The Galactic Center GeV Excess from a Series of Leptonic Cosmic-Ray Outbursts
Ilias Cholis, Carmelo Evoli, Francesca Calore, Tim Linden, Christoph Weniger, Dan Hooper
FFERMILAB-PUB-15-255-A
Prepared for submission to JCAP
The Galactic Center GeV Excess froma Series of Leptonic Cosmic-RayOutbursts
Ilias Cholis a Carmelo Evoli b Francesca Calore c Tim Linden d Christoph Weniger c and Dan Hooper a,e a Fermi National Accelerator Laboratory, Center for Particle Astrophysics, Batavia, IL 60510,USA b Instit¨ut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761,Hamburg, Germany c GRAPPA, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, Netherlands d University of Chicago, Kavli Institute for Cosmological Physics, 933 E. 56th St., Chicago,IL 60637 USA e University of Chicago, Department of Astronomy and Astrophysics, 5640 S. Ellis Ave.,Chicago, IL 60637 USAE-mail: [email protected], [email protected], [email protected],[email protected], [email protected], [email protected]
Abstract.
It has been proposed that a recent outburst of cosmic-ray electrons could accountfor the excess of GeV-scale gamma rays observed from the region surrounding the GalacticCenter. After studying this possibility in some detail, we identify scenarios in which aseries of leptonic cosmic-ray outbursts could plausibly generate the observed excess. Themorphology of the emission observed outside of ∼ ◦ − ◦ from the Galactic Center canbe accommodated with two outbursts, one which took place approximately ∼ yearsago, and another (injecting only about 10% as much energy as the first) about ∼ yearsago. The emission observed from the innermost ∼ ◦ − ◦ requires one or more additionalrecent outbursts and/or a contribution from a centrally concentrated population of unresolvedmillisecond pulsars. In order to produce a spectrum that is compatible with the measuredexcess (whose shape is approximately uniform over the region of the excess), the electronsfrom the older outburst must be injected with significantly greater average energy than thoseinjected more recently, enabling their spectra to be similar after ∼ years of energy losses. a r X i v : . [ a s t r o - ph . H E ] J un ontents Galprop
DRAGON
Over the past five years, an excess of GeV-scale gamma rays collected by the Large AreaTelescope (LAT) aboard the
Fermi satellite has been reported from the direction of the regionsurrounding the Galactic Center (GC) [1–10]. The consistency of the spectrum, angulardistribution, and overall normalization of the GeV excess with predictions of annihilatingdark matter (DM) has generated a great deal of interest. Specifically, the signal has beenshown to be distributed with approximate spherical symmetry about the GC, with a profilethat corresponds to a DM density that scales as ρ DM ∝ r − . , compatible with expectationsfrom simulations [12–17]. The spectral shape of the excess is also in good agreement withthat predicted from annihilating DM (see, for example, Figure 2 of Ref. [18]). For the case ofDM particles that annihilate predominately to ¯ bb , the observed spectrum is consistent witha dark matter mass in the range of 43–55 GeV at 68.3% CL [18] (or 36–51 GeV [8], 35–43GeV [7], as found by the authors of other recent analyses; see also Ref. [19]). Annihilationsto lighter quarks (or to a combination of quark species) can also provide a good fit, although A similar residual was also mentioned by the
Fermi team in 2009 [11]. No details were provided, how-ever, regarding its spatial morphology, systematic issues regarding astrophysical backgrounds, or possibleinterpretations. – 1 –or lower values of the DM mass [8, 18], whereas annihilations into hh , W + W − , ZZ , or hZ final states would point to higher DM masses [18–20].In addition to annihilating DM, two classes of astrophysical explanations have beenproposed for the GC GeV excess. The first posits that the signal is generated by a largepopulation of unresolved millisecond pulsars. This possibility is motivated in large part bythe spectral similarity between such sources and the observed GC excess [2, 21]. As recentanalyses have shown the excess to be spatially extended to a radial projected distance ofat least ∼ ◦ [6, 8, 9], such an interpretation is constrained by the results of point sourcesearches in the region. In particular, if enough millisecond pulsars were present within theinner 1.8 kiloparsecs (kpc) of the Galaxy to account for the observed emission, it was shownin Ref. [22] that there should exist ∼
60 such sources with L γ > erg/s (integrated above0.1 GeV), corresponding to a flux that is significantly greater than many of the gamma-raysources detected from the direction of the Inner Galaxy (see also Refs. [23, 24]). These andother works suggest that only up to 5–10% of the GeV excess emission could arise from anunresolved population of millisecond pulsars [22–25]. That being said, a debate is ongoingregarding how many near-threshold gamma-ray point sources could be present in the InnerGalaxy [26, 27]. Alternatively, this constraint could potentially be evaded if the luminosityfunction of the millisecond pulsars in the region surrounding the GC is significantly differentfrom that observed elsewhere in the Galaxy [22, 28, 29]. Although young pulsars could alsocontribute, as suggested in Ref. [30], the lack of excess emission from along the GalacticPlane strongly limits their role.The second explanation proposes that the excess emission may be generated as a resultof one or more recent cosmic-ray outbursts. In principle, this could be the result of ei-ther cosmic-ray protons interacting with gas via neutral pion production [31], or cosmic-rayelectrons undergoing inverse Compton scattering (ICS) [32]. As hadronic scenarios predicta gamma-ray signal that is significantly extended along the Galactic Plane and otherwisecorrelated with the distribution of gas [31], they are highly incompatible with the observedcharacteristics of the excess. A leptonic outburst, in contrast, could plausibly lead to a signalthat is more smoothly distributed and spherically symmetric. In light of these considera-tions, we consider a series of leptonic outbursts taking place over the past ∼ years tobe the most plausible of the astrophysical explanations proposed for the GC GeV excess.Previous work, however, has found it to be difficult to simultaneously explain the spectrumand spatial morphology of the GC excess [32]. In this paper, we revisit this possibility. Inparticular, given the apparent self-similarity of the spectral shapes of the excess emissionacross different regions of the Inner Galaxy [9], it is important to re-evaluate whether andunder what circumstances outburst scenarios could indeed account for such an observation.Our work extends the analysis of Ref. [32] in a number of significant ways. Firstly, we gobeyond semi-analytical solutions to the diffusion equation, allowing us to take into accountspatially dependent energy losses and variations in the cosmic-ray propagation conditions.Secondly, we consider the effects of re-acceleration, convection, and anisotropic diffusion,which were neglected previously. Thirdly, we systematically perform fits to the entire InnerGalaxy, spanning the 40 ◦ × ◦ region around the GC. And finally, we perform a systematicscan over source and propagation parameters.The present paper is organized as follows. In Sec. 2, we present a framework for modelingoutburst events in the GC and discuss the possible connection between such an event and thegamma-ray features known as the Fermi
Bubbles. In Sec. 3, we describe the data sets usedin our analyses and we provide technical details about the simulation of outburst models– 2 –sing cosmic-ray propagation codes. We present our results in Sec. 4, and summarize ourconclusions in Sec. 5.
The modeling of the Galactic gamma-ray foregrounds and backgrounds generated by dif-fuse emission processes in the Milky Way ( π -production, inverse Compton scattering, andBremsstrahlung), is generally done under the assumption that the distribution of cosmicrays can be approximated by a steady-state solution. Diffuse emission models derived underthis assumption have been unable to account for the observed characteristics of the GeVexcess [8–10].In some respects, the steady-state assumption appears to be well motivated. At present,the Milky Way’s supermassive black hole is in a quiescent state [33], and the star formationrate of the GC is ∼ . M (cid:12) /yr, near the average rate required to generate the stellar popula-tion of the nuclear bulge over a period of ∼ years [34]. There are a number of indications,however, that the intensity of cosmic-ray emission from the GC region may be far from uni-form in time, instead consisting in part of a series of energetic outbursts [35–39], relatedeither to outflows from the central black hole or to nuclear starburst events.The strongest indication in favor of such a scenario is represented by the gamma-rayfeatures known as the Fermi
Bubbles [39–41]. The diffuse gamma-ray emission associatedwith the Bubbles exhibits an hourglass-like morphology, extending up to ∼ ◦ ( ∼
10 kpc)above and below the Galactic Plane, and corresponds to a total gamma-ray luminosity of(0 . − × erg/s [42, 43]. The spectrum of the gamma-ray emission from the Bubblesis quite hard, dN/dE γ ∝ E − γ between E γ ∼ Fermi
Bubbles.Leptonic models are particularly attractive due to the fact that the same population of cosmic-ray electrons could generate the gamma-ray Bubbles through ICS, as well as the microwave“haze” observed by
WMAP and
Planck [43–46] via synchrotron emission. Intensity, spatialand spectral correlations between the gamma-ray and microwave signals have been shown tosupport this hypothesis [6, 40, 43, 46]. Various authors have proposed mechanisms for theorigin of these energetic electrons, including, for example, recent ( ∼ Fermi
Bubbles have been proposed [55], asso-ciated with long time scale ( ∼ Gyr) star formation in the GC, transferred away from theGalactic Plane by strong winds. Notably, this class of models predicts a large neutrino coun-terpart signal that could be probed by current and future neutrino telescopes [56, 57], andthat may already be in some tension with high energy ( O (30 – 1000) TeV) neutrino fluxmeasurements [58].Motivated by the possible connection with the Fermi
Bubbles, we focus in this work onthe case of cosmic-ray electron outbursts, from the inner parsecs or tens of parsecs aroundthe GC. A challenge in connecting the GC GeV excess with the
Fermi
Bubbles is that– 3 –he GeV excess is approximately spherically symmetric with respect to the GC, whereasthe Bubbles are strongly oriented perpendicular to the Galactic Plane. A common originfor these signals would require a scenario in which the ∼
10 GeV electrons responsible forthe GeV excess diffuse highly isotropically, whereas the higher energy ( ∼ − GeV)electrons responsible for the Bubbles propagate more rapidly and preferentially away from theGalactic Plane. Although we do not have a concrete proposal to account for these behaviors,one could imagine a scenario in which the magnetic fields of the GC efficiently isotropizethe diffusion of low-energy electrons, while allowing higher energy particles to propagate andescape the region more freely. Alternatively, an outburst responsible for the
Fermi
Bubblesmay have been forced by the surrounding environment to propagate away from the GalacticPlane, but left the GC region in a state that allowed the electrons of subsequent outburststo propagate more isotropically. For example, a large outburst of electrons injected with ajet-like morphology a few Myrs ago, may have not experienced strong diffusive re-accelerationwithin the inner tens of parsecs, allowing the electrons to maintain their original orientationand propagate preferentially away from the Galactic Plane while suffering relatively littleenergy losses. On the other hand, later outbursts, taking place ∼ . − i.e. many free parameters) to fit the morphology and spectrum of the GC GeVexcess. In the following section, we describe our modeling of cosmic-ray outbursts in anattempt to generate the observed characteristics of the GC GeV excess. In searches for DM annihilation products from the Inner Galaxy, the regions of optimalsensitivity (maximal signal-to-background) can extend up to tens of degrees away from thedisk, with the details depending on the specific emission profile [59–64]. Several analyses ofthe GC GeV excess have focused on what is known as the “Inner Galaxy” – a region includingthe surrounding tens of degrees around the GC and excluding portions of the Galactic Plane(where the backgrounds are highest). The fact that the GeV excess is observed to extendto relatively high latitudes has been interpreted in support of a DM interpretation of thissignal [6, 8, 9, 65], as has its approximate spherical symmetry with respect to the GC [8, 9].In contrast, many proposed astrophysical explanations for the GeV excess predict emissionthat is localized preferentially around the disk [30, 31].In this study, we make primary use of the results of Ref. [9], which characterized thespectrum and morphology of the GeV excess over the region of the Inner Galaxy. Whendiscussing the region of the GC and Galactic Plane (which are excluded from the InnerGalaxy studied in Ref. [9]), we draw from the results of Ref. [8]. In the following subsections,we describe separately the adopted data sets.– 4 – .1.1 The Inner Galaxy
For the Inner Galaxy analysis, we adopt the GeV excess data as derived in Ref. [9]. The excesswas characterized within the region | (cid:96) | < ◦ and 2 ◦ < | b | < ◦ , thus avoiding the criticalfew degrees in latitude where point-source subtraction and modeling of the diffuse Galacticgamma-ray emission are most difficult. Ref. [9] derived the spectral properties of the excessin the main region-of-interest by removing Galactic foregrounds and backgrounds. Care wasexercised in estimating the typical uncertainties of this removal by analyzing residuals alongthe Galactic Disk. The corresponding systematic uncertainties are represented by a fullcovariance matrix that encodes how uncertainties in the normalization and spectral slopeof the Galactic diffuse foreground components affect the determination of the GeV excessspectrum. In addition to determining the spectral properties of the excess in the main region-of-interest, Ref. [9] also characterized the morphology of the signal in several sub-regions,probing the emission spectrum up to | b | = 10 ◦ –20 ◦ , as well as its morphological properties.Exploiting the full spectral and morphological properties of the GeV excess emission is crucialfor shedding light onto the origin of the excess emission, and it leads to tighter constraintsthan would result from considering only the spectrum from within the main region-of-interest.In our analysis, we perform fits to the spectrum of the GC GeV excess as it was extractedin Ref. [9]. In particular, we will use the fluxes from the ten segmented regions within | (cid:96) | < ◦ and 2 ◦ < | b | < ◦ , from energies of 300 MeV to 500 GeV. When calculating the quality ofthe fits, we take into account the full covariance of the systematic errors [9]. Namely, the χ function is given by: χ = (cid:88) i =1 24 (cid:88) j,k =1 ( d ij − µ ij )(Σ ijk ) − ( d ik − µ ik ) , (3.1)where d ij ( µ ij ) denotes the measured (predicted) flux in region i and energy bin j , and Σ ijk isthe covariance matrix for energy bins j and k in region i . Note that, as in Ref. [9], we neglectthe covariance of the excess emission between different regions. When quoting p -values below,we always refer to the value of this χ function for the ten region fit, and assume it followsa χ k distribution with k = 240 − χ function. Explanations of the GeV excess must also confront the bright gamma-ray emission observedfrom the region within 1 ◦ − ◦ of the GC, which has been shown to be radially extended,spherically symmetric, centered within 0 . ◦ of the GC, and have a spectrum consistentwith the full extent of the gamma-ray excess [8]. When considering the emission from theregion immediately surrounding the GC (subsection 4.3.1), or from along the Galactic Plane(subsection 4.3.2), we will use the results of the analysis presented in Ref. [8]. To simulate the propagation of electrons in the GC and Inner Galaxy regions, we makeuse of the publicly available numerical codes
Galprop [66] and
DRAGON [67, 68]. Numerical– 5 –odes are necessary to properly model energy losses in this region, since they depend on thedistributions of gas, interstellar radiation field, and magnetic field.
Galprop
The GC GeV excess is discernible from throughout the innermost ∼ e.g. boron-to-carbon). Turning this around, the propagation conditionsin the Inner Galaxy are only weakly constrained by local cosmic-ray measurements, allowingus considerable freedom in selecting the parameters that describe diffusion, re-acceleration,convection, and energy loss processes for cosmic rays in the Inner Galaxy. We calculate thespectrum of gamma rays produced through either ICS with low-energy photons [69, 70], orthrough interactions with interstellar gas, giving rise to Bremsstrahlung emission [70–72].In our simulations, we assume that the cosmic-ray electron outbursts occur homoge-neously within a cylinder of 50 pc radius and 100 pc height, centered at the GC, and with aspectrum that takes the form of a power-law with an exponential cutoff: dN e dE e = N E − αe exp {− E e /E cut } . (3.2)We allow the N to vary freely, keeping in mind that the current power in gamma-ray emissionfrom the Fermi
Bubbles multiplied by an estimated age of 10 years suggests a total energy oforder ∼ erg. We allow the spectral index to take on values in the range of α = 1 − DRAGON , wealso consider E cut = 1 TeV). We clarify that this injection spectrum denotes the spectrum ofelectrons at ∼
50 pc from the GC, and not necessarily as they were injected from their originalsources (collections of supernovae or the central black hole). This allows us to consider valuessuch as α ∼ Galprop setup.The timescale that cosmic rays of a given rigidity remain within the region of the InnerGalaxy is set by the processes of diffusion and convection. For isotropic diffusion, the diffusioncoefficient is given as a function of rigidity, R , by: D ( R ) ≡ D xx ( R ) = D (cid:18) R GV (cid:19) δ , (3.3)where D is the diffusion coefficient at 3 GV and δ is the diffusion index. We assume thatany convection is perpendicular to the Galactic Disk, with a constant gradient dv c /dz .Cosmic ray electrons suffer from energy losses due to synchrotron, ICS, and Bremsstrah-lung emission. For the Galactic magnetic field (which determines the rate of synchrotronenergy losses), we assume a cylindrical symmetry with the following parameterization: B ( r, z ) = B e − r/r c e −| z | /z c , (3.4)where B is the magnetic field at the center of the Galaxy and r c and z c are the characteristicscale lengths. For the interstellar radiation field (which determines the rate of ICS), we use– 6 –he model employed within Galprop v54 (see Ref. [73]) but allow for the normalization ofthe emission from stars and dust grains to freely vary within a factor of 3 from the referenceassumptions, while the contribution from the cosmic microwave background is kept fixed.For Bremsstrahlung off interstellar atomic hydrogen (HI) and molecular hydrogen ( H ) gas,we adopt the gas distributions used within Galprop v54 [73].Cosmic ray electrons with energies below a few GeV can also also be diffusively re-accelerated. Diffusive re-acceleration is connected to spatial diffusion through the followingrelationship: D pp ( R ) = 43 δ (2 − δ )(4 − δ )(2 + δ ) R v A D xx ( R ) , (3.5)where v A is the Alfv´en speed [73, 74].In order to generate maps of the gamma-ray emission from an outburst of a givenage, τ , and duration, ∆ τ , we first inject cosmic-ray electrons from the GC and propagatethem until τ + ∆ τ / τ − ∆ τ /
2, and then finally take the difference of these two maps. Wehave examined the convergence of these results for both the assumed spatial grid and thetimestep employed in this work. We take ∆ τ = 1 × yr for all outbursts, have assumed acylindrical grid of ∆ r = 50 pc, ∆ z = 50 pc, and have set the Galprop timestep parametersto start timestep = 5 × yr, end timestep = 5 yr and timestep factor = 0 . Galprop v54 code [73, 75, 76] to produce ICS andBremsstrahlung gamma-ray templates for the Inner Galaxy at various energies. We studya wide variety of outburst injection and propagation assumptions in the Inner Galaxy (fordetails, see Appendix A).
DRAGON
In addition to
Galprop , we make use of the
DRAGON code in our study to test outburstscenarios in which propagation proceeds inhomogeneously and/or anisotropically (which weparameterize using two diffusion coefficients, D xx and D zz , which account for diffusion in thedirections parallel to and perpendicular to the Galactic Plane, respectively.). Furthermore, wefind that DRAGON performs better in terms of memory management and required computationtime than the publicly available version of
Galprop , allowing us to run the multi-dimensionalparameter scan, as discussed in the next section, in a reasonable amount of time. Finally, theuse of
DRAGON allows us to test different assumptions for the gas distributions in the InnerGalaxy.To simulate an outburst using
DRAGON , we assume as an initial condition the analyticsolution derived in Ref. [77] for spatially constant energy losses evaluated at the time t =10 years (which is much shorter than the outburst’s age). We then solve the time-dependentdiffusion equation for a total time corresponding to the age of the outburst, τ . We adopt aspatial grid step size of 50 pc and a constant time step of 1 kyr.While the GeV excess appears to be spherically symmetric with respect to the GC, thissymmetry is broken in outburst scenarios to some degree by energy losses that are highest nearor within the Galactic Plane. This is primarily the result of Bremsstrahlung, whereas ICSoccurs comparatively uniformly in space. By introducing anisotropic diffusion ( D zz (cid:54) = D xx ),we hope to identify scenarios that predict a more spherically symmetric gamma-ray signalfrom a given cosmic-ray outburst. – 7 –arameter Units Range Prior α δ D cm / s 0.1–20 lin D zz /D xx v A km / s 0–200 lin τ Myr 0.1–5 lin
Table 1 . The range of single outburst model parameters considered in our scans. For each parameter,we indicate whether we adopt logarithmic or linear priors in the Bayesian parameter scan.
In order to efficiently scan through the multi-dimensional space of propagation and sourceparameters (except for the overall normalization, which we obtain in the final χ minimiza-tion), we make use of the nested sampling algorithm implemented in MultiNest [78–80]. Wehave directly coupled
MultiNest to the cosmic-ray propagation code
DRAGON .Convergence is usually obtained after O (10 ) evaluations. Since some of the propagationand source parameters are either not well constrained or are highly degenerate, the choice ofpriors is critical for parameter estimation (although it does not affect our best-fit scenarios,which are always robustly identified). Our choice of priors (either logarithmic or linear) forthe different parameters in the scans is summarized in Tab. 1. Note that in this section, wefix the energy cutoff to a high value of E cut = 1 TeV, but will later consider lower values forthis quantity within the context of two outburst models. MultiNest provides as output a list of parameter points, with the density of pointsbeing a proxy for the posterior probability distribution function. The latter is given byBayes’ theorem: post( (cid:126)θ ) ∝ L ( (cid:126)θ | (cid:126)d )prior( (cid:126)θ ) , (3.6)where (cid:126)θ and (cid:126)d denote all model parameters and data, respectively, post(˜ θ ) and prior(˜ θ )denote the posterior and prior probability distribution functions, and the profiled likelihoodfunction is given by: − L ( (cid:126)θ ) = min norm χ . (3.7)As mentioned above, we allow the overall normalization of the injected excess to vary freelyin each parameter evaluation. When making statements about excess energetics, we alwaystake into account the uncertainty in the fitted normalization. In this section, we first present our results from the
MultiNest approach using
DRAGON , (seeSecs. 3.2.2 and 3.3). We will show in subsection 4.1 that models with a single outburst areunable to explain the overall morphology of the GC excess. In subsection 4.2 we considermodels with two outbursts of different ages, which enable us to obtain reasonable fits tothe gamma-ray data over most of the Inner Galaxy (neglecting for the moment the regionwithin 2 ◦ of the Galactic Plane). Finally, in subsection 4.3, we discuss the morphology of thegamma-ray emission from the regions within the innermost 2 ◦ around the GC and along theGalactic Plane. – 8 –
10 15 20 D [ × cm / s] τ [ M y r ] Model AModel B − − p - v a l u e v A [km / s] . . . . . α bu r s t Model AModel B − − p - v a l u e − D zz /D xx . . . . . α bu r s t Model AModel B − − p - v a l u e Figure 1 . The results of our scan over the parameters described in Tab. 1, for a single outburst. TheGC GeV excess is best-fit by an outburst with an age of about τ ∼ α < . v A >
100 km/s). Anisotropic diffusion does notsignificantly improve the overall fit.
In figure 1, we show the results of our scan over the parameters described in Tab. 1. From thedistribution of the models in the τ - D plane, we find that the best fits occurs for outburstswith an age of about τ ∼ D , confirming that the age of the outburst is mainly fixed by the energy loss time-scale (asopposed to the timescale for diffusion). From the second panel, we see that the spectrumof the excess is best reproduced for hard injection indices ( α < .
5) and that strong re-aceleration ( v A >
100 km/s) is favored. In the third panel, we learn that anisotropic diffusiondoes not significantly improve the overall fit.From the scan results, we focus now on the best-fit model, which we denote Model A,and compare these results to the best fit model obtained after bounding 2 < α < . α α NA NA 1.0 E cut , E cut , NA NA 60 GeV τ (Myr) 0.83 0.46 0.1 τ (Myr) NA NA 1.0 N (10 erg) 2.89 9.87 0.1 N (10 erg) NA NA 0.88 δ D (10 cm /s) 5.08 9.12 9.0 D zz /D xx v A (km/s) 176 122 150 B ( µ G) 11.5 11.5 11.7 r c (kpc) 10.0 10.0 10.0 z c (kpc) 2.0 2.0 0.5 dv c /dz (km/s/kpc) 0.0 0.0 0.0ISRF 1.0, 1.0 1.0, 1.0 1.8, 0.8 χ ( p − value) 277 (0.04) 317 (0.0004) 261 (0.14) Table 2 . Parameter values for our three main benchmark models. Models A and B include a singlecosmic-ray outburst, as described in Sec. 4.1 and shown in figure 2. These two sets of parametersrepresent the best-fits found when the injected spectral index is allowed to float freely (Model A)and when it is constrained to α ≥ α (cid:39) .
2, are strongly preferred by the data. Model C includes two outbursts, as described in Sec. 4.2.The two outburst model provides a significantly better fit to the GC GeV excess. The parametersgiven under the label “ISRF” denote the normalizations of the optical and infrared components ofthe interstellar radiation field, in units relative to the models of Ref. [81] for Models A and B, and ofRef. [73] (
Galprop v54 ) for Model C.
Galaxy. Although spectral indices around α ∼ α > p -value forModel A is 0.04, whereas it is 0.0004 for Model B). The hard injected spectral index and highAlfv´en speed characterizing Model A are responsible for the observed peak in the spectrumof the excess at an energy of a few GeV.Even these best-fit single outburst models, however, fail to provide a good fit to the ob-served characteristics of the GC GeV excess, providing p − values consistently lower than 0.04.With this in mind, we turn in the following subsection to models with multiple outbursts. As discussed in the previous subsection, a single outburst of cosmic-ray electrons does notappear to be capable of accounting for for the morphological characteristics of the GC GeVexcess, under-predicting the amount of emission from more than 8 ◦ away from the GC, and We note that the data points that we present are those directly from Ref. [9], and remind that withintheir error bars the data points within each window can move upwards or downwards in a correlated fashionbased on the covariance matrix. – 10 – . . . . . . . . × − I × − II − E d N d E [ G e V / ( c m ss r ) ] × − III × − IV − . . . . . . × − V × − VI − . − . . . . . . . . E d N d E [ G e V / ( c m ss r ) ] × − VII × − VIII E [GeV] − . − . . . . . × − IX E [GeV] × − X Figure 2 . The gamma-ray spectrum predicted from ten sub-regions of the Inner Galaxy, 40 ◦ × ◦ ,for the single outburst Models A (red) and B (black); see Tab. 2. These results are compared to theobserved characteristics of the GC GeV excess, as presented in Ref. [9]. Although neither of thesemodels provide a particularly good fit ( p -values of 0.04 and 0.0004, respectively), the hard spectralindex ( α = 1 .
2) of Model A allows it to better accommodate the data. – 11 –rom within the innermost 2 ◦ surrounding the GC. This motivates us to consider scenarioswith two cosmic-ray electron outbursts, which we take to have occurred 0.1 Myr and 1Myr ago. Since cosmic-ray electrons tend to propagate outwards from the center of theGalaxy, we expect that the cosmic-ray electrons from the older outburst will be responsiblefor generating the gamma-ray emission in the outer part of the excess, while the youngeroutburst will generate most of the excess gamma-ray emission in the inner several degrees.As we will discuss in Sec. 4.3, additional (and even more recent) outbursts may be requiredto accommodate the characteristics of the GeV excess observed from the innermost degreeor two surrounding the GC.In order to study scenarios featuring two cosmic-ray outbursts, we considered a variety ofmodels, with differing diffusion coefficients, Alfv´en speeds, convection velocities, interstellarradiation field densities, and magnetic field distributions. The details of this exercise aredescribed in Appendix A. Unlike in the case of models with a single outburst, we haverestricted our two outburst models to isotropic diffusion ( D xx = D zz ).From the wide range of two outburst models we have studied, the best-fit was found forthe model that we denote as “Model C”, whose characteristics are given in the last columnof Tab. 2. Note that the two outbursts have significantly different values for their cutoffenergies in this model, E cut = 20 GeV and 60 GeV, respectively. The higher energy cutofffor the older outburst is necessary to ensure that, after accounting for energy losses, thegamma-ray spectrum is approximately uniform over the region of the GeV excess. Note thatfor the single outburst models, as discussed in section 4.1, the exact value of the cutoff doesn’tstrongly impact the fit, as long as it is O (1) TeV. This is because of the rapid energy lossesexperienced by TeV-scale electrons (which scale as dE e /dt ∝ E e ). For the models consideredin this section, it is the combination of the two outbursts that fits the data, and we find thatvalues of E cut (cid:39) −
100 GeV for the older outburst yield the best fits. In figure 3, we show the gamma-ray emission from the combination of these two out-bursts in the 10 sub-regions of the GC GeV excess [9]. The best-fit model yields χ = 261,corresponding to a p -value of 0.14. As one can see, the flux from the younger outburst(dashed line) produces most of the emission at 2 ◦ − ◦ from the GC, while the 1 Myr oldoutburst accounts for the majority of the excess emission at larger angles.A few comments regarding the parameter choices of Model C are in order. In particular,the high value of the Alfv´en speed, v A = 150 km s − , may strike some readers as unconven-tional. Although local cosmic-ray measurements imply values for this quantity that are onthe order of a few tens of km/s, the conditions of the Inner Galaxy are not well understoodor constrained by observations. For this reason, we remain highly agnostic as to the valuesof this and other propagation parameters (see also Appendix A). Another notable charac-teristic of Model C is its very hard spectral indices, α = 1 .
1, 1.0, and spectral cutoffs at 20and 60 GeV. We find it plausible that such spectra might result under the conditions withinthe inner O (10) pc, where diffusive re-acceleration plays an important role (hardening thespectral index) and where there is strong turbulence and large amounts of energy in themagnetic field (causing the spectral cutoff from synchrotron energy losses). Finally, we notethat we require ∼ − ergs in cosmic-ray electrons to fit the GeV excess, similar tothe energetics required to generate the Fermi
Bubbles. In the single outburst models, the fit is most impacted by the regions I and II of the Inner Galaxy, whilein the two outburst models the spectral properties of the 1 Myr old outburst are most constrained by regionsIII-VIII, which prefer lower-energy cutoffs. – 12 – . . . . . . . . × − × − − E d N d E [ G e V / ( c m ss r ) ] × − × − − . . . . . . × − × − − . − . . . . . . . . E d N d E [ G e V / ( c m ss r ) ] × − × − E [GeV] − . − . . . . . × − E [GeV] × − Figure 3 . As in figure 2, but for the case of a model with two cosmic-ray outbursts (Model C, seeTab. 2). The dashed (dotted) line denotes the contribution from the younger (older) outburst, andthe solid line represents the sum of their contributions. This model provides the best fit of all thosewe have considered in this study, yielding a p -value of 0.14. In figure 4, we show the contributions in Model C from ICS (blue) and Bremsstrahlung(red), from the 0.1 Myr old (dashed) and the 1 Myr old (dotted) outbursts. The Bremsstrah-lung emission (which has been multiplied by a factor of 100 in order to be distinguishablefrom zero) is always subdominant to the ICS emission in this model (except in the regionnear the Galactic Plane, as discussed in Sec. 4.3).– 13 – . . . . . . . . × − I × − II − E d N d E [ G e V / ( c m ss r ) ] × − III × − IV − . . . . . . × − V × − VI − . − . . . . . . . . E d N d E [ G e V / ( c m ss r ) ] × − VII × − VIII E [GeV] − . − . . . . . × − IX E [GeV] × − X Figure 4 . As in figure 3, but showing separately the contributions from inverse Compton scattering(blue) and Bremsstrahlung emission (red) for our best-fit two outburst model (Model C, see Tab. 2).The Bremsstrahlung component has been multiplied by a factor of 100 in each frame in order to bedistinguishable from zero. Once again, the dashed and dotted lines denote the contributions from theyounger and older outburst, respectively.
In this section, we discuss aspects of the morphology of the gamma-ray emission from cosmic-ray outbursts that is not directly evident from the plots shown in the previous subsections,focusing on the regions that lie within 2 ◦ of the GC or along the Galactic Plane.– 14 – | b | [deg], at ‘ = 0 ◦ − − − − d N / d E [ / c m s r s G e V ] GeV excess emissionat E = 2 GeV Model AModel BModel CFermi Bubbles (extrapolated)HI + H2 (at z < . Figure 5 . The angular profile of the GC excess flux, as evaluated at 2 GeV, for single outburstModels A and B, and for the two outburst Model C (see Tab. 2). Single outburst models can fitthe morphology of the excess in the range between ∼ ◦ − ◦ away from the GC, while two outburstmodels can fit the excess morphology between ∼ ◦ − ◦ . Additional recent outbursts, and/or apopulation of centrally located millisecond pulsars, are required in order to account for the emissionobserved from the innermost degree surrounding the GC. For details on the observational data, seeRef. [18] and Ref. [82]. In figure 5, we show the angular profile of the gamma-ray emission predicted by the cosmic-ray outburst Models A, B and C. Models with a single outburst (Models A and B), can onlyfit the data between approximately 2 ◦ and 8 ◦ , while the two outburst model (Model C) canaccommodate the observed morphology of the GC GeV excess between approximately 1 ◦ and 15 ◦ from the GC. The main difficulty in accommodating the observed morphology of theexcess with cosmic-ray outbursts is that the data favor a concave profile, while individualoutbursts produce a convex distribution. The sum of the contributions from multiple carefullyplaced convex-profiled outbursts can, however, resemble the observed concave distribution.At angles greater that 15 ◦ from the GC, the excess signal is faint (and is difficult todistinguish from the Fermi
Bubbles) and has relatively little statistical impact on the fits.Of greater importance is the morphology of the excess within the innermost 1 ◦ , which ismeasured to continue rising to at least within (cid:39) . ◦ (or (cid:39) . ◦ without significantly impacting the emission from much further away fromthe GC (similar to how the 0.1 Myr old outburst does not contribute very much to theemission further away than 5 ◦ ). In order for the flux of the gamma-ray excess to continueincreasing to within 0 . ◦ of the GC, the most recent outburst(s) must have taken place veryrecently, within the past several hundred years.To explain the rising gamma-ray flux at (cid:39) . ◦ ( (cid:39) F radio F γ (cid:12)(cid:12)(cid:12)(cid:12) CR = ρ B ρ rad , (4.1)whereas DM annihilations to prompt photons leads to a lower ratio: F radio F γ (cid:12)(cid:12)(cid:12)(cid:12) DM = B e (cid:18) ρ B ρ B + ρ rad (cid:19) B e (cid:18) ρ rad ρ B + ρ B (cid:19) + B γ , (4.2)where B e and B γ are the fractions of the energy generated in DM annihilations that gointo electrons/positrons and prompt photons, respectively. If we keep the energy output ingamma rays (from prompt annihilations and/or ICS) fixed in these two scenarios, we findthe following ratio of power in synchrotron emission: F radio , CR F radio , DM = 1 + B γ B e (cid:18) ρ B + ρ rad ρ rad (cid:19) . (4.3)For the conservative case of ρ B (cid:28) ρ rad , and for annihilations to b ¯ b , this ratio is approximately2.5, and therefore radio constraints will be at least a factor of 2.5 times more stringentin the cosmic-ray outburst case than in the case of annihilating DM. If we consider themore conservative ( i.e. less restrictive) assumptions of Ref. [84], the constraint from radioobservations of the GC requires that the profile of the electron distribution must become flatbefore reaching the innermost r ∼ ◦ –2 ◦ , while a population of centrally located unresolved millisecond pulsarsis responsible for the majority of the emission from the innermost volume around the GC.Millisecond pulsars have often been discussed as a possible source of the GC GeV excess [2, 4,7, 22–25, 29], motivated by the fact that the gamma-ray spectra measured from this class ofobjects is similar to that of the observed excess. And while it has been argued that the lackof pulsar-like point sources detected and resolved by Fermi in the regions north and southof the GC significantly disfavor the possibility that millisecond pulsars could account for the– 16 –ntirety of the GeV excess [22] (see also Refs. [18, 29]), bright backgrounds in the innermostdegree around the GC make
Fermi much less sensitive to point sources in this region of thesky. To account for half of the GeV excess observed from within in the innermost degree (theother half coming from cosmic-ray outbursts), would require approximately 225 millisecondpulsars, of which we expect only ∼ L γ > erg/s (integrated above0.1 GeV). Observations from Fermi cannot, at this time, rule out the existence of a centrallyconcentrated millisecond pulsar population of this size.One argument that can be made against the existence of a large centrally concentratedpopulation of millisecond pulsars makes use of the connection between millisecond pulsarsand low-mass X-ray binaries (LMXBs) [22]. Most millisecond pulsars are thought to haveevolved from LMXBs, and the abundance of LMXBs in a region is expected to be highlycorrelated to the number of millisecond pulsars that are present. Focusing on the brightest(
L > erg/s) LMXBs, for which we are believed to have a complete inventory, we cancompare the number of bright LMXBs in globular clusters to the gamma-ray emission fromsuch systems (assumed to be produced by millisecond pulsars), and compare this ratio tothat found in the GC. If half of the GeV excess from the innermost degree around the GC wasproduced by millisecond pulsars, this exercise predicts that approximately 25 bright LMXBsshould also be present in the same region. In contrast, INTEGRAL (which has sensitivityin the direction of the GC well above the level required to detect such bright sources) hasdetected only one bright LMXB candidates in this region of the sky [85], suggesting thatrelatively little of the GeV excess originates from MSPs.
In figure 4, the gamma-ray emission is dominated by ICS over each of the regions of the skyshown, with Bremsstrahlung playing only a very subdominant role. Due to the nearly homo-geneous nature of the interstellar radiation field throughout the Inner Galaxy, the emissionfrom ICS is predicted to be approximately spherically symmetric with respect to the GC, inagreement with the observed morphology of the GeV excess. In the regions near the Galac-tic Plane, however, the Bremsstrahlung component is not necessarily negligible, as the gasdensities are much higher within the volume of the Galactic Disk. To explore this feature,we plot in figure 6 the ratio of ICS-to-Bremsstrahlung emission from either a single outburst(model A) or from a combination of two outbursts (model C), throughout the 20 ◦ × ◦ boxaround the GC, including the Galactic Disk where the gas density peaks. In the region within1 ◦ of the GC, the intensity of the Bremsstrahlung emission can be as large as 30-100% ofthe ICS component. Excluding the window around the GC, the region along the GalacticPlane can exhibit Bremsstrahlung-to-ICS ratios on the order of ∼ This will lead to agamma-ray morphology that is slightly elongated along the direction of the Galactic Plane.In Sec. IV of Ref. [8], constraints were placed on such departures from spherical symmetry,finding that axis ratios which depart by more than ∼
20% from unity along the Galactic Planeare strongly disfavored. Although this constraint is in mild tension with the results shownin figure 4, it does not significantly disfavor the models under consideration in this study. Iffuture observations were to further tighten this constraint, this could be used to test cosmicray outburst models of the GC excess. In figure 6, we show ratio maps as evaluated at both 1 and 5 GeV, in order to test for any energydependance. The similarity of these frames indicates that any such variation with energy is mild. – 17 – odA, 1 GeV . . ModA, 5 GeV . . ModC, 1 GeV . . ModC, 5 GeV . . Figure 6 . The ICS-to-Bremsstrahlung ratio map for the single outburst Model A (upper frames)and the two outburst Model C (lower frames); see Tab. 2. The maps cover the 20 ◦ × ◦ regioncentered at the GC, without masking the Galactic Plane. The color-bar is in log scaling. Even inthe pixels closest to the GC, the ICS-to-Bremsstrahlung ratio is always greater than unity, althoughthe Bremsstrahlung component can be as bright as ∼ − ◦ , and ∼
10% elsewhere along the Galactic Plane.
In this paper, we have presented a detailed study of the gamma-ray emission from leptoniccosmic-ray outbursts from the Galactic Center, and considered this as a possible sourcefor the GeV excess observed by the
Fermi -LAT. We solved the full cosmic-ray propagationequations numerically, allowing us to fully take into account spatially dependent energy losses,variations in the cosmic-ray propagation conditions, and effects including re-acceleration andanisotropic diffusion, all of which had been neglected previously. We performed a systematic– 18 –can over source and propagation parameters using nested sampling techniques, and fittedthe results to the spectrum and morphology of the GeV excess over the region of the InnerGalaxy.For models with a single outburst, as had been considered previously in the literature,we find that it is not possible to explain the overall morphology of the observed GeV excess.In particular, despite allowing many parameters to vary freely, we find that single outburstmodels uniformly underproduce the amount of excess emission that is observed from theinnermost regions, outermost regions, or both, throughout the Inner Galaxy. The best-fitmodels include very strong re-acceleration (Alfv´en speeds of around 175 km/s) and requireelectrons to be injected from the Galactic Center with an extremely hard spectral index,around α (cid:39) .
2. Dramatically worse fits were found for models with a softer spectral index,as are generally predicted from Fermi acceleration.Motivated by the inability of single outburst models to accommodate the observed char-acteristics of the GeV excess, we also explored models that included two leptonic outbursts.Focusing on models with an older (1 Myr) and younger (0.1 Myr) outburst, we find thatsuch scenarios could indeed provide much better fits to the data, provided that the injectionparameters of both bursts are adjusted appropriately. The older outburst in this scenarioaccounts for the emission at high Galactic latitudes, whereas the younger outburst dominatesthe excess emission at lower latitudes and closer to the Galactic Center. If the injected spectraof the two outbursts are tuned accordingly, it is possible that their spectra will be similar af-ter accounting for energy losses that take place during propagation, enabling the gamma-rayemission to exhibit approximately the same spectral shape at all relevant latitudes, consistentwith the observed characteristics of the GeV excess.Our best-fit was found for a model with two outbursts, yielding a fit to the data with a p -value of 0.14, which is only modestly lower than those values previously found for annihilatingdark matter models ( p (cid:39) . This model featuresinjected spectral indices of around 1.1 and 1.0, and spectral cutoffs around 20 and 60 GeV forthe younger and older outburst, respectively. The extremely hard spectral indices necessaryto produce the GC spectrum appear to be incompatible with models of first-order Fermiacceleration, as well as with the observed electron injection spectra observed from gamma-rayblazars [86]. However, these spectral indices could plausibly result from the strong diffusivere-acceleration and turbulence in the region surrounding the Galactic Center. The totalenergy required in cosmic-ray electrons from the younger and older outbursts are 1 × ergs and 8 . × ergs, respectively. If we take their durations to be ∼ years, forexample, this implies cosmic ray electron luminosities on the order of L ∼ erg/s and L ∼ erg/s, below the Eddington limit of Sgr A* [87].As discussed in sections 1 and 2, cosmic ray outbursts might originate from past activityof the central massive black hole or from a series of starburst events. Some hints suggestthat these processes may have occurred on time scales of Myrs ago. As for the massive blackhole, its activity is firmly constrained over the past ∼ years, while less robust constraintson the luminosity of Sgr A* exist for timescales of 10 − years [38].One other finding of this study is that, even in models with two outbursts, it is not In Ref. [18], p -values of 0.35 and 0.37 are quoted for dark matter annihilations to b ¯ b and c ¯ c , respectively.These values refer to the fitting of the spectrum over the entire | l | < ◦ , 2 ◦ < | b | < ◦ region, while the p -values quoted throughout this paper are derived from fitting the combination of the ten sub-regions, as shownin Figs. 2-4. When this procedure is followed for the same dark matter models, p -values of approximately 0.7are found. – 19 –ossible to generate the observed morphology of the excess emission from the innermost de-gree or two around the Galactic Center. To address this, the two outburst model could beaugmented with either additional recent outbursts, or with a centrally concentrated popu-lation of hundreds of unresolved millisecond pulsars. Although there exist constraints thatchallenge the viability of each of these scenarios, neither can be entirely excluded at thistime. In summary , we find that scenarios involving a series of leptonic cosmic-ray outburstsfrom the Galactic Center (perhaps augmented by a centrally concentrated population ofmillisecond pulsars) could potentially generate gamma-ray emission with a spectrum andmorphology that is able to account for the GeV excess observed by the
Fermi gamma-raytelescope. The approximately uniform spectral shape exhibited by the GeV excess is notgenerally expected from scenarios featuring multiple cosmic ray outbursts, however, and thisfeature of the observed excess can be accommodated only if the parameters of the model areselected rather carefully. If variations in the spectrum of the GeV excess across the InnerGalaxy were to be observed in the future, it would provide support for interpretations in-volving a series of cosmic-ray outbursts. Similarly, multiple outburst models do not generallyproduce a gamma-ray signal with radial morphology that follows a power-law (as e.g. pre-dicted by dark matter annihilation). The robust characterization of the latitude profile ofthe observed excess is thus crucial to discriminate between different possible origins.Although a series of leptonic outbursts as an explanation for the full GeV excess cannotbe ruled out observationally, none of the characteristic features of such bursts have beenobserved up to now. In contrast, the required injection and propagation parameters need totake extreme values and to be finely adjusted to reproduce all observational aspects of theexcess, making this scenario observational viable, but unlikely.
Acknowledgments.
We would like to thank John Beacom, Gianfranco Bertone, SeraMarkoff and Andrew Taylor for fruitful discussions. IC is supported by the US Depart-ment of Energy, and would like to thank the Korean Institute for Advanced Study (KIAS)for their hospitality during the progression of this work. CE acknowledges support fromthe “Helmholtz Alliance for Astroparticle Physics HA”, funded by the Initiative and Net-working Fund of the Helmholtz Association. FC is supported by the European ResearchCouncil through the ERC starting grant WIMPs Kairos, P.I. G. Bertone. TL is supportedby the National Aeronautics and Space Administration through Einstein Postdoctoral Fel-lowship Award No. PF3-140110. CW is P.I. of the VIDI research programme “Probing theGenesis of Dark Matter”, which is financed by the Netherlands Organisation for ScientificResearch (NWO). DH is supported by the US Department of Energy under contract DE-FG02-13ER41958. Fermilab is operated by Fermi Research Alliance, LLC, under ContractNo. DE- AC02-07CH11359 with the US Department of Energy. This work has made useof SciPy [88], PyFITS , PyMinuit , IPython [89], and HEALPix [90]. We acknowledge theUniversity of Chicago Research Computing Center for providing support for this work. http://code.google.com/p/pyminuit – 20 – Description of the two outburst models considered in this study
In this appendix, we describe the different parameter values we have considered regardingcosmic-ray diffusion, convection, diffusive re-acceleration, magnetic fields, and the interstellarradiation field. As the conditions within the Inner Galaxy need not resemble those of our localenvironment (which are constrained by measurements of cosmic-ray secondary-to-primaryratios), we consider models that span a rather wide range of assumptions.Considering many sets of values for the injection and propagation parameters allows usto accomplish two things. First, this procedure allows us to identify the range of models thatprovides the best fit to the data (this is how we identified Model C, as discussed in the maintext). Second, it helps us to address the question of how generic various features are for thegamma-ray emission from cosmic-ray outbursts.In figure 7, we show the gamma-ray emission predicted in models with two cosmic-rayelectron outbursts (a 0.1 Myr and a 1 Myr old), for 35 different propagation models, eachdescribed in table 3. For each of these propagation models, we have allowed the injection pa-rameters to float freely to their best-fit parameters, exploring approximately 3000 parametersets in total. From this figure, we see that while many of these propagation models dramat-ically fail to accommodate the observed features of the GeV excess, there exist several thatprovide a reasonably good fit to the data. This is confirmed in the last column of table 3,where we list the p -values for each best-fit two outburst model. References [1] L. Goodenough and D. Hooper,
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I 3.0 0.5 37.3 0 11.7 10 2.0 1.0, 1.0 0.065II 3.0 0.3 37.3 0 11.7 10 2.0 1.0, 1.0 0.084III 5.0 0.3 37.3 0 11.7 10 2.0 1.0, 1.0 0.082IV 9.0 0.3 37.3 0 11.7 10 2.0 1.0, 1.0 0.062V 9.0 0.3 37.3 0 54.7 5 1.0 1.0, 1.0 9 × − VI 30.0 0.3 37.3 0 54.7 5 1.0 1.5, 1.5 1.4 × − VII 30.0 0.5 37.3 0 109 5 1.0 1.0, 1.0 < − VIII 30.0 0.3 37.3 1000 109 5 1.0 1.0, 1.0 < − IX 30.0 0.3 37.3 0 11.7 10 2.0 1.0, 1.0 0.026X 30.0 0.5 37.3 0 11.7 10 2.0 1.0, 1.0 0.0082XI 30.0 0.5 37.3 0 23.4 10 1.0 1.0, 1.0 0.011XII 100.0 0.5 37.3 0 23.4 10 1.0 1.0, 1.0 0.0008XIII 100.0 0.5 37.3 0 11.7 10 1.0 1.0, 1.0 0.0002XIV 20.0 0.4 37.3 0 11.7 10 0.5 1.0, 1.0 0.017XV 20.0 0.45 50. 0 11.7 10 0.5 1.0, 1.0 0.015XVI 20.0 0.45 80. 0 11.7 10 0.5 1.0, 1.0 0.015XVII 20.0 0.3/0.5 37.3 0 11.7 10 0.5 1.0, 1.0 0.012XVIII 9.0 0.3 150. 0 11.7 10 0.5 1.2, 0.8 0.049XIX 15.0 0.3 200. 0 11.7 10 0.5 1.0, 1.0 0.016XX (C) 9.0 0.3 150. 0 11.7 10 0.5 1.8, 0.8 0.14XXI 9.0 0.3 200. 0 11.7 10 0.5 1.5, 1.5 0.083XXII 9.0 0.3 150. 0 11.7 10 0.5 3.0, 1.0 0.0002XXIII 9.0 0.3 150. 0 11.7 10 0.5 0.5, 0.5 0.0005XXIV 9.0 0.3 150. 0 5.8 10 0.5 0.3, 1.0 5 × − XXV 9.0 0.3 20. 0 11.7 10 0.5 1.0, 1.0 0.021XXVI 9.0 0.3 0. 0 11.7 10 0.5 1.0, 1.0 0.021XXVII 30.0 0.3 37.3 100 54.7 5 1.0 1.0, 1.0 0.078XXVIII 30.0 0.3 37.3 500 54.7 5 1.0 2.0, 1.0 0.025XXIX 30.0 0.3 150. 500 54.7 5 1.0 1.0, 1.0 0.043XXX 30.0 0.3 37.3 300 54.7 5 1.0 1.8, 0.8 0.057XXXI 9.0 0.3 37.3 0 209 2.5 1.0 1.5, 1.5 < − XXXII 9.0 0.4 37.3 0 11.7 10 1.0 1.0, 1.0 0.030XXXIII 9.0 0.3 37.3 0 74.9 2.5 1.0 1.0, 1.0 8 × − XXXIV 9.0 0.3 37.3 0 74.9 2.5 1.0 1.5, 1.5 5 × − XXXV 9.0 0.3 37.3 0 209 2.5 1.0 1.0, 1.0 < − Table 3 . The properties of the propagation models considered for the scenarios with two cosmicray outbursts. The two numbers in the “ISRF” column denote the normalizations of the optical andinfrared components of the interstellar radiation field, in units relative to the model given in Ref. [81].Finally, the “ p best ” column lists the highest p -value acquired after fitting the injected spectral indices,spectral cutoffs, and normalizations of the two outbursts (0.1 Myr and 1 Myr old) to the data. Notethat the Model XX in this table is identical to Model C discussed throughout the main body of thetext.[12] J. F. Navarro, C. S. Frenk, and S. D. M. White, The Structure of Cold Dark Matter Halos , Astrophys. J. (1996) 563–575, [ astro-ph/9508025 ].[13] J. F. Navarro, C. S. Frenk, and S. D. M. White,
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