4. CONCLUSIONANDDISCUSSION notabletosimulategalaxiesfromfirstprinciples,wearenot In this paper, we investigate the feasibility of the tSZ to- able to robustly quantify the impact of these gastrophysical mography method, which aims to extract the redshift infor- processesonr . Instead, we willadoptthe halomodeland mg mation of the tSZ effect. As explained in §2, future SZ testr byvaryingparametersofthehalooccupationdistribu- mg and galaxy surveys will measure the galaxy angular power tion (HOD) to study possible impactof these gastrophysical spectrumCg in eachredshiftbinandthecorrespondingtSZ- processes. The technical details and results are presentedin l theappendix. We findthatinsomecasesr barelychanges galaxiescrosspowerspectrumCtSZ- g. Giventhecrosscorre- mg l and in other cases it can change by ∼10%. Due to uncer- lation coefficient r(k,z) between the gas thermal energy and tainties in the dependenceof HOD parameterson these gas- the galaxy number density, we are able to reconstruct the trophysicalprocesses,wearenotabletodrawanyconclusive time resolved thermal energy distribution. So the key ques- observation.However,theseresultsdowarnusthepossibility tion in the tSZ tomographyis whether we can robustly pre- thatthese astrophysicalprocessescould significantlychange dictr(k,z),givenuncertaintiesingastrophysicalprocesses.To r and hence the cross correlation coefficient between the quantifytheeffectoftwodominantgastrophysicalprocesses, mg gaspressureandgalaxydistribution,despitethefactthatthey namely radiative cooling and SN feedback, we compare the have only minor impact on the cross correlation coefficient result of our adiabatic simulation against the one with cool- betweenthegaspressureandmatterdistribution. ing and feedback. We find that r(k,z) is insensitive to these Thegoodnewsisthat,evenforthisworstscenario,weare additionalgastrophysics. Theresultingdifferenceinr(k,z)is still able to carry out the thermal SZ tomography robustly, ∼1- 2%inmostrelevantkandzrange,muchsmallerthanthe with the help of the lensing tomography. SZ surveys of- ∼30%changeintheSZpowerspectrumandthegasdensity ten overlap with lensing surveys. For example, the South weighted temperature. This allows us to neglect the depen- Pole Telescope survey basically overlaps with the Dark En- dence of r(k,z) on these gastrophysicalprocesses and adopt ergySurvey(DES).Throughthe lensingtomographyof sur- ther(k,z)evaluatedfromadiabaticsimulationsastheinputof veys with redshift information, in principle we are able to thetSZ tomography. We thusshowthatthetSZtomography reconstruct the 3D matter distribution from the nearby uni- isfeasibleinreality. verse to z ∼ 2 where the furtherest source galaxies reside. Thereareanumberofimprovementswhichcanbemadein Masseyetal.(2007)haveclearlydemonstratedthefeasibility futurework.(1)Asaddressedintheendof§3,Weneedtorun ofthistechnique.GiventhecapacityofstageIIIandIVlens- moresimulationstoreducethecosmicvariance.Wemayalso ing surveys, the 3D matter distribution can be reconstructed needtocorrectforthealiaseffecttoimprovetheaccuracyof ratscalesk&4 hMpc- 1. (2)Inthispaper,toevaluater(k,z), 10Ifr(k)stronglydependsonthedetailedgastrophysics,thetSZtomog- wehaveapproximatedgalaxydistributionasdarkmatterdis- raphyprocedurewillbecomecircular. Inthiscase,weneedtoknowthede- tribution for simplification. In the limit that the stochastic- tailedgastrophysicstoreliablypredictr(k,z)fromsimulationsandthenapply backtoobservations,whilewecountonobservationstotellusthedetailed ity vanishes, ourapproximationbecomesexact. Thisshould gastrophysics. be the case at large scales. At small scales, non-vanishing 8 stochasticity could result in some systematical errors in the andfurtherinvestigationsareneeded.However,anoptimistic tSZ tomography. In the futurework, we will populatehalos news is that we DO demonstrate the robustness of the tSZ inoursimulationswithgalaxiesandquantifythepossiblesys- tomography,alternativelyusingtSZ surveywith dark matter tematics inducedby this simplification. (3) Anotherissue is instead of galaxies, while we can measure the clustering of theAGNfeedback,whichpotentiallyhaslargerinfluenceon darkmatterto highaccuracyon scalesof tSZ interest, using the tSZ effect, although we do not expect it to significantly lensingtomography. Wethuscanexpectausefulapplication modifyr(k,z). Weshallinvestigateitinthefutureandquan- oftSZtomographyinthefuturesurveysoftSZeffect. tifyitspossibleimpactonthetSZtomography. There’s also a possible limitation that may degrade tSZ tomography method using galaxies surveys. In our non- adiabatic simulation, extra gas dynamics may influence the 5. ACKNOWLEDGEMENT galaxy-matterrelation,resultingindeviationsandstochastic- WethankVolkerSpringelforprovidinghisnumericalsim- itiesinthecorrelationcoefficientr(k,z). Thoughwecan’tfig- ulationresultforcomparisonandKlausDolagforusefulcon- ureouttheexactinfluenceofgasdynamicsandgalaxyselec- versation. The simulations were done at Shanghai Super- tionfunction,givenlittleknowledgeofrecipesofgalaxyfor- computer Center by the supports of Chinese National 863 mationatpresent, we testthe possibleeffectswith a generic project(grantNo.06AA01A125). This work is supportedin halo occupation distribution (HOD) model in the appendix. partbytheNationalScienceFoundationofChina(grantNo. TheresultsofthisgenericHODmodelfavortSZtomography 10533030,10543004,10821302,10873027&10878001),the methods using galaxy survey in some aspects, while disfa- KnowledgeInnovationProgramofCAS (grantNo. KJCX2- voritinotheraspects. Thuswecan’tpindowntheinfluence YW-T05&KJCX3-SYW-N2)andthe973programgrantNo. of gas dynamics and selection function in galaxy surveys, 2007CB815401&2007CB815402. 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This assumption assures that there’s a deterministic map between the galaxy and matter mg 9 distribution,andthusthecorrelationcoefficientofpressureandgalaxyisequaltothatofpressureanddarkmatter. Galaxysurvey isactually anagencywhich introducesdarkmatterdistributionin the tSZ tomographymethod. Therefore,the uncertaintiesin galaxyclusteringof differentgalaxysamplesmay introducestochasticitiesin estimating the underlyingdarkmatter clustering features. Inoursimulations,whenweemergeextragastrophysics,darkmatterwouldbeinfluencedlittle,whilegalaxies,onthe otherhand,sufferfromthesegastrophysicalprocesses. Exactschemeofgastrophysicsisneededtomodelthegalaxyformation for both simulations and theories, while it’s beyond our knowledge so far. We thus have no idea how to incorporate detailed gasdynamicsinbothobservationsandsimulations,givenlittleknowledgeoftheunderlyinggastrophysics.Nevertheless,earlier studiesof galaxy-mattercorrelation(Seljak 2000; Guzik&Seljak 2001) showedthat r is approximateto unityup to k∼1- 2 hMpc- 1,bycrosscorrelatingdarkmatterwithgalaxysamplesselectedbydifferentselectioncriteria. Inanalternativepointof view,ifwecanmeasurematterpowerspectrum,wecancircumventthegalaxy-mattercrosscorrelationwhiledirectlycorrelate darkmatterandpressurepowerspectrumwhendoingtSZtomography. Seljak&Warren(2004)proposedto usefaintgalaxies as dark matter tracer, as they found the linear bias of these galaxies is nearly constant. Pen (2004) suggested to obtain the 3D dark matter clustering using the cross correlation between galaxies with distance information and projected weak lensing darkmattermaps. Morerecently,Baldaufetal.(2009)proposedthatbycombininggalaxy-clusteringandgalaxy-galaxylensing measurements,theycanreconstructthedarkmatterpowerspectrumatafewpercentaccuracyuptok∼1 hMpc- 1. However,wewouldliketoinvestigatethiscorrelationcoefficientr inamoregeneralway,say,byaHODmodelinsteadof mg lookingintoourtwospecifichydrodynamicsimulations.FollowingtheroutineofhalomodelinCooray&Sheth(2002),wecan calculatethe powerspectrumof darkmatteras wellasof galaxies. Giventhe halomodelandthe halooccupationdistribution model (HOD) for galaxies, we can predict the matter power spectrum P (k) and galaxy power spectrum P(k), as well as the m g crosspowerspectrumP (k).Thecrosscorrelationcoefficientisthussimplyderivedbyr (k)=P (k)/ P (k)P(k). Wewon’t mg mg mg m g describe the details of how to calculate the power spectra, for which readerscan refer to Cooray&Shpeth (2002) for a review. However,wepresentinthefollowingparagraphshowwepopulategalaxiesinhaloswithagenericHODmodel. Thoughit’shardforonetoexactlypredictthegalaxydistributioneitherbytheoriesorbysimulations,detailedsemi-analytic galaxyformationmodelsprovidesone with an experiencedway to describe the galaxydistribution in the existence of generic gastrophysics. Inthispaper,wearenotgoingtoaddressvariousgastrophysicsortherebyquantifytherelationbetweengalaxies anddarkmatter. Asasimpleapproximation,weillustratethemeangalaxynumberhN |MiinahalogivenmassM bytaking gal theparameterssuggestedbySheth&Diaferio(2001)andCooray(2004)fromthesemi-analyticresults,asfollows N for M ≤M≤M hN |Mi= 0 min B (A1) gal (cid:26)N0(M/MB)α for M>MB Here the parameters are Mmin =1011h- 1M⊙,MB = 4×1012h- 1M⊙, N0 = 0.7 and α= 0.8. We take this set of parameters as our fiducialmodel. Besides the mean occupationnumberof galaxies, we need to know as well the informationof the second momentum of the galaxies. The usually adopted Poisson probability distribution of galaxies deviates from the results from numericaldata(Berlindetal.2003),wethusinsteadusethebinomialdistributionsuggestedbyScoccimarroetal.(2001) hN (N - 1)|Mi1/2=β(M)hN |Mi, (A2) gal gal gal hereβ(M)=log M/MminM⊙forM<M0withM0=1013h- 1M⊙,andβ(M)=1otherwise.Inourfiducialmodel,weassumethe galaxiesinhalopstracethedarkmatterdistribution,whichfollowstheNFW(Navarroetal.1997)profile ρ ρ(r)= s , (A3) (r/r )(1+r/r)2 s s whereρ isthedensityatacharacteristicradiusr ,whichcanberelatedtothevirialradiusr withconcentrationc=r /r . A s s vir vir s usefulapproximationforconcentrationisgivenbyBullocketal.(2001) M - 0.13 c (M)=9 . (A4) m (cid:20)M (cid:21) ⋆ There will be great differences between the HOD parameters of galaxy samples selected by various selection criteria, e.g. luminosity, color, history and morphology. However, we take into account the selection criteria and extra gastrophysics by consideringtwogenericcases, whichareframedbychangingthenumberdensityofgalaxiesandchangingthemorphologyof galaxydistribution. MorespecificallyinourHODmodel,weintroducethesetwoeffectsbyvaryingthesixparametersofHOD adoptedhere,i.e. M ,M ,M ,N ,αandtheconcentrationofgalaxiesc . Inpractice,wecalculater bychangingoneofthe min B 0 0 g mg parameterswhilefixingtheothers.IntheexistenceofstarformationandSNfeedback,sinceit’shardforgastocoolandcondense into stars, we expectless galaxiesin the non-adiabaticsimulation than the adiabatic one. We set the parameterssuch that the averagegalaxynumberislessthanthatinthefiducialHOD model. We thusraiseeachtimeoneofthemassthresholds,M , min M andM astwiceasthatinthefiducialmodel,whilereduceN byahalf.Yangetal.(2005)foundc =1/3c byinvestigating B 0 0 g m galaxysurveys,andNagai&Kravtsov(2005)foundsimilarresultsinhydrodynamicalsimulations. Inspiredbytheseworks,we set the galaxy concentrationc as 1/3 or 2/3 as that of dark matter c . For α, we reduce it to 0.7 so as to reduce the average g m galaxynumbertotesttheeffect. Weshowtheresultofthecrosscorrelationcoefficientr inFig. 7. WefindthatreducingN doesn’tchangetheratall,and mg 0 changingM onlyinfluencerbyaround2percent,excepta5%droponscalesaroundk∼5hMpc- 1. Amoreextendedgalaxy 0 distribution in halos, i.e. c =1/3c would induce a 10% suppression of r at k∼7hMpc- 1 . However, the suppression is g m mg 10 FIG.7.—Thesensitivityofthedarkmatter-galaxycrosscorrelationcoefficientrmgtodifferentHODparameters. Wechangeoneoftheparametersonceata timewhilekeepothersfixedduringthecalculation. 2Mmin,2MBandreducingαto0.7greatlyinfluencermg. However,it’spossiblethattheycanceleachother partlyandreducethestochasticity.Encouragingly,changingN0,M0,cgwouldintroduceseveralpercentinfluenceonronscalesofinterestoftSZeffect. atmost5%onscalesofinterestoftSZ,say,k.3hMpc- 1. c =2/3c , asexpected,onlyreducer byatmost3%onsmall g m mg scales. ActuallyanamplitudeofgalaxynumberN won’tintroduceanydifferenceofgalaxyclustering. Whilegalaxynumberis 0 insensitivetoM orirrelevanttoc ,theonehalotermofgalaxypowerspectrum,i.e. onsmallscalesisindeedinfluenced. 0 g Thechangesin M , M andα, on theotherhand, leadto greatinfluencein r . 2M , i.e. increasingthe massthreshold min B mg min of host halo of central galaxy, reduces the average galaxy number by nearly a half, from 1.93×10- 2h3Mpc- 3 in the fiducial modelto9.93×10- 3h3Mpc- 3,while2M onlychangesthatby3%. Interestingly,theyinduceoppositeeffectsinr ,bothwith B mg amplitudesupto15%-20%onscalesofinterest.Thoughα=0.7reducesthegalaxynumberbylessthan2%,itsuppressesr by mg 10%aroundk∼1- 2hMpc- 1. Thusfromthissimplemodel,it’spossiblethatthestochasticitybetweendarkmatterandgalaxies islarge,andthereforetheeffectivenessoftSZtomographywoulddegrade. However,wehopethatthechangeinr wouldnot mg be so severe. Firstly, the parameters we adopt here are extreme, while in the non-adiabatic simulation, cooling and feedback influencemostlyinthecentralregionsofhalos,anditwouldtendtoinfluencesmallscalespowerspectrumbelowclusterscales. Secondly, they lead to opposite effects to r , it’s thus possible that the effects partly cancel out each other. In earlier works, mg Guzik&Seljak (2001) showedthatgalaxy-darkmatter correlationcoefficientsof differentgalaxysamples, which are selected byluminosity,colourandstarformationrate,convergetounityonscalesabove1hMpc- 1. Also,theyfoundinthefaintestand mostabundantsamples, r is unity up to k∼3hMpc- 1, which is also indicated by Seljak&Warren (2004) where they found smallhalostracedarkmatterwithconstantbias. Thisisencouraging,asinpractice,wecanselectfaintgalaxysampletodotSZ tomography.Afterall,thestochasticityofgalaxy-matterrelationisaninterestingissue,andneedtobefurtherinvestigated.