Constraining DM properties with SPI

1.1 MB·English

by  Alexey Boyarsky

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’E´cogia16,1290Versoix,Switzerland
n
GenevaObservatory,51ch.desMaillettes,CH-1290Sauverny,Switzerland
a
5E´colePolytechniqueFe´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 or-
der 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 re-
gion 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) sce-
nario 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 sce-
nario 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 produc-
tion 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;Sa´nchez-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(Attie´etal.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(Attie´etal.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
≈