Anisotropy Constraints on Millisecond Pulsars in the Diffuse Gamma Ray Background

Pulsars emerge in the Fermi era as a sizable population of gamma-ray sources. Millisecond pulsars (MSPs) constitute an older subpopulation whose sky distribution extends to high Galactic latitudes, and it has been suggested that unresolved members of
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    a  r   X   i  v  :   1   0   1   1 .   5   5   0   1  v   2   [  a  s   t  r  o  -  p   h .   H   E   ]   8   J  u  n   2   0   1   1 Mon. Not. R. Astron. Soc.  000 , 1–10 (2010) Printed 9 June 2011 (MN L A TEX style file v2.2) Anisotropies in the gamma-ray sky from millisecondpulsars Jennifer M. Siegal-Gaskins 1 ⋆ , Rebecca Reesman 1 , Vasiliki Pavlidou 2 † ,Stefano Profumo 3 , and Terry P. Walker 1 1 Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210 USA 2 Astronomy Department, California Institute of Technology, Pasadena, CA 91125 USA 3 Department of Physics and Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064 USA Accepted Received ; in srcinal form ABSTRACT Pulsars emerge in the  Fermi   era as a sizable population of gamma-raysources. Millisec-ond pulsars (MSPs) constitute an older subpopulation whose sky distribution extendsto high Galactic latitudes, and it has been suggested that unresolved members of thisclass may contribute a significant fraction of the measured large-scale isotropic gamma-ray background (IGRB). We investigate the possible energy-dependent contribution of unresolved MSPs to the anisotropy of the  Fermi  -measured IGRB. For observationally-motivated MSP population models, we show that the preliminary  Fermi   anisotropymeasurement places an interesting constraint on the abundance of MSPs in the Galaxyand the typical MSP flux, about an order of magnitude stronger than constraints onthis population derived from the intensity of the IGRB alone. We also examine thepossibility of a MSP component in the IGRB mimicking a dark matter signal inanisotropy-based searches, and conclude that the energy dependence of an anisotropysignature would distinguish MSPs from all but very light dark matter candidates. Key words:  gamma-rays: diffuse background; pulsars: general; methods: statistical 1 INTRODUCTION In the era of precision gamma-ray astronomy, with data of unprecedented quality from the Fermi Large Area Telescope( Fermi  -LAT, Atwood et al. 2009) and ground-based Atmo-spheric Cherenkov Telescopes, including H.E.S.S., VERI-TAS, and MAGIC, long-standing questions about the high-energy universe might soon be successfully addressed. Oneof these is the detailed nature and origin of the diffusegamma-ray emission. The gamma-ray sky is dominated atlow Galactic latitudes by a bright diffuse Galactic compo-nent, stemming dominantly from processes involving cos-mic rays such as inelastic hadronic collisions producing neu-tral pions, and inverse Compton and bremsstrahlung emis-sion from relativistic cosmic-ray electrons and positrons(see, e.g., Strong et al. 2000). At high latitudes, the dif-fuse gamma-ray background is customarily attributed toextragalactic gamma-ray emitters, such as blazars (e.g.,Stecker & Salamon 1996). Recent  Fermi  -LAT results, how-ever, indicate that resolved blazars only contribute a smallfraction of the observed emission (Abdo et al. 2009b; how-ever, see also Abazajian et al. 2010), in contrast to, e.g., ⋆ E-mail: †  Einstein (GLAST) Fellow the diffuse X-ray background (Brandt & Hasinger 2005;Hickox & Markevitch 2006, 2007). Since the discovery of periodic gamma-ray emis-sion from pulsars (Browning et al. 1971), the possibil-ity that this source class contributes non-negligibly tothe diffuse gamma-ray emission has been considered(Bhattacharya & Srinivasan 1991; Bailes & Kniffen 1992; Bhatia et al. 1997). Some of the brightest emitters in the Fermi  -LAT gamma-ray sky are in fact associated with pul-sating objects, often corresponding to pulsars observed atradio and X-ray frequencies (Abdo et al. 2009c). Comparedto its predecessor EGRET,  Fermi   is shedding light not onlyon young, powerful “ordinary” pulsars (with typical rota-tion periods of the order of 0.01-1 sec and ages ranging be-tween 10 3 and 10 6 yr) but also on a distinct class of periodicgamma-ray emitters with much shorter pulsating periods(on the order of a few milliseconds), i.e. millisecond pulsars(MSPs). The characteristic age  τ  c  of MSPs, extrapolatedfrom their period and period-derivative, indicates that theseobjects are much older than ordinary pulsars, with  τ  c  ∼ 10 10 yr (Abdo et al. 2009a). MSPs are thought to be associatedwith binary systems, the spin-up of the pulsar period beingfueled by accretion of mass and angular momentum from theneutron star companion (Phinney & Kulkarni 1994; Lorimer  2  Siegal-Gaskins et al. 2001). While the determination of the age of MSPs is a de-bated matter given the highly non-trivial nature of theirevolutionary history (see, e.g., Kiziltan & Thorsett 2010),the significantly longer lifetime of these objects comparedto that of ordinary pulsars might offset a birthrate that isnecessarily lower (given the binary nature of MSPs), and asa result the MSP contribution to the Galactic gamma-rayluminosity may not be small compared to that of ordinarypulsars.Despite the dramatic increase in the number of de-tected gamma-ray pulsars in the  Fermi   era, the bulk of thepulsar contribution to the gamma-ray sky very likely srci-nates from a large population of unresolved sources. For in-stance, Faucher-Giguere & Loeb (2010, hereafter F-GL10) examined models for the unresolved MSP population andfound that in some optimistic models the MSP contributionto the diffuse background could even be dominant at certainenergies. In their “viable model” the small set of MSPs de-tected by  Fermi   imply almost 50k unresolved MSPs. Thegamma-ray emission from ordinary pulsars is very likelyconfined to rather low latitudes (see, e.g., Harding 1981;Bhattacharya & Srinivasan 1991), reflecting the fact thatpulsars are born in the Galactic disk, and that ordinarypulsars are relatively young objects. On the other hand, theproduct of typical pulsar kick velocities and the character-istic age of MSPs implies a length-scale that is much largerthan the thickness of the Galactic plane, suggesting thatMSPs should have a broad latitudinal distribution. This isreflected in the observed latitudinal distribution of ordinaryversus millisecond pulsars detected by the  Fermi  -LAT (seeFig. 1 in Abdo et al. 2010a). In this respect, MSPs can con-tribute to the diffuse gamma-ray emission at high latitudeswhere the Galactic diffuse component is generally thought tobe comparable or sub-dominant with respect to an isotropicextragalactic background.Interestingly, however, measurements of the spectrumof the large-scale isotropic diffuse gamma-ray background(hereafter IGRB) by  Fermi   find that it is consistent witha power law at energies between 250 MeV and 50 GeV(Abdo et al. 2010b), while MSP spectra exhibit a strongcut-off feature at typical energies of a few GeV (Abdo et al.2009a). This implies that either the MSP contribution is sub-dominant with respect to the primary IGRB component atthese energies, or that a complicated combination of sev-eral components with peculiar spectral features – e.g., astar-forming galaxy component with a feature at ∼ 300 MeV(e.g., Fields et al. 2010), a MSP component with a featureat a few GeV, and a hard blazar component dominating athigher energies – combine in such a way that they appearas an overall almost featureless power-law – a contrived sce-nario, but one that cannot be excluded in principle. In eithercase, it appears that it will be difficult to detect spectrallythe presence of a MSP component in the IGRB, althoughit remains possible to put conservative constraints on theunresolved MSP gamma-ray emission based on IGRB mea-surements (see, e.g., F-GL10).A powerful tool to investigate the nature of diffuseemission is to explore the intensity variation of the emis-sion in the sky, e.g., via the calculation of an angularpower spectrum of anisotropies. Recent theoretical workhas generated predictions for the angular power spec-trum of the gamma-ray emission originating from sev-eral known and proposed source classes. These includeconfirmed extragalactic gamma-ray populations such asAGN (Ando et al. 2007; Miniati et al. 2007) and star- forming galaxies (Ando & Pavlidou 2009), as well as darkmatter annihilation and decay in extragalactic structures(Ando & Komatsu 2006; Miniati et al. 2007; Ando et al. 2007; Cuoco et al. 2008; Taoso et al. 2009; Fornasa et al. 2009; Ibarra et al. 2009; Zavala et al. 2010; Cuoco et al. 2010). In addition, since the distribution of dark mattersubhalos in our Galaxy is quite radially extended, gamma-ray emission from annihilation and decay in Galactic sub-structure appears remarkably isotropic on large angularscales, although the clustering of dark matter in subhalosleads to small-scale anisotropies. Consequently, these struc-tures may provide a substantial contribution to anisotropiesin the IGRB (Siegal-Gaskins 2008; Fornasa et al. 2009; Ibarra et al. 2009; Ando 2009). The combined use of spectral and anisotropy in-formation in the IGRB (the anisotropy energy spec-trum) could conceivably help reveal the presence of even a subdominant component in the diffuse emission(Siegal-Gaskins & Pavlidou 2009). In particular, it has beenshown that the anisotropy energy spectrum could be a sensi-tive probe of the presence of a dark matter component in theIGRB (Hensley et al. 2010; Cuoco et al. 2010). This tech- nique is also promising for detecting a subdominant MSPcontribution to the IGRB, since the emission from unre-solved MSPs is expected to feature much stronger anisotropythan the extragalactic component, due to the fact that MSPsare relatively few and nearby, compared to cosmological pop-ulations that may constitute the dominant contributors tothe IGRB intensity.Additional motivation to study the gamma-rayanisotropy properties of MSPs is provided by the poten-tial interference of MSPs with anisotropy-based searches fordark matter.  Fermi   data (Abdo et al. 2009a) indicate thatthe typical gamma-ray MSP spectrum is, in fact, uncom-fortably similar in its overall features to what is expectedfor the annihilation or decay of certain particle dark mattercandidates, especially if the dark matter is light ( m DM   few tens of GeV). Furthermore, although the amplitude of anisotropies from dark matter annihilation is uncertain, insome scenarios it is expected to be quite large, and thusit is conceivable that a MSP-induced modulation in theanisotropy energy spectrum of the IGRB could be confusedwith a similar modulation induced by dark matter.In this paper, we explore the potential of an angularpower spectrum measurement of the IGRB to probe theproperties of the Galactic MSP population. We demonstratethe power of this approach for an example class of MSPpopulation models by deriving constraints on those mod-els from the  Fermi   preliminary anisotropy measurement(Siegal-Gaskins et al. 2010; see also Vargas et al. 2010). The model prescriptions we adopt to describe the intensityand sky distribution of unresolved MSPs are summarized in § 2, and our procedure for generating simulated maps of theMSP gamma-ray emission is outlined in  § 3. In  § 4 we calcu-late the intensity spectrum and energy-dependent angularpower spectrum of the collective unresolved MSP emissionfor this class of models and discuss those properties in thecontext of other relevant source classes, including dark mat-ter. We compare the predicted anisotropy from MSPs to the  Millisecond pulsar gamma-ray anisotropy   3 preliminary  Fermi   measurement of the angular power spec-trum of the IGRB and obtain constraints on the propertiesof the MSP population in  § 5. We discuss our findings andconclude in  § 6. 2 MODELING THE MSP POPULATION The properties of the MSP population that affect the mea-sured anisotropy are the sky distribution of MSPs and theirflux distribution. The former is determined by the spatialdistribution of MSPs in the Galaxy, while the latter is de-termined, for a fixed spatial distribution, by the distribu-tion of MSP luminosities. In this study we adopt models forthe gamma-ray MSP population based on the semi-empiricalmodels of  F-GL10. We emphasize, however, that this work isa technique demonstration, and therefore its goal is to showthat MSPs could produce an observable anisotropy signalin  Fermi  -LAT data, and that an anisotropy analysis couldbe used to constrain the collective properties of the Galac-tic MSP population; not to perform a detailed study of theconsistency of a specific model with the data, nor to iden-tify which of several models is preferred by the data. Withthat purpose in mind, we fix the values of the parameterscontrolling the spatial and luminosity distributions of MSPsto those of “viable” model  MSP2 base   of  F-GL10, and dis-cuss the expected impact of variations in these parameterson our results in  § 6.We take the fiducial number of MSPs in the Galaxy N  MSP  = 49k, as in model  MSP2 base  . Since the observablesconsidered in our study (high-latitude intensity and angularpower) scale straightforwardly with  N  MSP  and the typicalflux of an individual high-latitude MSP,  F  1 , we also considerthe dependence of our results on these parameters in  § 5.Following F-GL10, we describe the MSP spatial distri-bution with a Gaussian function of radius for the surfacedensity projected on the Galactic plane, ρ ( r ) ∝ exp( − r 2 / 2 σ 2 r ) 0  < r <  100 kpc ,  (1)where  r  is the projected distance from the Galactic Centre inthe Galactic plane,  ρ ( r ) is the surface density of MSPs, and σ r , taken to be 5 kpc, characterizes the radial extent of thedistribution. The latitude distribution of MSPs is assumedto follow a simple exponential form, N  ( z  ) ∝ exp( −| z  | / | z  | ) 0  < z < ∞ ,  (2)with the scale height  | z  | = 1 kpc.Early work on gamma-ray pulsars (e.g., Arons 1996)identified the simple empirical relation  L γ   ∝   ˙ E   betweenthe pulsar’s gamma-ray luminosity  L γ   and the rate it losesrotational kinetic energy ˙ E   = 4 π 2 I  ⋆  ˙ P/P  3 , where  P   and ˙ P  are the period and time derivative of the period, respectively,and  I  ⋆  is the moment of inertia of the star. However, recentwork (see, e.g., F-GL10) has found that the luminosities of gamma-ray MSPs appear to obey the relation  L γ   ∝  ˙ E  . As inF-GL10, we define the MSP gamma-ray energy luminosity(energy per unit time) L γ   ≡ min { C   ˙ P  1 / 2 P  − 3 / 2 ,f  max γ   ˙ E  } ,  (3)where the proportionality constant  C   = 10 40 . 9 erg  s 1 / 2 and f  max γ   = 0 . 05 is the assumed maximum fraction of rotationalpower loss converted into gamma rays. The integrated pho-ton luminosity (photons per unit time)  L ph γ   above 100 MeVis obtained by assuming an energy spectrum with approx-imately equal power per decade of energy up to a cutoff energy of   E  max  ≃  3 GeV. For the model adopted in thiswork, it is notable that Eq. 3 results in the vast majority of MSPs being assigned luminosities according to the  L γ   ∝  ˙ E  relation.For the MSP population, a power-law distribution forthe rotation period  P   is assumed, N  ( P  ) ∝ P  − 2 1 . 5 ms  < P <  60000 ms ,  (4)and the magnetic field strength  B  is taken to follow a lognormal distribution, N  (log B ) ∝ exp( − (log B − log B  ) 2 / 2 σ 2log B ) ,  (5)with   log B   = 8 and  σ log B  = 0 . 2 with  B  in Gauss. Thespin-down rate ˙ P   is determined via the relation  B  = 3 . 2 × 10 19 ( P   ˙ P  ) 1 / 2 G.Departing from the F-GL10 prescription, we adopt anempirical prescription for the energy spectra of the MSPsbased on the spectra of eight MSPs detected by  Fermi  , re-ported in Abdo et al. (2009a). The differential energy spec- tra of the  Fermi  -detected MSPs are well-described by apower law truncated by an exponential cutoff,d N  d E   ∝ E  − Γ e − E/E cut ,  (6)where Γ is the spectral index and  E  cut  is the cutoff energy.We assume that each spectral parameter, Γ and  E  cut , is nor-mally distributed in the MSP population, with mean  Γ  and  E  cut  and standard deviation  σ Γ  and  σ E cut , respectively. Weuse the spectral parameters of the detected MSPs to iden-tify the maximum-likelihood values of these distribution pa-rameters, taking into account themeasurement uncertaintiesfor each pulsar. For this procedure we follow the methodol-ogy described in Venters & Pavlidou (2007), and obtain the maximum likelihood parameters [  Γ  ,σ Γ ] = [1 . 5 , 0 . 20] and[  E  cut  ,σ E cut ] = [1 . 9 GeV, 0 . 54 GeV]. It is notable that thedistributions are relatively narrow; in particular they implythat the vast majority of MSPs have cutoff energies between ∼ 1 and 3 GeV.We consider two models for the energy spectra of theMSP population. First we examine the simple case in whichevery MSP is assumed to have the same energy spectrum,which we denote the  Reference Model  . In this scenario weassign each MSP the maximum-likelihood average spectralparameters, Γ = 1 . 5 and  E  cut  = 1 . 9 GeV. We also examinethe impact on our results of allowing the spectral parametersto vary within the MSP population according to the distri-butions above, and refer to this as the  Spectral Variation Model  . 3 SIMULATIONS Monte Carlo realizations of the Galactic MSP populationwere generated by creating mock MSP catalogs with indi-vidual MSP parameters drawn from the distributions givenin § 2. To assess the statistical variation between realizations,ten Monte Carlo realizations were generated for each caseconsidered. The HEALPix package (Gorski et al. 2005) was  4  Siegal-Gaskins et al. used to generate maps of the all-sky gamma-ray intensityfrom unresolved MSPs for each mock catalog. Maps wereconstructed at HEALPix order 7 resolution which corre-sponds to a pixel size of   ∼  0 . 45 ◦ on a side. Each MSP wastaken to be a point source with no angular extent, and soits flux was assigned to a single pixel.As we are interested in the emission from unresolvedMSPs, we excluded from the sky maps the emission fromMSPs in our mock catalogs which would likely have beendetected by  Fermi  . For this purpose we assumed a flux sen-sitivity of 10 − 8 ph cm − 2 s − 1 ( E >  100 MeV; see, e.g.,Abdo et al. 2010a) and excluded individual MSPs exceed-ing this flux threshold. We note that assuming a uniformflux sensitivity for MSPs across the sky is a rough approx-imation, since point source sensitivity varies with angularposition due to exposure and foreground contamination, andalso depends on the individual source spectrum. Althoughwe emphasize that this approximation is inadequate for as-sessing individual source detectability or completeness, it issufficient for the purpose of removing bright sources whichare likely to be resolved and would otherwise bias our pre-diction for the statistical properties of the diffuse emission.Choosing to exclude these sources is conservative, since in-cluding bright MSPs would lead to a larger predicted inten-sity and a larger contribution to the anisotropy of the IGRBfrom MSPs, and as a result to stronger constraints on MSPpopulation models. With this criterion we find that  ∼ 10 of every 10k MSPs in our Reference Model are detectable overthe entire sky. Note that the parameter  N  MSP  correspondsto the total number of MSPs in the Galaxy and thereforeincludes the detectable sources, although our analysis is per-formed on maps of the unresolved sources only. 4 GAMMA-RAY EMISSION FROM THEUNRESOLVED MSP POPULATION4.1 Sky distribution of gamma rays from MSPs We examined the constraints obtainable on the MSP pop-ulation from the intensity and anisotropy properties of the Fermi- measured IGRB, so we selected high-latitude sky re-gions by excluding Galactic latitudes  | b | <  30 ◦ . This choicematches the latitude mask applied in the  Fermi   angularpower spectrum analysis (Siegal-Gaskins et al. 2010). Thelatitude dependence of the results was studied by compar-ing the results using a mask excluding | b | <  40 ◦ . The choiceto apply a very generous mask to the Galactic plane also en-ables comparison of the high-latitude contribution of MSPsto the  Fermi- measured IGRB intensity.An all-sky map of the gamma-ray intensity for one real-ization of the MSP population Reference Model defined in § 2with  N  MSP  = 49k is shown in Figure 1. Emission from in-dividual MSPs with fluxes above the detectability thresholdis not shown. MSPs outside of the latitude mask boundaries(marked by lines) are evident, implying a MSP contributionto high-latitude diffuse emission. 4.2 Intensity energy spectra The intensity energy spectrum of the gamma-ray emissionfrom MSPs outside each latitude mask for the Reference 0.1110100Energy (GeV) 1e-081e-071e-06    E    2    d   N   /   d   E   (   G  e   V  c  m   -   2   s   -   1   s  r   -   1    ) Fermi IGRBMSP Ref Model, |b|>30° MSP Ref Model, |b|>40° MSP Spectral Variation, |b|>40° Figure 2.  Average intensity energy spectra of the MSP ReferenceModel (solid black line) and Spectral Variation Model (red x’s) for | b |  >  40 ◦ . The intensity spectrum of the Spectral Variation Modeldiffers negligibly from that of the Reference Model. The averageintensity of the MSP Reference Model for  | b |  >  30 ◦ (dashed ma-genta line) is also shown. The collective high-latitude intensity of the MSPs is more than an order of magnitude smaller than the Fermi  -measured IGRB intensity (blue crosses) at all energies. Model is compared with the  Fermi- measured IGRB inten-sity spectrum (Abdo et al. 2010b) in Fig. 2. The normaliza- tion of the MSP intensity outside each mask was obtained byaveraging over 10 realizations, and the spectral parametersof each MSP in the Reference Model were fixed to the maxi-mum likelihood values. The average intensity of the emissionfrom unmasked MSPs is a factor of  ∼ 2 larger when exclud-ing only  | b |  <  30 ◦ than when excluding  | b |  <  40 ◦ , but inboth cases is more than an order of magnitude smaller thanthe IGRB at all energies. This model is consistent with themeasured IGRB, but the overall intensity does not providea meaningful constraint on the population.In the Reference Model we adopted the simplifying as-sumption that all MSPs share the same energy spectrum. Totest the validity of this assumption, in Fig. 2 we compare thecollective intensity spectrum of the MSP population, aver-aged over | b | >  40 ◦ , for the Spectral Variation Model and theReference Model. The differential intensity d N/ d E   of eachrealization of the Spectral Variation Model was obtained bygenerating a map for each logarithmic energy bin containingthe integrated intensity of each MSP, given its spectral pa-rameters, and then dividing the integrated intensity by theenergy bin size ∆ E  . The points shown represent the averageintensity outside the mask in each energy bin of 10 MonteCarlo realizations of the Spectral Variation Model. To goodapproximation, the collective intensity energy spectrum of the Spectral Variation Model matches that of the ReferenceModel, with a small deviation from the Reference Modelspectrum evident only at the highest energy bin ( E   ∼  3GeV).In Fig. 3 we compare the MSP intensity for | b | >  30 ◦ tothe Galactic diffuse emission for  | b |  >  30 ◦ (from the modelused in Cuoco et al. 2010). At these latitudes, the intensityof the Galactic diffuse emission from cosmic-ray interactionswith the interstellar gas and photon fields is comparableto that of the IGRB, and the MSP emission is subdom-inant with respect to both of these signals. However, the  Millisecond pulsar gamma-ray anisotropy   5  4 Figure 1.  MSP gamma-ray intensity integrated from 0.1 to 10 GeV for one realization of the Reference Model. The map is shownin Galactic coordinates with the boundaries of the latitude masks excluding  | b |  <  30 ◦ and  | b |  <  40 ◦ marked. For this figure the mapresolution was degraded to improve the visibility of MSPs and illustrate their sky distribution; however, all calculations were performedon the high-resolution maps as described in the text. 0.1110100Energy (GeV) 1e-081e-071e-06    E    2    d   N   /   d   E   (   G  e   V  c  m   -   2   s   -   1   s  r   -   1    ) Fermi IGRBGalactic Diffuse, |b|>30°MSP Ref Model, |b|>30°DM: 40 GeV bbDM: 8 GeV τ +  τ − Figure 3.  Average intensity spectra of the MSP Reference Modelfor  | b |  >  30 ◦ (dashed magenta line) compared with the IGRBintensity (blue crosses), the Galactic diffuse emission for  | b |  >  30 ◦ (dot-dashed red line, from Cuoco et al. 2010), and two benchmarkdark matter models. The two dark matter models correspond toan 8 GeV particle pair-annihilating preferentially into  τ  + τ  − at arate   σv   = 1  ×  10 − 26 cm 3 s − 1 (dotted yellow line), and to a 40GeV particle annihilating into  b ¯ b  with   σv   = 3  ×  10 − 26 cm 3 s − 1 (solid yellow line). Galactic diffuse emission is not expected to contribute sig-nificantly to the anisotropy on angular scales of     1 − 2 ◦ ,corresponding to multipoles  ℓ    100 (see, e.g., Cuoco et al.2010), and therefore MSPs could be a dominant contributorto the anisotropy of the high-latitude diffuse emission whileremaining a subdominant contributor to the intensity.Fig. 3 also compares the intensity spectrum of MSPs tothat of the high-latitude emission predicted for two exam-ple dark matter models, chosen because their energy spectrabear some resemblance to the collective MSP energy spec-trum. We do not resort to any specific particle physics setupin the choice of the models. Rather, we specify a dominantpair-annihilation final state, the particle mass, and the rateof pair-annihilation. One of the dark matter models corre-sponds to a dark matter particle with a mass of 8 GeV anda cross section   σv  = 1 × 10 − 26 cm 3 s − 1 , for which the dom-inant annihilation final state is a pair of   τ   leptons. Thismodel was chosen to align with that found in the analysis of Hooper & Goodenough (2010) to best fit a gamma-ray ex- cess claimed to exist in the innermost 2 degrees in the direc-tion of the Galactic Centre (see also Abazajian 2010 for aninterpretation of that signal as MSP emission). We also com-pare a second dark matter model, with a mass of 40 GeV anda pair-annihilation cross section   σv   = 3 × 10 − 26 cm 3 s − 1 ,for which the dominant annihilation final state is bottomquarks. This second model can be regarded as a prototypicallight bino-like dark matter candidate from the minimal su-persymmetric extension of the Standard Model, with a crosssection that would allow for thermal production of the cor-rect universal dark matter density. The intensity of the darkmatter emission for these two models corresponds to thatpredicted for the high-latitude signal from annihilation inGalactic dark matter subhalos in model A1 of  Ando (2009), assuming the particle properties for each model specifiedabove.Dark matter annihilation or decay in Galactic substruc-ture may generate a significant level of anisotropy in theIGRB with an energy dependence similar to that from MSPsdue to their similar energy spectra. Although the detailedshapes of the energy spectra of the dark matter modelsshown in Fig. 3 differ from that of the collective MSP emis-sion, the energy range at which both of these possible con-tributors become most prominent in the IGRB, as well astheir cutoff energies, are similar. Since an anisotropy analy-sis requires large photon statistics to robustly measure smallanisotropies, the number of energy bins in which a measure-ment can be made with  Fermi  -LAT is limited, and there-fore it may be difficult to localize features in the energydependence of the anisotropy. Consequently, there remainsthe possibility that a MSP-induced anisotropy in the IGRBcould be confused with a similar signal from dark matter an-nihilation. However, we stress that only a signal from verylight dark matter candidates is likely to exhibit a spectral
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