Consequences of a Dark Disk for the Fermi and PAMELA Signals in Theories with a Sommerfeld Enhancement
aa r X i v : . [ a s t r o - ph . H E ] S e p Preprint typeset in JHEP style - HYPER VERSION
Consequences of a Dark Disk for the Fermi andPAMELA Signals in Theories with a SommerfeldEnhancement
Ilias Cholis and Lisa Goodenough
Center for Cosmology and Particle Physics, Dept. of Physics, New York University,New York, NY 10003
Abstract:
Much attention has been given to dark matter explanations of the
PAMELA positron fraction and
Fermi electronic excesses. For those theories with a TeV-scale WIMPannihilating through a light force-carrier, the associated Sommerfeld enhancement providesa natural explanation of the large boost factor needed to explain the signals, and the lightforce-carrier naturally gives rise to hard cosmic ray spectra without excess π -gamma raysor anti-protons. The Sommerfeld enhancement of the annihilation rate, which at low rela-tive velocities v rel scales as 1 /v rel , relies on the comparatively low velocity dispersion of thedark matter particles in the smooth halo. Dark matter substructures in which the velocitydispersion is smaller than in the smooth halo have even larger annihilation rates. N-bodysimulations containing only dark matter predict the existence of such structures, for examplesubhalos and caustics, and the effects of these substructures on dark matter indirect detectionsignals have been studied extensively. The addition of baryons into cosmological simulationsof disk-dominated galaxies gives rise to an additional substructure component, a dark disk.The disk has a lower velocity dispersion than the spherical halo component by a factor ∼ e + e − , p ¯ p and γ -rays as measured by Fermi and
PAMELA in models where the WIMP annihilations are into a light boson. We findthat both the
PAMELA and
Fermi results are easily accomodated by scenarios in which adisk signal is included with the standard spherical halo signal. If contributions from the darkdisk are important, limits from extrapolations to the center of the galaxy contain significantuncertainties beyond those from the spherical halo profile alone.
Keywords:
PAMELA , Fermi , dark disk, Sommerfeld enhancement. ontents
1. Introduction 12. Definitions and Parameters 4
3. Results 7
4. Conclusions 21
1. Introduction
Much excitement has been generated over the past year or so by the release of results fromseveral cosmic ray experiments.
PAMELA [10, 26, 84, 9] has shown a sharp upturn in thepositron fraction above 10 GeV.
Fermi [7] has published results indicating an excess in thetotal electronic spectrum e + + e − in the 100 GeV to 1 TeV range, while HESS has shownthat the spectrum falls off sharply above 1 TeV [14, 15]. Thermal WIMPs would naturallypredict such features in the electron and positron measurements. However, conventionalWIMPs annihilate far too little to yield these large signals, which require boost factors of O (100) above the thermal cross-section of 3 × − cm s − [21, 31, 38, 19, 40, 34]. Moreover,the absence of an excess in antiprotons up to 100 GeV in the PAMELA [8] data essentiallyexcludes models of dark matter which would produce the e + e − excess through annihilationsto standard model modes, since such models also have significant hadronic branching fractions[35, 50].An appealing possibility is for the dark matter (DM) to annihilate into a new, light( m . p production. Such annihilation modes easily fit the PAMELA and
Fermi electron results [29, 32], while also producing an interesting synchrotron “microwave haze”signal, which has been observed by
WMAP [56, 57, 48], and additionally yielding a significant– 1 – -ray signal [49] from inverse Compton and final state radiation that may be observable at
Fermi . A key consequence of including a new force in the dark sector is that at low velocitiesan annihilation rate much larger than that expected for a thermal WIMP can result fromthe Sommerfeld enhancement [17, 76, 79] or from capture into bound “WIMPonium” states[62, 39, 90, 80]. Sommerfeld-enhanced annihilation rates increase like 1 /v rel at small v rel ,until the non-zero mass of the mediator φ cuts off the enhancement. As argued in [17], animportant feature of such low-velocity enhancements is the added significance of substructure.Because the velocity dispersion in the dark disk can be significantly lower than in the smoothhalo, up to ∼ ∼ /v rel . Add to this effect the increasein the annihilation rate due to the fact that the DM density in substructure is larger than inthe smooth halo, and the role of substructure can indeed be substantial.The spectra of positrons and electrons can be greatly affected by the presence of sub-structure. Because distant ( d > ∼ PAMELA and
Fermi , are dominated by nearby sources. If there is nearby substructure, the observed spectracan be modified [66]. Quantifying the effects, even for standard WIMPs, is a great challenge;the
Via Lactea
N-body simulation [46, 73, 47] argues for a 1% chance of a subhalo giving aboost of 10 for conventional WIMPs, while the
Aquarius simulation sees lower boosts [100].For WIMP scenarios with a Sommerfeld enhancement, quantifying the overall effect is moredifficult. The Sommerfeld enhancement saturates at some velocity, below which the 1 /v be-havior is no longer valid. If the saturation occurs at v ∼
200 km/s, as suggested in [99],then the annihilation cross-section is the same in the smooth DM halo, which has a velocitydispersion of ∼
220 km/s, as in substructure with lower velocity dispersion. If the saturationoccurs at much lower velocities, the annihilation cross-section in substructure with the lowestvelocity dispersion can be substantially larger than in the smooth halo. Since it is the smallesthalos that have the lowest velocity dispersion, the smallest substructure can play a dominantrole. This results in significant sensitivity to difficult-to-resolve small scales, and addition-ally to the unknown mass of the mediator φ , which cuts off the enhancement, and cutoffs inthe primordial power spectrum. Other complications, such as resonances in the annihilationcross-section for specific parameters, can make quantifying the effects nearly impossible.Nonetheless, it is important to attempt to understand what effects substructure can haveon the present signals, in the context of Sommerfeld-enhanced annihilation channels. In thispaper, we consider the effects of the addition of a “dark disk”, as proposed by [91, 92], tothe smooth spherical halo (see also [86]). Cosmological simulations that model only the darkmatter particles are neglecting the baryons that dominate the mass of the galaxy interiorto our location at the Solar Circle at R ⊙ = 8 . ∼ −
90 km/s.The presence of a dark disk in our galaxy can have consequences for the observed cosmicray (CR) spectra. The disk does not have the relatively large increase in mass density at theGalactic Center that the spherical halo has, and it extends only ∼ − Disk / Φ Einasto . The ratio increases with energy, indicating that therelative contribution to the highest energy positrons is larger for the disk, which has a largerrelative dark matter density in the Galactic Plane than a spherical halo.The spectral hardening of the electrons and positrons is not accompanied by a corre-sponding hardening of the proton and antiprotons, as the latter are not subject to the samelarge energy losses during propagation. In this way, substructure can change the relativenumbers of annihilation products of a given energy reaching our detectors on Earth. SeeFig. 1b. The ratio of high energy antiprotons to positrons for the Einasto profile is ∼ / m e m p ) , while their diffusion lengths are equal in the ultra-relativistic limit. Within an in-finitely large and homogeneous diffusion zone, antiprotons diffuse a distance m p m e (on average)larger than that of positrons with the same injected and observed energies. Models of ourGalaxy indicate a diffusion zone of half-width L between ∼ ∼
10 kpc. Protons andantiprotons of E ∼
100 GeV measured by
PAMELA propagate to us from the entire volumeof the assumed diffusion zone. On the other hand, electrons and positrons with energies ∼ Fermi originate from a region within ∼ − ∼ − O (1) contribution to the local e ± fluxesat E >
500 GeV, contributes significantly less to γ -rays and synchrotron radiation in regionswithin the inner 2 kpc. Thus, using local e ± fluxes to find DM annihilation rates results in For a discussion of the effects of a dark disk on neutrino signals, see [25]. Within the validity of approximating the Klein-Nishina cross-section σ K − N with the Tompson cross-section σ T , σ KN ≈ σ T . – 3 – φ e + / ( φ e + + φ e - ) Energy (GeV)(a) Φ Disk / Φ Spherical Halo
Monoenergetic Electronsm χ = 100 GeVm χ = 300 GeVm χ = 1.0 TeV 0 0.2 0.4 0.6 0.8 1 10 100 1000 F l u x R a t i o Energy (GeV)(b)m χ = 100 GeV m χ = 300 GeV m χ = 1.0 TeV Φ pbarDisk / Φ pbarSpherical Halo Φ e + Disk / Φ e + Spherical Halo ( Φ pbar / Φ e + ) Disk / ( Φ pbar / Φ e + ) Spherical Halo
Figure 1:
Left: The ratio of the local flux of positrons from DM annihilation for the Dark Disk profileto that for the spherical Einasto profile. Here we assume monoenergetic electrons from the annihilation χχ → e + e − , with m χ = 100 GeV, m χ = 300 GeV, and m χ = 1 . dot-dashed ) and antiprotons( dotted ) from DM annihilation for the Dark Disk profile to that for the spherical Einasto profile forthe annihilation channel χχ → W + W − , with m χ = 100 GeV, m χ = 300 GeV, and m χ = 1 . ¯ p / Φ e + ) Disk / (Φ ¯ p / Φ e + ) Einasto ( solid ). smaller fluxes of γ -rays and synchrotron radiation from the center of the Galaxy for dark disks.Because of this, a significant dark disk component in models with a Sommerfeld enhancementalleviates some the constraints coming from gamma ray measurements of the Galactic Centerand Galactic Ridge regions. As a consequence, while the Galactic Center places the tightestconstraints on the annihilation rate for cusped, spherical profiles, the Galactic high latitudes(specifically, the isotropic diffuse flux there) usually places the tightest constraints on theannihilation rate for the disk profiles.
2. Definitions and Parameters
In the ΛCDM cosmological simulations of [91], the formation of the stellar thick disk throughaccretion can be accompanied by the formation of a dark matter disk. The fits of the darkdisk (DD) component can be described by an exponential function and parametrized by halfmass scale lengths. We consider a disk component based on the fits of [91], and assume acylindrically symmetric dark disk density profile of the form ρ ( R, z ) = ρ exp (cid:20) .
68 ( R ⊙ − R ) R / (cid:21) exp (cid:20) − . | z | z / (cid:21) , (2.1)– 4 – ark disk haloEinasto DM halo H kpc L at z = D M d e n s it y H G e V (cid:144) c m L H a L dark disk haloEinasto DM halo H kpc L at R = D M d e n s it y H G e V (cid:144) c m L H b L Figure 2:
Left: Radial dependence of DM density for the Einasto profile and the Dark Disk profileat z = 0. Right: z dependence of DM density for the Einasto profile and the Dark Disk profile at r cylindrical = R ⊙ . where R / = 11 . z / = 1 . R ⊙ = 8 . R isthe cylindrical radial coordinate.For the spherically symmetric halo model (SH) we use the Einasto profile [53] followingMerritt et al. [83] ρ ( r ) = ρ exp (cid:20) − α (cid:18) r α − R α ⊙ r α − (cid:19)(cid:21) , (2.2)with α = 0 .
17 and r − = 25 kpc. r is the spherical radial distance from the Galactic Center(GC). We take the value of the local density for both the spherical and disk components tobe ρ = 0 . − [27]. (See discussion below.) In Fig. 2 we show the dependence ofthese DM profiles on R for z = 0, i.e. in the Galactic Plane, and on z for R = 8 . ρ DD /ρ SH range from 0.2 up to more than 1.5. Thehighest values of ρ DD /ρ SH occur in simulations when the thick stellar disk has a higher massdensity than the thin stellar disk. This can result from an increase in the mass included inthe stars of the thick disk, or from a thin disk heated by very massive, high-redshift mergers,which results in the thick disk getting populated by thin disk stars. Another, though lesslikely, cause is the occurrence of multiple prograte and low inclination mergers [91].Constraints have been placed on the total local dark matter density from estimations ofthe local dynamical mass density based on measurements of proper motions and parallaxesof stars. Using the Hipparcos data, [63] derived the local dynamical mass density to be0 . ± .
010 M ⊙ pc − (95% C.L.). Their estimate for the local density of the visible matteris 0 . ± .
017 M ⊙ pc − (67% C.L.). Assuming the mean values of these densities, weget 0 .
008 M ⊙ pc − , or equivalently 0 . − , for the local dark matter density with– 5 –n error that is larger than the mean value, thus allowing for a dark disk with ρ DD ∼ ρ SH , where ρ SH = 0 . − . [44] derived a smaller local dynamical mass density of0 . ± .
015 M ⊙ pc − using the Hipparcos data, but values of 0 . − . − are allowedwithin their uncertainties. Later analysis of the Hipparcos data by [64] and [72] agree withthe constraints imposed by [63]. Recently [27] suggested a value of 0 . − as the bestestimate of the local DM density. Also [95] calculated the value of the local DM density tobe 0 . ± . ± .
10 GeVcm − , while [103] set an upper limit of 1 GeVcm − for the localDM density. We use GALPROP, developed by Moskalenko and Strong [101], for the propagation of allcosmic ray particle species. For our calculation of the Inverse Compton Scattering of elec-trons and positron we use the Interstellar Radiation Field model of [89]. For consistency inour comparisons of the spectra from a dark disk and from a spherical halo, we assume thesame energy losses, diffusion coefficient, and diffusion zone . Our choices for the backgroundmodels and propagation parameters give local CR spectra that agree with local cosmic raymeasurements, including the proton, He, C, Fe, B/C, Be / Be , and sub-Fe/Fe spectra. Addi-tionally, for the annihilation channels with no DM contribution to antiprotons our backgroundantiproton-over-proton ratio ¯ p/p agrees with the PAMELA ¯ p/p data. See Fig. 9a.In placing constraints on ¯ p fluxes from DM, we use a background antiproton ratio thatis 15% smaller. The smaller ¯ p/p background ratio shown in Fig. 9b is consistent with thecurrent constraints from all CR data. For example, a reduction in the background ¯ p/p couldbe based on uncertainties in the interstellar medium (ISM) distribution of Hydrogen andnuclei, since antiprotons are produced by collisions of CR protons and nuclei with the ISMprotons and nuclei. We consider two classes of annihilation scenarios, the “eXciting Dark Matter” (XDM) scenario[17, 30, 58] in which annihilation proceeds through a single mediator φ , and annihilationproceeding through two mediators Φ and φ . Additionally we consider annihilation directlyinto muons, χχ → µ + µ − , and annihilation into W + W − , which then decay. We discuss theimplications of the W + W − channel for the PAMELA antiproton data.In the XDM scenario, we present results for annihilationi) into e + e − through a scalar (or vector) φ with mass 2 m e < m φ < m µ and a branchingratio (BR) of 0.9 (or 1),ii) into µ + µ − through a scalar (or vector) φ with mass 2 m µ < m φ < m π and BR=1,iii) into π + π − through a vector φ with mass m φ = 750 MeV (the form factor for φ → π + + π − peaks at m φ ≃
750 MeV [52]), Refer to [32] or [102] for the details of the diffusion parameters. We expect a ∼
10% decay of scalar φ → γ due to 1-loop diagrams. – 6 –v) and into e + e − , µ + µ − and π + π − with relative BR’s 1:1:2, through a vector φ withmass m φ ≃
650 MeV.For DM annihilation with subsequent cascade decay through two mediators, we show thecase χχ → ΦΦ, with subsequent decays Φ → φφ and φ → µ + µ − , where 2 m µ < m φ < m π and m Φ ∼ m φ , and additionally the similar cascade decay with φ → e + e − . The spectra ofthe final e + e − injected into the ISM from these cascade decays are given in Appendix A. In the absence of the Sommerfeld enhancement, the total dark matter annihilation rate Γ ann from the spherical halo and dark disk is given by (for Majorana fermions)Γ ann = 12 (cid:18) ρ SH + ρ DD m χ (cid:19) h σ ann | v |i . (2.3)If the Sommerfeld enhancement is present, Eq. 2.3 is modified toΓ ann = 12 (cid:18) ρ SH m χ (cid:19) h σ ann | v |i SH + 12 (cid:18) ρ DD m χ (cid:19) h σ ann | v |i DD + (cid:18) ρ SH · ρ DD m χ (cid:19) h σ ann | v |i mixed . (2.4) h σ ann | v |i SH and h σ ann | v |i DD are in general different, because the velocity dispersion of darkmatter is different in the spherical and disk components of the halo. We define the boost factor (BF) as the ratio of the thermally averaged annihilation cross-section needed to fit a set of data, h σ ann | v |i fit , to 3 × − cm s − , the expected thermallyaveraged annihilation cross-section for a WIMP with mass m χ ∼
500 GeV, BF = h σ ann | v |i fit × − cm s − . (2.5)
3. Results
As mentioned earlier, the effect of a dark disk on the electronic signal is to produce aharder spectrum at high energies relative to that produced by the spherical Einasto profile.See Fig. 1a. This is a generic result for the dark disk profiles of [91] and any spherical haloprofile, and it arises because lower energy electrons and positrons, which propagate to Earthover larger distances on average, are probes of regions closer to the Galactic Center wherethe spherical halo profiles peak significantly more than the disk profiles. See Fig. 2a. Thisalso explains why the boost factor needed to get equal fluxes of dark matter electrons andpositrons at energies much less than m χ is lower for the Einasto profile than for the dark– 7 – φ e + / ( φ e + + φ e - ) Energy (GeV)XDM µ + µ - Channelm χ = 2.5 TeV (a) Disk Only, BF = 11000Spherical Halo Only, BF = 2500BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)XDM µ + µ - Channelm χ = 2.5 TeV DM Contribution (b)
HESS Systematic Error BandHESS DataFermi High E DataFermi Low E DataDisk Only, BF = 1800Spherical Halo Only, BF = 940Background 0.01 0.1 1 10 100 φ e + / ( φ e + + φ e - ) Energy (GeV)XDM π + π - Channelm χ = 3.1 TeV (c) Disk Only, BF = 4500Spherical Halo Only, BF = 1700BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)XDM π + π - Channelm χ = 3.1 TeV DM Contribution (d)
HESS Systematic Error BandHESSFermi High E DataFermi Low E DataDisk Only, BF = 1400Spherical Halo Only, BF = 890Background
Figure 3:
Positron fraction Φ e + / (Φ e + + Φ e − ) (left) and total electronic flux weighted by E (GeV m − s − sr − ) (right), both as a function of energy. Results are shown for the sphericalEinasto profile ( dashed ) and for a dark disk ( solid ). Backgrounds are shown in dot-dashed . Top:XDM µ + µ − channel with m χ = 2 . π + π − channel with m χ = 3 . PAMELA postron fractio data are from [9], while the total electronic flux measured by
Fermi given in [7, 41]. disk, i.e. why the boost factors needed to fit
PAMELA are lower for the Einasto profile thanfor the dark disk.In Figs. 3-8 we show the positron fraction Φ e + / (Φ e + + Φ e − ) and the E -weighted totalelectronic flux (the sum of the fluxes of electrons and positrons) for a number of annihilationchannels. We show the fits of the spectra to the PAMELA , Fermi , and HESS data separatelyfor the dark disk and spherical halo scenarios. The locally measured spectrum, which iscomposed of both a disk and a spherical halo component, is expected to lie between the two For the mixed term ( ∝ ρ SH · ρ DD ), we take h σ | v |i mixed = h σ ann | v |i SH . – 8 – φ e + / ( φ e + + φ e - ) Energy (GeV)XDM e + e - , µ + µ - , π + π - χ = 1.6 TeV (a) Disk Only, BF = 840Spherical Halo Only, BF = 330BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)XDM e + e - , µ + µ - , π + π - χ = 1.6 TeV DM Contribution (b)
HESS Systematic Error BandHESSFermi High E DataFermi Low E DataDisk Only, BF = 270Spherical Halo Only, BF = 160Background
Figure 4:
Cosmic ray signals as in Fig. 3. XDM e + e − , µ + µ − and π + π − channels with relative BR’sof 1:1:2. spectra shown.For the relatively hard annihilation channels, such as χχ → µ + µ − shown in Fig. 7, theDM component of the spectrum arising from the dark disk is too peaked between 400 and800 GeV to give really good fits to the Fermi data, though the fits for the Einasto profile forthese channels are generally good. Those annihilation channels with much softer electron andpositron spectra, for example XDM to muons or to pions only or some combination of theseand XDM through a 2-step decay (see Figs. 3-5), give very good fits to the
Fermi data forboth the spherical and disk profiles. Therefore, any relative combination of disk and sphericalhalo contributions to the total e ± flux is allowed by the Fermi data. (See also Section 3.3for more discussion on this). Both DD and SH profiles give good agreement with the sharpfall-off in the spectrum seen in the HESS data, which suggests a power law of E − . ± . inthe range 700 - 3500 GeV [14].We note that for the flux of primary electrons we use an injection power-law of ∼ E − . ,which agrees with conventional assumptions about the primary electron spectrum in the GeVenergy range [77, 3, 71]. We introduce a break in the spectrum at ∼ PAMELA positron fraction data is well fit for the Einasto profile by all of the XDMannihilation channels shown. The implication for the positron fraction of the harder DM e + e − spectra for the disk profile is a steeper rise of this fraction. For the dark matter massesthat fit the Fermi data, m χ & PAMELA – 9 –ata. See, for example, Fig. 3. However, for low DM masses that agree with
PAMELA butnot
Fermi , the dark disk can improve the
PAMELA fits.Since the velocity dispersion of the particles in the dark disk traces that of the stars inthe stellar thick disk, the local values are much lower than for the spherical Einasto profile.Typical local values of the velocity dispersion for the disk are ∼
30 km / s and for the sphericalhalo are ∼
220 km / s. Therefore, we expect that if the annihilation cross-section scales as ∼ /v , then in the disk it can be up to ∼
10 higher than that of the particles in the sphericalhalo. This is assuming, of course, that the Sommerfeld enhancement isn’t saturated forvelocity dispersions greater than that in the disk and that we are not near a resonance. (InSection 3.3, we discuss the possibility, as suggested by [99], that the annihilation cross-sectionis close to saturation for σ v ≃
200 km / s.) In Fig. 6 we show an example of the PAMELA and
Fermi spectra for a combination of disk and Einasto profiles for a disk annihilation rate thatis 5 times the annihilation rate in the spherical halo.For a specific DM mass and annihilation channel, the ratio of the boost factor needed tofit the
PAMELA data over the boost factor needed to fit the
Fermi data, BF P AM /BF
F ermi ,is larger for the dark disk profile than for the Einasto profile. For example, for the XDM π + π − channel BF P AM /BF
F ermi = 3 . e ± of DD profiles compared to SH profiles,the ratio of boost factors BF P AM /BF
F ermi is expected to be higher for DD profiles thanfor SH profiles. In other words, if the fluxes of e + e − from the spherical halo and from thedark disk are equal at 100 GeV, then the flux due to the spherical component will be greaterfor E .
100 GeV, while the flux for the dark disk component will be greater for E & Fermi energies in a spherical halo to have improved fits to the data for a disk halo.On the other hand, the “flatness” of the
Fermi data (in E -weighted plots) can be used toconstrain the disk contribution to the overall flux. Additionally, WMAP data can put upperlimits on the saturated Sommerfeld enhancement [99], so the contribution of the dark diskfor various channels can be constrained in this way as well. See Section 3.3.
Non-Sommerfeld Enhanced Channels
The annihilation channels that proceed through one or more light mediators (see Figs. 3-6) require large boost factors to fit the
PAMELA and
Fermi data, and these can be explainedby the Sommerfeld enhancement. Annihilation channels directly to leptons also require largeboost factors, BF ∼
500 for the spherical halo, but in these scenarios there is no long rangeforce to give rise to the Sommerfeld enhancement needed to motivate such a large boost factor.A Breit-Wigner resonance, as suggested by [60], could account for the large annihilation rateneeded at z = 0 to produce the signals we observe today, while not impacting the thermalrelic density at freeze-out. Also, WIMPs produced non-thermally, such as those of [51, 23],can naturally have large annihilation cross-sections that give good fits to the PAMELA and
Fermi data. For this reason, we consider here such non-Sommerfeld enhanced scenarios inthe context of a dark disk. – 10 – φ e + / ( φ e + + φ e - ) Energy (GeV)XDM 2-step to e + e - Channelm χ = 1.5 TeV (a) Disk Only, BF = 540Spherical Halo Only, BF = 200BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)XDM 2-step to e + e - Channelm χ = 1.5 TeV DM Contribution (b)
HESS Systematic Error BandHESSFermi High E DataFermi Low E DataDisk Only, BF = 170Spherical Halo Only, BF = 110Background 0.01 0.1 1 10 100 φ e + / ( φ e + + φ e - ) Energy (GeV)XDM 2-step to µ + µ - Channelm χ = 3.8 GeV (c) Disk Only, BF = 3500Spherical Halo Only, BF = 1400BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)XDM 2-step to µ + µ - Channelm χ = 3.8 TeV DM Contribution (d)
HESS Systematic Error BandHESSFermi High E DataFermi Low E DataDisk Only, BF = 1500Spherical Halo Only, BF = 880Background
Figure 5:
Cosmic ray signals as in Fig. 3. Top: XDM e + e − through two steps with m Φ ∼ m φ and m φ < m µ . Bottom: XDM µ + µ − through two steps with 2 m µ < m φ < m π . Two-step annihilationsproceed through χχ → ΦΦ, followed by the decays Φ → φφ and φ → e + e − or φ → µ + µ − . In Fig. 7 we show the positron fraction and total electronic flux for dark matter annihi-lating monochromatically to muons, χχ → µ + µ − . Good fits to both the PAMELA and
Fermi data can be achieved with the spherical halo profile. As mentioned earlier, the disk profilegives an electronic spectrum that is too hard to fit well the
Fermi data, though it does a goodjob reproducing the positron fraction. Additionally, the ratio BF P AM /BF
F ermi is 4.5 for thedark disk, which suggests that a dark disk dominant scenario does not accurately explainboth signals simultaneously.While direct decay to taus, χχ → τ + τ − , gives good fits to the Fermi and
PAMELA data,the annihilation rate for the tau channel is highly constrained by the
Fermi diffuse γ -ray data[37, 85] and by the HESS measurements of gamma rays in the Galactic Ridge [12, 82]. (The– 11 – φ e + / ( φ e + + φ e - ) Energy (GeV)XDM e + e - , µ + µ - χ = 1.3 TeV (a) Halo + Disk, BF = 140BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)XDM e + e - , µ + µ - χ = 1.3 TeV DM Contribution (b)
HESS Systematic Error BandHESS DataFermi High E DataFermi Low E DataHalo + Disk, BF = 45Background
Figure 6:
Cosmic ray signals as in Fig. 3. XDM e + e − and µ + µ − channels with equal BR’s. Here acombination of dark disk and Einasto profile is assumed. The annihilation rate in the disk is taken tobe 5 times the rate in the spherical halo. φ e + / ( φ e + + φ e - ) Energy (GeV) µ + µ - Channelm χ = 1.5 TeV (a) Disk Only, BF = 3000Spherical Halo Only, BF = 750BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV) µ + µ - Channelm χ = 1.5 TeV DM Contribution (b)
HESS Systematic Error BandHESSFermi High E DataFermi Low E DataDisk Only, BF = 660Spherical Halo Only, BF = 320Background
Figure 7:
Cosmic ray signals as in Fig. 3 for annihilation of DM directly to µ + µ − for m χ = 1 . τ + τ − channel produces a high energy prompt component of gamma rays through the π screated in the decay chain.) Annihilation directly into e + e − gives a propagated spectrum e + e − with spectral index of -2 and a hard cut-off at E = m χ [29, 78]; such a spectrum doesnot fit the Fermi and HESS e − + e + data well. Annihilation through W + W − gives very poor fits to the Fermi and
PAMELA data. Mostof the electrons and positrons are produced in the decay chains of the hadrons and taus, sothey have relatively soft spectra. The resulting abundance of low energy e + e − pairs gives a– 12 – φ e + / ( φ e + + φ e - ) Energy (GeV)W + W - Channelm χ = 1.0 TeV (a) Disk Only, BF = 680Spherical Halo Only, BF = 250BackgroundPAMELA Data 100 10 100 1000 E d N / d E ( G e V m - s - s r - ) Energy (GeV)W + W - Channelm χ = 1.0 TeV DM Contribution (b)
HESS Systematic Error BandHESSFermi High E DataFermi Low E DataDisk Only, BF = 1600Spherical Halo Only, BF = 690Background
Figure 8:
Cosmic ray signals as in Fig. 3 for DM annihilation to W + W − with m χ = 1 . A n t i p r o t on f r a c t i on ( pba r / p ) Energy (GeV)
BackgroundPAMELA Data 1e-06 1e-05 1e-04 0.001 0.1 1 10 100 1000 A n t i p r o t on f r a c t i on ( pba r / p ) Energy (GeV)W + W - Channelm χ = 1.0 TeV(b) antiproton fraction Disk Only, BF = 150antiproton fraction Spherical Halo Only, BF = 25BackgroundPAMELA Data Figure 9:
Antiproton over proton ratio ¯ p/p as a function of energy. Left: Background for thepropagation and diffusion parameters used in our calculations. Right: For DM annihilating through W + W − with m χ = 1 . solid ) and for the Einasto profile ( dashed ). Herethe background ( dot-dashed ) is smaller than the standard background by 10%. Data of the antiprotonto proton flux ratio are from [42]. positron fraction that is much too flat and an e + + e − spectrum that doesn’t simultaneouslyagree with the Fermi data around 20 GeV and the data around 1 TeV. In Fig. 8 we show theresults for m χ = 1 TeV, the DM mass that best fits the Fermi data. The fit to the
Fermi data is significantly better for the dark disk profile than for the spherical halo. Additionally,the fit of the positron fraction to the
PAMELA data has a modest improvement. Thus,for the W + W − channel the favored scenario is one in which the dark disk is the dominantcontribution to the local DM density. – 13 –n additional benefit of the dark disk for the W ± channel is that the constraint placedon the annihilation rate by the PAMELA antiproton data is somewhat relaxed. In Fig. 9bwe show that a very similar antiproton ratio is obtained for the dark disk with BF = 150as for the Einasto profile with BF = 25. The factor of ∼ p ’s propagate to us with little energy loss from the entire diffusion zone,and the disk has a smaller average dark matter density throughout the diffusion zone, asdiscussed in Section 1. This, of course, depends greatly on the relative widths of the diffusionzone and the dark disk. For diffusion zones with widths much larger than the scale widthof the disk, the improvement becomes significant, i.e. factors are much larger than 1. Fornarrow diffusion zones such as the ∼ m χ = 1 . W + W − in the dark disk only scenario, the BF needed to fit the PAMELA positron fraction is still ∼ PAMELA ¯ p/p ratio. For comparison, in the spherical halo only scenario the BF for the PAMELA positronfraction is ∼
10 times larger than that allowed by ¯ p/p . However, since the fits to the
PAMELA data are so poor, the BF ’s should not be taken too seriously. In summary, the discrepancybetween the larger boost factors needed to fit the positron fraction data and the smallerboost factors needed to fit the antiproton fraction data can be reduced by including a diskcomponent. Thus, a higher branching ratio to hadronic channels is allowed for a dominantlydark disk profile. Constraints from the Fermi e + + e − flux The combined flux from a spherical halo and a dark disk can be written as dN total dE = dN prim dE + dN sec dE + BF SH (1 + M ) (cid:18) dN SH dE + RM dN DD dE + 2 M dN comb dE (cid:19) . (3.1)Here M is the ratio of the local DM densities, ρ DD /ρ SH , R is the ratio of the thermally-averaged annihilation cross-sections, R = h σ | v |i DD / h σ | v |i SH , and BF SH is the Sommerfeldenhancement in the spherical halo (i.e. for the velocity dispersion in the spherical halo).Using the e + + e − flux as measured by Fermi , we can constrain M , R , and BF SH . Sincethe Fermi data constrain the total flux of Eq. 3.1, constraints on any of these parameters canbe used as further input to constrain the rest of the parameters. We consider the followingvalues for the parameters:0 < M < ρ DD ≤ ρ SH , ρ SH + ρ DD = 0 . − . (3.2) The dN comb dE comes from the term ∝ ρ DD ∗ ρ SH in Eq. 2.4, where we assume that the relative velocitydispersion between the particles in the DD and the SH is ∼
200 km/s. – 14 –e take R max = 100 to be the maximum ratio of the annihilation cross-section of DMparticles in the disk halo to the spherical halo. This value could occur if we live close to aresonance, where the enhancement scales as 1 /v , since the velocity dispersion in the sphericalhalo is ∼
220 km / s and in the dark disk halo is ∼
30 km / s. This assumes that the Sommerfeldenhancement does not saturate at velocities above ∼
30 km / s. If saturation occurs forannihilation in the dark disk but not in the spherical halo, i.e. at a velocity between ∼ ∼
30 km/s, then R is the ratio of the saturated cross-section in the disk to the cross-section in the spherical halo.In our calculations we allow the normalization of the primary e − flux to vary in the range1 . − . × − GeV − m − s − sr − at 20 GeV, and the power law index ( dN/dE ∼ E − α )to vary in the range 3 . < α < .
35. Since the secondary e ± ’s contribute negligibly to thetotal e + + e − flux for E >
20 GeV, we hold them constant assuming dN sec /dE ∼ E − . anda normalization of 6 . × − GeV − m − s − sr − at 20 GeV.In column 4 of Table 1 we show the 95% C.L. values for R , the ratio of the annihilationcross-section of DM particles in the disk to the spherical halo, assuming that the local DMdensity receives equal contributions from the disk and the spherical halo. If the Fermi error-bars are correlated, the χ distribution test is not an exact measure to calculate confidencelevels. Since we don’t have any information apart from the errorbars, the mean values of thefluxes, and the energy binning, we consider each point as an independent normally-distributedvariable with mean and standard deviation given by the Fermi data. For all cases except theXDM electron channel, the channel with the hardest DM e ± spectra, the annihilation rate inthe disk can be at least 100 times the annihilation rate in the spherical halo, indicating thatthe Fermi data is consistent with the presence of a dark disk component in a non-saturatedresonance scenario. In column 5 of Table 1 we show the 95% C.L. for the maximum valueof M , the ratio of the local DM densities, assuming that the annihilation rate in the diskis 100 times the annihilation rate in the spherical halo. The Fermi e + + e − data does notconstrain the ratio of the local DM densities within our allowed range of values (0 to 1),except for the XDM electron channel. The results of columns 4 and 5 in Table 1 suggest thatfor the XDM channels giving the best fits to Fermi and
PAMELA , the dark disk may be adominant contributor to the local flux of e ± coming from dark matter annihilation. In thelatter case, the Sommerfeld enhancement in the spherical halo is typically smaller,by an orderof magnitude, than the enhancement when the spherical halo is the dominant contribution.In the limiting case, ρ DD /ρ SH ≪
1, the dark disk contribution can be significant to theflux of e ± locally only if the masses of the dark matter m χ and the mediator m φ and the valueof the coupling between the two λ are such that for a velocity dispersion of ∼
30 km / s theannihilation cross-section has a resonance [17, 88]. The results shown in Table 1 indicate thatXDM channels that produce e ± with a spectral index α > , where dNdE ∝ E − α , Since we want to place an upper bound on the Sommerfeld enhancement in the dark disk, we use a valuefor the velocity dispersion that is on the low side, but within the range suggested by simulations [91]. Such avalue is motivated in dominant thin disk scenarios. XDM to pions only, XDM 2-step to muons, and XDM to e ± , µ ± and π ± with relative ratios of 1:1:2 – 15 –hannel M χ (TeV) BF SH R max ( M = 1) M max ( R = 100) BF W MAP XDM to e ± −
160 90 0.9 210XDM to µ ± − π ± − . −
380 100 1 560XDM 2-step to e ± . −
250 100 1 260XDM 2-step to µ ± − µ ± − Table 1:
Table of allowed values of the boost factor for the Einasto profile for fits to the
Fermi data, BF SH , the maximum allowed values of R = h σ | v |i DD / h σ | v |i SH assuming M = ρ DD /ρ SH = 1 (95%C.L.), the maximum allowed values of M for R = 100, and the boost factors BF W MAP excluded by WMAP χχ → µ + µ − . are allowed within 95% C.L. to have a resonance in their cross-section. We clarify that in thecases of resonant Sommerfeld enhancement at z = 0 for a velocity dispersion of 30 km/s, itis the dark disk that gives the dominant contribution to the local e ± flux, while the sphericalhalo is constrained to be subdominant. See Table 1. At v rel ∼
200 km/s for m χ ∼ TeV, theSommerfeld enhancement is O (10).Since the PAMELA positron fraction has only 6 data points at energies E e ± >
10 GeV,it constrains the annihilation rates less than the
Fermi e + + e − flux. Additionally, the factthat the secondary e + ’s are still at least half of the total e + flux for E e + <
50 GeV allows forgreater uncertainty in the necessary DM annihilation BF ’s. Constraints from WMAP5 data
The high energy electrons and positrons produced by dark matter annihilation in the earlyuniverse heat and ionize the photon-baryon plasma and thus can give rise to an increase in theionization fraction of the plasma after recombination. Perturbations in the ionization historyaround the time of last scattering ( z ∼ ) can affect the temperature and polarization powerspectra of the CMB. [99] (see also [105]) calculated the effect of annihilating DM on the plasmaand placed constraints on the maximum annihilation cross-section using the WMAP f with which the energy from DM annihilationgets converted into the energy of the e ± products. In column 6 of Table 1 we present theupper bounds on the boost factor (assuming h σ ann | v |i = 3 × − cm s − ) for several XDM-type annihilation channels. For the XDM e ± , XDM µ ± , XDM π ± channels, we used the meanefficiency f provided in Table 1 of [99]. For XDM 1:1:2 the efficiency is a linear combinationof the efficiencies for XDM e ± , XDM µ ± and XDM π ± , while for XDM 2-step to e ± and The efficiency f depends on the annihilation channel, the redshift, and also has a weak dependence on themass of the DM candidate. – 16 –-step to µ ± , the efficiency is approximately equal to the efficiency of XDM e ± and XDM µ ± ,respectively.The values of BF W MAP presented in the table are upper limits on the allowed BF fromSommerfeld enhancement at z ∼ z ∼ z ∼ BF SH and R of Table 1 makes it clear that the CMB induced limits are low enough to constrain thecontribution of the dark disk for the various channels. We remind the reader that the BF’sgiven in Figs. 3-5 assume a local DM density of ρ = 0 . − , and that, as discussedin Section 2, the value could be different by a factor of ∼
2. If ρ = 0 . − , the BF’sneeded to fit the Fermi or PAMELA data would be a factor of ∼ / µ ± channel, the Fermi data excludes neither the spherical Einasto halonor the dark disk. The boost factor for the fit of the dark disk to the
PAMELA data is ∼ (Similar results hold for the XDM e ± channel whosefits to the data are not shown.) However, if the contribution to the local e ± flux of the darkdisk and the spherical halo are comparable at ∼
500 GeV, the fluxes at
PAMELA energiescome mainly from the spherical halo. Thus, even if the dark disk gives an O (1) contributionto the local flux at E ∼ GeV, those channels are not yet excluded by the CMB limits,but are in some tension.The XDM π ± and XDM 1:1:2 channels are within CMB constraints, assuming the factorof 2 discrepancy between the fitted BF and the CMB limit mentioned earlier. In the com-bination channels, less of the available energy from DM annihilation is dumped into e ± at z ∼ e ± and 2-step to µ ± , the PAMELA fits require BF’s within a factorof a few of the CMB limits, thus those channels aren’t excluded. We note that regardless ofthe specifics of the annihilation channel (and the needed DM mass and BF), the fits to the
PAMELA and
Fermi data constrain the spectrum of DM e ± to be similar in all cases, thuswe expect the CMB limits to be similarly constraining in all cases.Recently [55, 54] suggested stronger constraints on the Sommerfeld enhancement basedon relic density calculations. In [55] the authors suggest that for m χ = 1 TeV and m φ = 250MeV, the maximum allowed Sommerfeld enhancement is 90. In general, they find that for m χ > m φ ∼ ∼ Multiple species can both decrease the needed annihilation rates for the observed
PAMELA positronfraction by a factor of ∼ – 17 –ark disk, which has a velocity dispersion that is ∼ ∼
200 km/s, or even ∼
30 km/s, the effect of the Sommerfeld enhancementon the relic density is much smaller, since much higher annihilation cross-sections at earlierepochs are not allowed. Thus we still consider as a reference the constraints of
WMAP
Planck [1] will make these constraintssignificantly tighter.
The predictions for the photon signals from dark matter annihilation are highly dependent onthe distribution of the photon sources, i.e. the dark matter, in the Galaxy. Because photonstravel to our detectors from their place of origin without diffusion and without loss of energy,their fluxes sample their source distribution very well. Thus, not only is the relative densityof dark matter in the disk to that in the spherical halo ( ρ DD /ρ SH ) important for calculatingphoton fluxes, but so is the relative annihilation cross-section. Whether or not there is aSommerfeld enhancement and whether or not it is in saturation is quite relevant for photons.For a discussion of the range of reasonable values of the ratio of densities, see Section 2. Thereare two limiting cases: ρ DD /ρ SH ∼ ρ DD /ρ SH ≪ Microwave haze
Cosmic ray electrons and positrons with GeV-scale energies will synchrotron radiatethrough their interactions with the Galactic magnetic field, resulting in microwave frequencyradiation. The dark matter models that fit the
PAMELA and
Fermi electronic data predicta population of e ± with the right spectrum and spatial distribution [57, 65, 29], assuminga spherical DM halo) to account for the WMAP microwave “haze” found by [56] and laterconfirmed by [48]. We have calculated the effect of the addition of a dark disk on the syn-chrotron radiation in the haze region, i.e. the average value over longitudes − ◦ to +10 ◦ forlatitudes between 6 ◦ and 15 ◦ from the Galactic Plane.Here we assume locally equal, ρ DD /ρ SH ∼
1, and spatially uniform annihilation cross-sections for the DM particles in the dark disk halo and the spherical halo. This case triviallyapplies for annihilating DM models like Kaluza-Klein DM as presented in [96, 28] and studiedby [18, 65, 67] (among others), where the Sommerfeld enhancement cannot be used to explainlarge boost factors. As we have already discussed, the Sommerfeld enhancement may be at orvery close to saturation, so the assumption of equal annihilation cross-sections may be validfor theories that include the Sommerfeld enhancement as well. However, the annihilationcross-section may drop near the Galactic Center, since recent simulations in which baryonsare included [94, 93, 87, 59] suggest that the velocity dispersion grows toward the GC. [87] findthat the dependence of the velocity dispersion σ v on the distance from the Galactic Center r goes as σ v ∼ r − / . – 18 –or the dark disk and Einasto profiles of Eqs. 2.1 and 2.2, under the assumption of equallocal densities, the flux of synchrotron radiation for the spherical profile is ∼
20 times largerthan that for the dark disk within the region of the microwave haze. This value is nearlyconstant over the entire region. When averaging over the inner 2 kpc, the Einasto profilehas a number density squared value n that is ∼
60 times larger than that of the dark diskprofile, n SH /n DD ∼
60, and about ∼ ρ SH / ρ SH · ρ DD ∼
5. If we define ρ ′ SH to represent the DM density in the case wherethere is only a spherical halo, then averaging over the inner 2 kpc, assuming ρ ′ SH = ρ SH and ρ SH ∼ ρ DD , we find ( ρ SH + ρ DD ) /ρ ′ SH ≈ /
4. Since we measure the total local DM density,we need to normalize the total DM density locally to the same value, ρ SH + ρ DD = ρ ′ SH .For ρ SH ∼ ρ DD we get, ρ ′ SH = 2 ρ SH → ρ ′ SH = 2 ρ SH . Thus after averaging over the inner2 kpc ( ρ SH + ρ DD ) /ρ ′ SH ≈ /
3. If we also normalize the BF to fit the local e ± fluxes andinclude the effects of propagation of the highest energy e ± , then in non-Sommerfeld casesand saturated Sommerfeld cases we still need approximately the same BF to explain the WMAP haze, even including a dark disk. However, if the Sommerfeld enhancement for thedisk is significantly greater than for the spherical halo (and still within constraints), then thesynchrotron radiation from the e ± of DM origin can be decreased by a factor of ∼ Common variations of dark disk profile of Eq. 2.1 include [91]: ρ ( R, z ) ∼ exp (cid:20) − RR / (cid:21) exp (cid:20) − (cid:16) zz / (cid:17) (cid:21) and ρ ( R, z ) ∼ exp (cid:20) − RR / (cid:21) sech (cid:20) − zz / (cid:21) . (3.3)Taking z / = 1 . R / = 11 . z / = 2 . R / = 12 . z / = 1 . R / = 11 . Fermi ( γ -ray) haze [49] is the γ -ray counterpart of the WMAP haze,the previous discussion applies to the
Fermi haze as well.
Diffuse γ -rays In addition to producing synchrotron radiation of microwave wavelengths through inter-actions with the Galactic magnetic field, the population of e ± from dark matter annihilationcan produce gamma rays through inverse Compton scattering (ICS) off of the optical, infrared,and CMB photons in the interstellar medium. Another source of γ ’s from DM annihilationis prompt emission, gammas produced in the decay chains of the annihilation products. If for the dark disk the annihilation cross-section is close to a resonance, then the effect on the synchrotronradiation from the e ± of DM origin is even more prominent. – 19 – E I n t en s i t y ( G e V c m - s e c - s r - ) Latitude (degrees)XDM e + e - ChannelM χ = 1.2 TeV, ICS E = 10 GeV(a) Fermi Isotropic, E γ = 10 GeVTotal, BF = 130Spherical Halo componentDark Disk componentMixed term component 1e-07 1e-06 1e-05 1e-04 0 10 20 30 40 50 60 70 80 90 E I n t en s i t y ( G e V c m - s e c - s r - ) Latitude (degrees)XDM e + e - Channel, 1/v ∝ r M χ = 1.2 TeV(b) Fermi Isotropic, E γ = 10 GeVFermi Isotropic, E γ = 100 GeVICS E γ = 10 GeV, Disk Only, BF = 180ICS E γ = 10 GeV, Spherical Halo Only, BF = 90ICS E γ = 100 GeV, Disk Only, BF = 180ICS E γ = 100 GeV, Spherical Halo Only, BF = 90 Figure 10: γ -ray flux from ICS of DM e ± off the ISRF as a function of galactic latitude for the XDM e ± channel with m χ = 1 . E γ = 10 GeV, saturated Sommerfeld enhancement. Right:Sommerfeld enhancement with annihilation in the dark disk only and in the spherical DM halo onlyfor E γ = 10 and 100 GeV. The isotropic diffuse γ -ray flux measured by Fermi [6] at 10 GeV and 100GeV(extrapolated) is shown within 2 σ . In Fig. 10a we show the flux of 10 GeV ICS γ -rays for the XDM e ± channel, averagedover longitudes | l | < ◦ , as a function of latitude b . Here we assume equal local DM densitiesin the spherical and disk halos, and normalize the local e + + e − flux to fit the Fermi data. Wealso indicate in the plot the isotropic diffuse γ -ray flux at 10 GeV as measured by Fermi [6]with 2 σ errorbars. As recently suggested by [43], about ∼
20% of the isotropic diffuse γ -rayflux for E > . E − . . (See also [5].) Thus, it is possible that some of the isotropic diffuse γ -rayflux is coming from dark matter annihilation or decay. For the scenario shown in Fig. 10a,the galactic diffuse γ -ray flux for | b | > ◦ is very similar in magnitude to the isotropic diffuseflux. It is thus evident that constraints can be put on the annihilation rate of DM modelsfrom diffuse γ -rays at high latitude. On a related note, [104] and [4] have suggested that darkmatter substructure at z > γ -ray flux.[4, 2, 68] have shown that calculations of the extragalactic γ -ray flux from cosmological DMcan also put constraints on the annihilation cross-section of various DM models.Including a dark disk increases the galactic diffuse γ -ray flux at high latitudes ( | l | > ◦ )by 1/2 relative to the flux at low latitudes (near the GC), in cases where the main productionmechanism of 10 GeV γ -rays from dark matter annihilation is ICS of e ± off the InterstellarRadiation Field (ISRF). This is readily seen in Fig. 10a by comparing the flux from theSpherical Halo to the total flux; the flux from the SH falls off more quickly with latitude. E γ ∼
100 GeV is also relevant for probing DM, but
Fermi statistics are very low and CR contaminationis more significant around 100 GeV. The flux is based on the data shown in [4] and [43]. – 20 –herefore, a significant disk component of dark matter could result in stronger constraintscoming from the isotropic diffuse γ -ray flux than from the diffuse flux at the Galactic Center.For DM models producing a large fraction their total gamma rays at E γ = 10 GeV comingfrom prompt emission, the relative increase of high latitude flux to low latitude flux is smaller.We therefore conclude that a dark disk can’t be a significant issue in the search for γ -rayanisotropies due to DM clumps [47, 75, 97, 24, 74, 70, 98, 61] at high latitudes.We now consider a situation in which the Sommerfeld enhancement is not in saturation,so that the annihilation cross-sections in the disk and spherical halo may be quite different.We assume ρ DD /ρ SH ∼
1. In Fig. 10b, we show the γ -ray flux from ICS of DM e ± off theISRF as a function of galactic latitude for the two extreme cases, one in which the annihilationin the spherical halo is dominant, i.e. we ignore any dark disk contribution, and one in whichthe annihilation in the dark disk is dominant, i.e. we ignore any spherical halo contribution.We show results for the XDM e ± annihilation channel with M χ = 1 . Fermi isotropic flux at E γ = 10 GeV and 100 GeV. We normalize the annihilation rates inboth cases to give agreement with the local e + e − CR spectra. The diffuse γ -ray flux at highlatitudes is larger by a factor of 2(4) at E γ = 10(100) GeV in the disk-dominated scenariothan in the spherical halo-dominated scenario. Thus, even if the Sommerfeld enhancement iscurrently not saturated, the existence of a dark disk with a velocity dispersion of ∼
30 km/sdoes not significantly increase the high latitude diffuse γ -ray flux. Clearly, Fig. 10b indicatesthat the annihilation rate in the dark disk is highly constrained by the isotropic diffuse fluxat E γ = 10 GeV.Since the velocity dispersion increases with decreasing galactocentric distance, annihila-tion cross-sections toward the center of the Galaxy will be smaller in models of DM with aSommerfeld enhancement, provided the enhancement is not saturated. So, the constraints onthe BF needed to fit the local CR e ± fluxes coming from the γ -rays measured by HESS in theGalactic Ridge (GR) region and inner 0 . ◦ of the Galactic Center [12, 11, 13, 16] are lifted, aswas shown in [33]. While the existence of a dark disk boosts the DM signals of local origin,the e ± signals, while at the same time not dramatically increasing the γ signals from the GC,the real gain to be had in Sommerfeld-enhanced models comes from the effects of the increasein velocity dispersion toward the GC. For DM models without Sommerfeld enhancement, theconstraints placed on the local annihilation rates by the gamma ray fluxes in the GR and GCregions [20, 22, 36, 45] are similar with and without a dark disk component.
4. Conclusions
A self-consistent explanation of the
PAMELA and
Fermi signals as coming from dark matterannihilation requires a local annihilation rate in the spherical halo that is 2 to 3 orders ofmagnitude larger than that expected for a thermal WIMP. The large boost factor could arisefrom substructure in the halo (a local inhomogeneity), from the Sommerfeld enhancement,or from a Breit-Wigner resonance, or it could be an indication of a non-thermal WIMP. Weconsider effects of a dark disk component with a local density of 0 . − . The lower– 21 –elocity dispersion of dark matter particles in the dark disk can naturally lead to higherannihilation rates through the Sommerfeld enhancement, which scales as 1 /v rel . We showthat if the dominant dark matter contribution to the local high energy e ± flux comes from adark disk, rather than the spherical halo, we can get good fits to the Fermi and
PAMELA witha boost factor that is almost an order of magnitude smaller than what is needed for the fitsto the data when only the spherical halo profile is taken into account. Thus, the Sommerfeldenhancement for the spherical halo with a velocity dispersion of v rel ∼
200 km/s need not takethe extremely large values previously thought. We stress that many of our conclusions wouldhold to varying degrees for all flattened halo profiles, even those with kinematics similar tothe spherical halo, since all of the cosmic ray dark matter signals are highly dependent onthe spatial distribution of the dark matter.We show that the locally observed spectra of e + e − are harder for a dark disk profile thanfor a spherical profile. This can improve the fits to the Fermi data for those annihilationmodes with softer e + e − spectra. Some tension arises as a result of adding a disk componentto the dark matter density, because the discrepancy between the boost factors needed to fitthe PAMELA positron fraction and to fit the
Fermi data increases with the addition of a diskcomponent. The spherical halo gives differences of factors of ∼ −
3, while the dark disk givesdifferences of factors of up to ∼ e + e − flux is dominant at the lower energies measured by PAMELA , whilefor energies greater than 300 GeV, the
Fermi range, the dark disk contribution to the e + e − flux becomes comparable. That, however, depends on the assumptions about the contributionto the local DM density from the dark disk and the relative annihilation cross-sections in thedark disk and the spherical halo. Furthermore, the possibility of multi-component DM ornearby pulsars contributing significantly to the local spectra decreases the significance of adifference in the boost factors necessary to fit PAMELA vs Fermi .We find that for hadronic annihilation channels, if the annihilation in the disk is thedominant contribution to the local cosmic ray fluxes of dark matter origin (and if the diskis thicker than the diffusion zone), the data allows for an annihilation rate that is larger bya factor of ∼ PAMELA antiprotondata constrains the annihilation products to be mainly leptonic for masses of m χ ∼ e ± fluxes usingthe Fermi e + + e − data and CMB data. The Fermi data is consistent with a scenario in whichthe dark disk is the dominant contributor to the local e + e − flux for the XDM annihilationchannels we studied. With the data currently available, it is impossible to determine whetherthe local e + e − flux is produced dominantly in one halo component or approximately equallyin both the disk and the spherical halo. In fact, due to the many possible dark matter massesand annihilation modes in the production of the local e + e − , it will be very hard to tell thedifference even with better data. The constraints on the DM annihilation rate coming fromthe WMAP5 measurements of the temperature and polarization power spectra of the CMBdo not exclude any of the XDM annihilation channels studied in a disk-dominated scenario.However, because the dark disk produces fewer electrons and positrons at
PAMELA energies– 22 –han the spherical halo, there is more tension for the disk than for the spherical halo betweenthe BF needed to fit the
PAMELA data and the CMB constraint.We find that the contribution from the dark disk can be up to ∼ / e ± flux of dark matter origin for E e ± >
500 GeV. Additionally, we find that including a darkdisk has a minimal effect on the flux of synchrotron radiation from dark matter e ± in themicrowave haze region. This result is valid even for disks with relatively large scale heights( ∼ ∼ γ -ray flux at high latitudes by more than O (1). If the Sommerfeldenhancement is not saturated, so that the annihilation rate in the disk is much greater thanthe annihilation rate in the spherical halo, the contribution to the gamma ray flux from thedisk at 10 GeV exceeds the value of the isotropic γ -ray flux as measured by Fermi by a factorof ∼ E γ = 10 GeV. Acknowledgments
IC and LG are supported by DOE OJI grant
Appendix A: 2-Step e + e − spectra The probability distribution function of injected electrons created by the annihilationchannel χχ → ΦΦ → φ → (4 µ ± → ) 4 e ± can be calculated analytically or numerically. Inthe frame of φ , it is the convolution of the spectra of electrons and positrons in that frame withthe spectrum of φ as seen from the frame in which annihilation takes place. For simplicity inour analytic calculations, we consider Φ and φ to be scalars so that the decay products have anisotropic distribution in the rest frame of the parent particle. The difference in the e ± spectraresulting from a vector mediator is a spectral hardening of O (0 .
1) at the highest ∼
10% of theenergies. Since the deviations appear only at the highest energies where experiments havetheir largest errorbars, the spectra for the scalar mediator case can be considered a goodapproximation to the spectra for the vector mediator case. In both cascades described, themultiplicity of electrons (positrons) produced per annihilation is 4. Our results agree withthose of [81].For the ”direct” case, φ → e ± , the distribution in the approximation m χ ≫ m Φ ≫ m φ ≫ m e is given by: f ( E ) = 1 m χ ln (cid:16) m χ E (cid:17) Θ (cid:16) m χ − E (cid:17) . (1)– 23 –or the case of φ → µ + µ − and consequently muon decay, the spectrum’s given by : f ( E ) = h m µ W m φ (cid:16)
12 1 γ − γ ln (cid:16) γ γ (cid:17) − γ − γ ln (cid:16) γ W m φ E m µ (cid:17) · (cid:16) γ γ − − ln( γ γ (cid:17) + 1 γ ln (cid:16) γ W m φ Em µ (cid:17)(cid:17) + 1936 m µ W m φ (cid:16) γ − γ · (cid:16) γ γ − − ln (cid:16) γ γ (cid:17)(cid:17) − γ (cid:17) − (cid:16) m µ W m φ (cid:17) E (cid:16) γ − γ (cid:16) γ − γ + 2 γ γ (cid:17) − γ (cid:17) + (cid:16) m µ W m φ (cid:17) E (cid:16) γ − γ (cid:16) γ − γ + 4 γ γ (cid:17) − γ (cid:17)i Θ (cid:16) W γ m φ m µ − E (cid:17) + h m µ W m φ (cid:16) γ − γ − γ − γ ln (cid:16) γ W · m φ E m µ (cid:17) + 512 1 γ − γ ln (cid:16) γ W m φ E m µ (cid:17)(cid:17) − (cid:16) m µ W m φ (cid:17) E γ − γ ) γ + 1108 (cid:16) m µ W m φ (cid:17) E γ − γ ) γ i Θ (cid:16) E − W γ m φ m µ (cid:17) Θ (cid:16) W γ m φ m µ − E (cid:17) . where W = m µ , γ = m χ m φ (cid:16) − r − (cid:16) m φ m Φ (cid:17) s − (cid:16) m Φ m χ (cid:17) (cid:17) , and γ = m χ m φ − γ . References [1] Planck: The scientific programme. 2006.[2] Kevork N. Abazajian, Prateek Agrawal, Zackaria Chacko, and Can Kilic. ConservativeConstraints on Dark Matter from the Fermi-LAT Isotropic Diffuse Gamma-Ray BackgroundSpectrum. 2010.[3] A. A. Abdo et al. Discovery of TeV gamma-ray emission from the Cygnus region of the galaxy.
Astrophys. J. , 658:L33–L36, 2007.[4] A. A. Abdo et al. Constraints on Cosmological Dark Matter Annihilation from the Fermi-LATIsotropic Diffuse Gamma-Ray Measurement.
JCAP , 1004:014, 2010.[5] A. A. Abdo et al. The Spectral Energy Distribution of Fermi bright blazars.
Astrophys. J. ,716:30–70, 2010.[6] A. A. Abdo et al. The Spectrum of the Isotropic Diffuse Gamma-Ray Emission Derived FromFirst-Year Fermi Large Area Telescope Data.
Phys. Rev. Lett. , 104:101101, 2010.[7] Aous A. Abdo et al. Measurement of the Cosmic Ray e+ plus e- spectrum from 20 GeV to 1TeV with the Fermi Large Area Telescope.
Phys. Rev. Lett. , 102:181101, 2009.[8] O. Adriani et al. A new measurement of the antiproton-to-proton flux ratio up to 100 GeV inthe cosmic radiation.
Phys. Rev. Lett. , 102:051101, 2009. assuming m e = 0 and γ >
10 which are valid for the cases at hand. – 24 –
9] O. Adriani et al. A statistical procedure for the identification of positrons in the PAMELAexperiment. 2010.[10] Oscar Adriani et al. An anomalous positron abundance in cosmic rays with energies 1.5.100GeV.
Nature , 458:607–609, 2009.[11] F. Aharonian et al. Very high energy gamma rays from the direction of Sagittarius A*.
Astron.Astrophys. , 425:L13–L17, 2004.[12] F. Aharonian et al. Discovery of Very-High-Energy Gamma-Rays from the Galactic CentreRidge.
Nature , 439:695–698, 2006.[13] F. Aharonian et al. HESS observations of the galactic center region and their possible darkmatter interpretation.
Phys. Rev. Lett. , 97:221102, 2006.[14] F. Aharonian et al. The energy spectrum of cosmic-ray electrons at TeV energies.
Phys. Rev.Lett. , 101:261104, 2008.[15] F. Aharonian et al. Probing the ATIC peak in the cosmic-ray electron spectrum with H.E.S.S.
Astron. Astrophys. , 508:561, 2009.[16] H. E. S. S. collaboration: F. Aharonian. Spectrum and variability of the Galactic Center VHEgamma- ray source HESS J1745-290. 2009.[17] Nima Arkani-Hamed, Douglas P. Finkbeiner, Tracy R. Slatyer, and Neal Weiner. A Theory ofDark Matter.
Phys. Rev. , D79:015014, 2009.[18] Edward A. Baltz and Dan Hooper. Kaluza-Klein dark matter, electrons and γ ray telescopes. JCAP , 0507:001, 2005.[19] V. Barger, Y. Gao, Wai Yee Keung, D. Marfatia, and G. Shaughnessy. Dark matter and pulsarsignals for Fermi LAT, PAMELA, ATIC, HESS and WMAP data.
Phys. Lett. , B678:283–292,2009.[20] Nicole F. Bell and Thomas D. Jacques. Gamma-ray Constraints on Dark Matter Annihilationinto Charged Particles.
Phys. Rev. , D79:043507, 2009.[21] Lars Bergstrom, Torsten Bringmann, and Joakim Edsjo. New Positron Spectral Features fromSupersymmetric Dark Matter - a Way to Explain the PAMELA Data?
Phys. Rev. ,D78:103520, 2008.[22] Gianfranco Bertone, Marco Cirelli, Alessandro Strumia, and Marco Taoso. Gamma-ray andradio tests of the e+e- excess from DM annihilations.
JCAP , 0903:009, 2009.[23] Xiao-Jun Bi et al. Non-Thermal Production of WIMPs, Cosmic e ± Excesses and γ -rays fromthe Galactic Center. Phys. Rev. , D80:103502, 2009.[24] Jo Bovy. Substructure Boosts to Dark Matter Annihilation from Sommerfeld Enhancement.
Phys. Rev. , D79:083539, 2009.[25] Tobias Bruch, Annika H. G. Peter, Justin Read, Laura Baudis, and George Lake. Dark MatterDisc Enhanced Neutrino Fluxes from the Sun and Earth.
Phys. Lett. , B674:250–256, 2009.[26] P. Carlson. PAMELA science.
Int. J. Mod. Phys. , A20:6731–6734, 2005.[27] Riccardo Catena and Piero Ullio. A novel determination of the local dark matter density. 2009. – 25 –
28] Hsin-Chia Cheng, Jonathan L. Feng, and Konstantin T. Matchev. Kaluza-Klein dark matter.
Phys. Rev. Lett. , 89:211301, 2002.[29] Ilias Cholis, Gregory Dobler, Douglas P. Finkbeiner, Lisa Goodenough, and Neal Weiner. TheCase for a 700+ GeV WIMP: Cosmic Ray Spectra from ATIC and PAMELA.
Phys. Rev. ,D80:123518, 2009.[30] Ilias Cholis, Douglas P. Finkbeiner, Lisa Goodenough, and Neal Weiner. The PAMELAPositron Excess from Annihilations into a Light Boson.
JCAP , 0912:007, 2009.[31] Ilias Cholis, Lisa Goodenough, Dan Hooper, Melanie Simet, and Neal Weiner. High EnergyPositrons From Annihilating Dark Matter.
Phys. Rev. , D80:123511, 2009.[32] Ilias Cholis, Lisa Goodenough, and Neal Weiner. High Energy Positrons and the WMAP Hazefrom Exciting Dark Matter.
Phys. Rev. , D79:123505, 2009.[33] Ilias Cholis and Neal Weiner. MiXDM: Cosmic Ray Signals from Multiple States of DarkMatter. 2009.[34] Marco Cirelli and James M. Cline. Can multistate dark matter annihilation explain the high-energy cosmic ray lepton anomalies? 2010.[35] Marco Cirelli, Mario Kadastik, Martti Raidal, and Alessandro Strumia. Model-independentimplications of the e+, e-, anti-proton cosmic ray spectra on properties of Dark Matter. 2008.[36] Marco Cirelli and Paolo Panci. Inverse Compton constraints on the Dark Matter e+e-excesses.
Nucl. Phys. , B821:399–416, 2009.[37] Marco Cirelli, Paolo Panci, and Pasquale D. Serpico. Diffuse gamma ray constraints onannihilating or decaying Dark Matter after Fermi. 2009.[38] Marco Cirelli and Alessandro Strumia. Minimal Dark Matter predictions and the PAMELApositron excess.
PoS , IDM2008:089, 2008.[39] Marco Cirelli, Alessandro Strumia, and Matteo Tamburini. Cosmology and Astrophysics ofMinimal Dark Matter.
Nucl. Phys. , B787:152–175, 2007.[40] James M. Cline, Aaron C. Vincent, and Wei Xue. Leptons from Dark Matter Annihilation inMilky Way Subhalos.
Phys. Rev. , D81:083512, 2010.[41] Fermi Collaboration. 2009 fermi symposium, 2009.[42] PAMELA Collaboration. PAMELA results on the cosmic-ray antiproton flux from 60 MeV to180 GeV in kinetic energy. 2010.[43] The Fermi-LAT Collaboration. The Fermi-LAT high-latitude Survey: Source CountDistributions and the Origin of the Extragalactic Diffuse Background. 2010.[44] M. Creze, E. Chereul, O. Bienayme, and C. Pichon Centre de Donnees astronomique deStrasbourg IU Basel. The distribution of nearby stars in phase space mapped by Hipparcos: I.The potential well and local dynamical mass. 1997.[45] R. M. Crocker, N. F. Bell, C. Balazs, and D. I. Jones. Radio and gamma-ray constraints ondark matter annihilation in the Galactic center.
Phys. Rev. , D81:063516, 2010.[46] J. Diemand et al. Clumps and streams in the local dark matter distribution.
Nature ,454:735–738, 2008. – 26 –
47] Jurg Diemand, Michael Kuhlen, and Piero Madau. Dark matter substructure and gamma-rayannihilation in the Milky Way halo.
Astrophys. J. , 657:262, 2007.[48] Gregory Dobler and Douglas P. Finkbeiner. Extended Anomalous Foreground Emission in theWMAP 3-Year Data.
Astrophys. J. , 680:1222–1234, 2008.[49] Gregory Dobler, Douglas P. Finkbeiner, Ilias Cholis, Tracy R. Slatyer, and Neal Weiner. TheFermi Haze: A Gamma-Ray Counterpart to the Microwave Haze. 2009.[50] F. Donato, D. Maurin, P. Brun, T. Delahaye, and P. Salati. Constraints on WIMP DarkMatter from the High Energy PAMELA ¯ p/p data.
Phys. Rev. Lett. , 102:071301, 2009.[51] Bhaskar Dutta, Louis Leblond, and Kuver Sinha. Mirage in the Sky: Non-thermal DarkMatter, Gravitino Problem, and Cosmic Ray Anomalies.
Phys. Rev. , D80:035014, 2009.[52] S. I. Eidelman. Recent results from e+ e- –¿ hadrons. eConf , C0209101:WE08, 2002.[53] J. Einasto. Influence of the atmospheric and instrumental dispersion on the brightnessdistribution in a galaxy.
Trudy Inst. Astrofiz. Alma-Ata , 51, 87, 1965.[54] Jonathan L. Feng, Manoj Kaplinghat, and Hai-Bo Yu. Halo Shape and Relic DensityExclusions of Sommerfeld- Enhanced Dark Matter Explanations of Cosmic Ray Excesses.
Phys. Rev. Lett. , 104:151301, 2010.[55] Jonathan L. Feng, Manoj Kaplinghat, and Hai-Bo Yu. Sommerfeld Enhancements for ThermalRelic Dark Matter. 2010.[56] Douglas P. Finkbeiner. Microwave ism emission observed by wmap.
Astrophys. J. ,614:186–193, 2004.[57] Douglas P. Finkbeiner. Wmap microwave emission interpreted as dark matter annihilation inthe inner galaxy. 2004.[58] Douglas P. Finkbeiner and Neal Weiner. Exciting dark matter and the integral/spi 511 kevsignal.
Phys. Rev. , D76:083519, 2007.[59] Fabio Governato. Private communication, 2009.[60] Wan-Lei Guo and Yue-Liang Wu. Enhancement of Dark Matter Annihilation via Breit-WignerResonance.
Phys. Rev. , D79:055012, 2009.[61] Brandon S. Hensley, Jennifer M. Siegal-Gaskins, and Vasiliki Pavlidou. The detectability ofdark matter annihilation with Fermi using the anisotropy energy spectrum of the gamma-raybackground. 2009.[62] Junji Hisano, Shigeki. Matsumoto, Mihoko M. Nojiri, and Osamu Saito. Non-perturbativeeffect on dark matter annihilation and gamma ray signature from galactic center.
Phys. Rev. ,D71:063528, 2005.[63] Johan Holmberg and Chris Flynn. The local density of matter mapped by Hipparcos.
Mon.Not. Roy. Astron. Soc. , 313:209–216, 2000.[64] Johan Holmberg and Chris Flynn. The local surface density of disc matter mapped byHipparcos.
Mon. Not. Roy. Astron. Soc. , 352:440, 2004.[65] Dan Hooper, Douglas P. Finkbeiner, and Gregory Dobler. Evidence Of Dark MatterAnnihilations In The WMAP Haze.
Phys. Rev. , D76:083012, 2007. – 27 –
66] Dan Hooper, Albert Stebbins, and Kathryn M. Zurek. Excesses in cosmic ray positron andelectron spectra from a nearby clump of neutralino dark matter.
Phys. Rev. , D79:103513, 2009.[67] Dan Hooper and Kathryn Zurek. The PAMELA and ATIC Signals From Kaluza-Klein DarkMatter. 2009.[68] Gert Hutsi, Andi Hektor, and Martti Raidal. Implications of the Fermi-LAT diffusegamma-ray measurements on annihilating or decaying Dark Matter. 2010.[69] P. M. W. Kalberla, L. Dedes, J. Kerp, and U. Haud. Dark matter in the Milky Way, II. the HIgas distribution as a tracer of the gravitational potential. 2007.[70] Matthew D. Kistler and Jennifer M. Siegal-Gaskins. Gamma-ray signatures of annihilation tocharged leptons in dark matter substructure.
Phys. Rev. , D81:103521, 2010.[71] T. Kobayashi, Y. Komori, K. Yoshida, and J. Nishimura. The Most Likely Sources of HighEnergy Cosmic-Ray Electrons in Supernova Remnants.
Astrophys. J. , 601:340–351, 2004.[72] V. I. Korchagin, Terrence M. Girard, T. V. Borkova, D. I. Dinescu, and W. F. van Altena.Local Surface Density of the Galactic Disk from a 3-D Stellar Velocity Sample. 2003.[73] M. Kuhlen, J. Diemand, P. Madau, and M. Zemp. The Via Lactea INCITE Simulation:Galactic Dark Matter Substructure at High Resolution.
J. Phys. Conf. Ser. , 125:012008, 2008.[74] Michael Kuhlen. The Dark Matter Annihilation Signal from Dwarf Galaxies and Subhalos.2009.[75] Michael Kuhlen, Jurg Diemand, and Piero Madau. The Dark Matter Annihilation Signal fromGalactic Substructure: Predictions for GLAST. 2008.[76] Massimiliano Lattanzi and Joseph I. Silk. Can the WIMP annihilation boost factor be boostedby the Sommerfeld enhancement?
Phys. Rev. , D79:083523, 2009.[77] Malcolm S. Longair.
High Energy Astrophysics, Volume 2, pp. 296-301, 351-357 . CambridgeUniversity Press, 2002.[78] Dmitry Malyshev, Ilias Cholis, and Joseph Gelfand. Pulsars versus Dark Matter Interpretationof ATIC/PAMELA.
Phys. Rev. , D80:063005, 2009.[79] John March-Russell, Stephen M. West, Daniel Cumberbatch, and Dan Hooper. Heavy DarkMatter Through the Higgs Portal.
JHEP , 07:058, 2008.[80] John David March-Russell and Stephen Mathew West. WIMPonium and Boost Factors forIndirect Dark Matter Detection.
Phys. Lett. , B676:133–139, 2009.[81] Jeremy Mardon, Yasunori Nomura, Daniel Stolarski, and Jesse Thaler. Dark Matter Signalsfrom Cascade Annihilations.
JCAP , 0905:016, 2009.[82] Patrick Meade, Michele Papucci, and Tomer Volansky. Dark Matter Sees The Light.
JHEP ,12:052, 2009.[83] David Merritt, Julio F. Navarro, Aaron Ludlow, and Adrian Jenkins. A Universal DensityProfile for Dark and Luminous Matter?
Astrophys. J. , 624:L85–L88, 2005.[84] Silvio Orsi. PAMELA: A payload for antimatter matter exploration and light nucleiastrophysics.
Nucl. Instrum. Meth. , A580:880–883, 2007. – 28 –
85] Michele Papucci and Alessandro Strumia. Robust implications on Dark Matter from the firstFERMI sky gamma map.
JCAP , 1003:014, 2010.[86] Miguel Pato, Oscar Agertz, Gianfranco Bertone, Ben Moore, and Romain Teyssier. Systematicuncertainties in the determination of the local dark matter density. 2010.[87] Susana E. Pedrosa, Patricia B. Tissera, and Cecilia Scannapieco. The joint evolution ofbaryons and dark matter haloes. 2009.[88] Lidia Pieri, Massimiliano Lattanzi, and Joseph Silk. Constraining the Sommerfeldenhancement with Cherenkov telescope observations of dwarf galaxies. 2009.[89] Troy A. Porter and A. W. Strong. A new estimate of the Galactic interstellar radiation fieldbetween 0.1 microns and 1000 microns. 2005.[90] Maxim Pospelov and Adam Ritz. Astrophysical Signatures of Secluded Dark Matter.
Phys.Lett. , B671:391–397, 2009.[91] J. I. Read, G. Lake, O. Agertz, and Victor P. Debattista. Thin, thick and dark discs in LCDM.2008.[92] J. I. Read, L. Mayer, A. M. Brooks, F. Governato, and G. Lake. A dark matter disc in threecosmological simulations of Milky Way mass galaxies. 2009.[93] Emilio Romano-Diaz, Isaac Shlosman, Clayton Heller, and Yehuda Hoffman. DissectingGalaxy Formation: I. Comparison Between Pure Dark Matter and Baryonic Models.
Astrophys. J. , 702:1250–1267, 2009.[94] Emilio Romano-Diaz, Isaac Shlosman, Yehuda Hoffman, and Clayton Heller. Erasing DarkMatter Cusps in Cosmological Galactic Halos with Baryons. 2008.[95] P. Salucci, F. Nesti, G. Gentile, and C. F. Martins. The dark matter density at the Sun’slocation. 2010.[96] Geraldine Servant and Timothy M. P. Tait. Is the lightest Kaluza-Klein particle a viable darkmatter candidate?
Nucl. Phys. , B650:391–419, 2003.[97] Jennifer M. Siegal-Gaskins. Revealing dark matter substructure with anisotropies in the diffusegamma-ray background.
JCAP , 0810:040, 2008.[98] Jennifer M. Siegal-Gaskins and Vasiliki Pavlidou. Robust identification of isotropic diffusegamma rays from Galactic dark matter.
Phys. Rev. Lett. , 102:241301, 2009.[99] Tracy R. Slatyer, Nikhil Padmanabhan, and Douglas P. Finkbeiner. CMB Constraints onWIMP Annihilation: Energy Absorption During the Recombination Epoch.
Phys. Rev. ,D80:043526, 2009.[100] Volker Springel et al. The Aquarius Project: the subhalos of galactic halos.
Mon. Not. Roy.Astron. Soc. , 391:1685–1711, 2008.[101] A. W. Strong and I. V. Moskalenko. The GALPROP program for cosmic-ray propagation: newdevelopments. 1999.[102] Andrew W. Strong, Igor V. Moskalenko, and Vladimir S. Ptuskin. Cosmic-ray propagation andinteractions in the Galaxy.
Ann. Rev. Nucl. Part. Sci. , 57:285–327, 2007. – 29 –
Phys. Rev. , D81:083502, 2010., D81:083502, 2010.