Strong Optimized Conservative Fermi-LAT Constraints on Dark Matter Models from the Inclusive Photon Spectrum
Andrea Massari, Eder Izaguirre, Rouven Essig, Andrea Albert, Elliott Bloom, German A. Gomez-Vargas
YYITP-SB-14-23
Strong Optimized Conservative
Fermi -LAT Constraints on DarkMatter Models from the Inclusive Photon Spectrum
Andrea Massari, ∗ Eder Izaguirre, † Rouven Essig, ‡ Andrea Albert, Elliott Bloom, and Germán Arturo Gómez-Vargas
4, 51
C.N. Yang Institute for Theoretical Physics,Stony Brook University, Stony Brook, NY 11794-3840 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 6B9, Canada SLAC - KIPAC, Stanford University, Stanford, CA 94305-4085 Instituto de Fisíca, Pontificia Universidad Católica de Chile,Avenida Vicuña Mackenna 4860, Santiago, Chile Istituto Nazionale di Fisica Nucleare,Sezione di Roma “Tor Vergata”, I-00133 Roma, Italy
Abstract
We set conservative, robust constraints on the annihilation and decay of dark matter into variousStandard Model final states under various assumptions about the distribution of the dark matter inthe Milky Way halo. We use the inclusive photon spectrum observed by the
Fermi Gamma-ray SpaceTelescope through its main instrument, the Large-Area Telescope (LAT). We use simulated data tofirst find the “optimal” regions of interest in the γ -ray sky, where the expected dark matter signal islargest compared with the expected astrophysical foregrounds. We then require the predicted darkmatter signal to be less than the observed photon counts in the a priori optimal regions. This yieldsa very conservative constraint as we do not attempt to model or subtract astrophysical foregrounds.The resulting limits are competitive with other existing limits, and, for some final states with cuspydark-matter distributions in the Galactic Center region, disfavor the typical cross section requiredduring freeze-out for a weakly interacting massive particle (WIMP) to obtain the observed relicabundance. ∗ Contact Author: [email protected] † Contact Author: [email protected] ‡ Contact Author: [email protected] a r X i v : . [ h e p - ph ] M a r ontents
1. INTRODUCTION 32. EXPECTED DARK MATTER SIGNAL 52.1. Prompt radiation 52.2. Inverse Compton Scattering 93. DATA SETS AND METHODS 113.1. Event Selection 113.2. Simulated (Monte Carlo) Data Sets 123.3. ROI Choice 123.4. Optimizing the ROIs and Energy Ranges using Simulated Data 133.5. Illustration of Procedure 154. RESULTS AND DISCUSSION 174.1. Constraints on Dark Matter Annihilation 174.2. Constraints on Dark Matter Decays 215. CONCLUSIONS 24A. Constraints on DM Models invoked to explain γ Rays from Inner Galaxy 26B. Dependence of Optimal ROI and Energy Range on DM Profile and DM Mass 27C. Effect of Source Masking and Choice of Front-/Back-converting events on Limits 28D. Inverse Compton Scattering 31E. Details on the Simulated data sets 33F. Comparison of limits between simulated and real data 34References 352 . INTRODUCTION
The
Fermi Gamma-ray Space Telescope ( Fermi ), through its main instrument, the LargeArea Telescope (LAT) [1], has been surveying the γ -ray sky since August 2008 in the energyrange from 20 MeV to above 300 GeV (with detected events up to ∼ TeV). In addition to γ rays produced by known astrophysical sources, the Fermi -LAT can detect photons frompostulated decay or annihilation of dark matter (DM) to Standard Model (SM) particles.The possibility that DM can annihilate is particularly motivated by the “WIMP miracle” [2].Here one hypothesizes the existence of weakly interacting massive particles (WIMPs) withfew-GeV to few-TeV masses and weak-scale annihilation cross sections. These WIMPs wouldhave been in thermal equilibrium with the SM sector in the early Universe and they generallyproduce the observed relic abundance of DM from thermal freeze-out. This suggests thatWIMPs could still be annihilating today to SM particles. The annihilation could producevarious SM particles, which can either radiate photons, further decay to other SM particlesincluding photons, or inverse Compton scatter (ICS) off background light, producing high-energy γ rays. Those photons that arrive at the Fermi -LAT could then be used to inferproperties of the DM particles and their distribution around us.Many WIMP searches have been performed using
Fermi -LAT data. Analyses by the
Fermi -LAT Collaboration and outside groups have searched for monochromatic γ -raylines [3–9] and continuum γ -ray excesses in the diffuse spectrum from different targetregions e.g. , dwarf spheroidal galaxies [10–16], clusters of galaxies [17–19], the Galactichalo [13, 20–22], the Inner Galaxy [23–36], the Smith cloud [37, 38], and the extragalactic γ -ray background [39–42]. No undisputed signal of DM has been detected thus far, and thecross-section upper limits from these analyses for DM masses m DM (cid:46) GeV are approach-ing the typical cross section required during freeze-out for a WIMP to obtain the observedrelic abundance, namely (cid:104) σv (cid:105) relic ∼ × − cm s − .While DM is often thought of as being a stable particle, viable DM candidates onlyneed to be stable on cosmological time-scales. In particular, DM lifetimes of the orderof the age of the Universe or longer ( τ DM > s) can typically evade cosmological andastrophysical bounds more easily than annihilating DM, such as constraints from Big BangNucleosynthesis [43], the extragalactic γ -ray background [44], and re-ionization and theCosmic-Microwave-Background [45–50]. The more relaxed constraints on decaying DM are3 result of the DM decay rate being linear with ρ DM , as opposed to quadratic with ρ DM inthe case of annihilation.In this paper, we will provide conservative DM cross-section upper limits and decay-lifetime lower limits from the Fermi -LAT inclusive photon spectrum. The inclusive spectrumis presumably dominated by astrophysical foregrounds in the Milky Way, though DM couldcontribute to it. We make no attempt at subtracting foregrounds and simply require thatany putative DM signal contribute less than the observed flux. A similar idea has beenused in other papers to derive conservative constraints [20, 22, 23], where the DM signal ismaximized until saturating the observed flux. The approach in this paper differs from suchprevious analyses in several ways, resulting in stronger constraints on DM. Firstly, we restrictour regions of interest (ROIs) to have a particular symmetric shape determined by only afew free parameters, and we optimize over these parameters. Secondly, we also optimize theenergy range that we use for deriving the constraint. Thirdly, we optimize with respect tothe constraint itself and not, for example, the signal-to-noise ratio, and last, we optimize ourconstraints on 10 simulated data sets, not on the measured data. After finding the optimalROI on simulated data, we use the real data from that same ROI to find the constraint. Wederive constraints in this fashion for various DM-halo shapes and for various annihilationand decay final states. The resulting constraints, while being robust and conservative asno foregrounds have been subtracted, are competitive with other existing constraints andstronger than other conservative bounds obtained by [20, 22, 23].The paper is organized as follows. In §2 we discuss the calculation of the expected γ -ray flux from DM annihilation and decay. In §3 we discuss the event selection, method,simulated data sets, and ROI selection. §4 discusses the resulting constraints, while ourconclusions are in §5. In Appendix A we use our method to calculate the limits on DM-annihilation models that have been invoked to explain an excess of γ rays from the GalacticCenter (GC) and Inner Galaxy region. Appendix B presents the optimal ROIs together withthe corresponding count spectra for several DM channels. Appendix C discusses the effecton our results of source masking and choosing front-/back-converting events. Appendix Ddescribes the astrophysical assumptions affecting the results that include contributions fromICS. Finally, Appendix E provides more details on the simulated data sets that we use, andAppendix F compares the limits obtained from our simulated data sets with those derivedfrom real data. 4 . EXPECTED DARK MATTER SIGNAL Gamma rays from DM annihilation or decay to SM final states can be produced in twodominant ways. The first possibility, which we refer to as prompt , is from either final-stateradiation (FSR) produced by Bremsstrahlung by SM particles or from the decay of hadronsthat arise in hadronic final states. The second possibility is from electrons and positrons(produced either directly or at the end of a cascade decay chain) that inverse Compton scatteroff background ambient light, which primarily consists of starlight, the infrared backgroundlight, and the Cosmic Microwave Background (CMB). This ICS process boosts the energyof the background light to produce γ rays. Unlike prompt radiation, ICS depends on variousunknown astrophysical parameters discussed below. Although a sizable contribution to theenergy lost by the electrons propagating through the Galaxy consists of synchrotron radiationdue to acceleration by the Galactic magnetic field, we note that the synchrotron radiationdoes not make up a noticeable fraction of the γ rays in the energy range under study, aswe only consider DM particles with mass below 10 TeV [51, 52]. We thus do not include itin this study. Moreover, the DM signal can receive additional sizeable contributions due toGalactic substructure, particularly for annihilations [53], but we do not include this effect inour study. This makes our analysis more conservative and model independent in this regard.We now outline the calculation of the DM-initiated γ -ray flux. The differential flux, d Φ γ /d E γ , of prompt photons coming from DM annihilation withinthe Milky Way halo is given by dΦ γ d E γ = 18 π (cid:104) σv (cid:105) m d N γ d E γ r (cid:12) ρ (cid:12) J ann , (1)where (cid:104) σv (cid:105) is the thermally averaged DM annihilation cross section, m DM is the DM mass,and d N γ / d E γ is the photon spectrum per annihilation. We assume ρ (cid:12) = 0 . / cm is theDM density at the Sun’s location in the Galaxy [54, 55] , and r (cid:12) = 8 . is the distance A range of values between 0.2 and 0.85
GeV / cm are possible at present though [54–58]. Note that adifferent value for the local DM density would shift up or down our predictions for DM annihilation anddecay by a factor proportional to ρ and ρ , respectively for annihilations and decays. J -factor” is given by J ann ≡ (cid:90) ROI d b d (cid:96) d s cos br (cid:12) (cid:20) ρ ( r ( s, b, (cid:96) )) ρ (cid:12) (cid:21) , (2)which depends on the distribution of DM in the Milky Way halo, ρ ( r ) , where r ≡ r ( s, b, (cid:96) ) is the Galactocentric distance, given by r = (cid:112) s + r (cid:12) − sr (cid:12) cos (cid:96) cos b , where (cid:96) and b arethe Galactic longitude and latitude, respectively, and s is the line-of-sight distance. Theintegral is over a particular ROI. For decays we can replace (cid:104) σv (cid:105) ρ (cid:12) / m with ρ (cid:12) /τ m DM in Eq. (1), where τ is the DM decay lifetime, with the J -factor J dec ≡ (cid:90) ROI d b d (cid:96) d s cos br (cid:12) ρ ( r ( s, b, (cid:96) )) ρ (cid:12) . (3)Moreover, for decays the d N γ /dE γ should be interpreted as the photon spectrum for indi-vidual DM particle decays.We consider four popular DM density profiles: the Navarro-Frenk-White (NFW) [60, 61],Einasto [62, 63], Isothermal [64], and “contracted” NFW ( NFW c ) [56, 67] with slope valuestaken from [5]. ρ Isothermal ( r ) = ρ Iso0 r/r s,iso ) (4) ρ NFW ( r ) = ρ NFW0 r/r s (1 + r/r s ) (5) ρ Einasto ( r ) = ρ Ein0 exp {− (2 /α ) [( r/r s ) α − } (6) ρ NFW c ( r ) = ρ NFW c ( r/r s ) . (1 + r/r s ) . . (7)We set α = 0 . , r s = 20 kpc [63, 67], and r s,iso = 5 kpc [64]. The normalization ρ ( r (cid:12) ) = ρ (cid:12) fixes ρ Iso0 (cid:39) . , ρ NFW0 (cid:39) . , ρ Ein0 (cid:39) . , and ρ NFW c (cid:39) . in units of GeV/cm . Ourchoice of the functional form and parameters in Eq. (7) is a representative example of thepossibility that, due to adiabatic contraction from the inclusion of baryonic matter, the DMprofile might have a central slope steeper even than that of the NFW or Einasto profiles(although note that high-resolution observations of the rotation curves of dwarf and low-surface-brightness galaxies favor cored distributions [68, 69]). The four profiles are shown Another popular parametrization of a “cored” profile is the Burkert profile [65]. Adopting the best-fitparameters in [66] yields a distribution that is very close to the Isothermal one for radii (cid:46)
10 kpc . We willsee that the optimal ROIs for the Isothermal profile for DM annihilation are contained with this region,and therefore the limits for the two cored distributions would be very similar. Thus we not explicitlyconsider the Burkert profile in our analysis. .1 1 10 10010 - - - r [ kpc ] ρ D M [ G e V c m - ] IsothermalNFWEinastoNFW c r ⊙ = ρ ⊙ = - - - - - - - E γ / m DM d N / dLog ( E γ / m D M ) b b τ + τ - g gW + W - u ue + e - μ + μ - ϕ ϕ → e ϕ ϕ → μϕ ϕ → e + μ + π [ ] Figure 1 . Left:
Dark-matter density profiles versus distance from the Galactic Center (GC). Weuse the Isothermal (green), NFW (red), Einasto (blue), and a “contracted” NFW (NFW c , orange,with ρ ∝ /r . for r → ) profile. Right:
Prompt γ -ray spectra produced in the annihilation of1 TeV dark matter to e + e − , µ + µ − , τ + τ − , b ¯ b , W + W − , u ¯ u , gg ( g = a gluon), and φφ , where φ decays either only to e + e − (with m φ = 0 . GeV), or only to µ + µ − (with m φ = 0 . GeV), or to e + e − , µ + µ − , and π + π − in the ratio (with m φ = 0 . GeV). in Fig. 1 ( left ).The (prompt) photon spectra, d N γ / d E γ have been generated with Pythia 8.165 [70]or are based on formulas in [71–74]. They are the same as in
DMFIT [75] after the latestupdate described in [12]. We will consider the ten different final states e + e − , µ + µ − , τ + τ − , b ¯ b , W + W − , u ¯ u , gg ( g = a gluon), and φφ , where φ decays either only to e + e − (with m φ = 0 . GeV), or only to µ + µ − (with m φ = 0 . GeV), or to e + e − , µ + µ − , and π + π − inthe ratio (with m φ = 0 . GeV) (the latter ratio is motivated if φ is a dark photonthat kinetically mixes with the SM hypercharge gauge boson). Other SM final states are ofcourse possible but they would yield constraints very similar to the channels we consider inour analysis. The annihilation channels to φφ are motivated by DM models [76, 77] thatattempt to explain the rising positron fraction measured by PAMELA [78], Fermi [79], andAMS-02 [80, 81]; the φ can also facilitate an inelastic transition between the DM ground stateand an excited state [76, 82] to explain e.g., the 511 keV line anomaly [83]. For DM decays,the φ channels can be viewed as “simplified models” that can capture how the constraintschange when there is a cascade, e.g., [52]. We will sometimes refer to these scalar-mediatedprocesses as “eXciting Dark Matter” (XDM). These spectra are shown in Fig. 1 ( right ) inthe case of annihilating DM and m DM = 1 TeV. We do not consider other popular DM7andidates like axions and gravitinos.We note that the observed differential photon flux can also be written as dΦ γ d E γ ≡ d N γ t tot A eff d E γ ≡ E d N γ d E γ , (8)where we have now explicitly included A eff , the effective area (which is a function of energy), t tot , the LAT’s total live time, and E , the LAT’s exposure. Given the photon spectra, thenumber of photons from a DM annihilation signal in a spatial region Ω i , with J -factor J i ann ,and energy range [ E k , E k +1 ] is given by N i,kγ = 18 π r (cid:12) ρ (cid:12) m (cid:104) σv (cid:105) J i ann E i,k (cid:90) E k +1 E k d E γ d N γ d E γ , (9)where E i,k is the exposure averaged over Ω i and calculated at the midpoint of [ E k , E k +1 ] (since the variation of the exposure over a single energy bin is very small). For decays thepredicted counts are N i,kγ = 14 π r (cid:12) ρ (cid:12) m DM τ J i dec E i,k (cid:90) E k +1 E k d E γ d N γ d E γ . (10)The approximately homogeneously distributed DM in the Universe could provide an extra-galactic contribution to the observed photon flux. However, the observed γ -ray spectrumwill be different than that expected from Galactic DM interactions since the photons redshiftas they propagate to us and there is a finite optical depth — the result of interactions ofthe γ rays with low-energy photons that compose the extragalactic background light (EBL).This yields the following expected extragalactic photon intensity for decaying DM [84, 85] d Φ γ d E γ dΩ = 14 π Ω DM ρ c, τ m DM (cid:90) ∞ d z e − τ ( E γ ( z ) ,z ) H ( z ) d N γ d E γ ( E γ ( z ) , z ) . (11)Here, Ω DM (cid:39) . is the present DM energy density, ρ c, (cid:39) . × − GeV/cm isthe critical density today, E γ ( z ) = E γ ( z + 1) is the energy of the emitted photon, H ( z ) = H (cid:112) Ω m (1 + z ) + Ω Λ , where Ω m (cid:39) . and Ω Λ (cid:39) . are the total matterand cosmological-constant energy densities [86], respectively, and we assume a flat Universewith Ω m + Ω Λ = 1 . The optical depth is given by τ ( E γ , z ) , and we use the parameterizationsfound in [84]. We note that, for annihilating DM, the smooth extragalactic contribution is8ubleading compared to the Galactic one and we ignore it, whereas for decays it is a factorof order (cid:46) as large as its Galactic counterpart and we include it in our analysis. We include the flux generated by ICS for the cases where DM annihilates/decays to e + e − , µ + µ − , τ + τ − , as well as φφ channels. In all cases we end up with high-energy electronsand positrons. These propagate within the Galaxy and can lose energy through ICS offstarlight, infrared background light, or CMB photons, or via synchrotron radiation in theGalactic magnetic field. The ICS process ( e ±(cid:48) γ (cid:48) → e ± γ ) can produce high-energy γ raysthat are observed by the LAT. The synchrotron-radiation contribution in the Fermi -LATenergy range is subdominant for the DM masses and models that we consider [52] and thuswe neglect it when we derive our limits. However, it must be included when calculating theICS γ -ray signal, since it determines how fast the electrons and positrons cool. In fact, thecooling time is strongly dependent on the Galactic magnetic field, whose values at differentlocations in our Galaxy are not known very accurately. This leads to large uncertainties inthe ICS signal. Moreover, the calculation of the ICS signal requires additional assumptions;for example, we assume, as generally done, that the density of electrons and positronsafter propagation follows a steady-state solution. However, phenomena such as the Fermi bubbles [87], pointing to a dynamical event in the Milky Way’s recent history, might makethis assumption not fully justified. We also assume that the steady-state propagation of theelectrons/positrons only occurs inside a cylindrical region of the Galaxy that has a maximumradius R h and half-height z h . The steady-state diffusion equation is given by (e.g., [88]) − D xx ( E (cid:48) e ) ∇ d n e d E (cid:48) e − ∂∂E (cid:48) e (cid:20) b ( r, z, E (cid:48) e ) d n e d E (cid:48) e (cid:21) = (cid:16) ρ ( r,z ) m DM (cid:17) (cid:104) σv (cid:105) d N e d E (cid:48) e , annihil.
12 2 ρ ( r,z ) m DM τ d N e d E (cid:48) e , decays . (12)Here d n e / d E (cid:48) e ≡ d n e ( r, z, E (cid:48) e ) / d E (cid:48) e is the energy-dependent differential electron+positrondensity at a given point in the Galaxy, ( r, z ) , where r and z are the cylindrical coordinatesof the electron/positron in the Galaxy. The right-hand side of Eq. (12) is the source termand contains the DM density profile, ρ ( r, z ) (a function of cylindrical coordinates) andthe electron+positron energy spectrum, d N e / d E (cid:48) e ; also, there is a factor / for Majoranafermions, otherwise for Dirac fermions. The first term on the left-hand side accounts for9he spatial diffusion and is characterized by an energy-dependent coefficient, D xx ( E (cid:48) e ) = D (cid:18) E (cid:48) e E (cid:19) δ . (13)The second term is the energy-dependent loss and is given by b ( r, z, E (cid:48) e ) ≡ − d E (cid:48) e d t = 4 σ T m e E (cid:48) e (cid:34) u B ( r, z ) + (cid:88) i =1 u γi ( r, z ) R i ( E (cid:48) e ) (cid:35) , (14)where σ T = 8 πr e / , with r e = α em /m e , is the Thomson cross section, and u B ( r, z ) = B / is the energy density of the Galactic magnetic field B , chosen to have the form [89] B ≡ B ( r, z ) = B e − [( r − R (cid:12) ) /R b + z/z b ] , (15)where R b = 10 kpc and z b = 2 kpc. The u γi ( r, z ) are the energy densities of the three relevantlight components in the Galaxy, i.e.: CMB, infrared light, and starlight. The factors R i ( E (cid:48) e ) take into account relativistic corrections. The γ -ray differential flux at energy E γ , resultingfrom ICS off an electron is d Φ γ d E γ dΩ = α (cid:90) d s (cid:90) (cid:90) d E (cid:48) γ d E (cid:48) e f IC ( q, (cid:15) ) E (cid:48) γ E (cid:48) e d n e d E (cid:48) e ( r, z, E (cid:48) e ) d u γ d E (cid:48) γ (cid:0) r, z, E (cid:48) γ (cid:1) , (16)where s is the line-of-sight distance, and f IC ( q, (cid:15) ) ≡ q log q + (1 + 2 q )(1 − q ) + 12 ( (cid:15)q ) (cid:15)q (1 − q ) , (17) q ≡ (cid:15) Γ(1 − (cid:15) ) , (cid:15) ≡ E γ E (cid:48) e , Γ ≡ E (cid:48) γ E (cid:48) e m e . (18)We calculate the ICS contribution with GALPROP V50 [90]. We use a version of
GALPROP V50 that was modified by the authors of [91] to include various DM annihilation and decay finalstates. We fix δ = 0 . , E = 4 GeV, and take the cylindrical geometry to have a maximumradius R h = 20 kpc and a maximum half-height z h = 4 kpc. As mentioned above, thegreatest source of uncertainty is due to the Galactic magnetic field, B . To capture some ofthis uncertainty, we vary B between − µ G, when showing our results in §4. We fix thespatial diffusion coefficient parameter to be D = 4 . × cm /s ( . × cm /s)for B = 1 µ G ( µ G). (See Appendix D for sources for these values.) In Appendix D we10how how our results are affected when varying z h and R h , in addition to D and B ; wefind that the largest effect on the results comes from the variation of B .
3. DATA SETS AND METHODS
We aim to set conservative, robust constraints on the annihilation and decay of DM intovarious SM final states. We consider the inclusive photon spectrum observed by the
Fermi -LAT, and use simulated data to first find the “optimal” ROI in the γ -ray sky, i.e. the onethat yields the strongest constraint. We then require the DM signal to be less than theobserved photon counts. We note that our approach does not allow us to search for theexistence of a DM signal.In this section we describe the event selection, how we use the simulated data sets inour analysis, the ROI choice, and how we construct optimal upper bounds on the DMannihilation cross section and lower bound on the DM decay lifetime. We also provide adetailed example of our procedure. The data set used for this study consists of ∼ . years of Fermi -LAT data (fromAugust 2008 until June 2014) in the energy range . − GeV. We select photons usingthe
P7REP_CLEAN event-class selection [92], to minimize contamination by residual cosmicrays. We also require the zenith angle to be smaller than ◦ to remove photons originatingfrom the bright Earth’s Limb. Details on the Fermi -LAT instrument and performancecan be found in [1, 93]. All data reduction and calculation of the exposure maps wereperformed using the
Fermi -LAT
ScienceTools , version v9r34p1 [94]. As for the
Fermi -LATinstrument response functions (IRFs), we use
P7REP_CLEAN_V15 for both MCs and data.As described in Appendix C, the results shown in this paper are obtained after masking allknown point sources identified in the 5-year
Fermi catalog (3FGL) [95], using a PSF (pointspread function)-like masking radius, except for those photons coming from within the inner ◦ × ◦ square at the GC. Moreover, we include both front- and back-converting events. InAppendix C we show that, although this choice is generally optimal, our results are notsignificantly affected if we mask only the brightest sources, or no sources at all, and if we11nclude only front- or only back-converting events. For our study, we use 10 Monte Carlo (MC) data sets, each a statistically independent ∼ . -year representation of the γ -ray sky. The same event selection described above isapplied to MC data. We use the simulated data sets to select “optimized” ROIs, indepen-dent of the real data, as described below in §3.3. By finding optimal ROIs based on theMC simulations, we avoid the possibility of accidentally obtaining a strong constraint dueto statistical fluctuations in the data. We describe the details of the simulated data in Ap-pendix E. Note that the MC simulations contain photons with an energy range of 0.5 GeVto 500 GeV (as opposed to 1.5 GeV to 750 GeV in the data). We account for this differenceby extrapolating the MC data up to 750 GeV as described in Appendix E. We take the ROI for annihilating DM to have the dumbbell shape as shown in Fig. 2( left ). This shape depends on three parameters: the radius from the GC to the edge of theROI, R , the width in latitude of the Galactic Plane (GP) that is to be excluded from theROI, b , and the width in longitude of the GC region that is to be included in the ROI, (cid:96) . The motivation for choosing such shape is that the DM distribution is approximatelyspherically symmetric (hence the choice of a circular region, parametrized by R ), but theGalactic foregrounds are largest in the GP region, which we then remove. However, weinclude the GC in our ROI as this is where the DM signal peaks as well, dramatically so forcuspy profiles (since N γ, DM ∝ ρ ).For decaying DM, our choice of ROI will consist of the two high-latitude regions shown inFig. 2 ( right ), and depends on only one parameter: the width in latitude from the Galacticpoles to the edge of the ROI, ∆ b d . In contrast to annihilation, the decaying DM signal isexpected to be much less concentrated in the GC, since N γ, DM ∝ ρ DM .12 -60 ° -30 ° ° ° ° -120 ° -60 ° ° ° b ∆ l ∆ ° -60 ° -30 ° ° ° ° -120 ° -60 ° ° ° d b ∆ d b ∆ Figure 2 . Left:
The choice of ROI (shaded) in the γ -ray sky for dark-matter annihilation. TheROI depends on 3 parameters, as indicated. Right:
The choice of ROI for dark-matter decays(shaded), which depends on one parameter, as indicated.
A particular DM model or
Theory Hypothesis , T H = [ m DM , ρ, annihilation/decay , channel ] ,is characterized by the DM mass ( m DM ), the DM density profile ( ρ ), whether it is annihi-lating/decaying DM, and the annihilation/decay final state. Given any ROI and a photonenergy range, [ROI, ∆ E ], we obtain a constraint on either the DM annihilation cross section, (cid:104) σv (cid:105) , or decay lifetime, τ , for a given T H by requiring that the number of DM events, N γ, DM ,in [ROI, ∆ E ] does not exceed the observed value, N γ,O . More precisely, to set a limit witha confidence level (C.L.) of − α , we vary (cid:104) σv (cid:105) or τ until the probability that N γ, DM > N γ,O is α ; in equations, the bound on (cid:104) σv (cid:105) or τ is obtained by solving N γ,O (cid:88) k =0 Poisson ( k | N γ, DM ) = 1 − α, (19)where as usual Poisson( k | λ ) = λ k e − λ k ! . (20)For each T H , we find the optimal ROI and optimal photon energy range, [ROI, ∆ E ] O , whichprovides the best limit on (cid:104) σv (cid:105) or τ . If we simply scan over all [ROI, ∆ E ] in the data, thiswould subject our constraints to statistical fluctuations. Instead, we use the 10 simulateddata sets to find [ROI, ∆ E ] O as follows. For the i -th ROI and energy range, [ROI, ∆ E ] i ,and j -th simulated data set, we calculate the bound on the cross section or lifetime, (cid:104) σv (cid:105) i,j or τ i,j , as described above in Eq. (19). We then average the resulting expected limit across13he 10 simulations, i.e. (cid:104) σv (cid:105) i = 110 (cid:88) j =1 (cid:104) σv (cid:105) i,j , (21) τ i = 110 (cid:88) j =1 τ i,j . (22)We then find [ROI, ∆ E ] O by scanning over all [ROI, ∆ E ] i ’s and selecting the one that yieldsthe minimum (cid:104) σv (cid:105) i (maximum τ i ), i.e. (cid:104) σv (cid:105) = min i (cid:104) σv (cid:105) i , (23) τ = max i τ i . (24)We then use [ROI, ∆ E ] O on the real data to calculate the limits on (cid:104) σv (cid:105) or τ for the given T H .The ROIs used in our optimization are given in §3.3. We bin each simulated dataset into . ◦ × . ◦ rectangular pixels in Galactic latitude and longitude and N = 127 logarithmically-uniform energy bins between . − GeV. We then vary the ROI shapeparameters described in §3.3 in steps of . ◦ for R , steps ∼ ◦ for ∆ b and ∆ (cid:96) , and . ◦ for ∆ b d . For each choice of ROI, we scan over all ( N − N − / choices of adjacentbins in energy, assuming a minimum of three adjacent bins. In Appendix B we show asample of the resulting optimized ROIs and energy ranges.We note that for large enough N γ, DM or N γ,O , the statistical distributions are approx-imately Gaussian, and we would obtain a 95% C.L. bound by requiring N γ, DM < N γ,O +1 . (cid:112) N γ,O (cid:39) N γ,O . Even our smallest optimal ROIs with the highest optimal energy rangescontain at least O (10) photons. Our method thus does not produce constraints that aresusceptible to Poisson fluctuations of the number of events in [ROI, ∆ E ] O , and, as a con-sequence, our constraints are not expected to improve significantly with more data (somesmall improvements may arise from, e.g., a better rejection of backgrounds).We also note that since [ROI, ∆ E ] O was selected using simulated data, other choicesof [ROI, ∆ E ] may provide a stronger constraint on the data. Also, the simulated data isnot a perfect representation of the data. Indeed, there are certain regions in the sky wherethe simulations do not model the data perfectly, and the “expected” limits using MC data14ay differ from the limit obtained on the real data (see Appendix F). One notable exampleis in the GC and in the Inner Galaxy region, which has led to claims of a γ -ray excess,see Appendix A. However, an imperfect modeling of the sky does not affect the validityof our constraints. We use the simulations as a tool to pick [ROI, ∆ E ] in an unbiasedway. Even if the simulations were a totally inaccurate representation of the real data, itwould not invalidate our limits, although other choices of [ROI, ∆ E ] would provide strongerconstraints.We note that for prompt radiation we include the effects of the Fermi -LAT’s PSF, byperforming its convolution with the J-factors, using the
Fermi -LAT
ScienceTools . For theconstraints that include prompt and ICS, however, convolving the PSF for the DM signalcalculation is computationally intensive, so we do not account for these effects. To see byhow much this could potentially affect our limits, we constrained the ROIs to have a shapewhich is safe w.r.t. the PSF containment radius at the lowest energies considered. If theROI includes a portion of GC (i.e. ∆ (cid:96) > ◦ ), then we require the width of this window to beat least ◦ (i.e. ∆ (cid:96) > ◦ ); for the width of the top and bottom of the ROI shape (resemblingcrescents) we require that R < ∆ (cid:96) (so the ROI is a circle), R < (cid:112) ∆ b + ∆ (cid:96) (so the twocrescents have no tips), and R > ◦ + ∆ b (so the two crescents are thick enough). Theupper bounds thus obtained are only degraded by at most ∼ − with respect to theunconstrained-ROI case. This is a small number; especially in view of the fact that thelargest uncertainty for the DM ICS signal comes from the value of the magnitude of thelocal magnetic field, see Appendix D.We note that systematic effects of the PSF are not included in our analysis, as they aremuch smaller than the other sources of systematic uncertainty considered, such as in theICS signal and DM density profile. An illustration of our method is shown in Fig. 3. The left plot shows the count spectrumfrom one of the MC data sets for the ROI shown in the inset. The green triangles show thespectrum for a 1.5 TeV DM annihilating to b ¯ b , assuming isothermally distributed DM, withthe cross section set at the 95% C.L. upper limit. This limit is derived by requiring thatthe number of signal events in the optimal energy range from 68 GeV to 142 GeV (vertical15 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△△ △ △ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ●●● E [ GeV ] P ho t on C oun t s DM DM → b b , m DM = - -
10 0 10 20 - - - -
10 0 10 20 - - - -
10 0 10 20 - - MC counts 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] ExtrapolatedMC pointsOptimizedEnergy Range - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → b b , Isothermal ( ) Individual MC 95 % upper boundsMean MC 95 % upper bound Figure 3 . Left:
Count spectrum from one of the MC data sets for the ROI shown in the inset.The green points show the spectrum for 1.5 TeV DM distributed according to the Isothermal profile,annihilating to b ¯ b , with a cross section chosen such that the number of signal events in the energyrange from 68 GeV to 142 GeV (vertical brown lines) is larger than the number of events in the MCdata (at 95% C.L.), as given by Eq. (19). Since the simulated data only contains photons up to460 GeV, we extrapolate it to 750 GeV (red points), using a power-law fit to the photon spectrumabove ∼ . GeV. See Appendix E for more details.
Right:
The best cross-section limit averagedover all ten MC data sets is shown with a green solid line, while the individual cross-section limitsfor each of the 10 MC data sets are shown with dashed gray lines. As explained in §3.5, the averagecross-section limit is used as a figure of merit for our ROI/energy range optimization. brown lines) be larger than the number of events in the MC data as given by Eq. (19), wherewe set α = 0 . . The number of events in this ROI and energy range will fluctuate fromone MC data set to another, and we calculate the average cross-section limit for all ten MCdata sets. We show the best average cross-section limit as a function of DM mass with agreen solid line in Fig. 3 ( right ), together with the cross-section limit for the ten individualMC data sets (dashed gray lines). In Fig. 3, we masked all point sources and included bothfront- and back-converting events.We now have all the ingredients put in place for calculating constraints from the γ -ray sky observed by the Fermi -LAT. In the next section we give the 95% C.L. bounds onthe annihilation cross section (upper bound) and on the DM lifetime (lower bound) forannihilations and decays into various SM modes, respectively.16 . RESULTS AND DISCUSSION
In this section we give the results from the optimization procedure described in §3. Weemphasize that the constraints obtained in this study are conservative and robust, since theydo not depend on the modeling and subsequent subtraction of astrophysical foregrounds.In §4.1 (§4.2) we discuss the constraints on annihilating (decaying) DM. Additionally, inAppendix A we use our method to derive bounds on models invoked to explain a putative γ -ray excess at the GC [25–36]. The effect on our constraints due to different choices ofsource-masking, and due to the variation of ICS parameters is discussed in Appendix C andAppendix D, respectively. The constraints on the DM-annihilation cross section as a function of DM mass arepresented in Fig. 4 for annihilation to e + e − , µ + µ − , τ + τ − , and φφ , where φ decays eitheronly to e + e − (with m φ = 0 . GeV), or only to µ + µ − (with m φ = 0 . GeV), or to e + e − , µ + µ − , and π + π − in the ratio (with m φ = 0 . GeV). Fig. 5 shows the results forthe final states b ¯ b , W + W − , u ¯ u , and gg . In all cases we present the results for four differentassumptions about the DM profile ρ ( r ) introduced in §2.1. We note that each DM mass foreach spatial distribution and final state choice has been separately optimized, and an optimalROI, ROI o,i , and photon energy range, ∆ E o,i , were obtained to set the 95% C.L. constraint.In Appendix B we illustrate how the optimal ROI and energy range change for various DMdensity profiles and for different DM masses (see Figs. 9 and 10).The constraints disfavor the thermal WIMP cross section for low DM masses and forthe cuspiest profiles (mostly the NFW c profile). For those cases in which the final statescontain high-energy electrons, i.e. Fig. 4, there is a contribution from prompt radiation fromFSR as well as ICS. The latter, while more uncertain, considerably strengthens the bounds,especially for high DM masses. In Fig. 4 the shaded band denotes the constraint from ICS asthe magnitude of the Galactic magnetic field at our Solar System’s location, B is varied from1 µ G to 10 µ G and correspondingly the diffusion coefficient D from . × cm /s to Note that limits for the W + W − channel extend to m DM < m W . In this region, the W + W − final stateis not produced on-shell, but instead the annihilation is to a three- or four-body final state consisting ofleptons and/or quarks through off-shell W ± . (The expected cross-section in any concrete DM model forthe off-shell process would be highly suppressed compared to the on-shell process.) - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → e + e - IsothermalNFWEinastoNFW c thermal WIMP solid: prompt onlyshaded: w / ICS, B = ( ) μ Gfor bottom ( top ) of curve dSph galaxies ( - year P7REP ) - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → μ + μ - IsothermalNFWEinastoNFW c thermal WIMP solid: prompt onlyshaded: w / ICS, B = ( ) μ Gfor bottom ( top ) of curve dSph galaxies ( - year P7REP ) - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → τ + τ - IsothermalNFWEinastoNFW c thermal WIMP solid: prompt only dSph galaxies ( - year P7REP ) shaded: w / ICS, B = ( ) μ Gfor bottom ( top ) of curve - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → ϕ ϕ → e + e - IsothermalNFWEinastoNFW c thermal WIMP solid: prompt onlyshaded: w / ICS, B = ( ) μ Gfor bottom ( top ) of curve - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → ϕ ϕ → μ + μ - IsothermalNFWEinastoNFW c thermal WIMP solid: prompt onlyshaded: w / ICS, B = ( ) μ Gfor bottom ( top ) of curve - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → ϕ ϕ → e + e - , 2 μ + μ - , 2 π + π - IsothermalNFWEinastoNFW c thermal WIMP solid: prompt onlyshaded: w / ICS, B = ( ) μ Gfor bottom ( top ) of curve Figure 4 .
95% C.L. upper limits on DM annihilation cross section vs. DM mass from
Fermi -LAT’sinclusive photon spectrum for the indicated final states. Each plot shows constraints for the Isothermal(green), NFW (red), Einasto (blue), and NFW c (orange) DM density profiles. Solid lines show constraintsfrom the inclusion of only the prompt radiation from the annihilation, while the bands include the ICSoff background light, with the Galactic B-field varying within − µ G and D within D , min − D , max (bottom-top of band). When available, we show the limits from the P7REP analysis of 15 dwarf spheroidalgalaxies with a cyan dashed line [12]. For the XDM models we show the approximate regions (gray) inwhich annihilating DM could account for the PAMELA/Fermi/AMS-02 cosmic-ray excesses. The best-fitparameters from [96] are shown as black dots. - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → W + W - IsothermalNFWEinastoNFW c dSph galaxies ( - year P7REP ) thermal WIMP - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → b b IsothermalNFWEinastoNFW c dSph galaxies ( - year P7REP ) thermal WIMP - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → u u IsothermalNFWEinastoNFW c dSph galaxies ( - year P7REP ) thermal WIMP - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → g g IsothermalNFWEinastoNFW c thermal WIMP Figure 5 .
95% C.L. upper limits on DM annihilation cross section vs. DM mass from
Fermi -LAT’s inclusivephoton spectrum for the indicated final states. Each plot shows constraints for the Isothermal (green), NFW(red), Einasto (blue), and NFW c (orange) DM density profiles. Solid lines show constraints derived fromincluding only the prompt radiation produced in the annihilation process (i.e. final-state radiation or in thedecay of hadrons). When available, we show the limits from the 4-year P7REP analysis of 15 nearby dwarfspheroidal galaxies with a cyan dashed line [12]. . × cm /s (see §2.2). The propagation was performed as described in §2.2, i.e. overa cylindrical geometry with radius R h = 20 kpc and half-height z h = 4 kpc. With ICSincluded and for cuspy profiles, DM annihilation to leptonic final states, particularly forelectrons, can be probed well into the annihilation-cross-section regime of a thermal relicthat freezes out early in the Universe, (cid:104) σv (cid:105) relic ≈ × − cm /s. The inclusion of extraparticle content in DM annihilations, namely the particle φ , is motivated by the best fit tothe PAMELA, Fermi , and AMS-02 cosmic-ray positron and electron data [78–81], if thoseexcesses are interpreted as coming from DM annihilation. Fig. 4 shows the approximate19egions (shaded gray) in the cross-section–versus–mass plane, in which annihilating DMcould offer an explanation for these excesses. These regions are meant to be illustrative onlyand chosen so that they contain the parameter choices found in [96], shown with black dots.(See also [97].) The inclusion of ICS severely constrains the favored parameter regions forall profiles except isothermal, while including only the prompt signal challenges the favoredregions only for the cuspy NFW c profile.The constraints from [12], which, using 4 years of P7REP data, analyzed 15 dwarfspheroidal satellite galaxies (dSph) of the Milky Way to set robust constraints on DM,are shown in Fig. 4 with a cyan dashed line. Due to the dSph’s proximity, high DM content,and lack of astrophysical foregrounds, they are excellent targets to search for annihilatingDM. Moreover, the available data on the velocity distribution of the stars in the dSph al-lows one to predict rather accurately the expected γ -ray flux from DM annihilation. Thisprediction is not subject to the same uncertainties as the expected flux in the Milky Wayhalo, which suffers from large uncertainties in the DM density profile. Our constraints arestronger than the dSph constraints over much of the DM mass range and for several of theDM profiles that we consider, especially at high energies. For DM masses (cid:46) GeV, ourconstraints are stronger than the dSph constraints for the NFW c profile, and comparable instrength for the Einasto profile, although weaker for the NFW and isothermal profiles. Newresults using P8 data to perform a similar analysis of the dwarf galaxies are expected soonand are somewhat more stringent than the P7REP results.Notice that some of the ICS-inclusive limits are actually weaker than the ones with promptradiation only. This might seem puzzling, as for a given ROI and energy range, the signalthat includes prompt and ICS is obviously larger than the one with prompt only and shouldlead to more stringent constraints. However, our ROI and energy range used to derive thelimits from the data are dictated by the optimization of the average MC limit, such that theoptimized ROIs and energy ranges for prompt+ICS and prompt-only might differ from eachother. If one considers this along with the fact that the simulated data sets are not perfectrepresentations of the real sky, the limit that includes ICS can be weaker on occasion thanthe prompt-only limit.It is useful to compare our limits with those obtained from similar analyses in the liter-ature where no attempt was made to model the astrophysical foregrounds. These analysesusually differ in their choice of DM-profile parameters, their procedure for constructing the20imits (Gaussian error on flux versus Poisson limit on counts), their choice of propagationmodels for the ICS signal, and the data energy range utilized. Nevertheless, we can tryto single out the effect of our ROI and energy-range optimization method alone by rescal-ing these other results to compensate for the different choices mentioned above. In [21],the limit was also constructed by scanning over a few differently shaped and located ROIs.Consequently our results are only within a factor of 1-2 stronger than theirs, across all chan-nels. In [22], the construction of the bound is quite different from ours, and our results arearound 2 times more stringent than theirs. In [20], an optimization procedure is performedon ROIs that look very different from ours, and a less extensive optimization is done on theenergy window. For annihilations we improve on these limits by a factor of 1–20, dependingon channel and profile, and by a factor 2–4 when including ICS. In [23], the ROI is opti-mized using the signal-over-background ratio as a figure of merit. For harder spectra, ourimprovement is between a factor of 3–8, while for softer spectra, the improvement is a factorof 1–4. While a favorite target for the DM annihilation rate comes from the thermal freeze-outof a thermal relic, which gives the correct present-day abundance, for decaying DM nosuch “favored” lifetime exists — the DM lifetime only has to be larger than the age of theUniverse. One possible target comes from explaining the rising fraction in the cosmic-raypositron spectrum with DM decays to final states that produce high-energy electrons andpositrons, with the preferred DM lifetime being in the range − s, depending onthe precise final states and astrophysical assumptions [21, 52, 99–105]. Such lifetimes donot only have a phenomenological motivation, but also arise naturally for TeV-scale DMparticles that decay via a dimension-six operator generated near the scale of Grand UnifiedTheories (GUT’s), M ∼ GeV, namely τ ∼ π M m ∼ × s (cid:18) m DM (cid:19) (cid:18) M GeV (cid:19) . (25)For example, in [99] DM decaying via dimension-six operators in supersymmetric GUT’swere posited to explain the cosmic-ray data from PAMELA.21 m DM [ GeV ] τ [ s ] DM → e + e - solid: prompt shaded: w / ICS, B = ( ) μ Gfor top ( bottom ) of curve NFW m DM [ GeV ] τ [ s ] DM → μ + μ - solid: prompt shaded: w / ICS, B = ( ) μ Gfor top ( bottom ) of curve NFW m DM [ GeV ] τ [ s ] DM → τ + τ - solid: prompt shaded: w / ICS, B = ( ) μ Gfor top ( bottom ) of curve NFW m DM [ GeV ] τ [ s ] DM → ϕ ϕ → e + e - solid: prompt shaded: w / ICS, B = ( ) μ Gfor top ( bottom ) of curve NFW m DM [ GeV ] τ [ s ] DM → ϕ ϕ → μ + μ - solid: prompt shaded: w / ICS, B = ( ) μ Gfor top ( bottom ) of curve NFW m DM [ GeV ] τ [ s ] DM → ϕ ϕ → e + e - , 2 μ + μ - , 2 π + π - solid: prompt shaded: w / ICS, B = ( ) μ Gfor top ( bottom ) of curve NFW
Figure 6 .
95% C.L. lower limits on DM decay lifetime vs. DM mass from
Fermi -LAT’s inclusive photonspectrum for the indicated final states. Shown are constraints for the NFW profile (the other profiles arevirtually identical). Solid lines show constraints derived from including only the prompt radiation producedin the annihilation process (i.e. final-state radiation or in the decay of hadrons), while the bands include theICS off background light, with the Galactic B-field varying within − µ G and D within D , min − D , max (bottom-top of band, when visible). For some models we show the approximate regions (gray) in whichdecaying DM could account for the PAMELA/Fermi/AMS-02 cosmic-ray excesses. The best-fit parametersfrom [98] are shown as black dots. m DM [ GeV ] τ [ s ] DM → W + W - NFW m DM [ GeV ] τ [ s ] DM → b b NFW m DM [ GeV ] τ [ s ] DM → u u NFW m DM [ GeV ] τ [ s ] DM → g g NFW
Figure 7 .
95% C.L. lower limits on DM decay lifetime vs. DM mass from
Fermi -LAT’s inclusive photonspectrum for the indicated final states. Shown are constraints for the NFW profile (the other profiles arevirtually identical). The constraints are derived from including only the prompt radiation produced in theannihilation process (i.e. final-state radiation or in the decay of hadrons).
The results for DM decays to leptonic and φφ final states are included in Fig. 6, whereasthose decays to b ¯ b, u ¯ u, gg, W + W − are shown in Fig. 7. We only show the constraints forthe NFW profile, as the other profiles lead to virtually identical constraints. As in the casefor DM annihilation, we include ICS for decaying DM for the leptonic final states only.The additional ICS component, while very sensitive to the value of the Galactic magneticfield, can enhance the constraints significantly, as in the case for annihilating DM. Notethat the bounds from prompt radiation start to deteriorate near DM masses of 1.5 TeV dueto the maximum-energy selection of 750 GeV used in this study. Our constraints comparefavorably with existing constraints in the literature; for example, they are a factor of 2–323tronger compared to [20–22].While the DM decay lifetime can span an enormous range consistent with all astrophysicaldata, there are many scenarios that are being probed by the constraints presented in thisanalysis. In particular, Fig. 6 shows with a gray shaded parallelogram the approximatepreferred regions in which decaying DM can explain the cosmic-ray positron and electrondata. Black dots indicate the best-fit regions found in [98], although note that these results donot include the latest data release from AMS-02 [81] (a more careful analysis of the preferredregions is beyond the scope of this paper); nevertheless, we expect that the preferred regionswould not shift significantly, and our regions are meant to be taken as a useful but roughqualitative guide only. We see that decays to τ + τ − are thoroughly disfavored, but ourconstraints for other channels are not strong enough to probe the relevant parameter regions.
5. CONCLUSIONS
This paper presented a conservative method for setting constraints on γ rays originatingfrom DM annihilation and decay, which does not rely on modeling of astrophysical fore-grounds when setting a limit. Optimal regions in the sky and energy were obtained by usingsimulations of the γ -ray sky, and a constraint was found by only requiring that the DMsignal does not over-predict the observed photon counts.For models of both annihilating and decaying DM, this method allows us to constraintheoretically-motivated parameter regions. For example, for cuspy enough profiles (e.g., con-tracted NFW), our method is able to disfavor the thermal-relic cross section for some leptonicand hadronic final states. Also, for steep-enough profiles, our constraints disfavor variousannihilating DM scenarios designed to explain the PAMELA/ Fermi /AMS-02 cosmic-raypositron and electron data. For decaying DM, a wide range of lifetimes are excluded forvarious SM final states, including the preferred parameter regions for DM decaying to τ + τ − to explain the PAMELA/ Fermi /AMS-02 data. The conservative constraints obtained inthis study are often competitive with, and in many cases stronger than, other availableconstraints in the literature. 24 cknowledgements
We thank Luca Baldini, Philippe Bruel, Seth Digel, Miguel Sánchez-Conde, and DavidThompson for reading the manuscript and providing valuable comments, Neelima Sehgalfor providing the photon spectra for the various dark matter annihilation and decay fi-nal states, Warit Mitthumsiri for his work on the MC simulations, and Eric Charles, IliasCholis, Tongyan Lin, Michele Papucci, and Gabrijela Zaharijas for helpful correspondenceor discussion. We also thank all the members of the
Fermi -LAT collaboration who pro-vided valuable comments and assistance, including Alessandro Cuoco, Alex Drlica-Wagner,Guðlaugur Jóhannesson, Philipp Mertsch, Igor Moskalenko, and Matthew Wood.RE is supported by the Department of Energy (DoE) Early Career research programDESC0008061 and by a Sloan Foundation Research Fellowship. Research at PerimeterInstitute is supported by the Government of Canada through Industry Canada and by theProvince of Ontario through the Ministry of Research and Innovation. EI is partly supportedby the Ministry of Research and Innovation - ERA (Early Research Awards) program. AMis supported by the C.N. Yang Institute for Theoretical Physics (Stony Brook University)and NSF grant PHY1316617. The work of GAGV was supported by Conicyt Anillo grantACT1102. GAGV thanks for the support of the Spanish MINECO’s Consolider-Ingenio 2010Programme under grant MultiDark CSD2009-00064 also the partial support by MINECOunder grant FPA2012-34694.The
Fermi -LAT Collaboration acknowledges generous ongoing support from a number ofagencies and institutes that have supported both the development and the operation of theLAT as well as scientific data analysis. These include the National Aeronautics and SpaceAdministration and the Department of Energy in the United States, the Commissariat àl’Energie Atomique and the Centre National de la Recherche Scientifique / Institut Nationalde Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italianaand the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture,Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization(KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallen-berg Foundation, the Swedish Research Council and the Swedish National Space Board inSweden. Additional support for science analysis during the operations phase is gratefullyacknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National25’Études Spatiales in France.
A. Constraints on DM Models invoked to explain γ Rays from Inner Galaxy
In this appendix we address claims made by several groups in recent years regarding a γ -ray excess from ∼ MeV to ∼ GeV, peaking in the 1–3 GeV window, in the InnerGalaxy [25–36]. While modeling uncertainties are large and the excess may very well havea non-DM origin, we use our method to set constraints on DM scenarios that have beeninvoked to explain the excess. Since we perform no foreground subtraction, a priori we donot expect the limits derived with our method to disfavor the best-fit DM scenarios foundin the literature; nevertheless, it is worthwhile to perform a careful check.The best fit for WIMP DM found in [32, 33] is for ∼ − GeV DM annihilatingpredominantly to b ¯ b . Furthermore, the spatial distribution of the putative signal is best fitby a generalized NFW profile, ρ NFW, γ ( r ) = ρ ( r/r s ) γ (1 + r/r s ) − γ , (A1)with a χ best fit obtained for γ ≈ . , although any γ in the range ∼ . − . allows fora reasonable fit. Analyses by other groups give results that are broadly consistent with thefindings in [32, 33]. In [35], it was found that DM annihilating dominantly to b ¯ b but withsome admixture of τ + τ − also provides a good fit. Other annihilation channels may also bepossible [106].In Fig. 8 we show the results of our optimization procedure applied to generalized NFW,Eq. (A1), with parameters chosen from best fits found in [31, 33, 36] (which differ in partfrom the assumptions made in §4.1). The authors of [33] ([36]) exclude from their analysis aband around the GP defined by | b | < ◦ (2 ◦ ) , thus not specifying a specific DM distributionwithin this latitude. We therefore use our usual ROIs shown in Fig. 2, but mask a squarecentered on the GC of side ◦ (4 ◦ ) . We show DM annihilating to b ¯ b ( left plot ) and τ + τ − ( right plot ). Unsurprisingly, the bounds that we obtain on the annihilation cross sectionare still a factor of ∼ or more from probing the best-fit regions shown with open or closedcontours in Fig. 8. As a reference for the reader, adopting all the assumptions in [33], forannihilation into b ¯ b , and choosing m DM = 25 GeV, the optimal ROI found with our method26 - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → b b NFW c , w / γ = ρ ⊙ = / cm , | b |> ° or | l |> ° Calore et al. ( ) NFW c , w / γ = ρ ⊙ = / cm , | b |> ° or | l |> ° Daylan et al. ( ) NFW c , w / γ = ρ ⊙ = / cm Abazajian et al. ( ) dSph galaxies ( - year P7REP ) thermal WIMP ∼ σ σ σ - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → τ + τ - NFW c , w / γ = ρ ⊙ = / cm , | b |> ° or | l |> ° Calore et al. ( ) NFW c , w / γ = ρ ⊙ = / cm Abazajian et al. ( ) dSph galaxies ( - year P7REP ) thermal WIMP3 σ∼ σ Figure 8 . 95% C.L. annihilation cross section upper limits on DM annihilating to b ¯ b ( left ) and τ + τ − ( right ) for an NFW c profile with various inner slopes and local DM densities (note that theassumptions made in deriving these limits differ in part from those made in §4.1). Also shown arethe preferred regions from [31, 33, 36] for DM to fit the claimed Galactic-Center γ -ray “excess”. Theconstraints have been computed with the same model assumptions as the best-fit regions (includingmasking a square centered on the GC of side ◦ or ◦ for analyses that excluded a band around theGP with the same thickness – see text for details). We also show with a cyan dashed line the limitobtained from the 4-year P7REP analysis of 15 nearby dwarf spheroidal galaxies [12]. is determined by the following parameters: R = 2 ◦ , ∆ b = 1 . ◦ , ∆ (cid:96) = 0 . ◦ , while theoptimal energy range is . GeV (cid:46) E (cid:46) . GeV.
B. Dependence of Optimal ROI and Energy Range on DM Profile and DM Mass
The optimal ROI and photon-energy range are found separately for each choice of DMspatial distribution, mass, and final state. In this section, we briefly illustrate the genericfeatures of the optimal search region and its dependence on the theory hypothesis. Fig. 9shows the obtained ROI and energy range for DM annihilation to b ¯ b for each of the fourspatial distributions studied, and for a fixed DM mass of 25 GeV. For this final state, withthe exception of NFW c , where it is beneficial to look near the GC, the optimal regionsin the sky involve semi-circular regions, symmetric in latitude b , with the GC removed.Furthermore, we find narrower optimal energy ranges for NFW c -distributed DM.For the b ¯ b final state, the effect of varying the DM mass is addressed in Fig. 10, where theoptimal regions are shown for two different masses: 350 GeV and 7 TeV, assuming NFW c -distributed DM. As the DM mass is increased, the strongest optimal regions are obtained27 ● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●● ● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △ △△△ △△△ △△ △ △ △ △ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E [ GeV ] P ho t on C oun t s DM DM → b b , m DM =
25 GeV, NFW - - - - - - - - - - - - Data 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] OptimizedEnergy Range ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●● ●● ● ● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△△ △ △△△ △△ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E [ GeV ] P ho t on C oun t s DM DM → b b , m DM =
25 GeV, Isothermal - -
10 0 10 20 - - - -
10 0 10 20 - - - -
10 0 10 20 - - Data 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] OptimizedEnergy Range ● ● ● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●● ● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △ △△△ △△△ △△ △ △ △ △ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E [ GeV ] P ho t on C oun t s DM DM → b b , m DM =
25 GeV, Einasto - - - - - - - - - - - - Data 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] OptimizedEnergy Range ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△△△△△ △△△ △ △ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E [ GeV ] P ho t on C oun t s DM DM → b b , m DM =
25 GeV, NFW c - - - - - - - - - - - - Data 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] OptimizedEnergy Range
Figure 9 . Count spectrum for 25 GeV DM annihilating to b ¯ b for various DM density profiles. The vertical(brown) lines show the optimal energy range for each DM model assumption. The inset shows the optimalROI. Note that PSF-convolution effects were included for the DM signal. The quoted (cid:104) σv (cid:105) is the annihilationcross section that saturates the 95% C.L. from the data. by including semi-circular regions in latitude, in addition to a rectangular area around theGC. We note that finite-resolution effects were included, by convolving the instrument’s PSFwith the J-factors, in the DM signal for all of the results in Fig. 9 and Fig. 10. C. Effect of Source Masking and Choice of Front-/Back-converting events on Limits
In this appendix we investigate the effect on the DM-cross-section upper limits whenmasking known point sources and using front- and/or back-converting events.Masking known sources reduces the observed counts in an ROI and can strengthen the28 ●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●● ●●● ● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△ △ △ △ △ △ △ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E [ GeV ] P ho t on C oun t s DM DM → b b , m DM =
350 GeV, Einasto - - - - - - - - - - - - Data 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] OptimizedEnergy Range ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ● ● △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△ △ △△ △△△△△△ △△△ △ △ △▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽▽ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ E [ GeV ] P ho t on C oun t s DM DM → b b , m DM = - - - - - - - - - - - - Data 〈σ v 〉 = × - cm s - ℓ [ ° ] b [ ° ] OptimizedEnergy Range
Figure 10 . Count spectrum for 350 GeV ( left ) and 7 TeV ( right ) DM annihilating to b ¯ b , assuming anEinasto profile. The vertical (brown) lines show the optimal energy range for each DM-model assumption.The inset shows the optimal ROI. Note that PSF-convolution effects were included for the DM signal. Thequoted (cid:104) σv (cid:105) is the annihilation cross section that saturates the 95% C.L. from the data. DM constraints, assuming that the masking does not also remove much of a potential DMsignal. This is the case if the ROI is large, as it is expected to be for decaying DM, or forannihilating DM with shallow DM density profiles (e.g., isothermal). Since astrophysicalpoint sources at very large energies ( >
20 GeV) typically exhibit a small flux, their maskingis expected to improve the limits for lower DM masses. For very cuspy profiles the ROIs tendto be small and concentrated around the GC region, where the number of known sources isalso large; in this case, masking all the point sources would remove most of the DM signalas well and will thus not likely lead to stronger limits.The amount of sky that needs to be masked to remove a point source depends on the
Fermi -LAT PSF, which depends on the energy and on where the photon converts in thedetector. In particular, photons that convert to an e + e − pair in the front part of the Fermi -LAT (consisting of the first 12 layers of thin tungsten foil) have a better angular resolution(smaller PSF) than those photons that convert in the back (next 4 layers of thick tungstenfoils). For very cuspy profiles the choice of including only front- or only back-convertingevents, or both, could potentially have important effects on the constraints.We obtain the point-source coordinates from the 3FGL catalog [95] and exclude all thephotons contained in pixels whose center lies within an angular radius of θ ( E ) from anypoint source; here θ is an approximation of the energy-dependent P7REP 68% point-source29 m DM [ GeV ] 〈 σ v 〉 i / 〈 σ v 〉 F + B , s . m . DM DM → b b , Isothermal Back OnlyFront + BackFront Only Solid: all s.m.Dashed: 10 σ s.m.Dotted: no s.m. ( s.m. = sources masked ) m DM [ GeV ] 〈 σ v 〉 i / 〈 σ v 〉 F + B , s . m . DM DM → b b , NFW c Back OnlyFront + BackFront Only Solid: all s.m.Dashed: 10 σ s.m.Dotted: no s.m,. ( s.m. = sources masked ) Figure 11 . Ratio of expected cross section upper limits vs. DM mass from simulated MC datafor DM annihilation to b ¯ b for isothermal ( left ) and NFW c ( right ) profiles. The denominator ofthe ratio, (cid:104) σv (cid:105) F+B , s . m . , is the cross section upper limit obtained when masking all known pointsources in the 5-year Fermi -LAT point-source catalog, outside a ◦ × ◦ square centered at the GCand including front- and back-converting events. The numerators of the ratios, (cid:104) σv (cid:105) i , are the crosssection upper limits obtained when masking all known point sources outside the ◦ × ◦ GC square(solid lines), masking only those sources detected at more than σ (outside the same ◦ × ◦ GCsquare) (dashed lines), and masking no sources (dotted lines). In each case we either include bothfront- and back-converting events (blue lines), only front-converting events (red lines), and onlyback-converting events (green lines). containment angle, θ ( E )[ ◦ ] = (cid:113) c ( E/ − β + c , (C1)and the parameters for (front-, back-) converting events are c = (0 . , . , c =(0 . , . , and β = (0 . , . . No source masking is performed within the in-ner ◦ × ◦ square at the GC, where the density of sources is very high and the expectedDM signal peaks.The effect on the cross-section upper limits versus DM mass, when masking known pointsources, and when including front- and/or back-converting events, is shown in Fig. 11 onsimulated data sets. The left (right) plot assumes DM annihilation to b ¯ b for our choice ofan isothermal (NFW-contracted) density profile. We choose a shallow and cuspy profile tosee how the results depend on having either large or small optimized ROIs, respectively.For each DM mass, and for each choice of source masking and inclusion of front-/back-converting events, we optimize the ROI choice and derive the average limit obtained from30he ten simulated MC data sets. In Fig. 11, we show a ratio of expected cross section upperlimits versus DM mass: the denominator of the ratios, (cid:104) σv (cid:105) F+B , s . m . , is the cross section upperlimit obtained when masking all known point sources as described above and including front-and back-converting events; the numerators of the ratios, (cid:104) σv (cid:105) i , are the cross section upperlimits obtained when masking all known point sources (outside the ◦ × ◦ GC square) (solidlines), masking only those sources detected at more than σ (outside the same ◦ × ◦ GCsquare) (dashed lines), and masking no sources (dotted lines). In each case we either includeboth front- and back-converting events (blue lines), only front-converting events (red lines),or only back-converting events (green lines).We see from Fig. 11 that, at least for the two annihilation models considered in thissection, the expected limits are the same within O (10 − . Moreover, the strongestconstraints are generically obtained when masking all point sources. For DM masses below ∼ GeV and cuspy profiles, the inclusion of only front-converting events is expected toprovide the strongest constraints, but only marginally so. Above ∼ GeV, the inclusion ofboth front- and back-converting events is best, since the photons produced in the annihilationof DM have such high energies that the PSF effects are negligible, and the inclusion of asmuch data as possible leads to stronger expected limits.Based on this, we conclude that the effect of source masking and choice of front-/back-converting events is not large on our results. We also note that the inclusion of both event-conversion types and the masking of point sources (blue solid line in Fig. 11) is expectedto give constraints that are among the best. We thus make this our standard choice whenshowing the results in §4.
D. Inverse Compton Scattering
In this appendix we discuss how the results from §4 depend on the parameters in theICS computation performed in
GALPROP . The amount of ICS radiation depends sensitivelyon various key propagation parameters whose values are not known to a satisfactory degree.Here we describe the effect on our constraints from varying these parameters in order tocapture some of the systematic uncertainties associated with the DM-generated ICS signal.We study how different models of propagation impact our results. We use, as a start-ing point, the
Fermi -LAT results from [107], in which various propagation models are fit31 m DM [ GeV ] 〈 σ v 〉 i / 〈 σ v 〉 B = μ G DM DM → e + e - , NFW, R h =
20 kpc, z h = D B = μ G B = μ G B = μ G m DM [ GeV ] 〈 σ v 〉 i / 〈 σ v 〉 z h = k p c , R h = k p c , D , m i n DM DM → e + e - , NFW, B = μ G z h =
10 kpc z h = z h = z h = R h =
30 kpc ( darker ) R h =
20 kpc ( lighter ) Solid: D Dashed: D Figure 12 . Ratios of cross-section upper limits from simulated data on DM annihilation to e + e − for an NFW profile, including prompt and ICS radiations, for different values of the Galacticmagnetic field ( left ) and different combinations of other propagation model parameters ( right ).The magnetic field has the largest effect on our analysis. to cosmic-ray spectra for various choices of the region of containment of the cosmic rays(parametrized with a cylindrical geometry of half-height z h and radius R h ). In our study,we vary z h and R h , and two other important parameters that have a big effect on the DMICS signal, namely the Galactic magnetic field value in the Solar System, B , and the spatialdiffusion coefficient D . The values used in our study are:1. z h = 4 , , ,
10 kpc R h = 20 ,
30 kpc D = D , min , D , max , where D , min and D , max are the minimum and maximum valuesof D spanned by the various GALPROP models studied in [107] for a given ( z h , R h ).4. B = 1 , , µ GAs an illustration of the dependence of the DM ICS signal on these parameters, Fig. 12shows the constraint on DM annihilation to e + e − , assuming an NFW c DM profile. Thegreatest effect on the uncertainty of the DM ICS signal originates from the variation in themagnitude of B , as clearly shown in the left plot. Varying the other parameters (right plot)has less of an effect on the DM ICS signal. We are therefore allowed, when showing theresults in §4, to fix z h = 4 kpc and R h = 20 kpc ; whereas we show the variation of our results32ith B and correspondingly D = D , min = 4 . × cm /s for B = 1 µ G (parametersyielding the strongest constraints) and D = D , max = 6 . × cm /s for B = 10 µ G(parameters yielding the weakest constraints).
E. Details on the Simulated data sets
The optimization procedure described in §3.4 to find the optimal ROIs and energy ranges,[ROI, ∆ E ] O , is performed on ten simulated data sets, each a 5.84-year representation of the γ -ray sky. Here we provide a few more details on the simulations.The generation of mock Fermi -LAT observations was carried out with the gtobssim routine, part of the Fermi Science Tools package v9r34p1. Its output is a list of MC-simulated γ -ray events with relative spatial direction, arrival time and energy, distributedaccording to an input source model and IRFs.A number of model elements were put into gtobssim (see [108]). These include the Fermi -LAT Collaboration’s model of the diffuse Galactic component, the isotropic component(derived for Pass 7 Reprocessed Clean front and back IRFs), and the 3FGL source catalogfor point and small extended sources [95].In addition, the full-sky simulations were calculated through gtobssim with the actualpointing and livetime history (FT2 file) of the Fermi -LAT for the first 5.84 years of thescientific phase of the mission. The source model simulated did not contain the Earth’sLimb emission, which is negligible at energies above 1 GeV, compared to the celestial γ -raysignal, when a zenith angle < ◦ cut is applied. The gtobssim tool convolves the fluxcomponents mentioned with the Fermi -LAT’s response, i.e. PSF, energy dispersion, andeffective area.Ten instances of the MC gtobssim-generated data were run, each with an independentstarting seed and the same source model; thus obtaining ten statistically independent in-stances of the γ -ray sky. The same event selection criteria were used for the MC data setsas for the real data. One important difference between the simulated data sets and the realdata is the energy range. Each simulated data set was calculated in an energy range of0.5 GeV to 500 GeV (as opposed to 1.5 GeV to 750 GeV for the actual data). The upperbound of 500 GeV in the gtobssim simulations is the upper limit in the energy map of the g ll_iem_v05_rev1.fit i so_clean_front_v05.txt and i so_clean_back_v05.txt - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → b b IsothermalNFWEinastoNFW c solid: datadashed: average MC limitshading: population st. dev.of the 10 MC limits thermal WIMP - - - - - - - m DM [ GeV ] 〈 σ v 〉 [ c m s - ] DM DM → e + e - IsothermalNFWEinastoNFW c solid: datadashed: average MC limitshading: population st. dev.of the 10 MC limits thermal WIMP Figure 13 . Comparison between average MC-based expected (dashed) and real-data (solid) 95%C.L. annihilation cross section upper limits on DM annihilating to b ¯ b ( left ) and e + e − ( right ) for theIsothermal (green), NFW (red), Einasto (blue), and NFW c (orange) DM density profiles (we onlyconsider prompt photons). The population standard deviations of the limits from the 10 individualMC simulations are also shown as shadings around the dashed lines. interstellar diffuse model [108]. To deal with this mismatch, we simply fit a power-law curveto each of the ten simulated data spectra for . GeV < E <
GeV that we obtain foreach ROI, and extrapolate it to 750 GeV. (The lower value of 6.2 GeV is low enough tohave enough photons to perform a meaningful fit even for small ROIs, and high enough fora single featureless power law to provide a reasonable fit to the spectra. The upper value of460 GeV is low enough to avoid count leakages due to finite energy resolution on the sharp500 GeV input-energy cutoff.) We then populate each bin above 460 GeV with a randomnumber of events chosen from a Poisson distribution whose expectation value equals theextrapolated value in a given bin. The subsequent optimal ROI and energy range for eachtheory hypothesis T H is found using the original plus extrapolated spectra. F. Comparison of limits between simulated and real data
In this appendix we compare the results derived from the real data with those derivedfrom simulated data. Since our simulated data is of course not a perfect representation ofthe real data, we do not expect that the limits derived on the real data will agree perfectlywith the limits derived on simulated data.Fig. 13 compares the simulated and observed limit on DM annihilation to b ¯ b ( left plot )34nd e + e − , including only prompt photons, ( right plot ), for the four different DM densityprofiles introduced in §2.1. Since the simulated data used in this study consists of 10statistically independent realizations of the γ -ray sky, we present the arithmetic mean ofthe 10 limits (dashed lines) and the standard deviation of the population (shaded bands),as well as the observed limits (solid lines). We see that the limits derived using real versussimulated data agree over a wide range of masses and profiles. [1] LAT Collaboration
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