Dark matter profiles and annihilation in dwarf spheroidal galaxies: prospectives for present and ...

31 Pages·2010·2.2 MB·English

by  Celine Combet

Mon.Not.R.Astron.Soc.000,1–29(Xxxx) Printed6July2011 (MNLATEXstylefilev2.2)
Dark matter profiles and annihilation in dwarf spheroidal galaxies:
prospectives for present and future γ-ray observatories
I. The classical dSphs
1
1
0 A. Charbonnier1, C. Combet2, M. Daniel4, S. Funk5, J.A. Hinton2⋆, D. Maurin6,1,2,7⋆,
2 C. Power2,3, J. I. Read2,11, S. Sarkar8, M. G. Walker9,10⋆, M. I. Wilkinson2

ul 1LaboratoiredePhysiqueNucle´aireetHautesEnergies,CNRS-IN2P3/Universite´sParisVIetParisVII,4placeJussieu,Tour33,75252ParisCedex05,France
J 2Dept.ofPhysicsandAstronomy,UniversityofLeicester,Leicester,LE17RH,UK
3InternationalCentreforRadioAstronomyResearch,UniversityofWesternAustralia,35StirlingHighway,Crawley,WesternAustralia6009,Australia
5
4Dept.ofPhysics,DurhamUniversity,SouthRoad,Durham,DH13LE,UK

5W.W.HansenExperimentalPhysicsLaboratory,KavliInstituteforParticleAstrophysicsandCosmology,DepartmentofPhysicsandSLACNationalAcceleratorLaboratory,Stanfor
]
E 6LaboratoiredePhysiqueSubatomiqueetdeCosmologie,CNRS/IN2P3/INPG/Universite´JosephFourierGrenoble1,53avenuedesMartyrs,38026Grenoble,France
7Institutd’AstrophysiquedeParis,UMR7095CNRS,Universite´PierreetMarieCurie,98bisbdArago,75014Paris,France
H
8RudolfPeierlsCentreforTheoreticalPhysics,UniversityofOxford,1KebleRoad,Oxford,OX13NP,UK
. 9InstituteofAstronomy,UniversityofCambridge,MadingleyRoad,Cambridge,CB30HA,UK
h
10Harvard-SmithsonianCenterforAstrophysics,60GardenSt.,Cambridge,MA02138,USA
p
11InstituteforAstronomy,DepartmentofPhysics,ETHZu¨rich,Wolfgang-Pauli-Strasse16,CH-8093Zu¨rich,Switzerland
-
o
r
t
s AcceptedXxxx.ReceivedXxxx;inoriginalformXxxx
a
[

2 ABSTRACT
v Due to their large dynamical mass-to-light ratios, dwarf spheroidal galaxies (dSphs) are
2 promisingtargetsfortheindirectdetectionofdarkmatter(DM)inγ-rays.Weexaminetheir
1 detectability by present and future γ-ray observatories. The key innovative features of our
4 analysis are: (i) We take into account the angular size of the dSphs; while nearby objects
0 have higher γ ray flux, their larger angular extent can make them less attractive targets for
.
4 background-dominatedinstruments.(ii)WederiveDMprofilesandtheastrophysicalJ-factor
0 (which parameterises the expected γ-ray flux, independently of the choice of DM particle
1 model)fortheclassicaldSphsdirectlyfromphotometricandkinematicdata.Weassumevery
1 little about the DM profile, modelling this as a smooth split-power law distribution, with
: and without sub-clumps. (iii) We use a Markov Chain Monte Carlo (MCMC) technique to
v
i marginaliseoverunknownparametersanddeterminethesensitivityofourderivedJ-factorsto
X bothmodelandmeasurementuncertainties.(iv)WeusesimulatedDMprofilestodemonstrate
r thatourJ-factordeterminationsrecoverthecorrectsolutionwithinourquoteduncertainties.
a
Ourkeyfindingsare:(i)Sub-clumpsinthedSphsdonotusefullyboostthesignal;(ii)The
sensitivity of atmosphericCherenkovtelescopesto dSphswithin ∼20kpcwith coredhalos
canbeupto∼50timesworsethanwhenestimatedassumingthemtobepoint-like.Evenfor
thesatellite-borneFermi-LATthesensitivityissignificantlydegradedontherelevantangular
scalesforlongexposures,henceitisvitaltoconsidertheangularextentofthedSphswhen
selectingtargets;(iii)NoDMprofilehasbeenruledoutbycurrentdata,butusingaprioron
the inner dark matter cusp slope 0 6 γ 6 1 providesJ-factor estimates accurate to a
prior
factorofafewifanappropriateangularscaleischosen;(iv)TheJ-factorisbestconstrained
atacriticalintegrationangleα = 2r /d(wherer isthehalflightradiusanddisthe
c half half
distanceto the dwarf)andwe estimate the correspondingsensitivity of γ-rayobservatories;
(v)The‘classical’dSphscanbegroupedintothreecategories:well-constrainedandpromising
(UrsaMinor,Sculptor,andDraco),well-constrainedbutlesspromising(Carina,Fornax,and
LeoI),andpoorlyconstrained(SextansandLeoII);(vi)ObservationsofclassicaldSphswith
Fermi-LAT integrated over the mission lifetime are more promising than observationswith
the planned Cherenkov Telescope Array for DM particle mass .700 GeV. However, even
Fermi-LATwillnothavesufficientintegratedsignalfromtheclassicaldwarfstodetectDM
in the ‘vanilla’Minimal SupersymmetricStandard Model. Both the Galactic centre and the
‘ultra-faint’dwarfsarelikelytobebettertargetsandwillbeconsideredinfuturework.
Key words: astroparticle physics — (cosmology:) dark matter — Galaxy: kinematics and
dynamics—γ-rays:general—methods:miscellaneous
2 Charbonnier,Combet,Daniel etal.
1 INTRODUCTION This paper extends the earlier study of Walkeretal. (2011)
which showed that there is a critical integration angle (twice the
Thedetectionofγ-raysfromdarkmatter(DM)annihilationisone half-light radiusdivided bythedSph distance) wherewecan ob-
ofthemostpromisingchannelsforindirectdetection(Gunnetal.
tain a robust estimate of the J-factor (that parameterises the ex-
1978; Stecker 1978). Since the signal goes as the DM density
pectedγ-rayfluxfromadSphindependentlyofthechoiceofdark
squared, the Galactic centre seems to be the obvious location to
matter particle model; see Section 2), regardless of the value of
search for such a signal (Silk&Bloemen 1987). However, it is
thecentralDMcuspslopeγ.Here,wefocusonthefullradialde-
plaguedbyaconfusingbackground ofastrophysicalsources(e.g.
pendenceoftheJ-factor.WeconsidertheeffectofDMsub-lumps
Aharonianetal.2004).Forthisreason,thedwarfspheroidalgalax-
withinthedSphs,discusswhichdSphsarethebestcandidatesfor
ies(dSphs)orbitingtheMilkyWayhavebeenflaggedasfavoured
anobservingprogramme,andexaminethecompetitivenessofnext-
targetsgiven theirpotentiallyhigh DMdensitiesandsmallastro-
generationACTsasdarkmatterprobes.
physicalbackgrounds(Lake1990;Evansetal.2004).
Thispaperisorganisedasfollows.InSection2,wepresenta
Despitethegrowingamountofkinematicdatafromtheclas- studyoftheannihilationγ-rayflux,focusingonwhichparameters
sical dSphs, the inner parts of their DM profiles remain poorly critically affect the expected signal. In Section 3, we discuss the
constrained and can generally accommodate both cored or cuspy sensitivityof present/future γ-rayobservatories. In Section4, we
solutions (e.g. Kochetal. 2007; Strigarietal. 2007; Walkeretal. present our method for the dynamical modelling of the observed
2009). There are two dSphs—Fornax and Ursa Minor—that kinematics of stars in dSphs. In Section 5, we derive DM den-
show indirect hints of a cored distribution (Kleynaetal. 2003; sityprofilesfortheclassicaldSphsusinganMCMCanalysis,from
Goerdtetal.2006);however,inbothcasesthepresenceofacore whichthedetectionpotentialoffutureγ-rayobservatoriescanbe
is inferred based on a timing argument that assumes we are not assessed.WepresentourconclusionsinSection6.1
catching the dSph at a special moment. Theoretical expectations Thispaper includesdetailedanalyses frombothhigh-energy
remain similarly uncertain. Cusps are favoured by cosmological astrophysics and stellar dynamical modelling. To assist readers
modelsthatmodeltheDMalone,assumingitiscoldandcollision- fromthesedifferentfieldsinnavigatingthekeysections,wesug-
less (e.g. Navarro,Frenk&White 1996). However, the complex gestthatthosewhoareprimarilyinterestedinthehigh-energycal-
dynamicalinterplaybetweenstars,gasandDMduringgalaxyfor- culationsmaywishtofocustheirattentiononSections2,3and5
mationcoulderasesuchcuspsleadingtocoreddistributions(e.g. beforemovingtotheconclusions.Readersfromthedynamicscom-
Navarroetal. 1996; Read&Gilmore 2005; Mashchenkoetal. munitymayinsteadprefertoreadSections2,4and5.Finally,those
2008;Goerdtetal.2010;Governatoetal.2010;Coleetal.2011). whoarewillingtotrusttheunderlyingmodellingshould proceed
Cores could also be an indication of other possibilities such toSection5whereour mainresultsregardingthedetectabilityof
as self-interacting dark matter (e.g. Hogan&Dalcanton 2000; dSphsarepresentedinFigs.12, 15, 16and17.
Mooreetal.2000).
Knowledge of the inner slope of the DM profile is of crit-
ical importance as most of the annihilation flux comes from
that region. Lacking this information, several studies have fo- 2 THEDARKMATTERANNIHILATIONSIGNAL:KEY
cused on the detectability of these dSphs by current γ-ray ob- PARAMETERS
servatories such as the satellite-borne Fermi-LAT and atmo-
2.1 Theγ-rayflux
spheric Cherenkov telescopes (ACTs) such as H.E.S.S., MAGIC
and VERITAS, using a small sample of cusped and cored pro- Theγ-rayfluxΦ (photonscm−2 s−1 GeV−1)fromDMannihi-
γ
files (generally one of each). Most studies rely on standard core lationsinadSph,asseenwithinasolidangle∆Ω,isgivenby(see
and cusp profiles fitted to the kinematic data of the dSph of AppendixAfordefinitionsandconventionsusedintheliterature):
interest (Bergstro¨m&Hooper 2006; Sa´nchez-Condeetal. 2007;
Bringmannetal. 2009; Pierietal. 2009; Pierietal. 2009). Other
dΦ
authors use a ‘cosmological prior’ from large scale cosmological γ(E ,∆Ω)=Φpp(E ) J(∆Ω), (1)
simulations (e.g. Kuhlen 2010). Both approaches may be com- dEγ γ γ ×
bined, such as in Strigarietal. (2007) and Martinezetal. (2009) Thefirstfactorencodesthe(unknown)particlephysicsofDMan-
whorelypartiallyontheresultsofstructureformationsimulations nihilationswhichwewishtomeasure.Thesecondfactorencodes
to constrain the inner slope and then perform a fit to the data to theastrophysicsviz. thel.o.s.integral of theDMdensity-squared
derivetheotherparameters.Howeversuchcosmologicalpriorsre- oversolidangle∆ΩinthedSph—thisiscalledthe‘J-factor’.We
mainsufficientlyuncertainthattheiruseisinappropriateforguid- nowdiscusseachfactorinturn.
ing observational strategies. There have been only a few studies
(e.g.,Essigetal.2009)whichhavenot assumed strongpriorsfor
theDMprofiles. 1 TechnicaldetailsaredeferredtoAppendices.InAppendixA,wecom-
Inthiswork,werevisitthequestionofthedetectabilityofdark mentonthevariousnotationsusedinsimilarstudiesandprovideconversion
matterannihilationintheclassicalMilkyWaydSphs,motivatedby factorstohelpcompareresults.InAppendixB,weprovideatoymodelfor
ambitiousplansfornext-generation ACTssuchastheCherenkov quickestimatesoftheJ-factor.InAppendixD,wecalculateinamoresys-
tematicfashiontherangeofthepossible‘boostfactor’(duetoDMclumps
TelescopeArray(CTA).Werelysolelyonpublishedkinematicdata
within the dSphs)for generic dSphs. InAppendix E,we show that con-
toderivethepropertiesofthedSphs,makingminimalassumptions
volvingthesignalbythePSFoftheinstrumentisequivalent toacruder
abouttheunderlyingDMdistribution.Mostimportantly,wedonot
quadraturesumapproximation.InAppendixF,wediscusssometechnical
restrictoursurveyofDMprofilestothosesuggestedbycosmolog-
issuesrelatedtoconfidenceleveldeterminationfromtheMCMCanalysis.
icalsimulations.Wealsoconsidertheeffectofthespatialextentof InAppendixG,thereconstructionmethodisvalidatedonsimulateddSphs.
thedSphs,whichbecomesimportantfornearbysystemsobserved InAppendixH,wediscusstheimpactofthechoiceofthebinningofthe
bybackground-limitedinstrumentssuchasACTs. starsandoftheshapeofthelightprofileontheJ-factordetermination.
Darkmatterannihilationindwarfspheroidalgalaxiesandγ-rayobservatories:I. ClassicaldSphs 3
2.1.1 Theparticlephysicsfactor xγ
Theparticlephysicsfactor(Φpp)isgivenby: /dNγ 1 A(IvBe)r BagMe4, ,B Berrginsgtrmoamn ent eatl a. l(.1 (929080)8)
d
2 x
10-1
dΦ 1 σ v dN
Φpp(Eγ)≡ dEγγ = 4πh2amnn2χi × dEγγ , (2) Forneτn+gτo- et al. (2004)
10-2 gluons
where mχ is the mass of the DM particle, σann is its self- Z bosons
annihilationcross-sectionand σ v theaverageoveritsvelocity W bosons
ann
h i bb
distribution,anddN /dE isthedifferentialphotonyieldperan-
nihilation.Abenchmγark vγalueis σannv 3 10−26cm3s−1 10-3 cttc
(Jungmanetal. 1996), which wouhld resuilt∼in a×present-day DM ss mχ = 1 TeV
uu or dd
abundancesatisfyingcosmologicalconstraints.
10-4
differUenntliiakleatnhneihainlantiihoinlastpioenctrcuromss(dsNecti/odnEan(dEpa)r)tircelqeumireasssu,sthtoe 10-2 10-1 x = Eγ/mχ1
γ γ γ
adoptaspecificDMparticlemodel.Wefocusonawell-motivated Figure1.Differentialspectra(multipliedbyx2)ofγ-raysfromthefrag-
classof modelsthatarewithinreachofup-coming directandin- mentationofneutrinoannihilation products(hereforaDMparticlemass
directexperiments:theMinimalSupersymmetricStandardModel of mχ = 1 TeV). Several different channels are shown, taken from
(MSSM). In this framework, the neutralino is typically the light- Fornengoetal. (2004) and an average parametrisation Bergstro¨metal.
eststableparticleandthereforeoneofthemostfavouredDMcan- (1998)ismarkedbytheblacksolid line; thisis whatweadopt through-
didates (see e.g. Bertoneetal. 2005). A γ-ray continuum is pro- out this paper. The black dashed line is the benchmark model BM4
duced fromthe decay of hadrons (e.g. π0 γγ) resultingfrom (Bringmannetal.2008)whichincludesinternalbremsstrahlungandserves
→ toillustratethatverydifferentspectraarepossible.However,theexample
theDMannihilation.Neutralinoannihilationscanalsodirectlypro-
shownhereisdominatedbylineemissionandthereforehighlymodelde-
ducemono-energeticγ-raylinesthroughloopprocesses,withthe
pendent;forthisreason,wedonotconsidersucheffectsinthispaper.
formationofeitherapairofγ-rays(χχ γγ;Bergstro¨m&Ullio
→
1997),oraZ0bosonandaγ-ray(χχ γZ0;Ullio&Bergstro¨m
1998).Wedonottakeintoaccountsuc→hlineproductionprocesses 2004,2005).2ThisdependsinverselyontheDMparticlevelocity,
since they are usually sub-dominant and very model dependent andthusrequiresprecisemodellingofthevelocitydistributionof
(Bringmannetal.2008).Thedifferentialphotonspectrumweuse theDMwithinthedSph;wewillinvestigatethisinaseparatestudy.
isrestrictedtothecontinuumcontributionandiswrittenas:
2.1.2 TheJ-factor
dN dNi
γ(E )= b γ(E ,m ), (3) ThesecondterminEq.(1)istheastrophysicalJ-factorwhichde-
dEγ γ i dEγ γ χ pendsonthespatialdistributionofDMaswellasonthebeamsize.
i
X
Itcorresponds tothel.o.s.integration of theDMdensity squared
wherethedifferentannihilationfinalstatesiarecharacterisedbya oversolidangle∆ΩinthedSph:
branchingratiob .
i
Using the parameters in Fornengoetal. (2004), we plot the
continuum spectracalculatedfor a1TeVmassneutralino inFig. J = ρ2DM(l,Ω)dldΩ. (5)
1. Apart from the τ+τ− channel (dash-dotted line), all the an- Z∆ΩZ
nihilation channels in the continuum result in very similar spec- Thesolidangleissimplyrelatedtotheintegrationangleαintby
tra of γ-rays (dashed lines). For charged annihilation products,
∆Ω=2π (1 cos(α )).
int
internal bremsstrahlung (IB) has recently been investigated and · −
foundtoenhancethespectrumclosetothekinematiccut-off(e.g., The J-factor is useful because it allows us to rank the dSphs by
Bringmannetal.2008).Asanillustration,thelong-dashedlinein theirexpectedγ-rayflux,independentlyofanyassumedDMpar-
Fig. 1 corresponds to the benchmark configuration for a wino- ticle physics model. Moreover, the knowledge of the relative J-
like neutralino taken fromBringmannetal. (2008). However, the factors would also help us to evaluate the validity of any poten-
shapeandamplitudeofthisspectrumarestronglymodeldependent tialdetectionofagivendSph,becauseforagivenparticlephysics
(Bringmannetal. 2009) and, as argued in Cannonietal. (2010), modelwecouldthenscalethesignaltowhatweshouldexpectto
thiscontributionisrelevantonlyformodels(andatenergies)where seeintheotherdSphs.
thelinecontributionisdominantoverthesecondaryphotons. AllcalculationsofJ presentedinthispaperwereperformed
Wewishtobeasmodel-independentaspossible,andsodonot usingthepubliclyavailableCLUMPYpackage(Charbonnier,Com-
considerinternalbremsstrahlung.Intheremainderofthispaper,all bet, Maurin, in preparation) which includes models for a smooth
our results will be based on an average spectrum taken from the DMdensityprofileforthedSph,clumpydarkmattersub-structures
parametrisation(Bergstro¨metal.1998,solidlineinFig.1): insidethedSph,andasmoothandclumpyGalacticDMdistribu-
tion.3
dN 1 dN 1 0.73e−7.8x
γ = γ = , (4)
dEγ mχ dx mχ x1.5 2 Thiseffectdependsonthemassandthevelocityoftheparticle;there-
sultingboostofthesignalandtheimpactondetectabilityofthedSphshas
withx Eγ/mχ.Finally,inordertobeconservativeinderiving beendiscussed,e.g.,inPierietal.(2009).
≡
detectionlimits,wealsodonotconsiderthepossible‘Sommerfeld 3 InAppendixB,weprovideapproximateformulaeforquickestimatesof
enhancement’ of the DMannihilation cross-section (Hisanoetal. theJ-factorandcross-checkswiththenumericalresults.
4 Charbonnier,Combet,Daniel etal.
2.1.3 DMprofiles d J [dSph: d=20 kpc, r=1 kpc]
FortheDMhaloweuseageneralised(α,β,γ)Hernquistprofile alise 1 s
m
givenby(Hernquist1990;rDeh−nγen1993;rZhaαo1γ9−α96β): Nor0.8 J(αinγγt)==/J10(α.5max=5°)
ρ(r)=ρ 1+ , (6)
s r r
(cid:18) s(cid:19) (cid:20) (cid:18) s(cid:19) (cid:21) 0.6
where the parameter α controls the sharpness of the transition
from inner slope, limr→0dln(ρ)/dln(r) = γ, to outer slope 0.4
limr→∞dln(ρ)/dln(r) = β, and rs is a c−haracteristic scale. (Ji+1-Ji) / J(αmax), with ∆αint=0.25°
Inprinciple we could add an−additional parameter inorder toin- γ=0
troduce an exponential cut-off in the profile of Eq (6) to mimic 0.2 γ=1.5
the effects of tidal truncation, as proposed in e.g., the Aquarius
(Springeletal.2008)orViaLacteaII(Diemandetal.2008)sim- 0
0 1 2 3 4 5
ulations. However, the freedom to vary parameters rs, α and β αint [deg]
in Eq (6) already allows for density profiles that fall arbitrarily
steeplyat largeradius. Moreover, given thatour MCMC analysis Figure2.Finitesizeeffects:J asafunctionoftheintegrationangleαint
foradSphat20kpc(pointingtowardsthecentreofthedSph).Theblack
latershowsthattheouterslopeβisunconstrainedbytheavailable
solidlineisforacoredprofile(γ = 0)andthegreendashedlineisfora
dataandthattheJ-factordoesnotcorrelatewithβ,wechoosenot cuspyprofile(γ=1.5);botharenormalisedtounityatαint=5◦.
toaddfurthershapeparameters.
For profiles such as γ > 1.5, thequantity J fromthe inner
regionsdiverges. Thiscanbeavoidedbyintroducingasaturation alsoindicatedbythesymbolswhichshowthecontributionofDM
scaler ,thatcorrespondsphysicallytothetypicalscalewherethe shellsintwoangular bins—whereas the(green) hollow squares
sat
annihilation rate [ σv ρ(r )/m ]−1 balances the gravitational haveaspikydistributioninthefirstbin(γ =1.5),the(black)filled
sat χ
infallrateofDMpahrticiles(Gρ¯)−1/2(Berezinskyetal.1992).Tak- circles(γ =0)showaverybroaddistributionforJ.
ingρ¯tobeabout200timesthecriticaldensitygives The integration angle required to have asizeable fraction of
the signal depends on several parameters: the distance d of the
m 10−26cm3s−1
ρsat ≈3×1018 100GχeV × σv M⊙kpc−3. dteSgpraht,iothneainngnleersparroefidleessilroapbeleγs,inacnedtthhiessmcainleimraisdeisuscorns.taSmminaalltiinng-
(cid:16) (cid:17) (cid:18) h i (cid:19) (7)
backgroundγ-rayphotonsandmaximisesthesignaltonoise.Thus
Theassociatedsaturationradiusisgivenby
thetruedetectabilityofadSphwilldependonitsspatialextenton
ρ 1/γ thesky,andthusalsoond,γandrs.
r =r s r . (8)
sat s(cid:18)ρsat(cid:19) ≪ s
Thislimitisusedforallofourcalculations. 2.2.2 GenericdSphprofiles
AswillbeseeninSection5,theerrorsonthedensityprofilesofthe
MilkyWaydSphs arelarge,making itdifficulttodisentangle the
2.2 Motivationforagenericapproachandreferencemodels
interplay between the key parameters for detectability. Hence we
In many studies, the γ-ray flux (from DM annihilations) selectsome‘genericprofiles’toillustratethekeydependencies.
is calculated using the point-source approximation (e.g., Themostconstrainedquantityisthemasswithinthehalf-light
Bergstro¨m&Hooper 2006; Kuhlen 2010). This is valid so radiusr (typicallyafewtenthsofakpc),asthisiswheremostof
half
longastheinnerprofileissteep,inwhichcasethetotalluminosity thekinematicdatacomefrom(e.g.,Walkeretal.2009;Wolfetal.
ofthedSphisdominatedbyaverysmallcentralregion.However, 2010).FortheclassicalMilkyWaydSphs,thetypicalmasswithin
if the profile is shallow and/or the dSph is nearby, the effective rhalf 300pcisfoundtobeM300 107M⊙(Strigarietal.2008,
∼ ∼
sizeofthedSphontheskyislargerthanthepointspreadfunction — see also the bottom panel of Fig. 13). If the DM scale radius
(PSF)of the detector, and the point-source approximation breaks is significantly larger than this (r r ) and the inner slope
s half
≫
down. For upcoming instruments and particularly shallow DM γ &0.5,wecanapproximatetheenclosedmassby:
profiles, the effective size of the dSph may even be comparable
to the field of view of the instrument. This difference in the 4πρ r3 300pc 3−γ
rSaedcitailonex3t)e.nHteonfctehweesidgonanlotdaosessummaetttheratitnhetedrmSpshoifsadeptoeicntito-snou(srceee M300 ≃ 3−sγs (cid:18) rs (cid:19) ≈107M⊙. (9)
butratherderivesky-mapsfortheexpectedγ-rayflux. Theparameterρsisthusdeterminedcompletelybytheabovecon-
dition,ifwechoosethescaleradiusr andcuspslopeγ.
s
Table 1 shows, for several values of r and γ, the value re-
s
2.2.1 Illustration:acoredvscuspedprofile quiredforρ toobtaintheassumedM mass.Wefixα=1,β =
s 300
3butourresultsarenotsensitivetothesechoices.4Thevaluesofr
Fig. 2 shows J as a function of the integration angle α for a s
int
arechosentoencompasstherangeofr foundintheMCMCanal-
dSphat20kpc(lookingtowardsitscentre).Theblacksolidlineis s
ysis(seeSection5).Tofurtherconvinceourselvesthatthegeneric
foracoredprofile(γ =0)andthegreendashedlineisforacuspy
profile(γ =1.5);botharenormalisedtounityatα =5◦.Forthe
int
cuspyprofile, 100%ofthesignalisinthefirstbinwhileforthe 4 Fora different mass for the dSph, the results for J below have to be
∼
coredprofile,J buildsupslowlywithαint,and80%ofthesignal rescaledbyafactor(M3n0e0w/107M⊙)2sincethedensityisproportionalto
(w.r.t.thevalueforαint = 5◦)isobtainedforα80% 3◦.Thisis M300,whileJgoesasthedensitysquared.
≈
Darkmatterannihilationindwarfspheroidalgalaxiesandγ-rayobservatories:I. ClassicaldSphs 5
dSph is at d = 100 kpc. We note that our consideration of a
Table1.Therequired normalisation ρs tohaveM300 = 107M⊙ fora
γ = 0 smooth component withNFW sub-clumps isplausible if,
sampleof(1,3,γ)profileswithvaryingscaleradiusrs.
e.g., baryon-dynamical processes erase cusps in the smooth halo
butcannotdosointhesub-subhalos.ThetotalJ isthesumofthe
ρs(107M⊙kpc−3) smoothandsub-clumpdistributions.Thecentreisdominatedbythe
γ rs[kpc] 0.10 0.50 1.0
\ smooth component, whereas some graininess appears in the out-
0.00 224 25.8 16.02 skirtsof thedSph. Inthisparticular configuration, the‘extended’
0.25 196 18.6 10.22 signal from the core profile, when integrated over a very small
0.50 170 13.4 6.47 solid angle, could be sub-dominant compared with the signal of
0.75 146 9.5 4.06
NFWsub-clumpsthatithosts.Thediscussionofcross-constraints
1.00 125 6.7 2.52
betweendetectabilityofsub-halosoftheGalaxyvs.sub-clumpsin
1.25 106 4.7 1.54
thedSphisleftforafuturestudy.
1.50 88 3.2 0.92
Intheremainder of thepaper, wewillreplace forsimplicity
the calculation of J (α ) by its mean value, as we are pri-
subcl int
marilyinterested in ‘unresolved’ observations. Hence clumps are
profileswepresenthereareapossibledescriptionofrealdSphs,we not drawn from their distribution function, but rather J is
subcl
checked(notshown)usingtypicalstellarprofilesandpropertiesof calculatedfromtheintegrationofthespatialandluminhosity(aisa
theseobjects(i.e.,halflightradiusofafew100pc),thataflat 10 functionofthemass)distributions(seeAppendixB2).
km s−1 velocity dispersion profilewithinthe error bars isre∼cov-
ered.WealsostudybelowtheeffectofmovingthesedSphsfrom
adistanceof10kpcto300kpc,correspondingtothetypicalrange 2.3.1 RadialdependenceJ(θ)
coveredbytheseobjects.
Theradial dependence of J isshown inFig.4for four values of
γ (foranintegrationangleα = 0.01◦).Thedashedlinesshow
int
theresultforthesmoothdistribution;thedottedlinesshowthesub-
2.2.3 Sub-structureswithinthedSph
clumpcontribution;andthesolidlinesarethesumofthetwo.The
Structureformationsimulations inthecurrently favoured ΛCDM peakofthesignal istowardsthedSphcentre.Aslongasthedis-
(coldDMplusacosmological constant)cosmology findthatDM tributionof clumps isassumed tofollow thesmooth one, regard-
halosareself-similar,containingawealthofsmaller‘sub-structure’ lessofthevalueofγ,thequantity(1 f)2J (0)alwaysdomi-
sm
−
halosdowntoEarth-masshalos(e.g.Diemandetal.2005).How- nates(atleastbyafactorofafew)over J (0) .(Recallthatin
subcl
h i
ever,asemphasisedintheintroduction,suchsimulationstypically ourgenericmodels,alldSphshavethesameM .)Thescatterin
300
neglecttheinfluenceofthebaryonicmatterduringgalaxyforma- J (0)isabout4ordersofmagnitudeforγ [0.0 1.5],butonly
tot
∈ −
tion.ItisnotclearwhateffectthesehaveontheDMsub-structure afactorof20forγ [0.0 1.0].Beyondafewtenthsofdegrees,
∈ −
distribution.Forthisreason,weadoptamoregenericapproach.We J dominates.Thecrossingpointdependsonacombination
subcl
assesstheimportanceofclumpsusingthefollowingrecipe:5 ohfthecilumpmassfractionf,γ,r ,d,α .ThedependenceofJ
s int
onthetwolatterparametersarediscussedinAppendixC.Thera-
(i) we take a fraction f = 20% of DM mass in the form of
dialdependenceisasexpected:thesmoothcontributiondecreases
clumps;
fasterthanthatofthesub-clumpone,becausethesignalispropor-
(ii) thespatialdistributionofclumpsfollowsthesmoothone;
tionaltothesquaredspatialdistributioninthefirstcase,butdirectly
(iii) theclumpprofilesarecalculateda` laBullocketal.(2001)
proportionaltothespatialdistributioninthesecondcase.Halvingf
(hereafter B01), i.e. an ‘NFW’ profile (Navarro,Frenk&White
tomatchthefractionfromN-bodysimulationswouldhavea 25%
1996)withconcentrationrelatedtothemassoftheclumps.
effect on (1 f)2J , but decrease J by afactor 4, sothat
(iv) theclumpmassdistributionis M−a(a= 1.9),within − sm subcl
amassrangeMmin−Mmax =[10−6∝−106]M⊙. − tahneglceroinssF-oigv.er4.betweenthetwocomponentswouldoccuratalarger
Although these parameters arevery uncertain, they allow us
toinvestigatetheimpactofsubstructuresontheJ-factor.Theyare
2.3.2 Boostfactor
variedwithinreasonableboundsinSection2.3.2(andAppendixD)
to determine whether the sub-clump contribution can boost the Whetherornotthesignalisboostedbythesub-clumppopulation
signal. Note that a 20% clump mass fraction is about twice as is still debated in the literature (Strigarietal. 2007; Kuhlenetal.
largeasthefractionobtainedfromnumericalsimulations(see,e.g., 2008;Pierietal.2008;Pierietal.2009).Asunderlinedinthepre-
Springeletal. 2008). This generous fraction does not affect our vioussections,thesub-clumpcontributiontowardsthedSphcentre
conclusions,asdiscussedbelow. never dominates over the smooth one if the spatial profile of the
sub-clumpsfollowsthatofthesmoothdistribution,andiftheinte-
grationangleremainsbelowsomecriticalanglediscussedbelow.
2.3 JsmandJsubclforthegenericmodels Let us first define properly the parameters with respect to
which this boost is calculated, as there is sometimes some con-
Asanillustration,weshowinFig.3onerealisationofthe2Ddis-
fusionaboutthis.Here,wedefineitwithrespecttotheintegration
tribution of J from a generic core profile (γ = 0) with r = 1
s
kpc(sub-clumpparametersareasdescribedinSection2.2.3).The angleαint(thepointingdirectionisstilltowardsthedSphcentre):
(1 f)2J (α )+J (α )
5 MoredetailsabouttheclumpdistributionscanbefoundinAppendixB2. B(αint)≡ − smJsmi(nαtint) subcl int . (10)
Seealso,e.g.,Section2inLavalleetal.(2008)andreferencestherein,as Inmoststudies,theboosthasbeencalculatedbyintegratingoutto
weusethesamedefinitionsasthosegiveninthatpaper. theclumpboundary (i.e.,αainllt = Rvir/d).Buttheboostdepends
6 Charbonnier,Combet,Daniel etal.
1
-5]kpc1012 f=0.20 αint =0.01o
2 J [M1011 ssuubbccll:: ddPP//ddMV ∝∝ sMm-o1.o90th, rs as smooth (T1o-tfa)2l J
0.5 1010 Msub=[1.0e-06-1.0e+06] M sm
1e+10 subcl profile: NFW97+B01 <Jsubcl>
109
108
0 107
106 γ=0.0
1e+09 γ=0.5
[r=0.5 kpc]
105 γ=1.0 s
-0.5 γ=1.5
104
0 0.2 0.4 0.6 0.8 1
θ from dSph centre [deg]
Figure4.J asafunctionoftheangleθawayfromthedSphcentrefora
-1 1e+08 dSphat100kpcwithrs=0.5kpc(ρsisgiveninTable1).Theintegration
1-1 -0.5 0 0.5 1 angleisαint =0.01◦.Forthefourinnerslopevaluesγ,thevariouscon-
tributionstoJareshownassolid(total),dashed(smooth),anddottedlines
(sub-clumps).
0.5
1e+10 m
] / J+ Jsubcls 2.22 fss=uubb0cc.2ll::0 ddPP//ddVM ∝∝ sMm-o1.o90th, rs as smooth
0 2 J1-f)sm 11..68 Msubsucbcll =pr[o1f.0ilee-:0 N6-F1W.09e+7+06B]0 M1 r=0.1 kpc
-0.5 1e+09 ≡ [(Boost 11..42 rs=1 kγγγp===c001...050 s γγγγ====0110....5050
γ=1.5
1
0.8
-1 1e+08
0.6
-1 -0.5 0 0.5 1
1 10-2 10-1 1
α × (d / 100 kpc) [deg]
int
Figure5.Boostfactor asafunction ofαint ×(d/100kpc) forprofiles
sub-clumpsfollowsmooth(seeSection2.2.3):thedSphisatd=100kpc
0.5 (lines)ord=10kpc(symbols).
1e+10
crucially on α (the radial dependence of the smooth and sub-
int
clumpcontributionsdiffer,seeSection2.3.1).
0
WeplotinFig.5theboostfordifferentinnerslopesγ,where
a direct consequence of Eq. (C7) is the α d rescaling. For
int
1e+09 ×
r . 0.1kpc(regardlessofγ),orforγ & 1.5(regardlessofr ),
s s
thesignalisneverboosted.6 Forsmallenoughα ,B issmaller
-0.5 int
than unity, and if γ is steep enough, B (1 f)2. For large
≈ −
values,aplateauisreachedassoonasα d&R (takentobe3
int vir
kpchere).Inbetween,thevalueoftheboostdependsonr andγof
s
thesmoothcomponent.Goingbeyondthisqualitativedescriptionis
-1 1e+08
difficult,asthetoymodelformulaeofAppendix B2givesresults
-1 -0.5 0 0.5 1
Figure3.2Dview(xandyaxisareindegrees)ofJforthegenericdSph
with γ = 0 and rs = 1 kpc at d = 100 kpc (M300 = 107M⊙).
The sub-clumps are drawn from the reference model described in Sec- 6 Thedifferencebetweenthelevelofboostobservedforrs =0.1kpcor
tion 2.2.3, i.e. f=20%, sub-clump distribution follows smooth, and sub- rs =1kpccanbeunderstoodifwerecallthatthetotalmassoftheclump
clumpinnerprofileshaveNFWwithB01concentration.Fromtoptobot- isfixedat300pc,regardlessofthevalueofγorrs.Forrs=0.1kpc,ρs
tsoammepacnoelol:uαrisnctal=ei0s.t1a◦k,en0.f0o5r◦t,haentdhr0e.e01in◦t.eFgorarttihoensaankgeleosf(cJomispainrisuonnit,sthoef AOs(1J0sm9M∝⊙ρk2spwc−he3r)e,awsJhseurebas∝foρrs,rtshe=re1latkivpec,aρmsou∼ntOof(J10su7bMw⊙ithkprecs−p3e∼c)t.
M2 kpc−5). to Jsm is expected to decrease with smaller rs.This is indeed what we
⊙
observeinthefigure(solidvsdashedlines).
Darkmatterannihilationindwarfspheroidalgalaxiesandγ-rayobservatories:I. ClassicaldSphs 7
correcttoonlyafactorof 2(whichisinadequatetoevaluatethe d 0.9
boostproperly). ∼ cale 0.8
s
To conclude, the maximum value for sub-clump follows e
smoothis.2,andthisvalueisreachedonlywhenintegratingthe α r800.7
signalouttoRvir/d.Theboostcouldstillbeincreasedbyvarying 0.6
thesub-clumpproperties(e.g.,takingahigherconcentration).Con-
0.5
versely,ifdynamicalfrictionhascausedthesub-clumppopulation
tobecomemuchmorecentrallyconcentratedthanthesmoothcom- 0.4
ponent,thentheboostisdecreased.ThisisdetailedinAppendixD.
0.3
Forthe most realisticconfigurations, thereisnosignificant boost
whenaclumpmassfractionf=20%isused.Naturallythisresult 0.2
isevenmoretrueforthesmallerf foundinN-bodysimulationsso
0.1
wedisregardtheboostfortherestofthispaperandconsideronly
thesmoothcontribution. 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Inner slope γ
Figure6.Theconeangleencompassing80%oftheannihilationfluxatas
3 SENSITIVITYOFPRESENT/FUTUREγ-RAY functionoftheinnerslopeγ.Severaldifferentvaluesofrsanddistanced
OBSERVATORIES areshownforeachγ,allscaledby(1kpc/rs and100kpc/d).Thebest-fit
curveisalsoshown,correspondingtoEq.(16).
Major new ground-based γ-ray observatories are in the plan-
ning stage, with CTA (CTAConsortium 2010) and AGIS
(AGISCollaboration 2010) as the main concepts. As the designs where
oftheseinstrumentsarestillevolving,weadoptheregenericper-
formance curves (described below), close to the stated goals of X =log (PhotonEnergy/TeV), (14)
10
theseprojects.FortheLargeAreaTelescope(LAT)oftheFermiγ-
LS = log (Differential Sensitivity/ergcm−2s−1), LA =
raysatellite,theperformancefor1yearobservationsofpoint-like, 10
log (EffectiveArea/m2), and ψ is the 68% containment
highGalacticlatitudesourcesisknown(Fermi-LATCollaboration 10 68
radiusofthepoint-spread-function(PSF)indegrees.
2010),butnoinformationisyetavailableforlongerexposuresor
For the Fermi detector a similar simplified approach
for extended objects. We therefore adopt a toy likelihood-based
is taken, the numbers used below being those provided by
modelfortheFermisensitivity,tunedtoreproducethe1yearpoint-
Fermi-LATCollaboration(2010).Theeffectiveareachanges asa
sourcecurves.Wenotethatwhilstthisapproachresultsinapprox-
function of energy and incident angle to the detector, reaching a
imate performance curves for both the ground- and space-based
maximumof 8000cm2.Theeffectivetime-averagedareaisthen
instruments,itcapturesthekeydifferences(inparticularthediffer- ≈
ǫAΩ/4πandthedata-takingefficiencyǫ 0.8(duetoinstrument
encesincollectionareaandangularresolution)andillustratesthe ≈
dead-timeandpassagesthroughtheSouthAtlanticAnomaly).The
advantagesandlimitationsofthetwoinstrumenttypes,aswellas
pointspreadfunctionagainvariesasafunctionofenergy(witha
theprospectsforthediscoveryofDMannihilationindSphswithin
muchsmallerdependenceasafunctionofincidenceangle)varying
thenextdecade.
from10degreestoafewtenthsof adegreeovertheLATenergy
range. A rate of 1.5 10−5 cm−2 s−1 sr−1 (> 100 MeV) and
×
3.1 Detectormodels aphotonindexof2.1areassumedforthebackground. Thesensi-
tivityisthenestimatedusingasimplifiedlikelihoodmethodwhich
Thesensitivityofamajorfutureγ-rayobservatorybasedonanar- provides results within 20% of the sensitivity for a one year ob-
ray of Cherenkov Telescopes (FCA in the following, for ‘Future servationofapoint-likesourcegivenbyFermi-LATCollaboration
CherenkovArray’)isapproximatedbasedonthepoint-sourcedif- (2010).
ferentialsensitivitycurve(fora5σdetectionin50hoursofobser- Whilstbothdetectorresponsesareapproximate,thecompar-
vations)presentedbyBernlo¨hretal.(2008).Undertheassumption isonisstilluseful. Ourworkincorporatesseveral keyaspectsnot
thattheangularresolutionofsuchadetectorisafactor2betterthan considered in earlier studies, including the strong energy depen-
HESS(Funketal.2008)andhasthesameenergy-dependence,and dence of the angular resolution of both ground and space based
that theeffective collectionareafor γ-rays growsfrom104m2 at instruments in the relevant energy range of 1 GeV to 1 TeV and
30 GeV to 1 km2 at 1 TeV, the implied cosmic-ray (hadron and hencetheenergy-dependentimpactoftheangularsizeofthetarget
electron) background rate per square degree can be inferred and region.
thesensitivitythusadaptedtodifferentobservationtimes,spectral
shapesandsourceextensions.Giventhatthedesignofinstruments
suchasCTAarenotyetfixed,weconsider thatsuchasimplified
3.2 Relativeperformanceforgenerichalos
response,characterisedbythefollowingfunctionsisausefultool
toexplorethecapabilitiesofagenericnext-generationinstrument: UsingtheresultsfromSection2.2.2andthedetectorperformance
modelsdefinedabovewecanbegintoinvestigatethesensitivityof
future ACT arrays and the Fermi-LATdetector (over long obser-
LS= 13.1 0.33X+0.72X2, (11)
vationtimes)toDMannihilationindSphs. Thedetectabilityof a
− −
sourcedependsprimarilyonitsflux,butalsoonitsangularextent.
LA=6+0.46X 0.56X2, (12) The impact of source extension on detectability is dealt with ap-
−
proximately(ineach energybinindependently) byassuming that
ψ =0.038+exp (X+2.9)/0.61, (13) theopeningangleofaconewhichincorporates80%ofthesignal
68
−
8 Charbonnier,Combet,Daniel etal.
isgivenby 3-1] sm10-20 FCA HESS J=1012 M2 kpc-5
θ80 =qψ820+α280, (15) v> [c 20 h
where ψ = 1.25ψ isassumed for the FCA and interpolated σ 10-21
80 68 < 20 h
from values given for 68% and 95% containment for the LAT
200 h
Fermi-LATCollaboration(2010);hereα80isthe80%containment 10-22 200 h
angleofthehaloemission.Thevalidityofthisapproximation (at
the level of a few percent) has been tested (see Appendix E) by 10-23
convolving realistic halo profiles with a double Gaussian PSF as 1 yr
foundforHESS(Horns2005).An80%integrationcircleisclose Fermi
10-24
tooptimumforaGaussiansourceonaflatbackground(intheback-
10 yr
groundlimitedregime).Fig.6showsthe80%containmentradius
10-25
oftheannihilationfluxofgenerichalosasafunctionoftheinner 10-2 10-1 1 10
mχ [TeV]
slopeγ.Thisresultcanbeparametrisedas:
α80 =0.8◦(1−0.48γ−0.137γ2)(cid:18)1krspc(cid:19)(cid:18)100dkpc(cid:19)−1. (16) 3-1] s> [cm10-20 FCA>1 GeV J=1012 M2 kpc-5
v
It is clear that for abroad range of d, γ and rs the characteristic σ <10-21
angularsizeoftheemissionregionislargerthantheangularreso-
ltuotiaosnseosfstthheeinpsetrrfuomrmenatnscuenadseracfounnscitdioernatoiofnth.eItaisngthuelraerfosirzeecoriftitchael 10-22 >100 GeV
dSphaswellasthemassoftheannihilatingparticle. 10-23 Likelihood
Fig. 7 shows the relative sensitivity of Fermi and an FCA
within our framework as a function of the mass of the annihilat- Fermi
10-24
ing particle, adopting the annihilation spectrum given in Eq. (4),
withtheseveralpanelsillustratingdifferentpoints.FromFig.7top
(thecaseofapoint-likesignalfordifferentobservationtimes)itis 10-2510-2 10-1 1 10
clearthatFermi-LAThasaconsiderableadvantageforlowermass mχ [TeV]
aDnMFCpaArti(cil.ee.so(mveχr a≪5-110TyeeVa)romnisthsieotnimliefsectiamlee)foirnccoonmstpruarcitsioonnotof 3-1] sm10-20 FCA J=1012 M2 kpc-5
c
adeepACTobservationof200hours.Furthermore,Fermi-LATis > [
lessadversely affectedbytheangular extent of thetarget regions σ v10-21
(see Fig. 7 bottom), due to its modest angular resolution in the < 1o
energyrangewhereitislimitedbybackground, meaningthatthe
sourceextensioniswellmatchedtothePSFoftheinstrument.The 10-22 0.1o
middle panel of this figure illustrates the impact of different ap-
proachestotheanalysis.InthecasethatthereisaDMcandidate 10-23
Point-like
inferred from the discovery of supersymmetry at the LHC (quite
Fermi
possible on the relevant timescale) a search optimised on an as- 10-24
sumedmassandspectralshapecanbemade(solidcurves).How-
ever,allinstrumentsarelesssensitivewhenagenericsearchisun- 10-25
10-2 10-1 1 10
dertaken. Simpleanalyses usingallthephoton fluxabove afixed mχ [TeV]
energy threshold (arbitrarilyset to reduce background) are effec-
Figure 7. Approximate sensitivities of Fermi-LAT (blue lines), HESS
tiveonlyinarelativelynarrowrangeofparticlemass.Forexample
(black lines) and the FCA described above (red lines) to a generic halo
keeping only >100 GeV photons works well for ACTs for 0.3-3
withJ = 1012 M2 kpc−5,asafunctionofthemassoftheannihilating
TeVparticles;whereaskeepingallphotons>1GeVworksmoder- ⊙
particle andfortheannihilation spectrumofEq.(4).Top:Theimpactof
atelywellinthe0.1-0.2TeVrange,butismuchlesssensitivethan
observationtimeisillustrated:dashedlinesgivethe1yearand20hoursen-
thehigherthresholdcutovertherestofthecandidatedarkmatter sitivitiesforFermiandFCA/HESSrespectivelywhilethesolidlinesrefer
particle mass range. The features of these curves are dictated by to10year(200hour)observations.Middle:theimpactofanalysismeth-
theexpectedshapeoftheannihilationspectrum.FromEq.(4)the odsisconsideredfor5year(100hour)observations usingFermi(FCA).
peakphotonoutput (adoptingtheaveragespectrumforDManni- Solidlinesshowlikelihoodanalysesinwhichthemassandspectrumofthe
hilation)occursatanenergywhichisanorderofmagnitudebelow annihilatingparticleareknowninadvance,whiledashedanddottedlines
theparticlemass–effectivedetectionrequiresthatthispeakoccurs showsimpleintegralfluxmeasurementsabovefixedthresholdsof1GeV
(dashed)and100GeV(dotted).Notethatthe1GeVcutimpliesaccepting
within(orcloseto)theenergyrangeoftheinstrumentconcerned.
alleventsfortheFCA(wherethetriggerthresholdis 20GeV).Bottom:
The total annihilation flux from a dSph increases at smaller ≈
theimpactoftheangularextensionoftargetsources,asgivenbythehalo
distances as 1/d2 for fixed halo mass, making nearby dSphs at-
profileinFig.6isillustrated.Thesolidlinesreproducethelikelihoodcase
tractiveforDMdetection.However,asFig.7shows,theincreased
fromthemiddlepanelforapoint-like source,andwithvalues ofα80 of
angular size of such nearby sources raises the required detection 0.1◦(dashed)and1◦alsoshown.
flux.Fig.8illustratesthereductioninsensitivityforanFCAwith
respecttoapoint-likesourceforgenericdSphhalosasafunction
ofdistance,forinnerslopes,γ,ofzeroandoneandwithr fixedto
s
Darkmatterannihilationindwarfspheroidalgalaxiesandγ-rayobservatories:I. ClassicaldSphs 9
Spoint-like102 fwHoeerrmdeivwfoiedrerteht-heceavvleceluloolcacititetyythdseaissmterpipblreuotifiinoltenossc.wiSricpthueolcauirfitbcaiadnloslypc,toifnnogtraaianniygnigvpeaanrptipdcruSolpxah-r
/ Sγθ)(80 iemstaimtealyteetqhueaslencuomndbevresloocfitmyemmobmeernstta(rssq,u7aarenddvweiltohciintyeadcishpebrisniowne)
as:
γ=0
N
10 Vˆ2 = 1 [(V Vˆ )2 σ2], (17)
γ=1 h i N 1 i−h i − i
− i=1
Ideal background X
whereN isthenumber ofmember starsinthebin.Wehold V
Ring background h i
fixed for all bins at the median velocity over the entire sample.
1 Foreachbinweuseastandardbootstrap re-samplingtoestimate
theassociatederrordistributionfor Vˆ2 ,whichisapproximately
10 102 Distance [kpc] Gaussian.Fig.9displaystheresultinhgveilocitydispersionprofiles,
Vˆ2 1/2(R),whicharesimilartopreviouslypublishedprofiles.
Figure8.RelativeDMannihilationdetectionsensitivityfora100hourFCA h iIn order to relate these velocity dispersion profiles to dSph
observation,asafunctionofdSphdistancefordifferentinnerslopesγand masses,wefollowW09inassumingthatthedatasampleineach
withrs fixedto1kpc.Thesensitivity forarealistic approachusingθ80 dSphasingle,pressure-supported stellarpopulation thatisindy-
isgivenrelativetothesensitivitytoapoint-likesourcewiththesameflux.
namicalequilibriumandtracesanunderlyinggravitationalpoten-
Largervaluescorrespondtopoorerperformance(largervaluesofthemin-
tialdominated bydark matter.Implicitistheassumption that the
imum detectable flux). The assumed spectral shape is again as given by
orbital motions of stellar binary systems contribute negligibly to
Eq.(4)withmχ = 300GeV.Thissensitivityratiodependsonthestrat-
egyusedtoestimatethebackgroundlevelatthedSphposition.Thedashed themeasuredvelocitydispersions.8 Furthermore,assumingspher-
lines showtheimpactofusinganannulus between 3.5◦ and4.0◦ ofthe icalsymmetry,themassprofile,M(r),of theDMhalorelatesto
dSphcentreasabackgroundcontrolregion.Thesolidlineassumesthatthe (moments of) the stellar distribution function via the Jeans equa-
backgroundcontrolregionliescompletely outsidetheregionofemission tion:
fromthedSph.
1 d β(r)v¯2 GM(r)
(νv¯2)+2 r = , (18)
νdr r r − r2
1kpc,relativetotheassumptionofthefullannihilationsignaland where ν(r), v¯2(r), and β β(r) 1 v¯2/v¯2 describe
apoint-likesource.Evenforγ = 1,thepoint-likeapproximation the 3-dimensiornal density, rrad≡ial velocity≡disp−ersioθn, arnd orbital
leadstoanorderofmagnitudeoverestimateofthedetectionsensi- anisotropy,respectively,ofthestellarcomponent.Projectingalong
tivityfornearby( 20kpc)dSphs.Afurthercomplicationishowto the l.o.s., the mass profile relates to observable profiles, the pro-
∼
establishthelevelofbackgroundemissionarisingfromtheresidual jectedstellardensityI(R)andvelocitydispersionσ (R),accord-
p
non-γ-raybackground.Acommonmethodinground-basedγ-ray ingto(Binney&Tremaine2008,BT08hereafter)
astronomyistoestimatethisbackground fromanannulusaround
thetargetsource(see,e.g.,Bergeetal.2007).Thedashedlinesin σ2(R)= 2 ∞ 1 β R2 νv¯r2r dr. (19)
Fig.8showtheimpactofestimatingthebackgroundusinganan- p I(R) − r r2 √r2 R2
nulusbetween3.5◦ and4.0◦ fromthetarget.Thisapproachhasa ZR (cid:18) (cid:19) −
Noticethatwhileweobservetheprojectedvelocitydispersionand
modest impact on sensitivity and isignored in the following dis-
stellar density profiles directly, the l.o.s. velocity dispersion pro-
cussionsasitreducesthedetectablefluxbutalsoθ andleadstoa
80 files provide no information about the anisotropy, β(r). There-
smallimprovementinsomecases.
fore werequire an assumption about β(r); here we assume β =
constant, allowing for nonzero anisotropy in the simplest way.
Forconstant anisotropy, theJeans equationhasthesolution(e.g.,
4 JEANS/MCMCANALYSISOFDSPHKINEMATICS Mamon&Łokas2005):
∞
4.1 dSphkinematicswiththesphericalJeansequation νv¯2 =Gr−2βr s2βr−2ν(s)M(s)ds. (20)
r
Extensive kinematic surveys of the stellar components of dSphs Zr
have shown that these systems have negligible rotational sup- WeshalladoptparametricmodelsforI(R)andM(r)andthenfind
port (with the possible exception of the Sculptor dSph, see valuesoftheparametersofM(r)that,viaEqs.(19)and(20),best
Battagliaetal.2008).IfweassumethatthedSphsareinvirialequi- reproducetheobservedvelocitydispersionprofiles.
librium,thentheirinternalgravitationalpotentialsbalancetheran-
dommotionsoftheirstars.Inorder toestimatedSphmasses,we
consider herethebehaviour ofdSphstellarvelocitydispersionas 7 KinematicsamplesareoftencontaminatedbyinterlopersfromtheMilky
afunctionofdistancefromthedSphcentre(analogoustorotation Wayforeground.FollowingW09,wediscardallstarsforwhichthealgo-
curvesofspiralgalaxies).Specifically,weusethestellarkinematic rithmdescribed byWalkeretal.(2009)returnsamembershipprobability
lessthan0.95.
data of Walkeretal. (2009) for the Carina, Fornax, Sculptor and
8 Olszewskietal. (1996)and Hargreavesetal. (1996)conclude that this
SextansdSphs,thedataofMateoetal.(2008)fortheLeoIdSph,
assumptionisvalidfortheclassicaldSphsstudiedhere,whichhavemea-
and data fromMateo et al. (inpreparation) for the Draco, Leo II sured velocity dispersions of 10 km s−1. This conclusion does not
and Ursa Minor dSphs. Walkeretal. (2009, W09 hereafter) have necessarily apply to recently-∼discovered ‘ultra-faint’ Milky Way satel-
calculated velocity dispersion profiles from these same data un- lites,whichhavemeasuredvelocitydispersionsassmallas 3kms−1
∼
der the assumption that l.o.s. velocity distributions are Gaussian. (McConnachie&Coˆte´2010).
10 Charbonnier,Combet,Danielet al.
Figure 9. Velocity dispersion profile data for the 8 classical dSphs, obtained as described in the text (the impact of the binning choice is discussed in
AppendixH1).Thesolidlinescorrespondtothebest-fitmodelsfortheinnerslopewhenγisleftfree(dark),γisfixedto1(blue),andγisfixedto0(red).
Becauseofthelargedegeneraciesamongthehaloparameters(seeSection5.1foralist),wedonotlistthecorrespondingbest-fitparameters.Themotivation
forshowingtheseprofilesistoillustratethatourhalomodeliscapableofdescribingthekinematicdata,andthattheinnerprofileisnotconstrainedbythe
data.
4.1.1 StellarDensity thatasteeperouterslopeorasteeperinnerslopeforthelightprofile
leavesunchangedtheconclusions(seeAppendixH2).
Stellar surface densities of dSphs are typically fit by Plummer
(1911), King (1962) and/or Sersic (1968), profiles (e.g.,
Irwin&Hatzidimitriou 1995). For simplicity, we adopt here the
Plummerprofile:
L 1 4.1.2 Darkmatterhalo
I(R)= , (21)
πrh2alf [1+R2/rh2alf]2 FortheDMhalowefollowW09inusingageneralisedHernquist
whichhasjusttwofreeparameters:thetotalluminosityLandthe profile,asgiven by Eq.(6).Intermsof theseparameters, i.e,the
projected9 half-light radius rhalf. Given spherical symmetry, the densityρsatscaleradiusrs,plusthe(outer,transition,inner)slopes
Plummerprofileimpliesa3-dimensionalstellardensity(BT08)of: (α,β,γ),themassprofileis:
ν(r)=−π1 Zr∞ ddRI √Rd2R−r2 = 4π3rLh3alf [1+r2/1rh2alf]5/(222.) M(r)=4πZ0rs2ρ(s)ds= 43π−ρsrγs3(cid:18)rrs(cid:19)3−γ (23)
3 γ β γ 3 γ+α r α
Salilnrcaediwi(eaallsmsuemaseutrheadtdDSMphsdohmavienacteenstrtahlemgarsasv-ittoa-tliiognhatlrpaotitoesn&tial10at, 2F1(cid:20) −α , −α ; −α ;−(cid:18)rs(cid:19) (cid:21),
e.g.,Mateo1998),thevalueofLhasnobearingonouranalysis.
where F (a,b;c;z)isGauss’hypergeometricfunction.
2 1
Weadoptvaluesofr (andassociatederrors)fromTable1inthe
half Eq. (6) includes plausible halo shapes ranging from the
publishederratumtoW09;thesedataoriginallycomefromthestar
constant-density ‘cores’ (γ = 0) that seem to describe rotation
count study of Irwin&Hatzidimitriou (1995). We have checked
curvesofspiralandlow-surface-brightnessgalaxies(e.g.,deBlok
2010 and references therein) to the centrally divergent ‘cusps’
9 ForconsistencywithWalkeretal.(2009)wedefinerhalf astheradius (γ >0)motivatedbycosmologicalN-bodysimulationsthatmodel
ofthecircleenclosinghalfofthedSphstellarlightasseeninprojection. onlytheDMcomponent. For(α,β,γ) = (1,3,1)Eq.(6)isjust
Elsewherethisradiusiscommonlyreferredtoasthe‘effectiveradius’. thecuspyNFW(Navarro,Frenk&White1996,1997)profile.