"Discrepant hardenings" in cosmic ray spectra: a first estimate of the effects on secondary antiproton and diffuse gamma-ray yields
aa r X i v : . [ a s t r o - ph . H E ] J a n “Discrepant hardenings” in cosmic ray spectra: a first estimate of the effects onsecondary antiproton and diffuse gamma-ray yields. Fiorenza Donato ∗ Dipartimento di Fisica Teorica, Universit`a di Torino and INFN–Sezione di Torino, Via P. Giuria 1, 10122 Torino, Italy
Pasquale D. Serpico † LAPTh, UMR 5108, 9 chemin de Bellevue - BP 110, 74941 Annecy-Le-Vieux, France
Recent data from CREAM seem to confirm early suggestions that primary cosmic ray (CR)spectra at few TeV/nucleon are harder than in the 10-100 GeV range. Also, helium and heaviernuclei spectra appear systematically harder than the proton fluxes at corresponding energies. Wenote here that if the measurements reflect intrinsic features in the interstellar fluxes (as opposed tolocal effects) appreciable modifications are expected in the sub-TeV range for the secondary yields,such as antiprotons and diffuse gamma-rays. Presently, the ignorance on the origin of the featuresrepresents a systematic error in the extraction of astrophysical parameters as well as for backgroundestimates for indirect dark matter searches. We find that the spectral modifications are appreciableabove 100 GeV, and can be responsible for ∼
30% effects for antiprotons at energies close to 1 TeVor for gamma’s at energies close to 300 GeV, compared to currently considered predictions based onsimple extrapolation of input fluxes from low energy data. Alternatively, if the feature originatesfrom local sources, uncorrelated spectral changes might show up in antiproton and high-energygamma-rays, with the latter ones likely dependent from the line-of-sight.
PACS numbers: 98.70.Sa LAPTH-043/10
I. INTRODUCTION
A more accurate determination of primary cosmic rayspectra at the top of the atmosphere has obvious im-plications for the understanding of the acceleration andpropagation of Galactic cosmic rays. It is also crucialfor other fields of investigations in astroparticle physics,two notable examples being atmospheric neutrino stud-ies (e.g. [1]) and the calculation of the backgrounds forindirect dark matter (DM) searches (see for example [2]).In the specific case of indirect DM searches, an impor-tant implicit assumption is that fluxes measured at thetop of the atmosphere, at sufficiently high energies toavoid solar modulation effects, are representative of in-terstellar medium (ISM) spectra. Or, more correctly, oneoften assumes universality for the injection term and thepropagation properties, fitting the free parameters (likeinjection and diffusion power-law index) in such a way toreproduce the observed spectra. It is those “universal”interstellar spectra which in turn enter as source term ofsecondary yields (like antiprotons or diffuse gamma rays)due to inelastic collisions in the ISM. This assumption isusually supported by the apparent featureless nature ofthe observed cosmic ray fluxes (suggesting, at least inaverage, some universal mechanism for production andpropagation) as well as by the check a posteriori thatthe diffuse gamma-ray radiation of hadronic origin hasa spectrum consistent with the hypothesis, within theerrors. However, especially at energies larger than the ∗ Electronic address: [email protected] † Electronic address: [email protected]
TeV scale, inferring accurate spectra is challenging duethe scarce statistics and experimental difficulties, mak-ing the above arguments at best based on shaky observa-tional evidence. Also, for assessing uncertainties in sec-ondary yields, the usual practice is to fit primary datato some power-law parameterization and extrapolate tohigh energies. While this is a reasonable prescriptionfor most applications given the present level of under-standing, these simplified approaches and assumptionsmight hide a systematic error when searching for signa-tures showing peculiar energy features. For example, forantiprotons this is the case involving contributions fromDM annihilation [3] or production at the sources [4].Obviously, the standard prescriptions do not usuallyaccount for the possibility that a systematic departure(rather than statistical scattering) is present in the spec-tral shape of the fitting formula, which is mostly cali-brated on low energy data, neither of the possibility thatobserved spectra might not be fair representative of theinterstellar ones. Recent data from the CREAM balloon-borne experiment [5] seem to confirm earlier suggestions(see e.g. [6]) that cosmic ray spectra at few TeV/nucleonare harder than in the 10-100 GeV range, and that he-lium (He) and heavier nuclei fluxes are harder than theproton ( p ) flux at corresponding energies. Preliminarydata from PAMELA also suggest a hardening in p andHe spectra at a rigidity of about 250 GV, with a He spec-trum having an index ∼ . ∼ local phenomena/sources, sec-ondary yields which probe a large volume of the ISM,like antiprotons, might not show relevant departures fromnaive expectations, while the diffuse gamma-rays alongdifferent lines of sight might reveal different hardeningsreflecting the primary spectra present in different regionsof the ISM. Clearly, this provides an important test fortheories about the origin of the breaks. To the best ofour knowledge, present data in high energy astrophysicsare either unrelated to the hardenings discussed here orstill of too limited precision to provide a crucial test, butthe situation is likely to change in the near future.This article is structured as follows: in Sec. II we dis-cuss the input fluxes and parameterization used to pro-vide a first estimate of the effect. In Sec. III we presentthe results for antiprotons and γ -rays, finally in Sec. IVwe discuss some implications of our findings, and con-clude. II. INPUT FLUXES
In the present exploratory study, we refrain from theambitious goal of analyzing the whole body of cosmicray flux data in the 10 − GeV/n range. Rather welimit ourselves to provide a first assessment of the sys-tematic effect potentially introduced by deviations fromthe power law behaviour at high energy, in general withdifferent spectral indexes for different species. To thispurpose, we explore the effects of combining the fits of“low-energy” (namely in the range about 10-100 GeV/n)proton ( i = 1) and helium ( i = 2) flux data, φ Li , takenfrom AMS-01 [9] (in turn, to large extent consistent withwhat reported by other experiments), with the “high-energy” (above about 1 TeV/n) fluxes φ Hi inferred byCREAM [5]. We adopt broken power-laws to connectthe two sets, using the following flux parameterizations(differential fluxes with respect to kinetic energy per nu-cleon T ): φ ( T ) = φ L ( T )Θ( B − T ) + φ H ( T )Θ( T − B ) , (1) φ ( T ) = φ L ( T )Θ( B − T ) + φ H ( T )Θ( T − B ) . (2)The fluxes “ L ” are the best fit values taken from AMS-01 [9], rewritten in terms of kinetic energy T per nu-cleon (in GeV/n) instead of rigidity and asymptoticallydecreasing as ∼ T − . for p and T − . for He. In unitsof (GeV / n m s sr) − , they write φ L ( T ) = 1 . × (cid:16)q ( T + m p ) − m p (cid:17) − . , (3) φ L ( T ) = 5 . × p (4 T + m He ) − m ! − . . (4)The high energy fluxes “ H ” are taken from CREAM withthe following criteria: i) power-laws in T are assumed,with the spectral indexes fixed to the best-fit values re-ported in [5], i.e. 2.66 for p and 2.58 for He; ii) the protonspectrum normalization is taken from the first CREAMpoint in Fig. 3 of [5]; iii) the Helium spectrum normal-ization follows from imposing that at T = 9 TeV/nucleonthe proton to helium flux ratio is equal to 8.9 [5]. As aresult, in units of (GeV / n m s sr) − , φ H ( T ) = 7 . × T − . , (5) φ H ( T ) = 4 . × T − . . (6)The crossover energies B , B for the broken power-lawsare simply obtained by continuity, and are approximately B p = 1000 GeV, B He = 30 GeV/n for the parametersabove . A comparison with the predictions followingfrom the extrapolation of the AMS-01 fits (i.e. the φ Li ofEqs.(3,4)) to arbitrarily high energy will be presented toprovide an estimate of the impact of high-energy spectraluncertainty on the secondary yield flux. III. RESULTS
Discrepant hardenings of primary cosmic ray fluxes,possibly of non-universal nature, would obviously affectall the yields of e + , ¯ p and γ secondaries produced bycollisions in the interstellar medium (ISM). Here we donot discuss charged leptons simply because the primaryflux effects do not provide the major uncertainty in theflux shape (even fixing the average propagation parame-ters): very likely recent data [10–12] indicate that addi-tional sources of “primary” positrons exist for which theabove mentioned effects are expected to be sub-leading(see e.g. [13]). Additionally, energy losses make the rangeshorter and the computation of the actual flux at theEarth non-trivial, so it would be more difficult to dis-entangle the effects due to the break in primary spectrafrom a complicated interplay of effects involving the dis-creteness of local sources, inhomogeneities in the radia-tion field, etc. as illustrated for instance in [14]. Theeffect of universal primary CR hardening should be ap-preciable in the predicted shape of the antiproton or dif-fuse gamma-ray signal. Here we report a careful com-putation of the effect on the antiproton spectrum, wherethe impact is expected to be the largest in view of futurehigh-statistics results from AMS-02, and an estimate of Assuming a relative uncertainty in the flux normalization of thetwo experiments of < ∼ ∼
250 GV hinted to by PAMELA, ref. [7]
FIG. 1: Ratio of antiproton fluxes from hard sources (Eqs.(1,2)) to the same flux obtained with p and He extrapolatedfrom AMS data to all energies (see text for details). the effect on the hadronic gamma-ray diffuse background,of some interest for the interpretation of FERMI data.
A. Effects on secondary antiprotons.
The computation of the secondary ¯ p flux has been per-formed as described in Refs. [2, 3], to which we referfor all the details. The only component which we willmodify in the present calculation is the input p and Hespectra. We briefly remind that secondary ¯ p are yieldedby the spallation of cosmic ray proton and helium nu-clei over the H and He nuclei in the ISM, the contribu-tion of heavier nuclei being negligible. The frameworkused to calculate the antiproton flux is a two–zone dif-fusion model with convection and reacceleration, as wellas spallations on the ISM, electromagnetic energy lossesand the so–called tertiary component, corresponding tonon–annihilating inelastic scatterings on the ISM. Therelevant transport parameters are constrained from theboron-to-carbon (B/C) analysis [15] and correspond to:i) the half thickness of the diffusive halo of the Galaxy L ; ii) the normalization of the diffusion coefficient K and its slope δ ( K ( E ) = K βR δ ); iii) the velocity of theconstant wind directed perpendicular to the galactic disk ~V c = ± V c ~e z ; and iv) the reacceleration intensity param-eterized by the the Alfv´enic speed V a . The above pa-rameters show significant degeneracies when confrontedto B/C data [15]. Nevertheless, the impact on the sec-ondary ¯ p flux is marginal [2]. The fluxes presented be-low have been obtained for the B/C best fit propagationparameters, i.e. L = 4 kpc, K = 0 . Myr − , δ = 0 . V c = 12 . km s − and V a = 52 . − [15].We are interested in the effect of primary p and Hehardening at high energies on the ¯ p flux and thereforeconcentrate on the relative shape effect through antipro-ton flux ratios. Our results are reported in Fig. 1, where we plot the ratio of antiproton fluxes obtained with twodifferent primary spectra. The flux at the numerator hasbeen obtained with the spectra in Eqs.(1, 2), while in thedenominator we employ the fit to AMS data arbitrarilyextrapolated to the highest energies. The modificationof the antiproton flux clearly reflects in its shape. Theeffect of the hardening of primary spectra at hundredsof GeV/n starts to be visible on the antiproton flux ataround 100 GeV. It is near 15% at 200 GeV and reaches30% at 1 TeV. Given the weak dependence of the sec-ondary antiproton flux on the B/C selected transportparameters, our results can be considered nearly inde-pendent of the propagation model. If the hardening ofprimary nuclei will be confirmed at high energies, a spec-tral distortion of the secondary antiproton flux has tobe expected. This effect could be potentially observableby a future high precision space-based mission, such asAMS-02. B. Effects on hadronic diffuse gamma-rays.
The cosmic gamma ray flux observed in our Galaxyis expected to be mainly due to the inelastic scatteringof incoming CRs on the nuclei of the ISM. The involvedhadronic reactions produce gamma rays mostly via π decays. In addition to this hadronic component, othercontributions are expected —at different levels dependingon the specific model— to Inverse Compton and brems-strahlung radiation. The basic models for the productionof gamma rays from π decays, considered for exampleby the Fermi-LAT Collaboration [16], do not introducehigh-energy spectral breaks in the proton spectrum φ ,and account for nuclear effects (both in CR spectra andin target composition) in the π yield simply by rescal-ing the pp production via a constant “nuclear enhance-ment factor”, taken from the value at the reference en-ergy T ∗ ≡
10 GeV/n reported in [17]. This enhancementencodes the relative yield of gamma-rays from nucleus- p and nucleus-Helium collisions compared with that from p - p collisions via appropriate factors m ip , m iα , basicallyconstant at T >
10 GeV/n (the effects discussed in thispaper are only relevant at high energy, so it’s enough tofocus on quantities at
T >
10 GeV/n). This enhancementis defined as ǫ M ( T ) = X i m i φ i ( T ) φ ( T ) + X i m i φ i ( T ) φ ( T ) × r − r , (7)where the index i runs over all CR species (includingprotons, i = 1), r ≃ .
096 is the He/H fraction in theISM and φ i being the CR spectrum of the species i . Ifall the nuclei have roughly identical T dependence of theirspectra, as suggested in [5], one can write ǫ M ( T ) = 1 + m r − r + (cid:18) m + m r − r (cid:19) φ φ + k N φ N φ , (8)where φ N ( T ) is any nuclear-like CR flux, and k N is anormalization factor. In Fig. 2 we show ǫ M ( T ) for three T [GeV/n] ε M NaiveHe breakAll
FIG. 2: Enhancement factor, see Eqs. (7,8), for the threerepresentative cases described in Sec. III B. cases: i) the constant value ǫ M = 1 .
84, adopted for ex-ample in [16] (long-dashed, black); ii) the fluxes of p andHe are set to the broken power-law functions describedabove, while the last term k N φ N /φ is taken constantin energy and fixed so that ǫ M ( T ∗ ) = 1 .
84 (short-dashed,blue). iii) As in ii) for p and He, but assuming for nu-clei heavier than He a constant contribution to ǫ below200 GeV/n (so that ǫ M ( T ∗ ) = 1 . T . ,as suggested by CREAM data (solid, red).In Fig. 3, we show the result of computing the diffusegamma ray spectrum (via the kernel provided in [18] )using the AMS-01 spectral fits φ Li , extrapolated to arbi-trarily high energy (long-dashed, black curve). The fluxhas been multiplied by E . γ to underline the departurefrom identical power-law behaviour between photons andparent CR due to production cross section/multiplicityeffects. Instead, if one keeps ǫ M = 1 .
84, but introducesthe broken power-law spectrum for the protons only asfrom Eq. (1), around 300 GeV one would obtain ∼ . − . ≃ . ∼ .
01 and 0 .
02 fit errors, respectively, reported by theexperiments). The solid, red curve shows the effect of“discrepant hardenings” of the spectra, namely the T dependence of ǫ M . This constitutes the major distortionand is mostly due to He (as shown by the short-dashed, Note that we are only interested in the effects that different high-energy CR spectra have on the gamma-ray spectrum at E γ ≫ E γ [GeV] E γ . φ γ [ a . u . ] Naivep break p+He breakAll
FIG. 3: γ -ray spectrum: the standard departure from equalityof power-law with parent flux (dot-dashed, black), of addingthe hardening in the p spectrum at TeV scale as suggestedby CREAM (long-dashed, purple), of assuming the CREAMhardening for both p and He (short-dashed, blue), and of in-cluding the small effect of other nuclei as well (solid, red). blue curve); overall, the spectrum around 300 GeV is 30%higher with respect to naive expectations.The effect discussed here is already “an estimate of er-ror”: assessing the error on this quantity goes beyondour purpose. However, we can safely conclude that ourresults are not significantly affected by the statistical er-rors with which the normalizations or spectral indices areknown. We have checked this explicitly as follows: whilekeeping the Helium fluxes at low and high energies at thevalues described above, we have varied the normalizationof the p fluxes at low/high energy in such a way that the p to He ration varies within ± ± . the shape of secondary radiation are negligible (i.e. atmost at a few percent level). This is due to the fact thatthe (relatively small) effect of p flux renormalization andthe change in ǫ M anti-correlate, and tend to cancel eachother. On the other hand, the exact value of the spectralhardening is more important: Fig. 3 shows that morethan 1/3 of the hardening is due to the assumed “bestfit” spectral index difference of ± .
12 between low andhigh energy. This should be compared with the statistical errors of about ∼ ± .
01 and ± .
02 quoted by the AMSand CREAM collaborations, respectively.
IV. DISCUSSION AND CONCLUSIONS
In this article we have argued that departure at highenergy from a simple and universal power-law for all cos-mic ray spectra, as suggested by recent data, should causea spectral distortion in the spectra of secondary cosmicray yields (like diffuse photons and antiprotons) com-pared to the predictions obtained extrapolating the bestfits to low-energy data sets. We have illustrated this ef-fect using the best fit results of AMS-01 data at low ener-gies and the CREAM data at high-energy, finding effectsexceeding 10% above ∼
100 GeV, and reaching about 30%for photons around 300 GeV and for ¯ p close to TeV en-ergy; this figure is somewhat sensitive to the systematicerror on the spectral index at high energy as well as othereventual systematics which do not cancel out in ratios ofspecies (like p /He). If the hardening in nuclei data wouldbe due to local effect and not representative of the ISMaverage, the effect on the antiproton yield may be smallwhile the hadronic diffuse gamma-ray spectrum could bemodified differently according to the line-of-sight. No-tice that the effect of a possible harder nuclear spectraon atmospheric neutrinos was already estimated in [1].One might wonder how relevant is a high-energy effectof a few tens of percent in a field where data are usu-ally plagued by larger errors. We think that, at present,this level of accuracy is becoming crucial for at least acouple of reasons: First, space experiments like FERMIor the future AMS-02 [19] are introducing us to a newera of large exposures, which can reveal more subtle fea-tures than previous cosmic ray or gamma-ray experi-ments. Fermi data errors at E γ ≃
100 GeV are already ∼ ±
20% [20], and forecasts that have been presented sug-gest that AMS-02 (if performing close to specifications)will be certainly sensitive to effects of this magnitude, seefor example [21]. Second, both diffuse gamma-rays [22]and the combination of hadronic data [23] are consistent,at least at leading order, with a “standard” scenario forthe production and propagation of cosmic rays in theGalaxy. It is very likely that any departure from base-line models, if detectable, is going to be present at such asub-leading level. Modelling thus the astrophysical back-ground for indirect DM searches as a simple power law,as often done in the literature, might lead to wrong con-clusions about the evidence of a signal, or to a bias inthe inferred values of the parameters describing the new phenomena, should they be detected.Even in a conservative scenario, the detection of suchspectral signatures in secondary channels would providea way to check the interstellar nature of the spectral fea-tures in the cosmic ray flux at the Earth suggested by thepresent experiments. We believe that secondaries providean important handle for an empirical cross-check. Oneshould also consider the partial degeneracy of such effectswith the extraction of propagation parameters, in orderto fully exploit the statistical power of forthcoming datasets. Knowing better the primary flux shapes would al-low one to set strategies minimizing these effects. Lastbut not least, a multi-messenger approach would allowone to disentangle these features from alternative sourcesof spectral distortions: features similar to the ones dis-cussed in this article arise e.g. in models where high en-ergy ¯ p are produced in sources [4], but in that case alsoassociated signatures in secondary/primary “metals” [24](and possibly in high energy neutrinos [25]) are expected,which are absent for the process described here.While we are entering a much higher precision erain cosmic ray studies, it is important to keep in minda couple of points: i) that multi-messenger and multi-channel analyses are mandatory, if one is to gain somedeeper knowledge of cosmic ray astrophysics. ii) Thatany hope for the detection of new physics (not to speakof extracting new physics parameters) requires a morerobust understanding of the possible range of astrophys-ical yields. In that respect, a natural development ofthis initial investigation would be to (re)assess how theerrors on primary flux knowledge map into the predic-tions for secondaries (including their normalization), asmuch as possible in a parameterization-independent way. Acknowledgments
We warmly acknowledge David Maurin for the agree-ment to use the USINE code for the calculation of theantiproton flux. [1] G. D. Barr, T. K. Gaisser, S. Robbins and T. Stanev,“Uncertainties in atmospheric neutrino fluxes,” Phys.Rev. D , 094009 (2006).[2] F. Donato, D. Maurin, P. Salati, A. Barrau, G. Boudouland R. Taillet, “Antiprotons from spallation of cosmicrays on interstellar matter,” Astrophys. J. , 172(2001).[3] F. Donato, D. Maurin, P. Brun, T. Delahaye andP. Salati, “Constraints on WIMP Dark Matter from theHigh Energy PAMELA ¯ p/p data,” Phys. Rev. Lett. ,071301 (2009).[4] P. Blasi and P. D. Serpico, “High-energy antiprotons fromold supernova remnants,” Phys. Rev. Lett. , 081103(2009). [5] H. S. Ahn et al. , “Discrepant hardening observed incosmic-ray elemental spectra,” Astrophys. J. , L89(2010).[6] A. D. Panov et al. , “Elemental energy spectra of cosmicrays from the data of the ATIC-2 experiment,” Bulletinof the Russian Academy of Sciences: Physics , Vol. 04,494-497 (2007) [astro-ph/0612377].[7] Talk by O. Adriani at 35th International Conferenceon High Energy Physics, Paris 2010. Slides available at http://pamela.roma2.infn.it .[8] P. L. Biermann, J. K. Becker, J. Dreyer, A. Meli,E. S. Seo and T. Stanev, “The origin of cosmic rays: Ex-plosions of massive stars with magnetic winds and theirsupernova mechanism,” Astrophys. J. , 184 (2010). [9] M. Aguilar et al. [AMS Collaboration], “The Alpha Mag-netic Spectrometer (Ams) On The International SpaceStation. I: Results From The Test Flight On The SpaceShuttle,” Phys. Rept. , 331 (2002) [Erratum-ibid. , 97 (2003)].[10] O. Adriani et al. [PAMELA Collaboration], “An anoma-lous positron abundance in cosmic rays with energies 1.5-100 GeV,” Nature , 607 (2009).[11] A. A. Abdo et al. [Fermi LAT Collaboration], “Measure-ment of the Cosmic Ray e+ plus e- spectrum from 20GeV to 1 TeV with the Fermi Large Area Telescope,”Phys. Rev. Lett. , 181101 (2009).[12] M. Ackermann et al. [Fermi LAT Collaboration], “FermiLAT observations of cosmic-ray electrons from 7 GeV to1 TeV,” Phys. Rev. D , 092004 (2010).[13] P. D. Serpico, “On the possible causes of a rise with en-ergy of the cosmic ray positron fraction,” Phys. Rev. D , 021302 (2009).[14] T. Delahaye, J. Lavalle, R. Lineros, F. Donato andN. Fornengo, “Galactic electrons and positrons at theEarth:new estimate of the primary and secondary fluxes,”Astron. Astrophys. , A51 (2010).[15] D. Maurin, F. Donato, R. Taillet, P. Salati, “Cosmicrays below z=30 in a diffusion model: new constraints onpropagation parameters,” Astrophys. J. , 585 (2001).[16] A. A. Abdo et al. [Fermi LAT Collaboration], “FermiLAT Observation of Diffuse Gamma-Rays ProducedThrough Interactions between Local Interstellar Matterand High Energy Cosmic Rays,” Astrophys. J. , 1249(2009).[17] M. Mori, “Nuclear enhancement factor in calculation of Galactic diffuse gamma-rays: a new estimate withDPMJET-3,” Astropart. Phys. , 341 (2009).[18] S. R. Kelner, F. A. Aharonian and V. V. Bugayov, “En-ergy spectra of gamma-rays, electrons and neutrinos pro-duced at proton proton interactions in the very high en-ergy regime,” Phys. Rev. D , 034018 (2006) [Erratum-ibid. D , 039901 (2009)].[19] [20] A. A. Abdo et al. [Fermi-LAT collaboration], “TheSpectrum of the Isotropic Diffuse Gamma-Ray EmissionDerived From First-Year Fermi Large Area TelescopeData,” Phys. Rev. Lett. , 101101 (2010).[21] J. Casaus, “The AMS-02 experiment on the ISS,” J.Phys. Conf. Ser. , 012045 (2009).[22] A. A. Abdo et al. [Fermi LAT Collaboration], “FermiLarge Area Telescope Measurements of the DiffuseGamma-Ray Emission at Intermediate Galactic Lati-tudes,” Phys. Rev. Lett. , 251101 (2009).[23] G. Di Bernardo, C. Evoli, D. Gaggero, D. Grasso andL. Maccione, “Unified interpretation of cosmic-ray nucleiand antiproton recent measurements,” Astropart. Phys. , 274 (2010).[24] P. Mertsch and S. Sarkar, “Testing astrophysical modelsfor the PAMELA positron excess with cosmic ray nuclei,”Phys. Rev. Lett. , 081104 (2009).[25] M. Ahlers, P. Mertsch and S. Sarkar, “On cos-mic ray acceleration in supernova remnants and theFERMI/PAMELA data,” Phys. Rev. D80