Clumpiness of dark matter and the positron annihilation signal

14 Pages·2007·0.45 MB·English

A&A462,827–840(2007) Astronomy
DOI:10.1051/0004-6361:20065312 &
(cid:1)c ESO2007 Astrophysics
Clumpiness of dark matter and the positron annihilation signal
J.Lavalle1,J.Pochon2,P.Salati3,4,andR.Taillet3,4
1 CentredePhysiquedesParticulesCPPM,CNRS-IN2P3/UniversitédelaMéditerranée,13288Marseille,France
e-mail:[email protected]
2 LaboratoiredePhysiquedesParticulesLAPP,74941Annecy-le-Vieux,France
3 UniversitédeSavoie,73011Chambéry,France
4 LaboratoiredePhysiqueThéoriqueLAPTH,74941Annecy-le-Vieux,France
Received29March2006/Accepted20August2006
ABSTRACT
Context.Thesmall-scaledistributionofdarkmatteringalactichalosispoorlyknown.Severalstudiessuggestthatitcouldbevery
clumpy,whichisofparamountimportancewheninvestigatingtheannihilationsignalfromexoticparticles(e.g.supersymmetricor
Kaluza-Klein).
Aims.Wefocusontheannihilationsignalinpositrons.Weestimatetheassociateduncertainty,thatisduetothefactthatwedonot
knowexactlyhowtheclumpsaredistributedintheGalactichalo.
Methods.Weperformastatisticalstudybasedonanalyticalcomputations,aswellasnumericalsimulations.Westudytheaverage
andvarianceoftheannihilationsignalovermanyGalactichaloshavingthesamestatisticalproperties.
Results.Wefindthattheso-calledboostfactorusedbymanyauthorsshouldbeusedwithcaution,asi)itdependsonenergyandii)
itmaybedifferentforpositrons,antiprotonsandgammarays,afactwhichhasnotbeendiscussedbefore.Asanillustration,weuse
ourresultstodiscussthepositronspectrummeasurementsbytheHEATexperiment.
Keywords.cosmology:darkmatter–Galaxy:halo–ISM:cosmicrays
1. Introduction First, the gravitational collapse of primordial density inho-
mogeneities that leads to the formation of cosmic structures is
Most observations of cosmological interest can be accounted characterized by a small scale cut-off, due to several physical
for by assuming that our Universe contains a large amount of effects.Particlesenduringcollapsemayinteractelasticallywith
non-baryonic matter, usually referred to as dark matter. The other species or between themselves, which is responsible for
mean density of matter Ωm can be consistently estimated to diffusion. After their interactions become negligible, they be-
be Ωm ∼ 0.23 from many observations, whereas the bary- come free to move out of the collapsing region: this is known
onicdensityΩb inferredfromprimordialnucleosynthesis,from as free-streaming. A general discussion of these effects can be
Cosmic Microwave Background (CMB) anisotropies (Spergel found in Berezinsky et al. (2003). The resulting cut-off may
et al. 2006), Large Scale Structures and by direct observations strongly depend on the nature and propertiesof the wimp (see
of luminous matter is an order of magnitude lower, namely e.g.Boehmetal.2001;Hofmannetal.2001).Forinstance,the
Ωbh2 ≈0.0223. recent study by Profumo et al. (2006) gives protohalo masses
Among the several possible solutions to the dark matter rangingfrom3×10−9 M(cid:4) to3×10−1 M(cid:4).
problems,the hypothesisthat it could be made of a weakly in- Then,the structuresevolve,mergeandcanbe partiallydis-
teractingfundamentalparticleofanewkind(hereafterwimpfor rupted by tidal forces, so that the current cut-off in the spec-
weakly interactive massive particle) has received considerable trum of clump masses corresponds to the smallest surviving
attention.Thisispartlyduetothefactthatthishypothesiscanbe clumps. The situation is still unclear, as numerical simula-
tested experimentally.Inparticular,thedetectionofthe annihi- tions by Diemand et al. (2005) showed that clumps as small
lationproductsofsuchexoticparticleswouldbeagreatachieve- as 10−6 M(cid:4) could survive disruption, while analytical work by
ment,andanimportantfractionoftheastroparticlephysicscom- Berezinsky et al. (2006) showed that structures smaller than
munityisinvolvedinthatquest. 103M(cid:4)weredisrupted.Thepossibilitythattidalinteractionwith
However, assuming that wimps actually do exist (see e.g. stars may play an important role has also been hotly debated
Bertoneetal.2004foranicereviewondarkmatter),theirnature (Zhaoetal.2005b;Mooreetal.2005;Zhaoetal.2005a).
is unknown.Some constraintscan be inferredfromhighpreci- Different experts provide very different descriptions of the
sion cosmological observations such as the CMB, but several clumpinessofgalactichalo.Here,weconsiderthewiderangeof
particlephysicsmodelsprovidecandidateswhosepropertiesare possibilitiesasthestartingpointofouranalysis.Theamountof
consistent with these observations. Extensions of the standard clumpinessisofparamountimportanceasitenhancestheanni-
model of particle physics, such as supersymmetryand Kaluza- hilationrateofwimpsandincreasesthedetectionprospects.
Klein theories, naturally offer such candidates. The lack of in- In moststudies, clumpinessis takeninto accountbya gen-
formationaboutthenatureofthewimpmaytranslateintoavery eral,energy-independentmultiplicativenumbercalledtheboost
largeuncertaintyonitsspatialdistribution.Therearetwomain factor,bywhichthesignalcomputedfroma smoothdarkmat-
physicalreasonsforthat. terdistributionshouldbemultiplied.Thisisnotcorrectandwe
Article published by EDP Sciences and available at http://www.aanda.orgor http://dx.doi.org/10.1051/0004-6361:20065312
828 J.Lavalleetal.:Positronannihilationinaclumpyhalo
show in this paperthat the effects of clumpinesscannotbe de- It describes the probability that a positron produced at point x
scribedbysuchauniquenumber.Moreover,thisisastochastic withenergyE reachestheEarthwithadegradedenergyE.As
S
problem, in the following sense: general hypotheses about the wimpsareatrestwithrespecttotheMilkyWay,theenergyE
S
statisticalpropertiesofthedistributionofclumpsintheGalactic is equal to the parent particle mass m . At this stage, we keep
χ
halo can be made, butthe exactposition of every clump is un- our discussion as general as possible. Because our formalism
known.Insomecases,theexpectedsignalfromagiventypeof could easily be extended to any charged species – to antipro-
wimpcanbequitesensitivetotheprecisepositionoftheEarth tonsorantideuteronsforinstance–thepositronpropagatorwill
relativetothenearestclumps. be denoted more simply as G(x,E). The total positron flux at
The aim of this paper is to study the effect of halo clumpi- the Earth results from the integral over the galactic DM mass
nessontheannihilationsignal,focusingonthecaseofpositrons. distributionρ(x)
Takingadvantageofanalyticalcomputationsandnumericalsim- (cid:5) (cid:3) (cid:4)
ulations,weinvestigatethestatisticalpropertiesoftheannihila- ρ(x) 2
φ=S G(x,E) d3x. (4)
tion signal. We show that, at variance with the assumptions of ρ
DMhalo 0
most studies, the clumpiness factor depends on energy and is
not the same for positrons as for gamma rays. We also show ShouldtheDMhalobesmoothlydistributedwithmassdensity
thateveniftheaverageproperties(averagingbeingmeantover ρs, the positron flux would be given by relation (4) where the
alargenumberofrealizationsofourGalactichalo)ofaclumpy wimpdistributionisnowdescribedbyρs
halo may be well described by the usual boost factor, the de- (cid:5) (cid:3) (cid:4)
viations from this average may be very large and the ability to φ =S G(x,E) ρs(x) 2 d3x. (5)
predictasignalfromamodelmaybeconsequentlyreduced. s ρ
DMhalo 0
Theimportanceofclumpinessindeterminingthedarkmat-
In theliterature,theeffectsofclumpinesshavebeenaccounted
ter annihilation signal in positrons has been assessed by Baltz
forbyshiftingthe fluxφ upwards.Themultiplicativefactoris
&Edsjö(1999),andfurtherstudiedinHooperetal.(2004)and s
calledtheboost.Itactsasaconstantofrenormalizationbywhich
Hooper&Kribs(2004).Thepossibilitythatthepositronexcess
the flux φ generated by a smooth DM halo should be multi-
observed by HEAT could be due to a single nearby clump had s
pliedinordertotakeintoaccounttheenhancementofthewimp
been raised. The probability of such a situation was estimated
to be low (about10−4). Morerecently,thisproposalresurfaced annihilation rate inside substructures. That procedure has been
widelyusedinthepastbutisshowntobewronginthepresent
(Cumberbatch & Silk 2006). As an illustration of the methods
paper.Inthefollowingwediscussthemethodthatmustbefol-
developedinthiswork,weshowthattheoddsforsuchanoccur-
lowedinordertocorrectlycomputethesignalφattheEarth.
renceareevenlowerthanHooperetal’sestimations.
Weassumethatsubstructures–whosedensityprofileinside
theithclumpisδρ(x)–floatinsideasmootherbackgroundwith
i
2. Theeffectiveboostfactor mass density ρ(cid:9) which is a priori different from ρ introduced
s s
above.Thehalodensityρcanbewrittenas
We first consider the case of wimps annihilating into positrons (cid:6)
and electrons at a given energy – the source spectrum of ρ=ρ(cid:9) + δρ, (6)
positrons can be considered monoenergetic. In Kaluza–Klein s i
i
inspired models (Servant & Tait 2003), dark matter species
may substantially annihilate into electron-positron pairs with andeachclumphasamass
a branching ratio as large as ∼20%. The positron production (cid:5)
ratePe+ countsthenumberofannihilationstakingplaceperunit Mi = d3x δρi(x). (7)
volumeatsomepointx ithclump
(cid:3) (cid:4)
(cid:1) (cid:2) ρ(x) 2 BecausewimpannihilationinvolvesthesquareoftheDMmass
Pe+(x)=δ(cid:5)σann χχ→e+e− v(cid:7) m (1) density, the production of positrons inside the ith protohalo is
χ enhanced with respect to the situation where that substructure
where the δ term is equal to 1/2 for a Majorana particle, tak- would be diluted in the surroundingmedium.Should the latter
ingintoaccountthefactthattheseparticlesarenotdiscernible, be homogeneously spread with a mass density ρh (which will
whereasitisequalto1/4forDiracparticles,takingintoaccount correspondto fρs below),the enhancementwould be givenby
thefactthatthedensityofparticlesandantiparticlesisρ/2and theboostfactorBiwhichwedefineas
notρ.Thecontributionoftheinfinitesimalvolumed3xlocatedat (cid:5)
pointxtothefluxattheEarth–inunitsofcm−2s−1sr−1GeV−1 d3x δρ2(x)= M ×ρ B. (8)
i i h i
–oftheresultingpositronswithenergyEmaybeexpressedas ithclump
(cid:3) (cid:4)
ρ(x) 2 That relation does not mean that the annihilation signal scales
dφ=SGe+(x(cid:4),E ← x,ES) ρ d3x, (2) linearly with the clump mass. The boost factor Bi takes into
0 account the inner DM distribution so that various profiles for
wherethe quantitySdependsonthe massdensityofreference δρi can lead to very different valuesfor Bi. The relevantquan-
ρ0 andonthespecificfeaturesofthehighenergyphysicsmodel tityturnsouttobetheeffectivevolumeBiMi/ρh.Inthecaseof
atstake model(B) ofBertoneet al. (2005)wherethe DM clumpshave
(cid:3) (cid:4) beenaccretedaroundintermediate-massblackholes,theaverage
S= 4δπve+(E)(cid:5)σann(cid:1)χχ→e+e−(cid:2) v(cid:7) mρ0 2· (3) vraadluiuesfiosrotnhlayt∼cr1upccia.lRfealcattoiornis(6∼)t4ra×ns1l0at5eskpinct3oetvheenpoifsitthreonspfliukxe
χ
attheEarth
The velocity of the positron with energy E at the Earth is de-
noted by ve+. The Green function Ge+ is discussed in Sect. 3. φ=φ(cid:9)s + φr, (9)
J.Lavalleetal.:Positronannihilationinaclumpyhalo 829
whosecomponent consider that it is constant. The expression for the flux φ
(cid:6) simplifiesinto
φr = ϕi (10) φ=φ(cid:9)+S BcMc (cid:6) G. (13)
i s ρ i
0 i
isproducedbytheconstellationofDMprotohalosthatpervade
(iii) A fraction f of the total DM halo is in the form of sub-
the MilkyWay.ThesignalφattheEarthisthereforeenhanced
structuresembeddedinsideasmoothcomponentwithmass
by a factor of B ≡ φ/φs with respect to the situation where densityρ(cid:9).Intheintermediate-massblackholescenarioof
theDMhaloiscompletelysmoothwithmassdensityρs.Many Bertoneestal.(2005),thefraction f issosmallthatρ(cid:9) (cid:12)ρ .
clumpdistributionsarepossibleandleadtodifferentvaluesfor s s
On the contrary,in theDiemandetal. (2005)simulations,
theboostB.Thedistributioninsidewhichweareembeddedisof
a valueas largeas f ∼ 0.5is foundwith a preponderance
courseunique.Unfortunately,weknowlittleaboutit.Inorderto
of small-scale clumpswhich shouldtrace the smooth DM
predictthesetofplausiblevaluesfortheboostB,weareforced
densityasascertainedinBerezinskyetal.(2003).Themass
to consider the vast ensemble of all the possible DM substruc- densityρ(cid:9) couldbequitedifferentfromρ butitscontribu-
tureconfigurations.Ourlackofknowledgelimitsustoderiving tionφ(cid:9) tostheoverallsignalφissmall.Wsewillassumefor
trends for the boost. The analysis of how B is statistically dis- s
simplicitythat
tributed is postponedto Sect. 4.1. Instead we now focus on its
averagevalueBeff whichsufficeswhenitsvarianceissmall.To ρ(cid:9)s =(1− f)ρs, (14)
proceed,afewsimplificationsarehelpful.
where f is constant all over the Milky Way. The corre-
sponding flux ratio φ(cid:9)/φ – which should not exceed 1 in
(i) Wewillfirstassumethatclumpsarepracticallypoint-like. s s
anycase–isnowgivenbythefactor(1− f)2.
Thishypothesisis expectedtobe validwhenthepropaga-
(iv) AnumberN ofDMsubstructurespervadetheMilkyWay
tion distance is large compared to the size of the clump. H
halo.Inthisanalysis,wewillnotconsiderthefluctuations
AsthevolumeofthegalaxyfilledbytheDMsubstructures
of that number. The probability that one of those lies at
becomesnegligible,thehalodensityρbecomes
(cid:6) point xiscontrolledbythedistribution p(x).Thenumber
ρ=ρ(cid:9) + M δ3(x−x), (11) ofclumpsthatthevolumed3xcontainsonaverageis
s i i
i (cid:5)dn(cid:7)= N p(x) d3x. (15)
H
and thesmoothcomponentφ(cid:9) of the fluxis givenby rela- WeinferanaveragefluxattheEarth
s
tion (5) where the mass density ρs is now replaced by ρ(cid:9)s. (cid:7)(cid:6) (cid:8)
Moreover,the positronflux ϕ whichthe clumplocated at B M
i (cid:5)φ(cid:7)=(1− f)2 φ +S c c G , (16)
positionxi yields,simplifiesinto s ρ i
0 i
BM
ϕi =S ρi i Gi, (12) wheretheaveragesumovertheGreenfunctionsGi isgivenby
0 theintegral
whereGi ≡G(xi,E). (cid:7)(cid:6) (cid:8) (cid:5)
(ii) TheboostfactorBiatthesourceshouldvaryfromonepro- Gi = G(x,E) (cid:5)dn(cid:7). (17)
tohalotoanotherevenifthemass Mi isassumedconstant. i DMhalo
TheinnerregionsoftheMilkyWaypresumablycollapsed
For illustration purposes,we have chosen in our numericalex-
earlierthanitsoutskirts,draggingwiththemsubstructures
amples a particular clump distribution. Inspired by Diemand
whose concentrations are higher than for the galactic pe-
etal.(2005),wehaveassumedthatprotohalostracethesmooth
riphery.Wecouldexpecttohavelargervaluesof B inside
i
distributionofdarkmatterwith
thesolarcircle.However,clumpsthatmovenearthegalac-
tic center experience strong tides that could significantly ρ (x)
reshape them (Berezinsky et al. 2003). Clumps may par- p(x)= s , (18)
M
tiallyevaporatelikeglobularclustersthatexhibitcharacter- H
istictidaltails.Ifthateffectisdominant,theclumpmassis where M is the mass of the DM Milky Way halo. We stress
H
reduced and probablythe boostfactor too – if the density thatouranalysisdoesnotdependonthatspecificchoiceandis
profileofthesubstructurereadjustsitselfaccordingly.Itis completelygeneral.Consideringa differentdistribution p(x) –
thereforedifficulttopredicthowtheclumpboostfactor B withnorelationtothemassdensityρ inparticular–wouldnot
i s
varieswithposition.Tosimplifythediscussion,weassume qualitativelyaffectthemainconclusionsofouranalysis.
that all the clumps have the same mass M ≡ M and the Wederivetheeffectiveboost
i c
sameboostfactorBi ≡ Bc.Thefirsthypothesisissupported (cid:5)φ(cid:7) I
by numericalsimulationsthatindicate thatthe mass func- Beff(E)≡ φ =(1− f)2 + f Bc I1, (19)
tion of substructures is a self-similar power law of slope s 2
dn(M)/dlogM ∝ M−1 and is actually dominated by the wheretheintegralI isdefinedby
n
lightestclumps(Diemandetal.2005).Thelatterhypothe- (cid:5) (cid:3) (cid:4)
sisisapriorimorequestionable(Zhaoetal.2005b;Moore ρ (x) n
I (E)= G(x,E) s d3x. (20)
et al. 2005; Zhao et al. 2005a; Berezinsky et al. 2006). It n
ρ
isneverthelessareasonablechoiceinsofarastheeffectsof DMhalo 0
agranularDMdistributiononthefluxofpositronswillbe AlthoughtheboostatthesourceB isfixed,theboostofthesig-
c
showntobemostlylocal.TheactualvalueofBishouldnot nalatthe Earth Beff dependsonboththe natureandtheenergy
vary much in the solar neighbourhood and we can safely ofthecosmicrayspeciesthroughtheGreenfunctionG andthe
830 J.Lavalleetal.:Positronannihilationinaclumpyhalo
Fig.1.TheeffectiveboostfactorBeffasafunctionofthepositronenergyEinthecaseofa100GeVline.Afraction f =0.2oftheDMdistribution
isintheformofsubstructureswhoseindividualboostfactorB –relativetothesolarneighbourhooddensity–hasbeenvariedfrom3to100.An
c
isothermalhalo–panela)–andaNFWprofile–panelb)–areconsidered.Theyillustratetheinfluenceofthecentralprofileindex.Theincrease
ofBeffisnoticeableespeciallyaroundE∼10GeV.
integralsI andI .Asthefluxφ isshiftedupwardsasaresult 3. Thepositronpropagator
1 2 s
of clumpiness, it also experiences a spectral distorsion insofar
The departures of the positron flux from φ are expected to be
asBeffisenergydependent.Thispropertyhasneverbeennoticed large when the positron energy E is closse to the production
beforeandisillustratedinthetwopanelsofFig.1wherethecase
ofa100GeVpositronlineisfeatured.Afraction f =0.2ofthe valueES.Inthisregime,theparticlescannothavebeenproduced
faraway.Theymostlyoriginatefromaregionclosetothesolar
DMhalohascollapsedintheformofclumpswhoseboostfac-
systeminsidewhichthedistributionofclumpsmaysignificantly
torB varyfrom3to100.IntheDiemandetal.(2005)numerical
c fluctuate. That is why we have focusedour analysis on cosmic
simulations,suchavalueforthefractionwouldcorrespondtoa
minimummass scale of 104 M(cid:4). The latter lies typicallyatthe ray positrons whose propagationthroughoutthe galaxy is now
brieflysketched.
lowertipoftherangeofprotohalomasseswhichwehaveused
Themasterequationforpositronpropagationisthecontinu-
inourexamples.AsfortheboostB ,thevaluesquotedinthelit-
c
ityrelation
eraturevaryfromafew(Berezinskyetal.2006)uptoovertwo
ordersofmagnitude(Diemandetal.2005).Thereferencemass
∂ Jµ + ∂ JE = Q, (21)
densityρ hasbeensetequaltothesolarneighbourhoodvalueof µ E
0
ρ ((cid:4))=0.3GeVcm−3.Theincreaseoftheeffectiveboostfactor
s whereQdenotestheproductionrateofpositronsperunitofvol-
withpositronenergyisclearinbothpanels.Neartheline–inthe
umeandenergy.Thespace-timevectorcurrentisdefinedas
regionwhereEtendstotheinputenergyE –thepositronGreen
S
function G probes only a small region of the Milky Way halo dn
Jµ = (cid:5)x˙µ(cid:7). (22)
around the solar system. With our definition of ρ , the integral
0 dE
ratioI1/I2isunityandBeffmaybeapproximatedby∼f Bc.IfE
isnowvariedfromitsupperlimitE downwards,largerportions The time-component J0 = dn/dE ≡ ψ(x,E) denotesthe num-
S
of the halo come into play in the integralsI and I , decreas- berdensityofparticlesperunitofvolumeandenergy.Thespace
1 2
ingtheirratio.Thateffectisquiteobviousinpanela)wherean currentaccountsforthescatteringofcosmicraysupontheinho-
isothermalprofile is assumed with core radiusa = 0.5 kpc. mogeneitiesofthegalacticmagneticfieldswhichisdescribedas
core
The DMdensityρ within1 kpcofthe galacticcenteris larger adiffusionprocesswith
s
thaninthecaseofaNFWdistributionandtherelativeincrease
of I – where the square of ρ is relevant and not merely ρ J =−K(x,E) ∇ψ. (23)
2 s s
alone–withrespecttoI ismorepronounced.Itispossiblethat
1 The energy component JE depends on the energy loss rate b
the energy dependence of Beff that we have discovered could through
strengthen the case of DM particles as a plausible explanation
(cid:9)(cid:10) (cid:11) (cid:12)
ofthestillputativepositronexcessreportedbyHEAT.Intheex-
JE =ψ E˙ ≡−b(E) . (24)
amplefeaturedinFig.1,thelargestspectraldistortionisactually
obtainedforapositronenergyE ∼10GeV.
AboveafewGeV,positronenergylossesaredominatedbysyn-
chrotronradiationinthegalacticmagneticfieldsandbyinverse
This distorsion effect should not be present in the case of Compton scattering on stellar light and on CMB photons. The
gammarays,whosepropagationdoesnotdependonenergy.For energylossratebdependsonthepositronenergyEthrough
antiprotons,theGreenfunctionalreadyprobesasignificantpor-
tionoftheDMhalo,andweanticipateamilddependenceofthe E2
b(E)= · (25)
boostfactorontheenergy.
E τ
0 E
J.Lavalleetal.:Positronannihilationinaclumpyhalo 831
WehavesettheenergyofreferenceE to1GeVandthetypical where the connection between the energy E and the pseudo-
0
energy loss time is τ = 1016 s. The master Eq. (21) may be time t˜is given by relation (29). In the case of monochromatic
E
expandedinto positrons,theproductionrateis
∂ψ − ∇·{K(x,E) ∇ψ} − ∂ {b(E)ψ}=Q(x,E). (26) Q(x,E)=Pe+(x) δ(E−ES), (36)
∂t ∂E
and the positron space and energy density at the Earth may be
Inordertosimplifythediscussion,steadystate isassumedand expressedas
the space diffusion coefficient K is taken to be homogeneous
withtheenergydependence ψ(x(cid:4),E) = θ((cid:5)ES−E) (37)
(cid:3) (cid:4)
E α × d3xSGe+(x(cid:4),E ← xS,ES) Pe+(xS).
K(E)= K · (27)
0 E
0
Equation(2)isbasedonthisrelation.
Thediffusioncoefficientat1GeVisK =3×1027cm2s−1with Thediffusivehaloinsidewhichcosmicrayspropagatebefore
0
aspectralindexofα=0.6.ThemasterEq.(26)simplifiesinto escapingintotheintergalacticmediumispicturedasaflatcylin-
(cid:3) (cid:4) derwithradiusR = 20kpcandextendsalongtheverticaldi-
gal
∂ (cid:12)2 rectionfromz=−Luptoz=+L.Wehaveassumedhereahalf-
K (cid:12)α∆ψ + ψ + Q=0, (28)
0 ∂(cid:12) τ thickness of L = 3 kpc. Without any boundary condition, the
E
propagatorG˜ wouldbegivenbythe3Drelation(32).However,
where(cid:12) denotestheratioE/E0. cosmicraysmayescapeoutsidethediffusivehaloandG˜ should
Equation(28)maybesolvedwiththeBaltz&Edsjö(1999) account for that leakage. In spite of the boundary at R , we
gal
solutionoftranslatingtheenergyE intothepseudo-time haveassumedthatcosmicraydiffusionisnotlimitedalongthe
(cid:3) (cid:4) radialdirectionbuttakesplaceinsideaninfinitehorizontalslab
(cid:12)α−1
t˜(E)=τ v(E) = · (29) withthickness2L.Wehaveneverthelessdisregardedsourceslo-
E 1−α catedataradialdistanceRlargerthanR .Indeed,becausetheir
gal
energyis rapidlydegradedas theypropagate,positronsare ob-
Theenergylossesthatpositronsexperienceleadtoanevolution
served close to where they are produced. Our radial treatment
inthispseudo-timesothatthepropagationEq.(28)greatlysim-
is justified because positrons do not originate from far away
plifiesinto
(Maurin & Taillet 2003). Even in the case of antiprotons for
∂ψ˜ (cid:1) (cid:2) whichthegalacticpropagationrangeissignificantlylargerthan
− K ∆ψ˜ = Q˜ x,t˜ . (30) forpositrons,theeffectsoftheradialboundaryattheEarthare
∂t˜ 0
notsignificantascosmicrayspeciestendtoleakaboveandbe-
Thespaceandenergypositrondensityisnowψ˜ =(cid:12)2ψwhereas neath the diffusive halo at z = ±L instead of traveling a long
thepositronproductionratehasbecomeQ˜ =(cid:12)2−αQ.Noticethat distancealongthegalacticdisk.Theinfiniteslabhypothesisal-
bothψ˜ and Q˜ havethesame dimensionsasbeforebecause(cid:12) is lows the radialandverticaldirectionsto be disentangledin the
reducedpropagatorG˜ whichmaynowbeexpressedas
dimensionless.Withoutanyspaceboundarycondition,Eq.(30)
(cid:3) (cid:4)
maybereadilysolved.Ifadropisdepositedattheoriginofthe (cid:1) (cid:2) θ(τ˜) R2
co(cid:1)ordinat(cid:2)esatpseudo(cid:1)-t(cid:2)imet˜S =0 G˜ x,t˜← xS,t˜S = 4π(cid:1)K0τ˜ exp −(cid:2)4K0τ˜
Q˜ xS,t˜S =δ3(xS)δ t˜S , (31) ×V˜ z,t˜←z ,t˜ , (38)
S S
the subsequent diffusion in an infinite 3D space would result whereτ˜ = t˜−t˜ .Theradialdistancebetweenthesourcex and
into the density ψ˜ atposition x andpseudo-timet˜givenbythe S S
thepointxofobservationisdefinedas
well–knownGreenfunction
(cid:9) (cid:12)
ψ˜(cid:1)x,t˜(cid:2) ≡ G˜(cid:1)x,t˜←0,0(cid:2) R= (x−xS)2 + (y−yS)2 1/2. (39)
(cid:3) (cid:4)
= θ(cid:1)t˜(cid:2) (cid:13)4πK t˜(cid:14)−3/2 exp − r2 , (32) Shouldpropagationbefreealongtheverticaldirection,theprop-
0 4K t˜ agatorV˜ wouldbegivenbythe1DsolutionV tothediffusion
0 1D
Eq.(30)
wherer≡|x|.ThegeneralsolutionofEq.(30)maybeexpressed (cid:1) (cid:2) (cid:1) (cid:2)
withtheGreenfunctionG˜ as V˜ z,t˜←z ,t˜ ≡ V z,t˜←z ,t˜
S S 1D S (cid:3)S (cid:4)
ψ˜(cid:1)x,t˜(cid:2)=(cid:5)t˜St˜=S=0t˜dt˜S (cid:5) d3xS G˜(cid:1)x,t˜← xS,t˜S(cid:2) Q˜(cid:1)xS,t˜S(cid:2), (33) = √4θπ(τ˜K)0τ˜ exp −(z4−K0zSτ˜)2 · (40)
Buttheverticalboundaryconditionsdefinitelyneedtobeimple-
andtranslatesinto
(cid:5) (cid:5) mented.Whereverthesourceinsidetheslab,thepositrondensity
ES=+∞ vanishesatz = ±L.Afirstapproachreliesonthemethodofthe
ψ(x,E) = dE d3x
S S so-called electricalimagesandhasbeen implementedby Baltz
ES=E & Edsjö (1999). Any point-like source inside the slab is asso-
×Ge+(x,E ← xS,ES) Q(xS,ES). (34) ciatedwiththeinfiniteseriesofitsmultipleimagesthroughthe
ThepositronpropagatormaybeobtainedfromG˜ through boundaries at z = ±L which act as mirrors. The nth image is
locatedat
τ (cid:1) (cid:2)
Ge+(x,E ← xS,ES)= E E(cid:12)2 G˜ x,t˜← xS,t˜S , (35) zn =2Ln + (−1)n zS, (41)
0
832 J.Lavalleetal.:Positronannihilationinaclumpyhalo
and has a positive or negative contribution depending on at random and that the set of all their possible distribu-
whethernisanevenoroddnumber.Whenthediffusiontimeτ˜ tionsmakesupthestatisticalensemblethatweconsiderin
issmall,the1Dsolution(40)isaquitegoodapproximation.The this section. The aim of our analysis is to investigatehow
relevantparameteris stronglythefluxφ mayfluctuateasaresultoftherandom
r
natureof the wimp clump distribution.We will derivethe
ζ = L2 , (42) associatedcosmic-rayfluxvarianceσr definedas
4K τ˜
0 σ2 =(cid:5)φ2(cid:7) − (cid:5)φ (cid:7)2. (49)
r r r
andintheregimewhereitismuchlargerthan1,thepropagation
Thevarianceσ isanessentialtool.Becausethetotalflux
isinsensitivetotheverticalboundaries.Onthecontrary,whenζ r
φanditsrandomcomponentφ areshiftedwithrespectto
is much smaller than 1, a large number of images need to be r
eachotherbytheconstantquantity
takenintoaccountinthesum
(cid:1) (cid:2) (cid:6)+∞ (cid:1) (cid:2) φ−φr =(1− f)2 φs, (50)
V˜ z,t˜←zS,t˜S = (−1)n V1D z,t˜←zn,t˜S , (43) thevarianceoftheformerisgivenby
n=−∞ σ2 =(cid:5)φ2(cid:7) − (cid:5)φ(cid:7)2 =(cid:5)φ2(cid:7) − (cid:5)φ (cid:7)2 =σ2. (51)
and convergence may be a problem. A different approach is φ r r r
possible in that case. The 1D diffusion Eq. (30) looks like the The effective boost Beff discussed in Sect. 2 is an average
value around which the true flux enhancement B ≡ φ/φ
Schrödingerequation–inimaginarytime–thataccountsforthe s
fluctuateswiththevariance
behaviourofaparticleinsideaninfinitelydeep1Dpotentialwell
thatextendsfromz = −Ltoz = +L.Theeigenfunctionsofthe σ = σφ = σr· (52)
B
associatedHamiltonianarebotheven φ φ
s s
ϕ (z)=sin{k (L−|z|)} (44) Therefore,thedeterminationofσrleadsimmediatelytothe
n n
boostfluctuationsσ .
B
andodd (ii) We will furthermore assume that clumps are distributed
(cid:13) (cid:14) independently of each other. The problem is then greatly
ϕ(cid:9)n(z)=sin kn(cid:9) (L−z) (45) simplifiedbecausewejustneedtodeterminehowasingle
clumpisdistributedinsidethegalactichaloinordertode-
functionsoftheverticalcoordinatez.Thewave-vectorsk andk(cid:9)
n n rive the statistical properties of an entire constellation of
arerespectivelydefinedas suchsubstructures.Inparticular,the averagevalue(cid:5)φ (cid:7) of
(cid:15) (cid:16) r
therandomcomponentofthecosmicrayfluxisreadilyob-
1 π π
kn = n− 2 L (even) and kn(cid:9) =n L(odd). (46) tainedfromtheaverageflux(cid:5)ϕ(cid:7)producedbyasingleclump
throughtherelation
Theverticalpropagatormaybeexpandedastheseries (cid:5)φ (cid:7)=N (cid:5)ϕ(cid:7), (53)
r H
V˜ (cid:1)z,t˜←zS,t˜S(cid:2) = (cid:6)+∞ L1 (cid:17)e−λnτ˜ϕn(zS) ϕn(z) ewrheedr.eTNheHvdaerniaontecsetσhre–towtahlincuhmisbtehreocfruclcuiamlpqsutaontbietycfoonrstihde-
n=1 (cid:18) fluxfluctuations–mayalsobeexpressedas
+e−λ(cid:9)nτ˜ϕ(cid:9)n(zS) ϕ(cid:9)n(z) , (47) σ2r = NHσ2 = NH(cid:9)(cid:5)ϕ2(cid:7) − (cid:5)ϕ(cid:7)2(cid:12). (54)
where the time constants λ and λ(cid:9) are respectively equal to (iii) Thesetoftherandomdistributionsofonesingleclumpin-
n n
K0kn2 andK0kn(cid:9)2.Intheregimewhereζ ismuchsmallerthan1 side the domain DH formsthe statistical ensembleT that
–forverylargevaluesofthediffusiontimeτ˜ –justafeweigen- weneedtoconsider.Aneventfromthatensembleconsists
functionsneedtobeconsideredforthesum(47)toconverge. ofaclumplocatedatpositionxwithintheelementaryvol-
umed3x. Its probabilitydP will be assumed to followthe
smoothedDMmassdistributionρ sothat
s
4. Ananalyticapproachofthecosmicrayflux
ρ (x)
fluctuations dP= p(x) d3x= s d3x. (55)
M
H
4.1.Therandomfluxφranditsvariance ThedomainD overwhichourstatisticalanalysisisperformed
H
Thecosmicrayflux(13)attheEarthcontainstherandomcom- is so large that the total number NH of clumps that it contains
ponent is essentially infinite. That region DH behaves therefore like a
(cid:6) (cid:6) so-calledthermostatinstatisticalmechanics.Itencompassesthe
φ = ϕ =S BcMc G, (48) diffusivehaloandmayevenbemuchbigger.Itmaybethought
r i i
ρ of – but not exclusively – as the entire Milky Way DM halo.
i 0 i
Itsactualsizehasnoimportancebecauseitwilldisappearfrom
whichisproducedbytheconstellationofDMclumpsinsidethe the final results in the limit where the ratio 1/N is negligible.
H
MilkyWayhalo. TheonlyrequirementisthatN shouldbemuchlargerthanthe
H
typical number N of clumps that effectively contribute to the
S
(i) TheactualdistributionofDMsubstructuresisunique,asis signal φ at the Earth. The domain D contains the total DM
r H
the cosmic ray flux that it generatesat the Earth. We will
mass M –afraction f ofwhichliesin N identicalclumpsso
H H
neverthelessconsideritasoneparticularrealizationamong
that
anessentiallyinfinitenumberofdifferentpossiblerealiza-
tions. We furthermore assume that clumps are distributed N M = f M . (56)
H c H
J.Lavalleetal.:Positronannihilationinaclumpyhalo 833
WearenowreadytoderivetheprobabilitydistributionP(ϕ)as-
sociatedwiththesignalϕthatasingleclumpgenerates.Thesta-
tistical propertiesof the random variable ϕ{T} reflect those of
thestatisticalensembleT itself.Moreprecisely,theprobability
functionP(ϕ)isrelatedtothespacedistribution p(x)through
(cid:5)
P(ϕ) dϕ=dP= p(x) d3x. (57)
D
ϕ
The subdomain D over which the space distribution p(x)
ϕ
should be integratedin the previousexpressionyields a flux at
theEarthbetweenϕandϕ+dϕ(D isthustheunionofallD ).
H ϕ
In the case of positrons, the probability distribution P(ϕ) will
be shown in Sect. 4.2 to concentrate around a flux ϕ equal to
0.Theaveragevalue–overthestatisticalensembleT –ofany
functionF thatdependsonthefluxϕmaybeexpressedas
(cid:5) (cid:5)
(cid:5)F(cid:7)= F(ϕ) P(ϕ) dϕ= F {ϕ(x)} p(x) d3x. (58)
D
H
In particular,the flux that a single clumpyields on average
attheEarthisreadilyderivedfromtheintegral
(cid:5) Fig.2. Therelativevariance σ /(cid:5)φ(cid:7)of therandom component of the
r r
M B
(cid:5)ϕ(cid:7)= ϕ(x) p(x) d3x=S c c I , (59) positronfluxattheEarth–solidlines–anditshard-sphereapproxima-
D MH 1 tion–long-dashedcurves–asafunctionofthepositronenergyE for
H threedifferentvaluesoftheclumpmass M .Theinjectedpositronen-
c
where I has been defined in relation (20). The average value ergyE hasbeensetequalto100GeV.ANFWprofilewithtypicalscale
n S
oftherandomfluxφ impliesN clumpsandexpression(53)– 25kpchasbeenassumed.Atfixedclumpmass,thevarianceincreases
r H
withthehelpofrelation(56)–leadsto withEandmatchesitshard-sphereapproximationabove∼40GeV.As
thenumber of√clumps√isdecreased, thecurvesareshiftedupwardsby
(cid:5)φr(cid:7) = SfBcI1 = fB I1, (60) afactorof1/ NH ∝ Mc.TherelativevarianceσB/Beff oftheboost
φ φ c I factorisalsodisplayedbytheshort-dashedcurve.Inthelimitwherethe
s s 2 clumpboostfactor B islarge–avalueof100hasbeenassumedhere
c
andtoformula(19). –σB/Beffandσr/(cid:5)φr(cid:7)areapproximatelyequal.
Startingfromthedefinition(54),thevarianceσ maybede-
r
rivedinthesamespiritwith
σ2r = 1 (cid:5)ϕ2(cid:7) − 1 · (61) imseinntcrBea≡seφd/.φTshiesarelslaotipvreesveanrtieadncienσFiBg/.B2e.ffInofthtehelimfluitxwehnehraentchee-
(cid:5)φ (cid:7)2 N (cid:5)ϕ(cid:7)2 N individualclumpboostfactorB islarge–wehaveselectedhere
r H H c
avalueofB = 100–therandomcomponentφ ofthepositron
c r
With the help of relation (58), the mean square of the single fluxdominatesoveritssmoothcounterpart(1− f)2 φ sothat
s
clumpfluxmaybeexpressedas
(cid:5) σ σ /φ σ
S2M2B2 B = r s (cid:12) r · (65)
(cid:5)ϕ2(cid:7)= D ϕ2(x) p(x) d3x= ρ0McHc J1, (62) Beff (1− f)2 + (cid:5)φr(cid:7)/φs (cid:5)φr(cid:7)
H
wheretheintegralJ isdefinedas That is why the solid lines and short-dashed curves of Fig. 2
n
(cid:5) (cid:3) (cid:4) are similar. In Fig. 3, the black central curve features the ef-
Jn(E)= G2(x,E) ρsρ(x) n d3x. (63) fceacsteivBecbo=ost1f0a0ctoorfBtheffeopfanaeNlFbWofhaFliog.an1dfcroormreswphoincdhsittohthaes
DMhalo 0 been extracted. The 1-σ range of its fluctuations extends from
Somestraightforwardalgebraleadstotherelativevariance Bmin = Beff − σB up to Bmax = Beff + σB. At fixed clump
mass, thatrange opensup as E approachesthe injected energy
σ2r = MH J1 − 1 (cid:12) Mc J1· (64) ES = 100GeV. Thefluctuationsin thepositronsignalincrease
(cid:5)φr(cid:7)2 ρ0NH I21 NH fρ0 I21 issiganlisfiocparnotlpyojrutisotnbaelltoow1t/h√eNpos∝itro√nMlin.eT.hTahteisbowohsytvtahreiaflnuccetuσaB-
c c
Because the domain DH is so large – remember that both DH tion bandbroadensasthe clumpmass isincreasedfrom104 to
and the Milky Way DM halo encompassesthe diffusive halo – 106 M(cid:4).
wecansafelydroptheratio1/N inthepreviousexpression.
H
The positron propagator of Sect. 3 has been used in re-
lation (64) to derive the solid curves of Fig. 2. At fixed N , 4.2.ThefluxdistributionP(ϕ)ofasingleclump
H
the clump mass Mc is determined by Eq. (56) and the rela- ThepositronfluxatenergyE ≤ E thatasingleclumplocated
tive variance σ /(cid:5)φ (cid:7) increases with the positron energy E at S
r r at position x generatesat the Earth implies the propagatordis-
the Earth. This behaviourwill be explainedin Sect. 5 with the
cussedinSect.3
ha√rd-sphere√approximation.Theratioσr/(cid:5)φr(cid:7)isproportionalto
1/ N ∝ M , and weighted by an effective volume J /I2. B M
ThecuHrvesarecthereforeshiftedupwardswhentheclump1mas1s ϕ(x)=S cρ c Ge+(x(cid:4),E ← x,ES). (66)
0
834 J.Lavalleetal.:Positronannihilationinaclumpyhalo
Fig.4. The density of probability P(Φ) as a function of the reduced
Fig.3.TheeffectiveboostfactorBeff–blackline–isplottedasafunc- fluxΦ = ϕ/ϕ thatasingleclumpgenerates.ANFWhalohasbeen
tion of the positron energy E for an injected energy ES = 100 GeV. assumedwithmaaxscaleradiusof25kpc.ThedomainD overwhichthe
The1-σrangeofitsfluctuationsextendsfrom Bmin = Beff −σB upto probability isnormalized to unityistheMilky WayDHMhalo up toa
Bmax = Beff +σB.Atfixedclumpmass,thatrangeopensupas E ap- radiusof20kpc.Theinjectionenergyis E = 100GeV.Thesmaller
proachestheinjectedenergyE =100GeV.Italsowidenssignificantly S
S thepositronenergyEattheEarth,thelargertheprobabilitydensityfor
atfixedpositronenergyEwhenthenumberofclumpsisdecreased.
anon-vanishingflux.Thefullynumericalcalculations–solidcurves–
arecomparedtotheinfinite3Dapproximation(71)thatcorrespondsto
thelong-dashedlines.
andmaybeexpressedwiththereducedGreenfunctionG˜ as
B M τ (cid:1) (cid:2)
ϕ(x)=S cρ c E E(cid:12)2 G˜ x(cid:4),t˜← x,t˜S . (67)
0 0
When the substructure is very close to the Earth, the flux ϕ
reaches a maximal value ϕ that depends both on the clump
max
propertiesthroughtheeffectivevolumeB M /ρ andonthespe-
c c 0
cificfeaturesassumedfortheDMparticlethroughthefactorS.
Withoutanylossofgenerality,wecansignificantlysimplifythe
discussionbyconsideringtheratio
ϕ(x) G˜(x)
Φ(x)= = , (68)
ϕ G˜
max max
insteadofthefluxϕitself.Wethereforewouldliketoderivethe
density of probabilityP(Φ) associated with the reducedflux Φ
asitvariesfrom0to1.
InFig.4,thatdistributionispresentedforthreetypicalval-
uesofthepositronenergyE attheEarth.TheenergyE ofthe
S
positron line has been set equal to 100 GeV and a NFW DM
halohasbeenassumed.Thesolidcurvescorrespondtothefully
numerical calculation of P(Φ) based on relation (57). The do- Fig.5. Thedensityofprobability P(Φ),ΦP(Φ)andΦ2P(Φ)arefea-
main D over which the probability is normalized to unity is turedasafunctionofthereducedfluxΦ=ϕ/ϕ forapositronenergy
H max
theMilkyWayDMhalouptoaradiusof20kpc.Thatdomain attheEarthof50GeV.
encompassesthediffusivehalooutsideofwhichthecosmicray
densityvanishes.Mostoftheprobabilityisthereforecontained Φ2P(Φ) whose integrals from Φ = 0 up to Φ = 1 are respec-
in the low flux regionand the density P(Φ) divergesat Φ = 0. tivelyrelatedto(cid:5)ϕ(cid:7)and(cid:5)ϕ2(cid:7).
As the energy E increases towards ES, the region of the dif- When the positron energy E is close to the energy ES, the
fusive halo that is probed by the positron propagator shrinks. pseudo-timedifference τ˜ = t˜−t˜ is so small that the diffusion
That region correspondsto large values of the positron flux Φ. is no longer sensitive to the vertiScal boundariesat z = −L and
As its volume decreases when E approaches ES, fewer clumps z = +L. The Green functionG˜ can be safely approximatedby
areinvolvedinthesignalandthecorrespondingprobabilityde- theGaussianfunction(seeEq.(32))
creases.NoticeinFig.4howtheprobabilitydensityP(Φ)drops (cid:3) (cid:4)
whenE isincreasedfrom1GeVto50GeV.Thelowercurveis (cid:1) (cid:2) r2
reproducedinFig.5togetherwiththedistributionsΦP(Φ)and G˜ x(cid:4),t˜← x,t˜S ={4πK0τ˜}−3/2 exp −4K τ˜ · (69)
0
J.Lavalleetal.:Positronannihilationinaclumpyhalo 835
5. Thehard-sphereapproximation
In the limit where the infinite 3D approximation applies – ac-
tuallyforalargerangeofvaluesofthepositronenergyE atthe
Earth–wecanfurthersimplifythepropagatorGe+andsubstitute
thestepfunction
(cid:1) (cid:2) θ(r −r)
G˜ x(cid:4),t˜← x,t˜S = SV (72)
S
fortheGaussianform(69).Thedistancebetweentheclumpand
theEarthisr ≡ |x−x(cid:4)|.Accordingtothishard-sphereapprox-
imation, the Green functionG˜ reachesthe constantvalue 1/V
S
insidethesphereD ofradiusr andvolumeV –whosecenter
S S S
coincideswiththeEarth–andvanisheselsewhere.Bothexpres-
sions(69)and(72)arenormalizedtounity.Theintegraloverthe
full3DspaceofthesquareofthoseGreenfunctionsshouldalso
bethesame.Thisconditiontranslatesinto
(cid:5)
1
= G˜2d3x, (73)
V
S
Fig.6.Thepositrondiffusionlengthλ decreasesastheenergyEatthe andleadstothevolume
D (cid:19)√ (cid:20)
ElinarethcoarprpesropaocnhdesstothaedeinffeurgsiyonESleonfgtthheλlDineeq.uTahletolotnhge-tdhaischkendeshsoLrizoofntthael VS = 2πλD 3. (74)
diffusionlayers.Belowthatlimit,positronpropagationisnotsensitive
to the vertical boundaries and the infinite 3D approximation is valid. In spite of its crudeness, the hard-sphere approximation turns
This regime corresponds to large values of the parameter ζ – see the out to be quite powerfuland is an excellent tool to understand
definition(42). thesalientfeaturesofthestatisticalpropertiesoftheclumpdis-
tribution and of its flux. The associated density of probability
haslittletodowiththecurvesofFig.4orwithrelation(71).It
This regime corresponds to large values of the parameter ζ – isactuallyabimodaldistributionwith
definedinrelation(42)– oralter√nativelytosmallvaluesofthe
positron diffusion length λ ≡ 4K τ˜. The latter is featured P(Φ)= pδ(Φ−1) + (1−p) δ(Φ). (75)
D 0
in Fig.6asa functionof E forthreedifferentvaluesoftheen-
ergy at source. In the case where E = 100 GeV, the diffusion The reduced flux Φ takes the value of 1 inside the sphere DS
lengthλ exceedsthethicknessL bSelowanenergyof∼8GeV. and0outside.Theprobability pthataclumpliesinsidethedo-
AbovethDatlimit,positronpropagationisnotaffectedbythever- mainDS –fromwhichitmayyieldasignalattheEarth–isthe
ratio M /M of the DM mass M confinedin that sphere with
ticalboundariesandtheinfinite3D approximation(69)applies S H S
withareducedfluxΦthatonlydependsonthedistancerofthe respecttotheDMmass MH containedintheentiredomainDH.
In the limitwhereλ ∝ r is small, the DM distributionis ho-
clumptotheEarth D S
mogeneousinsidethesphereD –withconstantdensityρ ((cid:4))–
(cid:19) (cid:20) S s
Φ=exp −r2/λ2 . (70) andtheprobabilitypmaybeexpressedastheratio
D
M V ρ ((cid:4))
An analytic density of probability may be derived in that case p= S = S s · (76)
M M
with H H
√ ForaninjectedenergyE = 100GeV andapositronenergyat
P(Φ)=2πλ3 ρs((cid:4)) −lnΦ· (71) theEarthE =50GeV,wSefindaprobability p∼2×10−3when
D M Φ the statistical domain D is chosen to be the above-mentioned
H H
NFWhaloextendingupto20kpcfromthecenteroftheMilky
That relation corresponds to the long-dashed curves of Fig. 4
Way.
whereavalueof MH = 1.357×1011 M(cid:4) hasbeenfoundforthe Because p is vanishingly small and the number of clumps
masscontainedintheinner20kpcoftheMilkyWayDMhalo. N insidethedomainD exceedinglylarge,thelimitofPoisson
When the positron diffusion length λ is smaller than the slab H H
D statisticsisreached.Theprobabilitytofindnclumpsinsidethe
thickness L, relation (71) is an excellent approximation to the sphereD isthereforegivenby
densityofprobabilityP(Φ).Asanillustration,wefindavalueof S
λ =1.26kpcwellbelowL=3kpcwhenthepositronenergyE N n
D P(n)= S exp(−N ), (77)
isequalto50GeV.Thisexplainswhythesolidandlong-dashed n! S
redlinesofFig.4aresowellsuperimposed.AsEdecreases,the
diffusionlengthλ becomeslargerwithrespecttoLandthein- where NS ≡ pNH is the average number of clumps that con-
D
tributetothesignal
finite3Dpropagator(69)tendstooverestimatetheregionfrom
whichthesignaloriginatesaswellasthecorrespondingproba- V fρ ((cid:4))
bilitydensityP(Φ).Noticehowthelong-dashedapproximation (cid:5)n(cid:7)= NS = SMs · (78)
linesareshiftedupwardswithrespecttothesolidtruenumerical c
curvesinFig. 4.As E decreases,theapproximation(71)wors- Departures from the statistical law (77) in the case of a realis-
ensandthedisagreementwith thecorrectresultbecomesmore tic positron propagatorwill be discussed in Sect. 6.1 when the
pronounced. numberN oftheclumpsinvolvedinthefluxattheEarthislarge
S
836 J.Lavalleetal.:Positronannihilationinaclumpyhalo
whereastheoppositeregimewillbeaddressedinSect.6.2.The bboooosstt ((5500 GGeeVV)) bboooosstt ((6655 GGeeVV))
EEnnttrriieess 11000000 EEnnttrriieess 11000000
Poissondistribution(77)isassociatedwiththevariance MMeeaann 00..99999999 240 MMeeaann 00..99999966
RRMMSS 00..0044229933 220 RRMMSS 00..0066445588
300 χχ22 // nnddff 22..554499 // 55 χχ22 // nnddff 33..777799 // 77
KK 3399..9911 ±± 11..2277 200 KK 3399..9933 ±± 11..2277
σ2n =(cid:5)n2(cid:7) − (cid:5)n(cid:7)2 = NS. (79) 250 µµσσ 00.. 000044..99449922991122 ±± ±± 00 00..00..0000000011994455 116800 µσµσ 00..00 0066..669900992299 55±± ±±00 ..0000..0000110055228811
200 140
Ipcnoonstishttreeolnlhaatflirodun-xsopafhtesrutehbesatprEupacrrotutxhriem–sagtteihonene,rcatothenestr–irbauinstdipoormnoptphoaartrtitotφnhareloteofntttihhreee 110500 EEMfB sdcc= ll == == 0 5111.00200 000G5 GMeVesoVl 1168020000 EEMfB sdcc= ll == == 0 6111.50200 000G5 GMeVesoVl
number n of clumps lying inside the sphere DS. We therefore 50 40
anticipate that the relative variance σ /(cid:5)φ (cid:7) should be equal to 20
r r 0 0
therelativevarianceσn/(cid:5)n(cid:7)ofthePoissonlaw(77).Inthelimit 0 0.2 0.4 0.6 0.8 1B1M.C2 / B1e.4ff (1B.6eff =1 .280.1)2 0 0.2 0.4 0.6 0.8 1B1M.C2 / B1e.4ff (1B.6eff =1 .280.3)2
whereλ issmallwithrespecttoL–andwherethehard-sphere bboooosstt ((8800 GGeeVV)) bboooosstt ((9900 GGeeVV))
approximDationbecomesvalid–theintegralsJ1andI1simplify. 140 EMREMRnnMMeettaaSSrrnnii ee ss 0000.. ..991111990011990000990011 100 EMREMRnnMMeettaaSSrrnnii ee ss 00 .. 111111..990000550000330011
Ifthemassdensityofreferenceρ issetequaltoitssolarneigh- χχ22 // nnddff 66..448844 // 1166 χχ22 // nnddff 3344..4411 // 2266
bourhoodvalueρ ((cid:4)),theratioJ0 /I2 is1/V sothattheexact 120 KµKµ 00..9999 33999988.. 77±±66 00 ±±..00 1100..33225566 80 KµKµ 00..9999 33998844.. 77±±55 00 ±±..00 1100..66225555
s 1 1 S σσ 00..11009966 ±± 00..00002266 σσ 00..119933 ±± 00..000055
relation(64)simplifiesinto 100
60
80 Es = 100 GeV Es = 100 GeV
σ2r = Mc J1 = Mc = 1 · (80) 60 EMf dc= l = = 0 81.020 0G5 MeVsol 40 EMf dc= l = = 0 91.020 0G5 MeVsol
(cid:5)φr(cid:7)2 fρ0 I21 VSfρs((cid:4)) NS 40 Bcl = 100 20 Bcl = 100
20
We have therefore shown that in the hard-sphere regime, the 0 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
variance σr of the random flux φr is indeed given by the vari- BMC / Beff (Beff = 20.5) BMC / Beff (Beff = 20.6)
ance σn thatcharacterizesthe Poisson statistics (77).InFig. 2, Fig.7. One thousand different realizations of the distribution of DM
therelativevarianc√eσr/(cid:5)φr(cid:7)–solidcurves–anditshard-sphere substructures inside the galactic halo has been generated by Monte-
approximation 1/ NS – long-dashed lines – are presented to- Carlosimulation.TheinjectedenergyisES =100GeV.ANFWprofile
getherforcomparison.Abovea positronenergyattheEarthof hasbeenassumedwithtypicalscale25kpc.Amassfraction f =0.2is
40GeV,thecorrectcalculationanditshard-spherelimitdifferby intheformofclumpswithmass105 M(cid:4).Eachhistogramcorresponds
lessthan∼5×10−3.Theagreementisremarkable.Thediffusion toaspecificpositronenergyEattheEarth.Thenumberofrealizations–
lengthdoesnotexceed∼1.5kpcinthatcaseandthehard-sphere eachinvolving271,488clumps–isplottedasafunctionofthereduced
approximationsuccessfullydescribesthestatisticalpropertiesof boost η = B/Beff. Up to an overall factor of a thousand – that corre-
spondstothenumberofMonte-Carlorealizations–eachpanelfeatures
therandompositronfluxφr.TherelativevarianceσB/Beff ofthe anumericalestimateoftheprobabilitydensityP(η).
boostfacto√risalsowellreproducedbythehard-sphereapproxi-
mation1/ N andboththeshort-dashedandlong-dashedcurves
S
are barely distinguishable from each other at high positron en- expressions(53)and(54)which havebeenestablishedandnu-
ergyE. merically computed in Sect. 4.1. Therefore, the probability of
obtainingafluxφ attheEarthis
r
⎧ ⎫ (cid:3) (cid:4)
6. AMonte-Carloapproachtothecosmicrayflux ⎪⎪⎨ (cid:6) ⎪⎪⎬ 1 (φ −(cid:5)φ (cid:7))2
fluctuations P⎪⎪⎩φr = ϕi⎪⎪⎭= (cid:28) exp − r 2σ2r · (82)
i 2πσ2 r
r
6.1.ThelargeNS limitandthecentrallimittheorem
The probabilitythatthe totalpositronflux φ at the Earthis en-
WhentheaveragenumberNS ofclumpsthatareinvolvedinthe hancedbyafactorofBwithrespecttoacompletelysmoothDM
signalislarge,thePoissonstatistics(77)becomestheGaussian distributionis
distribution ⎧ ⎫
P(δ)= √ 1 exp(cid:19)−δ2/2NS(cid:20), (81) P{B≡φ/φs}= (cid:28)21πσ2 exp⎪⎨⎪⎩−(B−2σB2Beff)2⎪⎬⎪⎭, (83)
2πN B
S
where δ ≡ n − N denotes the departure of the number n of where the varianceσB is givenby relation(52).Finally the re-
substructures insidSe the sphere D√S from its average value NS. ducedboostη≡ B/Beff follow⎧sthesameG⎫aussianlaw
The associated variance is σn = NS. We therefore anticipate 1 ⎪⎨ (η−1)2⎪⎬
tGhaautsthsieanflulaxwφwr withillmaelsaonbvealruaen(cid:5)dφom(cid:7)laynddivsatrriibauntceedσac.cordingtoa P{η≡ B/Beff}= (cid:28)2πσ2 exp⎪⎩− 2σ2η ⎪⎭, (84)
r r η
In order to determine the distribution of probability P(φ )
r
that drives the random flux φr – generated by the entire con- with an averagevalue of (cid:5)η(cid:7) = 1 and a varianceση = σB/Beff
stellation of the clumps lying inside the reservoir D – we nottoodifferentfromσ /(cid:5)φ (cid:7)asshowninformula(65).
H r r
should compute the productof convolutionof the N distribu- To checkourtheoreticalpredictions,we rana Monte-Carlo
H
tions of probability P(ϕ) associated each with the flux ϕ of a simulationofthedistributionofDMsubstructuresintheMilky
single substructure– oralternativelywith itsreducedfluxΦ as Way halo. A thousand different realizations were generated at
was discussed in Sect. 4.2. In the large N regime, the central random assuming a NFW DM galactic halo with a fraction
S
limittheoremmaybeapplied.Thistheoremstatesthattheprod- f = 0.2 in the form of 105 M(cid:4) clumps. In Fig. 7, the number
uct of convolution is a Gaussian distr(cid:9)ibution with(cid:12)mean value ofrealizationsisplottedasafunctionofthereducedboostηfor
(cid:5)φ (cid:7) ≡ N (cid:5)ϕ(cid:7) and variance σ2 ≡ N (cid:5)ϕ2(cid:7) − (cid:5)ϕ(cid:7)2 . These are 4valuesofthepositronenergyattheEarth.Thesedistributions
r H r H