Cosmic Ray Protons in the Inner Galaxy and the Galactic Center Gamma-Ray Excess
CCosmic Ray Protons in the Inner Galaxy and the Galactic Center Gamma-Ray Excess
Eric Carlson ∗ and Stefano Profumo † Department of Physics and Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA (Dated: March 13, 2018)A gamma-ray excess over background has been claimed in the inner regions of the Galaxy, triggering someexcitement about the possibility that the gamma rays originate from the annihilation of dark matter particles. Wepoint out that the existence of such an excess depends on how the diffuse gamma-ray background is defined, andon the procedure employed to fit such background to observations. We demonstrate that a gamma-ray emissionwith spectral and morphological features closely matching the observed excess arises from a population ofcosmic ray protons in the inner Galaxy, and provide proof of principle and arguments for the existence of sucha population, most likely originating from local supernova remnants. Specifically, the “Galactic center excess”is readily explained by a recent cosmic-ray injection burst, with an age in the 1-10 kilo-year range, while theextended inner Galaxy excess points to mega-year old injection episodes, continuous or impulsive. We concludethat it is premature to argue that there are no standard astrophysical mechanisms that can explain the excess.
PACS numbers: 95.85.Pw, 96.50.S-, 96.50.sb, 95.35.+d
I. INTRODUCTION
The Galactic center is a promising location to search fornon-gravitational signals from particle dark matter such asgamma rays from dark matter pair annihilation. Any modelfor the density distribution of dark matter in the Galaxy pre-dicts a high concentration of dark matter in the Galactic cen-ter, with a resulting large number density of dark matter parti-cle pairs. Barring the possibility of a large, nearby dark matter“clump”, the Galactic center direction is the direction in thesky where the line-of-sight integral of the dark matter densitysquared is maximal. As a result, the Galactic center is thelocation where one of the brightest photon signals from darkmatter annihilation is expected.On the downside, the center of the Galaxy hosts a com-plex combination of “standard” astrophysical γ -ray sources.The region contains numerous resolved and many unresolved γ -ray point sources; in addition, the diffuse Galactic emis-sion is brightest in the center of the Galaxy, where the largestmacroscopic concentrations of gas, cosmic rays and inter-stellar radiation energy density are found. This dense en-vironment copiously sources γ rays from hadronic inelasticinteractions as well as from inverse Compton scattering andbremsstrahlung. Such complex background structure can behardly reconstructed from first principles, and non-trivial ex-trapolations and inference often, if not always, define how thepredicted background emission is calculated.The combination of such an appealing target with such atreacherous background has contributed to much debate aboutthe existence and nature of excess γ -ray emission from theGalactic center region. Ever since the years of the EGRETtelescope, claims of an excess diffuse γ -ray emission (extend-ing even beyond the Galactic center) have been made [1, 2],and proved premature, with several groups proposing a Galac-tic cosmic-ray spectra differing from local values [3, 4]. TheEGRET excess was subsequently shown to be systematic in ∗ [email protected] † [email protected] origin, relating instead to a miscalculation of EGRET’s sen-sitivity above a few GeV [5], and was later shown to be con-clusively unfounded [6] using data from the Fermi Large AreaTelescope (LAT) [7].Shortly after LAT data were made public, claims of a Galac-tic center Excess (GCE) have been put forward, pointing todiffering particle dark matter properties (including the pre-ferred mass, pair-annihilation rate, and annihilation pathway)depending on the background model employed in the analysis,see e.g. [8–10].Several immediate issues have been raised following theidentification of excess of γ rays over background and withassociations to new physics. These include the question of γ -ray point source modeling associated with the radio sourceSgr A*, see e.g. [11–14], and the role of unresolved popula-tions of γ -ray emitters such as millisecond pulsars [15] (seehowever [16]).One of the key elements in assessing the presence of a gen-uine γ -ray excess in the Galactic center region is, naturally,that of modeling γ -ray sources in the region. Critical to thisis the role of unidentified point sources, including populationmodels for unidentified source classes, and of sources whosespectrum and even source extension is unclear (for examplethe γ -ray counterpart to Sgr A*). A second key element is thediffuse γ -ray emission induced by Galactic cosmic rays. Ithas long been known [6, 17] that the key components of suchemission, in the 0.1-100 GeV range are (i) hadronic emissionfrom neutral pion decay produced by inelastic proton collisionwith the interstellar gas, (ii) inverse Compton up-scattering ofbackground interstellar radiation by cosmic-ray electrons andpositrons, and (iii) bremsstrahlung.We review below how the two key ingredients to the back-ground model employed to extract the Galactic center excesshave been handled in the three most recent and comprehen-sive analyses. What we believe is a crucial point to makeis that the general procedure, in those studies, has been toemploy background templates that make crucial assumptionsabout the Galactic diffuse emission. One of us had pointedout in Ref. [18], with Linden, that important systematic ef-fects in extracting a diffuse γ -ray excess originate from ne- a r X i v : . [ a s t r o - ph . H E ] J u l glecting the cosmic-ray density distribution and in utilizingtemplates where the diffuse hadronic emission, item (i) in thelist above, follows the morphology of the target gas density.In the present study, we point out that (a) very little is knownabout cosmic rays in the Galactic center region; that (b) moreor less young populations of cosmic rays are likely to inhabitthat region and to importantly contribute to the hadronic emis-sion in a way that would be completely missed by a currenttemplate analysis; and, finally, that (c) such a population(s) islikely to source the claimed γ -ray excess.Let us first briefly review three recent studies devoted to theGalactic center excess, Ref. [19], [20] and [21]. The studypresented in Ref. [19] focuses on the ◦ × ◦ region cen-tered around the Galactic center (GC, b = 0 , l = 0 ), andemploys the recommended LAT Collaboration diffuse back-ground model gal 2yearp7v6 v0 (we will comment be-low on the implicit assumptions included in this model), plusisotropic backgrounds, and known γ -ray sources in the secondyear Fermi catalogue (2FGL). The study confirms evidencefor a spherically symmetric extended source, as obtained inprevious studies [22], with a spectrum consistent both withemission from millisecond pulsars and with dark matter an-nihilation. Ref. [19] also attempts to assess systematic un-certainties in the background modeling, concluding that suchuncertainty is in the vicinity of the 20% level. In a follw-up paper [23] by the same authors, 20 cm templates tracingthe molecular gas distribution were added to the likelihoodanalysis and were found to significantly improve the fit whilestill robustly detecting an approximately spherically symmet-ric GCE counterpart.The analysis of Ref. [20] also considers a region of interestof ◦ × ◦ centered around the GC, and employs two choicesfor the energy range, photon source class, pixel size, and en-ergy binning. Ref. [20] then fits a variety of templates tothe observed γ -ray data. These templates include, in addi-tion to point sources, the recommended Galactic diffuse emis-sion model gal 2yearp7v6 v0 and isotropic backgroundmodel iso p7v6source , a template (MG) that intends tomap the bremsstrahlung emission associated with high-energyelectrons interacting with molecular gas clouds as traced bythe 20 cm radio map of the GC [24], a Galactic Center Excess(GCE) source, and a “new diffuse” component associated witha central stellar cluster, with varying spatial profiles.The two key findings of Ref. [20] are that (i) an extendedemission in the GC region associated with the GCE templateis present with any combination of templates and with bothchoices for the pixel and energy binning etc.; and that (ii)the fluxes and spectra associated with both the γ -ray emis-sion from the central point source Sgr A* and with the GC ex-tended emission are significantly affected by the choice of thebackground model, especially in the low-energy range. TheGC excess is found to have a spatial distribution consistentwith a profile ∝ r − . .The study of Ref. [21], which appeared less than 10 daysafter Ref. [20], focused on an “Inner Galaxy” region, whichmasks out the Galactic plane ( | b | < ◦ ) and includes a largeregion of several tens of degrees, and on a “Galactic center”region, defined by | b | < ◦ and | l | < ◦ . Both studies use a novel cut on photon events based on the CTBCORE variable,producing higher resolution maps. In the “Inner Galaxy” anal-ysis, Ref. [21] makes use of three templates (the Fermi col-laboration p6v11 Galactic diffuse model, an isotropic back-ground and a uniform-brightness template matching the Fermibubbles) plus a “dark matter” template of variable inner slope.In the “Galactic center” analysis, the templates used include aGalactic diffuse emission provided by the Fermi collaboration( gal 2yearp7v6 v0 , the same choice as Ref. [20]), a tem-plate tracing the 20 cm emission, along the lines of Ref. [20],an isotropic component, and all 2FGL point sources [25]. Asin Ref. [20] it is found that the isotropic component needed toprovide an optimal fit is considerably brighter than the extra-galactic γ -ray background.Ref. [21] indicates a strong preference for the existence ofa Galactic center excess, and finds a similar preferred spatialdistribution profile to Ref. [20] and, generically, a similar pre-ferred spectral shape. Ref. [21] points out that the excess isapproximately spherically symmetric. From both spectral andmorphological considerations, Ref. [21] argues that a popula-tion of unresolved millisecond pulsars (MSP) in the relevantGalactic region is strongly disfavored. Also, as pointed out inRef. [18], based on the population of resolved MSPs, the con-tribution from an unresolved population should account forless than ∼ − % of the γ -ray excess (see also [16]).It is apparent that a central issue to the determination ofthe existence of any diffuse γ -ray excess is whether or notthe background model for the Galactic diffuse emission ac-curately reproduces the expected γ -ray emission. All re-cent studies reviewed above employ a diffuse Galactic modelrecommended by the Fermi collaboration for use with Pass7 LAT data [26]. Interestingly, the Collaboration explicitly(and in bold face) discourages the use of one the most recentsuch model for Pass 7 reprocessed data “for analyses of spa-tially extended sources in the region defined in Fig. 1”, a re-gion which includes the Galactic center region (as noted inRef. [21]). While the key concern is the inclusion, in the re-processed data background model, of sources with extensionmore than 2 ◦ , it is also apparent that such background modelsare not designed with the purpose of establishing the existenceof a diffuse emission.One of the key issues with using the diffuse model recom-mended by the Fermi Collaboration for the purposes of estab-lishing a diffuse excess is the set of templates employed to re-produce the morphology of the hadronic and inverse-ComptonGalactic diffuse emission. Employing gas column-densitymap templates to reproduce the diffuse γ -ray intensity entirelyneglects the possibility of a significantly enhanced cosmic-rayabundance in the inner Galaxy, which almost certainly exists.Similarly, the inverse-Compton template is based, and sensi-tively depends, on specific choices for the input parametersin the Galprop code, most significantly source distribution,diffusive halo geometry and source spectrum (see e.g. [27]).Other quite relevant issues with the Fermi Collaborationrecommended diffuse model have been discussed and tack-led in the recent studies of Ref. [20] and [21]. These includea component of bremsstrahlung emission corresponding, andtraced by, molecular gas [20, 21]; a diffuse component with adensity profile tracing the Milky Way Central Stellar Cluster[20]; and the so-called Fermi bubbles [21], whose intensityhowever quite likely deviates from the uniform-brightness as-sumption of Ref. [21].With all the mentioned caveat in mind, in the present studywe show that simple Galactic cosmic-ray models exist thatnaturally explain the observed excess. The origin of such cos-mic rays is likely associated either with supernova remnantsin the inner Galactic region, or with past activity of Sgr A*, orboth. We demonstrate that there is no spectral or morpholog-ical preference for dark matter over such cosmic-ray models,whose existence in the inner Galaxy is more than plausible.Based on Occam’s razor principle, we argue that the Galacticcenter excess finds a much more compelling interpretation inthe context of cosmic-ray models for the inner Galaxy ratherthan in that of dark matter annihilation.
II. COSMIC-RAY PROTONS IN THE INNER GALAXYA. Morphological properties
There exist two key potential sources of cosmic rays in theinner Galaxy within the energy range relevant here: (i) super-nova remnants and (ii) past activity of the central supermas-sive black hole associated with the radio source Sgr A*. Forsimplicity, we assume that both sources injected cosmic raysat the center of the Galaxy ( l = 0 , b = 0 ) at one or morepoints in time in the past. We will assume both an impulsiveand a continuous injection for the sources, the former arguablymore plausible for Sgr A* or for isolated star-formation bursts,and the latter closer to what expected for a population of su-pernova remnants. We first feature a qualitative analytic dis-cussion (sec. II A 1), and we then present detailed results ob-tained with a full cosmic-ray propagation simulation with the Galprop package (sec. II A 2).
1. Analytic Estimates
In the case of an impulsive source, the spatial distributionof the protons after a time t i can be approximated as follows[28]: f ( r ) ∝ exp[ − r /R ( t i )] R ( t i ) , (1)where the diffusion radius R dif ( E, t ) = 2 (cid:115) D ( E ) t exp[ tδ/τ pp ] − tδ/τ pp , (2)with D ( E ) = D ( E/ δ , and where τ pp is the ap-proximately energy-independent proton cooling time. Fortimescales t (cid:28) τ pp , R dif ≈ (cid:112) D ( E ) t , indicative of a purelyBrownian process. Note that the particle spectrum clearly de-pends on position unless δ = 0 , since the quantity R dif , driv-ing the spatial dependence, depends on energy. In the nar-rowly peaked energy range of interest for the Galactic Center Name Type Age R diff (2 GeV) Im1 Impulse .5 Kyr 18 pc . ◦ Im2 Impulse 2.5 Kyr 40 pc . ◦ Im3 Impulse 19 Kyr 110 pc . ◦ Im4 Impulse 100 Kyr 250 pc . ◦ Im5 Impulse 2 Myr 1.13 Kpc . ◦ C1 Continuous 7.5 Myr 2.19 pc ◦ C2 Continuous > ∼ Gyr ∞ ∞
TABLE I. Properties of a few benchmark emission sources. excess, such effect is, however, limited. For example, to firstorder and for δ ∼ . , R dif ∝ E . which is less than afactor 1.5 difference from 10 GeV to 100 GeV.The counterpart to Eq. (1) for a continuous source is givenby f ( r ) ∝ erfc[ r/R dif ] r , (3)where erfc is the error-function, and with the same R dif as inEq. (2) but with t , this time, referring to the time at which thecontinuous cosmic-ray source started injecting particles. Inthe limit r (cid:28) R dif , the cosmic-ray flux saturates to a density ∝ /r .For a diffusion coefficient D ( E ) = D ( E p / δ , with D = 6 . × cm s − , we then consider a variety of impul-sive and continuous sources, with ages listed in Table I alongwith their physical and angular diffusion radii at 2 GeV, wherethe GCE peaks.In Figure 1 we show the projected density of cosmic-rayprotons for the putative impulsive and continuous sourceslisted in Table I. In particular, we show the evolution of a sin-gle impulsive source over the times from Table I as well as thecontinuous models C1 & C2 along with a representative su-perposition of impulsive sources ( Im4 + 10 × Im5 ) which wewill employ in what follows. The overall normalization is leftarbitrary for the sake of illustration. It is crucial to note thatthis is the cosmic-ray proton density, and that it must be mul-tiplied with the spatially varying target gas density in order toobtain spatial distribution of the γ -ray flux. As a guideline,we also show the prompt γ -ray flux for an annihilating darkmatter candidate following an NFW profile of inner-slope γ between 1.1 and 1.3 and scale-length r s = 24 kpc. Thesetwo values bracket the signal morphology resulting from theanalyses of Ref. [20, 21].As we will demonstrate in the next section, the Im4 + 10 × Im5 and the C1 models have the correct proton densities toreproduce the GCE after convolution with the gas profile andare reshaped to closely match the γ = 1 . profile . In theplot, the shaded region indicates the angular region of inter-est, bounded at low angular scales ( ≈ . ◦ ) by the point-spread function of Fermi-LAT, and at the approximate angularscales ( ≈ ◦ ) where statistical and systematic uncertainties The other values of γ shown will be used in a later discussion of dusttemplate modulation. -1 Angle From Galactic Center ψ [deg] C o s m i c − R a y P r o t o n D e n s i t y [ a r b i t r a r y / s r ] G a mm a − R a y F l u x
500 yr2 . γ = . γ = . γ = . γ = . ImpulsesSummed Impulses7 . . Stationary Cont . Squared NFW
FIG. 1. The density of cosmic-ray protons, at an energy of 2 GeV,projected along the line of sight as a function of the angular distancefrom the Galactic center in the spherically symmetric analytic diffu-sion approximation. Shown in dotted blue lines is the evolution ofan impulsive source after .5, 2.5, 19, 100, and 2000 Kyr from topto bottom. We also show our summed impulse model (thick black),a 7.5 Myr old continuously emitting source (thin black), and a sta-tionary continuous source (black dashed). After a convolution withthe gas density profile, the summed impulse and 7.5 Myr old con-tinuous models have γ -ray flux profiles which approximately matchthat of an annihilating dark matter candidate following an NFW ofinner slope γ = 1 . (shown in dashed red for several values of γ ).The shaded region shows the angular scales which are both abovethe Fermi-LAT point-spread function (lower-bound ≈ . ◦ ) andbright enough to be differentiated from the background (upper bound ≈ ◦ − ◦ ). currently render the excess invisible over backgrounds. It isimportant to note that the recent bursts (Im1, Im2, Im3 andIm4), or superposition thereof, provide highly concentratedpopulations of cosmic-ray protons in the Galactic center, pos-sibly yielding a bright, centralized, and spherically symmetric γ -ray emission.Note that the time-scales we employ in the present esti-mates are not accidental: for example, model Im5 is close tothe age of the Fermi bubbles, as estimated e.g. by Ref. [29]and Ref. [30] to be around 1-3 Myr, while model Im4 is alsoclose to another alternate age estimate for the bubbles, × yr, obtained by Ref. [31], as well as matching age estimatesof − yr for the supernovae remnant Sgr A East atthe Galactic Center. Also notice that for the time-scales listedabove we are never in the regime where t (cid:29) τ pp with theexception of the stationary continuous source, where protonsare replenished over the region of interest anyway.
2. Numerical Simulations
The hadronic γ -ray emission from π decay traces both thedensity of cosmic-ray protons and the spatial distribution ofthe target interstellar gas. While the discussion above showsthat with one or more burst injections, a variety of cosmic-raydensity profiles can be obtained (including highly centrallyconcentrated ones), the present discussion must include thetarget density for hadronic inelastic processes. We note againthat the template analyses of Refs. [20, 21] are predicated on auniform distribution of cosmic-ray protons, and therefore ne-glect any gradients introduced by sources and by a non-trivialcosmic-ray morphology in the region of interest such as thoseshown in Fig. 1.In order to simulate in detail the γ -ray emission from theregion and to assess the role of the cosmic-ray distribution,we employ the code Galprop v54.1.2423 [32] whichprovides a 3+1-dimensional numerical solution to cosmic-ray transport along with empirically calibrated semi-analyticalmodels of atomic, molecular, & ionized hydrogen (HI, HII,H + ) gas in the Galaxy, in addition to a sophisticated treatmentof pion production and decay.For simulations longer than 50 Kyr we employ a Galprop simulation consisting of a × kpc box centered on theGalactic plane with the x-axis defined by the Sun-GC line.The half-height along the z-direction is 4 kpc with a latticespacing of 200 pc along each axis. For shorter simulations,the box-size is reduced to a sufficiently large cube of dimen-sion 4 kpc with lattice spacing reduced to 50 pc. A source ofcosmic-ray protons is then defined as a narrow sub-grid Gaus-sian localized at the Galactic center. In the case of impulsivesource models, the Galprop code has been modified to in-ject protons in time following a δ -function centered at t = 0 .Cosmic-ray transport is then solved forward in time with the Galprop code, using ‘explicit-time mode’ with step sizes of ∆ t = 10 , yr for sources younger and older than 50 Kyr,respectively.As in the previous section, we assume an isotropic diffusiontensor with diagonal entries D ( E ) = D ( E/ +0 . and a diffusion constant D = 6 . × cm s − . For ourmorphological study of impulsive sources, we have explic-itly verified that the diffusion constant and the diffusion time(the “age” of the source) are approximately degenerate for thequantity D t diff held constant. This is expected in the limit-ing case of Eq. (1) where the diffusion time is much shorterthan the proton cooling timescale. In other words, holding thequantity D t diff constant will preserve the shape of the dif-fusion cloud, although the flux scales as D − . This impliesthat if the diffusion constant differs in the Galactic center ourresults will still hold, but diffusion timescales will change, aswill the energetics in the case of a continuous source.The region of interest under consideration here extends to ± . kpc at 10 degrees, while the height of the diffusion zoneis much larger and set to ± kpc. Unless this half-height is Available at http://sourceforge.net/projects/galprop/ reduced to h dif < ∼ kpc, variations in the height of the diffu-sion zone are also of negligible impact, and are thus not con-sidered. Diffusive reacceleration is incorporated using a Kol-mogorov spectrum for interstellar turbulence ( δ turb = 1 / )and an Alfv´en velocity of 30 km/s.At the small Galactic latitudes of interest, low-speed ( < ∼ km/s) convective winds out of the Galactic disk have beenconfirmed to be negligible, via explicit simulations, and areset to zero. Notably, recent studies [33–36] have presented ex-tensive multi-wavelength evidence for very fast ( > ∼ km/s)global outflows from the Galactic center region. Driven by in-tense and approximately constant star-formation, this energyindependent advective transport provides a good fit over ra-dio, GeV, and TeV observations and it is suggested that such acomponent could, in fact, dominate over diffusive transport. Adetailed model of outflows is beyond the scope of the presentstudy, but should not alter our overall conclusions. The nar-row energy range of the GCE implies that diffusive transportis effectively energy-independent, and spherically symmetricadvection should produce a comparable morphology in the in-ner galaxy, albeit with somewhat different time scales and en-ergetics. However, it should be kept in mind that at the level ofmorphological detail required for template analysis, such ef-fects are important and could significantly change the qualityof fit compared to templates derived using diffusion alone.As mentioned at the beginning of this section, a crucial in-gredient that a full cosmic-ray simulation allows us to test isthe role of the interstellar gas distribution in predicting themorphology of the diffuse γ -ray emission. In our simulations,the interstellar gas consists of three components: molecular,atomic, and ionized hydrogen. In Galprop , the first twocomponents are modeled as independent, cylindrically sym-metric distributions of seven galactocentric rings derived fromsurveys of HI & CO, where the latter is used as a tracer ofmolecular hydrogen [37]. These surveys are then combinedwith distance information derived from the line-of-sight ve-locity distributions and Galactic rotation curves to assign a gasdensity to each ring as a function of height. Finally columndensities from the analytic model are renormalized to agreewith the survey gas column densities, breaking the cylindri-cal symmetry and reproducing the observed asymmetric gasstructures. Using this gas model,
Galprop propagates thecosmic-ray protons and convolves the resulting density withthe gas model in order to produce a projected map of the γ -ray flux. The smallest resolvable scales are thus ultimatelylimited by the gas map resolution. In the case of HI and H ,this amounts to an angular resolution of 0.5 ◦ and 0.25 ◦ re-spectively. Notably, the latter is approximately of the samecharacteristic size as the Fermi PSF above a few GeV.Within the Galactic plane, the mass fraction of ionized hy-drogen is only a few percent when compared with the othertwo components. For the sake of comparison with the ‘inner-Galaxy analysis’ of Ref. [21], we focus on Galactic latitudes | b | > ◦ where the ionized Warm Interstellar Medium (WIM)makes up a significant portion of the diffuse γ -ray signal. In Galprop , the WIM is based on the commonly used NE2001model of Cordes & Lazio [38, 39] with scale-heights doubledto 2 kpc to ensure consistency with recent pulsar dispersion data as described in Gaensler et al 2008 [40].We emphasize that our gas model is nearly identical tothat used to derive the hadronic component of the Fermi-LAT Collaboration’s Galactic Diffuse Model, although theFermi diffuse model also includes inverse Compton scatteringand bremsstrahlung contributions from high-energy electrons,which are not of interest in testing possible issues with thehadronic component of the diffuse emission. For a thoroughdescription of the gas model we discussed above, see Ref. [41]and enclosed references. One difference of limited impor-tance in our implementation of the scale-factor X CO ( R ) . Thisparameter captures the ratio between the survey-derived in-tegrated CO line intensity and the H column density. Incontrast to the fixed value used by the Fermi-LAT team, wechoose this ratio to increase as a function of Galactic radius,in accordance with the findings of Ref. [42]. As this functionis nearly flat in the inner Galaxy, this change is not expectedto play a significant role.The gas model and diffusion setup are now defined and wethus proceed to a morphological comparison between central-ized proton sources and the measured Galactic center excess.In the analyses of Refs. [20, 21], the basic features of theexcess emission show an approximately spherical shape withflux approximately 3% of the brightness of the Fermi diffusemodel in the central ◦ × ◦ window [20] centered on the GC.We define three benchmark cases of interest:(i) a continuously emitting central source of high-energycosmic-ray protons, which has reached steady state over > ∼ year timescales,(ii) a continuous source which was started injecting protons7.5 Myr ago, a time-scale consistent with ages proposed forthe Fermi-bubbles, and(iii) a two-component impulsive source where protons wereinjected at ages of 19 Kyr, 100 Kyr, and 2 Myr, summed withfree relative normalizations.In what follows, we calculate the γ -ray emission profile ofour models as a function of the projected distance from theGalactic center. We then fit this profile to the GCE to deter-mine statistical compatibility and study the remaining spatialproperties.In Figure 2 we plot the projected γ -ray flux, integratedalong the line-of-sight and assuming a solar position of r (cid:12) = Angle From Galactic Center ψ [deg] -6 -5 -4 F l u x [ a r b i t r a r y / s r ]
100 Kyr+2 Myr Impulse: χ / − . o . f . =1 . . χ / − . o . f . =1 . α =1 . χ / − . o . f . =1 .
100 Kyr Impulse2 Myr Impulse100 Kyr+2 Myr7.5 Myr Continuous ContinuousNFW α =1 . NFW α =1 . Daylan et al
FIG. 2. Projected flux density at 2 GeV as a function of from a protonsource at the Galactic center. For non-dark matter lines, results arederived from a full
Galprop simulation of diffusion and subsequentneutral pion decay averaged over the north + south regions with theGalactic plane ( | b | ± ◦ ) masked out upon integration. In black weshow radial flux profiles for our summed impulsive (thick), a 7.5 Myrold continuous source (thin), and a steady-state continuous source(dashed). In blue-dashed and blue-dotted we show the individualimpulsive sources at 100 Kyr and 2 Myr. Finally, we show NFWprofiles with inner slopes 1.3 and 1.1 in solid and dashed red. Datapoints are taken from Daylan et al (2014) [21]. fit. Both of these parameters are included when counting de-grees of freedom. In case (iii), i.e. the summed impulsivemodel, we do not include the 19 Kyr component since its con-tribution is negligible outside of the masked region (althoughit could be important to match the Galactic center analysis inthe central few degrees, as we will show below). We thensacrifice an additional degree of freedom and allow the nor-malization of the 100 Kyr and 2 Myr components to float in-dependently. Thus the summed model includes 2 ages and 2normalizations. The energetics of the normalizations are as-sumed arbitrary at this point. We will explore how reasonablethe resulting normalization values we infer actually are in sec-tion III B where a concrete astrophysical scenario is discussed.The profile slope of the continuous source in steady-stateappears to be slightly too flat to match the observed emis-sion and does not provide a particularly good fit to the data( χ / d . o . f . = 6 . ). However, if this emission were initiatedat an age of O ( χ / d . o . f . = 1 . – compared to the α = 1 . NFWprofile where χ / d . o . f . = 1 . . Our best fitting model is the100 Kyr+2 Myr impulsive model with χ / d . o . f . = 1 . . We find that for the summed impulse model, the best-fit injectionluminosities have relative normalization 1:10, the larger cor-responding to an event at 2 Myr. Although this precise ratiodepends on the relative ages of the two components, this factdoes indicate that two events with relatively comparable ener-getics provides good agreement with the observed excess andmay indicate that events of similar nature and origin mighthave fueled the two cosmic-ray bursts needed to explain theobserved morphology.In Figure 3 we investigate the overall spatial distributionof the emission from a new population of cosmic-ray protonsinjected in the Galactic center region. The Figure shows the γ -ray flux associated with a central proton source for the bench-mark impulse times of 0.5, 2.5 and 19 Kyr (upper panels) andof 100 Kyr, 2 Myr and continuous (lower panels). We use alinear scale in the three upper panels to help the Reader visu-ally compare our results with what shown e.g. in Fig. 9, rightpanels, of Ref. [21]. To the end of emphasizing the emissionoutside the Galactic plane, we instead employ a logarithmicscale for the older bursts and continuous sources in the lowerpanels. In each case, the fluxes are rescaled such that the max-imum flux equals unity. The Galactic plane mask ( | b | < ◦ )is bounded by white lines (or is masked out) and referencereticles have been overlaid at radial increments of 2 ◦ .The top three panels show that a recent (from a fractionof a Kyr to tens of Kyr) impulsive cosmic-ray proton injec-tion event in the Galactic center region yields a highly spheri-cally symmetric and concentrated source, with morphologicalproperties very closely resembling and matching those foundin the Galactic center analysis of Ref. [21] (see their Fig. 9,right panels), as well as in the GCE source residuals shownin the bottom panels of Fig. 1 in Ref. [20], and in the resid-ual found in Ref. [19] and shown in Fig. 3. As long as theinjection episode is recent enough, the morphology primarilytraces the distribution of cosmic-ray protons, and is relativelyinsensitive to the details of the target gas density distribution— the diametrically opposite regime from what assumed inthe diffuse Galactic emission background models of Ref. [20 ? , 21].It is evident that the sub-Myr simulations show a signif-icant degree of spherical symmetry outside the masked re-gions. Also, an excess with the same morphological aspect asin in fig. 9, right panels, of Ref. [21] can be easily reproducedby young or very young sources, as shown in the three upperpanels. As the diffusion time increases to to several Myr, theemission profile becomes more elongated and spherical sym-metry is degraded. At higher latitudes ( | b | > ∼ ◦ ), most of thespherical symmetry is, however, restored as the molecular andatomic gas distributions fall off, and the ionized componentproduces a more isotropic emission. In the template analysesof Refs. [20, 21], a portion of this residual ridge emission mayalso be absorbed by the Fermi diffuse model, although it is dif-ficult to exactly pinpoint this effect without repeating the fullmaximum likelihood analysis. It is also evident that gas struc-ture is mostly washed out for recent impulsive sources, andthat it becomes increasingly more prominent for older sourcesand for the continuous emission cases. Finally, we note that ifa substantial portion of the inner excess is due to unresolved R e l a t i v e F l u x L o g ( R e l a t i v e F l u x ) G a l a c t i c L a t i t u d e [ d e g ] . Kyr Impulse . Kyr Impulse . Kyr Impulse
20 10 0 10 20Galactic Longitude [deg]201001020 G a l a c t i c L a t i t u d e [ d e g ] Kyr Impulse
20 10 0 10 20Galactic Longitude [deg]201001020 Myr Impulse
20 10 0 10 20Galactic Longitude [deg]201001020
Continuous
FIG. 3. Hadronic γ -ray flux density at 2 GeV from an approximately central source of high-energy protons integrated over the line-of-sight.We show impulsive sources of increasing age in all panels with the exception of the bottom-right which shows a continuously emitting sourcein steady state. For each map, the fluxes are normalized to the maximum. For the ease of comparing the morphology of the claimed GCE inRef. [21] and shown in their fig. 9, we employ a linear scale in the three upper panels. The three lower panels employ, instead, a logarithmicscale to enhance the features of the emission outside the Galactic plane region. Also overlaid are reference reticles in increments of 2 degreesand indicators of the Galactic plane mask | b | < ◦ . All maps have been smoothed by a Gaussian of width σ = 0 . ◦ to match Ref. [21]. millisecond pulsars, much of the Galactic ridge would remainat a lower relative luminosity.Quantitatively examining the angular profile for eachsource at a variety of different radii shows that within ± ◦ of the north and south Galactic poles, there is a high degreeof spherical symmetry with typical (positive) variations onthe order of 20% with respect to the flux at Galactic north.At larger angles, however, the flux rapidly rises as one ap-proaches the Galactic plane to values many times larger thanthe Galactic north flux. Although this does significantly illu-minate the Galactic plane, it is unclear how important a rolethis plays in the analysis of Daylan et al [21], where sphericalsymmetry was tested by scanning the axis ratio of the (now el-lipsoidal) dark matter template. Their analysis found a strongstatistical preference in both the inner Galaxy and Galacticcenter analyses for an axis ratio of approximately ± . .While this template distortion does provide a simple test, itsgeometry is not physically motivated and does not correctlyprobe the bar+sphere shape expected from a central hadronicsource.In Appendix C of Ref. [21], the authors examine the ex-cess in two regions: north/south, defined by angles within the 45 ◦ of the poles, and east/west, defined as the comple-mentary region dominated by the Galactic disk. While bothregions exhibit an excess, the E/W template shows a signifi-cantly enhanced peak of the signal compared to a flatter N/Sspectrum [21]. This seems to indicate that either the Fermi-bubbles template absorbs much of the excess N/S emission,or that the emission is, in fact, more extended along the disk,as is seen in our benchmark models with a central cosmic-ray proton source. In further testing the axis-ratio, Ref. [21],again, uses ellipsoidal projections of the NFW emission, thistime allowing the template to rotate (there is still no test for arectilinear disk component), finding a small statistical prefer-ence for an axis ratio of 1 to 1.3-1.4 elongated at an angle of ≈ ◦ counter-clockwise from the Galactic disk. It is possi-ble that this component of the excess is in fact a component ofan extended central molecular gas bulge, as advocated e.g. inRef. [43], which is oriented at ∼ ◦ CCW and is not modeledby the cylindrically symmetric
Galprop gas model and that,as a result, is therefore not included in Fermi Diffuse Galactictemplate.In Appendix 4 of Ref. [21] the hypothesis of an excess pro-ton density is tested by adding an additional template basedon the Schlegel-Finkbeiner-Davis dust map [44]. The gas-correlated dust map is then spatially modulated so that the re-sulting template is given by
Modulation = SFD(˜r) × (cid:82) l . o . s . ρ (˜r)g(˜r) (4)where the NFW profile’s inner slope was scanned to maxi-mally absorb the emission, preferring an inner slope γ = 1 . .The functional form for g ( | l | , | b | ) was then assumed to bethe product of a latitudinal linear × exponential function anda longitudinal Gaussian. This function was then fit over | b | < ◦ and | l | < ◦ to also maximally absorb residuals.It was found that the modulated dust absorbed a significantcomponent of the excess when an additional NFW templatewas omitted. However, when the NFW template was includedin the analysis, it absorbed nearly the entire excess and themodulated dust map appears uncorrelated with the excess. Itwas concluded that gas-correlated emission does not providea suitable description of the GCE. We disagree with this con-clusion for the following reasons:1. The morphology of the underlying population ofcosmic-ray protons which reproduces the GCE is shownby the 7.5 Myr continuous source shown in Figure 1and is clearly very different from any of the NFW pro-files shown. In the modulated dust template analysis,the functional forms chosen for g ( (cid:126)r ) would need to bedrastically different in order to reproduce distributionof protons matching that of Figure 1. In particular, anyanalysis must consider that the target gas density al-ready falls off as one moves away from the Galacticcenter, and that the dust map should be initially modu-lated by the expected proton density, not proportionallyto a projected NFW profile. For example, if one takes g ( r ) = 1 in Eq. 4, the resulting γ -ray template wouldfall off much faster than r − when integrating over un-masked regions as was done for Fig. 2. As can be seenin Fig. 1, within the inner few degrees of the Galacticcenter, our 7.5 Myr continuous hadronic source wouldcorrespond approximately to a dust profile modulatedby an NFW profile of inner slope γ ≈ . , whichwould then be required to steepen to more than γ = 1 . by ◦ in order to not severely overestimate the flux atlarge radii.2. As seen in Figure 3, the gas-correlated emission fromcosmic-ray populations younger than a few hundredKyr remains highly spherically symmetric at high lat-itudes. Only in substantially older sources ( > ∼ )does the gas structure of the Galaxy become completelydominant in shaping the γ -ray morphology. In particu-lar, this indicates that much of the dust structure liesat radii intermediate between the Earth and the Galac-tic center, whereas protons from a young cosmic-raysource have only reached the inner-most rings. Our sim-ulations take this 3-dimensional structure into accountusing gas velocity measurements to construct a modelof Galactic structure and indicate that a 2-dimensional map of the column-density simply cannot account for anon- uniform cosmic-ray density.3. If astrophysical in nature, the residual is likely to be theresult of several emission sources. A substantial emis-sion component in the inner few degrees naturally needsto be attributed to unresolved MSPs [45] which exhibitan approximately spherically symmetric, or slightly el-lipsoidal profile (see however [16]). Such an additionwould inevitably alter the preferred templates in the un-masked Galactic center analysis.To summarize this section, we have used the cosmic-raypropagation code Galprop to simulate the γ -ray emissionassociated with neutral pion decay as cosmic-ray protons froma central proton source diffuse and interact with interstellargas. Using a gas model identical to that of the Fermi-LATGalactic diffuse template, we studied a variety of continuousand impulsive proton injection histories. Under standard as-sumptions for the diffusion setup, it was shown that one canreasonably reproduce the spatial morphology of the observedGalactic center excess using source histories that are poten-tially correlated with past Galactic activity. Specifically, theradial flux profile can be very closely matched if a continu-ous proton source turned on within the past 5-10 Myr, or iftwo or more events of comparable energy occurred at agesof around 0.1 and 2 Myr, although these simple benchmarksonly represent a few possible scenarios. The spatial distribu-tion of these source’s γ -ray emission may be somewhat moreextended along the Galactic plane compared to the observedGCE, although without repeating the full likelihood analysis,a direct comparison is difficult. Indeed, a repeated likelihoodanalysis using the hadronic templates derived here is key tohelping rule out a hadronic origin for the GCE and will bestudied in detail in follow-up work. The spatially concentratedexcess found in the ‘Galactic center’ analysis of Ref. [21] isreproduced by young impulsive sources active from a frac-tion to a few Kyr ago in the center of the Galaxy, or perhapseven by efficient trapping of the 100 Kyr cosmic-ray popula-tion in sub-resolution molecular clouds at the GC. At Galacticlatitudes above 2-3 degrees emission from the Galactic ridgebecomes no longer dominant and at angles within ≈ ± ◦ of the Galactic poles, our sources exhibit a very high degreeof spherical symmetry while the projected gas structure is leftlargely unresolved relative to the steady-state Galactic diffusemodel. Finally, we discussed possible correlations of the GCEwith unmodeled gas components in the Galactic center as wellas pointing out important issues with the modulated dust tem-plate analysis in Ref. [21]. In the next sections we turn to astudy of the spectral characteristics of the GCE. B. Spectral Properties
Three independent recent analyses of the GCE have foundspectra which share a characteristic peak near 2 GeV, with lit-tle excess emission over background either below a few hun-dred MeV or above GeV. Although the location of the spec-tral peak is relatively robust, the shape of the excess is verysensitive to the modeling of point sources in the field, withadditional systematic uncertainties such as the Galactic dif-fuse emission, and with differing “regions of interest”, lead-ing to a large variation in the reported low and high energyspectral slopes. While most models are relatively well fit by ahard exponentially cut-off power law for the photon spectrum(and, as a result, reasonably well fit by dark matter models),we show below that a power-law proton spectrum with a breakat energies of ≈ GeV also provides good fits to the excessspectrum.A crucial feature of the differential γ -ray spectrum pro-duced through the inelastic scattering of astrophysical high-energy protons on interstellar gas, is a characteristic maximumflux at 100 MeV induced by the rapid downturn of the inclu-sive π production cross-section below 1 GeV. Importantly, inthe spectral energy distribution representation, E γ dN/dE γ ,this peak is shifted to ≈ where both pulsar spectra andthe GCE approximately peak. It is thus a remarkable and un-fortunate coincidence that the claimed GCE spectrally peaksat ≈ , where the likelihood of confusion with astrophys-ical sources is maximal.Here, we consider three reference spectral models for theunderlying cosmic-ray proton population, and thus for the re-sulting γ -ray spectrum. The first cosmic-ray spectrum weconsider is a power-law with an exponential cutoff (PLExp)where the proton spectrum at momentum p p is given by, n p ( p p ) ∼ p − Γp exp[ − p p /p c ] . (5)The second and third models have a broken power-law in-jection of protons of the following functional form: n p ( p p ) ∼ (cid:26) ( p p /p br ) − Γ : p p < p br ( p p /p br ) − Γ : p p > p br , (6)where we allow the second index to be arbitrary in onecase (BPL), and where we fix it to Γ = Γ + 1 in theother (BPLFix). The BPLFix model will later be motivatedby the possibility of proton acceleration by supernova rem-nants taking place inside dense and partially ionized molec-ular clouds. We then calculate the resulting γ -ray spectrumusing one of Galprop’s newest models, employing the de-tailed low-energy parameterizations of Dermer (1986) [46]with interpolation to the Monte-Carlo studies of Kachelrieß &Ostapchenko (2013) which better fit available collider data athigh energies [47] (see App. A for details).We then take data from the two analyses of Daylan et al(2014) [21], where different versions of the Fermi-LAT Galac-tic Diffuse Model were used to extract the GCE spectrum (us-ing the template from the
P6v11 and
P7v6 releases, respec-tively), from Abazajian et al (2014) [20], and from Gordanand Mac´ıas (2013) [19]. We perform a maximum likelihoodfit for each of our three spectral models, and compute the re-duced χ for f = N − M degrees of freedom where N is thenumber of data points and M =3 for PLExp and BPLFix and4 for the general BPL.It is crucial to note that Daylan et al [21] do not provide anestimate of the systematic uncertainties (which are expected to be relatively large), nor do we attempt to include any suchestimate. The error bars quoted in the analysis of Ref. [21]arise purely from counting statistics. Abazajian et al [20] doestimate the relative systematic error Galactic diffuse modelbased on variations in the spectral form chosen for the GCE,but they do not provide a specific number. Based on theirFig. 8, we estimate the error (conservatively small) as × − GeV/cm /s, and combine this in quadrature with the statisti-cal errors for each point. Gordan & Mac´ıas [19] provide themost rigorous test of systematic uncertainties related to theGalactic diffuse model by looking at residuals as a windowis scanned along the Galactic plane in regions with no con-taminating point sources. This results in an estimated ≈ standard deviation from Fermi’s diffuse background model.However, if we combine their statistical and systematic errorsin quadrature, the fit is very poorly constrained. We thereforeuse only systematic uncertainties (which are typically larger)for this case. Below we discuss the results of Figure 4, butone can already see from the substantial variations betweenthe four extracted spectra that estimating the systematic un-certainties is a highly non-trivial issue. We thus urge cau-tion when interpreting the reduced χ values we quote, whichshould be taken only as a rough indicator of fit quality.Figure 4 shows the best fits for each of the three spec-tral models. In the top-left panel is the Daylan et al analy-sis which uses the the non-reprocessed P6v11 diffuse model.The excess is very well fit by the PLExp model which closelymatches the prompt emission from a light dark matter candi-date. The BPL model also provides an exceptionally good fit,although the pre-break index is unphysically steep, at Γ = − . while the second index converges to a value Γ ≈ E br = 23 . GeV, effec-tively mimicking the PLExp model (the two lines are in facthardly distinguishable in the figure). Of more interest is ourBPLFix model, which provides a reasonable, though not opti-mal, fit to the data considering the underestimated error bars.The best-fit low-energy index Γ =2 is intriguingly equal to thecanonical value Γ ≈ expected from the theory of linear dif-fusive shock acceleration (DSA) thought to drive supernovaeand black-hole acceleration processes. Note that there existsystematic uncertainties arising in the low and high-energyranges from modeling of the inclusive pp → π + anythingcross section, as is discussed in App. A and Ref. [48]. Suchuncertainties can be as large as 15% below 1 GeV up to 40%above 100 GeV, and thus affect any conclusion of the precisevalues needed for the cosmic-ray proton injection spectrum.The top-right panel shows the Daylan et al P7v6 anal-ysis, which includes a Fermi-LAT model of the bubbles inthe Galactic diffuse template in addition to the independentFinkbeiner bubble template. Unlike the
P6v11 analysis,which used mismatched photon data from the P7 release, thismodel is appropriately calibrated to the full P7 event data.Compared to the P6v11 analysis, this approach yields a sub-stantial flattening of the spectrum, with all models providingequally good fits, with nearly identical γ -ray spectra. Γ isfound to vary between 1.65 and 2.13 and in both BPL mod-els Γ ≈ . . The similarity between the BPL and BPLFixmodels is remarkable, given the significant difference in their0 -1 E d N / d E [ G e V / c m / s / s r ]
1e 6
PLExp1: p c =2 . , Γ= − . , χ /
17 d . o . f . =1 . br =23 . , Γ = − . , Γ =17 . , χ /
16 d . o . f . =1 . br =26 . , Γ =2 . , Γ =Γ +1 , χ /
17 d . o . f . =5 . -1 E [GeV] E d N / d E [ G e V / c m / s / s r ]
1e 6
PLExp2: p c =166 . , Γ=2 . , χ /
17 d . o . f . =1 . br =13 . , Γ =1 . , Γ =2 . , χ /
16 d . o . f . =1 . br =11 . , Γ =1 . , Γ =Γ +1 , χ /
17 d . o . f . =1 . -1 E [GeV] E d N / d E [ G e V / c m / s ]
1e 8
PLExp3: p c =1 . , Γ= − . , χ /
17 d . o . f . =1 . br =40 . , Γ = − . , Γ =45 . , χ /
16 d . o . f . =1 . br =26 . , Γ =1 . , Γ =Γ +1 , χ /
17 d . o . f . =2 . -1 E [GeV] E d N / d E [ G e V / c m / s ]
1e 7
PLExp4: p c =28 . , Γ=1 . , χ /
17 d . o . f . =0 . br =56 . , Γ =2 . , Γ =23 . , χ /
16 d . o . f . =0 . br =13 . , Γ =1 . , Γ =Γ +1 , χ /
17 d . o . f . =0 . FIG. 4. Best-fit γ -ray spectra for various analyses for the excess emission in the Galactic center region. In each panel we show three models ofthe underlying proton spectrum: Solid lines show the hadronic γ -ray emission for a broken power law proton injection spectrum where bothindices and the energy of the spectral break are varied. Dot-dashed lines employ the same functional form, but with the break in the spectralindex fixed to ∆Γ = 1 . The dotted lines represent an exponentially cutoff proton spectrum. In clockwise order and from the top left, the panelsshow data from Daylan et al Pass 6V11 [21], Daylan et al Pass 7V6 [21], Gordan & Mac´ıas [19], and Abazajian et al [20]. Note that the toprow is normalized by the solid angle, while the bottom rows are integrated over the respective regions of interest. initial spectral index. This indicates a weak spectral depen-dence on Γ due to the natural ‘GeV-bump’ associated withpion decay. This is also observed for in the fits to the otheranalyses, where the initial index can have completely unphys-ical values γ > ∼ with only a very small change in the log-likelihood. Later we will show contour plots for the BPLFixmodel which indicate a strong covariance between the breakmomentum and the low-energy spectral index, and acceptablevalues of Γ over the large range 1.25-2.5.In the bottom-left panel we show spectra taken from the fullmodel of Abazajian et al (Figure 3), Ref. [20], with statisticalerrors added as discussed above. Even our conservative es-timate of the systematic error leads to large uncertainties inthe spectrum, and all of our models provide acceptable fits.Although the BPLFix model does not appear to fit the dataparticularly well, we encourage the reader to review Figure8 of Ref. [20] where a range of GCE spectra are shown de-pending on the spectral model used in the likelihood fit. The data shown here is for the measured residual – as opposed towhat results from a specific dark matter template – and cor-responds approximately to the most strongly peaked model.The “mean model” of Fig. 8 in Ref. [20] has a significantlysofter low-energy spectrum. The fit is also severely impactedby the asymmetrically small number of data points above thebump.Finally, in the lower-right panel we show data from Gordan& Mac´ıas (2013) which we found, again, to be well fit by allmodels, with a preference for a slightly hardened low-energyindex of Γ =1.73 for the BPLFix model and a break energy of13.7 GeV.Collectively, our results reveal two characteristic features:Firstly, in most cases there is a slight preference for the PLExpmodel; the BPL with free indices typically tend to convergetowards a PLExp form. One exception is the P7v6 fit fromDaylan where the BPLFix model is actually preferred. TheBPLFix models provide a reasonable fit throughout, with the1exception of Daylan et al’s
P6v11 which, however, does notinclude any treatment of systematic uncertainties. Second, fora flat p − proton spectrum, the γ -radiation from π decaysnaturally peaks at ≈ . − GeV. In order to shift the peak to thesehigher energies we prefer a slightly harder initial spectral in-dex Γ between approximately 1.6 to 2, although there is lowsensitivity to this parameter. The placement of the spectralbreak is typically near p br = 10 − GeV and provides aneffective control of the width of the spectral peak while thesecond index Γ controls the cutoff rate as is expected fromthe nearly flat π production cross-section above 1 GeV givenin Eq. (12). The preference for a slightly hardened spectralindex could arise naturally if the emission is a combination ofe.g. SNR accelerated protons with index ≈ and MSP emis-sion which can easily have Inverse-Compton spectra harderthan 1.5.As an additional cautionary note, we reiterate that the the-oretical predictions for the γ -ray spectra from proton-protoncollisions are affected by significant systematic uncertaintiesassociated to the modeling of the pp → π + anything pro-duction cross section. Such uncertainty feeds into the inferredspectral properties for the cosmic-ray populations associatedwith a given γ -ray emission. We discuss and evaluate quan-titatively such uncertainties in the App. A. For now, it is im-portant to note that any conclusion on the nature of the GCEbased on spectral considerations alone ought to include thissource of systematic uncertainty as well.In addition to the ‘GeV bump’ feature of the pion-decayspectrum, we point out the discussion of Section 4.2.3 inRef. [28], which describes the temporal evolution of the spec-trum of a cosmic-rays which are accelerated inside a molec-ular cloud, where large gas densities and magnetic fields cantrap low-energy protons on timescales of yr. For an im-pulsive accelerator and a cloud of very high density, highenergy-protons can suffer substantial energy losses and prop-agate in a more rectilinear fashion, allowing escape while thelow-energy protons remain inside. The cloud is thus illumi-nated with a spectral energy distribution peaked at a few GeVwith a steepened high-energy falloff at ages greater than years. The low-energy index remains virtually unchangedunless the source is very young and brehmstrahlung fromsecondary electrons is contributing strongly. By yr thecloud’s peak flux decreases by 2 orders of magnitude andbecomes part of the diffuse background. Although this pro-duces gas-correlated emission that could potentially be re-solved, very close to the Galactic center the spatial resolu-tion of Fermi-LAT is limited to scales larger than about 30pc, larger than most of the (many) molecular clumps orbitingin the central few parsecs. Such sources cannot thus be spa-tially differentiated from the central point source with γ -rayobservations. If the escaping high-energy emission is alreadysuppressed, as in our BPLFix model, this would appear as anadditional spectral break at approximately the same energy.This very scenario may be realized at the Galactic center forthe ∼ − year old supernova remnant, Sgr A East,which we discuss in detail later. Almost certainly, molecularclouds are trapping protons at the Galactic center on scales unresolvable by Fermi-LAT and effectively reproducing themorphology of a younger source.In summary, we proposed three models for the spectrum ofa new population of cosmic-ray protons which could explainthe GCE: an exponentially cutoff power law, and two brokenpower laws with free and fixed ( ∆Γ = 1 ) changes to the spec-tral index, respectively. We calculated the γ -ray spectra result-ing from inelastic collisions of the protons on interstellar gas,noting that nearly all physically reasonable proton injectionspectra exhibit a bump near ≈ GeV in the γ -ray E d N/ d E distribution. For each model we performed a maximum likeli-hood fit to each of the four GCE residuals and found good fitsin all cases over a broad range of parameter values. We con-cluded that the core spectral features of the GCE – namely ahard low-energy spectral index, a peak between 1-3 GeV, anda rapid decline above a few GeV – can be naturally producedby an additional population of cosmic-ray protons in the in-ner Galaxy . In the next section, we provide theoretical andphenomenological evidence that such a population is likely toexist in the Galactic center. III. PHYSICAL MODELS FOR THE GC EXCESS
In this section we demonstrate that the needed luminosityand spectral properties for the cosmic ray population we in-voke to explain the GCE have sound physical motivations. Inparticular, we explain in Sec. III A how the spectral breaks inthe cosmic-ray proton spectra we consider might have arisenin the Galactic center region, and related observational evi-dence; we then estimate in Sec. III B the energetics requiredby a cosmic-ray interpretation of the GCE, and argue that thetime-scales and energy scales are plausible and in line withobservations and theoretical expectations.
A. A Mechanism and Evidence For GeV Spectral Breaks
For half a century, the bulk of Galactic cosmic rays has beenthought to originate from supernova remnants (SNRs) whichinject 3-30% of the total supernova energy ( E SN ≈ erg)into protons and other light nuclei [49]. A detailed theoryof diffusive shock acceleration is still incomplete, but simpli-fied linear models predict that supernova shocks propagatingthrough an ionized gas precursor can accelerate protons andother nuclei up to eV with a resulting proton spectrumof p − at the source [50]. When combined with sophisticatedmodels of nuclear propagation through the Galaxy and solarsystem, this source spectrum successfully reproduces the lo-cally measured spectrum of cosmic-ray nuclei. Direct con-firmation of this acceleration model was provided only veryrecently (2013) by the Fermi-LAT collaboration following thedetection of γ radiation characteristic of π -decay in associa-tion with two known SNRs, IC443 and W44 [49].In order to postulate a viable astrophysical model for theGalactic center residual – i.e. without invoking new parti-cle physics – we require either a substantial reduction in the eV high-energy cutoff, or a strong spectral break near ≈
210 GeV which renders the signal invisible over that of the dif-fuse sea of background cosmic-rays where the γ spectrum isroughly ∝ E − . γ . In what follows, we describe recent propos-als that modify the canonical theory of DSA in the presenceof dense molecular clouds which surround the inner Galaxy,as well as actual realizations of this scenario as seen in re-cent Fermi SNR observations showing significant breaks at O (10 GeV) in the underlying proton spectrum. It is thus pos-sible to provide a natural explanation for the spectrum, en-ergetics, and morphology of the GCE requiring only the as-sumption of an enhanced central supernova activity over thepast few million years.In DSA, shock waves propagating through ionized interstel-lar medium compress the plasma and transfer kinetic energydownstream through either two-body collisions, or throughcollective electromagnetic effects if the collision cross sec-tion is very small. In the compressed zone preceding theshock front, resonant scattering of Alfv´en waves efficientlyaccelerates particles until their gyro-radius r g = cp/ ( eB ) ex-ceeds the width of the shock layer [51]. While this test par-ticle case assumes a fully ionized cosmic-ray precursor, theGalactic center is only partially ionized, with well over 80%of the gas content associated with neutral molecular hydro-gen in the inner 200 pc, which completely engulfs the regionof central starburst activity. Malkov, Diamond, and Sagdeev[52, 53] demonstrated that when the upstream edge of super-novae shocks interact with molecular clouds, ion-neutral col-lisions effectively damp a range of otherwise resonant Alfv´enwaves, severely deteriorating particle confinement within aslab of momentum space, and steepening the spectral indexof protons by precisely one at an energy given in Ref. [52] as p br /m p c ≈ B µ T − . n − n − / i , (7)where B µ is the magnetic field strength in units of µG , T isthe temperature of the ionized precursor in units of K ,and n , n i are the neutral and ionized gas density given inin units of cm − , respectively. Similar developments in non-linear DSA have shown that over 1-10 GeV the spectrum canbe as steep as E − depending on the shock speed and envi-ronment, flattening out again above a few TeV [54].The mechanism described above successfully reproducesat least 6 of the 16 current Fermi-LAT observations ofSNRs [55–60], although the uncertainties associated with es-timating the relevant environmental parameters are consider-able. The 10 remaining observations have not yet incorporatedthis model into the analysis. In Ref. [57], several SNRs ob-served by Fermi were shown to be interacting with molecularclouds based on radio observations of 1720 MHz OH maseremission, providing a strong indication of shocked H . Thespectra were then reproduced by fitting the underlying protondistribution according to an exponentially cutoff power-law,as we do above.SNRs interacting with highest density clouds were foundto have low cutoff energies and hard proton spectra with[ Γ , E c ] = [1.7,160 GeV] and [1.7,80 GeV] compared to thelow-density cases, where [2.4,1 TeV] and [2.45,1 TeV]. Foranother SNR, W44, an independent analysis found that the γ -ray emission was well fit by a hard proton spectrum of in- dex between 1.74 and 2 with a cutoff at p c ≈
10 GeV/c [56].While these examples provide a representative sample of theexpected range for the low-energy spectral index and cutoffenergies, we do not necessarily expect a hardened spectrumto be correlated with high gas densities. These SNR spectramatch the γ radiation expected from an exponentially cutoffproton spectrum quite well, possibly indicating that the the-ory of Ref. [52] is underestimating the true breaking strengthdue to ion-neutral damping, or that an additional cutoff mech-anism is at play. In either scenario, a more pointed spectralpeak is predicted, and as a result the fit to the residual GCEspectrum in Section II B is generally improved.The Galactic center hosts a zoo of high-energy astrophys-ical sources including several SNRs, resolved & unresolvedpulsars, pulsar wind nebulae, and the central black hole SgrA*. Most notably Sgr A East is a ∼ − year old and 10pc wide SNR rapidly expanding into the molecular cloud M–0.02–0.07, where a half-dozen sites show also show the 1720MHz maser emission from shocked H [61]. This complexencompasses the central black hole with most of the structureresiding within a few parsecs from Sgr A* ( < ∼ . ◦ ). Thisseparation is too small to be spatially resolved by Fermi-LAT,which has a maximal angular resolution of about a quarterdegree, hence it will appear as a point source, perhaps withminor spatial extension, whose spectrum cannot be differenti-ated from additional Galactic center sources An especially intriguing candidate for the recent injectionof cosmic-ray protons in the inner Galaxy is Sgr A East. Asan estimate of the expected flux from Sgr A East, we utilize asimilar object, SNR W44. The latter is observed to have a dif-ferential flux of ≈ . × − GeV/cm /s. Multiplying by thesquare of the distance ratio d /d ≈ (2 . / . we obtain a flux of × − GeV/cm /s, precisely in line withthe GCE residual and the Sgr A* flux reported by Abazajianet al within a ◦ × ◦ box centered on the GC [20]. (Notethat the the two Daylan et al fluxes reported in Figure 4 arenormalized by the solid angle of a thin annulus at ◦ fromthe GC). It remains to be assessed whether the spectral breakenergy near the Galactic center is compatible with the the re-sults of Section II B, and whether a reasonable supernova rateis compatible with the observed flux.The environment of Sgr A East has been studied in detail atradio and X-ray wavelengths. Unfortunately, the complicatedstructure and rapid gradients in density, temperature, and mag-netic field strength imply that there will be no single predic-tion for the spectral break energy predicted by Equation (7),but, rather, a range of values dependent on the particular prop-erties of the shocked region. Here we expect that nearly all ofthe supernova activity will take place very close to the Galac-tic center, with conditions not far removed from those of Sgr AEast. The goal of the current study is to determine whether the For reference, the template analysis of Daylan et al, which uses large pho-ton statistics and an event selection which optimizes PSF, finds the mostlikely position for the GCE to be centered within about 3 arcmin of SgrA*. The next generation of ground-based γ -ray telescopes is likely to re-solve these structures at energies above 50 GeV. . This cloud makes up5-10% of the total Galactic molecular gas and is comprisedof dense clumps of H as well as of a lower density ambi-ent component which completely fills the acceleration volumefor any centralized SNR. In the inner 15 pc, typical densitiescan vary from the ambient value of cm − up to the densemolecular clouds at cm − [43, 62], occasionally reachingeven higher densities. The warm ionized hydrogen is signif-icantly more extended and provides the precursor for shockacceleration. There is only weak power-law dependence ofthe break momentum on the density and temperature of theionized component ( n − . i and T − . ). Both of these compo-nents are reasonably well measured in the Sgr A* region usingX-ray observations with ion densities near cm − and veryhot plasma temperatures of K [63].The most important, and also the most uncertain factorin determining the break momentum, is the magnetic fieldstrength in the shock propagation region. Zeeman splitting ofOH molecules provides a measurement of the magnetic fieldstrength along the line of sight, and indicates very strong fieldsin the large non-thermal radio filaments and possibly molec-ular clouds which can be as high as 1-4 mG [61, 64] whileFaraday rotation measurements indicate that the surroundingmedium can be somewhat lower with a strength down to sev-eral hundred µ G. For an extensive review of magnetic fieldsin the Galactic center, we point the Reader to Ref. [64].Efficient trapping of very low energy precursors in the verydense molecular clouds implies that these will be the primaryacceleration sites for the resulting high energy cosmic-raypopulation, although a fraction will still originate from thesurrounding lower density and lower magnetic field regions.In this case, the lower densities of the ionized and molecu-lar components partially cancel the effect of the smaller mag-netic field on the break momentum, but some broadening ofthe spectral peak may be expected toward lower energies. Inorder to estimate the range of break momenta achievable atthe GC, we simply fix the least sensitive parameters to typ-ical values, and set n i = 10 cm − , n = 10 cm − , and T = 10 K, while varying of B between 0.5 mG and 4 mG.Doing this provides a break momentum between 0.79 and 51GeV/c with a nominal value of 12.7 GeV/c for a 2 mG fieldstrength.Without more accurate measurements and high-resolution3-dimensional models of the Galactic center environment, Interestingly, the same gas model in Ref.[43] finds a large gas bulge ex-tending to 450 pc which is rotated 13.5 ◦ CCW from the Galactic planewhen projected along the line of sight with an axis ratio of 3:1. Daylan etal found a slightly preferred fit at roughly an angle of ◦ ± CCW with anaxis ratio of . ± . , possibly indicative of gas correlated emission. it is extremely difficult to definitively compute the result-ing cosmic-ray spectrum. If, in fact, these large magneticfields are contained strictly to non-thermal radio filaments,or are much weaker then previously thought, as suggested inRef. [24], the predicted momentum break would be signifi-cantly smaller, and the breaking mechanism would be disfa-vored as an explanation for the GCE. It is also very likelythat current conditions at the Galactic center differ substan-tially from those of 1-10 Myr ago especially if the Fermi bub-bles formed on comparable timescales. Compounded with un-certainties in non-linear DSA in the presence of ion-neutraldamping, a conclusive statement is currently not possible.Nonetheless, the observation of break energies from ten toseveral hundred GeV in nearby SNR indicates that such sce-narios are not uncommon, and provide evidence that the de-scription advocated above is not unrealistic.In Figure 5 we show confidence intervals for the low-energyspectral index and break energy for the BPLFix and PLExpmodels of the proton spectrum as fitted to the two Daylanet al GCE residuals as well as that extracted by Gordon &Mac´ıas. We do not show the results of the fits to the Abaza-jian et al results due to the previously mentioned asymmetryin the number of points below and above the spectral peakwhich forces a very hard spectrum that clearly does not fit therapid falloff above 2 GeV seen in the other datasets. Whilethe residual found by Abazajian et al is indeed very hard atlow energies, when an additional GCE template and spectralform is included as part of the fit, the low-energy index soft-ens significantly becoming very similar to the other analyses.This behavior is clearly delineated in Fig. 8 of Ref. [20] andthe enclosed discussion.In the left panel, the shaded regions along the x-axis showthe range of the low-energy proton index which are compat-ible with Fermi-LAT observations of SNRs interacting withmolecular clouds taken from Refs. [56, 57], highlighting thecanonical index Γ = 2 predicted by linear DSA. In theshaded y-axis regions, we show expectations for the positionof the spectral break in conditions typical of the very densemolecular clouds (dark cyan) and in the ambient lower densityenvironment (darker+lighter cyan). It is promising that thesecontours are fully compatible with one-another when fittingto the BPLFix model. Clearly, if one assumes the BPLFixmodel, the parameter values are in line with those expectedfrom SNR interacting with molecular clouds in the Galacticcenter.In the right panel we show similar regions shaded along thex-axis representing the range of the spectral indices compat-ible with Fermi-LAT observations where fitting the underly-ing proton spectrum used an exponentially cutoff power-lawmodel [55, 58, 60]. Although these studies also indicate GeV-TeV scale cutoff energies, it is unclear how such cutoff scalesshould change in the Galactic center environment without atheoretical understanding of the cutoff mechanism itself. Incontrast to the BPLFix model, a PLExp spectrum reveals lesscompatibility among the three GCE residuals, with the main P6v11 analysis of Daylan et al requiring an unphysically hardspectral index. Interestingly, two of the GCE datasets show arapid upturn in the contour as the spectral index rises above4
Spectral Index Γ Sp ec t r a l B r e a k M o m e n t u m p b r [ G e V / c ] Compatible with Fermi − LAT SNRs E x p ec t e d A cce l e r a t i o n C u t o ff BPLFix Model
Daylan et al P7v6Daylan et al P6v11Gordon & Macias 2013
Spectral Index Γ C u t o ff M o m e n t u m p c [ G e V / c ] PLExp Model
FIG. 5. 1,2, and 3 σ confidence intervals for a broken power-law proton spectrum which steepens its index Γ by one above the break energy p br (left panel), and an exponentially cutoff power law (right panel), fit to three extractions of the Galactic center excess spectrum (excludingAbazajian et al). In the top panel, the bands shaded along the x-axis represent the range of low-energy spectral indices for SNRs interactingwith dense molecular clouds as measured by Fermi-LAT in Ref. [57]. The dark and dark+light shaded bands along the y-axis indicate spectralbreak momenta expected to occur in dense molecular clouds and more ambient molecular densities respectively. Also note that confidenceregions for the two Daylan et al spectra do not include any systematic errors and hence the true confidence contours are likely to be significantlymore extended. Γ = 2 . In this region, the fit is almost completely insensitiveto the cutoff energy up to at least ≈ TeV. Notably, a spectralindex softer than 2 is commonly invoked when modeling radioand γ -ray emission from AGN in the context of hadronic in-jection. Although the relatively low momentum cutoff wouldstill need to be explained, the insensitivity here could allowfor a variety of possibilities, and warrants additional study.To summarize, we find that the occurrence of a break inthe spectrum of cosmic-ray protons in the specific environ-ment of the Galactic center is well-motivated. Observationsof the γ -ray spectrum of several SNR with the Fermi LATpoint to cosmic-ray proton spectral features aligning preciselywith those needed to fit the spectrum of the GCE; the loca-tion of a spectral break in the accelerated cosmic-ray protonsin the presence of dense molecular clouds in the inner Galaxyalso falls squarely in the range that optimally fits the inferred γ -ray spectrum of the GCE. We thus conclude that the spectrawe invoked to fit the GCE are well motivated by both theoryand observation. B. SNe Rates and Starburst Histories
In this section we explore the energetics required to pro-duce the GCE with cosmic-ray protons injection at the centerof the Galaxy. In the previous section, we showed that the fluxmeasured from SNR W44 corresponds to the approximate lu-minosity needed to explain the GCE in the inner Galaxy. Atradii larger than 1 degree, the GCE signal decays rapidly asshown in Fig. 2. In Section 2 we showed that such a radialflux profile could be achieved rather naturally by the diffusion of protons injected at the Galactic center in several differentepisodes – for example, impulsive injection over 2-3 differentepochs ( ≈ , , and 10 yr) or continuously if the sourcewas turned on around 7.5 Myr ago. Previously, we ignored thenormalization of the flux and were only concerned with therelative normalization of the summed impulsive models Thisrevealed that the 100 Kyr + 2 Myr summed model preferredrelative normalizations of, respectively, 1:10. The energeticsof these long-timescale events is more constrained than formore recent outbursts.We compute the γ -ray flux due to protons assuming a nu-clear injection spectrum of index Γ = 2 breaking to Γ = 3 at 10 GeV. We find that the K and summed impul-sive model requires a total injection of O (10 ) erg into pro-tons with energies above 100 MeV in order to produce fluxcompatible with the GCE consistent with the very recent find-ings of [36]. For continuous sources only a few million yearsold, the required energy is approximately erg/s, or a few erg/century, while continuous sources in steady-state arean order of magnitude less and comparable to the rates neededto maintain the current molecular gas temperatures near theGalactic center [65].Stellar densities at the Galactic center are extremely highrising from a mass density of M (cid:12) / pc − at a radius of10 pc to over M (cid:12) pc − in the central parsec (comparedto the local density (cid:28) M (cid:12) / pc ). Measurements of the in-frared luminosity near the Galactic center provide an indirectprobe of the star formation rate. If this has not changed dra-matically over short stellar evolution timescales ( yr), theexpected supernova rates are 0.01-0.1 per century [66] eachinjecting (cid:15) p erg where (cid:15) p is the fraction of the supernova5energy channeled into proton acceleration, often taken to benear . [28]. This implies an average continuous injectionrate of − erg/century, compatible with the observedexcess signal. For impulsive sources, the same value of (cid:15) p would require bursts of 10-100 supernovae to occur within atimescale relatively short – to yr – with respect tothe diffusion timescale. While any realistic scenario wouldlikely be an admixture of continuous and burst-like injections,the supernova rates required to reproduce the observed GCEflux in either case are well within the possible histories of theGalactic center Region.Star formation rates within the central hundred parsecsof the Galaxy is a subject of hot debate. Over ∼
10 Gyrtimescales, several studies[33–35] suggest that the star for-mation rate has been approximately stable, with long-livedbulge stars formed during the Milky Way’s last major mergerevent and relatively quiescent activity since. On much shortertimescales the situation is less clear. Highly variable and in-tense star-formation producing tens to thousands heavy starsover a few Myr, cannot be ruled out. High ionization rates, se-vere shocks, and the large scale inflow/outflow accompanyingmolecular cloud collisions or cataclysmic events, such as star-bursts or activity from the central supermassive black hole,can trigger periods of rapid star-formation taking place insidethe densest molecular clouds . In contrast to self-collapsingmolecular clouds, such external compression mechanisms arebelieved to induce significantly heavier initial mass functions,producing O/B type stars which evolve over − yearsbefore going supernova [68]. While many of the Galactic cen-ter conditions can also inhibit star formation, observations in-dicate at least 100 high mass stars with ages estimated aroundseveral Myr, indicating that an era of high star-formation ratesmay have occurred ∼ yr ago which has since halted.It is notable that the orbital time period for a typical molec-ular cloud at a radius of 1 pc is years providing ampleopportunity for interactions with other clouds, or with the ac-cretion disk surrounding the central black hole [68]. Alterna-tively, this could be taken as possible evidence of intense su-pernovae or Sgr A* activity several million years ago in whichshocked molecular clouds became highly compressed, initi-ating star-formation. Supernovae bursts have also been pro-posed as a driver of the Fermi bubbles on Gyr timescales [66]and as a mechanism to explain the extremely hot plasma tem-peratures in the Galactic center where gas in excess of up to K are observed, hotter than the Galactic escape energy,implying extraordinary energy injection event(s) with total en-ergy erg and a lifetime of order yr in order to remaincontained near the Galactic center [63]. Such extreme eventshave comparable timescales and energetics to produce the sce-narios explored earlier.Another possibility which has been previously consideredis the injection of protons directly from the central blackhole [69–71]. Our morphological analysis of Section II A is For a recent review of massive star formation in the Galactic center, seeRef. [67] substantially blind to the spectrum over the very narrow en-ergy range under consideration. Spectrally, the situation ismore difficult as such low-energy cutoffs in the proton spec-trum do not seem typical of active galaxies . It is possible thata yet unknown mechanism is responsible for producing a cut-off proton spectrum from Sgr A*. Such a scenario was in factconsidered in Ref. [72]. In this case, the black hole is taken tobe in a quiescent state with a very hard proton spectrum Γ = 1 exponentially cut off at 5 GeV. Secondary electrons producedin the hadronic interactions are of low enough energy to pre-serve their spectral shape and emit very hard infrared and mil-limeter synchrotron spectra, matching radio observations ofSgr. A*. In such a scenario, the soft protons could diffuse tolarge radii while the hard synchrotron emission would be con-fined to the confined to the ultra high magnetic fields in theimmediate vicinity of the central black hole.A very recent result [36] examined the compatibility of ra-dio and GeV/TeV γ -ray observations with predictions fromtwo models of starburst galaxies based on the interactions ofcosmic-rays (of supernova origin) in the Central MolecularZone. Particularly careful attention was paid to the relevantastrophysical parameters, which are fully consistent with whatwe employed here. For each model, the average magneticfield strength, convective wind speed, ionized gas density, andfree electron absorption fraction were allowed to vary in or-der to find optimized fits to data. The results strongly favorion densities between 50 and 100 cm − and magnetic fieldsbetween 100 and 350 µ G throughout the entire ambient CMZcloud. While radio and TeV observations are well fit, a GeVexcess still persists. The addition of additional populations ofeither protons or electrons is then considered. In the case ofprotons, a soft spectral index Γ ≈ . and a supernova rateenhanced by a factor ∼ are found to be consistent, butare dismissed from further analysis based on the required SNerate. For electrons, the energetics are more compatible, but thespectral indices predicted for radio and γ -rays are found to beinconsistent with observations. We note that in our analysis,this is precisely the proton spectral index we predict above a ∼ GeV and that the required SNe rate is substantially re-duced due to our much harder
Γ = 2 low-energy spectrum(which also matches the GeV excess in much greater detailthan what considered in Ref. [36]). We find it remarkable thata completely independent analysis of the conditions requiredto fit starburst models to observations of the CMZ can nat-urally motivate an SNR explanation for the GCE at such adetailed level.To summarize, in this section we have presented obser-vational and theoretical evidence for spectral breaks in thecosmic-ray spectrum when protons undergo diffusive shockacceleration by supernovae remnants which are inside orstrongly interacting with partially-ionized molecular cloud Although many AGN spectra do have breaks in the γ -ray spectrum near5 GeV, this results from absorption in the so-called ‘broad-line region’within a few hundred Schwarzschild radii of the central black hole anddoes not provide a viable mechanism for extended emission peaked in theGeV range. Γ of precisely ∆Γ = erg) to occur ontimescales somewhat smaller than the age of the outburst, orthat quasi-continuous sources inject protons at a rate of order erg/s. Finally, we discussed evidence for sporadic in-creases in star-formation and supernovae rates in the Galacticcenter on the timescales relevant explain the extension of theGCE in terms of cosmic-ray diffusion and subsequent γ -raysof hadronic origin. Very large uncertainties plague each stepof such an analysis and estimate. Nonetheless, the combina-tion of spectral compatibility along with reasonable energeticsand a plausible Galactic history provide a crucial backgroundto any analysis of ∼ GeV γ -ray data at the Galactic center. IV. DISCUSSION AND CONCLUSIONS
We presented a case for high-energy cosmic-ray protons in-jected in the Galactic center region as a plausible explanationto the reported Galactic center γ -ray excess over the expecteddiffuse background. Our study focused on whether such anexplanation meets the required (i) morphology, (ii) spectrumand (iii) energetics.We demonstrated that cosmic rays injected on the order ofa mega-year ago explain the observed spherical symmetry re-ported from the “inner Galaxy” analysis of Ref. [21], whilea more recent (on the order of a few kilo-years old) episodewould possess the same morphology obtained for the inner-most portions of the Galaxy in the “Galactic center” analysisof Ref. [21]We showed that the γ -ray spectrum predicted by cosmic-rayproton energy distributions responsible for the emission ob-served from supernova remnants (such as broken power lawswith specific spectral indexes, and exponentially suppressedpower laws) provide excellent fits to the observed Galacticcenter excess. We pointed out that the preferred range for thebreak of the power law and for the spectral indexes inferredfrom the observed excess fall squarely in the ranges inferredfrom observations of supernova remnants, as well as in thegeneral range expected from theory considerations. We alsopointed out the importance of systematic effects in spectral re-construction due to hadronic cross sections impacting the pre-dictions for the γ -ray spectrum from inelastic proton-protoncollisions. Finally, we inspected the time-scales, spectrum and ener-getics we invoked to reproduce the morphology and spectrumof the Galactic center excess in the context of one or more ad-ditional populations of cosmic-ray protons in the region. Wedemonstrated that the existence of such populations is moti-vated by a variety of observational and theoretical reasons,which we reviewed in detail.In conclusion, with the present study we gave proof of exis-tence of a well-motivated alternative to dark matter annihila-tion or milli-second pulsars as an explanation to the reportedGalactic center γ -ray excess. Our results indicate that conclu-sively claiming a signal of New Physics from γ -ray observa-tions of the inner regions of the Galaxy must contend with avariety of additional astrophysical processes. In particular, wehighlighted that one or more previously unaccounted-for pop-ulations of cosmic-ray protons in the Galactic center couldpotentially produce a γ -ray emission with a spectrum, mor-phology and intensity closely resembling those of the Galacticcenter γ -ray excess. ACKNOWLEDGMENTS
We thank Andy Strong, Amy Furniss, Roland Crocker, Gud-laugur J´ohannesson, and Tim Linden for very helpful discus-sions. EC is supported by a NASA Graduate Research Fel-lowship under NASA NESSF Grant No. NNX13AO63H. SPis partly supported by the US Department of Energy, ContractDE-FG02-04ER41268.
A. UNCERTAINTIES ON π EMISSIVITIES
The γ -ray emissivity q π ( E π ) of secondary neutral pionsproduced through inelastic scattering of cosmic-ray protonson interstellar hydrogen is given by the following expression: q γ ( E γ ) = 2 (cid:90) ∞ E min q π ( E π ) (cid:112) E π − m π d E π , (8)where E min = E γ + m π / (4 E γ ) and the neutral pion produc-tion term, q π , is defined by, q π ( E π ) = 4 πn H (cid:90) ∞ m p j p (cid:16)(cid:113) E − m (cid:17) d σ pH → π (E p , E π )d E π d E p , (9)with n H the target hydrogen gas density, σ pp → π the inclu-sive π production cross section (p + p → π + anything), and j p ( p p ) the cosmic-ray proton density as a function of the pro-ton momentum, following recent results from Ref [73]. Notethat many references use instead a proton spectrum follow-ing E tot rather than p p or kinetic energy T p . Although theseasymptote to each other at E (cid:29) m p , the assumption can havea non-negligible impact on the low-energy γ -ray spectrumfor soft proton spectra Γ > ∼ . , where the low-energy pro-tons contribute heavily. Since the cross-section falls off veryrapidly below 1 GeV, this is negligible for the harder spectra of7interest here. Remarkably, this cross-section is still not knownto better than ± T p = 280 MeV up to a few GeV, resulting in an important sys-tematic uncertainty when using the γ -ray spectra to probe theunderlying spectrum of nuclear cosmic-rays, or vice versa asis the case here. Until improved laboratory measurements aremade available this remains a limiting factor in determiningthe global spectrum of diffuse Galactic protons using Fermi-LAT photon data [74, 75]. In this Appendix we demonstratethe systematic variations between four common models of thepion emissivity.The first model we consider is the simple δ -function ap-proximation for the cross section[28] as parametrized inRef. [76]; we then consider the three numerical models im-plemented in Galprop , which use cross-sections from Ka-mae et al (2006) [77], Dermer (1986) [46], and the modelused throughout this paper: a combination of Dermer (1986)near threshold and interpolated to Kachelrieß & Ostapchenko(2013) at higher energies [47], hereafter DKO.The simplest estimate of the pion emissivity is obtained inthe delta-function approximation, where proton-proton colli-sions are assumed to produce only pions and hence the wellknown inelastic cross-section is used as a proxy for the inclu-sive π cross section and q π ( E π ) = n H κ π σ inelpp (cid:18) m p + E π κ π (cid:19) (10) × j p (cid:115)(cid:18) m p + E π κ π (cid:19) − m , (11)with cosmic-ray proton density j p , and with κ π ≈ . themean fraction of the impinging proton kinetic energy trans-ferred to the secondary π per collision [76]. This has been ad-justed empirically to provide better excellent agreement withMonte Carlo simulations above a few GeV [78]. We take thefollowing approximation for the proton-proton inelastic crosssection [78] in millibarnes: σ inelpp ( E p ) ≈ (34 . . L + 0 . L ) (cid:32) − (cid:18) E th E p (cid:19) (cid:33) , (12)where L = ln( E p / and E th = m p + 2 m π + m π / (2 m p ) is the pion production threshold, below whichthe inelastic cross section is zero. This provides a reason-able estimate for many cases, but as can be seen in Figure 6,it does not provide an adequate representation of the near-threshold behavior ( T p < ∼ GeV). Besides integrating overthe full range of proton energies (as opposed to approximat-ing with a δ -function) the core difference between this sim-plified approach and the more sophisticated calculations is adetailed parametrization of the inclusive production cross sec-tion and pion multiplicities at low energies, and sometimesMonte Carlo interpolation at high energies.Below a few GeV, light hadronic states decaying through π ’s provide the main contribution, primarily from the ∆(1232) resonance. As the proton energy increases, heavier E d N / d E [ a r b . ] j p ( p p ) ∝ p − . BPLFix Γ =2 , Γ =3 p br =30 GeV Dermer + Kachelrie ß &Ostapchenko (2013)Dermer (1986)Kamae et al (2006) δ − function -1 E [GeV] ∆% FIG. 6. Model variations in the γ -ray spectral energy distribution forcosmic-ray proton spectra following (in black) a broken power-lawspectrum with Γ = 2 , Γ = 3 , and p br = 30 GeV (see Eq. (6))and, in red, a flat power-law of index 2.82 representative of the ‘sea’of Galactic protons. resonances become more important as well as secondary pho-tons from η decays. The Dermer model includes the ∆(1232) using Stecker’s isobar model [79] at low-energies with linearinterpolation between 3 and 7 GeV to the scaling model ofStephens and Badhwar [80]. We note that this model relieson cross-sections originally compiled by Stecker in Ref. [48].At higher energies, however, this model violates the Feynmanscaling hypothesis, where E d σ/ d p becomes independent ofthe center of mass energy s for s (cid:29) m . Kamae et al [77]instead relies on parameterizations of Monte Carlo simula-tions in addition to corrections for the ∆(1232) , the N (1600) cluster of resonances, diffractive processes, non-scaling ef-fects, and scaling violations which provides a better fit tohigh-energy observations than Dermer. The mixed DKO [47]model used in this paper combines simulation/parametrizationapproaches by interpolating to results from event generatorQGSJET-II at energies above GeV providing a better fitto available high-energy collider data. When fitting a protonspectrum to γ -ray data, Dermer provides the best fit below1 GeV, but underestimates the higher-energy spectrum. Ka-mae et al has the opposite behavior, matching above 1 GeV,but overproducing photons below. The mixed model providesgood fits in both regimes, and hence is the model of choicehere.In the top panel of Figure 6 we show in black the γ -rayspectrum resulting from the fixed broken power-law (BPLFix)model of Eq. 6 with Γ = 2 , Γ = 3 , and E br = 30 GeV aswell as the background Galactic protons in red following a flat8power law with index
Γ = 2 . for each of the four models.Note that the relative normalization for each of the Galprop models is correct while the δ -function case renormalized tomatch DKO at 2 GeV. In the lower panel we show the frac-tional variation in the spectral energy distributions of eachmodel with respect to DKO. The two most important factorsfor an analysis of the Galactic center excess are the positionand width of the spectral peak. The models which includethe detailed low-energy characterization of Dermer producethe sharpest peak while that of Kamae et al is slightly broad- ened and peaks at 50% higher energy for the BPLFix model.This implies that the π spectrum using Kamae et al requiresslightly softer low and high-energy spectral indices than thoseof Dermer in order to match the GCE with a broken power lawproton spectrum. Of general interest, but less importance toour analysis is the significant variation in the predicted Galac-tic background spectrum, where two distinctive peaks are seenin Dermer models compared to only one in the other two. Itis clear that the δ -function approximation does not accuratelycharacterize the spectrum below ≈ GeV, and is hence notsuitable for calculating spectra over GCE energies. [1] S. Hunter, D. Bertsch, J. Catelli, T. Digel, S. Dingus, et al.,Astrophys.J. , 205 (1997).[2] W. de Boer, C. Sander, V. Zhukov, A. Gladyshev, and D. Kaza-kov, Astron.Astrophys. , 51 (2005), astro-ph/0508617.[3] A. W. Strong, I. V. Moskalenko, and O. Reimer, Astrophys.J. , 962 (2004), astro-ph/0406254.[4] T. Kamae, T. Abe, and T. Koi, Astrophys.J. , 244 (2005),astro-ph/0410617.[5] F. W. Stecker, S. D. Hunter, and D. A. Kniffen, AstroparticlePhysics , 25 (2008), 0705.4311.[6] A. Abdo et al. (Fermi-LAT collaboration), Phys.Rev.Lett. ,101101 (2010), 1002.3603.[7] W. Atwood et al. (LAT Collaboration), Astrophys.J. , 1071(2009), 0902.1089.[8] L. Goodenough and D. Hooper (2009), 0910.2998.[9] D. Hooper and L. Goodenough, Phys.Lett. B697 , 412 (2011),1010.2752.[10] D. Hooper and T. Linden, Phys.Rev.
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