Mon.Not.R.Astron.Soc.418,1526–1556(2011) doi:10.1111/j.1365-2966.2011.19387.x Dark matter profiles and annihilation in dwarf spheroidal galaxies: γ prospectives for present and future -ray observatories – I. The classical dwarf spheroidal galaxies (cid:2) A. Charbonnier,1 C. Combet,2 M. Daniel,3 S. Funk,4 J. A. Hinton,2 (cid:2) (cid:2) D. Maurin1,2,5,6 C. Power,2,7 J. I. Read,2,8 S. Sarkar,9 M. G. Walker10,11 and M. I. Wilkinson2 1LaboratoiredePhysiqueNucle´aireetHautesEnergies,CNRS-IN2P3/Universite´sParisVIetParisVII,4PlaceJussieu,Tour33,75252ParisCedex05, France 2DepartmentofPhysicsandAstronomy,UniversityofLeicester,LeicesterLE17RH D 3DepartmentofPhysics,DurhamUniversity,SouthRoad,DurhamDH13LE o w 4W.W.HansenExperimentalPhysicsLaboratory,KavliInstituteforParticleAstrophysicsandCosmology,DepartmentofPhysicsandSLACNational n lo AcceleratorLaboratory,StanfordUniversity,Stanford,CA94305,USA a d 5LaboratoiredePhysiqueSubatomiqueetdeCosmologie,CNRS/IN2P3/INPG/Universite´JosephFourierGrenoble1,53avenuedesMartyrs,38026 ed Grenoble,France fro 6Institutd’AstrophysiquedeParis,UMR7095CNRS,Universite´PierreetMarieCurie,98bisbdArago,75014Paris,France hm 7InternationalCentreforRadioAstronomyResearch,TheUniversityofWesternAustralia,35StirlingHighway,Crawley,WA6009,Australia ttp 8DepartmentofPhysics,InstituteforAstronomy,ETHZu¨rich,Wolfgang-Pauli-Strasse16,CH-8093Zu¨rich,Switzerland ://m 9RudolfPeierlsCentreforTheoreticalPhysics,UniversityofOxford,1KebleRoad,OxfordOX13NP n 10InstituteofAstronomy,UniversityofCambridge,MadingleyRoad,CambridgeCB30HA ras .o 11Harvard–SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA x fo rd jo u rn Accepted2011July5.Received2011July1;inoriginalform2011April3 a ls .o rg a/ ABSTRACT t L e Due to their large dynamical mass-to-light ratios, dwarf spheroidal galaxies (dSphs) are ice s promisingtargetsfortheindirectdetectionofdarkmatter(DM)inγ-rays.Weexaminetheir ter U detectability by present and future γ-ray observatories. The key innovative features of our n iv analysisareasfollows:(i)wetakeintoaccounttheangularsizeofthedSphs;whilenearby ers objectshavehigherγ-rayflux,theirlargerangularextentcanmakethemlessattractivetargets ity L for background-dominated instruments; (ii) we derive DM profiles and the astrophysical J- ib ra factor(whichparametrizestheexpectedγ-rayflux,independentlyofthechoiceofDMparticle ry o model)fortheclassicaldSphsdirectlyfromphotometricandkinematicdata.Weassumevery n N o littleabouttheDMprofile,modellingthisasasmoothsplit-power-lawdistribution,withand v e m withoutsubclumps;(iii)weuseaMarkovchainMonteCarlotechnique tomarginalizeover b e unknownparametersanddeterminethesensitivityofourderivedJ-factorstobothmodeland r 1 1 measurement uncertainties; and (iv) we use simulated DM profiles to demonstrate that our , 2 0 1 J-factordeterminationsrecoverthecorrectsolutionwithinourquoteduncertainties. 5 Ourkeyfindingsareasfollows:(i)subclumpsinthedSphsdonotusefullyboostthesignal; (ii)thesensitivityofatmosphericCherenkovtelescopestodSphswithin∼20kpcwithcored haloes can be up to ∼50 times worse than when estimated assuming them to be point-like. Evenforthesatellite-borneFermi-LargeAreaTelescope(Fermi-LAT),thesensitivityissignif- icantlydegradedontherelevantangularscalesforlongexposures;hence,itisvitaltoconsider theangularextentofthedSphswhenselectingtargets;(iii)noDMprofilehasbeenruledout bycurrentdata,butusingapriorontheinnerDMcuspslope0≤γ ≤1providesJ-factor prior estimatesaccuratetoafactorofafewifanappropriateangularscaleischosen;(iv)theJ-factor (cid:2) E-mail:[email protected](JAH);[email protected](DM);[email protected](MGW) (cid:4)C 2011TheAuthors MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS γ-rayfromdarkmatterannihilation indSphs 1527 isbestconstrainedatacriticalintegrationangleα =2r /d(wherer isthehalf-lightradius c h h andd isthedistancefromthedwarf)andweestimatethecorrespondingsensitivityofγ-ray observatories;(v)the‘classical’dSphscanbegroupedintothreecategories:wellconstrained andpromising(UrsaMinor,SculptorandDraco),wellconstrainedbutlesspromising(Carina, Fornax and Leo I), and poorly constrained (Sextans and Leo II); and (vi) observations of classical dSphs with the Fermi-LAT integrated over the mission lifetime are more promis- ing than observations with the planned Cherenkov Telescope Array for DM particle mass (cid:2) 700 GeV. However, even the Fermi-LAT will not have sufficient integrated signal from theclassicaldwarfstodetectDMinthe‘vanilla’MinimalSupersymmetricStandardModel. BoththeGalacticCentreandthe‘ultrafaint’dwarfsarelikelytobebettertargetsandwillbe consideredinfuturework. Key words: astroparticle physics – methods: miscellaneous – galaxies: dwarf – galaxies: kinematicsanddynamics–darkmatter–gamma-rays:general. D o w n lo a d e d fro m etal.2007;Bringmann,Doro&Fornasa2009;Pierietal.2009a; 1 INTRODUCTION h Pieri, Lattanzi & Silk 2009b). Other authors use a ‘cosmologi- ttp Thedetectionofγ-raysfromdarkmatter(DM)annihilationisoneof calprior’fromlarge-scalecosmologicalsimulations(e.g.Kuhlen ://m themostpromisingchannelsforindirectdetection(Gunnetal.1978; 2010). Both approaches may be combined, such as in Strigari nra s Stecker1978).SincethesignalgoesastheDMdensitysquared,the et al. (2007b) and Martinez et al. (2009) who rely partially on .o x GalacticCentreseemstobetheobviouslocationtosearchforsuch the results of structure formation simulations to constrain the in- fo asignal(Silk&Bloemen1987).However,itisplaguedbyaconfus- ner slope and then perform a fit to the data to derive the other rdjo u ingbackgroundofastrophysicalsources(seee.g.Aharonianetal. parameters.However,suchcosmologicalpriorsremainsufficiently rn a 2004).Forthisreason,thedwarfspheroidalgalaxies(dSphs)orbit- uncertain that their use is inappropriate for guiding observational ls .o ingtheMilkyWayhavebeenflaggedasfavouredtargets,giventheir strategies.Therehavebeenonlyafewstudies(e.g.Essig,Sehgal rg potentiallyhighDMdensitiesandsmallastrophysicalbackgrounds & Strigari 2009) which have not assumed strong priors for the a/ t L (Lake1990;Evans,Ferrer&Sarkar2004). DMprofiles. e ic Despitethegrowingamountofkinematicdatafromtheclassical Inthiswork,werevisitthequestionofthedetectabilityofDMan- e s dSphs, the inner parts of their DM profiles remain poorly con- nihilationintheclassicalMilkyWaydSphs,motivatedbyambitious ter U strainedandcangenerallyaccommodatebothcoredorcuspysolu- plansfornext-generationACTssuchastheCherenkovTelescope n iv tions(seee.g.Kochetal.2007;Strigarietal.2007a;Walkeretal. Array(CTA).Werelysolelyonpublishedkinematicdatatoderive e rs 2009a,hereinafterW09).TherearetwodSphs–FornaxandUrsa thepropertiesofthedSphs,makingminimalassumptionsaboutthe ity Minor–thatshowindirecthintsofacoreddistribution(Kleynaetal. underlying DM distribution. Most importantly, we do not restrict L ib 2003;Goerdtetal.2006);however,inbothcases,thepresenceofa oursurveyofDMprofilestothosesuggestedbycosmologicalsim- ra ry coreisinferredbasedonatimingargumentthatassumeswearenot ulations. We also consider the effect of the spatial extent of the o n catching the dSph at a special moment. Theoretical expectations dSphs,whichbecomesimportantfornearbysystemsobservedby N o remain similarly uncertain. Cusps are favoured by cosmological background-limitedinstrumentssuchasACTs. v e m modelsthatmodeltheDMalone,assumingitiscoldandcollision- ThispaperextendstheearlierstudyofWalkeretal.(2011)which b e less(e.g.Navarro,Frenk&White1996a).However,thecomplex showedthatthereisacriticalintegrationangle(twicethehalf-light r 1 dynamicalinterplaybetweenstars,gasandDMduringgalaxyfor- radiusdividedbythedSphdistance)wherewecanobtainarobust 1, 2 mationcoulderasesuchcuspsleadingtocoreddistributions(e.g. estimateoftheJ-factor(thatparametrizestheexpectedγ-rayflux 01 5 Navarro,Eke&Frenk1996b;Read&Gilmore2005;Mashchenko, from a dSph, independently of the choice of DM particle model; Wadsley&Couchman2008;Goerdtetal.2010;Governatoetal. seeSection2),regardlessofthevalueofthecentralDMcuspslope 2010; Cole, Dehnen & Wilkinson 2011). Cores could also be an γ. Here, we focus on the full radial dependence of the J-factor. indication of other possibilities such as self-interacting DM (e.g. WeconsidertheeffectofDMsublumpswithinthedSphs,discuss Hogan&Dalcanton2000;Mooreetal.2000). whichdSphsarethebestcandidatesforanobservingprogramme, Knowledge of the inner slope of the DM profile is of critical andexaminethecompetitivenessofnext-generationACTsasDM importance as most of the annihilation flux comes from that re- probes. gion. Lacking this information, several studies have focused on This paper is organized as follows. In Section 2, we present a the detectability of these dSphs by current γ-ray observatories studyoftheannihilationγ-rayflux,focusingonwhichparameters such as the satellite-borne Fermi-Large Area Telescope (Fermi- critically affect the expected signal. In Section 3, we discuss the LAT) and atmospheric Cherenkov telescopes (ACTs) such as the sensitivityofpresent/futureγ-rayobservatories.InSection4,we HESS, MAGIC and VERITAS, using a small sample of cusped present our method for the dynamical modelling of the observed and cored profiles (generally one of each). Most studies rely on kinematicsofstarsindSphs.InSection5,wederiveDMdensity standard core and cusp profiles fitted to the kinematic data of profilesfortheclassicaldSphsusingaMarkovchainMonteCarlo the dSph of interest (Bergstro¨m & Hooper 2006; Sa´nchez-Conde (MCMC) analysis, from which the detection potential of future (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS 1528 A.Charbonnieretal. γ-rayobservatoriescanbeassessed.Wepresentourconclusionsin Inthisframework,theneutralinoistypicallythelighteststablepar- Section6.1 ticle and therefore one of the most favoured DM candidates (see Thispaperincludesdetailedanalysesfrombothhigh-energyas- e.g. Bertone, Hooper & Silk 2005). A γ-ray continuum is pro- trophysicsandstellardynamicalmodelling.Toassistreadersfrom duced from the decay of hadrons (e.g. π0 → γγ) resulting from thesedifferentfieldsinnavigatingthekeysections,wesuggestthat theDMannihilation.Neutralinoannihilationcanalsodirectlypro- thosewhoareprimarilyinterestedinthehigh-energycalculations ducemono-energeticγ-raylinesthroughloopprocesses,withthe may wish to focus their attention on Sections 2, 3 and 5 before formationofeitherapairofγ-rays(χχ→γγ;Bergstro¨m&Ullio movingtotheconclusions.Readersfromthedynamicscommunity 1997),oraZ0bosonandaγ-ray(χχ →γZ0;Ullio&Bergstro¨m mayinsteadprefertoreadSections2,4and5.Finally,thosewho 1998).Wedonottakeintoaccountsuchline-productionprocesses arewillingtotrusttheunderlyingmodellingshouldproceedtoSec- since they are usually subdominant and very model-dependent tion5whereourmainresultsregardingthedetectabilityofdSphs (Bringmann, Bergstro¨m & Edsjo¨ 2008). The differential photon arepresentedinFigs12,15,16and17(shownlater). spectrumweuseisrestrictedtothecontinuumcontributionandis writtenas 2SITGHNEALD:AKREKYMPAATRTAEMREATNENRISHILATION ddNEγγ(Eγ)=(cid:2)i bi ddNEγγi(Eγ,mχ), (3) D o w 2.1 Theγ-rayflux wbrhaenrcehtihnegdraiftfioerbeni.tannihilationfinalstatesiarecharacterizedbya nload Theγ-rayflux(cid:5)γ(photonscm−2s−1GeV−1)fromDMannihilation Using the parameters in Fornengo, Pieri & Scopel (2004), we ed inadSph,asseenwithinasolidangle(cid:6)(cid:7),isgivenby(seeAppendix plotthecontinuumspectracalculatedfora1-TeVmassneutralino fro m Afordefinitionsandconventionsusedintheliterature) inFig.1. h dd(cid:5)Eγγ(Eγ,(cid:6)(cid:7))=(cid:5)pp(Eγ)×J((cid:6)(cid:7)). (1) lγa-triAoapynascrt(hdafarnosnhmeeldsthlieinnτeth+s)eτ.−cFoocnrhtiacnnhunauermgle(ddreoastnu–nldtiahisnihlaevtdeiorlyninspeir)mo,dailulalcrtthss,peeiancntternarinhoai-fl ttp://mnra s The first factor encodes the (unknown) particle physics of DM bremsstrahlung(IB)hasrecentlybeeninvestigatedandfoundtoen- .o x annihilationwhichwewishtomeasure.Thesecondfactorencodes hancethespectrumclosetothekinematiccut-off(e.g.Bringmann fo the astrophysics viz. the line-of-sight (l.o.s.) integral of the DM etal.2008).Asanillustration,thelong-dashedlineinFig.1cor- rdjo u densitysquaredoversolidangle(cid:6)(cid:7)inthedSph–thisiscalledthe respondstothebenchmarkconfigurationforawino-likeneutralino rn a ‘J-factor’.Wenowdiscusseachfactorinturn. takenfromBringmannetal.(2008).However,theshapeandam- ls .o plitudeofthisspectrumarestronglymodel-dependent(Bringmann rg et al. 2009) and, as argued in Cannoni et al. (2010), this contri- at L/ 2.1.1 Theparticlephysicsfactor butionisrelevantonlyformodels(andatenergies)wheretheline e ic Theparticlephysicsfactor((cid:5)pp)isgivenby contributionisdominantoverthesecondaryphotons. este Wewishtobeasmodel-independentaspossible,andsodonot r U (cid:5)pp(Eγ)≡ dd(cid:5)Eγγ = 41π(cid:6)σ2amnn2χv(cid:7) × ddNEγγ , (2) cboenbsaidseerdIoBn.aInnatvheeraregmeasipnedcetrruomfttahkisenpafrpoemr, tahlleopuarrarmeseutrlitzsatwioilnl nivers ity cwtirohones,rse-asnmedcχtdiioNsntγh/aednEmdγa(cid:6)sσissaotnhnfveth(cid:7)deiisfDfetMhreenpataviraetlircaplgheeo,σtooavnnenyriisietilstdsvpseeellorfc-aaintnynnidihhiisillatartitibioounn-. Library o A benchmark value is (cid:6)σannv(cid:7) ∼ 3 × 10−26cm3s−1 (Jungman, n N Kamionkowski & Griest 1996), which would result in a present- ov e dayDMabundancesatisfyingcosmologicalconstraints. m b Unliketheannihilationcross-sectionandparticlemass,thedif- er 1 ferentialannihilationspectrum[dNγ/dEγ(Eγ)]requiresustoadopt 1, 2 aspecificDMparticlemodel.Wefocusonawell-motivatedclass 0 1 ofmodelsthatarewithinreachofupcomingdirectandindirectex- 5 periments:theMinimalSupersymmetricStandardModel(MSSM). 1Technicaldetailsaredeferredtoappendices.InAppendixA,wecomment on the various notations used in similar studies and provide conversion factorstohelpcompareresults.InAppendixB,weprovideatoymodel forquickestimatesoftheJ-factor.InAppendixD,wecalculateinamore Figure1. Differentialspectra(multipliedbyx2)ofγ -raysfromthefrag- systematic fashion the range of the possible ‘boost factor’ (due to DM mentationofneutrinoannihilationproducts(hereforaDMparticlemassof clumpswithinthedSphs)forgenericdSphs.InAppendixE,weshowthat mχ =1TeV).Severaldifferentchannelsareshown,takenfromFornengo convolvingthesignalbythepointspreadfunction(PSF)oftheinstrument et al. (2004), and an average parametrization (Bergstro¨m et al. 1998) is isequivalenttoacruderquadraturesumapproximation.InAppendixF,we markedbytheblacksolidline;thisiswhatweadoptthroughoutthispaper. discusssometechnicalissuesrelatedtoconfidencelevel(CL)determination TheblackdashedlineisthebenchmarkmodelBM4(Bringmannetal.2008) from the MCMC analysis. In Appendix G, the reconstruction method is whichincludesIBandservestoillustratethatverydifferentspectraarepos- validatedonsimulateddSphs.InAppendixH,wediscusstheimpactofthe sible.However,theexampleshownhereisdominatedbylineemissionand choiceofthebinningofthestarsandoftheshapeofthelightprofileonthe thereforehighlymodel-dependent;forthisreason,wedonotconsidersuch J-factordetermination. effectsinthispaper. (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS γ-rayfromdarkmatterannihilation indSphs 1529 (Bergstro¨metal.1998;solidlineinFig.1): available data and that the J-factor does not correlate with β, we dNγ = 1 dNγ = 1 0.73e−7.8x , (4) choFoosrepnroofitlteosasdudchfuarsthγer≥sh1a.5p,ethpearqaumanettietrys.Jfromtheinnerregions dEγ mχ dx mχ x1.5 diverges.Thiscanbeavoidedbyintroducingasaturationscaler sat with x ≡ Eγ/mχ. Finally, in order to be conservative in deriving that corresponds physically tothe typical scalewhere the annihi- detection limits, we also do not consider the possible ‘Sommer- lationrate[(cid:6)σv(cid:7)ρ(rsat)/mχ]−1 balancesthegravitationalinfallrate feld enhancement’ of the DM annihilation cross-section (Hisano, of DM particles (Gρ¯)−1/2 (Berezinsky, Gurevich & Zybin 1992). Matsumoto&Nojiri2004;Hisanoetal.2005).2 Thisdependsin- Takingρ¯ tobeabout200timesthecriticaldensitygives verselyontheDMparticlevelocity,andthusrequiresprecisemod- (cid:8) (cid:9) (cid:10) (cid:11) ellingofthevelocitydistributionoftheDMwithinthedSph;we ρ ≈3×1018 mχ × 10−26cm3s−1 M(cid:12)kpc−3. sat 100GeV (cid:6)σv(cid:7) willinvestigatethisinaseparatestudy. (7) Theassociatedsaturationradiusisgivenby (cid:10) (cid:11) 2.1.2 TheJ-factor ρ 1/γ r =r s (cid:13)r . (8) Thesecondterminequation(1)istheastrophysicalJ-factorwhich sat s ρ s sat dependsonthespatialdistributionofDMaswellasonthebeam D Thislimitisusedforallofourcalculations. o size. It corresponds to the l.o.s. integration of the DM density w n squaredoversolidangle(cid:6)(cid:7)inthedSph: lo a (cid:3) (cid:3) d 2.2 Motivationforagenericapproachandreferencemodels e d J = (cid:6)(cid:7) ρD2M(l,(cid:7))dld(cid:7). (5) Inmanystudies,theγ-rayflux(fromDMannihilation)iscalculated from (cid:6)Th(cid:7)es=ol2idπa[1ng−leciosss(iαminpt)l]y.relatedtotheintegrationangleαintby u2st0sei0en6pg;,itKnheuwhphleoicnihnt2c-0aso1seu0r)thc.eeTthaopitsaplrisoluxvmiamilniadotisosiotnyl(ooefn.gtgh.eaBdseStrphghestirison¨dmnoemr&pinrHaotfioeoldepbeiysr http://mn TheJ-factorisusefulbecauseitallowsustorankthedSphsbytheir averysmallcentralregion.However,iftheprofileisshallowand/or ras.o expected γ-ray flux, independently of any assumed DM particle thedSphisnearby,theeffectivesizeofthedSphontheskyislarger xfo physicsmodel.Moreover,theknowledgeoftherelativeJ-factors than the PSF of the detector, and the point-source approximation rd wouldalsohelpustoevaluatethevalidityofanypotentialdetection breaks down. For upcoming instruments and particularly shallow jou of a given dSph, because for a given particle physics model, we DMprofiles,theeffectivesizeofthedSphmayevenbecomparable rnals coothueldrdthSepnhss.calethesignaltowhatweshouldexpecttoseeinthe teoxttehnetfiofeltdheosfigvnieawldoofesthmeaitntsetrriunmteernmt.sTohfidsedteifcfteiroenn(cseeeinStehcetiroand3ia)l. a.org/ All calculations of J presented in this paper were performed Hence,wedonotassumethatthedSphisapointsourcebutrather t L usingthepubliclyavailableCLUMPYpackage(Charbonnier,Combet deriveskymapsfortheexpectedγ-rayflux. eice s &Maurin,2011)whichincludesmodelsforasmoothDMdensity te profileforthedSph,clumpyDMsubstructuresinsidethedSph,and r U asmoothandclumpyGalacticDMdistribution.3 2.2.1 Illustration:acoredversuscuspedprofile niv e Fig.2showsJasafunctionoftheintegrationangleαintforadSph rsity at 20kpc (looking towards its centre). The black solid line is for L 2.1.3 DMprofiles a cored profile (γ = 0) and the green dashed line is for a cuspy ibra FortheDMhalo,weuseageneralized(α,β,γ)Hernquistprofile profile(γ =1.5);botharenormalizedtounityatαint=5◦.Forthe ry o givenby(Hernquist1990;Dehnen1993;Zhao1996) cuspyprofile,∼100percentofthesignalisinthefirstbin,while n N (cid:4)r(cid:5)−γ(cid:6) (cid:4)r(cid:5)α(cid:7)γ−αβ forthecoredprofile,J buildsupslowlywithαint,and80percent ovem ρ(r)=ρs rs 1+ rs , (6) ber 1 1 where the parameter α controls the sharpness of the transition , 20 from inner slope, limr→0dln(ρ)/dln(r) = −γ, to outer slope 15 limr→∞dln(ρ)/dln(r) = −β, and rs is a characteristic scale. In principle,wecouldaddanadditionalparameterinordertointro- duceanexponentialcut-offintheprofileofequation(6)tomimic the effects of tidal truncation, as proposed in, for example, the Aquarius (Springel et al. 2008) or Via Lactea II (Diemand et al. 2008)simulations.However,thefreedomtovarytheparametersr , s αandβ inequation(6)alreadyallowsfordensityprofilesthatfall arbitrarilysteeplyatlargeradius.Moreover,giventhatourMCMC analysislatershowsthattheouterslopeβ isunconstrainedbythe 2Thiseffectdependsonthemassandthevelocityoftheparticle;theresult- ingboostofthesignalandtheimpactondetectabilityofthedSphshasbeen Figure2. Finitesizeeffects:Jasafunctionoftheintegrationangleαintfor discussed,forexample,inPierietal.(2009b). adSphat20kpc(pointingtowardsthecentreofthedSph).Theblacksolid 3InAppendixB,weprovideapproximateformulaeforquickestimatesof lineisforacoredprofile(γ =0)andthegreendashedlineisforacuspy theJ-factorandcross-checkswiththenumericalresults. profile(γ =1.5);botharenormalizedtounityatαint=5◦. (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS 1530 A.Charbonnieretal. Table 1. The required normaliza- barsisrecovered.Wealsostudybelowtheeffectofmovingthese tionρstohaveM300=107M(cid:12)for dSphsfromadistanceof10kpcto300kpc,correspondingtothe asampleof(1,3,γ)profileswith typicalrangecoveredbytheseobjects. varyingscaleradiusrs. ρs(107M(cid:12)kpc−3) 2.2.3 SubstructureswithinthedSph γ/rs(kpc) 0.10 0.50 1.0 Structureformationsimulationsinthecurrentlyfavoured(cid:14)CDM 0.00 224 25.8 16.02 (coldDMplusacosmologicalconstant)cosmologyfindthatDM 0.25 196 18.6 10.22 haloesareself-similar,containingawealthofsmaller‘substructure’ 0.50 170 13.4 6.47 haloesdowntoEarth-masshaloes(e.g.Diemand,Moore&Stadel 0.75 146 9.5 4.06 2005). However, as emphasized in the introduction, such simula- 1.00 125 6.7 2.52 tionstypicallyneglecttheinfluenceofthebaryonicmatterduring 1.25 106 4.7 1.54 galaxyformation.ItisnotclearwhateffectthesehaveontheDM 1.50 88 3.2 0.92 substructuredistribution.Forthisreason,weadoptamoregeneric approach.Weassesstheimportanceofclumpsusingthefollowing ofthesignal(withrespecttothevalueforα =5◦)isobtainedfor recipe5: D int o α80percent≈3◦.Thisisalsoindicatedbythesymbolswhichshowthe w contributionofDMshellsintwoangularbins–whereasthe(green) (i) wetakeafractionf =20percentofDMmassintheformof nloa hollowsquareshaveaspikydistributioninthefirstbin(γ =1.5), clumps; de d the (black) filled circles (γ = 0) show a very broad distribution ((iiii)i)ththeescplautmialpdpisrtorfiibleustioanreocfacllcuumlaptesdfoa`llloawBsutlhloecskmeotoathl.o(2n0e;01) from forTJh.eintegrationanglerequiredtohaveasizeablefractionofthe (hereinafterB01),thatis,an‘NFW’profile(Navarroetal.1996a) http tashingegnliaenlsndaeerreppedrneodsfiisrleaobnsllesoespvienercγaeltaphnaidsramtmhieentisemcrasiz:leethsreacdodiniusttsaamrnsci.neSadtminfargollbmianctthkeeggrrdoaStuipnohdn, wmiat(shivsc)roatnnhcgeeecnMlturammtiinpo–nmMramesalsxat=deids[tt1roi0bt−uh6te–io1mn0a6iss]sM∝o(cid:12)fMt.h−eac(laum=p−s;1.9),withina ://mnras.o x γ-ray photons and maximizes the signal-to-noise ratio. Thus, the Althoughtheseparametersareveryuncertain,theyallowusto fo rd truedetectabilityofadSphwilldependonitsspatialextentonthe investigatetheimpactofsubtructuresontheJ-factor.Theyarevar- jo sky,andthusalsoond,γ andrs. ied within reasonable bounds in Section 2.3.2 (and Appendix D) urn a todeterminewhetherthesubclumpcontributioncanboostthesig- ls .o nal.Notethata20percentclumpmassfractionisabouttwiceas rg 2.2.2 GenericdSphprofiles largeasthefractionobtainedfromnumericalsimulations(seee.g. at L/ AswillbeseeninSection5,theerrorsonthedensityprofilesofthe Springel et al. 2008). This generous fraction does not affect our eic MilkyWaydSphsarelarge,makingitdifficulttodisentanglethe conclusions,asdiscussedbelow. es te interplaybetweenthekeyparametersfordetectability.Hence,we r U selectsome‘genericprofiles’toillustratethekeydependencies. 2.3 J andJ forthegenericmodels niv Themostconstrainedquantityisthemasswithinthehalf-light sm subcl ers radius r (typically a few tenths of a kpc), as this is where most As an illustration, we show in Fig. 3 one realization of the ity ofthekihnematicdatacomefrom(e.g.W09;Wolfetal.2010).For 2D distribution of J from a generic core profile (γ = 0) with Lib theclassicalMilkyWaydSphs,thetypicalmasswithinrh∼300pc rs=1kpc(subclumpparametersareasdescribedinSection2.2.3). rary is found to be M ∼ 107M(cid:12) (Strigari et al. 2008, see also the The dSph is at d = 100kpc. We note that our consideration of a o 300 n bottom panel of Fig. 13 shown later). If the DM scale radius is γ =0smoothcomponentwithNFWsubclumpsisplausibleif,for N o significantlylargerthanthis(rs(cid:15)rh)andtheinnerslopeγ (cid:3)0.5, example, baryon-dynamical processes erase cusps in the smooth vem wecanapproximatetheenclosedmassby halobutcannotdosointhesub-subhaloes.ThetotalJ isthesum b e M300(cid:16) 43π−ρsγrs3 (cid:10)30r0spc(cid:11)3−γ ≈107M(cid:12). (9) oboyfutttshhkeeirsstmsmooofoottthhheacndodSmpsphuo.bncelunmt,pwdhiesrteraibsustoiomnes.gTrahienicneenstsreaipspdeoarmsiinnatthede r 11, 2015 Theparameterρ isthusdeterminedcompletelybytheabovecon- In this particular configuration, the ‘extended’ signal from the s dition,ifwechoosethescaleradiusr andcuspslopeγ. coreprofile,whenintegratedoveraverysmallsolidangle,could s Table1shows,forseveralvaluesofr andγ,thevaluerequired besubdominantcomparedwiththesignalofNFWsubclumpsthat s for ρ to obtain the assumed M mass. We fix α = 1, β = 3 ithosts.Thediscussionofcross-constraintsbetweendetectability s 300 but our results are not sensitive to these choices.4 The values of ofsubhaloesoftheGalaxyversussubclumpsinthedSphisleftfor r are chosen to encompass the range of r found in the MCMC afuturestudy. s s analysis (see Section 5). To further convince ourselves that the Intheremainderofthispaper,wewillreplaceforsimplicitythe genericprofileswepresenthereareapossibledescriptionofreal calculationofJsubcl(αint)byitsmeanvalue,asweareprimarilyin- dSphs, we checked (not shown) using typical stellar profiles and terestedin‘unresolved’observations.Hence,clumpsarenotdrawn properties of these objects (i.e. half-light radius of a few 100pc) fromtheirdistributionfunction,butrather(cid:6)Jsubcl(cid:7)iscalculatedfrom that a flat ∼10kms−1 velocity dispersion profile within the error 4ForadifferentmassforthedSph,theresultsforJbelowhavetoberescaled 5MoredetailsabouttheclumpdistributionscanbefoundinAppendixB2. byafactor(M3n0e0w/107M(cid:12))2sincethedensityisproportionaltoM300,while See also, for example, Section 2 in Lavalle et al. (2008) and references Jgoesasthedensitysquared. therein,asweusethesamedefinitionsasthosegiveninthatpaper. (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS γ-rayfromdarkmatterannihilation indSphs 1531 D Figure4. JasafunctionoftheangleθawayfromthedSphcentreforadSph o w aαtin1t0=0k0◦p.0c1w.Fitohrrtshe=fo0u.5rkinpnce(rρssloipsegvivaeluneisnγT,atbhleev1a)r.iTouhsecionntetgrirbautitoionnasntgoleJ nload areshownasthesolid(total),dashed(smooth)anddottedlines(subclumps). ed fro m theintegrationofthespatialandluminosity(asafunctionofmass) h distributions(seeAppendixB2). ttp ://m n ra 2.3.1 RadialdependenceofJ(θ) s.o x TheradialdependenceofJisshowninFig.4forfourvaluesofγ (for ford faonrinthteegrsamtiooonthandgilsetrαibinutt=ion0,◦.0th1e).dTohteteddaslihneedslsinheoswshthoewstuhbecrleusmulpt journa ls contribution,andthesolidlinesarethesumofthetwo.Thepeakof .o thesignalistowardsthedSphcentre.Aslongasthedistributionof arg/ clumpsisassumedtofollowthesmoothone,regardlessofthevalue t L ofγ,thequantity(1−f)2Jsm(0)alwaysdominates(atleastbya eice factorofafew)over(cid:6)Jsubcl(0)(cid:7).(Recallthatinourgenericmodels, ste alldSphshavethesameM300.)ThescatterinJtot(0)isaboutfour r U ordersofmagnitudeforγ ∈[0.0–1.5],butonlyafactorof20for niv γTh∈e[c0ro.0s–si1n.0g].poBienytodnedpeanfdeswotnenathcsoomfbdinegatrieoens,o(cid:6)fJstuhbecl(cid:7)clduommpinmataesss. ersity L fractionf,γ,r ,dandα .ThedependenceofJ onthetwolatter ib s int ra parametersarediscussedinAppendixC.Theradialdependenceis ry asexpected:thesmoothcontributiondecreasesfasterthanthatof on N thesubclumpone,becausethesignalisproportionaltothesquared o v spatial distribution in the first case, but directly proportional to em thespatialdistributioninthesecondcase.Halvingf tomatchthe be fractionfromN-bodysimulationswouldhavea25percenteffect r 1 1 on (1 − f)2J , but decreases J by a factor of 4, so that the , 2 sm subcl 0 cross-over between the two components would occur at a larger 15 angle(Fig.4). 2.3.2 Boostfactor Whetherornotthesignalisboostedbythesubclumppopulationis stilldebatedintheliterature(Strigarietal.2007b;Kuhlen,Diemand &Madau2008;Pieri,Bertone&Branchini2008;Pierietal.2009a). Figure3. Two-dimensionalview(x-andy-axesareindegrees)of J for the generic dSph with γ = 0 and rs = 1kpc at d = 100kpc (M300 = Asunderlinedintheprevioussections,thesubclumpcontribution 107M(cid:12)).Thesubclumpsaredrawnfromthereferencemodeldescribedin towards the dSph centre never dominates over the smooth one if Section2.2.3,thatis,f=20percent,subclumpdistributionfollowssmooth, the spatial profile of the subclumps follows that of the smooth andsubclumpinnerprofileshaveNFWwithB01concentration.Fromthe distribution,andiftheintegrationangleremainsbelowsomecritical toptobottompanel:αint=0◦.1,0◦.05and0◦.01.Forthesakeofcomparison, anglediscussedbelow. thesamecolourscaleistakenforthethreeintegrationangles(Jisinunits Letusfirstdefineproperlytheparameterswithrespecttowhich ofM2(cid:12)kpc−5). thisboostiscalculated,asthereissometimessomeconfusionabout this.Here,wedefineitwithrespecttotheintegrationangleα (the int (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS 1532 A.Charbonnieretal. 3 SENSITIVITY OF PRESENT/FUTURE γ-RAY OBSERVATORIES Major new ground-based γ-ray observatories are in the planning stage, with the CTA (CTA Consortium 2010) and AGIS (AGIS Collaboration2010)asthemainconcepts.Asthedesignsofthese instrumentsarestillevolving,weadoptheregenericperformance curves(describedbelow),closetothestatedgoalsoftheseprojects. FortheLAToftheFermiγ-raysatellite,theperformancefor1-year observationsofpoint-like,high-Galactic-latitudesourcesisknown (Fermi-LATCollaboration2010),butnoinformationisyetavail- able for longer exposures or for extended objects. We therefore adoptatoylikelihood-basedmodelfortheFermisensitivity,tuned toreproducethe1-yearpoint-sourcecurves.Wenotethatwhilstthis approach results in approximate performance curves for both the ground-andthespace-basedinstruments,itcapturesthekeydiffer- D Figure 5. Boost factor as a function of αint × (d/100 kpc) for different ences(inparticular,thedifferencesincollectionareaandangular ow innerslopesγ andwithsubclumpsfollowingthesmoothprofile(seeSec- resolution)andillustratestheadvantagesandlimitationsofthetwo n tion2.2.3):thedSphisatd=100kpc(lines)ord=10kpc(symbols). instrumenttypes,aswellastheprospectsforthediscoveryofDM load e annihilationindSphswithinthenextdecade. d fro m pointingdirectionisstilltowardsthedSphcentre): 3.1 Detectormodels http (1−f)2J (α )+J (α ) The sensitivity of a major future γ-ray observatory based on an ://m B(αint)≡ smJsm(inαtint) subcl int . (10) aCrhraeyreonfkoCvhAerrernakyo’)vitsealepspcroopxeims(aFteCdAbainsetdheonfotlhloewpionign,t-fsooru‘rFceutduirfe- nras.o Ithnemclousmtsptubdoiuensd,athrye(bio.eo.sαtahlla=sbReen/cda)l.cHuloawteedvebry,tihnetebgoroatsitndgepoeuntdtos fvearteionntisa)lpsreensseintitveidtybcyuBrveern(lfo¨ohrrae5tσal.d(e2te0c0t8io).nUinnd5e0rhthoeurasssoufmopbtsieorn- xfordjo cruciallyonα (theradialindtependviernceofthesmoothandsubclump thattheangularresolutionofsuchadetectorisafactorof2better urn druscisoingrn(cid:2)Wienttcyare,t0ilb.acpi1usnoltokdnintopseineicfvnqs(eγurFidrneeitigbifngsfaoc.erose5rdtsselo,tteeehfsspdeese.eq6ebonuoSfFoaoeγotuscir)tgto,ishfnooom,nr(rBaCf2dlo.7li≈3rf)ef.γ1eni(sr)o1(cid:3)e.utnh−gt1ehi.f5nαα)ni2(nier.ntetrF,×gsoBalrorddiplslareeersssgsmscγeaoa,vllfiwlanerlhgrsu)e.et,rhsFte,ahoneaar (tstpfhhhrheoaeaanmndpdsreeeot1snnhn0sceai4eantH,imndvdaEi2stnoeSyadluStetrhtc3ch(tu0erFasoteuGnaxnt)edhtkeVabenpaeesttctoeifkofdage1nlcrt.sookt.i2umvGd0nei20idfvf8caeero)tnarle1tlaetnenhTctpdateoetiVorbhtn,hsadeetsehargdvret2eheaiaestcmiiaogfsnponnalrbmitoeieγfmedi-inecrneafsonsey,tserrsmsrrupgegmieydcrce-oatrdnwrnaeatydss-l at Leicester Uals.org/ cpboleamttwepaeouennie,sntrhte.eaGcvohaielnudgeabsoefsyotohonendbatoshoαissitnqtdduea(cid:3)plietRnatdviivsre(otdnakersescnraitnpodtbioγen3oifskptdhciefhfisecmruelo)t.,oaIthns srtoeuscephxopanlssoetrh,eecthhCaeTrcaAacptiesarbnizioleitdtiyebesytofitfhxaeedgf,oewlnleoerwcicionnngseixfdute-ngrcetthnioaentrasst,uiiocshnaainusssitemrfuupmllietfioneotd:l niversity L thetoy-modelformulaeofAppendixB2giveresultscorrecttoonly LS=−13.1−0.33X+0.72X2, (11) ib afactorof∼2(whichisinadequatetoevaluatetheboostproperly). rary Toconclude,themaximumvalueforsubclumpfollowssmooth LA=6+0.46X−0.56X2, (12) on is (cid:2)2, and this value is reached only when integrating the signal N o outtoR /d.Theboostcouldstillbeincreasedbyvaryingthesub- ψ =0.038+exp−(X+2.9)/0.61, (13) ve vir 68 m cdlyunmamppicraolpferritciteison(eh.ga.stackaiunsgedahthigehseurbccolunmcepntpraotpiounla)t.iConontovebresecloym,ief where ber 1 1 muchmorecentrallyconcentratedthanthesmoothcomponent,then X=log (photonenergy/TeV), (14) , 2 10 0 theboostisdecreased.ThisisdetailedinAppendixD.Forthemost LS = log (differential sensitivity/ergcm−2s−1), LA = 15 realisticconfigurations,thereisnosignificantboostwhenaclump 10 log (effectivearea/m2), and ψ is the 68 per cent contain- massfractionf =20percentisused.Naturally,thisresultiseven 10 68 mentradiusofthePSFindegrees. more true for the smaller f found in N-body simulations so we For the Fermi detector, a similar simplified approach is taken; disregardtheboostfortherestofthispaperandconsideronlythe thenumbersusedbelowarethoseprovidedbyFermi-LATCollab- smoothcontribution. oration(2010).Theeffectiveareachangesasafunctionofenergy and the incident angle to the detector, reaching a maximum of ≈8000cm2.Theeffectivetime-averagedareaisthen(cid:17)A(cid:7)/4πand the data-taking efficiency (cid:17) ≈ 0.8 (due to instrument dead-time andpassagesthroughtheSouthAtlanticAnomaly).ThePSFagain 6The difference between the level of boost observed for rs = 0.1 and varies as a function of energy (with a much smaller dependence 1kpc can be understood if we recall that the total mass of the clump is as a function of incidence angle), from 10◦ to a few tenths of a OJfisxm(e1d0∝9toρMs23(cid:12)0w0khppecrc,e−ar3es)g,Jaswrudbhl∝eersesρaoss,ffthtohererrevslaa=ltuive1eokafpmcγo,uoρnrstr∼os.fFJOsou(rb1r0ws7i=tMh(cid:12)r0e.s1kpkpepcc−ct,3to)ρ.sJAs∼ms dsre−g1re(e>o1v0e0rMtheeVL)AaTndenaerpghyortoanngien.dAexraotfe2o.f11a.r5e×as1su0m−5ecdmfo−r2sth−e1 isexpectedtodecreasewithsmallerrs.Thisisindeedwhatweobservein background. The sensitivity is then estimated using a simplified thefigure(solidversusdashedlines). likelihoodmethodwhichprovidesresultswithin20percentofthe (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS γ-rayfromdarkmatterannihilation indSphs 1533 sensitivityfora1-yearobservationofapoint-likesourcegivenby Fig.7showstherelativesensitivityofFermiandanFCAwithin Fermi-LATCollaboration(2010). ourframeworkasafunctionofthemassoftheannihilatingparticle, Whilstbothdetectorresponsesareapproximate,thecomparison adoptingtheannihilationspectrumgiveninequation(4),withthe isstilluseful.Ourworkincorporatesseveralkeyaspectsnotcon- severalpanelsillustratingdifferentpoints.FromFig.7(toppanel, sideredinearlierstudies,includingthestrongenergydependence the case of a point-like signal for different observation times), it oftheangularresolutionofbothground-andspace-basedinstru- isclearthattheFermi-LAThasaconsiderableadvantageforlower mentsintherelevantenergyrangeof1GeVto1TeVandhencethe massDMparticles(mχ(cid:13)1TeV)onthetime-scaleforconstruction energy-dependentimpactoftheangularsizeofthetargetregion. ofanFCA(i.e.overa5–10yrmissionlifetime)incomparisontoa deepACTobservationof200h.Furthermore,theFermi-LATisless adverselyaffectedbytheangularextentofthetargetregions(see 3.2 Relativeperformanceforgenerichaloes Fig. 7, bottom panel), due to its modest angular resolution in the UsingtheresultsfromSection2.2.2andthedetectorperformance energyrangewhereitislimitedbybackground,meaningthatthe modelsdefinedabove,wecanbegintoinvestigatethesensitivityof sourceextensioniswellmatchedtothePSFoftheinstrument.The future ACT arrays and the Fermi-LAT detector (over long obser- middle panel of this figure illustrates the impact of different ap- vationtimes)toDMannihilationindSphs.Thedetectabilityofa proachestotheanalysis.InthecasethatthereisaDMcandidate sourcedependsprimarilynotonlyonitsflux,butalsoonitsangular inferredfromthediscoveryofsupersymmetryattheLargeHadron D o extent.Theimpactofsourceextensionondetectabilityisdealtwith Collider(LHC)(quitepossibleontherelevanttime-scale),asearch wn approximately(ineachenergybinindependently)byassumingthat optimized on an assumed mass and spectral shape can be made loa theopeningangleofaconewhichincorporates80percentofthe (solidcurves).However,allinstrumentsarelesssensitivewhena ded signalisgivenby genericsearchisundertaken.Simpleanalysesusingallthephoton fro (cid:12) fluxaboveafixedenergythreshold(arbitrarilysettoreduceback- m h vθw8ah0lue=reesψgi8vψ0e82=n0f+1o.r2α65828ψ0a,6n8dis9a5spsuemrceednftocrotnhteaiFnCmAenatnfdorinthteerpLoAlTate(Fdefr(rm1o5mi)- fGgmoreaorVsuAsn.CwdFT)oorsarkrfseeoxmrea0fomf.de3pce–ltr3eiav,tTeekeleoyVenwplpyieanlrigltniiconalnetlrhsye,elwa>0thi.1v1e0e–r0le0ya.G2snekTaVreereVoppwhirnaorgntaogannelgls,epbwhuoootfrtipkosasnmrstwiu>cecll1hel ttp://mnras.o LATCollaboration2010);here,α80isthe80percentcontainment lesssensitivethanthehigherthresholdcutovertherestofthecan- xfo angleofthehaloemission.Thevalidityofthisapproximation(at didate DM particle mass range. The features of these curves are rdjo the level of a few per cent) has been tested (see Appendix E) by dictatedbytheexpectedshapeoftheannihilationspectrum.From urn convolving realistic halo profiles with a double Gaussian PSF as equation (4), the peak photon output (adopting the average spec- als foundfortheHESS(Horns2005).An80percentintegrationcircle trum for DM annihilation) occurs at an energy which is an order .org is close to optimum for a Gaussian source on a flat background ofmagnitudebelowtheparticlemass–effectivedetectionrequires a/ (in the background-limited regime). Fig. 6 shows the 80 per cent that this peak occurs within (or close to) the energy range of the t L e containmentradiusoftheannihilationfluxofgenerichaloesasa instrumentconcerned. ice functionoftheinnerslopeγ.Thisresultcanbeparametrizedas ThetotalannihilationfluxfromadSphincreasesatsmallerdis- ste (cid:10) r (cid:11)(cid:10) d (cid:11)−1 tancesas1/d2forfixedhalomass,makingnearbydSphsattractive r Un α80=0◦.8(1−0.48γ −0.137γ2) 1kspc 100kpc . (16) flaorrsDizMe odfetseuccthionn.eaHrboywesvoeurr,caessFraiigs.e7s tshheowresq,utihreedindcerteeacsteiodnanflguux-. iversity Itisclearthatforabroadrangeofd,γ andr ,thecharacteristic Fig.8illustratesthereductionofanFCAsensitivityasafunction L angularsizeoftheemissionregionislargerthasntheangularreso- of the distance of a generic dSph. Inner slopes γ = 0 and γ = ibra lutionoftheinstrumentsunderconsideration.Itisthereforecritical 1havebeenconsideredandrs isfixedto1kpc.Thesensitivityis ry o toassesstheperformanceasafunctionoftheangularsizeofthe expressedrelativetothatobtained usingthefullannihilationsig- n N dSphaswellasthemassoftheannihilatingparticle. nalinthepoint-likeapproximation.Evenforγ =1,thepoint-like ov e approximationleadstoanorderofmagnitudeoverestimateofthe m b detectionsensitivityfornearby(∼20kpc)dSphs.Afurthercompli- er 1 cfraotimonthisehroewsidtouaelstnaobnli-sγh-rthayelbeavceklgorfobuancdk.gAroucnodmemmoinssimonetahroidsining 1, 20 1 ground-basedγ-rayastronomyistoestimatethisbackgroundfrom 5 anannulusaroundthetargetsource(seee.g.Berge,Funk&Hinton 2007).ThedashedlinesinFig.8showtheimpactofestimatingthe backgroundusinganannulusbetween3◦.5and4◦.0fromthetarget. Thisapproachhasamodestimpactonsensitivityandisignoredin thefollowingdiscussionsasitreducesboththedetectablefluxand θ andleadstoasmallimprovementinsomecasesonly. 80 4 JEANS/MCMC ANALYSIS OF DSPH KINEMATICS Figure6. Theconeangleencompassing80percentoftheannihilationflux 4.1 dSphkinematicswiththesphericalJeansequation asafunctionoftheinnerslopeγ.Severaldifferentvaluesofrsanddistance dareshownforeachγ,allscaledby1kpc/rsand100kpc/d.Thebest-fitting Extensive kinematic surveys of the stellar components of dSphs curveisalsoshown,correspondingtoequation(16). have shown that these systems have negligible rotational support (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS 1534 A.Charbonnieretal. 3-1] sm10-20 FCA HESS J=1012 M2 kpc-5 c v> [ 20 h σ 10-21 < 20 h 200 h 10-22 200 h 10-23 1 yr Fermi 10-24 10 yr 10-25 D 10-2 10-1 1 10 mχ [TeV] Figure 8. Relative DM annihilation detection sensitivity for a 100-hour ow FCAobservation,asafunctionofdSphdistancefordifferentinnerslopes n 3-1]m s10-20 FCA J=1012 M2 kpc-5 γθ80anidsgwivitehnrrsefilaxtievdettoo1thkepcs.enTshiteivsietynstiotivaitpyofionrt-laikreeasloisutricceapwpirthoatchheussaimnge loaded σ v> [c<10-21 >1 GeV flmbyuinxei.qmLuuaamrtigoendre(vt4ea)clutwaebsitlhceomflrruχexs=)p.oT3n0dh0etoGaspesoVuo.mrTeerhdipssepsreefoncrtsrmiatilavnsitchyeapr(aleatiriogsedaregvpaaeinlnudeasssoognfivtthhenee http from 1100--2232 >100 GeV Likelihood ttfsdhhrtaoreesamhbtdeeaSgtdchpykehlgiundcrsoeSeesunpdnthsrdht.eoocawoesnsattthirbmoealacirmtkeeggptriahoocenutnlboideafsccukocsgoninrmtrogopulalnerndetegallineyovnoneuu.llTtuasshitdebtehestetohwleidedSreelpngihni3oe◦.pna5ososasfiunteimdomne4i.s◦s.s0Ttihohoaneft ://mnras.oxford 10-24 Fermi (withthepossibleexceptionoftheSculptordSph,seeBattagliaetal. journ a 2008).IfweassumethatthedSphsareinvirialequilibrium,then ls .o 10-25 theirinternalgravitationalpotentialsbalancetherandommotions rg 10-2 10-1 1 10 mχ [TeV] oftheirstars.InordertoestimatedSphmasses,weconsiderhere at L/ 3-1] sm10-20 FCA J=1012 M2 kpc-5 tdhiestabnecheavfrioomurthoefddSSpphhcsetneltlraer(avnealolocgitoyudsitsopreortsaiotinonascuarvfeusnocftisopniroafl eiceste > [c gMaalatexoie&s).OSplsezceiwficsaklily(,2w00e9ucs)efothrethsteelClaarriknian,eFmoartnicaxd,aStacuolfpWtoarlkanerd, r Un σ v<10-21 1o StheextLaenosIdSdpShpsh,,tahnedddaatataoffroMmatMeoa,tOeolsezteawl.sk(iin&prWepaalkraetrio(2n0)0fo8r)tfhoer iversity 10-22 0.1o Draco,LeoIIandUrsaMinordSphs.W09havecalculatedveloc- Lib itydispersionprofilesfromthesesamedataundertheassumption rary 10-23 thatl.o.s.velocitydistributionsareGaussian.Herewere-calculate o n Point-like theseprofileswithoutadoptinganyparticularformforthevelocity N o 10-24 Fermi distributions.Specifically,foragivendSph,wedividethevelocity vem sampleintocircularbinscontainingapproximatelyequalnumbers b e 10-25 of member stars,7 and within each bin, we estimate the second r 11 10-2 10-1 1 10 mχ [TeV] velocitymoment(squaredvelocitydispersion)as , 20 1 (cid:2)N 15 (Fbilgaucrkeli7n.esA)papnrdotxhiemFaCteAsednessictirvibiteiedsaobfovthee(rFeedrmlini-eLs)AtToa(bglueenelriincehs)a,loHwESitSh (cid:6)Vˆ2(cid:7)= N−1 [(Vi−(cid:6)Vˆ(cid:7))2−σi2], (17) J =1012M2(cid:12)kpc−5,asafunctionofthemassoftheannihilatingparticle where N is the in=u1mber of member stars in the bin. We hold (cid:6)V(cid:7) andfortheannihilationspectrumofequation(4).Toppanel:theimpact fixed for all bins at the median velocity over the entire sample. ofobservationtimeisillustrated:dashedlinesgivethe1-yearand20-hour sensitivities for Fermi and the FCA/HESS, respectively, while the solid Foreachbin,weuseastandardbootstrapre-samplingtoestimate lines refer to 10-year (200-hour) observations. Middle panel: the impact theassociatederrordistributionfor(cid:6)Vˆ2(cid:7),whichisapproximately ofanalysismethodsisconsideredfor5-year(100-hour)observationsusing Gaussian.Fig.9displaystheresultingvelocitydispersionprofiles, Fermi(FCA).Thesolidlinesshowlikelihoodanalysesinwhichthemass (cid:6)Vˆ2(cid:7)1/2(R),whicharesimilartopreviouslypublishedprofiles. andspectrumoftheannihilatingparticleareknowninadvance,whilethe In order to relate these velocity dispersion profiles to dSph dashedanddottedlinesshowsimpleintegralfluxmeasurementsabovefixed masses,wefollowW09inassumingthatthedatasampleineach thresholdsof1and100GeV,respectively.Notethatthe1-GeVcutimplies acceptingalleventsfortheFCA(wherethetriggerthresholdis≈20GeV). Bottom panel: the impact of the angular extension of target sources, as 7KinematicsamplesareoftencontaminatedbyinterlopersfromtheMilky givenbythehaloprofileinFig.6,isillustrated.Thesolidlinesreproduce Wayforeground.FollowingW09,wediscardallstarsforwhichthealgo- thelikelihoodcasefromthemiddlepanelforapoint-likesource,withthe rithmdescribedbyWalkeretal.(2009b)returnsamembershipprobability valuesofα80of0◦.1(dashed)and1◦alsoshown. lessthan0.95. (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS γ-rayfromdarkmatterannihilation indSphs 1535 D o w n lo a d e d fro m h ttp ://m n ra s .o x fo rd jo u rn a ls .o rg a/ Figure9. VelocitydispersionprofiledatafortheeightclassicaldSphs,obtainedasdescribedinthetext(theimpactofthebinningchoiceisdiscussedin t L AppendixH1).Thesolidlinescorrespondtothebest-fittingmodelsfortheinnerslopewhenγ isleftfree(dark),γ isfixedto1(blue)andγ isfixedto0 e ic (red).Becauseofthelargedegeneraciesamongthehaloparameters(seeSection5.1foralist),wedonotlistthecorrespondingbest-fittingparameters.The es motivationforshowingtheseprofilesistoillustratethatourhalomodeliscapableofdescribingthekinematicdata,andthattheinnerprofileisnotconstrained ter U bythedata. n iv e rs dSphasingle,pressure-supportedstellarpopulationthatisindy- Note that while we observe the projected velocity dispersion and ity namicalequilibriumandtracesanunderlyinggravitationalpotential stellar density profiles directly, the l.o.s. velocity dispersion pro- L ib dtioomnsinoaftsetdelblayrbDiMna.ryImsypsltiecmitsiscothnetraibsusutemnpetgiolingitbhlaytttohethoermbietaalsumreod- wfileesrepqrouviriedeannoaisnsfuomrmptaitoinonaabboouuttβthaneisoa;nihseorter,opwye,βaasnsisuom.Tehβeraneifsoor=e, rary o n velocitydispersions.8 Furthermore,assumingsphericalsymmetry, constant,allowingfornon-zeroanisotropyinthesimplestway.For N o themassprofile,M(r),oftheDMhalorelatesto(momentsof)the constantanisotropy,theJeansequationhasthesolution(e.g.Mamon v e stellardistributionfunctionviatheJeansequation: &Łokas2005) m b e ν1ddr(νv¯r2)+2β(rr)v¯r2 =−GMr2(r), (18) νv¯r2=Gr−2βaniso(cid:3) ∞s2βaniso−2ν(s)M(s)ds. (20) r 11, 20 where ν(r), v¯2(r) andβ ≡ β(r) ≡ 1−v¯2/v¯2 describe the r 15 r aniso θ r three-dimensional density, radial velocity dispersion and orbital WeshalladoptparametricmodelsforI(R)andM(r)andthenfind anisotropy,respectively,ofthestellarcomponent.Projectingalong valuesoftheparametersofM(r)that,viaequations(19)and(20), the l.o.s., the mass profile relates to observable profiles, the pro- bestreproducetheobservedvelocitydispersionprofiles. jectedstellardensityI(R)andvelocitydispersionσ (R),according p to(Binney&Tremaine2008,hereinafterBT08) (cid:4) (cid:5) (cid:3) 2 ∞ R2 νv¯2r σ2(R)= 1−β √ r dr. (19) p I(R) R anisor2 r2−R2 4.1.1 Stellardensity StellarsurfacedensitiesofdSphsaretypicallyfittedbyPlummer (1911), King (1962) and/or Se´rsic (1968) profiles (e.g. Irwin & 8Olszewski,Pryor&Armandroff(1996)andHargreaves,Gilmore&Annan Hatzidimitriou1995).Forsimplicity,weadoptherethePlummer (1996)concludethatthisassumptionisvalidfortheclassicaldSphsstud- iedhere,whichhavemeasuredvelocitydispersionsof∼10kms−1.This profile: conclusion does not necessarily apply to recently discovered ‘ultrafaint’ M∼i3lkkymWs−a1y(sMatceClliotensn,awchhiiech&hCavoˆete´m2e0a1s0u)r.edvelocitydispersionsassmallas I(R)= πLr2(1+R12/r2)2, (21) h h (cid:4)C 2011TheAuthors,MNRAS418,1526–1556 MonthlyNoticesoftheRoyalAstronomicalSociety(cid:4)C 2011RAS
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