CLUMPY: a code for gamma-ray signals from dark matter structures

2 MB·English

by  A. Charbonnier

clumpy: a code for γ-ray signals from dark matter structures
Alde´eCharbonniera,Ce´lineCombetb,c,DavidMaurina,b,c,d
aLaboratoiredePhysiqueNucle´aireetHautesEnergies,CNRS-IN2P3/Universite´sParisVIetParisVII,4placeJussieu,Tour33,75252ParisCedex05,France
bDept.ofPhysicsandAstronomy,UniversityofLeicester,Leicester,LE17RH,UK
cLaboratoiredePhysiqueSubatomiqueetdeCosmologie,CNRS/IN2P3/INPG/Universite´JosephFourierGrenoble1,53avenuedesMartyrs,38026Grenoble,
France
dInstitutd’AstrophysiquedeParis,UMR7095CNRS,Universite´PierreetMarieCurie,98bisbdArago,75014Paris,France
2
1Abstract
0
We presentthefirstpubliccodeforsemi-analyticalcalculationoftheγ-rayfluxastrophysicalJ-factorfromdarkmatterannihila-
2
tion/decayintheGalaxy,includingdarkmattersubstructures. Thecoreofthecodeisthecalculationofthelineofsightintegralof
n
thedarkmatterdensitysquared(forannihilations)ordensity(fordecayingdarkmatter). Thecodecanbeusedinthreemodes: i)
a
JtodrawskymapsfromtheGalacticsmoothcomponentand/orthesubstructurecontributions,ii)tocalculatethefluxfromaspecific
halo(thatisnottheGalactichalo,e.g. dwarfspheroidalgalaxies)oriii)toperformsimplestatisticaloperationsfromalistofal-
3
2lowedDMprofilesforagivenobject.ExtragalacticcontributionsandothertracersofDMannihilation(e.g.positrons,antiprotons)
willbeincludedinasecondrelease.

]
EKeywords: Cosmicrays,Cosmology,DarkMatter,Indirectdetection,Gamma-rays
H
.
h
PROGRAMSUMMARY 1. Introduction
p
ProgramTitle:CLUMPY
-
oProgramminglanguage: C/C++
rComputer:PCandMac
t Despite the several astrophysical evidences pointing at the
sOperatingsystem:UNIX(Linux),MacOSX
existenceofdarkmatter(flatrotationcurvesofgalaxies,gravi-
aRAM: depends on the requested size of skymaps ( 40 Mb for a
[ ∼ tationallensing,“bulletcluster”,etc.),itsnaturestillevadesus.
500 500map)
× Thissearchhasbecomeoneofthemajortopicinbothparticle
1Keywords: darkmatter,indirectdetection,gamma-rays
physicsandastrophysicsandistackledusingeitherdirectorin-
vClassification: 1.1,1.9
8Externalroutines/libraries:CERNROOT1,Doxygen2(optional) directdetectionmethods.Fortheformer,thehopeistodirectly
2Natureofproblem: Calculationofγ-raysignalfromdarkmatteran- witness the interaction of a dark matter particle with a detec-
7nihilation(resp. decay). Thisinvolvesaparticlephysicstermandan tor. Theindirectapproachesaimatmeasuringtheendproducts
4
astrophysicalone.Thefocushereisonthelatter. ofdarkmatterannihilation/decay(e.g.,γ-rays,positrons,anti-
.
1Solution method: Integrationof theDMdensity squared (resp. den- protons). Thedetectionofγ-rayswassoonrecognisedtobea
0sity)alongalineofsight. ThecodeisoptimisedtodealwiththeDM promisingchannel[1,2]. Ifsuchasignalisyettobemeasured
2densitypeaksencounteredalongthelineofsight(DMsubstructures).
bythe existingγ-raysobservatories(FERMI,HESS, MAGIC,
1Asemi-analyticalapproach(calibratedonN-bodysimulations)isused
: VERITAS), the prospect may significantly improve with the
vforthespatialandmassdistributionsofthedarkmattersubstructures
forthcomingnext-generationinstruments,suchasCTA[3].
iintheGalaxy.
XRestrictions: Some generic dark matter annihilation spectra arepro- Having a modelling tool that can compute the expected γ-
rvidedbutarenotincludedinthecalculationsofarasitisassumedthat ray flux from dark matter annihilation/decay in a wide range
a
theparticlephysicsisindependentoftheastrophysicsoftheproblem. of astrophysical configurations and making it available to the
Runningtime:thisishighlydependentoftheDMprofilesconsidered, community should prove very useful. This is the motivation
therequestedprecisionǫandintegrationangleαint: for developing the code presented in this paper. This paper
about60mnfora5 5 maptowardstheGalacticcentre,with highlightsclumpy’smainfeaturesbutamorethoroughdescrip-
◦ ◦
• ×
α =0.01 ,NFWdarkmatterprofilesandǫ=10 2; tioncanbefoundinthedocumentationcomingwiththesource
int ◦ −
code. The paper is organised as follows. Section 2 presents
about2hforthesameset-uptowardstheanti-centre;
• the formulation of the generic γ-ray signal calculation that is
0.1to10DMmodelspersecond,dependingonintegrationangle performed by clumpy, with special emphasis on the contribu-
•
andDMprofile.
tionfromDMclumps. Thecodeisdescribedinsect.3,which
presentsthe structure,parametersandfunctionsthatare atthe
coreofclumpy. Section4dealswithtwoexamplesofthesci-
Emailaddress:[email protected](Ce´lineCombet)
1http://root.cern.ch encethatcanbeperformedwithclumpy. Weconcludeanddis-
2http://www.doxygen.org cussthefuturedevelopmentsofthecodeinsect.5.
PreprintsubmittedtoComputerPhysicsCommunications January24,2012
2. Calculating the γ-ray flux from dark matter annihila- squaredalongthelineofsight(l,ψ,θ),intheobservationalcone
tion/decay ofsolidangle∆Ω(seeAppendix Aandgeometry.hfordefi-
nitionsandthegeometryused)
The γ-ray flux dΦ /dE from dark matter annihilat-
γ γ
ing/decaying particles is expressed as the product of a parti- ∆Ω
J(ψ,θ,∆Ω)= ρ2(l(ψ,θ)) dldΩ , (3)
cle physicstermbyanastrophysicalcontribution. Foragiven Z Z
0 l.o.s
experimentwithaspatialresolutionα correspondingtoanin-
int whereρ(l(ψ,θ))isthelocalDMdensityatdistancelfromEarth
tegrationsolidangle∆Ω = 2π(1 cos α ), atenergyE and
− int γ inthedirectionψ,θ (longitudeandlatitudeinGalacticcoordi-
pointinginthedirection(ψ,θ),thefluxiswrittenas:
nates).
dΦ dΦPP Thisintegralisthecoreoftheclumpycodeandcanberather
γ γ
(Eγ,ψ,θ,∆Ω)= (Eγ) J(ψ,θ,∆Ω). (1) complicatedtoevaluate.Thedifficultycomespartiallyfromthe
dE dE ×
γ γ
steepnessofsomeDMprofilesconsideredbutalso,andmainly,
2.1. Annihilationanddecay:generalstatements from the existence of DM clumps, the contribution of which
must be added to the smooth Galactic DM background. Both
2.1.1. Annihilation:theparticlephysicsterm
componentsaredealtwithhereafter.
Itcanbegenericallyexpressedas
ddΦEγPP(Eγ)≡ 41π hσ2amnn2vi · ddNEγf Bf, (2) 2.1γ.3-r.ayDseccaanyianlsgodbaerkprmodauttceerdfromthedecayofDM.Formally,
γ χ Xf γ thefluxisstillgivenbyEq.(1)buttheastrophysicalandparticle
physicstermstakedifferentforms:
wherem is themassoftheDM particle, σ istheannihila-
χ ann
tion cross section, σ v the annihilation rate averaged over
h ann i The particle physics term now corresponds to the decay
theDMvelocitydistribution,B isthebranchingratiointothe •
f spectrum and no parametrisation of that term has been
finalstate f anddNγf/dEγ isthephotonyieldperannihilation. included in the code so far. The parametrisation from
Awidelyacceptedvaluefor σ v estimatedfrompresentday
ann Bertoneetal. [7]couldbeused.
h i
thermal relic density is σ v = 3 10 26 cm3s 1 [4]3. The
ann − −
h i ·
photonspectrumisdependentontheannihilationchannels.We Becausewearenowconsideringadecayreaction,theas-
•
restrictourselvestothepopularMinimalSupersymmetricStan- trophysics term is the integration of the density (rather
dard Model (MSSM). In this framework, the neutralino is the thatdensitysquaredasinEq.(3))alongthelineofsight,
lighteststablesupersymmetricparticleandisthefavouredDM namely
candidate.
Neutralinoannihilationcanproduceγ-raysinthreeways: i) ∆Ω
J(ψ,θ,∆Ω)= ρ(l(ψ,θ)) dldΩ . (4)
DM annihilates directly into two photons or Z0γ, giving rise Z Z
0 l.o.s
to monochromatic lines, ii) or it can annihilate into primary
products,thehadronizationanddecayofwhichproduceaγ-ray This quantity can be straightforwardly com-
continuum,iii)finally,ifchargedparticlesareproduced,inter- puted in CLUMPY by setting the flag parameter
nal bremsstrahlung(IB) can also contribute to the continuum. gDM IS ANNIHIL OR DECAYtofalse(seetable3).
The lines are very model-dependentand given the uncertain-
ties on the particle physicsmodels, we did not include any in 2.2. Darkmatterdistribution
clumpy. Theuserishoweverfreetoimplementtheirownfunc- Todate,thereisnoconsensustowhattheGalacticDMpro-
tionifneedbe. Thereexistsintheliteratureseveralparametri- file, ρ(r), should be. There is some dynamical evidence that
sationsfortheγ-raycontinuumandwehaveimplementedthat theGalacticDMhalomightbetriaxial[8]andnumericalsim-
of Bergstro¨m et al. [4], and Tasitsiomi and Olinto [5]. The ulations, such as the Aquarius[9] or the Via Lactea runs [10]
IBcontributionsisalsoverymodeldependent,andweprovide alsofindnon-sphericalhalos. Whenincluded,thebaryonstend
onebenchmarkmodelfromBringmannetal. [6]. Wereferthe howevertomakethehalosmorespherical[11]. Inthisfirstver-
readerto 3.3.7(spectra.h)formoredetailsonthesespectra. sion of clumpy we simplifythe problem(asoften done)using
§
sphericallysymmetricDMhalos,butallowingfortri-axialityis
2.1.2. Annihilation:theastrophysicalcontribution being considered for future releases. The total averaged den-
Theγ-raysbeingproducedfromtheannihilationofpairsof sityprofilesoftheDMhalosmeasuredinthesesimulationsare
darkmatter particles, the γ-rayflux is proportionalto the DM generallyfittedbyaformoftheZhaoprofile[12]:
densitysquared.ThesecondtermofEq.(1),termedastrophys-
ρ
ical contribution,correspondsto the integrationof the density ρ (r)= s , (5)
tot (r/r )γ [1+(r/r )α](β γ)/α
s s −
whereγ,αandβarerespectivelytheinner,transitionandouter
3Notethatavelocity dependent crosssectionshouldbeincluded inafu- slope of the profile, ρ and r a scale density and radius. The
turerelease ofthecodeinordertoaccount fortheSommerfeldeffectinthe s s
Navarro et al. [13], Moore et al. [14], and Diemand et al.
calculation. Thiseffectresultsinmuchlargercrosssectionsforsomeresonant
neutralinomassesandthereforeboostthesignalfortheseDMcandidates. [15] profiles fall in this category. Some other profiles having
2
alog-varyinginnerslope[16,17]arealsoincludedinclumpy. butions are assumed independent4. If N is the total number
tot
The reader is referred to 3.3.4 (profiles.h) for a detailed of clumps in a DM host halo, then the overall distribution of
§
description. clumpsiswrittenas:
SomeoftheseDMprofilesaresteepenoughintheinnerre-
d2N d (r) d (M)
gionstoleadtoasingularityoftheγ-rayluminosityatthecen- = N PV PM , (8)
tot
treofthehalo. However,acut-offradiusr naturallyappears, dVdM dV dM
cut
within which the DM density saturates due to the balance be- where the spatial and mass distribution are probabilities, re-
tween the annihilation rate and the gravitational infalling rate spectivelynormalisedas:
ofDMparticles. Thesaturationdensityreads[18]
dPV(r)dV =1 and Mmax dPM(M)dM =1.(9)
ρsat =3.1018(cid:18)100mGχeV(cid:19)× 10−26σcmv3s−1!M⊙ kpc−3. (6) ZV dV ZMmin dM
h i AnalysisofN-bodysimulationshaveshownthemassdistribu-
Plugged in any profile parametrisation, this saturation density tiontovaryasasimplepowerlaw
defines the cut-off radius below which the annihilation rate
d (M)
is constant. This is a very crude description, but this is not PM M−αM withαM 1.9. (10)
dM ∝ ≈
important as this cut-off matters only for the steepest profiles
(γ 1.5)whichare disfavouredby currentnumericalsimula- Given the mass and spatial distribution, the total number of
tion≥s. clumpsNtotcanbedeterminedintwoways.Ifthemassfraction
The hierarchicalscenario of galaxy formationin a Λ-CDM f ofclumpsandthemassofthehostMtot haloareknown,
cosmologyis characterisedby a highdegreeof clumpinessof
fM
theDMdistribution(supportedbyN-bodysimulations)sothat Ntot = tot ,
M
thetotalaverageddensitycomputedfromthesesimulationscor- h 1cli
respondsto with M1cl theaveragemassofoneclump. Anotherapproach
h i
is to know the number of clumps N in a given mass interval
0
ρtot(r)=ρsm(r)+hρsubs(r)i (7) [M1,M2]andtocalculatethetotalnumberofclumpsas
whereρsm(r)isthe“true”smoothcomponentand ρsub(r) rep- N = N0 .
creosmenptosntehnetsavareeradgiescduesnsesidtybefrloowm.the substructuresh. Thesei two tot RMM12 dPdMM(M)
Clumpyusesthefirstapproachwhendealingwithsubstructures
in individual host halos (that are not the Milky Way) and the
2.2.1. Theclumps
secondonefortheGalactichaloclumps(numericalsimulations
Whethersubstructuresinsubhalosarescaled-downversions
have found about 100 clumps with masses above 108 M in
ofsubstructureinmainhalosremainsanopenquestion[9]. In MilkyWay-likehalos). clumpyallowstoselectanyofthep⊙ro-
thisreleaseofclumpy, weonlyconsideronelevelofsubstruc-
filesgivenin 3.3.4(profiles.h)fortheclumpspatialdistri-
turewithinthehalounderscrutiny(Galactichaloorindividual §
bution, independentlyof the choice made for the total density
halo such as a dwarf spheroidalgalaxy),and the propertiesof
profile.
these sub-halos can be independentlychosen from that of the
parenthalo(seebelow).
2.2.2. Thesmoothcomponent
From the spatial distribution d (r)/dV one can define the
Clump individualproperties. Inthe code,allclumpshavethe PV
averageclumpdensity ρ usedinEq.(7)as:
same inner DM distribution, which can be any of the profiles h cli
discussedin 3.3.4.Themassoftheclumpgenerallysufficesto d (r)
§ ρ (r) = fM PV . (11)
determineallitsproperties,i.e.,itssizeRcl ,andonceaninner h subs i tot dV
vir
density profile is chosen, its scale radius r and scale density
s Foragiventotaldensityprofileandclumpspatialdistribution,
ρ . This is done using the so-called concentration parameter
s Eq.(7)thengivesthesmoothdensityprofileas
c . Parametrisations of the mass-concentration relation have
vir
been established from numericalsimulationsfor isolated field ρ (r)=ρ (r) ρ (r) . (12)
sm tot subs
halos[19,20]buttheconcentrationgenerallypresentsasignifi- −h i
cantscatter[21]. Theserelationshavebeenimplementedinthe Using this approach, rather than providing independently a
code, regardlessof that scatter or of the environmentand for- smoothandaclumpprofile,ensures(byconstruction)thatthe
mationhistoryofthehalosthatcanalsoaffectthevalueofthe totaldensityprofilesfollowstheonemeasuredinN-bodysim-
concentration. We refer the reader to 3.3.6 (clumps.h) and ulations.ThishasbeendiscussedinPierietal. [23].
§
theDoxygendocumentationformoredetails.
4KeepinmindthatsuchanassumptionfailsneartheGalacticcentre,where
Spatialandmassdistributionoftheclumps. FollowingLavalle
smallclumpsareexpectedtobedisruptedbytidalforces.Inthisdirection,the
et al. [22], sphericalsymmetryis assumedand the two distri- signalfromthesmoothcontributiondominatesanyway.
3
2.3. CalculatingtheannihilationJ-factor where (M)istheintrinsicluminosityofaclumpgivenby
L
Theastrophysicalcontributiontotheannihilationfluxisfor- (M) (ρ )2dV . (19)
cl
mallygivenbyEq.(3). TheJ-factorismoreexplicitlywritten L ≡Z
Vcl
as:
Theaverageofthedistancetoanypowerniswrittenas
J = ∆Ω lmax 1 ρ + ρi 2l2dldΩ. (13) ln = ∆Ω lmaxlndPVl2dldΩ. (20)
Z0 Zlmin l2  sm Xi cl CombhiniingZth0ese,ZiltmicnanbedVshownthattheaveragecontribution
whereρ isgivenbyEq.(12)andthesecondtermcorresponds
sm fromtheclumpsisexpressedas
to the sum of the inner densities squared of all clumps i con-
tainedwiththevolumeelement. Thelattershouldnotbecon- J = N ∆Ω lmax dPVdldΩ Mmax (M)dPMdM ,(21)
subs tot
fusedwith ρ (r) . Threetermsarisefromthisequation: h i Z Z dV Z L dM
h subs i 0 lmin Mmin
Inthesamespirit,inthecontinuumlimitthecross-productbe-
∆Ω lmax
J ρ2 dldΩ, (14) comes
sm ≡Z Z sm
0 lmin ∆Ω lmax
J =2 ρ ρ dldΩ. (22)
cross prod sm subs
∆Ω lmax 2 h − i Z0 Zlmin h i
Jsubs ≡Z Z  ρicl dldΩ, (15) The importance of the cross product can easily be identified
0 lmin Xi  wtohtaelnacvoenrasgideerdienngstihtyatptrhoeficlelu.mInptshpaatticaalsdei,strρibuti(orn) fo=llofwρst(hre)
andthecrossproduct h subs i tot
andρ (r) = (1 f)ρ (r),leadingdirectlyto J /J =
sm tot cross prod tot
∆Ω lmax 2f(1 f). Fo−r a typical clump mass fraction f− = 10%,
Jcross−prod ≡2Z0 Zlmin ρsmXi ρicldldΩ. (16) tJhis c−rosρs2pdrloddΩuc.tTahmisosuhnotwsstoth1a8t%wheonf athneavreerfaegreendcceluJm-fapcdtoer-
tot ≡ tot
scriptiRonis usedto integratethesignalalongthe lineof sight,
2.4. Statisticalconsiderations
this cross product is not negligible and should be taken into
Taking α = 1.9 as a canonical value for the slope of account. In Appendix B, it is however shown that on an in-
M
the mass distribution, with 100 clumps above 108 M as de- dividual basis, the cross product of one clump in an underly-
termined from N-body simulations, and a minimal the⊙oretical ing smoothGalactic componentcan besafely discarded. This
mass of 10 6 M for the smallest ones, the total number of willapplytotheclumpsthatneedtobestatisticaldrawninour
−
clumpsintheMi⊙lkyWayhalo(i.e.,for∆Ω = 4π)isestimated model(seebelow).
tobe 1014,whichiscomputationallyprohibitive. Evenifwe
∼ 2.4.2. Varianceoftheclumpcontributionσ2
are interestedin a5◦ 5◦ skymaponly(i.e., ∆Ω 6 10−3), clumps
thenumberofclumps×is 5 1010. Thisistoolar∼gea×number Alongagivenlineofsightandforagivenintegrationangle
∼ × α , the average flux between l and l given by Eq. (21)
torealiseastatisticaldrawingaccordingtothed /dV distri- int min max
V
P canalsobeexpressedas
butionandthennumericallyintegratetheJcomponentforeach
oftheseclumps. However,theirhighnumbermeansthatasta- J = N J , (23)
subs cl 1cl
tisticalaveragecanbesafelyperformed,aslongasthevariance h i h ih i
is small. The clumpy code computes the γ-ray flux from DM where hNcli is the average number of clumps in this volume
(given the distance interval and mass range considered), and
clumpsbyeitherrandomlydrawingclumpsgivenuser-defined
J theaveragefluxofoneclump(seealso[24]). Usingthe
spatialandmassdistributionsandinnerDMprofiles,orbycal- h 1cli
point-likeapproximation,aclumpofmassMatadistancelhas
culating the mean contribution whenever the variance of this
aflux
contributionissmallerthansomeuser-definedvalue.
(M)
J = L , (24)
1cl l2
2.4.1. Averagedquantitiesfromthecontinuumlimit
sothat,recallingthatthemassoftheclumpanditslocationare
Giventheclumpmassandspatialdistributionsgivenabove,
independentvariables,
it is possible to define average values for the primary random
variables of the problem (a clump mass and position) and the 1
J = . (25)
one deriving from them. The mean mass and the mean lumi- h 1cli hLi*l2+
nosityarerespectivelydefinedby
Thevarianceonthefluxofasingleclumpisthendefinedas
M = Mmax MdPMdM , (17) σ2 = J2 J 2 = 2 1 J 2 . (26)
h i Z dM 1cl h 1cli−h 1cli hL i*l4+−h 1cli
Mmin
Thevarianceofapopulationof N clumpsfollowsas
cl
= Mmax (M)dPMdM . (18) σ2 = N σ2 . h i (27)
hLi ZMmin L dM cl h cli 1cl
4
103
102 aTnatbil-ece1n:trEex(acmriptelreioonflRcrEitJacnludm<ptsh5e%as)sfoocriaate2◦n×2u◦msbkeyrmoafpclwumithpsantoindtreagwrattoiownaradngthlee
101 (instrument)ofαint=0.1◦.Actually,agivenlcritgives<ncl>numberofclumps:
100 nMclatossdrdaewcaisdeobta#inceldumfropmsaPolisso(nkdpicst)ributio<n#ocflmumeapnsv>alue<#nctlo>.draw
10-1 Ψ = 180o log Mcl (Galaxy) kcprict (5 5 ) (5 5 )
%) 10 M ◦× ◦ ◦× ◦
RE (Jcl 1100--23 Ψ = 90o −−456:::(cid:16)−−345⊙(cid:17) 543...645××111000111234 111...000××111000−−333 211...485××111000−−543 000
10-4 − − × × − × −
3: 2 7.0 1011 7.3 10 3 1.2 10 3 0
− −
1100--65 ααiinntt == 00..01o1 o Ψ = 0o −−−21::−−01 18..18×××11001100 45..08×××1100−−12 27..86××10−2 03
10-7 1 - 10 M 0:1 1.4×109 2.0 3.7×10+1 45
sun 1:2 1.7 108 6.7 1.0 10+2 117
10-08.01 1 100 2:3 2.2×107 1.6 101 8.7×10+1 99
l (kpc) × × ×
min 3:4 2.8 106 3.3 101 3.6 10+1 44
× × ×
4:5 3.5 105 5.8 101 1.0 10+1 7
Figure 1: Relative errorbetween theaverage andtrue J value from clumps 5:6 4.4×104 9.5×101 2.2× 1
whenthesignalisintegratedbetweenlminandlmax,accordingtoEq.(28),for 6:7 5.6×103 1.5×102 4.1 10 1 0
αMinctl =∈0[.10,11◦0](dMas⊙h.edTlhinee)s.igTnahlreiesdinirteecgtriaotnesdionvtehreαGinatla=cti0c.1p◦lan(seoalirdelcionne)sido-r 7:8 7.0××102 2.5××102 6.6××10−−2 0
ered: towards theGalactic centre (blue), towards theanti-centre (black) and 8:9 8.8 101 2.9 102 9.5 10−3 0
× × ×
towardstheGalacticEast(red). Forthelatterthegreendottedlineillustrates 9:10 1.1 101 2.9 102 1.3 10 3 0
−
× × ×
howtogetthecriticaldistance(here2kpc)belowwhichclumps(inthismass
range)mustbedrawniftheuserrequiresanaccuracyof5%. Alldistributions
(smooth, clumpdistribution andinnerDMclumpprofile)aretakenasNFW Threshold mass mthresh for the sub-clumps in an “far-away”
profiles. halo. The second case where the criterion is used is for the
sub-clumpsinahalolocatedawayfromus,e.g.,adSphgalaxy.
2.4.3. Criterionforusingtheaverageddescription Inthatcase, thedistanced totheclumpsisknownastheyare
For a given integration domain, there is a threshold mass alldistributedwithintheirparenthalo’sradiusandwecande-
above which the clumps are not numerous enough to be de- termine in a similar fashion the threshold mass above which
scribed by an averaged description. Conversely, for a given clumpsmustbedrawn.
mass decade there is a critical distance below which the aver-
ageddescriptionfailsandwhereclumpsshouldbestatistically
3. Descriptionofthecode
drawn.
The averageddescriptioncan be safely used, as long as the Clumpy is written in C/C++ and relies of the CERN ROOT
relative error RE made on the clump flux with respect to the librarythatispubliclyavailablefromhttp://root.cern.ch.
totalflux(i.e. includingthesmoothcontribution) Themainfeaturesofthecodearedescribedhereafteralongwith
brief descriptionsof its most importantfunctions. The code’s
√N σ
REJclumps = N J cl+J1cl (28) structure is standard, with separate directories for the various
clh 1cli smooth pieces of code: declarations are in include/*.h, sources in
issmallerthansomeuser-definedprescription.
src/*.cc,compiled libraries, objects and executablesare re-
spectivelyinthelib/,obj/,andbin/directories.
Critical distance l for the clumpsin the Galaxy. TheRE is
crit
plottedinFig.1asafunctionofthelowerlimitoftheintegra-
3.1. Codeexecutables
tionl . Ascanbeseen,foranyrequiredprecision(e.g.,5%in
min
The code has one executable that can take three main op-
Fig.1,taggedbythedottedgreenline),onecaninferthecorre-
tionsdependingwhethertheuseri)isinterestedinperforming
spondingdistance l below which the relative errorbecomes
crit
agenericanalysisontheGalactichalo,ii)isfocusingonalist
largerthantherequest.
ofspecifichalos(e.g. dSphgalaxies),oriii)wishestoperform
In practise, it is convenient to perform the calculation in
astatisticalanalysisonasingleobject.Thesemainthreeusages
termsofmassdecades:foreachdecade,wefind(bydichotomy)
the critical distance below which clumps must be drawn5. A ofthecodeallaccommodateseveralcases,thedetailsofwhich
lower limit of 10 3 kpc is set on l to avoid divergencesfor aregivenbelow.
− crit
a clumpsitting at the observer’slocation. An exampleforthe
3.1.1. Galacticmode:./bin/clumpy -g[option]
finalnumberofclumpstodrawbelowl isprovidedinTab.1.
crit
This calculates the J-factor for the Galactic halo, including
TheflowchartgiveninFig.2summarisesthevariousstepsim-
plementedinclumpytoobtaintheskymap(see,e.g.,Fig.3). boththesmoothandtheclumpcomponent.Fortheclumpcom-
ponent,themeanfluxorastochasticrealisationcanbechosen.
Note that a list of specific halos can be added to the calcula-
5ThismassdecadecorrespondstoatypicalclumpradiusRcl andwealso tion. Several options allow to select the quantity to be calcu-
checkthatthepoint-likehypothesisusedtoderivedthecontinuumlimitisap-
propriate(lcrit≫Rcl). lated/plotted from a text interface. A mandatory input is the
5
READ IN USER−DEFINED PARAMETERS
CLUMPS DEFINE FIELD OF VIEW SMOOTH
COMPUTE L_CRIT(M) GIVEN CLUMP DISTRIB
For a given mass decade, need to
determine the distance l_crit below
which the analytical clump estimation
fails and clumps need to be drawn
from dP/dV
COMPUTE NUMBER OF CLUMPS TO COMPUTE ANALYTICAL CONTRIBUTION COMPUTE SMOOTH CONTRIBUTION
DRAW IN THE FIELD OF VIEW OF THE CLUMP "CONTINUUM"
e FROM 0 to L_CRIT(M)
d
a
c
e
d DRAW CLUMPS
s
s
a
m ADD ALL CONTRIBUTIONS PIXEL BY PIXEL
h IDENTIFY CLUMPS BELONGING
c TO THE F.O.V
a
e
r
o
F
COMPUTE CLUMP FLUX AND
STORE IN MAP PIXELS
STORE ALL COMPONENTS IN FILES
AND/OR
DISPLAY WITH ROOT
Figure2:Flowchartofthedifferentstepsleadingtothecreationofaγ-rayskymap.
Table2:FormatoftheASCIIfiletodescribehalosprocessedby./bin/clumpy dsph -h[option].’Type’shouldtakeoneofthevaluesofgENUM TYPEHALOES,
namelyDSPH,GALAXYorCLUSTER.’long’,’lat’and’d’arethegalacticlongitude,latitude,anddistancetothehalo,zitsredshiftandRviritsvirialradius.Theprofile
parametersareρs,rs,’prof.’ (seeprofile.h)andtheshapeparameters. Inparticular,if’prof’=kZHAO,see 3.2),theprofilecorrespondstothe(α,β,γ)profile
§
(inwhichcasethenext-to-lastthreecolumnsareused).IfankEINASTOisusedinstead,onlythefirstshapeparameterisconsidered.
Name Type long. lat. d z R ρ r prof. α β γ
vir s s
- - deg deg kpc - kpc M kpc 3 kpc - - - -
−
⊙
parameter file clumpy params.txtdescribed in 3.2 and ta- ’option = -g4’ is the same as ’-g3’ but including a
§ •
ble3. Theoptionsarethefollowing: listofspecificobjects,suchasdSphgalaxies.
’option = -g1’ plots the total, smooth and averaged ’option = -g5’plotsa 2D skymapof the J-factor, us-
• •
clump density in 1D, given the user-definedtotal density ingtheaveragedclumpdescription.where
andthenumberofclumpsinagivenmassrange.
’option = -g6’issimilar ’-g5’,butaddsspecificha-
•
’option = -g2’ plots J + J + J 6, re- lostotheplots(e.g.dSph),toidentifytheircontrasttothe
sm subs cross prod
• spectivelydefinedbyEq.(14),Ehq.(2i1)ahndEq−.(22i)asa background.
functionoftheintegrationangleα .
int ’option = -g7’ and -g8 are respectively the same as
•
’-g5’ and ’-g6’ but with the clumps randomly drawn
’option = -g3’isthesameas’-g2’butasafunction
• fromtheiruser-definedmassandspatialdistributions.
of the angulardistanceto the Galacticcentre, fora given
integrationangle.
3.1.2. Halolistmode: ./bin/clumpy -h[option]
Thisusageofclumpyperformsoperationsthatarebasically
6DoesnotexistfordecayingDM. the identical to those performed in the ’Galactic’ mode (see
6
3.1.1), but for a list of halos that are not the Galactic halo. enum gENUM GAMMASPECT kSUSY BUB98,
§ • {
This mode was initially implemented for the study of dwarf kSUSY TO02, kSUSY BBE08 fortheγ-rayspectrum;
}
spheroidalgalaxies[25], butcan be extendedto anyDM halo
enum gENUM PROFILE kZHAO, kEINASTO,
(e.g., galaxyclusters). If onlythe smoothtotaldensity profile • {
kEINASTO N for the DM spatial distributions (smooth
is considered, the halo profiles are read from a simple ASCII }
and clumpy components). The corresponding shape
list of profiles along with their parameters, as exemplified in
Tab.2. Ifsub-clumpsareconsidered,theclumpyparameterfile parameters(e.g. (α,β,γ)forkZHAO,αforkEINASTOand
n forkEINASTO N) shouldbeprovidedbythe user in the
clumpy params.txtmustalsobeprovided(seetable3).
inputparameterfile(seebelow).
Again, furtheroptionsareaccessed forthismodebymeans
ofasimpletext-userinterface:
Thisenumerationsallowtheusertousetheirexplicitdenomina-
tionkXXX,ratherthanthecorrespondingintegernumber.These
’-h1’to’-h3’areidenticalto’-g1’to’-g3’buttrans-
• parametersaregatheredintheASCIIfileclumpy params.txt
posedtoanyhalothatisnottheGalacticDMhalo;
which is formatted as “Parameter name”, “Units”, “Value”.
’-h4’ to ’-h5’ create 2D skymaps of the halo under The function load parameters(file name) defined in
•
scrutiny using respectively an averaged description or a params.h reads in the input file and assigns the appropriate
statisticalrealisationforthesub-clumpswithinthishalo; valuesto the parameters. The user canchangeany“Value”in
thisfile7.
3.1.3. ’Statistical’mode:./bin/clumpy -s[option]
Confidence levels on the astrophysical factor J, for e.g. 3.3. clumpylibraries
dSphs,canalsobeestimatedfromanASCIIstatisticalfile.The Thefunctionsofclumpyarelocatedineleven“library”files,
formatforthelatterisquitesimilartothehalolistfile(seedoc- accordingly to their field of action. We briefly describe here-
umentation) as it requiresthe profile parameters, but it differs after the most important functions of these libraries. We are
in the sense that for a single halo, thousands of profiles ’se- omitting in this document the more minor functions, the de-
lected’byastatisticalanalysismustbeprovided(theirrespec- scriptionofwhichcanbefoundinthedocumentationprovided
tive χ2 value isnecessary, althoughnotused forall CL calcu- withthecode. Also,wedonotdetailherewhatargumentsare
lations). Such a statistical analysis can be the Markov Chain passedinthesefunctionsasitcanbeeasily retrievedfromthe
MonteCarlo(MCMC),asdiscussedinCharbonnieretal. [25]; documentation.
Walkeretal. [26]. Themostimportantoptionsinthestatistical
modeare: 3.3.1. integr.h
This library contains standard integration algorithms. The
’-s2’ plots the probability density functions (PDF) and
• Simpsonintegratorwithdoublingstepisthemostwidelyused
correlationsoftheparametersgiveninthestatisticalfile;
throughoutthecodetoensurethattheuser-definedprecisionis
’-s4’locatesthe x%confidenceintervalsonthePDFof reached. Depending on the function to integrate, either linear
•
eachparameters; orlogarithmicsteppingcanbeused.
’-s5’computes,foreachradiusfromthedSphcentre,the
• 3.3.2. geometry.h
medianandconfidencelevelsoftheDMdensityρ(r);
In geometry.hare gatheredfunctionsperforminggeomet-
’-s6’isthesameas’-s5’butfor J(α ),i.e. asafunc- ricaltasks(changeofcoordinates,etc.). Theyarenotessential
int
• tionoftheintegrationangle; forthepresentationofthecodeandgenerallyself-explanatory.
We refer the reader to the extensive Doxygen documentation
’-s7’issameas’-s5’butfor J(θ),i.e. asafunctionof andtoAppendix Aformoredetails.
•
theangletothecentreofthehalo.
3.3.3. inlines.handmisc.h
3.2. User-definedparameters
These contain some definitions (unit conversion and inline
Formostoftheoptionsusedabove,auser-defineparameter functions)andmiscellaneousformattingfunctionsrespectively.
fileisread. Table3givesthelistofthemainparameters. Each
parameteris a single word formattedas g (forglobalparame- 3.3.4. profiles.h
ter)+physicskeyword+parametername.Forinstance,COSMO Thislibrarycontainsallthe densityprofilesavailable in the
refersto cosmology-relatedparameters, TYPEdefinesthe type codeandoperationsonthose.Amongthefunctionsinthe“pro-
ofhalounderscrutiny(dSPhgalaxy,galaxyorgalaxyclusters) files” section of the code, some act in a very straightforward
andGALreferstotheMilkyWayhaloparameters.Notealsothat way (e.g. squaring the density) and are not included in the
the parameters ending in FLAG CVIRMVIR, FLAG SPECTRUM, present document. The reader is, once again, referred to the
andFLAG PROFILEshouldtakevaluesfromthefollowingenu- Doxygendocumentationforafulldescription.
merations:
enum gENUM CVIRMVIR kB01, kENS01, kJS00 for 7Some of these parameters are optional and used only for specific
• { }
theclumpconcentration; modes[options]
7
Table3:clumpyuser-definedinputparameterfile(clumpy params.txt).
Name Definition
gCOSMO RHO0 C Criticaldensityoftheuniverse[M kpc 3]
−
gCOSMO OMEGA0 M Present-daypressure-lessmatterde⊙nsity
gCOSMO OMEGA0 LAMBDA Present-daydarkenergydensity
gDM GAMMARAY FLAG SPECTRUM γ-rayspectrumflag[gENUM GAMMASPECT]
gDM MMIN SUBS Min.massofaDM(sub-)clump[M ]
gDM MMAXFACTOR SUBS Definesmax.massofclumpinhost⊙haloasM =Factor M
max × host
gDM RHOSAT Saturationdensity[M kpc 3]
−
⊙
gTYPE CLUMPS FLAG CVIRMVIR ClumpconcentrationflagintheparentTYPEhalo[gENUM CVIRMVIR]
†
gTYPE CLUMPS FLAG PROFILE Clumpinnerprofileflag[gENUM PROFILE]
gTYPE CLUMPS SHAPE PARAMS[0-2] Shapeparametersforsub-clumpsinnerprofile
gTYPE DPDM SLOPE Slopesubclumpmassfunction
gTYPE DPDV FLAG PROFILE Spatialdistributionsubclumps[gENUM PROFILE]
gTYPE DPDV RSCALE Scaleradiusforthesubclumpdistributionintheparenthalo[kpc]
gTYPE DPDV SHAPE PARAMS[0-2] Shapeparametersfortheclumpspatialdistribution
gTYPE SUBS MASSFRACTION Massfractionoftheparenthaloinsub-clumps
gGAL CLUMPS FLAG CVIRMVIR Concentrationflag[gENUM CVIRMVIR]
gGAL CLUMPS FLAG PROFILE Clumpinnerprofileflag[gENUM PROFILE]
gGAL CLUMPS SHAPE PARAMS[0-2] ShapeparameterfortheGalacticclumpsinnerprofile
gGAL DPDM SLOPE Slopetheclumpmassfunction
gGAL DPDV FLAG PROFILE Clumpnumberdistributionflag
gGAL DPDV RS Scaleradiusforclumps[kpc]
gGAL DPDV SHAPE PARAMS[0-2] ThreeshapeparametersfortheGalacticclumpspatialdistribution
gGAL SUBS M1 ReferencemassforgGAL SUBS N INM1M2[M ]
gGAL SUBS M2 ReferencemassforgGAL SUBS N INM1M2[M⊙]
gGAL SUBS N INM1M2 #ofclumpsin[M ,M ] ⊙
1 2
gGAL RHOSOL LocalDMdensity[GeVcm 3]
−
gGAL RSOL DistanceSun–Galacticcentre[kpc]
gGAL RVIR VirialradiusoftheGalaxy[kpc]
gGAL TOT FLAG PROFILE TotalDMprofileoftheMilkyWay[gENUM PROFILE]
gGAL TOT RSCALE ScaleradiusforDMhalooftheMilkyWay[kpc]
gGAL TOT SHAPE PARAMS[0-2] ShapeparameterfortheGalactictotaldensityprofile
gSIMU ALPHAINT DEG Integrationangle[deg]
gSIMU EPS Defaultprecisionfornumericalintegrations
gSIMU IS ANNIHIL OR DECAY ForannihilatingordecayingDM
gSIMU SEED Seedofrandomnumbergenerator
†TYPEcorrespondstothetypeofhalounderscrutiny(nottheGalactichalo):DSPH,GALAXYorCLUSTERarethevalues
allowedbygENUM TYPEHALOESinthisversion.
8
rhosimplyreturnsthevalueofthedensitybycallingany
Table4:StandardDMdensityprofilesparameters,followingEq.(29). •
ofthefunctionsabove,dependingonthevalueofthepro-
α β γ Reference
fileflaggiveninclumpy params.txt.
ISO 2 2 0 -
NFW97 1 3 1 Navarroetal.[13]
Fortwodensityprofilesρ andρ ,rho mixreturnseither
M98 1.5 3 1.5 Mooreetal.[14] • 1 2
thedifferenceρ ρ ortheproductρ ρ . Thisisneeded
DMS04 1 3 1.2 Diemandetal.[15] 1− 2 1 2
whencomputingEq.(12)forthesmoothdensityprofileor
Eq.(22)forthecrossproduct.
rho ZHAOreturnstheDMdensityofauser-definedprofile
•
havingthegenericfunctionalformgiveninZhao[12] 3.3.5. integr los.h
This is the core of the clumpy code as it contains the rou-
ρ
ρ(r)ZHAO = s . (29) tinesperformingtheintegrationalongthelineofsightdirection
(r/rs)γ [1+(r/rs)α](β−γ)/α (θ,ψ)andoverthechosenangularresolutionαint.Givenafunc-
tiong(l,α,β;θ,ψ)where(l,α,β)aresphericalcoordinatesinthe
Theisothermal,Navarroetal. [13],Mooreetal. [14],and (θ,ψ) direction(see Fig. 6), the integrationalongthe observa-
Diemandetal. [15]profilesareallfromtheZhaofamily.
tioncone,froml tol isgenericallywrittenas:
1 2
TheyaredefinedusingEq.(29)withthesetofparameters
oftable4. Theuserisrequiredtospecifytheseparameters 2π αint l2
I(θ,ψ)= dβ sinαdα g(l,α,β;θ,ψ)l2dl.(32)
intheASCIIinputfile. Z Z Z
0 0 l1
rho EINASTOandrho EINASTO N:ithasbeenarguedthat integrand l: returns the value of the function
• •
aprofilewithalogarithmicinnerslopegivesabetterfitto f(l,α,β;θ,ψ) g(l,α,β;θ,ψ)l2. A switch is includedto
≡
the simulated data. This so-called Einasto is generically integratedifferentfunctionsofthedensityprofileρ. Note
writtenas thathereρcanrefertoanyprofileordistribution,beitthe
density or the spatial distribution of clumps dP /dV de-
V
ρ(r) exp( Arα) . pendingonthenormalisationused. Thefunctionsthatcan
∝ −
beintegratedalongthelineofsightare:
In Navarroet al. [16] and Springelet al. [9] the Einasto
– f(l,α,β;θ,ψ)=ρwhencalculatingtheclumpsaver-
profileisdefinedas
agedcontribution J ofEq.(21).
subs
h i
– f(l,α,β;θ,ψ) = ρ2 in orderto evaluateJ such as in
α
2 r
ρEINASTO(r)=ρ exp 1 , (30) Eq.(14).
−2 −α" r−2! − #! – f(l,α,β;θ,ψ) = l2ρ is used to calculate the mass
fractionornumberfractionofclumpsintheintegra-
where r is the radius at which the profile’s slope
2 tionvolume;
dlnρ/dl−nr = 2, and ρ the corresponding density.
2
Both Navarro et−al. [16]−and Springel et al. [9] find – f(l,α,β;θ,ψ) = l3ρ is needed when estimating the
α 0.17tobeagoodfittothesmoothcomponentofsim- meandistanceoftheclumpsasinEq.(20).
∼
ulatedhalos. Springeletal. [9]furthernotesthatα 0.68 – f(l,α,β;θ,ψ) = l4ρ is needed when estimating the
∼
allowsagooddescriptionofthespatialdistributionofsub- varianceofdistanceoftheclumps.
structuresdP/dV.
– f(l,α,β;θ,ψ) = l 2ρ must be evaluated to evaluate
−
Merrittetal. [17]useanEinastor1/n profile,slightlydif- thevarianceonJasinEq.(26).
ferentfromtheaboveandgivenby
Note that if decaying rather than annihilating DM is be-
r 1/n ingconsidered,i.e. gDM IS ANNIHIL OR DECAYisfalse
ρEINASTO N(r)=ρ exp d 1 . (31) insteadoftrue,anyρ2isreplacedbyρintheabovefunc-
e − n· re! −  tions8.
A variation of this function, called integrand lrel, is
where n = 6 and d = 3n 1/3+0.0079/n. The scale
n − usedwheneverthesteepnessoftheprofilerequiresit(see
radius r in this relation correspond to the radius within
e
Doxygendocumentationformoredetails).
whichhalfofthehalomassisenclosed.
fn beta alpha:computesthethirdintegrandinEq.(32),
The normalisation density of all these profiles (ρ , ρ •
• or ρ ) can be done by i) either requesting a givensvalu−e2 namely the integration over l. The functions to inte-
e grateroughlybehaveaspowerlaws,withavaryingslope,
for the density at a given point or ii) by requesting
a given mass within a given radius. The functions
set par0 given rhoref and set par0 given mref 8Thismakessomeofthesefunctionsredundantfordecay,buttheyarestill
achievethenormalisationusingbothapproaches. requiredforannihilation.
9
and log-step Simpson integration schemes are used (see frac nsubs in m1m2returnsthefractionofclumpswith
•
integr.h). Some profiles can be very steep and several a mass in the range [m ,m ], given the total number of
1 2
testsandtricksareneededtoensurethenumericalstabil- clumps.
ity/optimisationoftheintegration. Thereaderisagainre-
ferred to the documentationfor an explicitdescriptionof jcrossprod continuum returns the value of the cross-
•
theintegrationstrategy. productintegrated along the l.o.s as defined by Eq. (22).
Thisisasimplecalltolos integral mix.
fn beta: computes the integration over the angle α in
•
Eq. (32). A simple linear Simpson integration scheme is jcrossprod interpolationinterpolatesthevalueof J
used. • in the pixelsof a skymap, providedthat J haspreviously
beenestimatedinsomelocationsofthemap. Thismakes
los integral:itisthefinalpartofthetripleintegration,
• use ofROOT interpolationfunctionbased on Delaunay’s
i.e. the integration over the angle β. This is again per-
triangles and allows to speed-up the calculation signifi-
formed quite easily with a linear Simpson scheme. This
cantlywhentheskymaphasa(too)largenumberofpixels.
functioniscalledwheneverasingledensityprofileistobe
dealtwith.
j smooth returns the J-factor when a single density
•
los integral mix: as seen previously, it may be nec- profile is used (e.g. ρtot). This is a simple call to
• essary to integrate a difference or a product of two los integral.
densities along the line of sight. A flag parameter
j smooth mix is the same as above but deals with a
switch rho is introduced to tell rho mix to return ei-
•
combination of densities, e.g. when computing Eq. (14)
ther ρ ρ (switch rho = 0) or ρ ρ (switch rho =
1 2 1 2
− where ρ = ρ ρ . This is a simple call to
1) (see profiles.h). Thisfunctionthenbehavesidenti- sm tot cl
− h i
los integral mix.
cally to los integral and uses the same flags with re-
specttothequantitytointegrate.
j sub continuum computes and returns the averaged J
•
generatedby the sub-clumpsin a halo, in a given line of
3.3.6. clumps.h
sight.
Allthefunctionsrelatedtotheclumpsmassandspatialdistri-
butionsaregatheredinthisfile. Here,weonlygiveanoverview
lum singlehaloreturnsthe totalintrinsicluminosityof
ofthemostimportantones. •
aDMhalo,givenitsinnerdensityprofileasinEq.(19).
add halo in map: when using the code to generate a
• skymap where clumps are randomly drawn from their mass singlehalo returns the total mass of a DM halo,
•
mass and spatial distribution, this addsthe J contribution givenitsinnerdensityprofile.
of the drawnclumpto theappropriatepixels. Theclump
isonlyaddedtoapixelifJcl(pixel)>ǫ×Jsm(pix0)where 3.3.7. spectra.h
pix0 is the pixel containing the centre of the clump and Thislibrarycontainsthedifferentparametrisationsoftheav-
ǫ is the requested integration precision gSIMU EPS. This
erageγ-rayspectra,
avoids the clump from being integrated “too far” which
would only be time consuming without changing the re- dN dNf
γ
B (33)
sult. dE ≡ dE f
γ Xf γ
dpdv XXXandset normprob dpdv: theformerfunction
• intheparticlephysicstermofEq.(2). Asmentionedin 2.1.1,
returnsthespatialdistributionoftheclumpsasdescribed
§
we have included the Bergstro¨m et al.[4] and Tasitsiomi &
in 2.2.1 and the second calculates the appropriate nor-
§ Olinto [5] parametrisations for the γ-ray continuum and the
malisation to make it a probability according to Eq. (9).
Bringmannetal. [6]expressionoftheinternalbremsstrahlung.
The functiondpdv XXX is written in several versions de-
Defining x E /m , the three parametrisations are given by
pending of the reference frame and coordinate system in γ χ
≡
theircorrespondingfunctionsbelow.
whichtheclumpsaredrawn(cartesianXXX=xyz,spherical
centredonthe galacticcentreXXX=rthetapsi,spherical
dNdE BERGSTROMreturns
centredonEarthXXX=lthetapsi,orcentredonEarth,in •
thecone∆Ω,inthedirection(ψ,θ)XXX=lalphabeta).
dN 1 exp( 7.8x)
γ
=0.73 − .
dpdm and set normprob dpdm: this is the equivalentof dE · m · x1.5
γ χ
•
the above but for the mass distribution of the clumps as
givenbyEq.(10). dNdE TASITSIOMIcomputestheγ-rayspectrumas
•
frac nsubs in foi returns the number fraction of
• dN 1 10 5 5 5
clumpsinagivenfieldofintegrationw.r.t. thetotalnum- γ = x0.5 x 0.5+ x 1.5 .
− −
berofclumpsinthehosthalo dEγ mχ · 3 − 4 − 2 12 !
10