A&A manuscript no. ASTRONOMY (will be inserted by hand later) AND Your thesaurus codes are: ASTROPHYSICS 12.03.1;12.03.3;12.03.4;12.04.2;12.12.1;11.03.1 Sunyaev-Zel’dovich Surveys: Analytic treatment of cluster detection J.G. Bartlett 0 Observatoire Midi-Pyr´en´ees, 14, Ave.E. Belin, 31500 Toulouse, FRANCE 0 0 January 5, 2000 2 n a Abstract. Thanks to advances in detector technology mation, studies focussed on the global properties of clus- J and observing techniques, true Sunyaev–Zel’dovich (SZ) ters provide information on the nature of dark matter; 4 surveys will soon become a reality. This opens up a new therelativeproportionsofhotgas,darkmatterandstars; 1 window into the Universe,inmany waysanalogousto the and on scenarios of structure formation, including con- 1 X–raybandandinherently well–adaptedtoreachinghigh straints on the universal density parameter, Ω . One ex- o v redshifts.Idiscussthenature,abundanceandredshiftdis- ample of the latter that comes to mind in anticipation of 7 tributions of objects detectable in ground–based searches observations with the new generation of X–ray satellites, 6 withstate–of–the–arttechnology.Anadvantageofthe SZ ChandraandXMM,is the useofthe redshiftevolutionof 2 1 approach is that the total SZ flux density depends only the cluster abundance to constrain Ωo (Oukbir & Blan- 0 on the thermal energy of the intracluster gas and not on chard 1992, 1997; Bartlett 1997; Henry 1997; Bahcall & 0 its spatial or temperature structure, in contrast to the Fan 1998; Borgani et al. 1999; Eke et al. 1998; Viana & 0 X–ray luminosity. Because ground–based surveys will be Liddle 1999a); another is the now classic cluster baryon / h characterized by arcminute angular resolution, they will fraction test (White et al. 1993). p resolve a large fraction of the cluster population. I quan- - o tify the resulting consequences for the cluster selection While clusters have been extensively studied in the r function;theseincludelessefficientclusterdetectioncom- optical and X–ray bands, observations based on weak t s paredtoidealizedpointsourcesandcorrespondingsteeper gravitationlensingandtheSunyaev–Zel’dovich(SZ)effect a : integratedsourcecounts.This implies, contraryto expec- (Sunyaev & Zel’dovich 1972) are just coming to fruition. v tations based on a point source approximation,that deep InthecaseofSZobservations,importantsamplesconsist- Xi surveysare better than wide ones in terms of maximizing ingofseveraltensofclusterspre–selectedintheX–rayare r the number of detected objects. At a given flux density beginning to permit cosmologists to capitalize on the po- a sensitivity and angular resolution, searches at millimeter tential of combined SZ/X–ray observations (Carlstrom et wavelengths(bolometers)aremoreefficientthancentime- al. 1996,1999).Full maturity of the field will be heralded ter searches (radio), due to the form of the SZ spectrum. bytherealizationofpurelySZ–basedskysurveys.Inwhat Possibleground–basedsurveyscoulddiscoverupto 100 wemightrefertoasthe“SZ–band”,onecanthenimagine ∼ clusters per square degree at a wavelength of 2 mm and performing cluster science analogousto what is now done 10/sq.deg. at 1 cm, modeling clusters as a simple self– in the X–ray,e.g., the construction of cluster counts, red- ∼ similar population. shift distributions, luminosity functions, etc., all viewed via the unique characteristicsof the the SZ effect. For ex- Key words:cosmicmicrowavebackground–Cosmology: ample, several authors have emphasized the advantages observations – Cosmology: theory – large–scale structure of the SZ effect, over similar X–ray based efforts, to con- of the Universe – Galaxies: clusters: general strainΩ viatheclusterredshiftdistribution,aswellasto o study cluster physicsoutto verylargeredshifts (provided the clusters are out there, the very question of Ω itself) o (Korolyov et al. 1986; Bond & Meyers 1991; Bartlett & 1. Introduction Silk 1994; Markevitch et al. 1994; Barbosa et al. 1996; Eke et al. 1996; Colafrancesco et al. 1997; Holder et al. Cosmologists have long appreciated the value of the Uni- 1999).SuchpureSZsurveyswillbeperformed:thePlanck verse’s biggest objects, galaxy clusters. Besides being a Surveyor will supply an almost full–sky catalog of several collection of galaxies well suited for studies of galaxy for- thousand clusters detected uniquely by their SZ signal; Send offprint requests to: J.G. Bartlett and advances in both detector technology and observing Correspondence to: [email protected] techniques now offer the exciting prospect of performing 2 J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection purelySZ–basedsurveysfromtheground,withbothlarge and redshifts of clusters detectable by ground–based sur- format bolometer arrays and dedicated interferometers. veys, with the particular aims of understanding optimal I discuss in this paper some aspects of the science ac- objectextractionandthe accessible science.Forexample, cessible to pure SZ surveys by examining the nature of one of the key questions facing any survey is one of ob- their cluster selection. Because of the close analogy with serving strategy: given a fixed, total amount of observing X–ray studies, it is useful for this purpose to compare time, should one “go deep”, with long integrations on a and contrast SZ–based cluster searches to those based on few fields, or instead “go wide”, covering more fields to X–rayobservations.Theredshiftindependence ofthesur- highersensitivity.Ifoneisouttomaximizethenumberof face brightness of a cluster (of given properties) means detected objects, the answer depends on the slope of the that SZ cluster detection is inherently more efficient than counts.Onegainsbygoingdeeperiftheintegratedcounts X–ray detection at finding high redshift objects. Equally are steeper than S−2, assuming that noise diminishes as ν importantis that althoughthe SZ effect and X–raysboth 1/√t; otherwise, a larger area yields more objects. “see” the hot intracluster medium (ICM), they do so in The cluster selection criteria of a survey may be com- significantly different ways.In particular,the well–known pactly summarized by a minimum detectable mass as fact that the SZ effect scales as the gas pressure implies a function of redshift – M (z). Together with a suit- det that the flux density, Sν, is simply proportional to the able mass function (we shall use the formalism of Press total thermal energy of the gas. This makes modeling es- and Schechter 1974), this quantity determines both the pecially simple, for this quantity depends only on the to- sourcecountsandredshiftdistributionsofthefinalsource tal gas mass and the effectiveness of gas heating during catalog. Thus, in very concrete terms, we must examine collapse, in stark contrast to the X–ray emission that de- M (z) and understand the influence of the observation- det pends also on the density and temperature distribution ally imposed restrictions on θ and i . Given a set of ob- c ν of the gas. This simplicity is an advantage because any servations, i.e., a map, one could imagine many different theoreticalinterpretationofsurveyresultsrequiresanad- algorithms to extract astrophysical sources, and M (z) det equatelymodeledrelationbetweentheobservableandthe will depend upon this choice. There is in principle an op- theoretically relevant quantity of cluster mass. timal method, one which preserves signal–to–noise over These remarks concern essentially the physics of the the entirerangeofsourcesurfacebrightnessandsize.Itis ‘emission’ mechanism itself. Of equal relavence is the na- characterizedbyadecreasingsurfacebrightnesslimitwith tureofobjectselectionimposedbythe eventualdetection object size – the greater number of object pixels permits algorithm used to extract sources from a set of observa- lower surface brightness detections. This algorithm is dif- tions (a map); and this in turn depends crucially, as for ficult to apply in practice,and more standardapproaches any survey, on the particular combination of sensitivity searchinsteadfor a minimum number ofconnectedpixels and angular resolution of the observations. The objects above a preset threshold, thereby establishing a fixed cut detected by Planck will not be the same as those selected on source surface brightness. Detection signal–to–noise is byground–basedsurveys,andthefinalcatalogsshouldbe no longer constant, rather increasing with θ , and these c viewed as complementary. Planck will produce a shallow methods loose large,and in–principle detectable, low sur- ( tensofmJy)large–areasurvey,whiletheground–based face brightness objects. All of this will be reflected in the ∼ instruments will perform deeper surveys (< 1 mJy) over resulting functions M (z). det smaller sky areas (several square degrees). Most clusters Throughout the discussion, we will be guided by the remain unresolved at the Planck resolution of 5 10 characteristics of two potential types of ground–basedin- ∼ − arcmins,andthis characterizesthe kindsofobjectsacces- struments: large format bolometer arrays, epitomized by sibletothissurvey,e.g.,thecountsandtheredshiftdistri- BOLOCAM (Glenn et al. 19981), and interferometer ar- butions. The higher angular resolution of future ground– rays optimized for SZ observations,as suggested by Carl- basedinstruments(ontheorderofanarcmin)willresolve strometal.(1999)2.BOLOCAMisa151–elementbolome- many clusters and impose different selection criteria that ter array under construction at Caltech for operation in willdefinethecountsandredshiftdistributionsofthefinal three bands – 2.1 mm, 1.38 mm (the null of the thermal catalog. SZ effect) and850µm. At the CaltechSubmillimeter Ob- This is a centralissue ofthe presentstudy were,moti- servatory, it is expected that the array will be diffraction vated by the possibility of ground–basedsurveys,I exam- limited to arcminute resolution, or better, and limited ∼ ine the detection of resolved clusters. While the detection in sensitivity by atmospheric emission (rather than de- ofunresolvedsourcesisprincipallydependentonobserva- tector noise). With its 9–arcmin field–of–view, one could tionalsensitivity,andthefinalselectionismoreorlessone imagine surveying a square degree to sub–mJy sensitiv- of apparent flux – S θ2i – the detection of resolved ν ∼ c ν sources is a more complicated cuisine involving individu- 1 http://phobos.caltech.edu∼lgg/bolocam/bolocam.html ally the characteristic source size, θc, and surface bright- 2 While writing, I became aware of another project – the ness,iν.Thespecificgoalofthepresentworkistoquantify Arcminute MicroKelvin Imager. See Kneissl R. 2000, astro- in terms of observing parameters the abundance, masses ph/0001106 J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection 3 ity in these bands. Carlstrom et al. (1999) have recently spectrum at T = 2.725 K (Mather et al. 1999). The o expoundedthe virtues of interferometrictechniques using spectral shape is embodied in the function j , ν telescope arraysspecifically designed for SZ observations. (kT )3 x4ex x o They have proposed the construction of such an array, jν(x) = 2(h c)2 (ex 1)2 tanh(x/2) −4 operating at a wavelength of 1 cm, and estimated that it p − (cid:20) (cid:21) (kT )3 would be capable, in the course of one year of dedicated 2 o f (2) observations, of covering 10 square degrees to a limit- ≡ (hpc)2 ν ∼ ing sensitivity of 0.3 mJy at arcminute resolution. In = (2.28 104mJy/arcmin2)f ∼ × ν summary,then,weareinterestedinconsideringSZobser- whiletheamplitudeisgivenbytheComptony–parameter vations at arcminute angular resolution and to sub–mJy kT sensitivityatbothcentimeterandmillimeterwavelengths. y dl n σ (3) Thepaperisorganizedasfollows:arapidreviewofthe ≡Z mec2 e T SZeffectisgiveninthenextsection,followedbyadiscus- an integralof the pressure alongthe line–of–sightatposi- sionofthe unique aspectsof SZ clusterdetection. Section tionθrelativetotheclustercenter.Here,T isthetemper- 3detailstheclusterpopulationmodelemployed,basedon atureoftheICM(really,theelectrons),m istheelectron e the Press–Schechter(Press&Schechter,1974)massfunc- rest mass, n the ICM electron density, and σ is the e T tion and the isothermal β–model. Since we shall focus on Thompson cross section. Planck’s constant is written in issues of cluster selection as imposed by survey parame- these expressionsash ,the speedoflightinvacuum asc, p ters, the cluster model will be restricted to the simple ex- and Boltzmann’s constant as k. These formulae apply in ample of a self–similarpopulation.The next section(Sec- the non–relativisticlimitoflowelectron(andphoton)en- tion4)introducestheprincipalfigures(Figures1,2and3) ergies; relativistic extensions have recently been made by of the present work by consideration of unresolved clus- severalauthors(e.g.,Rephaeli1995;Stebbins1997;Challi- ter detection; this case will also be used as a benchmark nor & Lasenby 1998;Itoh et al 1998;Pointecouteauet al. againstwhichtoexaminetheeffectsofresolveddetection. 1998; Sazonov & Sunyaev 1998). The spectral shape of Section 5 then develops the principle themes of resolved thedistortionisunique,becomingnegativeatwavelengths SZ cluster detection, starting with consideration of the larger than 1.4 mm (relative to “blank” sky) and pos- ∼ optimal, constant signal–to–noise method, and followed itive a shorter wavelengths. This offers a way of clearly by detailed study of cluster detection based on the stan- separating the effect from other astrophysical emissions. dard algorithm. A final discussion (Section 6) then more All of the physics is in the Compton y–parameter, an closely examines the number of detections to be expected apparentlyinnocuous–lookingexpression.Infact,itholds from ground–based surveys and gives a non–exhaustive thekeytoallofthepleasingaspectsoftheSZmechanism. list of some important issues still to be treated. Section 7 First of all, the conspicuous absence of an explicit red- concludes. shiftdependenceisthewell–knownresultthattheSZsur- Key results will be the M (z) curves presented in face brightness is redshift–independent, determined only det Figure 1, quantifying the nature of SZ detected clusters, by cluster properties. This should be contrasted to other and the conclusion that resolved source counts are lower emission mechanisms which all experience “cosmic dim- andsteeperthanexpectationsbasedonsimpleunresolved ming” [ι (1+z)−4] due to the expansion of the Uni- ∝ sourcecountcalculations,Figure2.Tothepoint,thelatter verse. This is countered in the SZ effect by the increasing implies that surveys at arcminute resolution gain objects energy density towards higher z of the CMB, the source withanobservingstrategyof“goingdeep”.Thecosmolog- of photons for the effect. icaldensityparameterisdenotedbyΩ 8πGρ/3H2,the Another very important aspect of the SZ mechanism o ≡ o vacuum density parameter by λ Λ/3H2 and the Hub- resides in the fact that its amplitude is proportional to o ≡ o ble constant by H h100 km/s/Mpc; unless otherwise the pressure,orthermal energy,ofthe ICM.This appears o ≡ indicated, h=1/2 and λ =0. most clearly when we consider the total flux density from o a cluster,found by integratingthe surfacebrightnessover the cluster face: 2. The Particular Value of the SZ Effect kT(M,z) S (x,M,z) = j (x)D−2(z) dV n (M,z)σ We beginby establishingour notationinrecallingthe ba- ν ν a m c2 e T sicformulasoftheSZeffect.Thechangeinsurfacebright- Z e M <T > (4) gas ness relative to the unperturbed cosmic microwave back- ∝ Theintegralis overthe entirevirialvolumeofthecluster. ground (CMB), caused by inverse Compton scattering in In this expression,D (z) is the angular–sizedistance in a the hot ICM, is expressed as a Friedmann–Robertson–Walker metric – iν(θ)=y(θ)jν(x) (1) D (z)=2cH−1 Ωoz+(Ωo−2)(√1+Ωoz−1) ang o Ω2(1+z)2 wherex hpν/kToisadimensionlessfrequencyexpressed (cid:20) o (cid:21) in terms≡of the energy of the unperturbed CMB Planck =cH−1D(z) (5) o 4 J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection where I introduce the dimensionless quantity D(z). We one finds a good T–M relation with rather small scatter see clearly that the final result is simply proportional (Evrard et al. 1996; Bryan & Norman 1998). Putting all to the total thermal energy of the ICM, dVnT. of this together, a relation of the form (6) between the This is extremely important, because it means that the observable,S ,andclustermassappearsquite reasonable ν R SZfluxdensityisinsensitive (strictly speaking, completely and rather robust; and in any case, the modeling uncer- so for the total flux density and for fixed thermal energy) tainties are always easier to understand than in the case toeitherthespatialdistributionoftheICMoritstempera- ofX–rays,due to the allimportantinsensitivity ofthe SZ turestructure,makingmodelingmuchsimplerthaninthe flux density to spatial/temperature structure of the ICM. case of X–ray emission. Consider that in X–ray modeling The conclusionis that the SZ flux density should be a one prefers the X–ray temperature over luminosity as a very good halo mass detector, in principle sensitive to all more robust indicator of cluster mass, but even the tem- halos withsignificantamounts ofhotgas andovera large perature has some sensitivity to the gas distribution, be- range of redshifts. All of these remarks concernto a large cause it is all the same an emissionweightedtemperature extent the total SZ flux density of Eq. (4), and therefore that is actually observed. We would expect the tempera- apply primarily to situations where the clusters are unre- ture appearing in the second line of Eq. (4), which is the solved.Itisstilltruethat,evenwhenaclusterisresolved, true mean electron energy, to demonstrate an even bet- theSZsignalisproportionaltothetotalthermalenergyof ter correlation with virial mass than the observed X–ray thegas,butnowonlyofthatportioncontainedwithinthe temperature.Simplescalingargumentsleadonetobelieve columndefinedbythebeam.Afterfirstoutliningtheclus- that this correlationshould be T T M2/3(1+z), ter population model employed, we shall tackle in detail virial ∼ ∼ from which we deduce the additionalcomplexities introducedby resolved cluster observations. S f (M,z)M5/3(1+z)D−2(z) (6) ν gas ∼ wherefgasisthegasmassfractioncontributedbytheICM 3. Modeling the Source Population to the total cluster mass. The central ingredient of a model for the cluster popula- The SZ mechanism therefore conveniently reduces all tionanditsevolutionisthemassfunction,n(M,z),which the potentialcomplexityof the ICM to just its totalther- gives the number density of collapsed, virialized objects malenergy, f <T >.Thisquantitymaynevertheless gas ∝ as a function ofmass andredshift. The exactform of this beinfluencedbyseveralfactors.Forexample,thegasmass function depends on the statistical properties of the pri- fraction in Eq. (6) has carefully been written as a general mordialdensityfluctuations.ForInflationary–typescenar- function of both mass and redshift. In simulations this ios,inwhichthesefluctuationsareGaussian,areasonable quantityis mostoftenconstant,the majorityofgasbeing expressionfor the mass function appears to be the Press– primordialandsimplyfallingintotheclusteratformation. Schechter formula (Press & Schechter 1974) Onecouldimagineotherpossibilities(e.g.,Bartlett&Silk 1994; Colafrancesco & Vittorio 1994) that would lead to a more important dependence on either mass or redshift, n(M,z)dM = 2<ρ>ν(M,z) dlnσ(M) e−ν2/2dM(7) π M dlnM M althoughmetallicityargumentsseemto requirethatmost r (cid:12) (cid:12) (cid:12) (cid:12) ofthe gasbe primordial,atleastinthe moremassivesys- The quantity ρ represents the(cid:12)comoving(cid:12)cosmic mass h i (cid:12) (cid:12) tems(Metzler&Evrard1994;Elbazetal.1995).Whileit density and ν(M,z) δ (z)/σ(M,z), with δ equal to c c ≡ appearsfromnumericalstudiesthatshockingduringclus- the critical linear over–density required for collapse and ter formation efficiently heats the ICM to 80%– 100% σ(M,z) the amplitude of the density perturbations on ∼ of the virial temperature (Metzler & Evrard 1994; Bryan a mass scale M at redshift z. Numerical studies ascribe & Norman 1998), additional sources of heating could in rather remarkable accuracy to the simple expression of principle change the temperature of the gas relative to Eq.(7)(Lacey&Cole1994;Ekeetal.1996;Borganietal. that of the potential, i.e., T = T . Such heating may 1999),andweshalladoptitinthefollowing.Moreexplic- virial 6 not always produce the most obvious effects – remember itly, δ (z,Ω ,λ ) and σ(M,z)=σ (M) (D (z)/D (0)), c o o o g g × that it is the total thermal energy of the gas that counts, with D (z,Ω ,λ ) being the linear growthfactor.It is es- g o o and understanding the change of this quantity with heat- sentially through D that the dependence on cosmology g ing in a gravitationalpotential requires careful modeling. (Ω ,λ )entersthemassfunction,withΩ beingthemore o o o Although models studied so far do not lead to a strong importantofthetwoasthedependenceonλ isrelatively o effect (Metzler & Evrard 1994), we shall at times be dis- weak (see, e.g., Bartlett 1997 for a detailed discussion). cussing rather low mass systems, for which these effects This dependence on Ω in the exponent means that the o are poorly understood theoretically and observationally. clusterabundanceasafunctionofredshiftis averysensi- Finally,the exactformofthe virialtemperature–massre- tive probe ofthe density parameter(e.g., Oukbir& Blan- lationdependsinpartonthedarkmatterprofileofthecol- chard 1992, 1997), and is the motivation for many efforts lapsing proto–cluster; once again, numerical experiments in all wavebandsto find clusters at high redshifts.As em- seem to indicate that this does not change too much, i.e., phasized by several authors (Barbosa et al. 1996; Eke et J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection 5 al. 1996; Colafrancesco et al. 1997; Bartlett et al. 1998; Holder & Carlstrom 1999; Mohr et al. 1999), the SZ ef- fect is particularly well positioned in this arena (see also below). It is clear that the important theoretical variables are cluster mass and redshift. Although redshift is directly measurable,the massappearinginEq.(7)mustbe trans- lated into an observational quantity suitable for the type ofobservationsunder consideration.As mentionedabove, oneofthepleasantfeaturesofthe SZeffectisthe simplic- ity of this relation. Using the simulations of Evrardet al. (1996)to normalize the T M relation, we can quantita- − tively express the total SZ flux density of a cluster (e.g., Eqs. 4 & 6) as 1/3 ∆ (z) S =(34mJyh8/3)f (x)f Ω1/3 NL M5/3 ν ν gas o 178 15 (cid:20) (cid:21) (1+z)D−2(z) (8) where the mass M15 M/1015 M⊙ refers to the cluster ≡ virial mass and f is possibly a function of both mass gas and redshift (see also Barbosa et al. 1996, but note that Fig.1. a)Detection mass as a function of redshift for un- thedefinitionthereofD(z)differsbyafactorof2).Evrard resolved (dashed lines), optimal resolved (solid lines) and (1997) finds f = 0.06 h−1.5, while Mohr et al. (1998) gas standard resolved (dot–dashed lines) detection satisfying find marginal evidence for a decrease in lower mass sys- q σ =1.5 mJy at a wavelengthof 2 mm. In the unre- tems(see alsoCarlstrometal.1999forrecentworkbased det pix solved case, this simply corresponds to the limiting total on SZ images); there is little information on any possible flux density. For optimal resolveddetection, q refers to evolution with redshift at present. Other quantities ap- det q in Eq. (14), while for standard resolved detection it pearinginthisequationarethe meandensitycontrastfor opt refers to q of Eq. (16). The pixel size has been taken to virialization, ∆ (z,Ω ,λ ) (= 178 for Ω = 1, λ = 0), st NL o o o o be θ /2, and for the standard routine a detection an- and the dimensionless functions f and D(z) introduced fwhm ν gleθ =1/2θ hasbeenassumed,asindicated.Inall in Eqs. (2) and (5). det fwhm cases these parameters correspond to 3σ detections (see Observations for which clusters are unresolved mea- text for more detail). The upper (red) curves in eachcase surethis totalflux density,andthereforethis is allthatis correspond to the open model with Ω=0.3. neededinordertocalculatethe unresolvedsourcecounts, as we will do in the next section. For resolved sources, on the other hand, the detection criteria are more com- to the SZeffect.The physicsofthe SZ effectappearsonly plicated. Contrary to the point source limit, the details whenwemaketheconnectionbetweentheseempiricalpa- of the cluster SZ profile now play an important role. I rameters and the theoretically interesting ones, namely, will employ a simple isothermal β–model to describe this mass and redshift, via relations of the kind yo(M,z) and profile: θc(M,z). As our principle goal in this work is to under- standtheselectioneffectsofresolvedSZclusterdetection, y j (x) i (θ)= o ν (9) the modelfor cluster evolutionwill be keptsimple: a con- ν (1+θ2/θc2)α stant gas mass fraction, fgas = 0.06h−1.5 (Evrard 1997), over cluster mass and redshift, and a core radius scal- The exponent α = 0.5(3β 1), where β is the ex- − ing with the virial radius R , i.e., x R /r = const. ponent of the three–dimensional ICM density profile: v v v c ≡ n (1+r2/r2)−3β/2, r being the physical core radius. Unless otherwise specified, this constant will be ∝ c c given a value of 10. One deduces from simple scaling Local X–ray observations indicate that β 2/3, a value ∼ arguments that I adopt throughout for the calculations. In this case, α = 1/2, a rather significant value, as will be discussed 1/3 178 shortly. This profile will be assumed to hold out to the R =(1.69h−2/3 Mpc)M1/3(1+z)−1Ω−1/3 v 15 o ∆ (z) virial radius, R , of the cluster. (cid:18) NL (cid:19) v The β–profile of Eq.(9) is empirically described by y , where the normalization is taken from the spherical col- o asortofcentralsurfacebrightness(actually,itisy j that lapsemodel.Thisscalingrelationisaboutasrobustasthe o ν has units of surface brightness, but it is simpler to work relation for cluster temperature; in fact, the two are es- with y ), and θ . In these terms, there is nothing specific sentially the same,since T M/R . Some dependence of o c v ∼ 6 J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection Fig.1. b)Detection mass at 1 cm for the listed parame- Fig.1.c)Detectionmassat1cmandforΩ =1arcmin2 pix ters. The curves are labeled as in the previous figure, but (θ =2arcmins).Thecurvesarelabeledasinthe pre- fwhm note the change in scale along the y–axis. vious figures. Relative to Figure 1b, the lower resolution resultsinsmallerdetectionmasses(noteagainthechange in ordinate scale). The unresolved detection curves are thenormalizationonmassandredshiftcouldappearifthe unaffected by the change in resolution. density profile around a peak forming a cluster changed significantly with these two quantities. In the following, we shall ignore this possibility, which numerical simula- tions seem to indicate is a small effect in any case. This effectsimposedbyresolvedclusterdetection,andabench- then fixes the relation markforcomparingmoredetailedmodels.It is important in the following that one does not forget the model depen- r (M,z)=R (M,z)/x (10) c v v dence of our results, which can be retraced to this point of FortheaxiallysymmetricsurfacebrightnessofEq.(9), the discussion. theintegraldefiningthetotalSZfluxdensitymaybewrit- ten 4. Unresolved Detections S (M,z) = j 2π dθθ y(θ) ν ν Z This section is dedicated to the simple case of unresolved = 2πj y (M,z)θ2(M,z) 1+x2 1 SZ detection, which will be used as a reference in the fol- ν o c v− lowing discussion of resolved detection. It also offers an (cid:16)p (cid:17) Using Eq. (8) for S (M,z) in this expression, we deduce introductionto the mainfigures,Figures1, 2 and3,sum- ν marizingtheessentialresultsofthepresentwork.Theyare ∆ (z) y (M,z) = (6.40 10−5h2)f Ω NL constructed for two representative cosmologies: a critical o gas o × 178 (cid:18) (cid:19) modelΩo =1,andanopenmodel(λo =0)withΩo =0.3. x2 For the counts andredshift distributions of Figures 2 and M15(1+z)3 v (11) 3, I have used a CDM–like power spectrum with “shape 1+x2v−1! parameter”fixedatΓ=0.25;bothmodelsarenormalized Together with the β–propfile (Eq. 9), Eqs. (10) and tothepresentdayabundanceofX–rayclusters–σ8 =0.6 (11) define our cluster evolution model. As mentioned, andσ8 =1.0forthecriticalandopenmodels,respectively it is self–similar, and we see the expected scaling (e.g., Blanchard et al. 1999; Borganiet al. 1999; Viana & r M1/3/(1+z)−1 and y M(1+z)3. This is most Liddle 1999b). c o ∼ ∼ probably an oversimplified description of the actual clus- Observations for which most clusters are unresolved ter population, but it nevertheless provides a ‘standard’ measure the total SZ flux density. One can then simply withwhichwemayunderstandthenatureoftheselection invertEq.(8)tofindthecorrespondinglimiting detection J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection 7 mass as a function of redshift, Mur(z,S ): maximum emission of the effect is around 750µm). The det ν resolved detection mass limits, to be shortly discussed, 3/5 S depend also on the angular resolution. Mduert(z,Sν) = (0.12×1015h−8/5M⊙) mJνy Source counts for the two cosmological scenarios are (cid:18) (cid:19) 1/5 given in Figure 2. These have been calculated using Eq. 178 (f f )−3/5Ω−1/5 (13)andthe appropriatedetectionmass.In orderto shed ν gas o ∆ (cid:18) NL(cid:19) somelightontheimportanceoflowmassobjectstothese D6/5(z)(1+z)−3/5 (12) results, the counts are presented in pairs, one curve for a low mass cut–off of 1013 M⊙ and one for a cut–off of Integrating the mass function over redshift and over 1014 M⊙. Note that the x–axis denotes the pixel noise, masses greater than this limit directly yields the source σ ,andnotalimitingsourcefluxdensity;inthepresent pix counts: situationofunresolveddetections,thisjustmeansthatthe dN ∞ dV ∞ dn corresponding limiting flux density is qdet σpix. (>S )= dz dM (M,z) (13) × ν The first thing to remark from Figure 2 is the large dΩ dzdΩ dM Z0 ZMdet(z,Sν) differencebetweenthetwocosmologicalmodels.Thepres- Thecorrespondingredshiftdistributionissimplyobtained ence of clusters at high redshift in a low–density model as the integrand of the z–integral. shows up in the integrated counts, as confirmed by the Figure 1 compares the various detection masses as corresponding redshift distributions shown in Figure 3, a function of redshift for observations at 2 mm, e.g., a where the huge difference in cluster abundance at large bolometer array, and at 1 cm, representative of an inter- redshift is evident. It is for this reason that the redshift ferometer;in eachcase the upper (red) curve corresponds distributionofSZsourcesisapotentiallypowerfultoolfor to the open model. For the moment, concentrate only on constrainingΩo (Barbosaetal.1996,Bartlettetal.1998). the the dashed lines, which give the result for unresolved This is of foremost importance and represents one of the detection, Eq. (12), at a flux density of S = 1.5 mJy. primary motivating factors behind this type of survey. ν These curves remain unchanged from Figure 1b to 1c, Thissituationofunresolvedsourcesappliesinpractice both at 1 cm but differing in angular resolution, because to missions such as the Planck Surveyor,as discussed, for resolution is irrelevant for point sources (ignoring source example, by Barbosa et al. (1996) and Aghanim et al. confusion issues). Observe that in all cases the detection (1997).The higher angularresolutionof possible ground– mass decreases with redshift beyond z 1. This remark- based surveys calls for examination of resolvedsource de- ∼ able behavior is directly attributable to the fact that the tection. SZ surface brightness is independent of distance. As al- ready emphasized, the distance appearing in Eq. (12) is 5. Resolved Detections the angular distance and not the luminosity distance, a factor of (1 + z)2 larger. At high z the redshift depen- In this, the principle section of this paper, we treat in de- dence thereforescalesas z−9/5,one powercomingfrom tailtheissueofresolvedSZclusterdetection.Thecontext ∼ theassumedredshiftscalingofthevirialtemperatureand will be one of arcminute resolution (pixel size) and sub– the rest from the decrease in angular distance (focusing) mJy sensitivity, as targeted by the up–coming ground– as 1/z.Aself–similarclustermodel,implicitly assumed based instruments. It is worth being very explicit about ∼ in this context by the constancy of f , thus predicts the nature of the observations:the simplest case to imag- gas thatSZobservationsaremoresensitivetoobjects atlarge, inecorrespondstothatofanimageproducedbyabolome- rather than intermediate, redshifts. This overall behavior ter array, such as BOLOCAM. In this case each point on wouldnotchangeevenifwebrokethe self–similaritywith the image, a ‘pixel’, represents a sample point of the sky a declining gas mass fraction with mass; such a depen- brightness, as transformed by the optics of the observing dence could only modify the rate of decrease with z. On system. The optical response may be divided into that the other hand, an explicit decrease in f with redshift ofthe telescope–plus–atmosphere(defining the projection gas stronger than (1+z)−3 would cause Mur to actually in- of the sky onto the focal plane) and the optics proper to det crease with redshift. It is perhaps not so surprising that the detector (which act on the focal–plane image). There atcloserange,smallz,thedetectionmassalsodrops;this is a difference between bolometer arrays and the familiar is simply due to the increasing angular size of the object example of a CCD camera working in the visible. For the creating an increase in total flux density (the source is latter,atmosphericseeingandtelescopeopticsprojectthe assumed to always remain unresolved in this discussion). sky onto the focal plane by convolving with a Gaussian, From the difference between Figures 1a and 1b,c, we and the camera itself then convolves this focal–plane im- see that, at a given sensitivity, the 2 mm observations age with a square top–hat, one centered on each pixel. probefartherdowninmass.Thisisnothingmorethanthe The difference withabolometer arraylies inthe factthat spectral shape of the SZ effect, described by the function the CCD camera defines sharp, well–defined pixel bound- j :thebiggestdecrementoccurspreciselynear2mm(the aries, while a bolometer array, with its set of cones, con- ν 8 J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection volves the focal–plane image with something closer to a Gaussian. This means that, unlike CCDs, the pixels of a bolometer array ‘overlap’ in the focal plane. This has lit- tleconsequencefortheensuingdiscussion,butitisallthe same worth keeping in mind. This picture is notcompletely accuratewhen it comes to interferometers. Such instruments actually directly sample the Fourier transform of the sky. The result may often be modeled by a real sky image convolved with an effective, synthesized beam, but this beam lacks sensitiv- ity on large scales, i.e., large spatial wavelengths on the sky(shortbaselines).Thus,theeffectivebeamcannotnot be precisely a Gaussian, and it is especially important to correctly model the loss of response on large scales for extended objects such as clusters.For the ensuing discus- sion, I adopt the bolometer picture, applying it at times rather indiscriminately to characterize ground–based ob- servations; a future work will consider the details specific to interferometric observations (see also the recent work of Holder et al. 1999). Forabolometerarray,theresponseoftheentireoptical Fig.2. a) Cluster integral source counts at 2 mm as a chain(atmosphere-telescope-detector)is often adequately functionofmappixel noiseforthe twocosmologicalmod- modeled as a bi–dimensional Gaussian (if one is lucky, els introduced in the text. The angular resolution and a symmetric one!), and for proper sampling, respecting sampling correspond to the situation of Figure 1a. Unre- Shannon, the sample period must be 2 – 3 times smaller solved,optimal resolvedandstandardresolvedcounts are than the beam FWHM. We will characterize a survey by shown, respectively, as the dashed, solid and dot–dashed thepixelsizeandsensitivityperpixelofitsimages–Ω , pix lines;theupper(red)curveineachcasecorrespondstothe a solid angle, and σ , a flux density. Note that because pix open model with Ω = 0.3. For unresolved detections, the the pixels ‘overlap’ in the focal plane, what precisely is limiting source flux density is simply q (pixel noise). meant by Ω is the square of the separation between det pix × Thelightdottedlinesinthebackgroundindicatethecrit- sample points, θ ; the concept is a bit more ambiguous pix ical slope of 2. The fact that the resolved counts are thaninthecaseofaCCDcamera.Thus,propersampling − means that the pixel scale Ω θ2 θ2 /4. It is steeper than this value implies that, down to low noise pix ≡ pix ≤ fwhm levels,deepintegrationsyieldmoreobjectsthanwideand also worth explicitly remarking that, in the following, I shallow ones. assumethatthe noiseis uncorrelated(frompixeltopixel) and uniform over the image. Given, then, a map of the SZ sky, we would like to Theprocedureinthefollowingisthenalwaysthesame: understand how to extract clusters and the nature of the quantifythedetectionalgorithmintermsofΩ andσ , pix pix selectionimposedbyourtechnique.Inadditiontotheob- and then translate this, via the isothermal β–model, into servational parameters Ω and σ , this will depend on a M (z;Ω ,σ ). I employ a notation where the im- pix pix det pix pix theformoftheextendedemissionofthesources,acompli- minently interesting independent variables of a function cationavoidedin the case ofunresolvedcluster detection; appearbeforethe“;”,andparameterizingonesafterward. this represents an important difference between the two Thus, as written, the detection mass is primarily a func- situations.Employingtheβ–modelintroducedpreviously, tion of redshift, parameterized by the survey properties Eq. (9), we see that a cluster SZ profile may be described Ω andσ .Thisfunctionteachesusaboutthe kindsof pix pix by a characteristic central surface brightness, y , and an objects we detect, and leads directly to the survey counts o angularsize, θ (the coreradius). When couchedin terms and the redshift distribution of our clusters, via Eq. (13). c ofthepurelyempiricalparametersofΩ ,σ ,y andθ , Theselatterquantitiesarethekeyindicatorsofthescience pix pix o c we have before us a rather classic and well–known prob- content of the survey. lemofAstronomy.Theonlydifferencewithgalaxiesinthe This procedure will be applied to two source extrac- optical is the form of the source profile. All physics spe- tion methods in the following, and the results compared cific to the SZ effect itself appears only in the relation of to those for an unresolved SZ survey. We will refer to the the empirical source descriptors – (y ,θ ) – to the theo- firstas“optimaldetection”,becauseitextractssourcesin o c reticallymeaningfulonesofclustermass,M,andredshift, such a way as to preserve the signal–to–noise across the z. entire range of detectable surface brightness and source J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection 9 size. This is achieved by lowering the surface brightness limit for large sources, possible due to the greater num- ber of covered pixels. The second method, routinely used by suchpackagesas SExtractor(Bertin & Arnouts 1996), searchesforaminimumnumberofconnectedpixelsabove apresetthreshold.Theimportantdifferencewiththefirst technique is the imposition of a fixed surface brightness limit, independent of source size. The signal–to–noise of the detections is no longer constant, but increases with sourcesize.Thistechniquemaybeconsideredsub–optimal in the sense that it loses in–principle detectable low sur- facebrightnesssources,afactwellappreciatedinthe case of optical galaxy surveys. 5.1. Optimal case Optimal detection selects all sources with a flux density S q N1/2σ ν opt pix ≥ (assumingspatiallyuncorrelatedanduniformnoise)where N is the number of pixels covered by the cluster and Fig.2. b) Same as the previous figure for an observation qopt represents a threshold, say qopt 3 5; in fact, ∼ − wavelength of 1 cm; curve types have the same meaning q = S/N, the signal–to–noise of the detection. Notice opt as before (the labels ‘Optimal’ and ‘Standard’ have been alsothat,asadvertised,thelimitingsurfacebrightnessde- removed for clarity). The resolution and sampling corre- creases with object size: < iν > Sν/N qoptσpix/√N. ∼ ∼ spond to the situation of Figure 1b. One extracts in this way all objects detectable at a given S/N, and for this reason we may refer to the method as optimal. The number of object pixels is simply found as N = πθ2 /Ω , where θ = R /D is angular virial vir pix vir v ang radius. This permits us to express the detection mass as q σ 3/4 Mdoeptt(z,Sν) = (0.19×1015h−2M⊙) omptJypix (cid:18) (cid:19) arcmin2 3/8 (f f )−3/4Ω−1/2 Ω ν gas o (cid:18) pix (cid:19) 178 1/2 D3/4(z)(1+z)−3/2 (14) ∆(z) (cid:18) (cid:19) As written, this criteria uses an aperture correspond- ing to the full angular size of the object – S is under- ν stood to be the total SZ flux density in Eq. (8). For re- solved sources,one would like to chose an aperture which optimizes the signal–to–noise ratio of the detection. In- terestingly, a 3D gas profile close to r−2, corresponding to a SZ surface brightness y θ−1, results in a con- ∝ stant signal–to–noise with aperture radius. A β–model with n (1+r2/r2)−3β/2 and β 2/3 exhibits this be- haviora∝t largeradici, for example:∼y(θ) (1+θ2/θ2)−1/2. Fig.2. c) Same as Figure 2b (λ = 1 cm) but now for ∼ c Inthiscase,thesignal–to–noiseofaSZdetectionincreases Ω = 1 arcmin2, i.e., the situation of Figure 1c. The pix fromthecenteroftheclusterimageouttothecoreradius, smaller detection masses at this lower resolution result in r , beyond which it turns over to a constant out to the c highercountswhencomparedtoFigure2b.Note thatthe virial radius. The situation is different for X–ray images, unresolved counts are the same here as in Figure 2b. where the surface brightness falls off more rapidly, diving under the background at large radii. From this we con- clude that the simple criteriagivenaboveprovidesin fact 10 J.G. Bartlett: Sunyaev-Zel’dovichSurveys:Analytictreatment of cluster detection Fig.3. a) Redshift distribution of the integrated counts Fig.3. b) Redshift distribution of the integrated counts for a flux density of q σ = 1.5 mJy at 2 mm. The for a flux density of q σ = 1.5 mJy at 1 cm, and for det pix det pix parameters are the same as those in Figure 2a, and the the same parameters as in Figure 2b. line types have the same meaning. The upper (red) curve ineachcasecorrespondstotheopenmodelwithΩ =0.3. o Note the large difference between the two cosmological is traceable to the fact that the flux density of a source is models apparent in all cases. dispersedoverfewer(noisy)pixelsthanwouldbethe case atahigherangularresolution.Thisindicatesthatlowres- olutionobservationsata givenwavelengthandsensitivity an optimum SZ detection (at least as long as β remains are to be prefered, at least for detection purposes. There close to 2/3, as appears to be the case locally). is, however,a limit set by eventual source confusion. ThedetectionmassEq.(14)isdisplayedinFigure1as thesolidlines.Comparedtothehypotheticalpointsource We have just seen from Figure 1 that low surface results, observations resolving clusters are less efficient brightness clusters are “resolved out” at high resolution. at detecting clusters, especially at intermediate redshifts. This leads to overall lower counts that are also much This is easy to understand as the effect of distributing a steeper than the equivalent for unresolved point sources. givenflux density overN pixels, each adding a noise with Generally speaking, the unresolved counts do not deviate variance of σpix, resulting in a total noise level over the too much from a Euclidean law, ∝ Sν−3/2; on the other object image of √Nσ . A point source, in contrast, is hand, the resolved counts can be much steeper. The ex- pix only subject to the noise of one pixel, σpix. Hence, high amplesshowninFigure2areinfactsteeperthanSν−2,in- resolutionatfixedsensitivity“resolvesout”acertainfrac- dicatedbythedottedlines,downtoessentiallythefaintest tionofobjects.Theconsequencesforthesourcecountsare flux levels attainable in immediately foreseeable observa- clear and will be discussed shortly. These curves retain tions. This is critical for optimizing an observing strat- the sameasymptoticbehaviorasbefore,namelya greater egy with a fixed amount telescope time, T. Consider the sensitivitytolowmassesathighredshift.Despite the fact common situation in which the final map noise decreases that the object covers a larger number of noisy pixels as with integration time as 1/√t; then, the solid angle cov- z decreases, the optimal method is able to take proper ered in time T, with individual field integrations of du- advantage of the greater total flux density to detect low ration t, scales with sensitivity as T/t σ2 . Hence, ∼ ∼ pix massobjectslocally,justasinthecaseofunresolvedpoint if the integrated source counts are steeper than σ−2, one pix sources.Weshallseethatthisdoesnotfollowforthestan- gains objects by “going deep”, integrating longer on each dard detection routine (the dot–dashed lines), due to its individual field, rather than “going wide”, with shorter additionalsurfacebrightnessconstraint(discussedbelow). integrations covering a larger total solid angle. The im- BycomparingFigures1band1c,whichdifferonlyintheir portantconclusionto drawfromFigure2 is thenthat the angular resolution, we note that for a given sensitivity, way to optimize the number of detected objects in a sur- lower resolution observations are the more effective. This vey with arcminute resolution is by “going deep”, down