Table Of ContentFitting the Fermi-LAT GeV excess: on the
importance of the propagation of electrons from
dark matter
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J Thomas Lacroix∗
9 UPMC-CNRS,UMR7095,Institutd’AstrophysiquedeParis,98bisboulevardArago,75014
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Paris,France
E-mail: [email protected]
]
E
H
AnexcessofgammaraysatGeVenergieshasbeendetectedintheFermi-LATdata. Thissignal
.
h
comes from a narrow region around the Galactic Center and has been interpreted as possible
p
- evidenceforlight(30GeV)darkmatterparticles. Focussingonthepromptgamma-rayemission,
o
r previous works found that the best fit to the data corresponds to annihilations proceeding into
t
s b quarks, with a dark matter profile going as r−1.2. We show that this is not the only possible
a
[ annihilationset-up.Morespecifically,weshowhowincludingthecontributionstothegamma-ray
spectrumfrominverseComptonscatteringandbremsstrahlungfromelectronsproducedindark
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v matterannihilations,andundergoingdiffusionthroughtheGalacticmagneticfield,significantly
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affects the spectrum for leptonic final states. This drastically changes the interpretation of the
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4 excessintermsofdarkmatter.
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FrontiersofFundamentalPhysics14
15-18July2014
AixMarseilleUniversity(AMU)Saint-CharlesCampus,Marseille,France
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
FittingtheFermi-LATGeVexcess: ontheimportanceofthepropagationofelectronsfromdarkmatter
ThomasLacroix
1. Introduction: theGalacticCentergamma-rayexcess
AnexcessofgammaraysfromtheGalacticCenter(GC)hasbeendetectedintheFermi-LAT
databetweenroughly0.1and10GeV[1,2,3,4,5]. Thisexcesshasaspatialextensionsmallerthan
10◦×10◦,andissphericallysymmetric. Itwasobtainedbysubtractingtothedataknownsources
andatemplateforthebackgrounddiffuseemissionprovidedbytheFermiCollaboration. Although
this background modelling procedure has been debated, the picture that has emerged seems to be
robust. Thereisavarietyofastrophysicalexplanationsfortheexcess,butaninterpretationinterms
ofdarkmatter(DM)isneverthelesspossible.
Thebestfittotheexcessquotedintheliteraturehasbeenobtainedfor30GeVDMannihilating
intobb¯ (Fig.1,topleftpanel),withacrosssectionof2×10−26 cm3 s−1. Thedataalsopointtoa
density profile ρ ∝r−1.2. A mixture of 90% leptons and 10% b quarks gives a relatively good fit
(Fig.1,toprightpanel),whilethefitisbadforafinalstatecontaining100%leptons(Fig.1,bottom
panel). Heretheterm“leptons”referstodemocraticannihilationintoleptons,i.e.acombinationof
thee+e−,µ+µ−,τ+τ− finalstates,with1/3oftheannihilationsintoeachofthesechannels.
Consequently, final states of DM annihilation containing only leptons have not been consid-
ered viable when interpreting the GeV excess in terms of DM. However, these conclusions were
obtainedtakingintoaccountonlythepromptgamma-rayemission,namelythefinal-stateradiation
(FSR)single-photonemission,andtheimmediatehadronizationanddecayoftheDMannihilation
products into photons. Nevertheless, electrons and positrons are also by-products of DM anni-
hilations, and they produce gamma rays via inverse Compton (IC) scattering off photons of the
interstellar radiation field and bremsstrahlung. It had been argued by the authors of Refs. [6, 7]
that the contributions of these secondary emissions to the gamma-ray spectrum should not be ne-
glectedandshouldleadtocorrections. Ontopofthat,diffusionofelectronsandpositronsthrough
the Galactic magnetic field must be taken into account when modelling these emissions. Here we
showthatthesecontributionsdonotjustgivecorrectionsbuttheytotallychangetheinterpretation
oftheexcessintermsofDM.
2. DiffusionofelectronsandpositronsfromDM
In order to compute the gamma-ray spectrum from DM-induced electrons and positrons, we
solvethediffusion-lossequationofcosmicrays. Assumingasteadystate,theequationreads[8]
∂
K∇2ψ+ (b ψ)+q=0, (2.1)
tot
∂E
whereψ((cid:126)x,E)isthecosmic-rayspectrumafterpropagation,K isthediffusioncoefficient: K(E)=
K (E/E )δ withE =1GeV,b (E)isthetotalenergylossrate(IC,synchrotron,bremsstrahlung)
0 0 0 tot
andq((cid:126)x,E)isthesourcetermforDMannihilations,proportionaltoρ2. Tosolvetheequation,we
usethesemi-analyticmethoddescribedinRef.[8]. Inthisapproach,thespectrumofelectronsand
positronsafterdiffusionisgivenby
κ (cid:90) ∞ dn
ψ((cid:126)x,E)= I˜(λ (E,E )) (E )dE , (2.2)
(cid:126)x D S S S
b (E) dE
tot E
2
FittingtheFermi-LATGeVexcess: ontheimportanceofthepropagationofelectronsfromdarkmatter
ThomasLacroix
m =30GeV,b m =10GeV,l+b
DM DM
promptonly promptonly
) )
1− 1−
s s
2− 2−
m m
c c
V V
(Ge 10-7 (Ge 10-7
Eγ Eγ
d d
n/ n/
d d
2Eγ 2Eγ
10-1 100 101 10-1 100 101
Eγ(GeV) Eγ(GeV)
m =10GeV,leptons
DM
promptonly
)
1−
s
2−
m
c
V
(Ge 10-7
Eγ
d
n/
d
2Eγ
10-1 100 101
Eγ(GeV)
Figure 1: Best fit to the spectrum of the residual extended emission in the 7◦×7◦ region around the GC,
for 30 GeV DM annihilating into 100% b quarks (top left panel), and 10 GeV DM annihilating into 90%
leptonsand10%bquarks(toprightpanel),and100%leptons(bottompanel). Thebest-fitcrosssectionis
∼2×10−26cm3s−1. Thedatapointsaretakenfrom[3].
whereκ =(1/2)(cid:104)σv(cid:105)(ρ /m )2,with(cid:104)σv(cid:105)theannihilationcrosssection,ρ theDMdensityin
(cid:12) DM (cid:12)
the Solar neighborhood, and m the DM mass. I˜ is the so-called halo function that contains all
DM (cid:126)x
theinformationondiffusion, throughthediffusionlengthλ (E,E )thatdependsontheinjection
D S
energy E and the energy after propagation E. The halo function is convolved with the injection
S
spectrum dn/dE. The most difficult step of the resolution is to compute the halo function in the
context of a cuspy DM profile. For that we used the method relying on Green’s functions. The
halo function is thus given by the convolution over the diffusion zone (DZ) of the Green’s func-
tion G((cid:126)x,E;(cid:126)x ,E )≡G((cid:126)x,(cid:126)x ,λ (E,E )) of the diffusion-loss equation and the square of the DM
S S S D S
density[8]
(cid:90) (cid:18)ρ((cid:126)x )(cid:19)2
I˜(λ (E,E ))= d(cid:126)x G((cid:126)x,E;(cid:126)x ,E ) S . (2.3)
(cid:126)x D S S S S
ρ
DZ (cid:12)
Thedifficultywiththisintegralistwofold. Firstofall,todealwiththesteepnessoftheDMprofile,
we used logarithmic steps. Second, the halo function must boil down to (ρ/ρ )2 when diffusion
(cid:12)
becomes negligible, that is when λ → 0 (i.e. E → E ). The problem is that in this regime, G
D S
becomes infinitely peaked, and the sampling of the integral must be carefully chosen in order not
tomissthepeak. ThesolutionistodefinedifferentregimesforGdependingontheratioofλ and
D
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FittingtheFermi-LATGeVexcess: ontheimportanceofthepropagationofelectronsfromdarkmatter
ThomasLacroix
thedistancetotheGC,asdescribedindetailin[9]. OnceI˜hasbeencomputedwiththisdedicated
treatment of diffusion on small scales, one can compute the IC and bremsstrahlung gamma-ray
fluxesfromelectronsandpositrons.
3. FittingtheFermi-LATGeVexcesswiththeleptonicchannels
Weusedfixedvaluesforthequantitiesdescribingtheinterstellarmedium(themagneticfield
and the gas density) that correspond to average values for the losses in the GC region. In Fig. 2
weshowthatfordemocraticannihilationintoleptons,thecontributionsfrompromptemission,IC
and bremsstrahlung are of the same order of magnitude, and they combine to give an excellent fit
totheexcess,withabest-fitcrosssectionof(cid:104)σv(cid:105)=0.86×10−26cm3s−1. Thisisaveryimportant
result, since it means that 30 GeV DM annihilating into bb¯ is not the only possible annihilation
set-up,andthatDMcanannihilateintoleptons. ThisresultisdiscussedinmoredetailinRef.[10].
We also show in Fig. 3 (left panel) that including the contributions from IC and bremsstrahlung
doesnotsignificantlyaffectthespectrumforthebb¯ channel,exceptatlowenergies.
mDM=10GeV,B=3µG,ngas=3cm−3,leptons
prompt
bremsstrahlung
IC
) prompt+IC+brem
1−
s
2−
m
c
V
Ge
E(γ10-7
d
n/
d
2Eγ
10-1 100 101
Eγ(GeV)
Figure2: Spectrumoftheresidualextendedemissioninthe7◦×7◦ regionaroundtheGC.Thedatapoints
are taken from [3]. The prompt (black dashed), IC (green dashed-dotted) and bremsstrahlung (red dotted)
emissions from 10 GeV DM democratically annihilating into leptons add up to give a very good fit to the
data,asshownbytheblacksolidline,withabest-fitcrosssectionof(cid:104)σv(cid:105)=0.86×10−26cm3s−1.
Therefore, the effect of including the IC and bremsstrahlung contributions is maximal for
democratic annihilation into leptons. However, the authors of Ref. [11] used the AMS data to
set constraints on the leptonic channels, which exclude annihilations into e+e− and impose that
the branching ratio into µ+µ− should not exceed 25%. We do not discuss the validity of these
constraintshere, butweshowinFig.3(rightpanel)thebestfittotheexcessobtainedwithafinal
stateofDMannihilationcontaining25%µ+µ−and75%τ+τ−. Withsuchbranchingratios,thefit
is marginally good, with an effect of the secondary contributions to the gamma-ray spectrum less
significantthanfordemocraticannihilation.
Finally, a very important test of the leptonic scenario is the morphology of the emission. In-
deed, due to spatial diffusion, the IC and bremsstrahlung emissions are more extended than the
prompt component, and their morphology depends on the observed energy. This may allow one
4
FittingtheFermi-LATGeVexcess: ontheimportanceofthepropagationofelectronsfromdarkmatter
ThomasLacroix
mDM=30GeV,B=3µG,ngas=3cm−3,b mDM=10GeV,B=3µG,ngas=3cm−3,1/4mu+3/4tau
promptonly promptonly
prompt+IC+brem prompt+IC+brem
) )
1− 1−
s s
2− 2−
m m
c c
V V
(Ge 10-7 (Ge 10-7
Eγ Eγ
d d
n/ n/
d d
2Eγ 2Eγ
10-1 100 101 10-1 100 101
Eγ(GeV) Eγ(GeV)
Figure3: BestfitstotheFermiresidualwiththegamma-rayspectrumfromannihilationsof30GeVDM
particles into 100% bb¯ (left panel), and 10 GeV DM annihilating into 25% µ+µ− and 75% τ+τ− (right
panel). Thebest-fitcrosssectionis∼2×10−26cm3s−1fortheleftpanel,and∼1×10−26cm3s−1forthe
rightpanel.
to set constraints on the annihilation channel. Shown in Fig. 4 (left panel) is the gamma-ray flux
plotted against latitude, for the best-fit parameters corresponding to the spectrum of Fig. 2, and
E =0.1 GeV. Above a few degrees the morphology should be compatible with the one found in
γ
the literature, and corresponding to prompt emission. However, between O(0.1) and O(1)◦, the
fluxprofileisshallowerthanthatofthepromptcomponent,whichmightleadtoatensionbetween
the flux from the leptonic channels and the morphology from the literature. However, 0.1 GeV is
belowthelowestenergydatapoint(around0.3–0.4GeV),andthesecondaryemissionsdominateat
lowenergy. Itturnsoutthatat1GeV,thetensionbetweenthetotalfluxfromtheleptonicchannels
andthemorphologyofthepromptemissionismuchweaker,asshowninFig.4(rightpanel).
10-2 mDM=10GeV,B=3µG,ngas=3cm−3,Eγ=0.1GeV 10-2 mDM=10GeV,B=3µG,ngas=3cm−3,Eγ=1.0GeV
prompt prompt
10-3 prompt+IC+brem 10-3 prompt+IC+brem
) )
1sr− 10-4 1sr− 10-4
21ms−− 10-5 21ms−− 10-5
Vc 10-6 Vc 10-6
Ge Ge
dn(dEdΩγ10-7 dn(dEdΩγ10-7
2Eγ 10-8 2Eγ 10-8
10-9 10-9
10-2 10-1 100 101 102 10-2 10-1 100 101 102
b(deg) b(deg)
Figure4: Gamma-rayfluxfromDMannihilatingexclusivelyintoleptonsdemocratically, asafunctionof
latitudeb,forgamma-rayenergiesof0.1GeV(leftpanel)and1GeV(rightpanel).
Conclusion
The Fermi-LAT GeV excess is a strong case for DM, and annihilations of 30 GeV particles
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FittingtheFermi-LATGeVexcess: ontheimportanceofthepropagationofelectronsfromdarkmatter
ThomasLacroix
into bb¯ provide the simplest set-up a priori. However, it is very important to take into account all
relevantemissionprocessesandspatialdiffusion. Includingalltheseprocessesdrasticallychanges
the interpretation of the excess in terms of DM. Therefore, we showed that bb¯ is not the only
possible channel and DM can in fact annihilate into leptons. Finally, the morphology below ∼1◦
atlowenergiescanhelptodiscriminatebetweentheleptonicandbb¯ channels.
Acknowledgments
I thank Céline Bœhm and Joseph Silk for a fruitful collaboration on this project. This research has
beensupportedatIAPbytheERCProjectNo.267117(DARK)hostedbyUniversitéPierreetMarieCurie
(UPMC)-Paris6andatJHUbyNSFGrantNo.OIA-1124403.ThisworkhasbeenalsosupportedbyUPMC
andSTFC.Finally,thisprojecthasbeencarriedoutintheILPLABEX(ANR-10-LABX-63)andhasbeen
supported by French state funds managed by the ANR, within the Investissements d’Avenir programme
(ANR-11-IDEX-0004-02).
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