Strong support for the millisecond pulsar origin of the Galactic center GeV excess
aa r X i v : . [ a s t r o - ph . H E ] M a r Strong Support for the Millisecond Pulsar Origin of the Galactic Center GeV Excess
Richard Bartels, ∗ Suraj Krishnamurthy, † and Christoph Weniger ‡ GRAPPA Institute, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, Netherlands (Dated: 4 February 2016)Using γ -ray data from the Fermi
Large Area Telescope, various groups have identified a clearexcess emission in the Inner Galaxy, at energies around a few GeV. This excess resembles remarkablywell a signal from dark-matter annihilation. One of the most compelling astrophysical interpretationsis that the excess is caused by the combined effect of a previously undetected population of dim γ -ray sources. Because of their spectral similarity, the best candidates are millisecond pulsars.Here, we search for this hypothetical source population, using a novel approach based on waveletdecomposition of the γ -ray sky and the statistics of Gaussian random fields. Using almost seven yearsof Fermi -LAT data, we detect a clustering of photons as predicted for the hypothetical populationof millisecond pulsar, with a statistical significance of 10 . σ . For plausible values of the luminosityfunction, this population explains 100% of the observed excess emission. We argue that otherextragalactic or Galactic sources, a mismodeling of Galactic diffuse emission, or the thick-diskpopulation of pulsars are unlikely to account for this observation. Introduction.
Since its launch in 2008, the
Fermi
Large Area Telescope (LAT) has revolutionized our un-derstanding of the γ -ray sky. Among the major successesare the detection of more than 3000 γ -ray sources [1],the discovery of the Fermi bubbles [2], some of the moststringent limits on dark-matter annihilation [3] and, mostrecently, the detection of cross-correlations between theextragalactic γ -ray background and various galaxy cata-logs [4].One of the most interesting γ -ray signatures identifiedin the Fermi -LAT data by various groups [5–16], is anexcess emission in the Inner Galaxy at energies around afew GeV. This excess attracted great attention becauseit has properties typical for a dark-matter annihilationsignal. This Galactic center excess (GCE) is detectedboth within the inner 10 arcmin of the Galactic Center(GC) [7, 9, 10] and up to Galactic latitudes of more than10 ◦ [13, 15, 17, 18]. It features a remarkably uniformspectrum and approximately spherical symmetry [13, 15].Proposed diffuse emission mechanisms, like leptonic orhadronic outbursts [19–21] or cosmic-ray injection in thecentral molecular zone [22], potentially explain part ofthe excess emission. However, it is challenging to explain all of the above aspects of the GCE with these mecha-nisms alone.Probably the most plausible astrophysical interpreta-tion for the GCE is the combined emission from a largenumber of unresolved millisecond pulsars (MSPs) in theGalactic bulge region [10, 12, 23, 24]. MSPs feature aspectrum compatible with the GCE emission [15], anda large unresolved component can naturally explain theuniformity of the GCE spectrum in different regions ofthe sky. Recently, it was shown that the spatial distribu-tion of MSPs that were spilled out of disrupted globularclusters can explain the morphology of the GCE [25].Such MSPs from disrupted globular clusters have alsobeen suggested as the source behind the GeV throughTeV emission in the inner few parsec of the GC [26]. Fur- ther possible support for the MSP hypothesis might comefrom Chandra observations of low-mass x-ray binaries(which are progenitor systems of MSPs) in M31, whichshow a centrally peaked profile in the inner 2 kpc [27, 28],as well as the recent observation of extended hard X-rayemission from the Galactic Center by
NuSTAR [29].It was claimed that an interpretation of 100% ofthe GCE emission in terms of MSPs would be alreadyruled out: a sizeable fraction of the required 10 –10 MSPs should have been already detected by the
Fermi -LAT [30, 31], but no (isolated) MSP has been identifiedso far in the bulge region. This conclusion depends cru-cially, however, on the adopted γ -ray luminosity of thebrightest MSPs in the bulge population, on the effectivesource sensitivity of Fermi -LAT, and on the treatment ofunassociated sources in the Inner Galaxy [25, 32]. A real-istic sensitivity study for MSPs in the context of the GeVexcess, taking into account all these effects, was lackingin the literature up to now (but see Ref. [33]).In this Letter, we close this gap and present a noveltechnique for the analysis of dim γ -ray sources and ap-ply it to Fermi -LAT observations of the Inner Galaxy.Our method is based on the statistics of maxima in thewavelet-transformed γ -ray sky (in context of Fermi -LATdata, wavelet transforms were used previously for theidentification of point source seeds [1, 16]). We searchfor contributions from a large number of dim MSP-likesources, assuming that they are spatially distributed assuggested by GCE observations. Our method has sev-eral advantages with respect to previously proposed tech-niques based on one-point fluctuations [34], most notablythe independence from Galactic diffuse emission modelsand the ability for candidate source localization.
Modeling.
We simulate a population of MSP-likesources, which we hereafter refer to simply as the cen-tral source population (CSP), distributed around the GCat 8.5 kpc distance from the Sun. The CSP is taken tohave a spatial distribution that follows a radial power lawwith an index of Γ = − . r = 3 kpc [13, 15]. As a reference γ -ray energy spec-trum, we adopt the stacked MSP spectrum from Ref. [35], dNdE ∝ e − E/ .
78 GeV E − . . The γ -ray luminosity func-tion is modeled with a power law, dNdL ∝ L − α , with index α = − . L min = 10 erg s − and L max = 10 –10 erg s − ,respectively. Luminosities are integrated over 0.1–100GeV. Our results depend little on L min . Given thatonly about 70 MSPs have been detected in γ rays upto now [33], L max is not well constrained. The γ -ray lu-minosity of the brightest observed MSP is somewhere inthe range (0 . × erg s − [33, 35], depending onthe adopted source distance [25, 32]. Diffuse emission ismodeled with the standard model for point source ana-lysis gll iem v06.fits and the corresponding isotropicbackground. Data.
For our analysis, we use almost seven years ofultraclean
Fermi -LAT P8R2 data taken between August4 2008 and June 3 2015 (we find similar results for sourceclass data). We select both front- and back convertedevents in the energy range 1–4 GeV, which covers thepeak of the GCE spectrum. The region of interest (ROI)covers the Inner Galaxy and spans Galactic longitudes | ℓ | ≤ ◦ and latitudes 2 ◦ ≤ | b | ≤ ◦ . The data arebinned in Cartesian coordinates with a pixel size of 0 . ◦ . Wavelet peaks.
The wavelet transform of the γ -raydata is defined as the convolution of the photon countmap, C (Ω), with the wavelet kernel, W (Ω), F W [ C ](Ω) ≡ Z d Ω ′ W (Ω − Ω ′ ) C (Ω ′ ) , (1)where Ω denotes Galactic coordinates [38] [note that R d Ω W (Ω) = 0]. The central observable for the currentanalysis is the signal-to-noise ratio (SNR) of the wavelettransform, which we define as S (Ω) ≡ F W [ C ](Ω) p F W [ C ](Ω) , (2)where in the denominator the wavelet kernel is squaredbefore performing the convolution. If the γ -ray flux var-ied only on scales much larger than the extent of thewavelet kernel, and in the limit of a large number ofphotons, S (Ω) would behave like a smoothed Gaussianrandom field. Consequentially, S (Ω) can be loosely in-terpreted as the local significance for having a source atposition Ω in units of standard deviations.As the wavelet kernel, we adopt the second memberof the mexican hat wavelet family, which was shown toprovide very good source discrimination power [39] andwhich was used for the identification of compact sourcesin Planck data [40]. The wavelet can be obtained bya successive application of the Laplacian operator to atwo-dimensional Gaussian distribution with width σ b R . − − ℓ , Gal. longitude [deg] − − b , G a l.l a t i t ud e [ d e g ] − − FIG. 1. SNR of the wavelet transform of γ rays with energiesin the range 1–4 GeV, S (Ω). The black circles show the po-sition of wavelet peaks with S ≥
2; the red circles show theposition of third Fermi-LAT catalog (3FGL) sources. In bothcases, the circle area scales with the significance of the sourcedetection in that energy range. The dashed lines indicate theregions that we use for the binned likelihood analysis, wherelatitudes | b | < ◦ are excluded because of the strong emis-sion from the Galactic disk. The subset of 3FGL sources thatremains unmasked in our analysis is indicated by the greencrosses. Here, σ b = 0 . ◦ corresponds to the Fermi -LAT angu-lar resolution at 1–4 GeV, and R is a tuning parameter.We find best results when R varies linearly with latitudefrom R = 0 .
53 at b = 0 ◦ to R = 0 .
83 at b = ± ◦ . Thiscompensates to some degree the increasing diffuse back-grounds towards the Galactic disk, while optimizing thesource sensitivity at higher latitudes [40].The resulting SNR of the wavelet transform S (Ω) isshown in Fig. 1. As expected, the Galactic diffuse emis-sion is almost completely filtered out by the wavelettransform, whereas bright sources lead to pronouncedpeaks. We adopt a simple algorithm for peak identifi-cation: we find all pixels in S (Ω) with values larger thanin the four adjacent pixels. We then clean these resultsfrom artifacts by forming clusters of peaks with cophe-netic distances less than 0 . ◦ , and only keep the mostsignificant peak in each cluster.In Fig. 1, we show the identified wavelet peaks withpeak significance S >
2, as well as all 3FGL sources forcomparison [1]. For sources that are bright enough inthe adopted energy range, we find a good correspondencebetween wavelet peaks and the 3FGL, both in terms ofposition and significance (we compare the significance ofwavelet peaks S with the 1–3 GeV detection significancefor sources).It is worth emphasizing that for the adopted spheri-cally symmetric and centrally peaked distribution of theCSP, most of the sources would be detected not directlyat the GC but a few degrees away from the Galactic disk.This is simply due to the much weaker diffuse emissionat higher latitudes. Our focus on latitudes | b | ≥ ◦ , thus,avoids regions where source detection becomes less effi-cient, due to strong diffuse foregrounds, without signifi-cant sensitivity loss for the source population of interest. Before studying the statistics of thewavelet peaks in detail below, we remove almost all peaksthat correspond to the known 3FGL sources based on a0 . ◦ (1 ◦ for √ T S ≥
50) proximity cut. However, in orderto mitigate a potential bias on L max , we do not maskpeaks that correspond to 3FGL sources that are likelypart of the CSP. We identify such MSP candidate sources by requiring that they (i) are tagged as unassociated,(ii) show no indication for variability and (iii) have aspectrum compatible with MSPs. The last criterion istested by performing a χ fit of the above MSP referencespectrum to the spectrum given in the 3FGL (0.1–100GeV; five energy bins). Only the normalization is leftfree to vary. We require a fit quality of χ / DOF ≤ . p value ≥ . | ℓ | = 12 ◦ –60 ◦ is significantlysmaller, with an average of 3 .
1. It is tempting to inter-pret this excess of MSP candidate sources in the InnerGalaxy as being caused by the brightest sources of theCSP, above the less-pronounced thick-disk population ofMSPs [41, 42]. However, we emphasize that the statusof these 13 sources is currently neither clear nor quali-tatively decisive for our results. Whether we mask themplays a minor role in the detection of the CSP below (butit does affect the inferred values for L max ; see Supplemen-tal Material). Statistical analysis.
In Fig. 2 we show a histogramof the wavelet peaks in our ROI. We bin the peaks ina two-dimensional grid, which spans the projected anglefrom the Galactic Center 2 ◦ –17 ◦ and wavelet peak signif-icances in the range 1–10. The bin edges are as indicatedin the figure. As expected, photon shot noise gives riseto a large number of peaks with low significances S ≤
S ≥ ◦ – 5 ◦ ◦ – 8 ◦ ◦ – 11 ◦ ◦ – 14 ◦ ◦ – 17 ◦ Spatial and significance bins10 − N u m b e r o f p e a k s i n S ( Ω ) FIG. 2. Histogram of observed peaks in S (Ω), in bins ofprojected radial distance from the GC and SNR values (blackdots with statistical error bars). We show the expectationvalue for the case of a negligible CSP as blue bars, whereasthe expectation values for the best fit are shown in red. Poisson likelihood analysis of the wavelet peak distribu-tion. The likelihood function is given by L = n r Y i =1 n s Y j =1 P ( c ij | µ ij ( L max , Φ )) , (3)where n r and n s are, respectively, the numbers of radialand peak SNR bins, c ij is the observed and µ ij the ex-pected number of peaks, and P is a Poisson distribution.The expectation values depend directly on the maximalluminosity, L max , as well as on the number of simulatedsources, n . To ease comparison with the literature, wedetermine n as a function of Φ , which denotes the meandifferential intensity of the CSP at b = ± ◦ , ℓ = 0 ◦ , and2 GeV. In the case of the GCE, this value was found tobe Φ GCE5 = (8 . ± . × − GeV − cm − s − sr − at95 .
4% C.L. [15].
Results.
In Fig. 2, we show the expectation valuesthat we obtain when neglecting contributions from theCSP (and any other nondiffuse emission). This corre-sponds to good approximation to the case where the GCEis of truly diffuse origin, including the case of DM anni-hilation or outburst events. We find that the observednumber of wavelet peaks with S < S > Maximum γ -ray luminosity, L max [erg s − ]10 − − − I n t e n s i t y , Φ [ G e V − c m − s − s r − ] O b s . M S P s i nd i s k GCE intensity dNdL | L ≤ L max ∝ L − . Inner Galaxywavelet analysis
FIG. 3. Constraints on the maximum γ -ray luminosity of theCSP, L max , and the population averaged intensity at b = ± ◦ , ℓ = 0 ◦ , and 2 GeV, Φ , as derived from our wavelet analysis.We show 68 . . .
7% C.L. contours. We alsoindicate the values of Φ where the source population canexplain 100% of the GCE (horizontal gray band, 95 .
4% CL),and as vertical orange lines the luminosity of the brightestobserved nearby MSPs. nonzero contribution from the CSP is favored at thelevel of at least 10 . σ (when quoting the statistical sig-nificance, we conservatively take into account bins with S < =(9 . ± . × − GeV − cm − s − sr − and for the max-imum luminosity L max = (7 . ± . × erg s − . Ascan be seen in Fig. 2, we obtain in this case a very goodfit to the data.Our preferred range of the maximum γ -ray luminosi-ties reaches up to L max ≤ . × erg s − (at 95 . γ -rayluminosity of the brightest individually observed nearbyMSPs as given in Ref. [35] (we only show objects wheresecond γ -ray pulsar catalog [33] distances are available;see Ref. [25] for a detailed discussion about distance un-certainties). Furthermore, for the adopted slope of theluminosity function, α = 1 .
5, the best-fit value for thetotal differential intensity of the CSP Φ is consistentwith the CSP accounting for 100% of the GCE emission. Discussion and conclusions.
We found corroboratingevidence for the hypothesis that the GCE is caused bya hitherto undetected population of MSP-like sources.We performed a wavelet transform of the γ -ray emissionfrom the Inner Galaxy, which removes Galactic diffuseemission and enhances point sources, and we studied the statistics of the peaks in this transform. We detectedwith 10 . σ significance a suppression (enhancement) oflow- (high-) significance wavelet peaks, relative to theexpectations for purely diffuse emission. We showed thatthis effect is caused by the presence of a large numberof dim point sources. The spatial distribution of waveletpeaks in the Inner Galaxy is compatible with a centrallypeaked source distribution, and the inferred cutoff of the γ -ray luminosity function of these sources agrees with theobservation of nearby MSPs. This source population can,for reasonable slopes of the luminosity function, accountfor 100% of the GCE emission.For the purpose of this Letter, which introduces a noveltechnique, we kept our analysis as simple as possible.In general, one might worry that our results could beaffected by the presence of extragalactic and Galacticsources, by the thick-disk population of MSPs and youngpulsars, by the details of masking and unmasking 3FGLsources, by the details of the adopted γ -ray luminosityfunction, and by unmodeled substructure in the Galac-tic diffuse emission that is not removed by the wavelettransform. We address all of these points in the Supple-mental Material and show that it is rather unlikely thatthey affect our results qualitatively, although quantita-tive changes in the obtained best-fit values for Φ and L max are possible. In particular, we showed that thewavelet signal expected from the thick-disk populationof MSPs is an order of magnitude weaker than what weactually observed and that interpretations related to un-modeled gas remain on closer inspection unlikely.The prospects for fully establishing the MSP interpre-tation within the coming decade are very good. Our re-sults suggested that upcoming γ -ray observations withimproved angular resolution (planned or proposed γ -ray satellites like GAMMA-400 [43], ASTROGAM, andPANGU [44]) will allow us to detect many more of thebulge sources and study their distribution and spectra.For current radio instruments, it remains rather chal-lenging to detect a MSP population in the bulge [25],but prospects for next-generation instruments are good.We thank for useful discussions with David Berge,Francesca Calore, Ilias Cholis, Jennifer Gaskins, DanHooper, Simona Murgia, Tracy Slatyer, Ben Safdi andJacco Vink. We thank the Fermi
Collaboration for pro-viding the public
Fermi data as well as the
Fermi
Sci-ence Tools ( v10r0p5 ). SURFsara is thanked for use ofthe Lisa Compute Cluster. S.K. and C.W. are part ofthe VIDI research programme “Probing the Genesis ofDark Matter”, which is financed by the Netherlands Or-ganisation for Scientific Research (NWO). R.B. is partof a GRAPPA-Ph.D. program funded by NWO. Partof this work was performed at the Aspen Center forPhysics, which is supported by National Science Foun-dation Grant No. PHY-1066293.
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SUPPLEMENTAL MATERIAL
In the Supplemental Material we discuss the possible impact of various systematic effects on our results. Thisincludes a control region analysis, a discussion of various types of γ -ray sources, substructure in diffuse emission, athick-disk population of MSPs, sphericity and the role of negative wavelet peaks. A. Null results in control regions
In order to estimate the effect of various systematic uncertainties, it is useful to apply our analysis on controlregions along the Galactic disk (in the case of Galactic diffuse emission this was first systematically done in Ref. [15]).Potentially unresolved substructure in the Galactic diffuse emission (e.g. in the form of giant molecular clouds, seebelow), and contributions from various Galactic and extragalactic source populations could be responsible for thedetected wavelet signal in the inner Galaxy, but would in general also affect other regions in the Galactic disk. Tothis end, we focus on (partially overlapping) control regions along the Galactic disk, which are of the same size as theinner Galaxy ROI, but displaced by ∆ ℓ = ± k ◦ with k = 1 , , , − − − − T S v a l u e < S < < S < < S < FIG. S-1. We show, as function of the central longitudinal position of the control regions (for ℓ = 0 ◦ the main ROI), thesignificance of a CSP detection. We assume that the CSP is centered in each of the ROIs, and we refit the parameters L max and the number of sources. We indicate how different ranges of the wavelet peak significances contribute to the detection. All
In Fig. S-1 we show the
T S value for a detection of the CSP for the main and the different control ROIs along theGalactic disk. We leave L max and Φ free to vary in each region independently. In the main ROI that covers theinner Galaxy we find the significant detection of a CSP that was discussed in the main text. As shown in the plot,this high significance is supported by the low, intermediate and high-significance SNR peaks of the wavelet transformseparately. The directly adjacent regions also show relatively large T S values, which is either caused by the partialoverlap of these control regions with the main ROI, or by a CSP that is more disk-like than assumed in our analysis.We will address the latter point below. However, in the outermost six control regions we find no significant detectionof a CSP, for any of the considered values for L max and Φ (the large T S values at ℓ = 80 ◦ are caused by one extremelybright source that generates fake peaks in its tails). This observation makes it already extremely unlikely that ourfindings are driven by a mismodelling of the local Galactic diffuse emission, or by extragalactic sources. We willaddress this in more detail below. Maximum γ -ray luminosity, L max [erg s − ]10 − − I n t e n s i t y , Φ [ G e V − c m − s − s r − ] GCE intensity1 < S < < S < < S < < S < FIG. S-2. Similar to Fig. 3 in the main text, but showing limits derived for different ranges of the wavelet peak significancesseparately. The parameter degeneracies present in some of the fits cancel when fitting the entire significance range at once. Weonly show 68 .
7% and 95 .
4% CL constraints for clarity. Maximum γ -ray luminosity, L max [erg s − ]10 − − I n t e n s i t y , Φ [ G e V − c m − s − s r − ] GCE intensity ψ < ◦ ◦ < ψ < ◦ ◦ < ψ < ◦ ◦ < ψ < ◦ ◦ < ψ FIG. S-3. Similar to Fig. 3 in the main text, but showing limits as derived for different spatial bins separately.
B. Consistent wavelet signal in separate bins
It is instructive to see how wavelet peaks with different significances contribute to the constraints on the luminosityfunction that we showed in Fig. 3 in the main text. To this end, we show in Fig. S-2 the limits that we obtainseparately from peak significances in the range S = 1–3, S = 3–5 and S = 5–10, respectively. All three constraintsare mutually consistent to within 1 σ , leading to a consistent interpretation of the peaks shown in Fig. 1 in the maintext. In all cases we find some degeneracy in the ( L max , Φ ) plane. High significance peaks predominantly provide astringent upper limit on L max , whereas low significance peaks mostly constrain the overall luminosity of the modelledsource population.To show that our assumption on the spatial distribution of the CSP is consistent with the data, we show Fig. S-3 Maximum γ -ray luminosity, L max [erg s − ]10 − − − I n t e n s i t y , Φ [ G e V − c m − s − s r − ] GCE intensity dNdL | L ≤ L max ∝ L − . dNdL | L ≤ L max ∝ L − . dNdL | L ≤ L max ∝ L − . FIG. S-4. Similar to Fig. 3 in the main text. We show the 68 .
7% and 95 .
4% CL contours for luminosity functions with spectralindices of 1 .
2, 1 . . the result obtained for the five different spatial bins independently. Ring 1–5 correspond to r ∈ [ i ◦ , i + 3 ◦ ] with i = 2 , , , ,
14, respectively. We find constraints on L max and Φ that are mostly consistent to within 1 σ .Finally, we checked that the identified wavelet peaks are symmetrically distributed in the north, south, east andwest parts of our main ROI. Only at S >
C. Mild dependence on the MSP luminosity function
Theoretically, α is not well constrained and can plausibly range from 1 . α ∼ . .
7% and 95 .
4% CL contours for different luminosity functions, respectively with spectral indices of 1 . .
7. For a fixed intensity (Φ ), hardening (softening) of the luminosity function corresponds to an enhancement(suppression) of the number of sub-threshold point sources, which explains the direction in which the best fit regionmoves. We note that we obtain very similar T S values for all slopes that we considered.
D. The role of unmasked 3FGL sources
In the present analysis, we make use of the 3FGL, the third
Fermi source catalogue, which is based on the firstfour years of
Fermi pass 7 data. One important ingredient in our analysis is the masking of 3FGL sources. Thesesources are of Galactic and extragalactic origin and leaving them unmasked would inevitably induce sizeable signalsin our search for a sub-threshold source population in the bulge. However, as discussed in the main text, we keepunassociated sources with MSP-like spectra unmasked. These sources could be part of the bulge MSP populationthat we are looking for, and masking them would bias our results. The 13 sources that pass our MSP cuts are listedin Tab. I.In general, falsely masking unassociated sources that actually belong to the bulge MSP population would push L max to lower values, whereas falsely unmasking foreground sources would push it to large values. This is illustratedin Fig. S-5. We show the case where we mask all unassociated sources, as well as the case where we adopt a weakercriterion for the spectral fit, leaving around 20 sources unmasked. We find that in both cases the best-fit value for L max moves in the expected direction, but the results remain consistent to within 1 σ . Furthermore, the significanceof our wavelet detection that we quote in the main text (where we include S < . σ when we0 ℓ [ ◦ ] b [ ◦ ] χ / dof √ T S S L [10 erg / s]J1649.6-3007 -7.99 9.27 1.07 5.6 7.4 7 . +5 . − . J1703.6-2850 -5.08 7.65 0.48 2.4 5.0 3 . +2 . − . J1740.5-2642 1.30 2.12 0.37 6.4 3.5 14 . +10 . − . J1740.8-1933 7.43 5.83 0.77 1.9 2.3 3 . +2 . − . J1744.8-1557 11.03 6.88 0.40 3.7 3.4 5 . +3 . − . J1758.8-4108 -9.21 -8.48 0.90 5.6 3.8 4 . +3 . − . J1759.2-3848 -7.11 -7.43 0.35 4.6 5.6 5 . +4 . − . J1808.3-3357 -1.94 -6.71 0.40 6.9 6.3 8 . +5 . − . J1808.4-3519 -3.15 -7.36 0.41 4.6 4.4 5 . +3 . − . J1808.4-3703 -4.68 -8.19 0.22 4.9 5.3 4 . +3 . − . J1820.4-3217 0.74 -8.17 1.04 5.7 1.7 7 . +5 . − . J1830.8-3136 2.35 -9.84 0.54 5.9 6.0 5 . +3 . − . J1837.3-2403 9.85 -7.81 0.28 4.0 3.0 4 . +3 . − . TABLE I. List of the 13 unassociated 3FGL sources with MSP-like spectra, which we leave unmasked in our analysis. If theGeV excess is caused by dim point sources, it is likely that some or most of them are part of the CSP. The last four columnsshow the goodness-of-fit of the reference MSP spectrum, the 3FGL significance in the 1–3 GeV band, the corresponding peakof the wavelet SNR, and the γ -ray luminosity (assuming 8 . ± Maximum γ -ray luminosity, L max [erg s − ]10 − − − I n t e n s i t y , Φ [ G e V − c m − s − s r − ] GCE intensity StandardAll 3FGL masked p < .
05 masked
FIG. S-5. Similar to Fig. 3 in the main text. We show the effect of masking all mask all sources, and to 10 . σ when keeping 20 sources unmasked. The (un-)masking of 3FGL is hence not decisivefor our qualitative findings, although quantitative results can be affected.It is interesting to note that the faintest 3FGL source in the inner Galaxy ROI that passes our MSP-spectrum cut,has a luminosity of L = 3 . × erg s − if placed at 8.5 kpc distance. This is a good, though rough, indication forthe de facto sensitivity threshold of Fermi -LAT for the detection of sources with MSP-like spectra in the bulge region.Lastly, one can use the sources in Tab. I to compare the sensitivity of the 3FGL with our wavelet analysis. Averagingover the 13 sources, we find a ratio of S / √ T S ≃ . ± .
4, indicating that the sensitivity of the wavelet method issimilar to the 3FGL sensitivity in a comparable energy range. However, the scatter exceeds the one expected fromstatistical fluctuations alone, which can be attributed to differences in the systematics that affect the 3FGL and thewavelet analysis.1 − − − N u m b e r o f F G L s o u r ce s Dotted: allSolid: p MSP ≥ . p PSR ≥ . FIG. S-6. Latitude profile of sources, for different ROIs along the Galactic disk, as function of their longitudinal center (dottedlines). The ROIs have a size of 24 ◦ × ◦ , with the Galactic disk, | b | < ◦ , masked. The different colors correspond to differentsource categories. In the case of unassociated sources, the solid (dashed) lines show only sources that pass the spectrum cutfor MSP-like (young pulsar-like) sources. N u m b e r o f F G L s o u r ce s UnassociatedPSRGLCSNR & PWNExtragalactic
FIG. S-7. Stacked histogram of wavelet peak values S that correspond to 3FGL sources in the inner Galaxy ROI. We show thecontribution from different source categories separately. Unassociated sources generate mostly low-significance wavelet peaks,whereas for example source that are marked as pulsar in the 3FGL only generate peaks with a high significance (see discussion). E. The role of various source populations
In Fig. S-6, we show the number of 3FGL sources in our inner Galaxy ROI as well as in various same-sized ROIsthat are displaced along the Galactic disk in steps of ∆ ℓ = ± ◦ . We show (identified and associated) extragalacticsources, various Galactic source classes and unassociated sources classes separately. We also show for the unassociatedsourced how many sources pass our MSP cut. It is apparent that the number of unassociated sources strongly peaksin the inner Galaxy ROI, however with a clear asymmetry towards negative values of ℓ , and another peak around ℓ ≈ ◦ . However, after applying the MSP cut, predominantly sources in the inner Galaxy survive. In Fig. S-7, on2the other hand, we show a histogram of the wavelet peak significance that our analysis attributes to the 3FGL sourcesin the inner Galaxy ROI. Again, unassociated sources play a major role and produce wavelet peaks down to values of S ∼
1. On the other hand, 3FGL sources that are identified as pulsars appear only with S > Extragalactic sources.
As shown in Fig. S-6, the number of extragalactic sources in the individual ROIs along theGalactic disk fluctuates around values of about ∼
13. No significant suppression is observed in the inner Galaxy ROI,which would have indicated that it is more challenging to identify or associate extragalactic sources in this region.We will use here a simple argument to show that extragalactic sources cannot play a significant role for our results.The average number of S = 3–5 wavelet peaks in the above control regions along the Galactic disk is 20; in the innerGalaxy ROI it is 42 (we exclude here all > σ upward fluctuation above theexpected 20. This makes it extremely unlikely that extragalactic sources contribute significantly to our results in theinner Galaxy. Similar arguments can be made using wavelet peaks in the range S = 1–2. Supernova remnants and pulsar wind nebulae.
As apparent in Fig. S-6, only a very small number of sources alongthe Galactic disk at latitudes | b | > ◦ (and almost none at | b | > ◦ ) are identified with supernova remnants or pulsarwind nebulae. In our ROIs their number is much less than the number of extragalactic sources, and their distributionis centrally not peaked, indicating that sources at these latitudes are mostly local. Sources in this category aretypically more easily detectable at higher and lower energies than the energy range used in our analysis, and wouldbe most likely listed in the 3FGL and hence masked if they were abundant and significant. We consider it hence asextremely unlikely that sources of this category significantly affect our results in the inner Galaxy. Young and millisecond pulsars.
An interesting feature of pulsars in the 3FGL is that they always induce largewavelet signals in our analysis, as shown in Fig. S-7. This makes indeed sense, since the identification of a γ -ray sourceas pulsar requires the measurement of its pulsation, and hence a large enough number of photons. Furthermore, thepulsar energy spectrum often peaks close the energy range of our analysis. Already from Fig. S-7 it is obvious thata large fraction of the unassociated sources, which mostly appear with lower significance peaks, must in fact bepulsars. Interestingly, as shown in Fig. S-6, identified pulsars do not show a centrally peaked distribution, whereasthe unassociated sources clearly do. This behaviour is expected, given that with increasing distance the pulsaridentification becomes more challenging. Globular clusters.
The γ -ray emission from globular clusters that is not masked in our analysis, either becausethe globular clusters did not enter the 3FGL, or because they happen to be among our 13 unmasked unassociatedsources, could in principle contribute to the detected signal. Since their total emission is usually due to several MSPswhich appear as a single source for Fermi -LAT, their presence could bias L max towards larger values. However, giventhe simulation results from Ref. [25], we expect this effect to be small, and leave a more detailed discussion to futurework. Unassociated sources.
The peak of unassociated sources in the inner Galaxy, as shown in Fig. S-6, appears clearlyasymmetric, with a second peak at ℓ ≈ ◦ . As discussed above, a large fraction of these sources is expected tobe young or millisecond pulsars. Obviously, the 3FGL unassociated sources do not directly contribute to our resultssince they are masked (except the 13 MSP candidates). However, the unassociated sources are extremely abundanteven down to S ∼
1, and their probable nature can be used as an indicator for what source population dominatesjust below threshold.In Fig. S-6, we see that our MSP cut removes most of the unassociated sources, but leaves an excess of 13 unassoci-ated sources in the inner Galaxy, as discussed above. What is more, if we slightly modify the spectral criterion, using dN/dE ∝ e − E/ E − , which is somewhat more pulsar-like (softer index, lower cutoff), this behaviour changes andwe instead find excesses that are more correlated with the peaks of the unassociated sources away from the innerGalaxy. Although the statistical significance of this finding is rather difficult to quantify without a detailed study(which we leave for future work), this result is indicative. It suggests that a large fraction of the inner Galaxy unas-sociated (and sub-threshold) sources are likely MSPs, whereas unassociated sources in other parts of the disk have alarger fraction of young pulsars. The latter point is further supported by the fact that similar structures can be foundin the longitudinal distribution of identified pulsars.In summary, we expect that our wavelet signal is dominated by whatever source class is responsible for most of theunassociated sources towards the inner Galaxy. Very likely, these are millisecond and young pulsars, with a somewhathigher MSP/young pulsar ratio than in the rest of the disk. Since these sources appear in general both in the Galacticdisk as well as in the bulge, it is important to study whether the excess/suppression of wavelet peaks in the innerGalaxy points to a disk population, a bulge population, or to a combination of both. This will be discussed in the3 − − ℓ , Gal. longitude [deg] − − b , G a l.l a t i t ud e [ d e g ] − . − . − . − . − . . . . . . . FIG. S-8. Similar to Fig. 1 in the main text (note the different color scale), but showing the transform of the diffuse BG modelonly, without Poisson noise being applied. Outside of the Galactic disk, | b | > ◦ , which we exclude from our analysis, thesignificance of wavelet peaks remains below 0 .
5. The variance is below 0 .
1, which shows that even 1 σ peaks in the wavelettransform are unlikely to be strongly affected by the Galactic diffuse emission. section G below. F. Possible caveats concerning the Galactic diffuse emission
In our Monte Carlo studies, we use the standard
Fermi diffuse model for pass 8 data analysis. The wavelettransform of this model, without applying Poisson noise , but using the same exposure as in our main analysis, isshown in Fig. S-8. Outside of the masked Galactic disk at | b | > ◦ , we do not find any excesses with a significancelarger than about 0 . σ . The main effect of such variations would be to offset the significance of random statisticalfluctuations and sub-threshold point sources towards higher or lower values. This would not significantly affect peakswith a large SNR. However, it can potentially be important for low-significance peaks, because the collective shift ofa large number of peaks, even by a small amount, could become statistically relevant. But since the variance of theSNR values shown in Fig. S-8 is below 0.1, we do not expect that the details of the modeling of diffuse emission whendoing MCs is going to affect our results.The Fermi diffuse models might not actually contain all relevant small-scale gas structures, and the effect of thesemissing structures on our results is not straightforward to estimate without a detailed analysis and modeling of thepower-spectrum of gas at small scales. It is hence rather important that our non-detection of strong wavelet signalsalong the Galactic disk in Fig. S-1 largely excludes that mismodeling of local gas is the cause for the detected signaltowards the inner Galaxy, since it would affect other parts of the disk as well. This is in particular true since there isrelatively little molecular gas in our main ROI, compared to the control regions [46]. Thus, gas-related effects shouldbe larger in the control regions than in the main ROI.If one insists on a gas-related interpretation, our results hence suggest that the wavelet signals are caused byunmodeled gas in the Galactic bulge, at a height of 0.3–1 . γ -ray emissivity at 1 GeV is around 3 × − s − GeV − per hydrogenatom [47]. This implies that dense gas clouds with masses around 3 × M ⊙ would be at 1 GeV roughly as brightas MSPs with a luminosity of L = 7 × erg s − .[48] Interestingly, giant molecular clouds are known objects ofthat mass, and they can be dense enough to appear at GC distance point-like for Fermi . However, the scale heightof known giant molecular clouds is at the level of a few 10 pc (they usually intersect with the Galactic disk) insteadof the required ∼ O (10–100) K km s − , which is not seen in current observations [46]. If observed, such CO emission should be distributed4 − − − − T S v a l u e Solid: thick-disk MSP incl.Dotted: thick-disk MSP × < S < < S < < S < FIG. S-9. Similar to Fig. S-1, but including a thick-disk MSP population calibrated to local bright high-latitude MSPs asadditional background. Only for illustration, we also show the effect of a 10 × more dense thick-disk population (which is incontradiction with local observations). Note that we unmask 3FGL as described in the main analysis when deriving the waveletpeaks. north/south symmetric, as our wavelet peaks are too.If one could show that a large number of such giant molecular clouds (or other structures with similar mass anddensity) can form and be transported to kpc heights in the Galactic bulge, while hiding from all observations, theinterpretation of the identified wavelet peaks in terms of unmodeled gas would remain a possibility. However, asof now, and for all of the above reasons, we regard gas-related interpretations of our results as rather unlikely andspeculative. G. Potential impact of thick-disk population
As argued above, the most relevant Galactic background in our ROI are expected to be pulsars, and in particularthe MSP thick-disk population that reaches up to high latitudes. We will now show that a thick-disk population ofMSPs (or other sources with a similar luminosity function) cannot be responsible for the observed signal.In most cases, the thick-disk population of MSPs is modeled as a cylindrically symmetric exponential distribution,with a scale height in the range 0.5–1 kpc and a scale radius of a few kpc, which is only poorly constrained bydata (see e.g.
Ref. [41]). We will adopt here a distribution with a scale height of 1 kpc and a scale radius of 5 kpc,which was previously used to argue against the MSP-origin of the
Fermi
GeV excess [31]. The distribution reads n ∝ exp ( − R/R s ) exp ( − | z | /z s ), with R s = 5 kpc and z s = 1 kpc. We will address below how the results change whenother parametrizations are adopted.As γ -ray luminosity function, we adopt an inverse power-law with L min = 10 erg s − , L max = 7 × erg s − andindex α = 1 .
5. We fix the overall normalization of the disk source density such that the number of bright MSPs athigh latitudes, | b | > ◦ , is consistent with the number of such MSPs listed in the 3FGL. As flux threshold for brightMSPs we adopt a flux that corresponds to a γ -ray luminosity of 10 erg s − at 3 kpc distance (9 . × − erg s − cm − in our energy range). We find 31 MSPs above that threshold flux and note that since the number of unassociatedhigh-latitude bright nonvariable sources with a curved enough spectrum is small, this number cannot increase by morethan 50% when more unassociated sources are identified as MSPs) [1]. For the present scenario, we find that the totalnumber of thick-disk sources with γ -ray luminosity above 10 erg s − is ∼ erg s − ). This implies, as already argued in Ref. [31], that a thick-disk population with the adoptedgeometry cannot be responsible for the Fermi
GeV excess. However, it also trivially implies that the number of5wavelet peaks caused by thick-disk sources in the inner 2 kpc is about an order of magnitude below what is predictedby our best-fit bulge population, and hence an order of magnitude below what is actually observed. This still leavesthe possibility that thick-disk MSPs on the line-of-sight towards the inner Galaxy, outside of the inner ∼ . . × − erg s − cm − (this corresponds roughly to (4–7) × erg s − when the sources are put at 8 . < S < γ -ray emission similar to the level of the Fermi
GeV excess, just with a morphology that is incompatible with theobservations. A population with a scale radius of 1–2 kpc would indeed commonly be referred to as bulge population.Such a population would be very similar to the bulge population that we put forward in the main part of the paperas explanation for the
Fermi
GeV excess, with the main difference being that our population fits better the excessmorphology.Second, one could increase the number of MSPs in a ring-like region around the Galactic bulge, excluding the inner2 kpc, such that these additional ring-like distributed sources will enhance the number of foreground sources withoutaffecting the number of sources in the Galactic bulge. In this case, however, the wavelet signal should clearly be moreextended along the Galactic disk than what is shown in Fig. S-1, since such a ring would not be centrally peakedand extend to longitudes of at least ∼ ◦ . For illustration, we here quote the relative number of wavelet peaks oneexpects in the control regions along the disk and the main ROI produces by such a ring (1 kpc scale height, 5 kpcscale radius, the inner 2 kpc radius excluded): ∆ ℓ = {± , ± , ± , ± , } and N peaks ∝ { . , . , . , . , . } .Moreover, in order to avoid a conflict with the above calibration with bright high-latitude sources, the ring shouldbe further constrained to lie within . × denser population)is added as an additional background component. For simplicity, we assume that the thick-disk population causesdeviations of the expectation values µ ij in Eq. (3) from the null hypothesis that are proportional to the deviationscaused by the best-fit bulge population. We adjust the normalization of these deviations such that the number ofadditionally predicted 3 < S < H. Further discussions