Robust implications on Dark Matter from the first FERMI sky gamma map
IIFUP-TH/2009-25 CERN-PH-TH/2009-238
Robust implications on Dark Matterfrom the first
Fermi sky γ map Michele Papucci a and Alessandro Strumia bc a Institute for Advanced Study, Princeton, NJ 08540 b Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italia c CERN, PH-TH, CH-1211, Gen`eve 23, Suisse
Abstract
We derive robust model-independent bounds on Dark Matter (DM) annihilationsand decays from the first year of
Fermi γ -ray observations of the whole sky. Thesebounds only have a mild dependence on the DM density profile and allow the follow-ing DM interpretations of the PAMELA and Fermi e ± excesses: primary channels µ + µ − , µ + µ − µ + µ − or e + e − e + e − . An isothermal-like density profile is needed forannihilating DM. In all such cases, Fermi γ spectra must contain a significant DMcomponent, that may be probed in the future. Contents
Fermi sky map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Computing γ ’s from DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fermi data 15 a r X i v : . [ h e p - ph ] F e b Introduction
Recently the
Fermi collaboration released the first sky map of γ -rays up to energies below afew hundred GeV [1, 2, 3]. From a particle physics point of view its main interest resides inthe possible presence of a DM signal over the astrophysical background. Excitingly enough,although no clear excess is present, a few theorists claim possible hints [4]. In this paper we donot address these issues, that will need a full understanding of the data (released photon datastill contain a non-negligible contamination from mis-identified hadrons at energies around andabove 100 GeV), a proper modeling of the astrophysical backgrounds, subtraction of identifiedpoint-sources, and maybe more statistics.Here we take a different approach: using the available data we derive robust bounds onDM annihilations and decays, by only demanding the DM-induced γ -ray flux to be belowthe observed flux. Since DM is neutral, γ ’s are only produced at higher order in the QEDcoupling from various processes: we only include those contributions that can be computed ina model-independent way.We do not attempt to subtract the astrophysical and instrumental backgrounds, nor pointsources: many of these subtractions can be performed only by assuming some astrophysicalmodel, reducing the robustness of the bounds. Whenever possible, we will also comment onand try to quantify the residual uncertainties of our results. Progress on understanding thebackgrounds can only render our bounds more stringent, presumably by order one factors. ADark Matter signal could be lurking just below our bounds.An interesting application of our results is checking whether the DM interpretations of the e ± excesses observed by PAMELA [5], Fermi [6] and ATIC [7] give an Inverse Compton (IC)photon flux compatible with γ -ray observations. Indeed, according to DM interpretations, the e ± excess should be present everywhere in the DM halo (rather than only locally, as if the e ± excess is due to a nearby astrophysical source such as a pulsar), giving rise to an unavoidableassociated γ -ray signal: e ± produced by DM loose essentially all their energy by Compton up-scattering ambient light, giving rise to a photon flux at the level of Fermi sensitivity. Being adiffuse signal, one can consider regions of the sky that have smaller astrophysical uncertainties.By doing so, we find that many DM interpretations of the e ± excesses are already excluded bythis bound (future improvements will not change this conclusion); moreover, in some cases theexpected effect is at the level of the observed flux (future improvements should allow to testit). The plan of the paper is the following: in Section 2 we present technical details of ouranalysis; in Section 3 we present results for DM annihilations and in Section 4 for DM decays.Results are summarized in the Conclusions. We consider DM annihilations (parameterized by the DM DM cross section σv ) or decays(parameterized by the DM decay rate Γ = 1 /τ ) into the following set of primary DM particles:2 e, µ, τ, e, µ, τ, q, b, t, h, W (1)2here 2 e stands for e + e − and 4 e stands for V V , where V is an hypotethical new light particle,whose mass we take to be m ∼ M the DM mass. For the Higgs boson, we assume a mass m h = 115 GeV.Concerning all the details not specified here, we follow [9]. We consider the following Milky Way DM tentative density profiles ρ ( r ) [10]: ρ ( r ) ρ (cid:12) = (1 + r (cid:12) /r s ) / (1 + r /r s ) isothermal, r s = 5 kpc( r (cid:12) /r )(1 + r (cid:12) /r s ) / (1 + r/r s ) NFW, r s = 20 kpcexp( − r/r s ) α − ( r (cid:12) /r s ) α ] /α ) Einasto, r s = 20 kpc, α = 0 . , (2)keeping fixed the local DM density ρ ( r = r (cid:12) ≈ . ρ (cid:12) = 0 . / cm . NFW andEinasto profiles are favored by N -body simulations, isothermal-like profiles by observations ofspiral galaxies [11].When assessing the residual uncertainties of our bounds, we will also consider the possibilityof a disk-like component for the DM [12]. For this “Dark Disk” we will assume a profile ρ disk ( r, z ) ∝ exp( r/r d ) sinh( z/z d ) and we will vary r d between 5 and 15 kpc, z d between 0.2and 10 kpc, and the fraction of DM in the Dark Disk at solar position from 0 to 50%.Regarding the diffusion of e ± in the Milky Way, we consider the min, med, max propagationmodels of [13] characterized by the following astrophysical parameters:Model δ K in kpc /Myr L in kpcmin 0.85 0.006 1med 0.70 0.011 4max 0.46 0.076 15 . (3)The diffusion coefficient K = K E δ is assumed to be constant inside a cylinder with height 2 L centered on the galactic plane and radius 20 kpc, and infinitely large outside.When assessing the dependence of our bounds on the size of the diffusion zone, we willconsider also the more realistic case in which the diffusion coefficient depends on the distancefrom the galactic plane [14], K ( E, z ) = K E δ exp( | z | /z h ). where z h effectively determines thethickness of the diffusion zone, that dies off gradually instead of turning off abruptly at | z | = L . Fermi sky map
We divide the
Fermi γ -ray sky, parameterized by galactic longitude (cid:96) and latitude b ( (cid:96) = b = 0corresponds to the Galactic Center, GC) into several regions, depicted in Fig. 1. We extract thephoton spectrum within each region from the first public release of the Fermi γ -ray data [1].The details of the extraction are summarized in the Appendix. The Fermi collaborationpublished the energy spectra below 100 GeV in a few regions [2]; we checked that our procedurereproduces the
Fermi results at low and intermediate energies. We do not subtract pointsources, but we exclude the region most polluted by astrophysical sources, the galactic plane,by restricting ourselves to | b | > ◦ . Since the signal we are seeking to bound is not uniformly3 (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
20 0 10 20 45 90 135 180 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:123) in degrees G a l ac ti c l a tit ud e b i nd e g r ee s IC bound on Σ v (cid:72) DM DM (cid:174) Μ (cid:43) Μ (cid:45) (cid:76) in 10 (cid:45) cm (cid:144) sec for M (cid:61) L (cid:61) Figure 1:
Subdivision of the sky and example of a bound from each different region. In paren-thesis, the same bound is computed neglecting both the diffusion of e ± and the finite volume ofthe Milky Way diffusion halo. distributed on the sky, we consider a finer grid around the GC, where DM (but also astrophysicalsources) concentrate. Conversely we use regions with increasingly larger areas towards thetwo poles, where the signal is expected to be smaller. Furthermore, we separately considernorth-west, south-west, north-east and south-east regions, because the one less polluted byastrophysical sources will offer the best sensitivity.As the DM density profile is unknown, we do not know which region is most sensitive to DM.A robust constraint is obtained by demanding that the minimal computable DM- γ spectrumin each region, Φ DM i in energy bin i , does not exceed the measured γ flux Φ exp i at 3 σ , for anyenergy bin and any region. Fig. 1 shows an example of the bounds on the DM annihilationcross sections for each region. Fig. 2a compares the Fermi data in the single region that givesthe stronger bound for this particular model with the model prediction at its bets-fit point forthe PAMELA and
Fermi e ± excesses.Various regions give comparable bounds. Thereby one can do slightly better, still maintain-ing the absolute robustness of the bounds, by combining all regions in a global fit. We impose4 (cid:45) (cid:45) (cid:45) (cid:45) Photon energy in GeV E d (cid:70) Γ (cid:144) d E i n G e V (cid:144) c m s ec s r DM DM (cid:174) Μ (cid:43) Μ (cid:45) , M (cid:61) Σ v (cid:61) (cid:180) (cid:45) cm (cid:144) s (cid:45) (cid:60) b (cid:60) (cid:45) (cid:60) (cid:123) (cid:60) L (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) Photon energy in GeV E d (cid:70) Γ (cid:144) d E i n G e V (cid:144) c m s ec s r DM (cid:174) Τ (cid:43) Τ (cid:45) with M (cid:61)
6. TeV and
Τ (cid:61) (cid:180) sec F E R M I FSRIC
Figure 2:
Left:
Fermi data compared with an example of best-fit DM annihilation signal.Photons above ≈
100 GeV can be still contaminated by hadrons. The dotted line shows InverseCompton computed neglecting e ± diffusion and the finite volume of the diffusion halo. Right: Fermi preliminary extra-galactic data [3] compared with an example of best-fit DM decay signal. the 3 σ bound, χ <
9, where χ = min e (cid:88) i (Φ DM i ( E i (1 + e )) − Φ exp i ) δ Φ Θ(Φ DM i − Φ exp i ) + e δe , (4)where Θ( x ) = 1 if x > E i ); the χ must be marginalized (minimized in Gaussian approximation) overthe energy-scale free parameter e ; the last term in the χ accounts for the δe ≈ Fermi uncertainty on the energy scale; that only has a minor effect on the bounds.Typically, such global bound is a factor of few stronger than the bound obtained demandingthat no single point is exceeded at more than 3 σ . Furthermore, there are conceptual advantages.The ‘single point’ bound depends on how we choose the energy and angular binning and canbe fully dominated by the single bin where a downward statistical fluctuation happened in thetotal rate. On the other hand, the global fit does not depend on the binning (in the limit whereit is dense enough); e.g. one can even split one bin into two coincident bins without affectingthe global fit. γ ’s from DM Since DM has no electric charge, γ -ray production from DM annihilations or decays occurs athigher order in the electromagnetic coupling from many different processes with comparablerates: i) bremsstrahlung from charged particles and π decays; ii) virtual emission, iii) loopeffects; iv) astrophysical processes involving other particles produced by DM. We only considerthe following two sources of γ -rays that can be robustly computed in a model-independent way:5. FSR γ , i.e. Final State Radiation emitted by the primary DM annihilation or in subsequentdecays. With a slight abuse of terminology, we will include in this contribution also thephotons from hadronic decays. This gives photons with the largest E γ ∼ M .Primary channels such as 2 τ, q, W give rise to π s that decay as π → γ giving a γ yieldlarger than channels (such as 2 e or 2 µ ) that only produce γ from bremsstrahlung. Modelsinvolving new neutral light particles give the smallest γ yield [15, 16]. We do not considerelectroweak bremsstrahlung, that cannot be computed in a model-independent way: one needsto know in which electroweak multiplets the DM lies. The large corrections found in [17] if M (cid:29) πv are only present for those decay or annihilation channels that arise thanks to anon-vanishing Higgs vev v . All the channels we consider can be realized, in appropriate DMmodels, as effective operators that do not involve the Higgs (e.g. as effective operators where aDM pair couples to a vector leptonic current, or to the W, Z field-strength squared), such thatelectroweak bremsstrahlung remains one higher order effect, as in [18].At lower photon energies, the DM γ flux is dominated by the second source of γ -rays thatwe consider [9, 19]2. IC γ , i.e. photons from Inverse Compon. DM gives rise to e ± that loose most of theirenergy by up-scattering galactic ambient light (CMB and starlight, partially rescatteredby dust): this Inverse Compton e ± γ → e (cid:48)± γ (cid:48) process gives rise to γ (cid:48) s with energy E γ (cid:48) ∼ E γ ( E e /m e ) ∼
30 GeV.The IC energy loss process competes with energy losses due to synchrotron radiation in thegalactic magnetic fields. The rates of these two processes are respectively proportional to u γ ( (cid:126)x )and to u B ( (cid:126)x ) = B /
2, the energy densities in photons and in magnetic fields. We take thevalues from [20, 21], and in particular we assume B ( r, z ) ≈ µ G · exp( − r/
10 kpc − | z | / . (5)In this case u γ (cid:29) u B everywhere: Inverse Compton is dominant, and it can be reliably computedas essentially all the e ± energy goes into IC, irrespectively of the precise galactic maps of u γ and of u B .Only large deviations from the maps we adopted can affect our bounds, so we commentabout this possibility. Both galactic radiation and magnetic fields are better known for ourneighborhood than for other regions of the Galaxy, with the magnetic field being the mostuncertain. The most realistic worry is that magnetic fields in the Inner Galaxy might be intenseenough that one there has u B ∼ u γ , weakening our bounds. We checked the dependence onthe magnetic field uncertainties by varying the scales in Eq. (5) on which the galactic magneticfield changes both radially and vertically, while keeping its value at solar position fixed. Wefind that a factor of 2 variation in these scales changes the IC γ fluxes at high latitudes by lessthan 10 ÷
20% and in regions closer to the GC ( | b | < ◦ ) by 60%. Since these regions arerelevant in our global fit, our bounds can be relaxed at most by a factor of ∼ . ÷ γ , following [9] we take into account the diffusion of e ± , with charac-teristic diffusion length of about λ ∼ λ/r (cid:12) ∼ ◦ : even if DM annihilations are concentrated close to the GC, IC γ are not. Thereby Fermi γ -ray data are more relevant than HESS data both for the energyrange and for the angular range they observe. 6 xcludedby FSR10 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) DM mass M in GeV Σ v i n c m (cid:144) s ec DM halo model: isothermal sphere V (cid:174) eV (cid:174)Μ V (cid:174)Τ e ΜΤ qb W h t Excludedby FSR10 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) DM mass M in GeV Σ v i n c m (cid:144) s ec DM halo model: NFW V (cid:174) eV (cid:174)Μ V (cid:174)Τ e ΜΤ qb W h t Figure 3:
Fermi full-sky bounds on Final State Radiation γ -rays, for the DM annihilationmodes indicated along the curves. Excludedby IC10 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) DM mass M in GeV Σ v i n c m (cid:144) s ec DM halo model: isothermal sphere V (cid:174) eV (cid:174)Μ V (cid:174)Τ ΜΤ q b Wh t Excludedby IC10 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) DM mass M in GeV Σ v i n c m (cid:144) s ec DM halo model: NFW V (cid:174) eV (cid:174)Μ V (cid:174)Τ ΜΤ q b Wh t Figure 4:
Fermi full-sky bounds on Inverse Compton γ -rays, for the DM annihilation modesindicated along the curves.
102 3 4 5 71100.33 L in kpc Σ v w it h r e s p ec tt o N F W , L (cid:61) c isoTNFWEinasto Figure 5:
Example of how the
Fermi IC γ bound on σv changes as function of the height L ofthe diffusion volume for different DM profiles. We here assumed DM annihilations into µ + µ − with M = 1 . , but this plot would be almost the same for other DM models. Assuming DM annihilations, fig. 3 shows the
Fermi all-sky global bounds on FSR γ as functionof the DM mass M and the DM cross section σv . Fig. 4 shows the corresponding bounds onIC γ , assuming L = 4 kpc.In both cases the left (right) panels holds for the isothermal (NFW) DM profile. We seethat the Fermi bounds only have a mild dependence on the DM profile: although we do notknow where DM is,
Fermi observed all the sky, so that it is no longer possible to hide DMwith an appropriate density profile.The diffusion volume of e ± is assumed to be a cylinder that extends away from the galacticplane up to | z | < L . Fig. 5 shows a typical example of how the IC bounds depend on L and onthe DM profile. The main result is that if L is as small as 1 kpc, DM can produce a significantfraction of its e ± outside of the diffusion volume, that negligibly contribute to the IC γ signal.Indeed they escape away, as the e ± mean free path is one or two orders of magnitude longerthan the Milky Way, and the probability of entering into the diffusive halo is small. We alsochecked the importance of the diffusion zone thickness with the more realistic model describedin Sect. 2.1, in which diffusion exponentially dies off on a length scale z h . Solving the diffusionequation in an ideally infinite volume, we find that the IC γ flux is closer to the large L ≈ z h = 2 ÷ L , especially since the energy dependence of thediffusion coefficient plays only a sub-leading role in the IC predictions.The possibility of a thin diffusion volume will be relevant for our later discussion, so that itbecomes important to settle this issue. As far as we know, it is disfavored by various arguments:a) global fits of charged CR propagation models favor L ≈ (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) Μ , isothermal profile GR (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA and FERMI Ν IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) Μ (cid:43) Μ (cid:45) , isothermal profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA and FERMI Ν CMB IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) Τ (cid:43) Τ (cid:45) , isothermal profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA and FERMI Ν CMB IC freeze (cid:45) out10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) Μ , NFW profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radioGC (cid:45) VLT PAMELA and FERMI Ν IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) Μ (cid:43) Μ (cid:45) , NFW profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radioGC (cid:45) VLT PAMELA and FERMI Ν CMB IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) Τ (cid:43) Τ (cid:45) , NFW profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radioGC (cid:45) VLT PAMELA and FERMI Ν CMB IC freeze (cid:45) out
Figure 6:
Bounds on DM annihilations into leptonic channels . The
Fermi bounds aredenoted as FSR γ (continuous blue line) and IC γ (red curves, for L = 1 , , from upper tolower). Other bounds are described in the text; their labels appear along the corresponding linesonly when these bounds are significant enough to appear within the plots. Cosmological freeze-out predicts σv ≈ − cm / sec (lower horizontal band) and connections with the hierarchyproblem suggest M ∼ (10 ÷ . The region that can fit the e ± excesses survives onlyif DM annihilates into e ’s or µ ’s and DM has an isothermal profile. All bounds are at σ ;the green bands are favored by PAMELA (at σ for 1 dof ) and the red ellipses by PAMELA,FERMI and HESS (at and σ , 2 dof, as in [9]). time [22]; c) more realistic boundary conditions as described above; and presumably d) the factthat Fermi observes 100 GeV γ rays also away from the GC suggests that L is not small.Fig.s 6 and 7 show again the Fermi bounds at 3 σ (the IC γ bounds is plotted for a few valuesof the height of the diffusion volume, L = 1 , , e ± excesses and with various other 3 σ bounds already considered in previous papers [15, 23, 9]:- The GC- γ (blue continuous curves) and GR- γ (dot-dashed blue curves) bounds refer to theHESS observations [24, 25] of the photon spectrum above ≈
200 GeV (so that it constrainsFSR γ and heavier DM, rather than IC γ and lighter DM) in the ‘Galactic Center’ region( √ (cid:96) + b < . ◦ ) and in the ‘Galactic Ridge’ region ( | (cid:96) | < . ◦ and | b | < . ◦ ). In these9 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) bb , isothermal profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA Ν CMB IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) W (cid:43) W (cid:45) , isothermal profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA Ν CMB IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) hh , isothermal profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA Ν IC freeze (cid:45) out10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) bb , Einasto profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA Ν CMB IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) W (cid:43) W (cid:45) , Einasto profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA Ν CMB IC freeze (cid:45) out 10 (cid:45) (cid:45) (cid:45) (cid:45) DM mass in GeV Σ v i n c m (cid:144) s ec DM DM (cid:174) hh , Einasto profile GC (cid:45)Γ GR (cid:45)Γ dS (cid:45)Γ FSR (cid:45)Γ GC (cid:45) radio PAMELA Ν IC freeze (cid:45) out
Figure 7:
Bounds on DM annihilations into non-leptonic channels . These channels canfit the PAMELA e + excess, but not the Fermi e + + e − excess. The Fermi bounds on FSR γ exclude non-leptonic DM interpretations of the PAMELA e + excess, even for an isothermalDM profile. Non-leptonic branching ratios must be small. regions the DM density ρ ( r ) is uncertain by orders of magnitude, such that one gets strongbounds assuming NFW-like DM profiles and negligible bounds assuming isothermal-likeprofiles.- The dS- γ bound (dashed blue curves) refers to the HESS and VERITAS observations ofvarious dwarf spheroidal galaxies [26, 27, 28].- The ν bounds (black curves) refer to the SuperKamiokande (SK) observations of neutrinofrom regions around the Galactic Center [29, 23, 9, 30].- The GC-radio bound (red dashed curves) refers to radio observations of the Sgr A ∗ blackhole at the dynamical center of the galaxy and depends on the extremely uncertain localDM density [31, 15]. 10 The cosmological CMB bound (red dashed curves) refers to the contribution δτ to theoptical depth of CMB photons due to DM re-ionization of H and He .In conclusion, DM interpretations of the e ± excesses survive only if DM annihilates into 2 µ ,4 µ or 4 e and if DM has a quasi-constant isotermal-like density profile. The GC- γ and GR- γ bounds already disfavored solutions involving NFW or Einasto profiles [15, 9], but this neededextrapolating these profiles down to small scales not probed by N -body simulations. Now anunseen excess would be present at larger scales where N -body simulations are under control andfavor these profiles. Furthermore, channels involving τ are now disfavored even for an isothermalprofile. In view of the FSR- γ FERMI bound, non-leptonic channels (fig. 7) can similarly atmost have a small sub-dominant branching ratio, so that solutions involving Minimal DarkMatter or the supersymmetric wino [33] are now firmly excluded.The allowed solutions predict that a sizable fraction of the photons observed by
Fermi around 100 GeV must be due to IC γ from DM e ± . The Fermi bound on IC γ becomes weakerif the diffusive volume of our galaxy is thin, L ≈ γ and (to a lesser extent) IC γ aremoved towards the Galactic Plane, where the astrophysical γ background is higher. However,to relax the conclusions on the DM profile, one needs this fraction to be large, of order unity,especially if the dark disk has a thickness z d not much smaller than r (cid:12) .On the other hand, the Fermi bound can be made stronger subtracting from the γ spectrathe hadrons misidentified as γ and the identified astrophysical point-like sources. Interpretations of the e ± excesses in term of DM decays (rather than annihilations) attractedinterest because they were not in tension with γ -ray observations [35] (see also [36]). Indeed,the space-time density of DM decays is ρ/τ M while the space-time density of DM annihilationsis σv ( ρ/M ) /
2. Thereby, HESS observations of γ -rays from the Galactic Center gave significantconstraints on DM annihilations if r ρ ( r ) is large for r →
0. On the contrary, even for a NFWprofile, r ρ ( r ) remains small such that DM decays were not significantly constrained.Fig. 8 shows, in the mass-lifetime plane, the new Fermi
FSR- γ and IC- γ bounds on DMdecays, together with previous bounds from SK neutrino observations (for simplicity we do notplot the previous HESS γ bounds, as they are now subdominant) and with the regions favoredby interpretations of the e ± excesses in terms of DM decays. We see that such interpretationsare now constrained. Also in the case of DM decays, the remaining viable channels are 2 µ ,4 µ or 4 e . Channels like τ ’s, producing π and other mesons decaying into photons, are now We adopt the computation by Cirelli et al. [32] (not performed for all DM channels we consider), whoplotted the WMAP bound at 1 σ , δτ < . Fermi e ± excesses is compatible with the CMB bound because we plot the WMAP bound at 3 σ , δτ < . ρ (cid:12) , while all other curves actually constrain σv ρ (cid:12) . Ref. [34]claims that ρ (cid:12) is larger than the ρ (cid:12) = 0 . / cm assumed here and close to 0 . / cm : in such a casethe CMB bound would be relatively less stringent by a factor of 1 . DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) Μ , isothermal profile PAMELA and FERMI Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) Μ (cid:43) Μ (cid:45) , isothermal profile PAMELA and FERMI Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) Τ (cid:43) Τ (cid:45) , isothermal profile PAMELA and FERMI Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) Μ , NFW profile PAMELA and FERMI Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) Μ (cid:43) Μ (cid:45) , NFW profile PAMELA and FERMI Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) Τ (cid:43) Τ (cid:45) , NFW profile PAMELA and FERMI Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) bb , NFW profile Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) W (cid:43) W (cid:45) , NFW profile Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ DM mass in GeV D M li f e (cid:45) ti m e Τ i n s ec DM (cid:174) hh , NFW profile Ν IC (cid:45)Γ exG (cid:45)Γ FSR (cid:45)Γ
Figure 8:
Bounds on DM decays . In the upper rows we consider the leptonic channels thatcan fit the e ± excesses. In the lower row we consider the ‘traditional’ channels. Fermi around100 GeV away from the Galactic Center should be due to DM.The bound denoted as ‘exG- γ ’ is obtained demanding that the cosmological γ flux fromDM decays does not exceed the extra-galactic isotropic flux observed by Fermi [3]. Thiscosmological flux is expected to be comparable to the galactic flux:Φ cosmo Φ galactic ∼ ρ cosmo R cosmo ρ (cid:12) R (cid:12) ∼ ρ cosmo = Ω DM ρ cr ≈ . − GeV / cm and R cosmo ∼ /H ≈
13 Gyr. The isotropiccosmological γ flux is d Φ γ dE γ = c π (cid:90) da e − τ aH ( a ) · dN γ ( E in γ = E γ /a ) dV dt dE in γ (7)where the first term inside the integral generalizes the usual line of sight integrand ds , to thecosmological geometry described by the Hubble rate H ( a ) = H (cid:113) Ω Λ + Ω m /a , as function ofthe scale factor a of the universe, which gives the E in γ = E γ /a redshift . The last term isthe usual space-time density of γ sources, equal to Γ( ρ DM ( a ) /M ) dN γ /dE in γ in the case of DMdecays. We can neglect absorption of γ , as the optical depth is τ (cid:28) γ .The flux of photons generated by cosmological Inverse Compton scatterings of e ± fromDM annihilations on Comic Microwave Background with energy density u γ ( a ) = πT /
15 andspectrum dn γ /dE = E /π / ( e E/T −
1) at temperature T = T /a can be written as: d Φ IC γ dE γ = 9 cm e Γ ρ πH M (cid:90) da/a (cid:113) Ω Λ + Ω m /a (cid:90)(cid:90) N e ( E in e ) 1 u γ dn γ dE in γ dE in e E in4 e dE in γ E in γ f IC , (8)where N e ( E ) = (cid:82) ME dE (cid:48) dN e /dE (cid:48) and the function [40] f IC = 2 q ln q + (1 + 2 q )(1 − q ) + 12 ( (cid:15)q ) (cid:15)q (1 − q ) (9)describes IC scattering γ ( E in γ ) e ( E in e ) → eγ ( E out γ = E γ /a ) at E in e (cid:29) m e in terms of the dimen-sionless variables (cid:15) = E out γ E in e , Γ = 4 E in γ E in e m e , q = (cid:15) Γ(1 − (cid:15) ) . (10) E out γ lies in the range E in γ /E in e ≤ (cid:15) ≤ Γ / (1 + Γ). The non-relativistic (Thompson) limit corre-sponds to Γ (cid:28)
1, so that (cid:15) (cid:28) ≤ q ≤ Fermi isotropic γ data (according to the preliminary analysis in [3]),compared to the cosmological flux generated by the best-fit DM → τ + τ − model. In such a case This bound was considered in [37, 38] in the case of DM decays and in [39] in the case of DM annihilations.In such a case the larger suppression, ( ρ cosmo /ρ (cid:12) ) is counteracted by DM clumping in structures and galaxies,an effect which cannot be computed reliably. Thereby the cosmological signal allows to probe DM decay models (not considered in this paper) thatgive hard photon emission only at 400 GeV − Fermi reach, get partiallyred-shifted down to the
Fermi range. e ± excesses can have a much smaller FSR γ , but have a very similar IC γ . Thereby allsuch models predict an IC γ at the level of the Fermi observations. Depending on how
Fermi extracted the isotropic component of their sky map, it might be correct to include in it notonly the extra-galactic DM γ flux but also the galactic DM γ flux from the direction where it isminimal, thereby strengthening the bound [38]. We presented robust model-independent bounds on DM annihilations and decays obtaineddemanding that the computable part of the DM-induced γ flux be below the observed flux,as recently observed by Fermi in the full sky. Fig. 1 shows an example of bounds from thedifferent regions of the sky we consider. Our results in fig.s 3 to 7 for DM annihilations and infig. 8 for DM decays are based on a full-sky global fit, and thereby only have a mild dependenceon the DM density profile.We show the bounds on Final State Radiation γ separately from the bounds on InverseCompton γ , as the latter can be weakened if one allows for significant variations in astrophysicswith respect to the models we consider: i) increasing the magnetic fields until synchrotronenergy losses dominate over IC; this is presumably allowed only in the inner regions of theGalaxy, and would reduce our full-sky IC bounds by up to 1 . ÷
2. ii) IC bounds can bereduced by a factor of few if the e ± diffusion zone is very thin ( L ∼ ∼
50% of the local DM density canalso be invoked to weaken these bounds.On the other hand, subtraction of astrophysical backgrounds (such as identified point-likesources) and of mis-identified hadrons, still present in the
Fermi data we fitted, can onlystrengthen our bounds, presumably by a factor of few.Present data are enough to make progress on testing DM interpretations of the e ± excessesobserved by PAMELA, Fermi , ATIC. At the light of the new
Fermi γ bounds, the remainingallowed DM interpretations involve DM annihilations or decays into µ + µ − , V V → µ + µ − µ + µ − or V V → e + e − e + e − primary channels. τ ’s in the final staes are now disfavored even in the DMdecay case. Moreover, for DM annihilation, a quasi-constant isothermal-like density profile isneeded. This profile is not favored by DM simulations, that suggests that DM is concentratedaround the Galactic Center. In such a case, a viable interpretation may be obtained e.g.assuming that DM annihilates into intermediate particles V with a lifetime longer than a fewkpc [41] or that DM decays.Even in these cases, the expected DM signal is at the level of the observed flux so that itwill be interesting to improve the sensitivity with forthcoming cleaner data and more statistics.According to [4], present Fermi γ data already suggest the presence of a ‘ Fermi haze’ excesswith an angular and energy spectrum compatible with the expected IC DM excess.14 cknowledgments
We thank P. Meade, T. Volansky, M. Cirelli, P. Panci, P.D. Serpico for discussions. We verifiedthat our bounds reduce to the ones in the recent paper [38] if we neglect diffusion of e ± in theMilky Way and the finiteness of the diffusion volume and restrict our Fermi full-sky map tothe three regions presented by
Fermi below 100 GeV. The work of MP is supported in partby NSF grant PH0503584.
A Extraction of the
Fermi data
We used the
Fermi γ -ray data from [1]. We selected events from the ‘diffuse’ class, whichhave tighter cuts to reject the CR background. We considered the first 64 weeks of data (upto MET 280417908). After removing the Earth albedo, the data was binned in 0 . ◦ × . ◦ ingalactic latitude and longitude and in 16 logarithmically spaced bins in energy. We computedthe effective area and exposure times and used them to convert the binned events into thephoton flux. We then combined the bins in the larger areas of Fig. 1. We assigned systematicuncertainties according to [42] for the energy bins below 100 GeV: 10% at 100 MeV, 5% at 0.5GeV, progressively increasing to 20% at 10 GeV and above.In this analysis we considered also events around and above 100 GeV: while these are stillsignificantly contaminated by CR and the systematics of the publicly available tools are notcompletely studied, we feel that they are still usable for setting bounds the way we proceedin this paper. 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