On discrepancy between ATIC and Fermi data
OOn discrepancy between ATIC and Fermi data
Dmitry Malyshev ∗ CCPP, 4 Washington Place, Meyer Hall of Physics, NYU, New York, NY 10003 (Dated: November 15, 2018)Either ATIC or Fermi-LAT data can be fitted together with the PAMELA databy three components: primary background ∼ E − . , secondary background ∼ E − . ,and an additional source of electrons ∼ E − γ a Exp( − E/E cut ). We find that the bestfits for ATIC + PAMELA and for Fermi + PAMELA are approximately the same, γ a ≈ E cut ∼
500 GeV. However, the ATIC data have a narrow bump between300 GeV and 600 GeV which contradicts the smooth Fermi spectrum. An interpre-tation of the ATIC bump as well as the featureless Fermi spectrum in terms of darkmatter models and pulsars is discussed. ∗ Electronic address: [email protected]; On leave of absence from ITEP, Moscow, Russia, B. Cheremushkinskaya25 a r X i v : . [ a s t r o - ph . H E ] J u l The question of interpretation of Fermi-LAT [1], HESS [2, 3], and ATIC data [4] can besplit into two parts: general properties of the flux and the presence of features. If one takes intoaccount PAMELA data [5], then both Fermi and ATIC require the existence of an additional fluxof electrons and positrons complementary to the standard primary and secondary backgrounds.We will consider the following form for the additional flux F a ∼ E − γ a e − EEcut . (1)The general properties of the flux will be parameterized by the index γ a and the exponentialcutoff E cut . Significant deviations from this form will be considered as “features”.We will assume the primary background ∼ E − γ p , the index γ p ≈ . − . − . . ∼ E − . [11, 12] with the dust. We will assume the secondary background ∼ E − γ s , γ s ≈ . γ a ≈ E cut ∼
500 GeV, i.e., these experiments are consistent with each other from thepoint of view of general properties of the flux parameterized by Eq. (1). It should be noted thatwithout PAMELA data, the ATIC bump is better fitted with a harder additional flux, γ a ≈ . γ p ≈ . − . χ = 0 .
4, than the best fit for ATIC + PAMELA, χ = 1 . (cid:248)(cid:248) (cid:137)(cid:137) (cid:243)(cid:243) FermiATICConcordance
200 400 600 800 10001.61.82.02.22.4 E cut (cid:72) GeV (cid:76) Γ a FIG. 1: The results of fitting the primary and secondary backgrounds together with an additional sourceof electrons and positrons to Fermi + PAMELA and to ATIC + PAMELA. As everywhere else in thepaper, for PAMELA, only
E >
10 GeV points are used. The best fit values and the parameter rangesare given in Table I. The star (cross) represents the best fit to Fermi + PAMELA (ATIC + PAMELA)with the reduced chi-squared χ
2r best = 0 . χ
2r best = 1 . χ
2r best + 1 and χ
2r best + 2 for Fermi (ATIC) + PAMELA. The concordance model has χ = 0 . .
0) forFermi (ATIC) + PAMELA. kpc. For the purposes of flux calculation, the DM distribution can be viewed as homogeneousand constant [14] (unless there is a significant contribution from a local DM substructure suchas a clump [14, 15, 16]). The flux from a homogeneous source is [10] F DM = c π b ( E ) (cid:90) M DM E Q DM ( ˜ E ) d ˜ E, (2)where ˙ E ≡ − b ( E ) is the energy losses. At E >
10 GeV the energy losses are due to InverseCompton Scattering and synchrotron radiation in the galactic magnetic field, thus b ( E ) = b E [10]. Q DM is the source function for e + e − produced by annihilating or decaying DM, Q DM = dNdEdV dt .The cutoff energy E cut is the energy where the integral on the right hand side of Eq. (2) issaturated. For energies E (cid:28) E cut the integral is insensitive to the variations of the lower limitand F DM ∼ /b ( E ). Thus an index γ a ≈ γ a < γ a > γ a < E cut ∼
500 GeV, the DM mass and the shape of the spectrum depend on the DM
Data γ a E cut E F ( E ) χ γ p γ s (GeV) (GeV m − s − sr − )ATIC + PAMELA 1 . ± .
15 480 ±
200 50 ± . − . . ± .
10 500 ±
150 44 ± . − . .
05 450 47 1.3 3.29 3.63TABLE I: Numerical values for the fits presented in Fig. 1. The normalization is given for E = 100GeV. The ranges of parameters correspond to the ranges of reduced chi-squared χ in the fifth column.The error bars for Fermi are computed as square root of systematic plus statistical errors squared. γ p and γ s are the indices of the primary and secondary backgrounds respectively. The χ for the concordancemodel is computed using Fermi, ATIC, and PAMELA ( >
10 GeV) points. The best fits are found byvarying 7 parameters: 2 indices and 2 normalization constants for primary and secondary backgroundstogether with the index, the normalization and the cutoff of the additional flux in Eq. (1). model. In general, DM annihilation is followed by a sequence of decays leading to electrons andpositrons together with other stable particles in the end. Models with many steps in the decayprocess have smooth e + e − spectrum and a large DM mass M DM (cid:29) E cut [14, 18]. These modelsare favored [19, 20] by featureless Fermi spectrum. Models with decay channels through W , Z gauge bosons and quarks have stricter constraints due to absence of significant deviations fromthe expected backgrounds for anti-protons [21] and diffuse gamma rays from the Galactic center[22, 23]. The flux of gamma rays from DM clumps assuming b ¯ b annihilation channel was alsoestimated [24]. DM models with small DM mass M DM < ∼
500 GeV (e.g., [25, 26]) seem to be intension with Fermi data due to absence of significant step-like features below or around 500 Gev.DM models with few decay steps have a sharper cutoff and M DM > ∼ E cut [14, 18, 27]. Thesemodels give reasonable fits to the Fermi data for M DM < ∼ M DM they, generally, fit better the ATIC data [18]. These models may have additionalconstrained due to final state radiation from the Galactic center and the Galactic ridge [18, 27].Local clumps may produce additional features at high energies. The presence of a large localclump is disfavored by Fermi but consistent with ATIC [14]. Furthermore, in models with manydecay steps, a significant contribution from a local clump may be necessary to fit the ATIC bump.Thus, the Fermi data favor dark matter with many decay steps and a large DM mass whileATIC requires either a DM model with few decay steps or a significant local substructure in DMdensity distribution.Let us now turn to pulsars. In the calculation of electron and positron fluxes, pulsars can beconsidered as point-like instantaneous sources [28], Q ∼ δ ( x − x ) δ ( t − t ). The main reason isthat the typical propagation time ( > ∼
100 kyr) is much larger than the characteristic time scale (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232)(cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:232)
ATIC (cid:236)
HESS2008 (cid:159)
HESS2009 (cid:159)
FermiConcordance modelExtra sourceBackgrounds10 50 100 500 1000 5000501002005001000 E (cid:72)
GeV (cid:76) E d N d E (cid:72) G e V m (cid:45) s (cid:45) s r (cid:45) (cid:76) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) Total positron ratioExtra sourceSecondary background (cid:232)
PAMELA10 20 50 100 200 500125102050100 E (cid:72)
GeV (cid:76) (cid:215) F e (cid:43) F e (cid:43) (cid:43) F e (cid:45) FIG. 2: The concordance model corresponding to the triangle in Figure 1 with the parameters given inTable I. The upper (lower) edge of the band is the best fit for ATIC (Fermi) + PAMELA points withenergies
E >
10 GeV. HESS 2008 and HESS 2009 points fit the concordance model, if multiplied by afactor ∼ . when a pulsar loses most of its rotational energy and the electrons and positrons are released tothe interstellar medium (ISM).For every pulsar, we will consider two energy scales [13, 28]: the cutoff in the injectionspectrum of electrons and positrons from the pulsar into the ISM, E inj . cut , which can be betweenfew hundred GeVs and tens of TeVs (see, e.g., [29] and references in [28]), and the cooling break E br = b t , obtained by integrating the energy losses ˙ E = − b E during the propagation time t , where t is approximately the age of the pulsar. At E ≈
100 GeV the energy losses can beestimated as b ≈ . − s − ≈ − Myr − [30]. At energies E > ∼
100 GeV the coefficient b slightly decreases with the energy [30], since the Thompson approximation to inverse Comptonscattering between electrons and the starlight becomes inapplicable.For pulsars with age t < ∼
10 Myr, the cooling break is E br > ∼
20 GeV. In the ATNF catalogthere are several hundred pulsars within 3 kpc from Earth and an age t <
10 Myr [31]. Belowapproximately 300 GeV, the corresponding flux is well approximated by the flux from a continuousdistribution of pulsars in the Galactic plane [28]. An index γ a ≈ γ inj < ∼ E > ∼
300 GeV, the flux receives contributions only from young pulsars t < ∼ d < ∼ E inj . cut and E cut , there are two possibilities for the e + e − flux from pulsars: • E inj . cut ∼ E cut , then the observed spectrum is naturally flat if we assume that the injectionspectrum from pulsars is flat. This possibility is favored by the Fermi data [13]. • E inj . cut (cid:29) E cut , then the cutoff in the observed spectrum is due to the cooling break whichis much sharper than an exponential cutoff. One should also expect a series of steps due toconsecutive cooling break cutoffs from different pulsars [28]. This possibility is consistentwith ATIC but may be in tension with Fermi.To summarize, both Fermi and ATIC require an additional source of electrons and positronswith an index γ a ≈ E cut ∼
500 GeV. However the presence of abump at high energies in ATIC data contradicts the smooth spectrum of Fermi and HESS. Thesources that produce featureless spectrum include the DM models with M DM > E inj . cut ∼ E cut . Thesources that can produce ATIC bump include DM models with M DM ∼ E cut and few decay stepsor DM models with M DM (cid:29) E cut and a significant contribution from a local clump. Pulsars with E inj . cut (cid:29) E cut may give a sharp cutoff in the observed e + e − flux and, possibly, step-like featuresdue to a series of cooling break cutoffs from the youngest nearby pulsars. This possibility is intension with the Fermi data but consistent with the ATIC. Acknowledgments.
The author is thankful to Mirko Boezio, Ilias Cholis, Joseph Gelfand, Lisa Goodenough, MichaelKuhlen, Neal Weiner, and Weiqun Zhang for valuable discussions. This work is supported inpart by the Russian Foundation of Basic Research under grant RFBR 09-02-00253 and by theNSF grants PHY-0245068 and PHY-0758032. [1]
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