Extragalactic gamma-ray background radiation from dark matter annihilation

22 Pages·2010·1.28 MB·English

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed4February2010 (MNLATEXstylefilev2.2)
Extragalactic gamma-ray background radiation from dark
matter annihilation
Jesu´s Zavala1,2⋆, Volker Springel1 and Michael Boylan-Kolchin1
1Max-Planck-Institut fu¨r Astrophysik, Karl-Schwarzschild-Straße 1, 85740 Garching bei Mu¨nchen, Germany
0 2MPA/SHAO Joint Centerfor Astrophysical Cosmology at Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China
1
0
2 4February2010

b
e
F ABSTRACT
If dark matter is composed of neutralinos, one of the most exciting prospects for its
4
detectionliesinobservationsofthe gamma-rayradiationcreatedinpairannihilations

between neutralinos, a process that may contribute significantly to the extragalactic
]
O gamma-raybackground(EGB)radiation.Wehereusethehigh-resolutionMillennium-
C II simulation of cosmic structure formation to produce the first full-sky maps of the
expectedradiationcomingfromextragalacticdarkmatterstructures.Ourmapmaking
.
h procedure takes into account the total gamma-ray luminosity from all haloes and
p their subhaloes, and includes corrections for unresolved components of the emission
- aswellasanextrapolationtothedampingscalelimitofneutralinos.Ouranalysisalso
o
r includes a proper normalization of the signal according to a specific supersymmetric
t model based on minimal supergravity. The new simulated maps allow a study of the
s
a angularpowerspectrumofthegamma-raybackgroundfromdarkmatterannihilation,
[ which has distinctive features associated with the nature of the annihilation process

and may be detectable in forthcoming observations by the recently launched FERMI
3
satellite. Our results are in broad agreement with analytic models for the gamma-
v
8 ray background, but they also include higher-ordercorrelations not readily accessible
2 in analytic calculations and, in addition, provide detailed spectral information for
4 each pixel. In particular, we find that difference maps at different energies can reveal
2 cosmiclarge-scalestructureatlowandintermediateredshifts.Iftheintrinsicemission
. spectrumischaracterizedbyanemissionpeak,cosmologicaltomographywithgamma
8
0 ray annihilation radiation is in principle possible.
9
Key words: cosmology:dark matter – methods: numerical
0
:
v
i
X
r 1 INTRODUCTION generically predict theexistenceof a lightest supersymmet-
a ricparticle(LSP),whichappearsasaplausibledarkmatter
ThenatureofdarkmatterintheUniverseisoneofthemost
candidate. Among the possible LSPs resulting from SUSY
fascinating and important mysteries of astrophysics today.
theories, two are strongly motivated and havebeen studied
Based on theevidencegathered duringthelast decades,we
indetail:thegravitino,whichisthespin-3/2superpartnerof
know that a suitable particle physics candidate has to be
thegraviton,andthelightestneutralino,thespin-1/2Majo-
neutralandstable(orwithalifetimecomparabletotheage
rana fermion that appears in the Minimal Supersymmetric
oftheUniverse)tobethemajorcomponentofdarkmatter,
Standard Model (MSSM) as the lightest mass eigenstate of
and that its interactions with ordinary matter have to be
a mixture of the superpartners of the neutral gauge bosons
very weak in order to be consistent with constraints from (thefermionsB˜0 andW˜0)andtheneutralSUSYHiggs(H˜0
cosmic structureformation. and H˜0). Although gravitinos are interesting viable candiu-
Amongthedifferenttheoriesthatproposeasolutionto d
datestobethemajorcomponentofdarkmatter(seeforin-
the dark matter problem, supersymmetric (SUSY) exten-
stance Feng (2005); Steffen (2006) for recent reviews), they
sions of the standard particle physics model with R-parity
can be classified as extremely weakly interacting particles
conservation are particularly attractive (see for example
(Steffen 2007) and are therefore very hard to detect exper-
Martin1998,forapedagogicalintroductiontoSUSY).They
imentally. On the other hand, the character of neutralinos
as Majorana fermions (they are their own antiparticle) and
theirsomewhatstrongerinteractionsasjustweaklyinteract-
⋆ e-mail:[email protected]
(cid:13)c 0000RAS
2 J. Zavala, V. Springel and M. Boylan-Kolchin
ing massive particles (WIMPs) make them more promising Within our Galaxy, the annihilation signal is ex-
candidates for detection, possibly in thenear future. pected to be particularly strong in the high density re-
Being WIMPs with expected masses of the order gions around the galactic centre (Bertone & Merritt 2005;
of 100GeV, neutralinos have all the properties required Jacholkowska et al. 2006). However, since other astrophys-
to be the cold dark matter (CDM) in the prevailing ical sources contribute prominently to the gamma-ray flux
ΛCDM cosmogony, which is at present the most success- there,itmaybedifficulttodisentangleapossibleneutralino
ful model of structure formation (e.g. Komatsu et al. 2009; annihilation signal from the rest of “ordinary” sources.
Springel et al.2006).Significanteffortsarethereforeunder- A more promising approach may lie in an analysis of
taken to find experimental proof for the existence of neu- the gamma-ray radiation produced in the smooth Milky
tralinos. These experiments are usually classified as direct Way halo as a whole (Stoehr et al. 2003; Fornengo et al.
andindirect searches(forrecentreviews ondirect andindi- 2004; de Boer et al. 2004; Springel et al. 2008). Particular
rect dark matter searches in the Galaxy see Bertone 2007; attention has also been given to substructures within the
Hooper 2007; Spooner2007; deBoer 2008). MW halo (e.g. Siegal-Gaskins 2008; Ando 2009), which
Direct dark matter search experiments look for the re- have been suggested to dominate the annihilation signal
coil of ordinary matter by scattering of WIMPs in labo- observed at Earth (Berezinsky et al. 2003), even though
ratories on Earth. Some of the experiments that have been they have recently been shown to be more difficult to de-
lookingforsignalsare:CDMS(Akeribet al.2004),XENON tect compared with the signal from the galactic centre
(Angle et al.2008),ZEPLIN(Alneret al.2005)andDAMA (Springelet al.2008).Searchesinothernearbygalaxieshave
(Bernabei et al. 2000, 2008). The latter is the only experi- also been considered, such as Andromeda (Falvard et al.
mentthathasreportedapositivesignalsofar,afindingthat 2004; Lavalle et al. 2006) and other galaxies in the Local
hasnotyetbeenconfirmedbyanotherexperiment,however. Group (Wood et al. 2008). Clusters of galaxies may also be
The interaction rate of neutralinos with ordinary nu- interesting targets for searches (Jeltema et al. 2009) due to
clei fundamentally depends on two factors, the flux of neu- theirlarge content of darkmatter, even thougha varietyof
tralinos on Earth and the interaction cross section of neu- other possible sources of high energy gamma-rays in clus-
tralinos with a given nuclei. The latter can be computed ters, such as cosmic rays generated in structure formation
theoretically within the context of the SUSY model, but shocks, active galactic nuclei and supernovae, can lead to
the former depends strongly on the phase space distribu- confusion and may make it challenging to cleanly separate
tion of dark matter at the Solar circle, which needs to be thedark matter signal from theconfusing sources.
predicted by cosmological models of structure formation. On still larger scales, we expect that traces of dark
In standard analysis, a constant dark matter density and matter annihilation should be present in the extragalac-
a simple Maxwell-Boltzmann distribution for the particle tic gamma-ray background (EGB) radiation, which con-
velocitiesisoftenassumed,properlymodifiedtoincludethe tains gamma-rays coming from all possible cosmic sources.
motionoftheSunaroundtheGalaxyandthatoftheEarth In the present work, we will focus on these extragalactic
aroundtheSun(Freese et al.2001).However,itisnotatall gamma-rayphotons.TheEGBhasbeenmeasuredbydiffer-
clear that this simple assumption is justified, because cold ent gamma-ray satellites, in particular in the energy range
dark matteris expected toproducea large numberof over- between 1MeV and 30GeV by COMPTEL and EGRET
lapping cold streams locally, and may leave residual struc- (Stronget al. 2004). In the near future, our understanding
ture in energy space. High-resolution N-body simulations of the EGB will greatly improve by the Large Area Tele-
of structureformation combined with an explicit modelling scope aboard the recently launched FERMI satellite (for-
of thelocal stream density evolution provideonepromising merlyknown asGLAST,Atwood et al. 2009),which covers
new approach to provide a more accurate characterization anenergyrangebetween20MeV and300GeV andfeatures
ofthepossiblesignal(Vogelsberger et al.2008,2009).Other a greatly improved sensitivity compared with its predeces-
dynamicallocalSolarSystemeffectsmayalsobeimportant sorEGRET.Severalsourcesofdifferentastrophysicalorigin
(Lundberg& Edsj¨o 2004). are known to contribute significantly to the EGB, includ-
Indirect dark matter searches look for the products ingblazars(Andoet al.2007a;Kneiske & Mannheim2008),
of WIMP annihilations generated in cosmic structures. typeIasupernovae (Strigari et al. 2005), cosmic raysaccel-
Such possible annihilation products include, for example, erated at cosmological structure formation shocks (Miniati
positrons(Baltz & Edsj¨o1999),neutrinos(Berezinsky et al. 2002; Miniati et al. 2007;Jubelgas et al. 2008), among oth-
1996) and gamma-ray photons. Interestingly, the satel- ers (see Dermer 2007; Kneiske 2008, for recent reviews).
lite PAMELA has recently reported an anomalously high Of course, dark matter annihilation also contributes to the
positron abundance in cosmic rays at high energies EGB, and it is in this component that we focus on in this
(Adrianiet al. 2009). Although this signal could be ex- work.
plained by a number of astrophysical sources, dark matter Using standard analytic approaches to describe cos-
annihilations offer a particularly attractive explanation un- mic structure formation, like halo mass functions and clus-
dercertainscenarios(Bergstro¨m et al.2008),andthispossi- tering amplitude as a function of redshift, it is possible
bilityhascreatedaflurryofactivityinthefield.Inarelated to make predictions for the relative gamma-ray flux com-
experiment, the FERMI satellite has measured an excess ing from haloes and their subhaloes as a function of mass
in the total flux of electrons and positrons in cosmic rays and redshift (Ullio et al. 2002; Taylor & Silk 2003) and to
(Abdoet al. 2009), which, although is smaller than those use this information to estimate the contribution of dark
reportedbytheballoon experimentsATICandPPB-BETS matter annihilation to the EGB. Both, the isotropic and
(Chang et al. 2008; Torii et al. 2008) is nevertheless signifi- anisotropic components of this radiation have been pre-
cant. dictedinthiswayinpreviousworks(Els¨asser & Mannheim
(cid:13)c 0000RAS,MNRAS000,000–000
EGB radiation from dark matter annihilation 3
2005; Ando& Komatsu 2006; Cuoco et al. 2007, 2008; in a given volume V is given by (e.g. Gondolo & Gelmini
deBoer et al. 2007; Fornasa et al. 2009). In particular, 1991):
Ando& Komatsu (2006) first pointed out that the density
squareddependenceofdarkmatterannihilationshouldleave hσvi
a specific signature in the angular power spectrum of the n= 2m2 ρ2χ(x)d3x, (1)
EGB. We also note that Siegal-Gaskins & Pavlidou (2009) χ ZV
haveshownthattheenergydependenceoftheangularpower wheremχ andρχ arethemassanddensityofthedarkmat-
spectrum of the total diffuse gamma-ray emission can be ter species, and hσvi is the thermally averaged product of
used to disentangle different source populations, including theannihilation cross section and theMøller velocity.
that associated with dark matter annihilation. Since the particle physics details related to the intrin-
However, it is not clear how accurately these analytic sic nature of dark matter are encoded in the term outside
approachescapturethenon-linearstructuresresolvedinthe the integral in Eq. (1), it is convenient to remove this de-
newest generation of high-resolution cosmological simula- pendence in analyzing the effect of dark matter clustering
tions.Also,thepreviousanalysishavesofarbeenrestricted on the annihilation rate. For this purpose, it is customary
to statistical statements about the power spectrum of the to introduce the ratio of gamma ray emission coming from
EGB, or its mean flux. For a full characterization of the a given halo of volume V to the average emission of a ho-
signal, it would be useful to have accurate realizations of mogeneous distribution of thesame amount of dark matter
maps of the expected gamma-ray emission over the whole within thisvolume:
sky.Suchmapscontaininprinciplethefullinformationthat 1
f (V)= ρ2(x)d3x (2)
canbeharvested,andallow,forexample,correlationstudies h ρ¯2V χ
h ZV
withforegroundlarge-scalestructureintermsofgalaxiesor
where ρ¯ is theaverage density of dark matter particles in-
dark matter. h
side V.
In the present paper, we therefore focus on techniques
For instance if the region of interest is a dark matter
topredicttheextragalacticgamma-raybackgrounddirectly
halo, then Eq.(2) can be solved analytically byassuming a
from cosmological N-body simulations. To this end, we use
specificdensitydistribution.Inparticular,forthespherically
the Millennium-II simulation. With 1010 particles in a ho-
mogeneously sampled volumeof(100h−1Mpc)3,itisoneof symmetricNFWprofile(Navarro et al.1997),fh(V)isgiven
by(Taylor & Silk 2003):
the best resolved structure formation simulations to date.
We use a ray-tracing technique to accumulate the signal c3 1−1/(1+c3)
f (c )= ∆ ∆ , (3)
from all haloes and subhaloes over the past light-cone of h ∆ 9[ln(1+c )−c /(1+c )]2
(cid:2) ∆ ∆ (cid:3)∆
a fiducial observer, positioned at a plausible location of the
MilkyWayinthesimulationbox.Throughcarefulaccount- where c∆ =r∆/rs is theconcentration parameter, rs is the
ing for unresolved structure, we can extrapolate our signal scaleradiusoftheNFWprofile,andr∆isthelimitingradius
to the damping scale limit and obtain the first realization of thehalo, connected toits enclosed mass M∆ through:
of realistic full sky maps of the expected gamma-ray radia- 4 4
M = πρ¯ r3 = π∆ρ r3, (4)
tion from dark matter annihilation. The maps can be used ∆ 3 h ∆ 3 crit ∆
bothtocheckearlieranalyticestimates,andtodevelopnew
where∆istheoverdensityparameter.Differentvaluesfor∆
strategies for identification and exploitation of the annihi-
are used in the literature, with the two main choices being
lation component in the EGB. For example, we show that ∆ = 200 (Navarroet al. 1997), and ∆ ∼ 178Ω0.45 for a
difference maps at different energies could be used to un- m
ΛCDM model with Ω +Ω =1 (Ekeet al. 2001).
m Λ
cover nearby cosmic large-scale structure, which could be
Forsphericallysymmetrichaloes,theintegralinEq.(2)
correlatedwithotherprobesoflarge-scalestructuretomore
can also be written as the scaling law (Springel et al. 2008)
unambiguously identify the origin of this background com- L′ ∝ V4 /r , where V is the maximum rotational
ponent. h max half max
velocity of the halo and r is the radius containing half
half
This paper is organized as follows. In Section 2, we
of thegamma-ray flux.For a NFW halo, thelatter formula
brieflyreviewtheanalyticformulaeassociatedwiththecon-
reducesto
tributiontotheEGBfromdarkmatterannihilation.InSec-
tion3,wedescribetheSUSYmodelchosenforouranalysis. L′ = ρ2 (r)dV = 1.23Vm4ax, (5)
We then present the N-body simulation used, and the ef- h Z NFW G2rmax
fect of dark matter clustering in the annihilation’s gamma where r is the radius where the rotation curve reaches
max
ray flux in Section 4. In Section 5 we describe our method its maximum. We will use Eq. (5) for our analysis in this
to generate simulated sky maps of the EGB radiation from paper. The most recent high-resolution N-bodysimulations
darkmatterannihilation,andweanalysetheenergyandan- supportaslightlyreviseddensityprofilethatbecomesgrad-
gular power spectra of the signal. We give a summary and uallyshallowertowardsthecentre(Navarro et al.2009),de-
our conclusions in Section 6. viating from the asymptotic power-law cusp of the NFW
profile.ForthisEinastoprofile,thecoefficient1.23inEq.(5)
changesto1.87,aninsignificantdifferencecomparedtoother
uncertaintiesintheanalysisofthedarkmatterannihilation
2 DARK MATTER ANNIHILATION RATES
radiation.
AND THE EGB
WecanextendEq.(2)tolargervolumescontainingsev-
Ifdark matteris madeof particles that can annihilate with eralhaloesofdifferentsizes.Ifweassumethatalldarkmat-
each other, then the number of annihilations per unit time terparticlesinagivencosmicvolumeV belongtovirialized
B
(cid:13)c 0000RAS,MNRAS000,000–000
4 J. Zavala, V. Springel and M. Boylan-Kolchin
haloes, then the ratio of gamma-ray emission coming from allredshiftsalongtheline-of-sightforafiducialobserver,lo-
all haloes contained in V with masses larger than M cated at z=0, for all directions on its two-dimensional full
B min
to the emission produced by a smooth homogeneous distri- sky. The values we want to calculate for each pixel of such
bution of dark matter, with average density ρ¯ , filling this a sky map are that of the specific intensity, the energy of
B
volumeis given by: photonsreceivedperunitarea,time,solid angleandenergy
1 ∞ range:
f(M >M )= dN (M)ρ¯2V(M)f (V(M))
min ρ¯2V VB h h 1 dr
B B ZMmin (6) Iγ,0 = 4π ǫγ,0(Eγ,0(1+z),z)(1+z)4, (11)
Z
where dN (M) is the number of haloes in the volume V
VB B wheretheintegralisoverthewholelineofsight,ristheco-
withmassesintherange[M,M+dM],whichcanbewritten
movingdistanceandE istheenergymeasuredbytheob-
as γ,0
serveratz=0.Notethatǫ isevaluatedattheblueshifted
γ,0
dNVB(M)=VBdnd(MM)dM =VBddnlo(gMM)dlogM, (7) tehneercgoysm(1o+logzi)cEalγ,r0eadlsohnifgtitnhge.line-of-sight tocompensatefor
withdn(M)/dlogM denotingthedifferentialhalomassfunc-
tion. Thus:
f(M >M )= 3 DARK MATTER ANNIHILATION AND THE
min
1 ∞ dn(M) SUSY FACTOR
ρ¯2V(M)f (V(M))dlogM. (8)
ρ¯2 dlogM h h In the present section we describe the particular SUSY
B ZMmin(cid:18) (cid:19)
model that we used to compute the gamma-ray emissiv-
In the literature, f(M > M ) has been called
min ity produced bydark matterannihilation. In particular, we
the “flux multiplier” (Eq. 18 of Taylor & Silk 2003) or
provide a closer examination of the SUSY factor given in
the “clumping factor” (Eq. 6 of Ando& Komatsu 2006).
Eq.(10).
The limiting lower mass in Eq. (8) is the minimum mass
A detailed particle physics model for the nature of the
scale for virialized haloes surviving today, which is roughly
neutralinoas dark matter, and its mass and interactions, is
given by the free streaming length of dark matter parti-
needed to compute f . We restrict our analysis to the
cles: ∼ 10−6M⊙ for mχ = 100GeV (Hofmann et al. 2001; framework of the minSiUmSaYl supergravity model (mSUGRA,
Green et al. 2004), see section 4.1 for a discussion on this
for reviews see Nilles 1984; Nath 2003). The mSUGRA
cutoff mass. Although many of the small scale haloes have
model is popular, among other reasons, due to its relative
suffered from tidal truncation or were disrupted, it is pos-
simplicity,whichreducesthelargenumberofparametersin
sible that they contribute significantly to the gamma-ray
the general MSSM (& 100) to effectively four free parame-
emission produced by dark matter annihilation.
ters, m , m , tanβ, A and the choice of sign for µ. The
Eqs. (2) and (8) allow us to study the gamma ray flux 1/2 0 0
parameters m , m and A are the values of the gaugino
comingfromindividualhaloesorgroupsofhaloesinagiven 1/2 0 0
andscalarmassesandthetrilinearcoupling,allspecifiedat
region of the Universe at a given redshift (both can be ap-
the GUT scale, tanβ is the ratio of the expectation values
plied toanyredshift,provided thatρ ,ρ¯ andρ¯ arecom-
χ h B in the vacuum of the two neutral SUSY Higgs (H˜0 y H˜0),
puted at such redshift). u d
and finally,µ is theHiggsino mass parameter.
Following Eq. (1), we define the gamma ray emissivity
The general 5-dimensional parameter space of the
(energyemittedperunitenergyrange,unitvolumeandunit
mSUGRA model is significantly constrained by various re-
time) associated with dark matterannihilation as
quirements: consistency with radiative electroweak symme-
dN hσvi ρ 2 try breaking (EWSB) and experimental constraints on the
ǫ =E γ χ , (9)
γ γdE 2 m low energy region, which are given by current lower mass
γ (cid:20) χ(cid:21)
boundsforthelightestSUSYHiggsafterEWSB,m ,aswell
h
where dN /dE is the total differential photon spectrum
γ γ as concordance between experimental and theoretical pre-
summed over all channels of annihilation. Of all the quan-
dictions for the decay b →sγ and the anomalous magnetic
tities in this equation, only ρ depends on the spatial dis-
χ momentofthemuon(g −2)/2.Importantly,itisnormally
µ
tribution of dark matter, the rest is related to its intrinsic
also assumed that the major component of dark matter is
properties. For the analysis of the latter, since we consider
the LSP of the theory, then the LSP must be neutral and
dark matter to be made of neutralinos within a SUSY sce-
Ω = Ω , which reduces the allowed regions in the pa-
nario, we convenientlydefinethe so called SUSYfactor1: LSP DM
rameterspaceconsiderably.FortypicalmSUGRAscenarios,
dN hσvi the LSP that satisfies these conditions is the lightest neu-
f = γ . (10)
SUSY dEγ m2χ tralino. Its relic density Ωχ needsto beproperly calculated
by solving the Boltzmann equation, with the parameters of
The signal that we are ultimately interested in is the
thetheorychosensuchthatthedesireddarkmatterdensity
contributionofneutralinoannihilationtotheEGB,ormore
results.
specifically, the spatial distribution of the gamma-ray radi-
Observational constraints on the abundance of dark
ation comingfrom darkmatterannihilation integrated over
matter,Ω ,come from varioussources, includingthecos-
DM
mic microwave background (CMB), galaxy clusters, super-
1 Thistermhasbeenextensivelyusedpreviouslyintheliterature, novae,Lyman-αforestdata,andweakgravitationallensing.
for example in Fornengoetal. (2004), fSUSY is related to the Forthepurposesof theanalysis in thissection, wetakethe
integralquantityΦSUSY definedthere. 1σ bound for the amount of dark matter in the Universe
(cid:13)c 0000RAS,MNRAS000,000–000
EGB radiation from dark matter annihilation 5
Ourgoalinthissectionistoanalyzethesupersymmet-
ricfactorf predictedbythemSUGRAmodelinthedif-
SUSY
ferentallowedregionsdepictedinFig.1.Todoso,wedefine
several“benchmarkspoints”whichfulfillalltheexperimen-
tal constraints discussed above and which can be taken as
representative of the different allowed regions in parameter
space that we just described. The benchmark points were
selectedfollowingtheanalysisofBattaglia et al.(2004)and
Gondolo et al.(2004),butslightlymodifiedtobeconsistent
withthecosmologicalconstraintofEq.(12).Weusethenu-
mericalcodeDarkSUSY(Gondolo et al.2004,2005)withthe
interface ISAJET (Baer et al. 2003) to analyse these bench-
markpoints.Table1liststhe11selected benchmarkpoints
with their corresponding values for the five mSUGRA pa-
rameters,thevalueofthelightestneutralino’smass,itsrelic
density,andthenameoftheregiontheybelongtoinFig.1.
Fig. 2 shows the energy spectrum of f computed
SUSY
for four of the benchmark points presented in Table 1: A
(black dashed line), E (red dotted line), K (green dash-
dotted line), and L (blue solid line), which are represen-
tatives of the regions CA, FP, RAF and B, respectively.
The main features of Fig. 2 are determined by the photon
Figure 1. Sketch showing the allowed parameter space of the
mSUGRAmodelwhentheconstraintontherelicdensityaccord- spectrumdNγ/dEγ whichreceivescontributionsfromthree
ingto Eq. (12) (dark blueregion) is imposed. Theexperimental mechanisms of photonproduction (Bringmann et al. 2008):
constraint on the mass of the lightest SUSY Higgs, mh, is also (i) gamma-ray continuum emission following the decay of
shown for reference. The grey region inthe lower rightcorner is neutralpionsproducedduringthehadronization of thepri-
wheretheLSPischarged,whereasinthegreyregionintheupper maryannihilationproductsatthetree-level;thismechanism
left corner there is no radiative EWSB. The abbreviated names is dominant at low and intermediate energies (in Fig. 2 it
forvariousimportantregionsintheparameterspacearemarked
dominates the spectrum in most of the energy range); (ii)
inthefigurewithcapitalletters,seetextfordetails.
monoenergetic gamma-ray lines for neutralino annihilation
intwo-bodyfinalstatescontainingphotons,whichisallowed
in higher orderperturbation theory;themost relevant final
according to the fiveyear results of WMAP data combined
states containing these lines are χχ → γγ and χχ → Z0γ,
with distance measurements from type Ia supernovae and
where Z0 is the neutral Z boson; for non-relativistic neu-
Baryon Acoustic Oscillations (Komatsu et al. 2009):
tralinos the energy of each outgoing photons in these pro-
0.11096ΩDMh2 60.1177. (12) cesses is Eγ = mχ and Eγ = mχ(1−m2Z0/4m2χ), respec-
tively (in Fig. 2 these lines are prominent for benchmark
The general picture for the allowed regions in the pa- point E (red line) and negligible in the rest of the cases);
rameter space of the mSUGRA model resulting from the (iii)whenthefinalproductsofannihilation arecharged,in-
differentconstraintscanbequalitativelydescribedinthe2- ternal bremsstrahlung (IB) needs to be considered, it leads
dimensionalplane(m0−m1/2),asshowninFig.1.Theblue to the emission of an additional photon in the final state;
areainthefigurerepresentstheallowed regionaccordingto the IB contribution to the spectrum is typically dominant
Eq. (12) for the observational bounds on the abundance of at high energies resulting in a characteristic bump (bench-
dark matter. mark points A and L have an important contribution from
ThedifferentregionsinthemSUGRAparameterspace IB at energies close to m , for the other two points the
χ
thatsatisfy theseconstraintshavebeenstudiedabundantly contribution from IB is negligible). Processes (ii) and (iii)
in the past (e.g. Baer & Bal´azs 2003; Battaglia et al. 2004; are subdominant relative to process (i), but they display
Feng 2005; B´elanger et al. 2005) and have received generic distinctive spectral features intrinsic to the phenomenon of
names, marked with capital letters in the figure: (B) bulk annihilation.
region (low values of m0 and m1/2), (FP) focus point re- Additionally, the annihilation of neutralinos into elec-
gion (large values of m0), (CA) co-annihilation region (low trons and positrons contributes indirectly to the high
m0 and mχ . mτ˜1, where τ˜1 is the lightest slepton) and energy photon spectrum when low energy background
(RAF) rapid annihilation funnel region (for large values of photons, such as those in the cosmic microwave back-
tanβ and a specific condition for mχ2). The regions in grey ground (CMB), are scattered to higher energies via inverse
(light)representzoneswheretheLSPischarged(lowerright Compton (IC) scattering by these electrons and positrons
corner)andwherethereisnoradiativeEWSBsolution(up- (Profumo & Jeltema 2009; Belikov & Hooper 2009). In the
perleft corner). Thedashed line is shown as areference for case of CMB photons, the energy peak of this IC photon
theexperimental constraints for mh. emission (EγIC) is approximately independent of redshift
andisgivenby:EγIC ≈EγCMB(Ee±/me±c2)2,whereEγCMB
and Ee± are the average energies of the CMB photons
2 Therelationwhichshouldholdis:2mχ≈mA,whereAisthe and e± respectively. For neutralino masses in the 100 GeV
CP-oddHiggsboson;seeforexampleFeng(2005)fordetails. range,thisenergy peak lies in theX-rayregime (∼10keV)
(cid:13)c 0000RAS,MNRAS000,000–000
6 J. Zavala, V. Springel and M. Boylan-Kolchin
Modela A C D E F G H I J K L
m 550 435 520 399 811 396 930 395 750 1300 450
1/2
m0 114 94 113 2977 4307 118.5 242.5 193 299.5 1195 299
tanβ 7 11 10 30 30.4 20 20 35 35 46 47
sign(µ) + + − + + + + + + − +
mχ 226.4 175.1 214.9 154.2 337 160.4 394.2 160.6 315.5 566.1 184.9
Ωχh2 0.115 0.114 0.116 0.118 0.112 0.113 0.118 0.118 0.114 0.116 0.111
Regionb CA B CA FP FP B CA B CA RAF B
Table 1. Selected benchmark points in the mSUGRA parameter space. For our main analysis, we will focus on point ‘L’, which
predictsanannihilationfluxclosetothemaximumpossiblefortheconstrainedmSUGRAmodels.
a TheletterstorepresenteachpointwerechosentofollowthenotationofBattagliaetal.(2004)andGondoloetal.(2004)
b BforBulkregion,FPforFocus Pointregion,CAforCo-annihilationregionandRAFforRapidAnnihilationFunnelregion
tionfunnelregion(RAF,greendash-dottedline),andfinally
theco-annihilationregion(CA,blackdashedline).Thislast
feature could be of importance for constraining the allowed
mSUGRA parameter space even more: in principle, a pre-
cise measurement of the gamma-ray flux coming from dark
matter annihilation could discriminate between the charac-
teristicregionsFP,B,CAandRAF,consideringforexample
thatthedifferenceinf betweentheCAandFPregions
SUSY
isaroundtwoordersofmagnitude.However,suchinferences
will onlybecome possible iftheremaining sourcesof uncer-
taintycanbereducedtolessthantwoordersofmagnitude.
This is a demanding goal, given the limited knowledge we
haveonsomeoftheotherastrophysicalfactorsthatplayan
important role in the production of these gamma-rays, as
well as theobservational difficulties in properly subtracting
from an observed gamma-ray signal the contributions from
astrophysicalsourcesunrelatedtodarkmatterannihilation.
Nevertheless, ‘dark matter astronomy’ remains an interest-
ing possibility in thelight of theresults shown in Fig. 2.
Inoursubsequentanalysis,wewillnowchooseonepar-
ticular benchmark point and adopt its f value for the
SUSY
restofourwork,keepinginmindtheresultsabove.Ourcho-
Figure2.Supersymmetricfactorasafunctionofγ-rayemission senbenchmarkpointisthemodelL(thatwehighlightwith
energyforaselectionofthebenchmarkpointsdescribedinTable athickbluesolid linein Fig. 2),which givesan upperlimit
1: A, E, K and L, which are shown with black (dashed), red onf forthebenchmarkpointsinthebulkregionandis
SUSY
(dotted), green(dash-dotted) andblue(solid)lines,respectively.
closetothemaximumvalueoff for allthebenchmark
SUSY
points analyzed. The latter is the main reason to choose
this particular model because it is desirable that f ,
SUSY
(Profumo & Jeltema 2009). Unless the neutralino mass is and hence the predicted gamma-ray flux, has a large value
larger than 1 TeV and with a hard e± spectrum, the con- amongthedifferenttheoreticalpossibilities. Thisoptimistic
tribution of this mechanism is not significant in the energy choiceasfarasprospectsforfuturedetectionsareconcerned
range relevant to the present study (& 0.1 GeV). We note then also demarcates the boundary where non-detections
that we are currently working on the analysis of the X-ray can start to constrain the mSUGRA parameter space. But
extragalactic background radiation from dark matter anni- there are other reasons as well. The mass of the neutralino
hilation and we will present our results in a futurework. in thismodel is “safely” larger than thelower mass bounds
In Fig. 2, we have shown only 4 of the 11 benchmark coming from experimental constraints (m > 50GeV ac-
χ
points we considered; the rest show very similar features cording to Heister et al. 2004), but low enough to be de-
and lie in between these 4 cases. Fig. 2 also indicates that tectableintheexperimentsavailableinthenearfuture.Also,
thenormalization of thespectrum of f typically varies a high value of the parameter tanβ seems to be favored by
SUSY
by three orders of magnitude among the different regions other theoretical expectations (e.g. Nu´n˜ez et al. 2008). We
of the allowed parameter space. Also there appears to be a notethatalthoughbenchmarkpointLdoesnothavepromi-
trendbetweenthedifferentallowedregionsinthemSUGRA nent gamma-ray lines in its spectrum, it does have an im-
model: f has the largest value for benchmark points portantIBcontributionandservesthepointofshowingthe
SUSY
on the focus point region (FP, red dotted line), followed implications of a peak in the annihilation energy spectrum
by the bulk region (B, blue solid line), the rapid annihila- for theEGB.
(cid:13)c 0000RAS,MNRAS000,000–000
EGB radiation from dark matter annihilation 7
We note that recent analyses on the cosmic-ray imum circular velocity of a (sub)halo and the radius where
anomalies as measured by PAMELA and FERMI favour thismaximumisattained.Wealsouser ,theradiuswhich
1/2
higher neutralino masses (∼ 1 TeV) when these anoma- encloses half of the mass of a subhalo, and r , the radius
200
lies are explained mainly by dark matter annihilation (e.g. wherethemeandensityofamainhalois200timesthemean
Bergstro¨m et al. 2009). However, it is important to keep in background matter density.
mind that other astrophysical sources, such as pulsars and Even with the large particle number available in the
supernovae remnants, are expected to have a meaningful MS-IIsimulation, itis important tonotethat adirecteval-
contributiontothesolutionofthecosmicrayanomalies.At uation of halo luminosities by estimating local dark matter
present,theseexperimentsareonly suggestive of dark mat- densities at the positions of each simulation particle (for
ter particles in the TeV range, not definite. Our goal is to example based on the SPH kernel interpolation technique)
give an account of the general features of the gamma-ray would be seriously affected by resolution limitations. This
spectrum from annihilation and their imprint on the EGB. is because most of the emission originates over a tiny ra-
For this purpose, our fiducial model with mχ ∼200 GeV is dial region, close to the very centre of a halo, where the
adequate.Modelswithhighermasseswouldshifttheenergy density is easily underestimated even for otherwise well re-
scale to higher values, but our general conclusions would solvedstructures.Infact,itrequiresoforder108−109 par-
remain unaltered. ticles to obtain a converged estimate of the emission from
the smooth part of a cuspy dark matter halo in a brute-
force approach (Springel et al. 2008).On theotherhand,it
isnumericallyvery much easiertoresolvetheexistenceofa
4 GAMMA-RAY LUMINOSITY FROM
halo or subhalo, and to accurately estimate its total mass,
ANNIHILATION IN HALOES AND
as well as its primary structural properties such as V
max
SUBHALOES: THE ASTROPHYSICAL
and r . Our strategy will therefore be to adopt an ana-
max
FACTOR
lytic fit for the density profile of each detected dark matter
Inthissectionweanalysetheastrophysicalpartoftheanni- haloandtopredictitsexpectedemissionbyintegratingthis
hilation gamma-rayflux,arisingfrom theclusteringofdark profile. In other words, we will use the scaling relation (5)
matterintohaloesandsubhaloesacrosscosmictime.Tothis to compute L′h for main haloes and subhaloes alike. Since
end we analyse the dark matter haloes in one of the most Vmax and rmax can bemeasured reasonably accurately even
recent state-of-the-art cosmological N-body simulations. for poorly resolved haloes, this gives us a reliable account-
Specifically, we use the “Millennium-II” simulation ing for the total emission of all haloes down to a fairly low
(MS-II) of Boylan-Kolchin et al. (2009) which has the masslimit.Importantly,wecancompletelyavoidinthisway
same particle number (21603) and cosmological parameters tointroducestrongmass-dependentresolutioneffectsinour
(Ω = 0.25, Ω = 0.75, h = 0.73, σ = 0.9 and n = 1, measurements of luminosity as a function of halo mass.
m Λ 8 s
where Ω and Ω are the contribution from matter and For simplicity, we shall assume that all haloes have a
m Λ
cosmological constant to the mass/energy density of the NFW structure, where the local logarithmic slope of the
Universe, respectively, h is the dimensionless Hubble con- density profile tends asymptotically to −1 for small radii,
stant parameter at redshift zero, n is the spectral index even though the Aquarius project, a recent set of simula-
s
of the primordial power spectrum, and σ is the rms am- tions aimed at following the formation of MW-sized haloes
8
plitude of linear mass fluctuations in 8h−1Mpc spheres at atdifferentmassresolutions,demonstratedthattheEinasto
redshiftzero)astheMillenniumsimulation(MS-I)(Springel profile provides an even better fit (Navarro et al. 2009), a
2005) but with a box size that is 5 times smaller, equal to findingthatisalsosupportedbytheindependent‘GHALO’
L = 100h−1Mpc on a side, thus having a mass resolution simulation (Stadelet al. 2009). In the Einasto profile, the
of 6.89×106h−1M⊙, 125 times smaller thanin MS-I.Typ- locallogarithmicslopeofthedensityprofilechangeswithra-
ical Milky-Way sized haloes are resolved with several 105 diusaccording toa power law with exponent αsh (typically
particles, while clusters of galaxies have about 50 million calledshapeparameter).AccordingtotheAquariusproject,
particles. αsh is of the order 0.16−0.17 and varies from halo to halo
For the dynamical evolution of the MS-II, a fixed co- (Navarroet al. 2009). Previously, Gao et al. (2008) found a
moving gravitational softening length of ǫ = 1h−1kpc dependenceon halo mass for thevalueof theshapeparam-
(Plummer-equivalent) was used. The MS-II was processed eter (going from 0.16 for galaxy-haloes to 0.3 for massive
on-the-fly to find haloes using a friend-of-friends (FOF) al- clusters). However, for the purposes of our work, the NFW
gorithm,followedbyapost-processingsteptoidentifygravi- profile is a good enough approximation and using it sim-
tationally bounddarkmattersubstructuresdowntoalimit plifies comparisons to a number of results in the literature.
of 20 particles within each group based on the SUBFIND In any case, we note that using the Einasto profile (with
algorithm (Springel et al. 2001). A given FOF halo is de- a typical shape parameter of 0.17) instead would increase
composedbySUBFINDintodisjointsubgroupsofself-bound ourresultsforthefluxesby∼50%.Thisvaluechangesonly
particles. The most massive of these subgroups represents veryslightly overtherangeof valuesforαsh foundfor dark
the “main” or “background” halo of a given FOF halo (see matterhaloes.Suchanuncertaintyonthedarkmatterden-
Fig.3ofSpringelet al.2001).Forthereminderofthiswork sityprofileisconsiderablylowerthanothersinouranalysis,
we will refer to these subgroups as main haloes or simply namely,thephotonspectrum from annihilation (see Fig. 2)
haloes,andtotherestoftheidentifiedsubgroupsofagiven and the contribution from unresolved haloes and subhaloes
FOF halo as subhaloes of that main halo. SUBFIND also (see sections 5.3 and 5.4).
computes several properties for each subgroup; the ones we Nevertheless, it is important to take into account the
willuseextensivelyinthisworkareV andr ,themax- impact of finite numerical resolution in the measured val-
max max
(cid:13)c 0000RAS,MNRAS000,000–000
8 J. Zavala, V. Springel and M. Boylan-Kolchin
uesofV andr forthesmallest haloesthatcanbere-
max max
solved.AshasbeenshownbySpringelet al.(2008),theval-
uesofr areincreasinglyoverestimatedforsmallerhaloes
max
whereasforV theoppositeholds.Apartialcorrectionfor
max
thesenumericaleffectscanbeobtainedbytryingtoaccount
fortheeffectofthegravitational softening ǫforhaloeswith
values of r close to ǫ. Assuming adiabatic invariance of
max
orbitsbetweensoftenedandunsoftenedversionsofthesame
halo, oneobtains to leading order:
r′ =r [1+(ǫ/r )2]−1
max max max
V′ =V [1+(ǫ/r )2]1/2, (13)
max max max
where the primed quantities refer to the case without soft-
ening.
Ingeneral,thevalueofr tendstobemuchmoreaf-
max
fectedbythesofteningthanV .Thiscanbeclearlyseenin
max
theupperpanelofFig.3whereweshowtheV -r rela-
max max
tionforallmainhaloes oftheMS-II.Notethattherelation
has some intrinsic scatter, but we here only show themean
valuesofr inlogarithmicbinsofV .Theredsolidlines
max max
show thecorrelation for different redshifts equalto z=0.5,
1.0, 1.5 , 2.1, 3.1, and 4.9, after the softening correction of
Eqs. (13) has been applied, whereas the thicker line is for
z=0.Theblacksolid lineisanextrapolation oftheresults
found by Neto et al. (2007) at z =0 for the MS-I. There is
aclearupwardsdeviationfromthepower-lawbehaviourfor
haloes with particle number below Np ∼3600 (correspond- Figure 3. Upper panel: Relation between rmax and Vmax for
ing to a mass of ∼2.5×1010h−1M⊙, or Vmax ∼60kms−1 allmain haloes inthe MS-IIat different redshifts. Thered solid
at z = 0). This deviation indicates that the softening cor- lines are based on measurements corrected for softening effects
rection is only able to correct part of the resolution effects. with Eqs. (13). The blue dashed lines show the values we adopt
We can be certain that this is indeed a numerical effect by for calculating the expected luminosity of a halo; here an addi-
lookingatthecorrespondingresultsfortheAquariushaloes tionalcorrectionforrmax wasappliedforsmallhaloes suchthat
they extend the power lawbehaviour measuredforwellresolved
(Springel et al. 2008), which demonstrate with simulations
structures.Theblacksolidlineisanextrapolation oftheresults
of the same initial conditions but drastically different mass
foundbyNetoetal.(2007)forthe MS-I,whilethe blackdotted
resolutions that the r -V relation is indeed a power
max max lineisthepowerlawobeyedbysubhaloes,seeFig.7.Thethicker
lawoveralargedynamicrange,atleastforV assmallas
max linescorrespond to z =0, whilethe restof the lines arefor red-
∼1.5kms−1.Thisisthecaseforboth,mainhaloesandsub-
shifts: 0.5, 1.0, 1.5, 2.1, 3.1 and 4.9. Bottom panel: The relation
haloes (see Fig. 26 of Springel et al. (2008)). We note that between concentration c200 andVmax.Thelinestylesandcolors
haloes with Np ∼ 1000 or less are affected importantly by arethesameasintheupperpanel.
two body relaxation. The discrete representation of a dark
matter halo with this low number of particles will undergo
dependence can be understood as a change in the mass-
two-bodyencounterswhereasarealdarkmattersystemwith
concentration relation of haloes, which gets flatter for in-
∼100 GeV particles is essentially collisionless. Generically,
creasingredshift(Zhao et al.2003;Gao et al.2008).Wees-
thetwo-bodyrelaxationprocesstendstoreducetheconcen-
timate the concentration of the main haloes following the
tration of objects (e.g. Diemand et al. 2004). Wethink this
description given inSpringel et al. (2008),wherea measure
is themain mechanism responsible for theupturnshown in
of the concentration is obtained in terms of the mean over-
Figs. 3 and 7, even though a detailed study of this effect is
density within r relative tothe critical density ρ :
out of thescope of this work. max crit
Sinceanoverestimateofrmaxcanresultinanunderesti- δ = ρ¯(rmax) =2 Vmax 2, (15)
mateofthecomputedluminosityforagivenhalo,wefurther V ρ H(z)r
crit (cid:18) max(cid:19)
correct the r values of small haloes phenomenologically
max where H(z) is the Hubble rate as a function of redshift.
in order to allow use of all haloes in the MS-II down to the
The overdensity δ can be related to the usual concentra-
minimumresolvedhalomassof1.4×108h−1M⊙,andtoget tion parameter c V =r /r of the NFW profile using the
the correct mean halo luminosity over the full mass range. 200 200 s
characteristic overdensity:
For this purpose, we force the values of r for all haloes
max
with less than 3600 particles to follow the power law δ = 200 c3200 = ρs =7.213δ .
c 3 ln(1+c )−c/(1+c ) ρ V
V′ α(z) 200 200 crit
r′′ =A(z) max (14) (16)
max kms−1
(cid:18) (cid:19) Using Eq. (16) we can compute c200 for a given value of
where A(z) and α(z) are the parameters of the power law. V andr .ThelowerpanelofFig.3showstherelation
max max
These parameters actually depend very mildly on redshift between c and V for the corrected and uncorrected
200 max
(with A(0) ≈ 0.04 and α(0) ≈ 1.32 at z = 0). The redshift valuesofr asbluedashedandredsolidlines,respectively,
max
(cid:13)c 0000RAS,MNRAS000,000–000
EGB radiation from dark matter annihilation 9
Figure 4.Total luminositycomingfrommainhaloes per differ- Figure5.Fluxmultiplierf(Mh>Mmin)forthemainhaloesin
ential mass interval as a function of mass. The thin blue lines theMS-IIasafunctionofMmin.Thesolid-blueanddashed-black
are for different redshifts as in Fig. 3, while the thick solid line linesareanalogoustotheonesinFig.4.Thesolid-blacklineisa
shows the z = 0 result. The dashed lines show the run of our theoretical estimateasdescribedinthetext.
extrapolations asdiscussedinthetext.
Atintermediatemassranges,F (M )isclearlywellap-
h h
withtheredshiftincreasingfromtoptobottom.Thechange proximated by a power law:
inslopeofthepowerlawfollowedbythebluedashedlinesis
F (M ,z)=A (z)Mαh(z). (19)
roughly in agreement with the results of Zhao et al. (2003) h h h h
and Gao et al. (2008). Our goal is to fit the parameters of this power law so that
an extrapolation can be done down to the cutoff mass for
neutralinos. For the neutralino mass corresponding to the
4.1 Gamma-ray luminosity of haloes down to the
modelwe havechosen, thefree streaming mass is of theor-
damping scale limit
der of 10−7h−1M⊙ (Hofmann et al. 2001), however, acous-
Following the formulation in section 2, we here analyze the tic oscillations due to the coupling between cold dark mat-
flux multiplier for large volumes, Eqs. (6) and (8), for the ter and the radiation field in the early Universe, can also
MS-II. Recall, the flux multiplier gives the ratio of the γ- produce a damping in the power spectrum of density per-
rayfluxcomingfromallhaloes insidethesimulated volume turbations(e.g. Loeb & Zaldarriaga 2005).Thecutoff mass
withmasseslargerthanaminimummassM totheemis- of the smallest haloes that can be formed is determined by
min
sionproducedbyahomogeneousdistributionofdarkmatter the strongest of these effects. Taking the recent results of
filling the box of volume V with an average density ρ¯ . Bringmann (2009) (see their Fig. 3), this cutoff mass for
B B
Weobtainthisdimensionlessfluxmultiplierbydefining mχ = 185 GeV lies in the range 10−9−10−4M⊙. We will
first the function take a fiducial value of 10−6h−1M⊙ for our extrapolation,
noting that the value of the minimum mass for bound neu-
L
Fh(Mh)= M¯ ∆loghM , (17) tralino dark matter haloes is a source of uncertainty in our
hP h results.
wherethesumisoveralltheluminosities Lh ofhaloeswith Weobtain theparameters ofthepower law in Eq.(19)
massesinthelogarithmicmassrange:logMh±∆logMh/2, by fitting the function Fh(Mh) between two mass limits,
where ∆logMh is a fixed logarithmic bin size; M¯h is the withthelowerlimitchosenasMlim,min=6.89×108h−1M⊙,
mean value of the halo mass in the given bin. Using this correspondingtohaloeswith100particles(belowthisnum-
definition we can approximate Eq. (8) as ber the mass and abundance of haloes is not reliable), and
1 ∞ F (M ) the higher limit set equal to the last logarithmic mass bin
f(Mh >Mmin)∼ ρ¯2V hln10h dMh. (18) with more than 500 haloes, such that uncertainties from
B B ZMmin countingstatistics are avoided. Wefind that for thesemass
In this sense, the function F (M ) is just the total lumi- ranges, the parameters of the power law fits change only
h h
nosity of haloes in a mass range, per unit mass range. The slightly with redshift; in fact for z < 2.1, α ≃ −1.05 with
h
function F (M ) is shown in Fig. 4 for all main haloes in lessthan2%variation,andA ≃6.92×1011 withless than
h h h
theMS-II.Thedifferentbluelinesarefordifferentredshifts, 50%variation.TheblackdashedlinesinFig.4showthere-
as in Fig. 3. sultingextrapolation ofthepowerlaw downto107h−1M⊙.
(cid:13)c 0000RAS,MNRAS000,000–000
10 J. Zavala, V. Springel and M. Boylan-Kolchin
The blue lines in Fig. 5 give results for the flux multi-
plier, Eq. (18), as a function of M for the main haloes
min
in the MS-II for different redshifts, as in in Fig. 4. The
black dashed lines show the extrapolation of the flux mul-
tiplier down to Mmin = 10−6h−1M⊙ using Eq. (19). The
flux multiplier has been previously computed analytically
byTaylor & Silk(2003).Wefollow theirprocedureandcal-
culate an analytical estimate of f(M > M ) for the cos-
min
mology corresponding to the MS-II. We first use Eq. (3) to
compute the individual flux multiplier f (c ) of an NFW
h ∆
halo with a concentration c given by the redshift mass-
∆
concentrationrelation,computedasintheanalyticalmodel
ofEke et al.(2001)andwithachoiceof∆=178Ω0.45 ∼95.
m
Usingthevalueoff (c )wethencomputethefullfluxmul-
h ∆
tiplierbasedonEq.(8),assumingamassfunctioncalculated
with the formalism of Sheth et al. (2001). The result is in-
cluded in Fig. 5 with black solid lines.
Weseethattheseanalyticestimatesroughlyagreewith
theactualnumericalresultsfromtheMS-IIovertheresolved
massrange.However,smalldifferencesintheslopeoff(M >
M ) at the minimum resolved mass range (1.39−6.89×
min
108h−1M⊙),producealargedifferencefortheextrapolated
valuesatthedampingscalelimit,whichamountstoafactor
of up to 3 − 5. We note that for redshifts z > 2.1, our Figure 6. Differential cumulative flux multiplier |df(Mh >
extrapolation down to the cutoff mass is not accurate any Mmin)/dlogMmin| as a function of minimum mass. The line
stylesandcolorsareasinFig.5.
moreandtheobtainedvaluesareclearlyoverestimated.This
is a reflection of the difficulties to still reliably fit a power
law to F(M) at these high redshifts, where the population
ofhaloesovertheresolvedmassrangebecomeseversmaller.
Another interesting question that arises from this sta-
tisticalanalysisis:Whichmassscalecontributesmosttothe
total flux of gamma rays coming from dark matter annihi-
lations? Theanswer is presented in Fig. 6,which shows the
differential cumulative flux, which is really just the loga-
rithmic derivative of the cumulative flux shown in Fig. 5,
as a function of the minimum halo mass. From the fig-
ure we see that less massive haloes contribute increasingly
more to the total gamma-ray flux coming from dark mat-
ter annihilation. There is a clear change in behaviour for
a given mass scale Mmin ∼ 1014h−1M⊙ at z = 0, going
down to Mmin ∼ 1012h−1M⊙ at z = 2.1. Fig. 6 also re-
veals an artificial downturn for the minimum mass range
resolved (1.39−6.89×108h−1M⊙). The analytical expec-
tations, shown as black solid lines, qualitatively agree with
the results from the simulation. However, the quantitative
differences in theextrapolated region are large.
4.2 The effects of substructure
Substructures within CDM haloes produce a total γ-ray
luminosity which is dominant over the smooth halo com- Figure 7.Relationbetweenrmax andVmax forthesubhaloesof
the 10 most massive clusters inthe MS-II at z =0. These main
ponent for an external observer (e.g Taylor & Silk 2003). haloes have Vmax values in the range ∼850−1250kms−1. Line
Springel et al.(2008)showedthatforaMWhalo,substruc-
stylesandcolorsareasinFig.3
tures with masses larger than 105M⊙ within r200 emit as
muchas76%oftheemissionofthesmoothhalo.Also,using
their set of simulations of thesame halo at different resolu- basis the most massive haloes in the MS-II that contain
tionstheyextrapolatedthecontributionfromsubstructures significant resolved subhaloes and are therefore suitable for
down to masses of 10−6M⊙, at which point they found the suchan analysis. Tothisendwe follow ananalogous proce-
total cumulativeluminosity tobe232 times larger than the dureto theone used in theprevious subsection.
contribution from thesmooth halo component. Fig.7showsther −V relationforthepopulation
max max
In this subsection, we analyse the contribution of sub- of subhaloes inside the10 most massive clusters of theMS-
structures to the γ-ray flux of the main haloes using as a II at z = 0, the line styles and colors are the same as in
(cid:13)c 0000RAS,MNRAS000,000–000