aa r X i v : . [ a s t r o - ph ] D ec Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 27 October 2018 (MN L A TEX style file v2.2)
Dark Matter Annihilation in Substructures Revised
Lidia Pieri , , Gianfranco Bertone and Enzo Branchini Istituto Nazionale di Astrofisica - Osservatorio di Padova,Vicolo dell’Osservatorio 5, 35122 Padova, Italy Istituto Nazionale di Fisica Nucleare - Sezione di Padova,Via Marzolo 8, 35131 Padova, Italy Institut d’Astrophysique de Paris, UMR 7095-CNRS,Universit´e Pierre et Marie Curie, 98bis boulevard Arago, 75014 Paris, France Department of Physics, Universit`a di Roma Tre,Via della Vasca Navale 84, 00146, Rome, Italy
27 October 2018
ABSTRACT
Upcoming γ -ray satellites will search for Dark Matter annihilations in Milky Waysubstructures (or ’clumps’). The prospects for detecting these objects strongly dependon the assumptions made on the distribution of Dark Matter in substructures, and onthe distribution of substructures in the Milky Way halo. By adopting simplified, yetrather extreme, prescriptions for these quantities, we compute the number of sourcesthat can be detected with upcoming experiments such as GLAST, and show that,for the most optimistic particle physics setup ( m χ = 40 GeV and annihilation crosssection σv = 3 × − cm s − ), the result ranges from zero to ∼ hundred sources, allwith mass above 10 M ⊙ . However, for a fiducial DM candidate with mass m χ = 100GeV and σv = 10 − cm s − , at most a handful of large mass substructures can bedetected at 5 σ , with a 1-year exposure time, by a GLAST-like experiment. Scenarioswhere micro-clumps (i.e. clumps with mass as small as 10 − M ⊙ ) can be detected areseverely constrained by the diffuse γ -ray background detected by EGRET. Indirect Dark Matter [DM] searches are based on the detec-tion of secondary particles and radiation produced by theself-annihilation of DM particles (Bergstr¨om 2000, Bertoneet al. 2005a).Although the predicted annihilation flux is typically af-fected by large astrophysical uncertainties, the detection ofmultiwavelenght photons, neutrinos or anti-matter from re-gions with high DM density would be of paramount impor-tance for the identification of DM particles. In fact, accel-erator searches of Physics beyond the Standard Model atthe Large Hadron Collider, will not necessarily unveil thenature of DM, even if new particles are discovered, due tothe difficulties associated with the reconstruction of the cos-mological abundance of the newly discovered particles (e.g.Baltz et al. 2006a, Nojiri et al. 2005). At the same time,DM particles could have small enough couplings to nucle-ons, to lead to null searches in direct detection experiments(see e.g. Mu˜noz 2003 and references therein).In the framework of indirect DM searches, severalstrategies have thus been devised, in order to obtain conclu-sive evidence from astrophysical observations. For instance,one could search for peculiar features, such as lines or sharpcut-offs, in the γ -ray spectrum. Although for commonlystudied DM candidates there are no tree level processes fordirect annihilation into photons, loop-level processes to γγ and γZ may produce detectable lines at an energy equal to the DM particle mass (see e.g. Bergstr¨om & Ullio 1997, Ullio& Bergstr¨om 1998, Gounaris et al. 2003, Bergstr¨om et al.2005a). Other spectral features may help distinguishing theDM annihilation signal from ordinary astrophysical sources(Bergstr¨om et al. 2005b, Bergstr¨om et al. 2005c; see alsothe discussion in Baltz et al. 2006b). Alternatively, one cansearch for annihilation radiation from regions characterizedby large concentrations of DM, but very few baryons, suchas DM substructures in the Milky Way [MW hereafter] halo,including dwarf galaxies (Baltz et al. 2000, Tasitsiomi etal. 2003, Pieri & Branchini 2003, Evans et al. 2003, Tyler2002, Colafrancesco et al. 2007, Bergstr¨om & Hooper 2006)and DM mini-spikes around Intermediate Mass Black Holes(Bertone 2006, Bertone et al. 2005b, Horiuchi & Ando 2006,Fornasa et al. 2007, Brun et al. 2007). Finally, DM an-nihilation features can be detected in the energy spectrumand angular distribution of the cosmic γ -ray background(Bergstr¨om, Edsj¨o & Ullio 2001, Ando & Komatsu 2006).In the popular Cold DM scenario, gravitational insta-bilities lead to the formation of a wealth of virialized struc-tures, the DM haloes, spanning a huge range of masses, fromthe largest clusters of galaxies of ∼ M ⊙ down to Earth-size clumps of ∼ − M ⊙ (Green et al. 2004, Green etal. 2005). Although the detectability of individual DM sub-structures, or ”clumps” has been widely discussed in liter-ature, the number of detectable clumps with a GLAST-likeexperiment, at 5 σ in 1 year and for a WIMP DM parti-cle is highly uncertain, ranging from ∼ < c (cid:13) L. Pieri, G. Bertone & E. Branchini et al. 2004) to more than 50 (Baltz 2006b) for large masshaloes, while for microhaloes (i.e. clumps with a mass assmall as 10 − M ⊙ ) the predictions range from no detectableobjects (Pieri et al. 2005) to a large number of detectableobjects, with a fraction of them exhibiting a large propermotion (Koushiappas 2006). The apparent inconsistency ofthe results published so far, is actually due to the differ-ent assumptions that different groups adopt for the physicalquantities that regulate the number and the annihilation”brightness” of DM clumps. In particular, even in the con-text of the benchmark density profile introduced by Navarro,Frenk and White 1996 [NFW], the results crucially dependon the substructures mass function, their distribution withinthe halo host and their virial concentration c ( M, z ) which isa function of mass and of collapse redshift of DM clumps.The paper is organized as follows: in Sec. 2, we describethe model we have adopted for the smooth component ofthe Galactic halo, and introduce the eight different modelsfor DM substructures that will be discussed in the rest ofthe paper. In Sec.3 we estimate the contribution to the γ -ray flux due to the smooth Galactic halo, unresolved DMclumps, and resolved (detectable) clumps. In Sec.4, we studythe prospects for detection of substructures with upcomingexperiments such as GLAST, and in Sec.5 we discuss theresults and present our conclusions. High resolution N-body experiments indicate that a largefraction of the mass within Dark Matter haloes is in the formof virialized subhaloes in all resolved mass scales. Their an-nihilation signal, which adds to that of the smooth Galacticcomponent could be significant (Stoher et al. 2003, Die-mand et al. 2006, Diemand et al. 2007a). A precise model-ing of both the smooth DM distribution (the diffuse galacticcomponent) and the subhalo population within (the clumpycomponent) is therefore mandatory to assess the possibilityof detecting DM subhaloes through their annihilation sig-nal. High resolution numerical simulations enable to studygravitationally bound subhaloes with M SH ≥ − M Halo ,where M Halo is the mass of the host, and therefore cannotresolve substructures in a MW-size halo all the way downto ∼ − M ⊙ . In fact, the smallest substructures of inter-est must be studied within host haloes with mass ∼ . M ⊙ and only at very large redshift (Diemand et al. 2006). At z = 0 and within a MW-host the smallest subhaloes thatwe resolve have masses ≥ M ⊙ (Diemand et al. 2007a)As a consequence the spatial distribution, mass functionand internal structure of Galactic subhaloes can only beinterpolated from the results of several numerical experi-ments spanning a large range of masses and redshifts usingself-similarity arguments. This interpolation is affected bya number of uncertainties that we account for by exploringdifferent models that meet the numerical constraints. The recent “Via Lactea” high resolution simulation (Die-mand et al. 2007a) shows that the density profile of a MW-sized DM halo is consistent to within 10% with the NFW profile that we adopt here: ρ χ ( r ) = ρ s (cid:0) rr s (cid:1) (cid:0) rr s (cid:1) , r > r min ρ χ ( r ) = ρ χ ( r min ) , r ≤ r min (1)where r is the distance from the halo center. This profiledepends on two free parameters, the scale density, ρ s , andthe scale radius, r s , that are related to each other by thevirial mass of the halo, M h . The latter is the mass enclosedin a sphere with radius r vir within which the mean density is200 times above critical. A different definition of r vir wouldnot change the DM profile nor our results. . Finally, we adopta small core radius r min = 10 − kpc.An important shape parameter that characterize thedensity profile is the virial concentration defined as theratio between the virial radius and the scale radius, c ≡ r vir ( M h ) /r s . Theoretical considerations corroborated by nu-merical experiments show that a relation exists between themass of a halo, its collapse redshift, z coll , and concentrationparameter c . The collapse redshift is defined as in Bullock etal. 2001, as the epoch in which a mass scale M h breaks intothe nonlinear regime, i.e. when σ ( M h ) D ( z coll ) ∼
1, where σ ( M h ) is the present linear theory amplitude of mass fluc-tuation on the scale M h and D ( z coll ) is the linear theorygrowth factor at the redshift z coll . The two models proposedby Bullock et al. 2001 [B01] and Eke et al. 2001 [ENS01]are consistent for masses larger than ∼ M ⊙ and in thispaper we use the concentration parameter by B01 to modelthe diffuse Galactic component. On the contrary, for smallermasses ENS and B01 predictions become very different andwe will have to consider both of them to model the subhaloesannihilation signal.Since a sizable fraction of the mass in the MW is inform of virialized subhaloes, there is not a unique way todetermine ρ s , and r s of the host MW halo. For this rea-son we adopted two different procedures. In the first onewe have computed ρ s , and r s as if the total mass of the sys-tem, including the clumpy component, were in fact smoothlydistributed in the Galactic halo. In the second we have con-sidered the total mass of the system to determine the con-centration parameter but have used the diffuse componentalone to relate ρ s to r s . Having checked that both proce-dures give similar predictions for the probability of subhalodetection (Section 3), in the following we will only discussmodels based on the second approach. For the Milky Way wehave used a virial mass M h = 10 M ⊙ and a concentrationparameter c vir ∼ . To account for the presence of a population of DM subhaloesand investigate their effect on the annihilation signal, weneed to specify their mass spectrum, spatial distribution anddensity profile. c (cid:13) , 000–000 ark Matter Annihilation in Substructures Revised Subhalo ModelsModel c ( M ) z coll c name > M ⊙ − M ⊙ − M ⊙ − M ⊙ B ref,z B01 B01 0 63B z B01 DMS05 0 80B z , σ B01 DMS05-5 σ ref,z c B01 B01 B01 ∼ . z c B01 DMS05 DMS05 ∼ z c , σ B01 DMS05-5 σ DMS05 ∼ . z ENS01 DMS05 0 80ENS z c ENS01 DMS05 DMS05 ∼ Table 1.
Halo model parameters. Column 1: Model name. Col-umn 2: Halo density profile for
M > M ⊙ . Column 3. Halodensity profile benchmark for M = 10 − M ⊙ , used as normaliza-tion. Column 4: Collapse redshift model, used as normalization.Column 5: Concentration parameter at 10 − M ⊙ . For all the mod-els we have adopted an NFW profile. High resolution N-body experiments show that the massfunction of both isolated field haloes and subhaloes is wellapproximated by a power lawd n ( M ) / dln( M ) ∝ M − α , (2)with α = 1, independently of the host halo mass, overthe large redshift range z = [0 , M = [10 − , ] M ⊙ (Jenkins et al. 2001, Moore et al.2001 Diemand et al. 2004, Gao et al. 2005, Reed et al.2005, Diemand et al. 2006). Self-similarity is preserved atthe present epoch down to the smallest masses if subhaloessurvive gravitational disturbances during early merger pro-cesses and late tidal disruption from stellar encounters.Analytical arguments have been given against (Zhao etal. 2005) or in support of this hypothesis (Moore et al. 2005,Brezinsky et al. 2006). Moreover, currently resolved massfunctions in numerical experiments suffer from dynamicalfriction at the high mass end which could steepen the halomass function. Since changing the halo mass function slopemight have a non-negligible impact on our analysis, we adopta power-law index α = 1 as a reference case but also exploretwo shallower subhalo mass functions with α = 0 . α =0 .
95. All plots in this paper refer to the reference case α = 1and discuss the effect of adopting shallower slopes in thetext.As far as the spatial distribution of subhalos inside ourGalaxy is concerned, we follow the indications of the numer-ical experiment of Reed et al. 2005 and assume that thesubhalo distribution traces that of the underlying host massfrom r vir and down to a minimum radius, r min ( M ), withinwhich subhaloes are efficiently destroyed by gravitationalinteractions. We explicitly assume spherical symmetry andwe ignore the possibility, indicated by some numerical ex-periments, that the radial distribution of subhalos might bemore extended than that of the dark matter. Folding these indications together we model the numberdensity of subhaloes per unit mass at a distance R from theGC as: ρ sh ( M, R ) = AM − θ ( R − r min ( M ))( R/r
MWs )(1 +
R/r
MWs ) M − ⊙ kpc − , (3)where r MWs is the scale radius of our Galaxy and the ef-fect of tidal disruption is accounted for by the Heavisidestep function θ ( r − r min ( M )). To determine the tidal ra-dius, r min ( M ), we follow the Roche criterion and computeit as the minimum distance at which the subhalo self gravityat r s equals the gravity pull of the halo host computed atthe orbital radius of the subhalo. As a result r min ( M ) is anincreasing function of the subhalo mass, implying that nosubhaloes survive within r min (10 − M ⊙ ) ∼
200 pc.To normalize eq. 3, we again refer to numerical simula-tions that show that [5-10]% of the MW mass is distributedin subhaloes with masses in the range 10 − M ⊙ (Die-mand et al. 2005, Diemand et al. 2007) In the followingwe use the optimistic value of 10 % and note that assuming5%, instead, would decrease the probability of subhalo de-tection by a factor 2. With this normalization about 53% ofthe MW mass is condensed within ∼ . × subhaloeswith masses in the range [10 − , ] M ⊙ , whose abundancein the solar neighborhood is remarkably high ( ∼
100 pc − ).The remaining 47% constitutes the diffuse galactic compo-nent that is assumed to follow a smooth NFW profile. Wedo not account here for the presence of mass debris streamsresulting from tidal stripping since these structures, char-acterized by a mild density contrast, would not contributesignificantly to the annihilation flux. Finally, we need to specify the density profile for the sub-structures. Constraints from numerical models only appliesto limited mass ranges at very different epochs. At z = 0 Die-mand et al. 2007b find that the velocity profile of Galacticsubhaloes above 4 × M ⊙ in the “Via Lactea” simulationare well fitted by the NFW model. This result is also validfor the much smaller substructures with masses in the range[10 − , × − ] M ⊙ that populate a parent halo of 0 . M ⊙ at z = 86 (Diemand et al. 2006). A large fraction of thesesmall substructures do not survive the early stage of hierar-chical merging and late tidal interaction with stellar encoun-ters (Zhao et al. 2005). The ∼ survivors suffer fromsignificant mass loss. Presumably this modifies their originalNFW density profile that, however, seems to be preserved inthe innermost region where most of the annihilation signaloriginates from (Kazantzidis et al. 2004).Yet these constraints from numerical experiments donot uniquely define the subhalo density profiles. Therefore,instead of relying on a single model profile we will exploreseveral of them in an attempt of bracketing the theoreticaluncertainties.All models that we have considered, and that are listedin Table 1, assume that subhaloes have the same NFW den-sity profile as their massive host but with different concen-tration parameters. These models have been flagged withthe following prescriptions: c (cid:13) , 000–000 L. Pieri, G. Bertone & E. Branchini • B -models assume the c ( M ) relation of B01 for M > M ⊙ ; • ENS -models, use the ENS01 model for
M > M ⊙ ; • In models flagged as z the low mass extrapolation ofthe concentration parameter is normalized to that of fieldhaloes of 10 − M ⊙ measured in the numerical simulation ofDiemand et al. 2005 (DMS05) at z = 26 and linearly ex-trapolated at z = 0, i.e. c (10 − M ⊙ , z = 0) = c (10 − M ⊙ , z =26) × (1 + 26). The underlying assumption is that, as inthe Press-Schechter approach, all existing haloes have justformed, and the (1 + 26) scaling is required to account forthe change in the mean density between z = 26 and z = 0; • Models flagged as z c assume instead that, once formed,subhaloes that survive to z = 0 do not change their densityprofile. Therefore for each subhalo of mass M we determineits collapse redshift z c (defined as in B01 for M > M ⊙ )and compute its concentration parameter accordingly, as c ( M, z = z c ) = c ( M, z = 0) / (1 + z c ), where c ( M, z = 0)is computed as in the z z
0, rescaled for the smallestmasses collapse redshift z c ( M = 10 − M ⊙ ) = 70 suggestedby DMS05 and used by Koushiappas 2006. In Fig. 1 we showthe collapse redshift as a function of the mass adopted in the z c -models (thin line); • Models flagged 5 σ assume that all existing subhaloeswith mass M = 10 − M ⊙ form at the 5 σ peaks of the densityfield, which has the effect of increasing the normalization to c (10 − M ⊙ , z = 0) = 400; • Models flagged as ref, z ref, zc are identical tothe corresponding z zc models for M > M ⊙ . In-deed, they use a na¨ıve extrapolation of the B01 model atlow masses, both for the collapse redshift (thick line in Fig.1) and for the concentration parameter (filled circles in Fig.2). Though this extrapolation is not supported by numericalsimulations, it intuitively reflects the theoretcal flattering ofthe σ ( M ) curve at low masses.The c ( M ) profile for each of the z c ( M ) relation is notdeterministic. Instead, for a fixed mass, the probability of agiven value for c ( M ) is well described by a lognormal distri-bution P ( c ( M )) = 1 √ πσ c ¯ c ( M ) e − (cid:0) ln( c ( M )) − ln(¯ c ( M ))4 σc (cid:1) , (4)where the mean ¯ c = c ( M ) is the concentration parameter ofB01 or ENS and the dispersion σ c = 0.24 does not dependon the halo mass (B01). In the following, we include thislognormal scatter in all models described in Table 1.We will not consider in this paper the possibility thatsubhaloes might contain sub-substructres (Strigari et al.2007, Diemand 2007a) and that their concentration (andscatter) might depend on the distance from the GalacticCentre (Diemand et al. 2007b). Indeed, such features arefound in numerical simulations capable of resolving largesub-haloes ( M > M ⊙ ) but there is no evidence whetherthey also apply to much smaller haloes, with masses downto 10 − M ⊙ . Figure 1.
Collapse redshift, z c , as a function of the halo massfor model B01 (points) for M > M ⊙ . The thin line shows theextrapolation at low masses normalized to the DMS05 value for z c (10 − M ⊙ ). The thick line shown a na¨ıve extrapolation of theB01 model at low masses. γ -RAY FLUX FROM ANNIHILATION IN DMCLUMPS The photon flux from neutralino annihilation in galactic sub-haloes, from a direction in the sky making an angle ψ fromthe Galactic Center (GC), and observed by a detector withangular resolution θ , can be factorized into a term dependingonly on particle physics parameters, d Φ PP /dE γ and a termdepending only on cosmological quantities, Φ cosmo ( ψ, θ ): d Φ γ dE γ ( E γ , ψ, θ ) = d Φ PP dE γ ( E γ ) × Φ cosmo ( ψ, θ ) (5) The first factor of Eq.5 can be written as: d Φ PP dE γ ( E γ ) = 14 π σ ann v m χ · X f dN fγ dE γ B f (6)where m χ denotes the Dark Matter particle mass and dN fγ /dE γ is the differential photon spectrum per annihi-lation relative to the final state f , with branching ratio B f . Although the nature of the DM particle is unknown,we can make an educated guess on the physical parame-ters entering in the above equation. The most commonlydiscussed DM candidates are the so-called neutralinos, aris-ing in Super-symmetric extensions of the Standard Modelof particle physics [SUSY], and the B (1) particles, first ex-citation of the hypercharge gauge boson in theories withUniversal Extra Dimensions [UED] (see Bergstr¨om 2000,Bertone et al. 2005b and references therein). Typical val-ues for the mass of these candidates range from ∼
50 GeVup to several TeV. The annihilation cross section can be as c (cid:13) , 000–000 ark Matter Annihilation in Substructures Revised Figure 2.
Concentration parameters as a function of halomass at z = 0 computed for the ENS z (solid), B z (dotted)and the B z , σ (dashed) model described in the text. Filled cir-cles show the B01 model na¨ıvely extrapolated at low masses(B ref,z model). Figure 3.
Exclusion plot in the ( σ ann v, m χ ) plane. The solid linecorresponds to our best case particle physics scenario, adoptedfor all the z0 models. Models above the dashed (long dot-dashed,dot-dashed) curve violate the EGRET extra galactic background(EGB) flux constraints in scenario ENS z c (B z c ,B z c , σ ) for a massfunction slope α = 1. For comparison, we show with filled (empty)dots a scan of SUSY models with relic density within 2 (5) stan-dard deviations from the WMAP+SDSS value. The dotted linecorresponds to the UED model. Supersymmetric models were ob-tained using the DARKSUSY package (Gondolo et al. , 2004) Figure 4. Φ cosmo as a function of the angular distance fromthe Galactic center, for the MW smooth component (solid thickline) and for unresolved clumps in model B z c , σ (solid thin line),B z c (dot-dashed), ENS z c (long dot-dashed), B z , σ (long dashed),B ref,z c (open circles), B z (dotted), B ref,z (filled circles) andENS z (dashed). The mass function slope is α = 1. high as σ ann v = 3 × − cm s − , as appropriate for ther-mal relics that satisfy the cosmological constraints on thepresent abundance of Dark Matter in the Universe. How-ever, we note that the annihilation cross section can be muchsmaller, as the appropriate relic density can be achievedthrough processes such as co-annihilations (Bergstr¨om 2000,Bertone et al. 2005a). This can be seen from Fig. 3 thatshows the range of σ ann v and m χ allowed in the UED andSUSY models: solid (empty) circles correspond to modelswith relic density within 2 (5) standard deviations fromthe WMAP+SDSS suggested value Ω DM h = 0 . +0 . − . ,where as usual Ω DM is the DM density in units of the crit-ical density, and h is the Hubble parameter in units of 100km s − kpc − , h = 0 . +0 . − . (Tegmark et al. 2006).In order to optimize the prospects for detection, weadopt here a very low value for the particle mass, m χ = 40GeV, together with a high annihilation cross section σ ann v =3 × − cm s − . As for the nature of the DM particle, weassume here a 100% branching ratio in b ¯ b and the dN b ¯ bγ /dE γ functional form of Fornengo et al. 2004. The results can berescaled for any other candidate, although in most cases thephoton spectrum arising from annihilations yields similarresults. The contribution of unresolved substructures to the annihi-lation signal is given byΦ cosmo ( ψ, ∆Ω) = Z M dM Z c dc Z Z ∆Ω dθdφ Z l . o . s dλ c (cid:13) , 000–000 L. Pieri, G. Bertone & E. Branchini [ ρ sh ( M, R ( R ⊙ , λ, ψ, θ, φ )) × P ( c ) ×× Φ cosmo halo ( M, c, r ( λ, λ ′ , ψ, θ ′ , φ ′ )) × J ( x, y, z | λ, θ, φ )] (7)where ∆Ω is the solid angle of observation pointing in thedirection of observation ψ and defined by the angular res-olution of the detector θ ; J ( x, y, z | λ, θ, φ ) is the Jacobiandeterminant; R is the galactocentric distance, which, insidethe cone, can be written as a function of the line of sight ( λ )and the solid angle ( θ and φ ) coordinates and the pointingangle ψ through the relation R = p λ + R ⊙ − λR ⊙ C ,where R ⊙ is the distance of the Sun from the Galactic Cen-ter and C = cos( θ ) cos( ψ ) − cos( φ ) sin( θ ) sin( ψ ); r is theradial coordinate inside the single subhalo located at dis-tance λ from the observer along the line of sight defined by ψ and contributing to the diffuse emission.The expressionΦ cosmo halo ( M, c, r ) =
Z Z ∆Ω dφ ′ dθ ′ Z l . o . s dλ ′ (cid:20) ρ χ ( M, c, r ( λ, λ ′ , ψ, θ ′ φ ′ )) λ J ( x, y, z | λ ′ , θ ′ φ ′ ) (cid:21) ; (8)describes the emission from such a subhalo. Here, ρ χ ( M, c, r )is the Dark Matter density profile inside the halo.By numerically integrating Eq. 7, we estimate the con-tribution to Φ cosmo from unresolved clumps in a 10 − sr solidangle along the direction ψ , for each substructure model con-sidered. The result is shown in Fig. 4. In the same figure,the solid thick line corresponds to the contribution from theMW smooth halo component described in Sec.2.1 , which iscomputed according to Eq.8, with the distance to the ob-server λ = R ⊙ .We summarize the properties of the smooth subhalocontribution in Table 2. In the second and third column weshow, for each model, the contribution to Φ cosmo in unitsof GeV cm − kpc sr towards the Galactic center ( ψ = 0 ◦ )and the angle ψ d beyond which the smooth subhalo contri-bution starts dominating over the MW halo foreground. Inthe fourth column we show the boost factor for each model,computed as the ratio of the integral over the MW volume ofthe density squared including subhaloes to the same integralfor the smooth MW only: b = R MW dV ( ρ MWsmooth + ρ sh ) R MW dV ρ MWsmooth (9)To set the two remaining parameters that determinethe intensity of the annihilation flux, namely the particlemass m χ and the annihilation cross section σ ann v we adoptthe most optimistic combination allowed by the SUSY andUED models shown in Fig. 3 that do not not exceed thecurrent EGRET upper limits for the annihilation flux above3 GeV and within a solid angle of 10 − sr. The latter re-ceives contribution from two distinct components: the firstone is of Galactic origin, dominates for ψ < ◦ and is char-acterized by a power-law photon spectrum, that leads, uponextrapolation at high energies (Bergstr¨om et al. 1998) tothe following parametrization dφ gal − γ diffuse d Ω dE = N ( l, b ) 10 − E − . γ γ cm s sr GeV , (10)where l and b are the galactic latitude and longitude. Thenormalization factor N depends only on the interstellar Figure 5.
Number of photons above 3 GeV, in 1 year in asolid angle of 10 − sr, as a function of the angle ψ from theGC. From top to bottom, the lines correspond to the B z c , σ ,B z c , ENS z c , B z , σ , B z and ENS z model. Empty (filled) circlesshow the B ref,z c (B ref,z ) model. σ ann v = 3 × − cm s − , m χ = 40 GeV and BR b ¯ b = 100% have been used, correspondingto the best value Φ PP = 2 . × − cm kpc − GeV − s − sr − .The solid thick line shows the EGRET diffuse expected Galacticand extragalactic background computed along l = 0. The massfunction slope is α = 1. Results for subhalo modelsModel Φ cosmo0 ψ d b Φ PP − B ref,z . × − z . × − . z , σ . × − . ref,z c . × − z c . × − . z c , σ . × − . z . × − . z c . × − . Table 2.
Results for the halo models. Column 1: Modelname. Column 2: Φ cosmo [ GeV cm − kpc sr] toward theGalactic Center. Column 3: Angle at which the subhalodiffuse contribution dominates over the MW smooth fore-ground [degrees]. Column 4: boost factors. Column 5: Φ PP − [10 − cm kpc − GeV − s − sr − ]. The value for the z0 and theB ref,z c models correspond to our best case particle physics sce-nario. Values for the zc models except the B ref,z c are normalizedto EGRET data. c (cid:13) , 000–000 ark Matter Annihilation in Substructures Revised matter distribution, and is modeled as in Bergstrom et al.1998.The second one is extragalactic, dominates at ψ > ◦ ,and for its photon spectrum we use an extrapolation fromlow energy EGRET data, following Sreekumar et al. 1998: dφ extra − γ diffuse d Ω dE = 1 . × − E − . γ cm s sr GeV . (11)The thick line in Fig. 5 shows the EGRET photonflux as a function of ψ . The EGRET flux is computed ac-cording to Eqs. 10 and 11 extrapolated above 3 GeV,within a field of view of 10 − sr in 1 year of observation.The flux is computed along l = 0, that is away from theGalactic plane, where the EGRET flux is minimum. TheGalactic and extragalactic contributions are clearly visi-ble. The other curves show the predictions for all mod-els in Table 1, obtained when using our best case particlephysics scenario described in Sec. 3.1 that corresponds toΦ PP = 2 . × − cm kpc − GeV − s − sr − . All the valuesof σ ann v and m χ that would result in the same Φ PP arerepresented with a solid line in Fig.3.With this choice of parameters all models flagged with zc (but the B ref,z c ), for which the halo properties are com-puted at the collapse redshift, exceed experimental dataclose to the GC and for ψ > ◦ , where the EGRET flux isassumed to have an extragalactic origin. We do not regardthe mismatch at small angles as significant, because of thelimited angular resolution of the original EGRET data. Onthe contrary, we decrease the values of Φ PP to bring the zc -models into agreement with data at large angles from theGC. The values of Φ PP adopted in each model are listed inthe last column of Table 2. We note that a smaller Φ PP cor-responds to assuming a larger particle mass or a smallercross section. The [ σ ann v, m χ ] phase space parameter al-lowed for SUSY (circles) or UED (dotted line) models isshown in Fig. 3. In the same figure the three dashed anddot-dashed lines correspond to the EGRET constrained val-ues of Φ PP for the zc models. The particle physics modelsabove the corresponding line are thus excluded by EGRETdata. Besides the diffuse signal produced by annihilation in boththe subhalo population and the smooth MW component, weconsider here the contribution from individual subhaloes,that we regard as Poisson fluctuation of the underlyingmean distribution of subgalactic haloes that could be de-tected as isolated structures. To estimate their flux we con-sider 10 independent Monte Carlo realizations of the clos-est and brightest subhaloes, in a cone of ∼ ◦ pointingtoward the Galactic Center. To do this we generate, foreach mass decade, the positions of those subhaloes that haveΦ cosmo > h Φ cosmo B z ( ψ = 50 ◦ ) i ∼ × − GeV cm − kpc sr, where the brackets indicate the mean annihilation flux. If N <
100 such objects are found, then we still include theremaining 100 − N nearest subhaloes in that mass range.Adding contribution from an increasing number of indi-vidual haloes monotonically increases the chance of subhalodetection within the angular resolution element of the de-tector. To check whether our procedure is robust and thenumber of detectable haloes has converged we reduced the number of Monte Carlo-generated haloes by 70 % and foundthat probability of subhalo detection indeed remains con-stant.To summarize, for each model, the total contribution toΦ cosmo is given by the sum of three terms: the diffuse contri-bution coming from unresolved haloes, corresponding to themean contribution of the clumpy component computed withEq.7, the contribution of the diffuse smooth Galactic com-ponent and that of individual nearby subhaloes, both com-puted using Eq.8. Figs.6 and 7 show the three contributionsto Φ cosmo for the models B ref,z c and ENS z , respectively.In each figure, the contribution from unresolved clumps isshown in the upper left panel, the one from the diffuse MWin the upper right, the one from resolved clumps in the lowerleft, and the sum of all contributions in the lower left panel.The smooth MW halo contribution falls rapidly with thedistance from the GC, while the diffuse subhalo contribu-tion keeps a high value even at large angular distances. Thesingle halo contribution is almost completely hidden by theoverwhelming diffuse foreground, while it is nicely resolvedas a standalone component.The same procedure described in this section for a conepointing to the GC has been repeated for two other regions:a cone pointing to the Galactic anticenter and a cone tothe Galactic pole b = 90 ◦ . The purpose is to compute theannihilation flux and to evaluate the number of detectablesubhalos over the whole sky. In absence of strong features in the annihilation spectrum,the best chances to detect the annihilation signal within ourGalaxy is to observe some excess on the γ -ray sky either dueto diffuse emission or to resolved sources that have no astro-physical counterpart. However, the requirements for signaldetection are different in the two cases since the smoothannihilation flux, that contributes to the signal in the firstcase, adds to the noise in the second.To determine the probability of halo detection, we con-sider a 1 year effective exposure time performed with aGLAST-like satellite. We note that, given the 2.4 sr field ofview and the all-sky survey mode of GLAST, such an expo-sure will be achieved in about 5 years of actual observationtime.The prospects for detecting γ -rays from DM annihila-tions are evaluated by comparing the number n γ of expectedsignal photons to the fluctuations of background events n bkg .To this purpose we define the sensitivity σ as: σ ≡ n γ √ n bkg (12)= √ T δ ǫ ∆Ω R A eff γ ( E, θ i )[ dφ signal γ /dEd Ω] dEd Ω qR P bkg A effbkg ( E, θ i )[ dφ bkg /dEd Ω] dEd Ωwhere T δ defines the effective observation time and φ bkg isthe background flux. The quantity ǫ ∆Ω is the fraction of sig-nal events within the optimal solid angle ∆Ω correspondingto the angular resolution of the instrument and is usually ∼ .
7. We set it equal to 1 to get the most optimistic val-ues. The effective detection area A eff for electromagnetic or c (cid:13) , 000–000 L. Pieri, G. Bertone & E. Branchini
Figure 6.
Map of Φ cosmo (proportional to the annihilation signal) for the B ref,z c model, in a cone of 50 ◦ around the Galactic Center, asseen from the position of the Sun. Upper left: smooth subhalo contribution from unresolved haloes. Upper right: MW smooth contribution.Lower left: contribution from resolved haloes. Lower right: sum of the three contributions. hadronic particles is defined as the detection efficiency timesthe geometrical detection area. In the following we makethe realistic assumption that all hadronic particles will beidentified, so that the background will be composed by pho-tons only. We also assume that all photons will be correctlyidentified, which is somehow optimistic, since there will bea small amount (a few percent) of irreducible backgrounddue, e.g., to the backsplash of high energy photons.We use A eff = 10 cm , independent from the energy E andthe incidence angle θ i , and an angular resolution of 0 . ◦ .Both these values are rather optimistic, since the expectedGLAST angular resolution approaches 0 . ◦ only at about 20GeV, while the on-axis effective area is quoted to be max-imum ∼ × cm above 1 GeV and decreases by ∼ ◦ .As anticipated, different annihilation signals need to becompared with different background noises. For the detec-tion of the diffuse annihilation flux the background is con-tributed both by Galactic (Eq. 10) and extragalactic astro-physical sources (Eq. 11) measured by EGRET. In the caseof individual subhaloes, the annihilation photons producedin the smooth Galactic halo and in the unresolved clumpycomponent contribute to the background rather than to thesignal. We first study the sensitivity σ of such a GLAST-like obser-vatory to the annihilation flux from the smooth DM profileand from the diffuse contribution of unresolved subhaloes.Both signals are computed above 3 GeV, and the astrophys-ical background noise is obtained from Eqs.10 and 11 spec-ified along l = 0.The result is shown in Fig.8, where we plot the statisti-cal significance of the detection as a function of ψ for each ofthe models listed in Table 2.2. 1 σ detections of the annihila-tion signal is expected at ψ < ◦ for all the models labelled zc . The chances of observing the diffuse annihilation flux aresignificantly higher in the direction of the Galactic Centeralong which models labelled z0 predict a signal detectabil-ity as high as 5 σ . Yet, these predictions should be takenwith much care since the measured astrophysical γ -ray fluxabove 3 GeV in the direction of the GC, which constitutesthe background, is known with large uncertainties. Subhaloes can also be detected through the annihilation fluxproduced by individual, nearby clumps that would appear as c (cid:13) , 000–000 ark Matter Annihilation in Substructures Revised Figure 7.
Same as in Fig. 6 for the ENS z model. bright, possibly extended, sources, as shown in the bottomright panels of Figs. 6 and 7. In this case the signal is pro-duced within the individual haloes of our Monte Carlo real-izations, while the background is contributed by the smoothastrophysical background plus the diffuse annihilation fluxproduced by the Galaxy and its subhaloes.For each halo in the 10 Monte Carlo realizations, and foreach virial concentration model, we • assign to it an arbitrary concentration parameter c ( M ); • calculate the annihilation signal; • find the value of the concentration parameter that guar-antees a 5 σ detection in 1 year exposure time, c σ ( M ); • identify the probability of detection of the clump withthe probability P ( > c σ ) that such a clump has a concentra-tion as high as c σ ( M ), assuming the lognormal distributiondescribed in Eq.4.The total number of detectable subhaloes is then sim-ply given by P i P i ( > c σ ), where the sum is performed overall haloes in the realization. Results are obtained by aver-aging over all over the 10 Monte Carlo realizations and theprocedure is repeated for all models listed in Table 1.The number of haloes that can be detected in 1 yearwith a significance above 5 σ in cone of view with angularopening of 50 ◦ towards the GC is shown in Fig.9 for the z0 models and in Fig. 10 for the zc models, as a function of thesubhaloes mass. Had we assumed a deterministic relationfor c ( M ) the number of events would have decreased by afactor ∼ b = 90 where these twoeffects interplay in a most favourable way for the detection.Table 3 lists the number of haloes that can be detectedwith a significance larger than 5 σ in a cone of 50 ◦ aroundthe Galactic Center (first column), the Galactic pole (sec-ond column) and the Galactic anticenter (third column), foreach model, for our reference mass function slope α = 1 andthe Φ PP values listed in Table 2. Hereafter, each error is thestandard deviation obtained averaging over the 10 MC rep-resentations. The effect of decreasing the number of haloesfar from the GC is compensated by the lower foreground dueto the diffuse subhalo contribution to the annihilation flux.The best compromise is found around the Galactic poles.In Table 4 we show the total number of haloes that canbe detected with a GLAST-like satellite in the whole skywith a significance larger than 5 σ , with a mass functionslope α = 1 (first column) and Φ PP values listed in Table 2. c (cid:13) , 000–000 L. Pieri, G. Bertone & E. Branchini
Figure 8.
Statistical significance, as a function of the anglefrom the GC ψ , for the detection of the DM annihilation fluxfrom diffuse subhaloes plus the MW smooth component (along l = 0), for the different models explored: B z (dotted line ),ENS z (short dashed), B ref,z (filled circles), B ref,z c (empty cir-cles), B z , σ (long dashed), B z c (long dot-dashed), ENS z c (dot-dashed), B z c , σ (solid). The mass function slope is α = 1. Werefer to Table 2 for the values of Φ PP used in this figure.Number of detectable haloes ( α = 1)Model N σGC N σ N σ B ref,z . ± .
45 0 . ± .
43 0 . ± . z . ± .
45 0 . ± .
43 0 . ± . z , σ . ± .
34 0 . ± .
34 0 . ± . ref,z c . ± .
60 23 . ± .
28 18 . ± . z c . ± .
92 2 . ± .
82 2 . ± . z c , σ . ± .
05 0 . ± .
08 0 . ± . z . ± .
12 0 . ± .
10 0 . ± . z c . ± .
40 0 . ± .
37 0 . ± . Table 3.
Number of haloes detectable, at 5 σ in 1 year of effec-tive observation with a GLAST-like satellite, in a 50 ◦ f.o.v. conetowards the GC (column 1), the Galactic pole (column 2) andthe anticenter (column 3). The subhaloes mass function slope is α = 1. We refer to Table 2 for the values of Φ PP used in thistable. The remaining two columns show the effect of adopt-ing a mass function with power-law index α = 0 .
95 (sec-ond column) and α = 0 . Total number of detectable haloesModel N σtot ( α = 1) N σtot ( α = 0 . N σtot ( α = 0 . ref,z . ± .
00 3 . ± .
30 3 . ± . z . ± .
97 3 . ± .
30 3 . ± . z , σ . ± .
09 3 . ± .
17 3 . ± . ref,z c . ± .
96 132 . ± .
15 125 . ± . z c . ± .
67 104 . ± .
78 119 . ± . z c , σ . ± .
56 10 . ± .
36 96 . ± . z . ± .
89 0 . ± .
58 0 . ± . z c . ± .
48 23 . ± .
17 30 . ± . Table 4.
Total number of haloes detectable over the whole sky, at5 σ in 1 year of effective observation with a GLAST-like satellite,for a mass function slope α = 1 (column 1), α = 0 .
95 (column2) and α = 0 . PP used in this table. whose main contribution is given by small sub-haloes, de-pends on the model explored.As expected, we can observe how this effect is largerfor those models whose overall contribution to Φ cosmo islarger, that is for those models whose concentration param-eters have been computed at the collapse redshift. The effectis reduced for the other models, as well as for the B ref,z c one,for the following reason: when using α = 1, the Φ PP valuefor the zc models (but the B ref,z c one) has been decreasedin order to respect the EGRET EGB limit, while when us-ing α = 0 . PP . This is also true for α = 0 .
95, except for the B z c , σ model, where we have to useΦ PP = 0 .
84 cm kpc − GeV − s − sr − . In fact, the z ref,z c model experience just a minor increase (compati-ble within the error bars) of the number of detectable halos;the B z c and ENS z c models reach this stability for α ≤ . PP allowed value gets our best value; theB z c , σ model keeps on showing a large effect when changingmass function slope, because it is allowed to have the bestvalue Φ PP only when α = 0 . ref,z c model, we expect that aGLAST-like experiment could detect ∼ −
130 subhaloeswith masses above 10 M ⊙ over all sky, for all the massfunction slopes considered in this analysis. In all the modelswhose concentration parameters are computed at z = 0 thenumber of detectable events is compatible with zero withinthe errors, whatever slope is used. Accordingly to the afore-mentioned discussion, the effect of changing the mass func-tion slope is dramatic in the B z c (B z c , σ ) model, for whichthe total number of events ranges from ∼
10 ( ∼
0) for α = 1to ∼
120 ( ∼ α = 0 .
9. A large effect ( ∼ ∼ z c model too. The prospects for detecting γ -rays from the annihilation ofDM particles in substructures of the MW have been inves- c (cid:13) , 000–000 ark Matter Annihilation in Substructures Revised tigated by a number of authors (e.g. Stoehr et al. 2003,Pieri & Branchini 2004, Koushiappas et al. 2004, Odaet al. 2005, Pieri et al. 2005). In this work we confirmthat substructures can provide a significant contributionto the expected Galactic annihilation signal, although theactual enhancement depends on the assumptions made onthe clump properties, which are affected by large uncertain-ties. Indeed, given the assumed substructure mass function dN/dM ∝ M − , the contribution to the total γ -ray fluxby subhaloes of different masses depends on the annihila-tion signal produced within each clump which is dictated bythe internal structure. Numerical experiments have shownthat the total annihilation signal is dominated by the high-est mass subhaloes both in a Galactic halo at z=0 (Stoehret al. 2003) and in 0.1 M ⊙ halo host at z=75 (Diemand etal. 2006). However, the recent results of the high resolution’Via Lactea’ simulation (Diemand et al 2007a) indicate thatthe annihilation luminosity is approximately constant perdecade of substructure mass while analytical calculations(Colafrancesco et al. 2006) tend to find that the signal isdominated by small mass subhaloes. Indeed this is also thecase with our model predictions. Fig. 11 shows the expectedcontribution of the unresolved haloes to the total annihila-tion flux as a function of the subhalo mass, integrated oneach mass decade. In all the models explored the annihila-tion signal is dominated by the smallest clumps, as a resultof the decrease of the virial concentration with the subhalomass, as shown in Fig. 2. Under optmistic assumptions onthe particle physics parameters of DM particles, a GLAST-like experiment might detect such a DM annihilation flux,but only in a few pixels around the Galactic Center.It should be noticed, however, that estimates of the an-nihilation signal from the Galactic Center are affected bythe poor knowledge of the DM profile in the innermost re-gions of the Galaxy, which is usually obtained by extrap-olating over many orders of magnitude the results of nu-merical simulations. The presence of a Supermassive BlackHole at the center of the Galaxy makes things even morecomplicated, as it may significantly affect the distributionof DM within its radius of gravitational influence , leadingto the formation of an overdensity called ”spike” (Gondoloand Silk, 1999). Spikes require however rather fine-tunedconditions to form (Ullio et al. 2002) and any overdensity isin any case severely suppressed by the interaction with starsand DM self-annihilations (Merritt et al. 2002, Bertone &Merritt 2005).In alternative, one could look for an annihilation signalfrom individual DM substructures, such as dwarf galaxiesor even smaller, ’baryon-less’, clumps. We have shown that,depending on the assumptions made on the properties ofclumps, only large haloes with M > M ⊙ can be detectedand identified with a GLAST-like experiment, which is con-sistent with the analyses of Stoher et al. 2003 and Koushi-appas et al. 2004. The number of detectable haloes rangesfrom 0 to more than a hundred, depending on the model.Adopting a shallower subhalo mass function increases thenumber of detectable subhaloes in those models in which thediffuse annihilation signal is dominated by the unresolved,low mass haloes.In any case, scenarios leading to a large number ofdetectable small-scale clumps appear to be severely con-strained by the γ -ray background measured by EGRET. Figure 9.
Number of events detectable in 1 year a 5 σ with aGLAST-like experiment in a 50 degrees cone towards the GC forthe models ENS z (dot-dashed), B z (solid), B z , σ (dashed) andB ref,z (long dot-dashed), assuming the lognormal distributionfor the concentration parameter. We refer to Table 2 for the usedvalues of Φ PP . The mass function slope is α = 1. Figure 10.
Number of events detectable in 1 year a 5 σ with aGLAST-like experiment in a 50 degrees cone towards the GC forthe models B ref,z c (dashed) ENS z c (solid), B z c (long dot-dashed)and B z c , σ (dot-dashed), assuming the lognormal distribution forthe concentration parameter. We refer to Table 2 for the usedvalues of Φ PP . The mass function slope is α = 1.c (cid:13) , 000–000 L. Pieri, G. Bertone & E. Branchini
In particular, the model of Koushiappas 2006 is similarto our B z c model, as far the cosmological term is considered,while the particle physics contribution corresponds to ourbest case scenario Φ PPB z . Although nearby haloes would bebright, and observable, in this case, we have shown that theassociated diffuse emission produced by all the remaining,unresolved, clumps in the Milky Way, would far exceed the γ -ray background measured by EGRET. The chances to de-tect the proper motion of clumps are thus very low, as thelowest mass detectable subhaloes, (M=10 M ⊙ ), are typi-cally found at a distance greater than 0 . ∼ . ′ yr − , well below the GLASTangular resolution of a few arcminutes.We have made use of simplified and extreme scenar-ios for the subhaloes concentration parameter models. Moreaccurate scenarios, though not supported by numerical sim-ulations for small mass haloes, could lead to different dif-fuse foreground levels and to both more or fewer detectablehaloes.One may wonder why we preferentially expect to indi-vidually detect the more massive subhaloes, while the un-resolved annihilation signal is mainly contributed by smallmass clumps. The reason is that the volume over whichindividual haloes can be detected decreases rapidly withthe halo mass. To see this, let us consider the maximumdistance D MAX at which a clump can be detected. Thisdistance depends on the halo luminosity which, in turns,depends on the halo mass and concentration. For a NFWprofile: D MAX ∝ M . c ( M ) . . On the other hand, as dis-cussed e.g. by Koushiappas 2006, given a subhalo mass func-tion dN/dM ∝ M − , the number of detectable haloes permass decade is: dN/dLog ( M ) ∝ D M − ∝ M . c ( M ) . .Assuming a simple scale-free virial concentration c ( M ) ∝ M − γ , we see that if γ > γ th = − /
9, then the numberof detectable haloes per mass decade halo indeed increaseswith the subhalo mass. The c ( M ) relation for some of ourmodels is shown in Fig. 2 along with the γ th = − / z and ENS z , thathave γ > γ th do indeed predict that the probability of sub-halo detection increases with the mass (see Fig. 9). On theother hand, the slope of the c ( M ) relation for M < M ⊙ is sligthly steeper than γ th . Therefore we expect a bimodalprobability that peaks at high masses with a secondary max-imum for the smallest subhaloes. Indeed, this is what weobserve in Figure 9.In conclusion, we have studied the prospects for indirectdetection of Dark Matter in MW subhaloes with a GLAST-like satellite. We have chosen 8 different models for the con-centration parameter, which span the phase space the the-oretical uncertainties on the Dark Matter halo properties,as well as 3 different values for the subhaloes mass functionslope. For each model, we have computed the diffuse emis-sion from unresolved subhaloes, as well as the γ -ray fluxfrom individual, nearby haloes.We found that for models with concentration param-eter computed at z = 0 the detection of individual haloesappears challenging, while the diffuse emission from unre-solved clumps dominates the MW smooth emission for skydirections > α = 1, in all the zc models except theB ref,z c , the diffuse emission from unresolved clumps exceedsthe EGRET constraints in a portion of the DM parame- Figure 11.
Contribution to Φ cosmo from substructures of massM integrated over the mass decade, for the different B mod-els explored and computed toward the Galactic Center. Solidand empty circles correspond to the B ref,z and B ref,z c modelsrespectively. Lines show the B z c , σ (dot-dashed), B z c (dashed),B z , σ (dotted) and B z (solid) models. The mass function slopeis α = 1. ter space relevant for SUSY models, as shown in Fig. 3.Adopting DM models compatible with the EGRET data,one may still hope to detect individual haloes, like e.g. inthe B z c model.The B ref,z c is our best case model for all the values ofthe mass function slope, though it should be stressed thatit is not supported by numerical experiments but only bytheoretical considerations. The success of the B ref,z c modelis due to the fact that the diffuse emission expected fromsubhaloes is dominated by small mass haloes while the largemass haloes are most favourably detected as spare sources.A functional form for the concentration parameter whichflattens at low masses, as it is the B ref,z c one, will in factdecrease the diffuse emission, thus allowing a larger value forΦ PP and consequently increasing the chances of detectionfor large mass haloes.In general, adopting the most optimistic set of parame-ters for the DM particle compatible with EGRET (see Table2), the number of detectable subhaloes over all sky (at 5 σ ,with a GLAST-like experiment and a 1-year exposure time),for the 8 x 3 models we have studied, ranges between 0 and120. Yet, it should be noticed that the numbers listed inTable 2 are obtained with a very optimistic value for theParticle Physics involved in the process. If we assume amore realistic model for the Dark Matter particle ( m χ =100 GeV , σv = 10 − cm s − ) instead, we find that at mostonly a handful of detectable haloes are found (at most ahandful over all sky for the most optimistic B ref,z c model).In all the models explored, small mass subhaloes are al-ways below the threshold for detection, and their presencecould be revealed only through the enhancement of the dif- c (cid:13) , 000–000 ark Matter Annihilation in Substructures Revised fuse foreground emission. The different predicted ratio be-tween diffuse emission and number of detected haloes forthe models we have considered, would provide precious in-formation on the underlying cosmology, in case of positivedetection. We are indebted to J. Bullock for help and discussion. Wethank J. Diemand, S. Koushiappas and G. Tormen for usefuldiscussions, comments and suggestions.
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