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Mon.Not.R.Astron.Soc.000,1–15(2007) Printed5January2009 (MNLATEXstylefilev2.2) CERN-PH-TH/2007-202 Constraining DM properties with SPI Alexey Boyarsky1⋆, Denys Malyshev2,3, Andrey Neronov4⋆, Oleg Ruchayskiy5 9 0 1CERN,PH/TH,CH-1211Geneve23,Switzerland 0 2BogolyubovInstituteforTheoreticalPhysics,Kiev,03780,Ukraine 2 3DublinInstituteforAdvancedStudies,31FitzwilliamPlace,Dublin2,Ireland 4INTEGRALScienceDataCenter,Chemind’Ećogia16,1290Versoix,Switzerland n GenevaObservatory,51ch.desMaillettes,CH-1290Sauverny,Switzerland a 5EćolePolytechniqueFe´de´raledeLausanne,InstituteofTheoreticalPhysics, J FSB/ITP/LPPC,BSP720,CH-1015,Lausanne,Switzerland 3 ] h p - o ABSTRACT r Using the high-resolution spectrometer SPI on board the International Gamma-Ray Astro- t physics Laboratory (INTEGRAL), we search for a spectral line produced by a dark matter s a (DM)particlewithamassintherange40keV <MDM <14MeV,decayingintheDMhalo [ oftheMilkyWay.TodistinguishtheDMdecaylinefromnumerousinstrumentallinesfound intheSPIbackgroundspectrum,westudythedependenceoftheintensityofthelinesignal 2 ontheoffsetoftheSPIpointingfromthedirectiontowardtheGalacticCentre.Afteracritical v 2 analysis of the uncertaintiesof the DM density profile in the inner Galaxy,we find that the 2 intensityoftheDMdecaylineshoulddecreasebyatleastafactorof3whentheoffsetfrom 9 the Galactic Centre increases from0◦ to 180◦. We find that such a pronouncedvariationof 4 the line flux acrossthe sky is notobservedfor any line, detected with a significance higher . than3σintheSPIbackgroundspectrum.PossibleDMdecayoriginisnotruledoutonlyfor 0 1 theunidentifiedspectrallines,havinglow(∼ 3σ)significanceorcoincidinginpositionwith 7 theinstrumentalones.Intheenergyintervalfrom20keVto7MeV,wederiverestrictionson 0 theDMdecaylineflux,impliedbythe(non-)detectionoftheDMdecayline.Foraparticular v: DMcandidate,thesterileneutrinoofmassMDM,wederiveaboundonthemixingangle. Xi Keywords: methods:dataanalysis–techniques:spectroscopic–Galaxy:halo–darkmatter; r a 1 INTRODUCTION pact halo objects (MACHOs) – constitute dominant fraction of mass in the halo (Alcocketal. 2000; Lasserreetal. 2000; Alard DarkmatterintheUniverse 1999). The only non-baryonic DM candidate in the SM can- There is a vast body of evidence, suggesting that the large frac- didates – (left-handed) neutrino – is ruled out from the large tionofmatterintheUniverseexistsintheformoftheDarkmat- scale structure (LSS) considerations (see e.g. Bondetal. 1980; ter(DM).However,whilethetotaldensityoftheDMismeasured Hannestad&Raffelt2004;Crottyetal.2004). withaveryhighprecision(ΩDMh2 = 0.105+−00..000079,Spergeletal. What are the properties of a successful DM candidate? 2007),littleisknownaboutitspropertiesapartfromthis.Thepos- First of all, this particle should be massive. Many extensions sibilitythattheDMiscomposedoftheStandardModel(SM)par- of the SM present the DM candidates with the masses rang- ticleshasbeenruledout foralongtimealready. Indeed, theDM ing from 10−10 eV (massive gravitons, Dubovskyetal. 2005) cannot be made out of baryons, as producing such an amount of and 1∼0−6 eV (axions) to hundreds of GeV (WIMPs) and ∼ baryonic matter would require drastic modifications of the sce- evento1013GeV(WIMPZILLA,Kuzmin&Tkachev1998,1999; nario of the Big Bang nucleosynthesis (BBN), which otherwise Chungetal.1999).ForareviewofparticlephysicsDMcandidates successfully describes the abundance of light elements (see for seee.g.Bergstrom(2000);Bertoneetal.(2005);Carretal.(2006). example Dar 1995). Recent microlensing experiments rule out Secondly, there should exist mechanisms of DM production thepossibility that another type of baryonic DM– massive com- withthecorrectabundances.Theproductionmechanisminpartic- ular determines the velocity distribution of particles in the early Universe.Thisvelocitydistributioncan,inprinciple,beprobedex- ⋆ OnleaveofabsencefromBogolyubovInstituteforTheoreticalPhysics, perimentally.Namely, ifduringthestructureformationepoch the Kiev,Ukraine DM particles have velocities, comparable to the speed of sound 2 A. Boyarskyet al. in the baryon-photon plasma, they “erase” density fluctuations at Bertoneetal. (2007); Zhangetal. (2007) (see also the book scales, smaller than the distance, they have traveled (called the byKhlopov1997). free-streaming length). To differentiate various models in accor- dancewiththisproperty,theDMcandidateswiththenegligibleve- locitydispersion(and,correspondingly, free-streaming)arecalled cold DM (CDM), while those with the free-streaming of the order of 1Mpc are considered to be warm (WDM).1 It is pos- SterileneutrinoDM ∼ sible to constrain the free-streaming length of a particular DM It was noticed long ago that the right-handed (or as it is often candidate by probing the structure of the Universe at galaxy-size calledsterile)neutrinowiththemassinthekeVrangewouldrep- scales.ThiscanbedonethroughtheanalysisoftheLyman-αfor- resent a viable DM candidate (Dodelson&Widrow 1994). Such estdata(Huietal.1997).Lyman-αanalysisputsanupper bound a neutrino would interact withthe rest of the matter only via the on the free-streaming of the DM particles (Hansenetal. 2002; quadraticmixingwithleft-handed(active)neutrinosandtherefore Vieletal. 2005; Seljaketal. 2006; Vieletal. 2006; Vieletal. (althoughnotstable)couldhavecosmologicallylonglife-time.At 2007).Itshouldbenotedhoweverthatcurrentlyexistinginterpreta- thesametime,itcouldbeproducedintheearlyUniversewiththe tionoftheLyman-αdataismodel-dependent,as,apartfromanum- correctabundances(Dodelson&Widrow1994;Shi&Fuller1999; berofastrophysicalassumptions(seeHuietal.1997)andcompli- Shaposhnikov&Tkachev2006).Oneofthedecaychannelsofthe catedhydrodynamicsimulations,itreliesonaprioriassumptions unstablesterileneutrinos includesemission ofphotons of theen- aboutthevelocitydistributionoftheDMparticles. ergyequaltohalfofthesterileneutrinorestenergy.Thispotentially A way to differentiate between CDM and WDM models providesapossibilitytoobservethedecaysofDMsterileneutrinos would be to compare the numerical simulations of the DM dis- viadetectionofacharacteristicspectrallineinthespectraofastro- tribution in the Milky Way-type galaxies with the actual obser- physicalobjectswithlargeDMconcentration. vations.However, theresolutionoftheN-bodysimulationsisnot RecentlythisDMcandidatehasattractedmuchattention(see yet sufficient to answer the questions about e.g. the DM density e.g.Shaposhnikov(2007)andreferencestherein).Itwasfoundthat profiles in dwarf satellite galaxies. Moreover, most of the simu- averymodestandnaturalextensionoftheSMby3right-handed lationsincludeonlycollisionlessDMparticles,anddonot model neutrinos(makingtheSMmoresymmetricasallSMfermions,in- thebaryonsandtheirfeedbackonthegalaxystructureformation. cludingneutrino,wouldhavenowtheirleftandrighthandedcoun- TheseproblemsarenotsolvedevenfortheCDMsimulations,and terparts)providedaviableextensionofthetheory,capableofsolv- WDMsimulationshaveadditionalseriousdifficulties.Fromanob- ingseveral“beyondtheSM”problems.Firstofall,suchanexten- servationalpointofview,ithasbeenarguedforsometimealready sionmakes neutrinosmassiveandthusperhaps providesthesim- thatthereisadiscrepancybetweenCDMsimulationsandobserva- plestandthemostnaturalexplanationofthephenomenonof“neu- tions(seee.g.Moore1994;Mooreetal.1999;Klypinetal.1999; trino oscillations” (see e.g. Foglietal. (2006); Strumia&Vissani Bodeetal.2001;Avila-Reeseetal.2001;Goerdtetal.2006)Ithas (2006); Giunti (2007) for reviews). The smallness of neutrino beenclaimedrecentlythatanumberofrecentobservationsofdwarf masses in this model (called νMSM in Asaka&Shaposhnikov satellite galaxies of the Milky way and Andromeda galaxy seem 2005)isachievedbytheusualsee-sawmechanismwithMajorana to indicate the existence of the smallest scale at which the DM massesofright-handedneutrinosbeingbelowelectroweakscale.2 exists (Gilmoreetal. 2006, 2007; Gilmore 2007; Koposovetal. Secondly, if two heavier sterile neutrinos (N and N ) 2007). However, this statement and the interpretation of the ob- 2 3 are almost degenerate in mass and have their masses between servations are still subject to debate (Klimentowskietal. 2007; (100)MeVand (20)GeV,theνMSMprovidesthemechanism Penarrubiaetal. 2007; Strigarietal. 2007; Simon&Geha 2007). O O ofgeneratingthebaryonasymmetryoftheUniverse.Thirdly,the ThereforeitistooearlytosaywhatkindofDMmodelsisfavoured lighteststerileneutrinoN canhavearbitrarymassandarbitrarily bycomparingsimulationsandobservations. 1 weakcouplingwiththe(active)neutrinosector.Atthesametime, UsuallyitisalsonecessaryfortheDMcandidatetobestable. itcanbeproducedintheearlyUniverseinthecorrectamounts.It Forthemostpopular DMcandidate–weaklyinteractingmassive representsthereforetheDMparticleintheνMSM.Thus,altogether particles(WIMPs),thisisrelatedtothefactthattheparticlesof ∼ theνMSMrepresents(arguably)thesimplestextensionoftheSM, electroweakmass,havingweakstrengthinteractionwithSMmatter capableofexplainingthreeimportantquestions:originandsmall- (requiredtoproducethecorrectamountofDM),woulddecaytoo nessofneutrinomasses,baryonasymmetryintheUniverseandthe fastandwouldnotbe“dark”.If,however,theDMparticleinteracts existenceoftheDM. withtheSMmoreweaklythanWIMPs,itcouldwellhaveafinite (althoughcosmologicallylong)lifetime. There exist several unstable (decaying) DM candidates e.g. gravitino (Borganietal. 1996; Baltz&Murayama 2003; Roszkowskietal. 2005; Cerdenoetal. 2006; Cembranosetal. ExistingrestrictionsonsterileneutrinoDMparameters. 2006; Lolaetal. 2007). In this paper we will concentrate mainly on one candidate, the sterile neutrino (although our results Whatarethecurrentrestrictionsonparameters(massandmixing) will be applicable for any type of decaying DM). Constraints of sterile neutrino DM? First of all sterile neutrino mass should on the decaying DM were analyzed in deRujula&Glashow (1980); Berezhianietal. (1987); Doroshkevichetal. (1989); Berezhianietal. (1990); Berezhiani&Khlopov (1990); 2 The fact that the νMSM does not introduce any new scale above the 1 Theleft-handed neutrino wouldrepresent hotDMinthisterminology, electroweakone,makesthistheoryespeciallyappealingfromthepointof i.e.theDMwiththefree-streaminglength≫1Mpc. viewofitsexperimentalverification/falsification. ConstrainingDM propertieswithSPI 3 satisfytheuniversalTremaine-Gunnlowerbound:3M &300 DM − 500eV.4 Next,asthesterileneutrinopossessesthe(two-body)radiative decaychannel:N ν+γ,theemittedphotonwouldcarrytheen- 1 → ergyE =M /2.Alargefluxofsuchphotonsisexpectedfrom γ DM thelargeconcentrationsoftheDMsterileneutrinos,likegalaxies orgalaxyclusters. Recently an extensive search of the DM decay line in the region of masses M . 20 keV was conducted, using the DM data of Chandra (Riemer-Sørensenetal. 2006; Boyarskyetal. 2006d; Abazajianetal. 2007) and XMM-Newton (Boyarskyetal. 2006a,b,c; Watsonetal. 2006; Boyarskyetal. 2007). The region of soft X-ray (down to energies 0.2 keV) was explored by Boyarskyetal. (2007) with the use of the wide field of view spectrometer (McCammonetal. 2002). The non-observation of the DM decay line in X-ray, combined with the first principles calculation of DM production in the early Universe (Asakaetal. 2007), implies that the Dodelson&Widrow (1994) (DW) scenario can work only if the sterile neutrino mass is below 4 keV (Boyarskyetal. 2008). If one takes into account recent lower bound on the mass of sterile neutrino DM in the DW scenario M > 5.6 keV (Vieletal. 2007), it seems that the pos- DM sibility that all the DM is produced via DW scenario is ruled Figure1.ComparisonofsensitivitytowardsthesearchofthenarrowDM decaylinefordifferent instrumentswiththewideFoV.Diagonalstraight out(Boyarskyetal.2008).Thepossibilitythatonlyfractionofthe linesshowtheimprovementofsensitivity(byafactor,markedontheline) DMisproduced viaDWmechanismremainsopen(Palazzoetal. ascomparedwiththeHEAO-IA4lowenergydetector(LED),takenasa 2007). reference. There are other viable mechanisms of DM production, in- cludinge.g.resonantoscillationproductioninthepresenceoflep- tonasymmetries(Shi&Fuller1999).SterileneutrinoDMcanbe When the preparation of this paper was at its final producedbythedecayoflightinflaton(Shaposhnikov&Tkachev stage, Yu¨kseletal. (2008, hereafter Y07) published their work, 2006) or in a similar model with the different choice of parame- whichused theresultsof Teegarden&Watanabe(2006,hereafter ters(Kusenko2006;Petraki&Kusenko2007).Thesemechanisms T06)toplacerestrictionsontheparametersofsterileneutrinoDM are currently not constrained and remain valid for DM particles intherange40 700keV.WediscussitinmoredetailsinSec- − withthemassesinthekeVrangeandabove. tion6. The search for the DM decay line signal produced by ster- ileneutrinos withmassesabove 20 keV iscomplicated bythe ∼ SPIspectrometer absence of the focusing optics telescopes (similar to Chandra or XMM-Newton) in the hard X-ray and γ-ray domain of the spec- Theabsenceofthefocusingopticssignificantlyreducesthesensi- trum. For example, the existing restrictions in the 20 100 keV tivityof thetelescopesoperating inthehard X-ray/softγ-rayen- − mass range (Boyarskyetal. 2006a,c) are derived from the obser- ergyband. Most of theinstrumentsoperating inthisenergy band vationsofdiffuseX-raybackground,withthehelpofnon-imaging usecollimatorsand/orcodedmaskstodistinguishsignalsfromthe instruments,HEAO-I(Gruberetal.1999).Thecurrentstatusofas- sourcesontheskyfromtheinstrumentalbackground.Contraryto trophysicalobservationsinsummarizedinRuchayskiy(2007). thefocusingopticstelescopes,boththesourceandbackgroundsig- In this paper we use the spectrometer SPI on board of IN- nalsarecollectedfromtheentiredetector,whichsignificantlyin- TEGRAL satellite to place restrictions on parameters of decay- creasestheirreduciblebackground. ing DM in the mass range 40 keV 14 MeV. This range Thefocusingopticsenablestosignificantlyreducetheback- − of masses is interesting, for example, the sterile neutrinos, pro- groundonlyinthestudiesofpointsources.Ifthesourceunderin- duced in the early Universe in the presence of large lepton vestigationoccupiesalargefractionofthesky(e.g.theentireMilky asymmetries (Shi&Fuller 1999) or through the inflaton de- Waygalaxy),theperformanceofthefocusingandnon-focusingin- cay(Shaposhnikov&Tkachev2006).Itisalsorelevantforthecase strumentswiththesamedetectorcollectionareaare,infact,com- ofgravitinoDM(Pagels&Primack1982;Bondetal.1982). parable. In the case of an extended source, emitting a narrow spec- tralline,anefficientwayofreductionofinstrumentalbackground 3 In its simplest form the Tremaine-Gunn bound comes from the is via the improvement of the spectral resolution of the instru- fact that for the fermions there is a maximal density in the phase ment(inthecaseofabroadcontinuumbackgroundspectrum,the space (Tremaine&Gunn 1979; Dalcanton&Hogan 2001) and therefore number of background counts attheenergyof thelineispropor- theobservedphase-spacedensityinvariousDMdominatedsystemsshould tionaltothespectralresolution∆E).Thebestpossiblesensitivity belessthatthis(massdependent)bound. isachievedwhenthespectralresolutionreachestheintrinsicwidth 4 AstrongerlowerboundfromLy-α(Seljaketal.2006;Vieletal.2006; ofthespectralline(seeFig.1forthecaseofwideFoVinstruments Vieletal. 2007) can be obtained in the case of the particular production mechanisms – the Dodelson-Widrow scenario (Dodelson&Widrow andBoyarskyetal.(2007)forthecaseofnarrowFoVinstruments). 1994).Forotherpossibleproductionmechanisms(e.g.Shi&Fuller1999; In the case of the line produced by the DM decaying in the Shaposhnikov&Tkachev2006)theLy-αconstraintsshouldbereanalyzed. MilkyWayhalo,thelinewidthisdeterminedbytheDopplerbroad- 4 A. Boyarskyet al. R H Figure3.TheeffectiveareaoftheSPIdetectorforanon-axissource,asa functionofthephotonenergy.Theplotisproducedbycollectivetheon-axis effectiveareasofthe17SPIdetectorsfromtheinstrumentalcharacteristics files. R det 2 THEEXPECTEDSIGNALFROMTHEDMDECAYIN THEHALOOFTHEMILKYWAY. Theexpected surface brightness of theDM decay lineinagiven Figure2.ThegeometryoftheSPIFoV. directionontheskyisafunctionoftheangulardistanceφbetween thegivendirectionontheskyandthedirectiontowardstheGalactic center(GC).ItcanbecalculatedbytakingtheintegraloftheDM eningbytherandommotionoftheDMparticles.Thevelocitydis- densityprofileρDM(r)alongthelineofsight(“columndensity”) persionoftheDMmotioninthehaloisabouttherotationvelocity ∞ oftheGalacticdisk,v 200km/s.ThismeansthatDopplerbroad- (φ)= dzρ r2 2zr cosφ+z2 , (4) eningoftheDMdecay∼lineisabout SDM Z DM„q ⊙− ⊙ « 0 ∆E v 10−3 . (1) wherer⊙ ≃ 8.5kpcisthedistancefromtheSolarsystemtothe E ∼ c ≃ GC.Angleφisrelatedtothegalacticcoordinates(b,l)via Thus,theoptimalspectralresolutionofaninstrumentsearchingfor cosφ=cosb cosl. (5) theDMdecaylineproducedbytheMilkyWayDMhaloshouldbe Thus, the galactic center corresponds to φ = 0◦, the anti-center ∆E 10−3E. φ = 180◦,andthedirectionperpendiculartothegalacticplaneto ≃ Such optimal spectral resolution isalmost achieved withthe φ=90◦.TheexpectedDMfluxisgiventhenby spectrometer SPIonboard ofINTEGRALsatellite,whichhasthe dF (φ) Γ E maximal spectral resolving power of E/∆E 500 and works DM = DM γ (φ), (6) ≃ dΩ 4πM SDM inthe energy range 20 keV –8 MeV (Vedrenneetal.2003).SPI DM isa“codedmask”typeinstrumentwithanarrayof19hexagonal whereΓDMistheDMdecayrate shaped Ge detectors (of which only 17 are operating at the mo- Ingeneral, thesurface brightnessFDM(φ)isvariable across ment). thetelescopeFoV.Thisisespeciallytrueforawidefieldofview TheSPItelescope consists of acoded maskinscribed intoa (FoV) instruments (like SPI). In order to calculate the detector circle of the radius R = 39 cm, placed at the height H = countrate,onehastointegrateflux(6)overtheFoVandoverthe mask 171cmabovethedetectorplaneandofthedetector,whichhasthe (effective)detectorareaandthendividebytheenergyofthepho- shape of a hexagon inscribed into a circle of the radius Rdet tons,Eγ =MDM/2: ≃ 15.3cm(seeFig.2).Theportionoftheskyvisiblefromeachpoint A (E α,β)dF hoafsthteheSrPefIodreeteacntgourl(atrhdeisaom-ceatellredfullycodedfieldofview,FCFOV) R=FZoZVdαdβ eff Eγγ| dΩDM`φ(α,β)´, (7) where (α,β) are the angular coordinates in the FoV, A is the eff R R ΘFCFOV =2arctan» maskH− det–≈16◦ , (2) teifofnec(tαiv,eβa)r.eaatenergyEγforthephotons,comingfromthedirec- TheeffectiveareaoftheSPIdetector(whichisdeterminedby whiletheportionoftheskyvisiblebyatleastsomeofthedetectors thetransparencyofthemaskandthequantumefficiencyofthede- (thepartiallycodedfieldofview,PCFOV)is tector)changeswiththephotonenergy.Foranon-axispointsource, Θ =2arctan Rmask+Rdet 35◦ . (3) dF/dΩ(α,β)=f0δ(α)δ(β), PCFOV » H –≈ theintegralofEq.(7)reducestof A ,whereA (E )is 0 eff,on eff,on γ The solid angle spanned by the cone with this opening angle is thedetectoreffectiveareaforanon-axissource.Itsdependenceon energyE isshownonFig.3.5 Ω =2π 1 cos(Θ /2) 0.29(seeFig.2).Widefield γ PCFOV “ − PCFOV ”≃ ofviewmakestheSPItelescopesuitableforthestudyofthevery extendedsources,liketheMilkyWayDMhalo. 5 The on-axis effective area is calculated by summing the energy- ConstrainingDM propertieswithSPI 5 2.1 ModelingtheDMhalooftheGalaxy TheDMhalooftheGalaxyhasbeenextensivelystudied(seee.g. 2 SPI effective area [cm ] at 100 keV Kravtsovetal.1998;Klypinetal.2002;Battagliaetal.2005).Var- 25 160 iousDMprofiles,usedtofitobservedvelocitydistributions,differ 20 themostintheGCregion. 140 It was shown in Klypinetal. (2002); Battagliaetal. (2005) 15 g]g] 120 that the DM halo of the MW can be described by the Navarro- ee 10 Frenk-White(NFW)profile(Navarroetal.1997) dd gle [gle [ 5 100 ρ (r)= ρsrs3 , (11) anan 0 80 NFW r(r+rs)2 r r ntente -5 60 withparameters, given inTable 1.Therelationbetween virial pa- cece rametersandρs,rs caneasilybefound(seee.g.theAppendixA ---10 OffOff 40 ofBoyarskyetal.2007). -15 To explore the uncertainty of the DM density profile in the 20 inner part oftheGalaxy, wealsodescribe theDMdistributionin -20 theMWviaanisothermalprofile(Bahcall&Soneira1980): -25 0 v2 1 ρ -25-20-15-10 -5 0 5 10 15 20 25 ρ (r)= h = 0 . (12) iso 4πG r2+r2 1+(r/r )2 N c c OOffff--cceenntteerr aannggllee [[ddeegg]] ThefollowingparametersofisothermalprofilereproducetheDM contribution to the (outer parts of) Galaxy rotation curve v = h 170 km/sec and r = 4kpc (Boyarskyetal. 2006c, 2007) (i.e. c Figure 4. Dependence of the effective area onthe off-axis position ofa 2 2 ρ = 1.2 106 keV vh 4kpc ).Theseparametersare (point)source. 0 × cm3h170km/si h rc i consistent withthose, from favoredNFWmodels of Klypinetal. (2002);Battagliaetal.(2005),i.e.forφ > 90◦ thedifferencebe- Inthegeneralcaseofextendedsources,evaluationofthede- tweenisothermalmodelandNFWwithpreferredparameterswas tector count rate (7) analytically is not possible because of the completely negligible (less than 5%) – c.f. FIG.5. Both types of complicateddependenceoftheeffectiveareaontheoff-axisangle modelsprovidethelocalDMdensityatthepositionoftheSunto (shownonFig.4).Inthesimplestcaseofanextendedsourcewith beρDM(r⊙)≃0.22GeV/cm3,whichisclosetotheexistingesti- a constant surface brightness dF (φ)/dΩ = f = const, the mates(Kuijken&Gilmore1989c,a,b,1991;Gilmoreetal.1989). DM ext integralofEq.(7)reducestothemultiplicationbythesolidangle The DM flux from a given direction φ, measured by an ob- Ω 0.29andtheeffectivearea,averagedovertheFoV: server on Earth (distance r⊙ 8.5kpc from the GC), is given PCFOV ≃ by ≃ 1 Aeff,ext(Eγ)= ΩPCFOVFZoZVdαdβAeff(Eγ|α,β) Siso(φ)=ρ0Rrc2 ×8< πa2r+ctaanrctan“Rr⊙Rcosφ,”, ccoossφφ><00 , ≈κ(Eγ)Aeff,on(Eγ). (8) “r⊙|cosφ|” (13) : Thenumericalfactorκ(Eγ)dependsontheenergyandhastobe whereR= r2+r2 sin2φandρ r 1.5 1028keV/cm2. calculated via a numerical integration over the energy dependent q c ⊙ 0 c ≃ × TheuncertaintyoftheDMradialdensityprofileintheinner off-axis response map of the SPIdetector. A reasonably accurate Galaxystemsfromthedifficultyofseparationbetweenvisibleand numericalapproximationtoκ(E )isgivenby γ DMcontributionstotheinnerGalaxyrotationcurve.6 Inorderto κ(E) 0.165(E/keV)0.11 . (9) getthemostconservativelimitonthecolumndensityoftheDMin ≈ thedirectionoftheGC,onecanassumethefollowing“rigidlower Onecanseethatκ 1inalltheenergyinterval.Thisisexplained bound”:whiletheDMoutsidether isdescribedbythe“maximal ≪ ⊙ bythefactthatthedetectorareavisiblefromagivendirectionon disk” model (model A of Klypinetal. 2002), for r 6 r DM 2 ⊙ thesky strongly decreases withthe increase of the off-axis angle densityremainsconstant (sothat thetotalDMmasswithinr is ⊙ of this direction, so that the sky-averaged effective area is much thesameasinthemodelA ofKlypinetal.2002).Thisgives 2 smallerthantheon-axiseffectiveareaofthedetector.Substituting M keV (9), (8) into(7) onefinds that for anextended source of constant ρmin 3.9 106 ⊙ =0.146 106 . (14) surfacebrightnessthedetectorcountrateis DM ≃ × kpc3 × cm3 The surface brightness profile on the “constant density” model cts 1keV R =2.73 10−5 (10) is shown in black dashed line on the Fig.5. One can see that ext × s » Eγ – the difference between the maximal (φ = 0◦) and the minimal Aeff,ext(Eγ) (dFDM/dΩ)ext φ = 180◦) column densities is 3.4 (as compared to 6 for ∼ ∼ ×» 150cm2 –»10−15erg/(cm2ssr)– 6 WhenquotingresultsofKlypinetal.(2002),wedonottaketheeffects ofbaryoncompressiononDMintoaccount.WhiletheseeffectsmakeDM dependenton-axiseffectiveareasofeachofthe17operatingdetectorsof distributioninthecoreoftheMWdenser,anysuchcomputationisstrongly SPI,extractedfromtheinstrument’scharacteristicsfiles. modeldependent. 6 A. Boyarskyet al. Table1.Best-fitparametersofNFWmodeloftheMWDMhalo.Max.diskmodelmaximizesamountofbaryonicmatterintheinner3kpcoftheMWhalo (MDM/(Mdisk+Mbulge)=0.4forthemodelA2andMDM/(Mdisk+Mbulge)=0.14inthemodelB2). References Mvir[M⊙] rvir[kpc] Concentration rs[kpc] ρs[M⊙/kpc3] Klypinetal.(2002),favoredmodels(A1orB1) 1.0×1012 258 12 21.5 4.9×106 Klypinetal.(2002),Max.diskmodelsA2 0.71×1012 230 5 46 0.6×106 Klypinetal.(2002),Max.diskmodelsB2 0.71×1012 230 10 23 3.1×106 Battagliaetal.(2005) 0.8+1.2×1012 255 18 14.2 11.2×106 −0.2 ThelowerboundontheDMdecaylinerateinSPIpointingstoward theinner Galaxy iscalculated bysubstituting thecolumn density =1028keV/cm2(seeFig.5)intoEqs.(10),(16) S cts (φ) sin22θ M 4 Fmin ≃3.0×10−6cm2s»102S8DkMeV/cm2–»10−10 –»1kDeMV– (17) Theapproximationoftheconstantsurfacebrightnessworkswell, iftheextendedsourcehasacoreoftheangulardiameterexceeding thesizeoftheSPIpartiallycodedFoV(Θ 17◦ maximal PCFOV ≈ off-axisangle).Takingisothermalprofiletheangularsizeoftheflat coreoftheextendedsourceis φ =arctan(r /r ) 25◦ , (18) core c ⊙ ≃ whichsatisfiesthisconstraint. 3 STRATEGYOFSEARCHFORTHEDMDECAYLINE WITHSPI TheMWhalocontributiontotheDMdecaysignalrepresentsthe all-skysource.Indeed,astheresultsofSection2.1show,thevari- Figure5.ExpectedcolumndensityforvariousDMprofiles:favoredNFW abilityofthesignalovertheskymaybeaslowasthefactor 3. profile(redthick solidline); NFWprofilewiththemaximaldisk(model ∼ A2,seeTable1)–bluesolidline;cored(isothermal)profile–greenthick ThismakesthestrategyofsearchoftheDMdecaysignaldifferent dashedline;constantdensitywithinr⊙–blackdashedline). from any other types of astrophysical sources: the point sources, diffusesources(e.g. 10◦Gaussianprofilefore+e−annihilation ∼ region,Kno¨dlsederetal.(2005))oreventhesearchforDManni- isothermal model). For comparison we show on Fig. 5 expected hilationsignal(seee.g.Tasitsiomietal.2004;Boehmetal.2004; DM flux (6) for various profiles. The minimal column density is Diemandetal.2007;Sańchez-Condeetal.2006;Carretal.2006). ofcoursetheoneinthedirectionofanti-center: (φ = 180◦) Theproblemgetsexacerbatedbythefactthatduringitsmo- S ≃ 0.33 1028 keV/cm2.Weseethatevenfortheminimalprofile tion,SPIisirradiatedbythechargedhigh-energy particles(parti- S(φ<×30◦)>1028keV/cm2. clesfromEarthradiationbelt,Solarwind,cosmicTeVphotons).As aresult,thematerials(evendetectorsthemselves)usedforSPIcon- structionstarttoradiateindifferentenergyregions(seesubsection 2.2 DMdecaylinecountrate 3.2).AsaresultanySPIspectrumconsistsofabroadcontinuum, In the case of the Majorana sterile neutrinos of mass M the whichisacombination oftheskyandinstrumental backgrounds, DM DMdecaywidthisgivenby(Pal&Wolfenstein1982;Bargeretal. andofasetoftheinstrumentalbackgroundlines(Attieétal.2003; 1995):7 Diehletal.2003;Jeanetal.2003;Weidenspointneretal.2003).In ordertodetectaspectrallineproducedbyanastrophysicalsource Γ 1.3 10−32 sin22θ MDM 5 s−1. (15) one has to be able to (a) separate the continuum and line contri- DM ≃ × »10−10 –»1keV– butionstothespectrumand(b)separatetheinstrumentalandsky Substituting(15)to(6)wefind signalcontributionstothelinesfound. Onecanexpectthreeapriorisituations: dF DM(φ) 8.3 10−15erg/(cm2ssr) dΩ ≃ × (I) DM decay line is strong (its equivalent width much larger (16) thanthespectralresolution)andatitspositionthereareno sin22θ M 5 (φ) otherstronglines(ofeitherinstrumentalorastrophysicalori- DM SDM gin).Suchaline,duetoitspresenceinanySPIspectrumand ×»10−10 –»1keV– »1028keV/cm2– itslowvariabilityovertheskycaninprinciplebeconfused withsomeunknowninstrumentalline. 7 ThequotedvalueofΓ isfortheMajoranasterileneutrino.Incaseof (II) DMlineisweak( 3 4σ detectionoverthecontinuum) DM ∼ − Diracparticlethisvalueis2timessmaller (c.f.Pal&Wolfenstein1982; butitspositionalsodoesnotcoincidewithanyinstrumental Bargeretal.1995). line. ConstrainingDM propertieswithSPI 7 (III) DM decay line coincides with some instrumental line. To information. There exist various “background tracers” (Gedetec- beablefindsuchalineweneedtomodelSPIinstrumental torssaturationrates,anti-coincidence shieldrates,ratesofcertain background. background lines, see Jeanetal. (2003); Teegardenetal. (2004); Teegarden&Watanabe(2006)andrefs.therein). Tobeabletoworkeffectivelywithallthesesituations,weneedto findthewaytoseparatethesourceandbackgroundcontributions. 3.3 Searchingforthelines TobeabletodetectstrongDMline,whichisnotcloseinposition 3.1 Imaging toanyinstrumentalline(caseIabove),weusedthemodificationof To distinguish source and background contribution to the signal, themethodofbackgroundsubtraction,describedinTW06.TW06 one often uses imaging capabilities of an instrument. If the size looked for γ-ray lines, assuming different types of sources, from of a point or even an extended source on the sky is smaller than the point sources to the very diffuse sources (10◦ Gaussian, 30◦ the size of the SPI FoV one can (at least, to some extent) use flat,etc.)TW06showedthatthestrongbackgroundlineat198keV the imaging capabilities of the SPI instrument. In this case the canbeusedasabackgroundtracer,ifbackgroundobservationsare coded mask, placed above the detector, partially screens the in- matchedcloseintimetothecorresponding“source”ones.Thisal- dividual detectors from the source, so that the source at a given lowedTW06tocancelallstronginstrumentallineswiththepreci- positionontheskyproduces different count ratesindifferent de- sionbetterthan1%.TW06detectednoemissionlineinsuchback- tectors. One can find the source flux by comparing the ratios of groundsubtractedspectrum(apartfromthe511keVand1809keV) the actual count rates in different modules of the detector to the withthesignificanceabove3.5σ. ones predicted by the degree of screening of the modules by the WeadoptthefollowingmodificationoftheTW06method: mask (see Dubathetal. 2005; Skinner&Connell 2003). It is a – AstheDMdecaysignalremainsnearlyconstantwithincen- challenge, however, to use the imaging capabilities of the SPI to tral30 50◦,themethodofTW06,ifapplieddirectly,couldcancel separatetheastrophysicalsignalfromtheinstrumentalbackground mosto−ftheDMsignal.8Wethereforesubtractthedata(renormal- ifthesizeoftheextendedsourceiscomparabletothesizeofthe izedbythestrengthof198keVline)inthedirectionawayfromthe SPIFoV(seee.g.Kno¨dlsederetal.2005;Allain&Roques2006; GC(off-GCangleφ > 120◦)fromtheON-GCdataset(theangle Weidenspointneretal. 2007, and refs. therein). Therefore for our φ613◦). analysiswedidnotuseanyimagingcapabilitiesofSPI,andtopro- – Intheresulting“ON–OFF”spectrumweperformthesearch duce spectra fromsome point inthe sky wejust collectedall the forthelinewiththesignificancehigherthan3σ. photons,arrivingintheSPIFoV. This procedure allows to eliminate strong instrumental lines withtheprecisionbetterthanfewpercents.Atthesametimeany strongDMlinewouldremaininthe“ON–OFF”spectrum.Indeed, 3.2 SPIbackgroundmodeling even for the flattest profile (Section 2.1), the strength of the DM signal in the OFF dataset is at least 60% weaker that of the ON Intheabsence ofimaging, theseparationof theinstrumental and dataset.Thereforewesee,thatthemodification,describedabove, astrophysicalcontributionstothelinespectrumrequiressomesort is indeed well suited for searching of the strong DM decay line of background modeling (see e.g. Weidenspointneretal. 2003; (caseI). Teegardenetal.2004; Teegarden&Watanabe2006).Namely, for However,thismethoddoesnotworkwellfortheweak(3 4σ) thebackgroundmodelingwecanusethefactthatforanyDMdis- − lines,orforthelines,whosepositioncoincideswithsomeinstru- tributionmodeltheintensityoftheDMdecaylinechangesbyafac- mentalline(casesII-IIIabove).Indeed,inthiscaseitisnotpos- tor>3betweenthepointingstowardstheGalacticcenter(φ 0◦) sible to tell whether the remaining line is the residual of the in- ∼ andanti-center(φ 180◦,seeSec.2.1).Ontheotherhand,ifthe strumentaloneorhastheastrophysicalorigin.Belowwewilluse ∼ lineisofpurelyinstrumentalorigin,thereisnoa-priorireasonwhy analternativemethodofanalysisofthedetectedlines,suitablefor thestrengthofthelineinthebackgroundspectraofthepointings casesIIandIII. towardse.g.theGalacticAnti-centershouldbedifferentfromthe strength of the line in the spectra of the pointings toward e.g the GC.Thus,onepossiblewaytodistinguishbetweentheDMdecay 3.4 Analyzingacandidateline andinstrumentaloriginofthelineistostudythevariationsofthe line’sstrengthdependingonitsskyposition(inthesimplestcase– Havingdetectedanumberoflineswiththesignificanceof3σand onthe“off-GC”angleφ,ofthepointing,Eq.(5)). above,weshoulddecidewhichofthemcanbeconsideredas“DM Thesituationbecomesmorecomplicatedduetothefactthat decaylinecandidates”.Tothisendwedothefollowing. the instrumental background (and thus the intensity of the in- a) Wecomparelinefluxforeachoftheselineswiththeflux strumental lines) experiences great variability in time (depend- ofthesamelineinthe“ON”spectrum.Wedecidethatthelineis ing on the position in orbit, solar flares and the solar activity a“DMlinecandidate”ifthecancellationofthefluxbetweenON period, degradation of the detectors, etc., c.f. Jeanetal. 2003; andOFFdatasetswasworsethan10%.9 Teegardenetal. 2004). As observations of different parts of the skycanbesignificantlyseparatedintime,oneneedstouse“back- 8 ForexampleforthemostconservativeDMdistributionmodel,thediffer- ground tracers” to find the correct spatial dependence of the line enceofDMsignalsatφ=0◦andφ=30◦ismere8%. intensity(Jeanetal.2003;Teegarden&Watanabe2006).Without 9 In principle, the DM line in ON–OFF spectrum should not cancel by somesort of “renormalization” procedure, which correctstheab- morethan∼ 40%,whilethebackgroundinstrumentallineshouldcancel solute value of the line flux using a measurement of a specific betterthan1%.Thusthechoiceofthethresholdtobearound10%ensures characteristicsof theSPIinstrument asa“calibrator”of the flux, that noDMdecay line was thrown away whilemostoftheinstrumental theφdependence foranyofthedetectedlinescontainsnouseful linesdisappeared. 8 A. Boyarskyet al. b) Forany“DMcandidateline”weconstructits“spatialpro- file”(asdescribed indetailsin thenext Section) tocheck for the ON-OFF possibility of it to be a DM decay line (we also construct distri- 0.003 ON spectrum × 0.01 bution of the line flux over the sky for all the unidentified lines fromWeidenspointneretal.2003).Sincethecolumndensityofthe V] ke 0.002 DtuhaMelddiienrcertchetaeiosdeniroteofcwtthiaoerndlittnohewesaGtrradelnatghctethicGwaCintthsih-tcoheuenlitdnecrb,reeoahnsieginshgheorauntlhgdalneseφteh.aaWtgoerfaddino- ate [cts/sec/ 0.001 notmakeanyspecificassumptionabouttheDMdensityprofileand nt r donottrytofitthecandidatelinespatialprofiletoanyparticular ou 0 C model,butratherlookifthereisageneraltrendofdecreasingin- tensityofthelinewiththeincreasingoff-GCangle. -0.001 50 100 150 200 250 E [keV] 4 DATAREDUCTION Figure 6. Comparison of ON–OFF spectrum (thick solid line) with the 4.1 ONdataset 0.01× theONspectrum afterthesliding window (thin dashedline). It canbeseenthattheinstrumentallinesaresubtractedwiththeprecisionbet- Duringitsalmost 5yearsinorbitINTEGRALhasintensivelyob- terthan1%. served the inner part of the Galaxy (Galactic Center, Galactic BulgeandtheinnerpartoftheGalacticPlane)andcollectedabout 4.2 ON–OFFdataset T 10MsofexposuretimeintheGCregion.Inouranalysis exp ∼ oftheinnerGalaxyweusedthepubliclyavailabledata(asofJuly Most of the lines found in the continuum subtracted background 2007)fromallINTEGRALpointingsatwhichtheangleofftheGC spectrum are of the instrumental origin. To remove them, we wasat most 13◦ and for whichtheSPIexposure timewas larger matchedeachScWintheONdatasetwiththepointingawayfrom than1ksec.Thiscriteriaselects5355pointings(or“ScienceWin- the GC (galactic coordinate φ > 120◦) – OFF pointing. As de- dows”,ScW),withtotalexposuretimeof12.2Ms,spreadoverthe scribedbyTW06,the198keVlinecanserveasgoodbackground periodfromFebruary2003,tillApril,25,2006.Wecallthisdataset tracer if the time duration between ON and OFF observations is "ON"dataset. 6 20 days. We were able to match 3688 ON-OFF pairs. For For each of the analyzed ScWs, we have extracted photon eachON–OFFpairweintroducednormalizingcoefficientnforthe (event) lists from spi-oper.fitsfiles and applied additional OFF spectrum in such a way that the strong instrumental line at energy correction to convert the channel number into photon en- 198keVcancelscompletelyaftersubtractionoftheOFFspectrum ergy, using spi gain cor tool from standard Offline Analysis multiplied by the factor n from the ON spectrum. After that we Software (OSA). We have binned the events into narrow energy subtracted(renormalized)OFFScWfromthecorrespondingScW bins of the size ∆E = 0.5 keV to generate the background from theON dataset. Thisallowed us to remove most prominent bin countsspectraineachScW,eachrevolutionand,subsequently,in instrumental lines with the precision better than 1% (c.f. Fig. 6). theentiredataset. To avoid contributions of strong astrophysical sources (such as Wethenappliedthe“slidingspectralwindow”methodtothe e.g. Crab) we threw out all pairs with negative total flux at 20– linesearch(asdescribede.g.inTW06toproduceacontinuumsub- 40keVrangeaftersubtraction.Takingaverageover2456remain- tractedspectrumofthe"ON"dataset.Namely,ateachgivenenergy ing“good”pairswereceivedthespectrumalmostfreefromback- E ,onedefinesanenergyintervalE 2∆E <E <E +2∆E, groundatenergiesabove200keV.Atlow(<200keV)energieswe 0 0 0 − where ∆E is the SPI spectral resolution at a given energy, as a foundcontinuumcomponent, whichcanbefittedwiththesimple ”linesignal”energyband.Forthe(energydependent)∆Eweused powerlaw: approximateformulafromSPI/INTEGRALgroundcalibrationof E α FWHM(Attieétal.2003): F(E)=F (20) 0»100keV– ∆E(E)=F1+F2√E+F3E (19) Parametersofthisbackgroundwerefoundtobe wEhiesrienFk1eV=.F1o.r54E,F=21=034k.e6V·1F0W−3H,MF3 =26.3.0k·eV10.−4 andenergy F0 =(4.95±0.05)×10−5cts/s/cm2 (21) ≈ α= (2.264 0.003) ForeachenergybincenteredatanenergyE wehavedefined − ± 0 thetwoadjacentenergyintervals,E 4∆E <E <E 2∆E Thiscontinuumrepresentstheresidualcontributionfromalltheset 0 0 andE +2∆E <E <E +4∆E,a−ndpostulatedthatthe−sumof oftheastrophysicalsourcespresentintheGalacticBulge. 0 0 thecountratesinthesetwoadjacentenergybandsgivesthemea- sure of the continuum count rate in the energy band around E . 0 4.3 Systematicerror Subtractingthesumofthecountratesintheadjacentenergybands fromthecountrateinthe”linesignal”energyband,wehavecal- Toestimatethesystematicerrorofour“ON-minus-OFF”dataset, culatedthecontinuumsubtractedcountrateatagivenenergyE . wecomputedbackgroundaroundthe“tracerline”of198keV.We 0 Doingsuchprocedureatalltheenergies20keV<E <8MeV, found that it does not vanish. Thus, we estimated the systematic 0 wehaveproduceda“continuumsubtracted”SPIbackgroundspec- error as the error in thenormalization coefficient n which would trum.Inthisspectrumwewereabletoidentifymostoftheknown makethebackgroundzerowithinsystematicuncertainty.Thiscor- instrumentallines(Weidenspointneretal.2003). rectionδncanbefoundasfollows.Letnbethecoefficient,needed ConstrainingDM propertieswithSPI 9 Phase Revolutions(start-stop) 180 1 042–092 096–140∗ 160 2 140-205 140 209-215∗ 3 215-277 120 4 282-326 5 330-395 deg] 100 6 400-446 φ [ 80 Table2.Splittingrevolutionsintophasesincorrespondencewithannealing 60 phasesandbreakageofthedetectors(revolutions,markedwiththe∗). 40 20 tocancelfluxinthe198keVlineinONandOFFspectra: 0 0 100 200 300 400 500 600 n=F /F (22) ON OFF Revolution whereF ,F –fluxesin198keVlineinONandOFFScWs ON OFF Figure7.Positionontheskyasafunction ofrevolution over6yearsof correspondingly. Theremainingnon-zero δF fluxintheadjacent INTEGRAL observations. The periods of annealing phases are shown in to line position in ON-OFFspectrum, determines the uncertainty solidverticallines.Twodashedlinesindicatetherevolution,duringwhich ofthecoefficient: 2NDand17thof19SPIdetectorshavefailed. δF δn= (23) F ON We found, that average value of δn is equal to δn = 1.1 0.009 h i h i · 10−3, n 1.So,oursystematicerroroffinalON–OFFspectra atenerghyEi∼is1.1 10−3FOFF(E) 1.1 10−3FON.Weaddthis 0.008 · ≈ · systematic uncertainty to the flux of ON–OFF spectrum in every V] energybin. ke 0.007 4.4 Obtaining3σrestrictions ate [cts/sec/ 0.006 Attheenergiesatwhichnolinesweredetected(i.e.the“continuum unt r 0.005 o subtracted”countratedidnotdeviatebymorethan3σfromzero) C weobtainedthe3σ upperlimitonthepossiblefluxfromtheDM 0.004 decay.Above 200keVthefluxintheenergybiniszerowithin ∼ statisticalerrors,therefore3σupperlimitfluxisgivenbystatistical 0.003 plussystematicerrors.Below200keVweputstatisticalrestrictions 0 20 40 60 80 100 120 140 160 180 φ [deg] abovepowerlawcontinuumflux(20),describedintheSection4.2. UsingEq.(17)onecanderivetherestrictiononthesterileneutrino Figure8.ScatterofthefluxdatapointsforthelineatE=1068keVasa mixingangle,impliedbythisupperlimit.Oneshouldalsotakeinto functionoftheoff-GCangle. accountthatthesubtractionoftheOFFobservationsledtothere- ductionof theexpected DMsignal. Takingthemost conservative “minimal”model,describedinSection2.1,weseethatthesubtrac- tionoftheOFFsignalleadstoabout40%decreaseoftheexpected possibletodistinguishbetweentheinstrumentalandDMdecayori- DMsignal.10Theresulting3σboundisshownonFig.10. ginofalinebyfittingF(φ)withaknownprofilecalculatedfrom the radial DM density profile. However, the details of the radial DM density profile inthe inner Galaxy are highly uncertain, and 4.5 PossibleDMcandidates thispreventsusfromdirectlyfittingthemodelprofiletothedata. WeadoptedasimplecriterionwhichselectsaDMdecaycandidate WhenanalyzingON–OFFspectrum,wefoundthatalmostalllines, line:theratiooffluxes presentinONspectrumcancelwithprecisionbetterthanfewper- cents. We found 21 lines (see Table 3) that did not cancel by at F(0◦) = > 3. (24) least90%,(includingknownlinesat511keVand1809keV).Apart R F(180◦) Rmin ≃ fromthese2linesallotherlinesaredetectedwithlowsignificance 3 4σ. where min istheratiooftheDMdecaylinefluxesfromtheGC − As discussed in Sections 3.3–3.4, we took all these lines as andtheRGalacticanti-centerinthe“minimalDMcontent”modelof possible DM candidates and analyzed the dependence of theline DMdistribution. fluxesF(φ)ontheoff-GCangleφofthepointing.IftheDMdis- Sincetheobservationsatdifferentoff-GCanglesaredonedur- tributionintheinner partoftheGalaxywereknown, itwouldbe ingdifferenttimeperiods,toproperlystudythedependenceofthe lineflux on the off-GC angle φ one should take into account the timevariabilityof the response of the SPIdetectors. Several fac- 10 Toestimatethis,wetookthemaximalcolumndensityforOFFobserva- torshavetobetaken intoaccount. First,theSPIinstrument goes tionsatφ=120◦−17◦. throughaso-called“annealing”phase–heatingofthedetectorsto 10 A. Boyarskyet al. 6.1×10-2 1021 6.0×10-2 2m] 6.0×10-2 1020 ne flux [cts/sec/c 55..89××1100--22 τLife-time [yr] 1019 EXCLUDED Li 5.8×10-2 5.8×10-2 1018 0 20 40 60 80 100 120 140 160 180 20 100 1000 7000 Off-center angle φ [deg] Eγ [keV] Figure9.Dependenceoftheintensityofthepositronannihilation lineat Figure11.Life-timeoftheradiatively decayingDMasafunctionofthe E =511keVontheoff-GCangle.Thesolidlineshowsfittothedatain emittedphotonenergy.Regionbelowthecurveisexcluded. theformconst+Ne−φ2/(2σ2). 5 RESULTS WeanalyzedthespectrumofSPIandfoundthatnoneofthestrong recoverfromaradiativedamage.11Next,twoofthe19SPIdetector (i.e.detectedwithsignificanceabove5σ)linescanbeinterpreted have“died”.12Thefaileddetectorsalsoaffecttheresponseoftheir asthatofthedecayingDM.Thisconclusionwasbasedonthefact neighbors. Tomarginalizetheeffectsofthechangingresponseof thatvariabilityoftheselinesovertheskyislessthan10%(when theSPIdetector,wesplittheentiredatasetinto7periods,asshown movingfromGCtotheanti-center,seeFig.8).Atthesametimefor onFig.7.TheintervalsaresummarizedintheTable2.Asbothde- anyrealisticDMmodelsuchavariabilitywouldbegreaterthanat tector failuresoccurred soon after theend of anannealing phase, least60%.Thus,weexcludethepossibilitythatoneofthespectral wechose toignorerevolutions136 through 140and209 through lines,detectedintheSPIbackgroundspectrumisaDMdecayline. 215. The period 096–140 does not cover theessential part of the The non-detection of a DM decay line in the entire energy skyandthereforeweskipit,leavingonly6periods. rangeoftheSPIdetectorhasenabledustoputanupperlimitonthe Foreachoftheperiods,shownintheTable2,weplotthedis- parametersoftheDMparticles.Inparticular,the3σ upperbound tributionofthelinefluxasafunctionoftheoff-GCangleφ.The onthemixingangleofthesterileneutrinoDMinthemassrange resultsaresummarizedonFig.13,p.16.Onecanseethatnoneof 40keV–7MeVisshownonFig.10,p.12. theselinesexhibitscleartrendofdecreasingfromφ=0◦towards OurresultsareapplicabletoanydecayingDM.Tothisendwe φ = 180◦. For each line (and each phase) we also compute the alsopresenttherestrictionsontheDMlife-time(withrespecttothe averagefluxF¯,standarddeviationσ fromtheaverage,minimum F radiativedecay)asafunctionoftheenergyofemittedphoton.The (F )andmaximum(F ).Ouranalysisshowsthat(a)95 100% min max − correspondingexclusionplotisshownonFig.11.Forexample,the ofallpointsliewithin3σ fromtheaverage(thus,thedataiscon- F gravitinocandecayintotheneutrinoandphoton(similarlytothe sistent with having flat spatial profile) and (b) the scatter of the caseofsterileneutrino)insupersymmetrictheorieswithbrokenR- data(F F )ismuchlessthanitsmeanvalue F .Therefore, max− min h i parity.Suchaninteractionisgeneratedviatheloopeffects(seee.g. noneofthemcannotoriginateentirelyfromaDMdecay.Thecor- Borganietal. 1996; Lolaetal. 2007). The restrictions on Fig. 11 respondingnumbersforeachlineandeachphasearesummarized improve existingbounds on thelife-timeof such a gravitinoDM inTable4,page17. byseveralordersofmagnitude(c.f.Borganietal.1996). The positron annihilation lineat E = 511keV illustratesa Topresentourresultsintheformlessdependentonaparticu- situation, when a line of astrophysical origin is superimposed on larmodelofDMdistributionintheMW,weshowthe3σsensitivity top of the strong instrumental line. In this case, the data can be towardsthelinesearchonFig.12.Note,thattheseresultsshould fittedbytheconstant,plussomefunction,depending onassumed beusedwithcare,asthesensitivitydependsontheassumedspatial shapeofthesource.Fig.9showsthedependenceofthefluxofthe profileofthesource(becausetheeffectiveareadecreaseswiththe 511keVlineontheoff-GCangleoftheSPIpointing.Onecansee off-axisangle,seediscussioninSection2).Theresults,presented that for the pointings with theoff-GC angle lessthan 20◦ (about onFig.12arevalidforanextendedsourcewiththesurfacebright- the size of the PCFOV of SPI) the 511 keV line flux contains a nesswhichvariesontheangularscaleslargerthan(orcomparable contributionfromaskysourceatthepositionoftheGC,whilefor to)thesizeoftheSPIfieldofview(blacksolidline).Thisplotis thepointingatlargeroff-GCanglestheastrophysicalsourceisnot analogoustotheFig.9ofTeegarden&Watanabe(2006)(TW06). visibleandtheonlycontributioncomesfromtheinstrumentalline, However,adirectcomparisonoftheFig.9ofTW06andFig.12is whosefluxdoesnotdependontheoff-GCangleofthepointing. notpossible,sinceTW06haveassumedadifferentmorphologyof theextendedsource(10◦Gaussian).Explicitlytakingintoaccount thedependenceoftheeffectiveareaoftheSPIdetectorontheoff- axis angle (see Section 2), one can find that in order to make a 11 FordetailsseeSPIUserManual: direct comparison between the two figures, one has to “re-scale” http://isdc.unige.ch/Instrument/spi/doc/spi um. theresultsof Fig.9of TW06by an(energydependent) factor of 12 Detector#2atrevolution140anddetector#17atrevolutions214-215. 1.5.Thisfactorconvertsthesensitivityfortheline,producedby ≈