Dark matter profiles and annihilation in dwarf spheroidal galaxies

31 Pages·2011·3.06 MB·English

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
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Accepted2011July5.Received2011July1;inoriginalform2011April3 a
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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
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n
lo
a
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e
d
fro
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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
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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