A DM Signal from Unbiased and Anti-Biased Distributions of Subhalos in the Milky Way

The 111 and 129 GeV -ray lines from annihilations in the Milky Way dark matter halo, dark disk and subhalos

Abstract

Recently a series of indications have been put forward suggesting the presence of two -ray lines at 110-130 GeV (centered at 111 and 129 GeV). Signals of these lines have been observed toward the Galactic center, at some galaxy clusters and among some of the unassociated point sources of the 2 years Fermi catalogue. Such a combination of signals could be generated by dark matter annihilations in the main dark matter halo, its substructures and nearby galaxy clusters. We discuss in this work the consistency between the number of events observed at the line energies in the sky and the predictions using results from the Via Lactea II numerical simulation and extrapolations below its mass resolution, taking into account that the annihilation cross-section to the lines can be estimated from the Galactic center signal. We find that some extrapolations to small substructures can naturally account for the point sources signal, although the hypothesis of background only cannot be rejected. We also study the morphology of the -ray sky at the 2 lines energies, testing different Galactic diffuse background models to account for interstellar medium uncertainties and different assumptions on the DM diffuse component profile. We find from template fits that within reasonable diffuse background uncertainties the presence of a spherical halo component is preferred with cuspier dark matter halo profiles being preferable even from the full sky fit. We finally check the impact of a dark disk component suggested by cosmological simulations that include baryons and find that thin dark disks can not be disfavored, thus possibly accounting for the preferentially closer to the Galactic disk distribution of the point sources lines signal.

FERMILAB-PUB-13-080-A-PPD

I Introduction

The possible identification by the Fermi Gamma-ray Space Telescope of a signal compatible with the monochromatic photon emission due to pair annihilations of cold dark matter (DM) particles has recently been one of the most debated topics. Originally, Bringmann et al. (2012); Weniger (2012) suggested the detection of a line at GeV with a 3.3 significance Weniger (2012) in a wide window toward the Galactic center (GC). A similar signal has been indicated by Su and Finkbeiner (2012a) at GeV with a 5.0 significance. A pair of lines with energies of and GeV can alternatively explain -ray excess at 5.4 significance Su and Finkbeiner (2012a). Similarly the line signal at 130 GeV has also been found by Tempel et al. (2012) with Rajaraman et al. (2012); Buchmuller and Garny (2012); Boyarsky et al. (2012) suggesting the presence of 2 lines at 110 and 130 GeV. Both line signals are in agreement with constraints from line searches of the Fermi collaboration Ackermann et al. (2012) and indicate a preference for dark matter (DM) annihilation rather than decay Weniger (2012); Su and Finkbeiner (2012a); Kuhlen et al. (2012); Rao and Whiteson (2012) (for a recent review on DM line searches read Bringmann and Weniger (2012)). The Fermi collaboration, motivated by the results of Bringmann et al. (2012); Weniger (2012); Su and Finkbeiner (2012a); Tempel et al. (2012); Rajaraman et al. (2012); Buchmuller and Garny (2012); Boyarsky et al. (2012), has performed an analysis oriented toward the GC and some of those results are presented in Charles et al. (2012); Albert et al. (2012). Similar to Finkbeiner et al. (2012); Whiteson (2012), no obvious systematic error has been found to account for the amplitude of the line signal measured by Weniger (2012); Su and Finkbeiner (2012a) (see though Whiteson (2013)). The Fermi results confirm a line-like signal at E130 GeV at , or E135 GeV at 3.3 significance after reprocessing the data to take into account the shift of the reconstructed energy with time. Yet, some part of the amplitude may be related to limb photons Charles et al. (2012). Adding information on the performance of the instrument’s energy reconstruction decreases the significance of the signal Albert et al. (2012). Thus a conclusive answer on whether the line signal is a systematic error identified toward the GC or a signal of DM annihilations has not been provided yet.

A monochromatic gamma-ray flux is expected in most scenarios in which DM is in the form of weakly interacting massive particles (WIMPs) since two-body annihilation final states containing photons arise at the 1-loop level. At the same time, in such a framework, it is foreseen that tree-level WIMP annihilations into other SM final states, in turn hadronizing and/or decaying into , , s and s, would give sizable yields with continuum energy spectrum on top of the monochromatic -ray yield. Yet, no clear indication of -ray excess, other than the lines, has been found toward the inner few degrees of the GC, leading to the extraction of constraints on the continuum Buchmuller and Garny (2012); Buckley and Hooper (2012); Cohen et al. (2012); Cholis et al. (2012a); and thus motivating further discussions for the particle physics origin of the lines Buckley and Hooper (2012); Weiner and Yavin (2013); Fan and Reece (2012).

Another aspect of a WIMP annihilation signal is that one should expect to see line signals at the same energy and annihilation cross-section toward other dark matter targets. In Hektor et al. (2012a), a 130 GeV line signal toward known galaxy clusters has been suggested. There have been also indications for two lines at 111 and 129 GeV in unassociated point sources which would imply DM annihilating in substructures Su and Finkbeiner (2012b); Hektor et al. (2012b) 1 (see also Hooper and Linden (2012) for an alternative interpretation).

We assume that the line signal from the unassociated point sources is indeed of DM origin, with the same annihilation cross-section to the lines as is estimated from the GC. We then confront that signal with predictions from cosmological simulations such as Via Lactea Diemand et al. (2007a, b); Kuhlen et al. (2008). In section II, we discuss the -ray data that we use and our general assumptions for the background and the DM density distribution in the Galaxy. As a general reference we use the total number of -ray events with energy GeV and GeV. These events are taken to be of both DM and diffuse/point source -ray background origin. The comparison of the observed -ray data with the predicted contribution from substructures within the Galaxy’s virial radius is done in section III. Our aim is to conclude on whether the line signal at the point sources can be physically associated to the same energy line signal toward the GC and under what assumptions on the substructure distribution. On the CDM simulation side we use VLII subhalo distribution data Kuhlen et al. (2008) and also extrapolate the VLII simulation mass function to smaller subhalo masses.

The observed non-isotropic distribution in the sky of the DM line(s) signal associated to the point sources, could be explained by the presence of a strong dark disk. Such a dark disk would also have an impact on the diffuse distribution of the DM originated line photons. Independently, in the context of self-interacting DM, the formation of dark disks has been suggested to explain the relatively large amplitude of the line signal in the inner kpc of the Galaxy Fan et al. (2013a, b). To study the diffuse -ray sky at the energies of the 2 lines, one needs predictions on both the Galactic diffuse backgrounds and on the DM diffuse contribution. In section IV, we test that possible contribution to the two -ray lines from DM annihilations in the Galaxy and from the diffuse backgrounds by doing a template fit. The importance of template fits is that one can take into account the different morphologies of the various diffuse components. We study the impact on the significance of a DM signal on the 4 sky (including the GC) of different assumptions for the Galactic diffuse background, related to physical properties of the interstellar medium. We also test different assumptions (and thus different templates) on the main spherical halo density profile, on the significance of the dark disk component to the local dark matter density and its thickness and finally on the contribution of the dimmer DM subhalos that would also add to the diffuse -ray sky flux. We consider that the main contribution from the brighter DM structures has been already observed by Su and Finkbeiner (2012b) and exclude them from the -ray fits. We also derive upper limits on the diffuse emission from annihilations in the main DM halo, and give our conclusions in section V.

Ii Gamma-Ray data, Diffuse Background and DM distribution Assumptions

The Fermi Large Area Telescope publicly available events are categorized in different classes based on the expected level of cosmic ray (CR) contamination. In this work we use the ULTRACLEAN events class which is the cleanest -ray events sample. There are 686(744) photons with energy between GeV and 611(668) photons with energy between GeV in the 4 yr (4.4 yr) full sky Fermi ULTRACLEAN class -ray data, with the quoted energy ranges representing the relevant energy dispersion for these lines 2. In section III where we compare with the findings of Su and Finkbeiner (2012b), we use the 4yr sample since it approximates their events sample, while in section IV we use the slightly longer period of 4.4 yr .

The emission of the diffuse Galactic -ray background above 100 GeV is dominated by the contribution, i.e. the decay of mesons produced by inelastic collisions of CRs with the interstellar medium (ISM) gas and by the up-scattering of low energy photons of the interstellar radiation field (ISRF) from high energy CR electrons (inverse Compton scattering). The morphology of these components on the sky is different mainly because of the different distributions of the ISM gas density and the ISRF energy density in the Galaxy. Moreover the energy loss of CR electrons and protons during their propagation in the Galaxy is different. The bremsstrahlung radiation off CR electrons at these energies is completely subdominant but is included in our code.

To compute the diffuse -ray background, we use the DRAGON package Evoli et al. (2008); http://www.desy.de/ maccione/DRAGON/ (); Cholis et al. (2012b) with a new ISM gas model Tavakoli (2012) that ensures good agreement with -ray spectral data between 1 and 200 GeV in the full sky and subsections of it Tavakoli et al. (2013). We ignore the contribution of the ”dark gas” (not related to DM substructures) whose uncertainties are significant in the inner in latitude Grenier et al. (2005); Collaboration (2012). Based on the relevant uncertainties (see Tavakoli et al. (2013)), we allow for different assumptions on the ISM gas and the ISRF which influence the and the inverse Compton -ray emissivities respectively.

In the case where there are 2 lines as has been indicated by Rajaraman et al. (2012); Su and Finkbeiner (2012a), the energy of these lines is centered at and GeV Su and Finkbeiner (2012a). The lines come from either the combination of lines or from the lines.

In Cholis et al. (2012a), five individual modes/channels of DM annihilation: , , , and have been studied. The limits on the DM annihilation cross-sections based on their contribution to the continuum -rays spectrum in the , observation window have been derived. Typically, DM models have sizable branching ratios into more than one of these channels. Yet apart from the channel and mainly the channel, in all the other annihilation channels to SM particles with a continuum spectrum, the -ray DM signal at 111 and 129 GeV can not be explained/mimicked by the continuum spectrum. Thus it originates from the annihilation into and . For and , the final state radiation (FSR) and virtual internal bremsstrahlung (VIB) can contribute to the line signal as discussed in Bringmann et al. (2012); Cholis et al. (2012a).

For simplicity we assume that the DM induced -rays with energy GeV and GeV come from the annihilation of a 129 GeV DM particle into and respectively. Alternatively, these -ray lines could come from and for the case of a 142 GeV DM particle. The relevant ratio of the luminosity of two lines is taken to be 1/2 for the 111/129 GeV lines as suggested in Su and Finkbeiner (2012b), thus for the case of 129 GeV DM particle, the annihilation cross-sections to and are assumed to be the same.

For the DM distribution we assume that it is a combination of a spherically symmetric ”main” DM halo and a dark disk (DD). For the main halo we assume a spherical Einasto DM profile:

(1)

using Merritt et al. (2005) with kpc. The values of result in a more (less) cuspy DM distribution. The density normalization parameter is set in terms of the local DM density, after including a contribution of the DD.

The profile of the DD component is assumed to be described by Read et al. (2008):

(2)

where and are the half mass scale lengths in the Galactic plane and perpendicular to the Galactic plane, respectively and kpc. Here is the cylindrical radial coordinate.

The ratio of the local DM density in the dark disk to the local DM density in the spherical halo typically ranges between 0.2-1.5 Read et al. (2008), with the higher ratios being related to higher mass densities in the thick stellar disk rather than in the thin stellar disk. The thick stellar disk can be populated by thin stellar disk stars, if the thin stellar disk gets heated by very massive, high-redshift mergers. Another cause could be multiple pro-grate and low inclination mergers Read et al. (2008).

In the template analysis performed below we will restrict to the case:

(3)

fixing Catena and Ullio (2010); Salucci et al. (2010):

(4)

Regarding the dark disc thickness, some authors Kalberla et al. (2007) have suggested thicker disks, while thinner and less significant dark disks can also be the case. Keeping in the parametrization of eq. 2 fixed Read et al. (2008), we will test the half mass scale length values of 0.5, 1.0, 1.5, 3.0 kpc.

In the standard model for cosmology, cold DM structures form hierarchically, with small DM halos collapsing first and subsequently merging into larger and larger objects. Since tidal disruption may only be partially effective, massive DM halos, such as the halo of our own Galaxy, are expected to contain a vast population of subhalos, with mass spanning from a tiny seed mass up to a fraction of the hosting halo mass. The minimum mass is essentially associated to the free-streaming scale of DM particles, in turn depending on their temperature of kinetic decoupling in the early Universe. For WIMPs the minimum mass can be as small as about Green et al. (2004); Profumo et al. (2006), much lighter than the dwarf galaxy scale, possibly to the smallest environment which can host stellar populations and hence a luminous counterpart. Because of the highly non-linear nature of the merging process, up to now the only efficient technique to model in detail DM halos is the use of numerical N-body simulations with large populations of substructures found in such studies. We will assume as primary reference in our analysis results from Via Lactea II (VLII)Diemand et al. (2008), one of the highest resolution simulations up to date of a Milky Way-sized CDM halo (virial mass ), with over one billion DM ”particles” and nominal mass resolution of about 4100 (numerical effects appear to enter well above this scale, possibly affecting the subhalo mass spectrum up to about ). In our analysis we will discuss both the DM pair annihilation associated to individual DM substructures as well as the collective effect from the whole subhalo population. In both respects, the resolution of the simulations appears insufficient to properly model the expected signals. Our approach will then be to use the simulation results to properly calibrate the necessary extrapolations to smaller masses: tuning, at a given Galactocentric radius, the subhalo pericenter distribution and applying a recipe for taking into account tidal stripping effects. We derive a model which reproduces fairly well the subhalo mass function and the distribution in halo concentration as a function of radius in the VLII simulation, and we use it as a prediction below its resolution (some details about our approach are given in Appendix A).

The general trends in the DM subhalo distribution can be understood from the fact that more massive objects are more prone to tidal stripping than the less massive ones, because they typically have smaller average density, reflecting the fact that they collapsed later in the cosmic history at a lower averaged background density. As a result, when going toward the center of the host halo, the average subhalo density increases and the average mass decreases. Also, in the inner part of a DM halo the tidal forces become stronger, possibly making the subhalos spatial distribution to be anti-biased with respect to host density profile. This tendency has been found in some numerical simulations, see, for example Kuhlen et al. (2008); Springel et al. (2008), while it has also been claimed that unbiased or anti-biased distributions may just stem from selection effects Diemand et al. (2007b); Diemand et al. (2008); Diemand and Moore (2011); Anderson et al. (2010). The model in our extrapolation from the VLII results gives an unbiased distribution. To bracket uncertainties we consider also an anti-biased distribution which is instead taken from Kuhlen et al. (2008). The other parameters entering most critically in our analysis are the spectral index for the subhalo mass function and the minimum subhalo mass . In Fig. 1, we plot as a function of Galactocentric radius, the average DM squared density associated to the full subhalo population for two choices of the spectral index, i.e. and , three sample values of the minimum subhalo mass, and for the unbiased (solid lines) and anti-biased (dashed lines) distributions. For a comparison the density squared of the smooth DM halo component, Eq. 1, with , is also given.

Figure 1: Average DM squared density associated to subhalo population as a function of Galactocentric radius , for the case of an unbiased distribution (solid lines) and an anti-biased distribution (dashed lines) and for a few assumptions on the values for subhalo mass function and minimum subhalo mass (see the text for details). Also shown is the density squared of the smooth halo component.

Iii DM substructures in the Milky Way and the 2 -ray lines

In Su and Finkbeiner (2012b), 16 unassociated point sources have been identified with at least one 100-140 GeV photon for SOURCE event class . Among those photons, there are 9 photons in the ULTRACLEAN event class sample with energies 111 and 129 GeV. Those photons from unassociated point sources may imply a signal of DM annihilation in close by substructures. That ansatz can be compared to predictions from cosmological simulations.

The 2 yr point source catalogue Nolan et al. (2012) has 575 unassociated point sources. In Su and Finkbeiner (2012b), the 9 ULTRACLEAN photons at 111 and 129 GeV lie within a / radius for FRONT/BACK converted events covering an area of 0.07/0.28 square degrees. Thus the 575 unassociated point sources cover at most deg2 out of deg2 and would give a conservative upper estimate of 5.06 photons out of the 3. The probability that 9 or more ULTRACLEAN class photons out of the photons of the 4 sky fall within the area covered by the 575 unassociated point sources is 4. Yet there may also be a bias toward discovering point sources around single high energy events Drlica-Wagner (2012). While by itself not a strong statistical deviation, the coincidence in energy with the GC signal inclines us to test for alternative possibilities and whether some of these photons come from unassociated point sources that are DM subhalos.

The number of photons that we receive from a single subhalo with luminosity and line of sight (los) distance from us, for channel or is given by:

(5)

where is the DM particle mass and for . and are the detector’s exposure time and effective area for photon’s energy of 129 GeV, respectively. In this work, we use for the averaged Fermi-LAT exposure after years . For this section, we consider two values of annihilation rates: cm s, a value derived assuming our default smooth component DM density profile (no DD) and fitting the monochromatic signal in the region &  Cholis et al. (2012a); and cm s, which fits better the whole sky region (see section IV).

To quantify the possible impact of substructures in the line photons on the sky we ask the following questions:
A) How many subhalos give 2 or more photons in to the 2 -ray lines?
B) How many photons (in the two lines energies) do we get from all the subhalos that give a more than 0.1 photons?
The difference between the number of photons from the entire subhalo population received and the answer to question (B) is a proxy for the diffuse gamma-ray flux to the two lines from DM substructures gravitationally bound in the main DM halo. We will refer to these photons as ”DM substructure diffuse”.

In Table 1 we consider first the single subhalo sample from the VLII simulation Diemand et al. (2008) (no extrapolation below the mass resolution at this level) and compute answers to the questions formulated above, averaging over results obtained for 100 random choices for the position of the observer, all at fixed Galactocentric distance (the average is performed to wipe out fluctuations involving effects of nearby subhalos or voids in a single random choice). We then turn our analysis to the extrapolated subhalo populations focussing on the unbiased and anti-biased distribution and considering two possible extrapolations for lowest subhalo mass and subhalo mass spectral index, bracketing extreme possibilities, case I corresponding to and case II to . The number of line photons received is computed including only subhalos within the virial radius since we find that for VLII subhalos only of the total photon comes from substructures lying outside it. On the other hand there is a further uncertainty one should be careful about: by tuning our subhalo model to the VLII results we are fixing the normalization of subhalo number density (above mass resolution) for the Milky Way halo according to that specific realization. This is a quantity which actually has a certain scattering among different halo realizations and different simulations. For instance if we wanted to follow the results of the Aquarius simulation Springel et al. (2008) we should increase the normalization of the subhalo number density by about a factor of 2, shifting results in Table 1 by the same factor.

As a further test, in Table 2 we report the total number of monochromatic photons expected from our entire subhalo population under a set of different assumptions for spectral index and cutoff mass. The results are again shown for unbiased and anti-biased distributions.

Simulation Assump. Q.A Q.B
VLII 0 (0) 0.213 (0.024)
unbiased - case I 0.0198 (0.00344) 0.473 (0.0874)
unbiased - case II 0.0139 (0.0024) 0.342 (0.0618)
anti-biased - case I 0.0746 (0.0176) 1.24 (0.296)
anti-biased - case II 0.0898 (0.0196) 1.62 (0.361)
Table 1: Relevance of substructures for detection of the monochromatic photons referring to questions A and B as posed in the text, using cross section which fits GC (fits the whole sky). Answers are provided in case of the subhalo sample from the VLII simulation itself and from extrapolations of it in case of unbiased or anti-biased distributions with the parameter choice (case I) and (case II). Changing the overall normalization of the subhalo number density would shift the results provided in the table accordingly, e.g., by a factor of about 2 if adopting the normalization of the Aquarius simulation Springel et al. (2008).

If the photons from unassociated point sources are from DM annihilation in substructures, their number (9 ULTRACLEAN events) probes the number of photons from the brighter substructure subsample. Considering that the number of photons originating from subhalos that emit more than 0.1 photons in the two lines (i.e. the results of Question B in Table 1), is indicative of such number of photons, we compare the probability of having observed 9 (or more) photons for a DM signal calculated in the VLII sample (0.213 photons), in extrapolation for unbiased distribution and case I (0.473 photons), for anti-biased distribution and case I (1.24 photons), for unbiased distribution and case II (0.342 photons), and for anti-biased distribution and case II (1.62 photons). These probabilities are for the background plus the DM signal stemming from VLII sample, for the background plus the DM signal in the extrapolation for unbiased (anti-biased) distribution in case I and for background plus DM signal in the extrapolation for unbiased (anti-biased) distribution in case II. Using the normalization from Aquarius simulation would increase the number of subhalos and received photons by a factor of 2, shifting the probabilities to, respectively, and . Thus the most conservative VLII assumptions case is marginally favorable than the just background case. When extrapolating below the mass resolution, probabilities increase further, reaching relevant levels in optimistic extrapolations. On the other hand, using the cross section which fits the whole sky (see section IV), we don’t see much differences in the p-values from having just a background signal. Also, from Question A, there are no subhalos expected to give more than 2 photons. One must keep in mind that lowering the value of cross section by a factor of 3, by going from which fits GC better to the one which fits the whole sky better, does not simply reduce all the values in Table I by the same factor. This is because the photons produced by each subhalo will decrease; so that some subhalos which previously gave more than, say, 0.1 photons; will now give less. The photons coming from such subhalos are not included anymore. (Similarly with the numbers of subhalos which give more than 2 photons.) However, in Table II, all values do lower by the same factor, because they are the number of photons coming from all subhalos.

The differences between the numbers of photons that originate from all DM subhalos (Table 2) and the numbers of photons that originate from DM subhalos that contribute lines photons or more (Q.B) are conservative probes to the diffuse contribution from the DM subhalos at 111 and 129 GeV. The VLII sample gives DM substructure diffuse component photon, extrapolation for the unbiased (anti-biased) distribution in the case I () photons, while in case II () photons. An upper (rough) limit to the DM substructure diffuse 111 and 129 GeV photon component can be derived by considering it approximately isotropic and then counting the 111 and 129 GeV photons laying above times 7.46 (the ratio of 4 to the area of the sky with ). There are 40 1115 GeV and 30 1296 GeV photons above , i.e. an upper estimate of the 111 and 129 GeV photons in the isotropic component is 522 photons; thus significantly larger than the 1.0, 4.99 (2.66), or 95.7 (85.4) DM substructure diffuse component photons predictions from VLII and unbiased (anti-biased) case I and II. In section IV we have a more model-dependent estimate of the isotropic 111 and 129 GeV -rays component, which though decreases the isotropic component photons down to 190-230.

While for the anti-biased distribution we find more events from fewer sources than the unbiased distribution, since the subhalo concentration has a sharper dependence on Galactocentric radius with higher luminous subhalos closer to the GC, here the trend is reversed when summing over the whole population of dim sources.

Index ” m () unbiased anti-biased
2.0 1.0 96 87
2.0 1.0 20.8 20.4
1.9 1.0 16.3 10.2
1.9 1.0 5.46 3.90
1.9 2.0 4.02 2.99
Table 2: Number of 111 and 129 GeV lines photons contributing to the DM subhalo diffuse lines component, for various choices of subhalo distributions, using cross section which fits GC (using cross section which fits whole sky instead, will scale all values by a factor of ). We show results for different subhalo mass function spectral index and lower mass cut-off , and for unbiased and anti-biased distributions.

Iv Diffuse -ray lines emission from dark matter annihilation

As discussed in II the -ray lines centered at 111 and 129 GeV that are observed in the sky originate from combination of sources. In this part we mask out the contribution from the 16 point sources detected at Su and Finkbeiner (2012b) each with one photon at energies between 100-140 GeV. We also mask out the extended sources (galaxy clusters) where a similar excess of 100-140 GeV -rays has been observed Hektor et al. (2012a). We use a mask of in radius for each of the 16 point sources of Su and Finkbeiner (2012b) and a radius mask for the targets of Hektor et al. (2012a). The remaining contribution to the 111 and the 129 GeV lines may come from the diffuse -rays emission due to DM annihilations in the main halo and its dark disk, from background -rays produced in the Milky Way, from DM annihilation at small scale substructures in the Milky Way (that we have not yet identified as point sources) and from other isotropically distributed extragalactic astrophysical sources. DM annihilation in extragalactic structures and CR contamination will give an additional isotropic component. In Fig. 2 we show the 4 sky after implementing our mask on the 16 point sources and the 6 extended ones.

Figure 2: -ray events (ULTRACLEAN class) with energy of 1115 and 1296 GeV after 4 yrs of collection by Fermi-LAT. We mask out the 16 point sources of Su and Finkbeiner (2012b) and the 6 extended sources of Hektor et al. (2012a) (see text for more details). We present -ray events in Mollweide projection using HEALPix Gorski et al. (2005).

The DM annihilation rate in any part of the Galaxy is given by:

(6)

with the annihilation cross-section taken to be the same for both the DM particles in the dark disk, the spherical halo and the substructures. For the case of Sommerfeld enhancement these cross-sections are in general different because of the dependence of the annihilation cross-section to the velocity dispersion of dark matter Arkani-Hamed et al. (2009); Lattanzi and Silk (2009); Hisano et al. (2004, 2005) and the fact that for the DM particles in the DD the dispersion is suppressed by a factor of 5-6 compared to that in the spherical halo components Read et al. (2008); with subhalos having even lower velocity dispersions. Thus for Sommerfeld enhanced models the dark disk contribution to CRs and -rays can be much more significant Cholis and Goodenough (2010) (see also Slatyer et al. (2012) for a discussion on the impact of subhalos).

In fitting to the 2 -ray line full sky data we probe the prompt -ray DM annihilation component of the spectrum which is directly related to the annihilation rate in eq. 6.

Figure 3: The -ray templates as can be used in eq. 8. Top left: the Galactic diffuse background template including the , inverse Compton scattering and bremsstrahlung components at energies of 1115 and 1296 GeV. We show the ”Back A” model. Top right: the diffuse DM spherical halo component assuming an Einasto with profile. Bottom left: diffuse emission from the combined spherical and dark disk DM distributions, assuming a maximal dark disk (), which results in showing the ”SphDM”+ ”DarkDisk” + ”MixedDM” combined template. We use Einasto profile with for the spherical and kpc for the scale hight of the dark disk. Bottom right: the DM diffuse subhalo template ”SubDM” for unbiased distribution with together with the spherical DM halo ”SphDM” template (Einasto with ) . We use Mollweide projection. To demonstrate the different morphologies, each template is normalized to 1 at the GC. Dark blue (dark grey) color refers to a flux suppressed by a factor of compared to the GC in each template. In eq. 8 we use the calculated flux values from the DRAGON package.

We use the masked full sky data with energies GeV and GeV. We perform a maximum likelihood fit calculating the log-likelihood based on Dobler et al. (2010):

(7)

where is the model of linear combination of templates at pixel , and is the map of observed counts which is just the single GeV and GeV -ray Fermi masked map. Our diffuse -ray model is composed of 6 templates with 4 free parameters:

(8)

The Back template comes from our DRAGON run and is kept fixed modulo a normalization for a specific set of assumptions on the ISM gas and ISRF energy densities, the SphDM refers to the term in eq. 6 that is proportional to . The DarkDisk template refers to the term in eq. 6 proportional to and the MixedDM template to the term. The contribution of dim Galactic DM subhalos to the diffuse -rays is included in the ”SubDM” (related to the term in eq. 6), where as described earlier we have masked out the brightest possibly detected members. refers to the ratio of local DM density of the DD over the spherical halo given in eq. 3. The Iso template includes the contribution of DM extragalactic annihilations, the extragalactic background from other sources and the possible CR contamination. We also multiply the model map by the total FRONT+BACK-converted ULTRACLEAN photons exposure map and multiply by the mask. In Figure 3 for specific choices we show 4 different templates. On top left a specific model (”Back A”) for the Galactic diffuse background is shown at the energies of interest. On top right we plot the ”SphDM” template for an Einasto DM spherical halo profile with , and on bottom left we show the combined DM spherical and dark disk for (”SphDM”+ ”DarkDisk” + ”MixedDM”). The impact of adding the DM subhalos contribution (”SubDM”) is given in the bottom right. We use HEALPix Gorski et al. (2005) with Nside = 128 which represents closer the angular resolution of Fermi LAT at these energies 5.

DM profiles / Backgrounds Back ph. DM ph. Iso ph. TS
Ein. () / Back A 1.5 (4.5) 5.73 1146 40 (121) 214 9.1
Ein. () / Back A 2.2 (7.1) 5.55 1146 43 (138) 207 6.1
Ein. () / Back A 2.7 (8.5) 5.38 1157 41 (127) 201 2.8
Ein. () / Back B 1.6 (4.8) 5.87 1134 44 (129) 219 11.9
Ein. () / Back C 1.5 (4.6) 5.81 1144 39 (124) 217 9.2
Ein. () / Back D 1.3 (4.3) 6.05 1137 36 (115) 226 7.8
Table 3: The values of relevant parameters for various assumptions on the Galactic diffuse background and the distribution of DM spherical halo ignoring the contribution of a DD () and subhalos. Second column gives the best fit annihilation cross-section assuming equal annihilation cross-sections to the two lines; = in units of cm s. is the calculated isotropic flux at GeV and GeV in units of GeV cm s sr. Columns 4-6 refer to the GeV and GeV photons in the Background, DM and Isotropic diffuse components as predicted by the fitted values of , and of eq. 8. Last column gives the TS for detection of a DM signal. Values in parentheses refer to 3 upper limits on the DM annihilation cross-section.
DM profiles / Backgrounds Back ph. DM ph. Iso ph. TS
Ein. (); DD / Back A 2.8 5.72 1143 43 213 8.7
Ein. (); DD / Back A 2.6 5.69 1144 42 212 8.0
Ein. (); DD / Back A 2.5 5.64 1146 43 210 7.7
Ein. (); DD / Back A 2.4 5.60 1145 45 209 7.6
Ein. (); DD / Back A 4.2 5.56 1143 49 208 5.6
Ein. (); DD / Back A 4.7 5.40 1154 43 201 2.4
Ein. (); DD / Back B 3.1 5.91 1130 48 221 11.5
Ein. (); DD / Back C 2.8 5.79 1141 43 216 9.0
Ein. (); DD / Back D 2.5 6.04 1135 38 225 7.5
Table 4: The values of relevant parameters for various assumptions on the Galactic diffuse background, the distribution of DM spherical halo and the DD assuming the maximal DD contribution () and ignoring the contribution of subhalos. Second column is as in Table 3, is in units of GeV cm s sr and is in units of cm s. Columns 4-6 refer to the GeV and GeV photons in Back, DM and Iso components predicted by the fitted values of , and . Last column gives the TS for detection of DM signal.

We also calculate the significance of a DM contribution from the diffuse analysis by the test statistic, where

(9)

allows for the DM to contribute, while in we set the DM diffuse component to zero and refit the other two diffuse components. Our results are shown in Tables 3-5.

Ignoring first both the contribution of a dark disk and the DM subhalos, we find that the more cuspy DM profiles for the main/spherical halo that lead to less DM contribution to the diffuse -ray spectrum away from the Galactic center, provide a larger positive fit to the 4 sky (see Table 3). Yet the significance of that is not very large (TS = 9.1/9.2 and 11.9 in the best cases). An even more cored (flat) Burkert DM profile for the main halo in the inner kpcs does not give a better fit to the 111 and 129 GeV lines distribution in agreement with the findings of Su and Finkbeiner (2012a) and Weniger (2012) performed in subsections of the sky.

Using the model of for the Einasto DM density profile that provides the best fit, we also test different diffuse background models (”Back A”-”Back D”) to account for uncertainties in the interstellar medium gas distribution and interstellar radiation field distribution. We find that in all cases a DM component is preferred by the fit at 2-3 significance for the cuspier DM models (1-sided since in our fits we allow for the DM component to be even negative), and accounting for about 35-45 photons (cross-sections of cms) 6. As with ”Back A” the cuspier DM profiles are preferred for the tested backgrounds. Our ”Back A” is the same as the reference model described in Tavakoli et al. (2013), which was shown to provide a good agreement to the 4 sky and in energies between 1 and 200 GeV and has also been cross-checked to local CR measurements. Model ”Back B” assumes a different distribution for the molecular hydrogen gas component that is dominant at lower latitudes and toward the GC where many of the 111 and 129 GeV photon excess have been claimed (see Tavakoli et al. (2013) for more details). ”Back C” and ”Back D” Galactic diffuse models assume, respectively, an enhanced ISRF energy density distribution toward the disk and a minimal metallicity gradient Porter et al. (2008). The latter assumption affects both the morphology and the spectrum of the ISRF and as a result the -rays produced via Compton up-scattering of these photons by high energy CR electrons. These background models are discussed in further detail in Tavakoli et al. (2013). They have not been constructed to study just the Galactic -ray background at 110-130 GeV, but instead the general uncertainties in the Galactic diffuse -rays between 1-200 GeV, in the full sky and in subsections of it.

Since in our fits we also allow for an isotropic component we can calculate the isotropic flux at these energy ranges, taking into account also the Galactic diffuse background uncertainties. We find that photons can be accounted by that component. This gives an isotropic flux of 5.6-6.1 GeV cm s sr which is in agreement with the extragalactic isotropic -ray flux of Abdo et al. (2010) described by measured between 200 MeV and 100 GeV. The photons of the isotropic component can also be used to set approximate limits on the contribution from DM halos at extragalactic distances.

We find that adding a dark disk component does not significantly change the fit to the data from the case of having only a spherical DM halo contribution. That is for any choice of diffuse background or spherical DM halo shown Table 4. Thus a significant DD contribution is not preferred by the current data; while yet it can not be excluded either. We give the likelihood fits for the maximal DD contribution.

DM profiles Back ph. DM ph. Iso ph. TS
Ein. (); Unbiased Sub. Dist. () 1.4 0.0 5.34 1146 53(16) 199 8.6
Ein. (); Unbiased Sub. Dist. () 1.5 0.0 5.70 1146 40(0.36) 213 9.1
Ein. (); Anti-Biased Sub. Dist. () 1.4 0.0 5.39 1146 52(14) 201 8.8
Ein. (); Anti-Biased Sub. Dist. () 1.5 0.0 5.70 1146 40(0.41) 213 9.1
Ein. (); Unbiased Sub. Dist. () 2.0 0.0 5.07 1149 61(23) 189 5.2
Ein. (); Unbiased Sub. Dist. () 2.4 0.0 5.54 1146 46(0.59) 207 6.1
Ein. (); Anti-Biased Sub. Dist. () 2.1 0.0 5.10 1149 60(21) 190 5.5
Ein. (); Anti-Biased Sub. Dist. () 2.4 0.0 5.54 1146 47(0.66) 207 6.1
Ein. (); DD ; Unbiased Sub. Dist. () 2.4 1 5.04 1144 64(28) 188 7.8
Ein. (); DD ; Unbiased Sub. Dist. () 2.8 1 5.769 1143 43(0.69) 212 8.7
Ein. (); DD ; Anti-Biased Sub. Dist. () 2.6 1 5.13 1143 64(25) 192 8.2
Ein. (); DD ; Anti-Biased Sub. Dist. () 2.8 1 5.69 1143 43(0.77) 212 8.7
Table 5: The values of relevant parameters for various assumptions on the distribution of DM spherical halo, the DD and dim Galactic DM subhalos with different choices of the minimal subhalo mass. Units for and are as in Tables 3 and 4. In parentheses we give the photon number associated to the subhalo diffuse component only. The subhalo mass spectral index is assumed to be -2. In all cases we use our reference ”BackA” as Galactic diffuse background.

Our fits give a relatively flat behavior in the parameter with a slight preference for . More specifically for a DD with kpc (thick DD) with we get a lower test statistic by about compared to the cases of having only the spherical DM halos contributing, i.e. the data show a slight preference for having only a spherical DM halo at these energies. That is for any of our background assumptions. Yet thin DD ( kpc) give about the same TS as the only spherical DM halo contribution for any of our background models. Having tested also the 4 yr sample, that gave a slight preference for with a kpc DD, we conclude that the current small number of photons at the two energy lines does not provide us with enough detail to discriminate a thin DD component from the disk-like distribution of the Galactic background -rays. With more -ray data we expect that we will be able to further disentangle the different -ray morphologies.

Even for the case where we have a maximal DD contribution, i.e. when most of the DM photons are due to the spherical and the mixed terms of eq. 6. In those cases, the equivalent DM annihilation cross-section that accounts for the DM template normalization is maximal. That is the case since the total number of DM photons is about the same while the term in eq. 6 is minimal. Since the DD term can not be excluded this is a way of allowing for higher annihilation cross-sections, by up to a factor of .

In Table 5 we also study the impact of the diffuse DM subhalo template ”SubDM” of eq. 8 with and without a DD component. To bracket the uncertainties we used both the unbiased and the anti-biased Galactic subhalo distributions (see Appendix). We also tested the subhalo mass extrapolations down to and . For the cases where the extrapolation is down to the DM subhalo diffuse component is subdominant contributing only a few line photons. On the contrary, for extrapolations down to that component can contribute up to () of the Galactic DM line photons, with the remaining coming from the spherical main halo (spherical DD components). With the current data the difference in the TS fit between the unbiased and the anti-biased distributions is very small (see Table 5). Yet, even adding a strong subhalo term/template, our method can discriminate between different assumptions for the cuspiness of the spherical halo ”SphDM”. In agreement with all our previously discussed tests, a preference toward cuspier halos is found. In Table 5 we compare between and Einasto profiles that differ only in the inner few degrees from the GC.

Having tested the DM case both with and without a dark disk and including/excluding the contribution of Galactic bound dim DM subhalos we have consistently found a thermally averaged cross-section = for the two lines that is in the range of 1.5-4.5 cms. These values are a factor of 9-3 smaller than the suggested values from analyses coming when concentrating only toward the GC Weniger (2012). Fitting the entire 4 sky can dilute the DM signal from the GC and thus suggest a smaller annihilation cross-section. Yet, we note that a strong annihilation annihilation cross-section to the lines should be seen at the diffuse spectrum at high latitudes when including the contribution from the dim subhalos with masses down to the free streaming scale. Our fits do not suggest such a case. Taking the subhalos to have masses down to , we get from our fits shown in Table 5 15-20 photons (30 including a DD in the fit) from annihilations just in the dim subhalos. That number of photons from the diffuse subhalo component can be explained by a thermally averaged cross-section of 1.5-2.5 cms. Given that the velocity dispersion in bound substructures is smaller than in the GC or locally that may be expected. Alternatively explained, thermally averaged cross-section of cms down to the smallest subhalos would give 100 photons on the 4 sky just on the two lines and just from the dim Galactic subhalos. Of these photons would be at with the isotropic component predicting photons in the two energy ranges (, GeV). Thus the lines would have to be observed at high latitudes as well. If not found at these latitudes, then either the is smaller in the subhalos, in general, or subhalos are not formed down to masses of . We note that in the above numbers we have not included the possible and more model dependent contribution form extragalactic DM annihilations.

V Discussion and Conclusions

Recently, Su and Finkbeiner (2012b) has found indications for the 111 and the 129 GeV lines in the 2 yr Fermi unassociated point sources catalogue that would be indicative for DM annihilation in substructures. We compare the compatibility of the findings of Su and Finkbeiner (2012b) with the results from VLII cosmological simulations and extrapolations of it. In that process we have assumed the same annihilation cross-section to the Galactic center lines signal Weniger (2012); Su and Finkbeiner (2012a); Tempel et al. (2012); Rajaraman et al. (2012); Buchmuller and Garny (2012); Boyarsky et al. (2012) given that the point sources lines signal and the Galactic center line(s) signal have been observed at the same -ray energies. We find that just considering VLII assumptions we do not get enough line photons from the brightest Galactic DM subhalos to claim agreement with the signal seen by Su and Finkbeiner (2012b). Yet completions of the VLII simulation results do give a number of line photons from the brightest subhalos that in the most optimistic cases is in good agreement with (but still below) the lines events number found by Su and Finkbeiner (2012b) (9 events in their ULTRACLEAN sample) (see discussion in section III). These most optimistic extrapolations to very small substructures () predict that many diffuse line photons that are in some tension with the -ray flux at high latitudes, but still can not be excluded in the most conservative manner. The same applies when we compare their predictions to the isotropic -ray flux which for the energies of the lines we have calculated to be centered at GeV cm s sr, with 4 sky fit values being in the range of GeV cm s sr (accounting for systematic/model uncertainties in the evaluation of the isotropic component).

Most of the 111 and the 129 GeV line photons on the -ray sky are not from DM annihilations but rather from the Galactic diffuse and the isotropic diffuse background (shown in Fig. 2). By doing a template fit to the 4 sky, we test how robust the DM signal hypothesis is for different physical assumptions on the Galactic diffuse -ray background flux and on the DM halo profile (see discussion on section IV). We find a positive fit of having a DM spherical halo component. Our DM annihilation component hypothesis is preferred to the case without a DM component at a test statistic significance of up to 12; with the values depending on the exact DM halo assumptions. More concentrated DM profiles give a larger significance to a DM signal. We also find that our results on the positive signal of DM annihilation are weakly dependent on Galactic diffuse background uncertainties related to either the uncertainties in the distribution of the interstellar medium gas or to the energy density in the interstellar radiation field.

Extending our set of tests on the DM distribution in the Galaxy, we include a dark disk component that could explain the non-isotropic distribution on the sky of the point sources of Su and Finkbeiner (2012b). Our fits can not strongly favor or disfavor a significant disk-like DM component, even though there is a small preference toward thinner dark disks. We also study the contribution of the dimmest DM subhalos in the Milky Way to the diffuse gamma-ray sky at energies around 111 and 129 GeV and find a preference toward an anti-biased distribution of the subhalos within the main spherical DM halo. Yet our analysis is somewhat constrained in its power by the small number of photons on the sky at the energies of the 2 lines. As more statistics are being accumulated, a better understanding of the morphology of the -ray sky will be achieved allowing for such a template analysis to further disentangle the background -ray sky from any possible DM component.

Finally using the various combinations of backgrounds and DM distributions, we find the thermally averaged annihilation cross-section to be smaller than what has been originally suggested, with values ranging between cms. While the full sky fits are not optimal for a DM signal toward the GC, they include high latitudes which probe also the contribution from smaller substructures. A suppressed flux at high latitudes to the lines either indicates a smaller overall cross-section or a suppressed contribution from smaller substructures compared to the GC.


Acknowledgments

The authors would like to thank Alex Drlica-Wagner, Carmelo Evoli, Andrew Hearin, Dan Hooper, Ran Lu, Meng Su and Gabrijela Zaharijas for valuable discussions. PU acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). In this work the authors have used the publicly available Fermi-LAT data and Fermi Tools archived at http://fermi.gsfc.nasa.gov/ssc/. We also acknowledge the use HEALPix.

Appendix A DM Signal from Unbiased and Anti-Biased Distributions of Subhalos in the Milky Way

We fit the subhalos in the VLII simulation Diemand et al. (2008); http://www.ucolick.org/~diemand/vl/data.html () using an NFW profile Kuhlen et al. (2008):

(10)

We read out the scale radius, , and scale density, , in terms of the peak of circular velocity and the corresponding radius (see, e.g. equation 9-10 in Kuhlen et al. (2008)). Their luminosities are given by:

(11)

where is their tidal radius. By using this formula, combined with eq. (5), we find the results in the first row of Table 1.

In deriving the unbiased distribution of subhalos, first, we calculate the gravitational potential from the host halo, cutting the profile at its virial radius and distinguish bound subhalos from the unbound ones. There are 13510 bound subhalos, 9372 of them are within the simulated galaxy’s virial radius, Furthermore, there are only 9 unbound subhalos within the virial radius and all of them are beyond 250 kpc from GC. We also calculate each subhalo’s pericenter, , to obtain the pericenter probability distribution function (PDF) as a function of Galactocentric radius, , e.g . For unbound subhalos, we identify their tidal radius as their virial radius, and their tidal mass as their virial mass. According to the virial theorem, the average density inside the virial radius should be the same for all halos. We find that this is indeed the case in VLII simulation: the average density inside the virial radius of unbound subhalos is , where is the overdensity relative to the critical matter density for spherical collapse for and is the critical matter density. For bound subhalos, the tidal radius is usually much smaller than virial radius because of tidal stripping. We define the tidal concentration of a subhalo, , as its average density within its tidal radius divided by its critical matter density; for unbound subhalos, . To determine the virial concentration PDF for unbound subhalos, we use the Bullock model Bullock et al. (2001) as parametrized in eq. (2) of Zechlin et al. (2012) which relates the median virial concentration of a subhalo, , with its virial mass, . We find that this parametrization can fit unbound subhalos’s virial concentrations fairly well. We take the mass PDF of unbound subhalos to be a power law, and vary the spectral index . In fitting the mass and concentration PDF, we use the unbound subhalos because they are not tidally stripped. Hence, in this model, the Galactocentric radial dependence of mass and concentration will appear later, after they are tidally stripped by their host. To find a subhalo’s minimum concentration at some Galactocentric radius , we apply the Roche criteria: for a subhalo in circular orbit, the subhalo’s self-gravity at should be equal to the differential gravity pull of the host halo computed at . As the subhalo’s orbit is not exactly circular, the tidal force is strongest at its pericenter, so the concentration is determined by the tidal forces at its pericenter Tormen et al. (1997):

(12)

where refers to the VLII host density profile. We then refine the previous estimate by taking into account the pericenter radius distribution for subhalos, . Subhalos initially with are left intact, whereas subhalos initially with are tidally stripped until is reached, e.g.:

(13)

It is a fair approximation that the scale and do not change in this process Kazantzidis et al. (2004). By following these steps; a subhalo’s mass, concentration and luminosity after tidal stripping are completely determined by its pericenter, mass and concentration before tidal stripping. We then calculate the average mass, concentration and luminosity as a function of galactocentric radius, after tidal stripping and compare them against the VLII simulation. We find that luminosity and mass can be fitted very well, with best fit parameters . The here is the minimum subhalo mass before tidal stripping. Our model also fits the tidal concentration very well near the host’s center, although slightly deviates near its virial radius. This might indicate that our treatment of dynamical effects is oversimplified. Especially, the minimum concentration calculated by Rochi criteria is only achieved after several pericentric passages Hayashi et al. (2003); Gan et al. (2010); Kazantzidis et al. (2004), whereas in our model, we assume that all of them are already above minimum concentration. However, the procedure is validated by very good fits to the luminosities. Besides, it is conceptually simple and can be easily generalized to other host and subhalo mass profiles and redshifts.

Regarding the subhalos number density profile, we consider both the unbiased and anti-biased cases. As mentioned in the section III, the strong tidal force near the GC could make the subhalos spatial distribution to be anti-biased with respect to the host density profile. Indeed, for VLII itself, we find that the deviation from unbiased, starts to happen around . However, we find that the subhalo spatial distribution for VLII is more unbiased with respect to VLI Diemand et al. (2007a); http://www.ucolick.org/~diemand/vl/data.html (), which has a lower mass resolution. This might indicate that the anti-bias is a result of numerical effect. The strong tidal force could strip subhalos until they are below or near the resolution limit, hence undetectable as subhalos. Also, VLII subhalos which are selected by mass, show a more anti-biased tendency than the ones which are selected by maximum circular velocity, confirming Diemand et al. (2007b); Diemand et al. (2008); Diemand and Moore (2011). Keeping these in mind, we take the unbiased distribution for subhalos number density defined as:

(14)

With this normalization, the total mass and number of subhalos with tidal mass bigger than between and in VLII simulation is within 5% from the value calculated with our model using best fit parameters.

The Aquarius simulation Aq-A-1 Springel et al. (2008) is another simulation which has parameters similar to VLII. Specifically, it has nominal mass resolution, , host halo mass, , and host halo radius, . However, their number of subhalos with mass bigger than is approximately twice VLII’s. We discuss the modification to our results in Table 1 and Table 2 when we use the Aquarius normalization.

From this procedure, we can find the number density of subhalos per unit luminosity after tidal stripping, . The number of photons that we receive from a single subhalo with luminosity and line of sight (los) distance from us, for channels or is given by eq. (5). By folding eq. (5) with , we can obtain the values in Tables 1 and 2. On the other hand, the contribution to the host’s from subhalos is given by:

(15)

where is the average luminosity of the entire subhalos population after tidal stripping.

The anti-biased distribution (where is less concentrated than ) is taken from appendix A of Kuhlen et al. (2008). The normalization is such that the total mass in subhalos with masses between and is of . In the original paper, the authors normalize to of . However, if we only include subhalos within the virial radius, the mass fraction is only . In this model, the authors encapsulate the tidal force from the host halo by adding radial dependence to the relation, so that a subhalo closer to the GC has a higher concentration on average.

As a further refinement, we also add the Galactic disk to the VLII halo. For the Galactic disk model, we follow Catena and Ullio (2010), for NFW parameter and our distance to the GC, of . We spherically average the Galactic disk. The relative difference in density between the averaged version and the original version is substantial only in the disk plane. There is no noticeable difference to the pericenter PDF and the only modification to unbiased distribution is that we replace in (12). For the anti-biased one, the effect of host tidal forces have been taken into account by adding radial dependence to the relation. Therefore, in eq. (12), we replace . We also use the same pericenter distribution as in the unbiased case.

Footnotes

  1. There have been16 unassociated point sources in the 2yr Fermi catalogue detected at the level of 4 or more. In oder to reach that significance a spectrum at low energies is also needed. DM does not give a strong spectrum signal at low energies so the actual spectrum of these sources (if they are DM subhalos) may be contaminated by other close-by point sources or the diffuse -ray component. That may also result in a non-trivial selection effect.
  2. Our 4.4 yr sample refers to the data from 04 August 2008-03 January 2013.
  3. This value is obtained assuming isotropic distribution. The actual distribution of the 111 and 129 GeV photons is not isotropic but rather peaks on the Galactic disk. Our results do not differ that much though since the distribution of the 575 unassociated point sources also peaks on the Galactic disk.
  4. Su and Finkbeiner (2012b) suggest a p-value of for having 13 SOURCE events given a background described by a power-law.
  5. The angular resolution of the Fermi LAT instrument is different at the energies that we study by about a factor of 2 between the front and the back-converted converted photons. With Nside=128 our pixels include entirely the front-converted photons and are slightly smaller than the containment angle for back-converted events. Choosing Nside=256 would have resulted in photon events being spread in multiple pixels thus making them correlated; while Nside, would be underestimating the ability of the instrument to resolve -ray structures at these energies. We have checked that our best fit results are consistent within 2 for Nside of 128, 64, 32, 16 and 8.
  6. Since we use the entire 4 sky and have selected in advance the energy range of the -ray events based on the works of Weniger (2012); Su and Finkbeiner (2012a); Tempel et al. (2012); Rajaraman et al. (2012); Buchmuller and Garny (2012); Boyarsky et al. (2012) there is no look elsewhere effect accounted for.

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