SEARCH FOR DARK MATTER SATELLITES USING FERMI-LAT

The Aquarius (Springel et al. 2008) and Via Lactea II (VL-II; Diemand et al. 2007) projects are currently the highest resolution numerical simulations of dark matter substructure at the scale of Galactic halos. As part of the Aquarius project, six independent simulations of a Milky Way mass dark matter halo were generated. The VL-II project provides an additional independent simulation. These simulations model the formation and evolution of a Milky Way size dark matter halo and its satellites having over one billion ∼10³ M_☉ particles. Each simulation resolves over 50,000 satellites within its respective virial radius, which is defined as the radius enclosing an average density 200 times the cosmological mean matter density. Each bound satellite has associated with it a position with respect to the main halo, a velocity, a tidal mass, M_tidal, a maximum circular velocity, V_max, and a radius of maximum circular velocity, $R_{V_{\rm max}}$.

Generally, the γ-ray flux from annihilating dark matter in a satellite (ϕ_WIMP) can be expressed as a product of the line-of-sight integral of the dark matter distribution in a satellite (J-factor) and a component depending on the particle physics model (Φ^PP) for WIMP annihilation (Strigari et al. 2007)

$$\begin{equation} \phi _{\mathrm{WIMP}}(E) = J \times \Phi ^{\rm PP}(E) \end{equation} \tag{ 1 }$$

where

$$\begin{eqnarray} J &= \int _0^{\Delta \Omega } \left\lbrace J(\theta) \right\rbrace d\Omega \nonumber \\ &= \int _0^{\Delta \Omega }\left\lbrace \int ^{\ }_{{\rm l.o.s.}}\rho ^2[r(D,l,\theta)] dl \right\rbrace d\Omega, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} J &\approx \frac{1}{D^2}\int _V{{\rho (r)}^2dV}, \end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray} {\Phi }^{\rm PP}(E) &=\frac{1}{4\pi }\frac{\langle \sigma v \rangle }{2M^2_{\rm WIMP}} {\sum _f{\frac{dN_f}{dE} B_f} }. \end{eqnarray} \tag{ 4 }$$

Equation (2) represents the line-of-sight integral along l through the satellite dark matter density profile ρ, where D is the distance to the satellite center, θ is the offset angle relative to the center, and $r(\theta,D,l) = \sqrt{D^2 + l^2 - 2Dl\cos {\theta }}$ is the distance from the satellite center. The solid angle integral is performed over ΔΩ = 2π(1 − cos θ). For satellites at large distances from the Earth, the J-factor can be approximated by Equation (3) where the volume integration is performed out to the satellite tidal radius (Tyler 2002). In Equation (4), 〈σv〉 is the velocity-averaged annihilation cross section, M_WIMP is the mass of the dark matter particle, and the sum runs over all possible pair annihilation final states with dN_f/dE and B_f representing the photon spectrum and branching ratio, respectively.

For this paper, it was important to model the dark matter distribution within the satellites themselves. A Navarro–Frenk–White (NFW) profile (Navarro et al. 1997) with scale radius r_s and scale density ρ_s was used to approximate the dark matter distribution in the satellites:

$$\begin{equation} \rho (r) = \frac{\rho _s}{(r / r_s) (1+r/r_s)^2}. \end{equation} \tag{ 5 }$$

An NFW profile was uniquely defined for each satellite from the values of V_max and $R_{V_{\rm max}}$ using the relations (Kuhlen et al. 2008)

$$\begin{eqnarray} r_s & = \frac{R_{V_{\rm max}}}{2.163} \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \rho _s & = \frac{4.625}{4\pi G}{\left(\frac{V_{\rm max}}{r_s}\right)}^2, \end{eqnarray} \tag{ 7 }$$

where r_s is in  kpc and ρ_s is in M_☉  kpc⁻³. The existence of dark matter substructure within the satellites themselves would increase the J-factor, but this contribution is expected to be no greater than a factor of a few (Martinez et al. 2009; Anderson et al. 2010). Thus, we took a conservative approach and did not include this enhancement when calculating J-factors.

Although the NFW model is widely used for its simplicity and is broadly consistent with the dark matter distribution in satellites (Springel et al. 2008), a few caveats are important. First, simulations with increased resolution have revealed that more scatter exists in the central dark matter density profile than is implied by the NFW profile (Navarro et al. 2008). In fact, these simulations indicate that the central dark matter density in satellites is systematically shallower than the r⁻¹ central density implied by the NFW model. However, these deviations occur at a scale of ≲ 10⁻³ of the halo virial radius (Navarro et al. 2008), and thus do not strongly affect the predicted flux given the LAT's angular resolution. Second, satellites are, in nearly all cases, more severely tidally truncated than the r⁻³ outer density scaling of the NFW profile. For satellites with large r_s, using the NFW profile will result in a slight (<10%) overestimation of the predicted flux.

The satellite mass functions for the Aquarius and VL-II simulations are complete down to ∼10⁶ M_☉. However, theoretical arguments suggest that the mass function of satellites may extend well beyond this resolution limit, perhaps down to Earth-mass scale (∼10⁻⁶ M_☉). Therefore, it is likely that the satellites resolved in the Aquarius and VL-II simulations are only a small fraction of the total number of bound dark matter satellites present in our Galaxy. We refer to these unresolved dark matter satellites with M_tidal < 10⁶ M_☉ as "low-mass" satellites.

To estimate the contribution of low-mass satellites to the LAT search, we extrapolated the distribution of satellites in VL-II down to 1 M_☉. Assuming a power-law mass function for satellites, dN/dM_tidal∝M^−1.90_tidal (Madau et al. 2008; Springel et al. 2008), we calculated the number of satellites at a given M_tidal within 50 kpc of the Galactic center. These low-mass satellites were distributed within this 50 kpc radius in accordance to the radial distribution described in Madau et al. (2008). The cut at 50 kpc is conservative based on the null detection of the Segue 1 dwarf spheroidal with a tidal mass of ∼10⁷ M_☉ at ∼28 kpc from the Galactic center (Geha et al. 2009).

In order to model the internal dark matter distribution of low-mass satellites with an NFW profile (using Equations (6) and (7)), we fit relationships between M_tidal, V_max, and $R_{V_{\rm max}}$. For VL-II satellites within 50 kpc of the Galactic center and with M_tidal > 10⁶ M_☉, we found that

$$\begin{equation} V_{\rm max} = V_0 \left(\frac{M_{\rm tidal}}{M_\odot } \right)^{\beta } \end{equation} \tag{ 8 }$$

with V₀ = 10^{−1.20 ± 0.05} km s⁻¹, β = 0.30 ± 0.01, and a log-normal scatter of $\sigma _{V_{\rm max}} = 0.063 \mathrm{\,km} \mathrm{\,s} ^{-1}$. Additionally, we found that

$$\begin{equation} R_{V_{\rm max}} = R_0 \left(\frac{M_{\rm tidal}}{M_\odot }\right)^{\delta } \end{equation} \tag{ 9 }$$

with R₀ = 10^{−3.1 ± 0.4} kpc, δ = 0.39 ± 0.02, and a log-normal scatter of $\sigma _{R_{V_{\rm max}}} = 0.136 \mathrm{\,kpc}$. Using these relationships, we randomly generated low-mass satellites consistent with the VL-II simulation down to a tidal mass of 1 M_☉.

The VL-II simulation, extrapolated as described above, provides a theoretical model for the population of Milky Way dark matter satellites from 10¹⁰ M_☉ to 1 M_☉. A simple estimate of the detectable fraction of these dark matter satellites can be obtained from the 11-month limits on the WIMP annihilation flux from dwarf spheroidals (Abdo et al. 2010e). No γ-ray signal was detected and the strongest limits on the annihilation cross section result from the analysis of the Draco dwarf spheroidal, which has a J-factor (integrated over the solid angle of a cone with radius 05) of ∼10¹⁹ GeV² cm⁻⁵. The central mass of Draco is well known from stellar kinematics (Strigari et al. 2008; Walker et al. 2009; Wolf et al. 2010); however, the total dark matter mass for Draco is less certain due to the lack of kinematic measurements in the outer regions of the halo. A conservative lower bound on the total dark matter halo mass of Draco was taken to be 10⁸ M_☉. Based on this lower bound, we determined what fraction of the satellites have a larger detection potential than Draco, i.e., what fraction of satellites have a J-factor greater than that of Draco.

For an NFW profile, the J-factor is roughly described by

$$\begin{equation} J \propto \frac{r^3_s{\rho }^2_s}{D^2} \propto \frac{M^{0.81}}{D^2}. \end{equation} \tag{ 10 }$$

using the relationships that r_s∝M^0.39 (from Equations (6) and (9)) and ρ_s∝M^−0.18 (from Equations (6)–(8)). Equation (10) makes it possible to compare the relative astrophysical contribution to the γ-ray flux for different halos based on their tidal mass and distance from Earth (Figure 1). The choice of particle physics annihilation model merely scales all satellites by the same constant factor.

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Figure 1 serves as a guide for evaluating the detectability of low-mass satellites. While the total number of satellites increases with decreasing mass, the J-factors of these low-mass satellites tend to decrease. This means that low-mass satellites, while dominating the local volume in number, are a subdominant contributor to the γ-ray flux at the Earth. Using the procedure discussed in Section 5, we verify that extending the VL-II mass function to low mass has a minimal effect (<5%) when setting upper limits on 〈σv〉; consequently, we do not consider these satellites in our primary analysis. For low-mass satellites to dominate the γ-ray signal, a mechanism must be invoked to either increase the concentration for low-mass satellites or decrease the slope of the mass function. Of course, the above statements do not preclude the possibility that there could be a low-mass satellite with a high J-factor very near to the Earth.

In the context of the CDM theory, several dark satellite galaxies with no associated optical emission could be detectable by the LAT. In addition to motivating our satellite search, Figure 1 allowed us to narrow our focus to those satellites with the best prospects for detection. Using Equation (10), we omitted satellites with J-factors more than an order of magnitude less than the lower bound on the J-factor for Draco. This greatly reduced the number of satellites for which the full line-of-sight integral in Equation (2) was calculated.

Fermi has been operating in sky-scanning survey mode since early 2008 August. The primary instrument on board Fermi is the LAT, designed to be sensitive to γ-rays in the range from 20 MeV to >300 GeV. The LAT has unprecedented angular resolution and sensitivity in this energy range, making it an excellent instrument for detecting new γ-ray sources (Atwood et al. 2009).

Our data sample consisted of "Diffuse" class events from the first year of LAT data collection (2008 August 8 to 2009 August 7), which overlapped substantially with the data used for the 1FGL (Abdo et al. 2010b). To reduce γ-ray contamination from the bright limb of the Earth, we rejected events with zenith angles larger than 105° and events taken during time periods when the rocking angle of the LAT was greater than 47° (the nominal LAT rocking angle was 35° during this time period). Due to calibration uncertainties at low energy and the current statistical limitations in the study of the instrument response functions (IRFs) above 300 GeV, we accepted only photons with energies between 200 MeV and 300 GeV. This analysis was limited to sources with absolute Galactic latitudes (|b|) greater than 20°, since the Galactic diffuse emission complicates source detection and the analysis of spatial extension at lower Galactic latitudes. We modeled the diffuse γ-ray emission with standard Galactic (gll_iem_v02.fit) and isotropic (isotropic_iem_v02.txt) background models.⁵⁴ Throughout this analysis, we used the LAT ScienceTools⁵⁵ version v9r18p1 and the P6_V3_DIFFUSE IRFs.⁵⁶

The 1FGL is a collection of high-energy γ-ray sources detected by the LAT during the first 11 months of data taking (Abdo et al. 2010b). It contains 1451 sources, of which 806 are at high Galactic latitude (|b| > 20°). Of these high-latitude 1FGL sources, 231 are unassociated with sources at other wavelengths and constitute the majority of the sources tested for consistency with the dark matter satellite hypothesis. However, the 1FGL spectral analysis, including the threshold for source acceptance, assumed that sources were point-like with power-law spectra. This decreased the sensitivity of the 1FGL to both spatially extended and non-power-law sources, which are characteristics expected for dark matter satellites. In an attempt to mitigate these biases, we augmented the unassociated sources in the 1FGL with an independent search of the high-latitude sky.

We performed a search for γ-ray sources using the internal LAT Collaboration software package, Sourcelike (Abdo et al. 2010d; Grondin et al. 2011). Sourcelike performs a fully binned likelihood fit in two dimensions of space and one dimension of energy. When fitting the spectrum of a source, Sourcelike fits the fraction of counts associated to the source in each energy bin independently. The overall likelihood is the product of the likelihoods in each bin. This likelihood calculation has more degrees of freedom than that performed by the LAT ScienceTool, gtlike, which calculates the likelihood from all energy bins simultaneously according to a user-supplied spectral model. In this analysis, we used 11 energy bins logarithmically spaced from 200 MeV to 300 GeV.

Using Sourcelike, we searched for sources in 2496 regions of interest (ROIs) of dimension 10° × 10° centered on HEALPix (Górski et al. 2005) pixels obtained from an order four tessellation of the high-latitude sky (|b| > 20°). Each ROI was sub-divided into 01 × 01 pixels, and for each pixel the likelihood of a point source at that location was evaluated by comparing the maximum likelihood ($\mathcal {L}$) of two hypotheses: (1) that the data were described by the standard LAT diffuse background models without any point sources (H₀), and (2) that the data were described by the existing model with an additional free parameter corresponding to the flux of a source at the target location (H₁). Utilizing the likelihood ratio test, we defined a test statistic (TS)

$$\begin{equation} \mathrm{TS} = - 2 \ln \left(\frac{\mathcal {L}(H_0)}{\mathcal {L}(H_1)} \right). \end{equation} \tag{ 11 }$$

After generating a map of the test statistic over the entire high-latitude sky, we iteratively refit regions around potential source candidates more carefully. The flux normalizations, spectral indices, and emission centroids of candidate sources with TS > 16 were refined while incorporating the flux normalizations of diffuse backgrounds and other candidate point sources with TS > 16 within the ROI as free parameters in the fit. After refitting, only candidate sources with TS > 24 were accepted into the list of source candidates.⁵⁷ Finally, to avoid duplicating sources in the 1FGL, we removed candidate sources with 68% localization errors overlapping the 95% error ellipse given for 1FGL sources.

Our search of the high-latitude sky revealed 710 candidate sources, of which 154 were not in the 1FGL (36 of these candidate sources subsequently appear in the Second LAT Source Catalog; Abdo et al. 2011). We did not expect to recover all 806 high-latitude 1FGL sources, since the 1FGL is a union of four different detection methods and external seeds from the BZCAT and WMAP catalogs (Abdo et al. 2010b). However, since Sourcelike fits each energy bin independently, we expected to find source candidates that were excluded from the 1FGL, either because they had non-power-law spectra or they had hard spectra with too few photons to pass the 1FGL spectral analysis. We sacrificed some sample purity for detection efficiency in our candidate source list because stringent cuts on spatial extent and spectral shape were later applied. We obtained a final list of 385 high-latitude unassociated LAT sources and source candidates by combining the 231 unassociated sources in the 1FGL with these 154 non-1FGL candidate sources.

To check for consistency with the source analysis of the 1FGL, we performed an unbinned likelihood analysis with gtlike assuming that the unassociated sources were point-like with power-law spectra. Our fitted fluxes and spectral indices are in good agreement with those in the 1FGL for the 231 unassociated 1FGL sources. The values are plotted in Figure 2, where it can be seen that the unassociated LAT sources span a wide range of fluxes and spectral indices. This wide range was taken into account when designing selection criteria for candidate dark matter satellites. The strong correlation between spectral index and flux is due to the improvement of the point-spread function (PSF) of the LAT with increasing energy and the relatively soft spectral dependence of the Galactic diffuse background. It is apparent that there are more non-1FGL source candidates in this sample with very hard spectra (spectral index ∼1.0) and very low fluxes (∼10⁻¹⁰ photons cm⁻² s⁻¹). In the Appendix, we show that these source candidates are very likely spurious.

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The LAT has the potential to resolve some dark matter satellites as spatially extended γ-ray sources. While the bulk of satellites are not spatially resolvable by the LAT, spatial extension is an important feature for distinguishing large or nearby satellites from point-like astrophysical sources (see Section 2). Assuming that the spatial and spectral distribution of γ-rays produced from dark matter annihilation factorize, the shape of the projected dark matter distribution (Equation (2)) is convolved with the LAT PSF. For an NFW dark matter distribution (Equation (5)) with scale radius r_s at a distance D, the angular extent of a satellite can be characterized by the parameter α₀ = r_s/D. Approximately 90% of the integrated J-factor comes from within the angular radius α₀ (Strigari et al. 2007) and the LAT is sensitive to the spatial extension of satellites with α₀ > 05.

We used the likelihood ratio test, as implemented by Sourcelike, to test sources for spatial extension. We defined a test statistic for extension as

$$\begin{eqnarray} \mathrm{TS}_\mathrm{ext}^{} &= - 2 \ln \left(\frac{\mathcal {L}(H_\mathrm{point})}{\mathcal {L}(H_\mathrm{NFW})} \right) \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &= \mathrm{TS} _\mathrm{NFW}-\mathrm{TS} _\mathrm{point}, \end{eqnarray} \tag{ 13 }$$

where TS_point was the test statistic of the candidate source assuming that it had negligible extension (α₀ much smaller than the LAT PSF) and TS_NFW was the test statistic of the candidate source when α₀ was fit as a free parameter. In both cases, the position of the source was optimized during the fit.

We sought to define a cut on the value of TS_ext to eliminate 99% of point sources over the range of spectral indices and fluxes found in the unassociated LAT sources. This cut was labeled TS⁹⁹_ext. While the point and extended hypotheses are nested and TS_ext is cast as a likelihood ratio test, it is unclear whether this analysis satisfied all of the suitable conditions for the application the theorems of Wilks (Wilks 1938) or Chernoff (Chernoff 1954). Therefore, we relied on simulations to parameterize TS⁹⁹_ext as a function of source flux and spectral index.

To evaluate TS⁹⁹_ext over the pertinent range of source fluxes and spectral indices, the unassociated LAT sources were bracketed with 10 representative power-law models (Table 1 and blue stars in Figure 2). For each of the 10 representative models, 1000 independent sources were simulated at random locations in the high-latitude sky using the LAT simulation tool, gtobssim, and the spacecraft pointing history for our one-year data set. To accurately incorporate imperfect modeling of background point and diffuse sources, we embedded the simulated point sources in one year of LAT data and calculated TS_ext for each simulated source. We defined TS⁹⁹_ext for each representative model as the smallest value of TS_ext that was larger than that calculated for 99% of simulated point sources. The values of TS⁹⁹_ext for all 10 models were calculated independently of the TS > 24 detection cut (Table 1).

We used a bilinear interpolation to estimate the value of TS⁹⁹_ext for any point in the space spanned by the grid of flux and spectral index. Since each measurement of source flux and spectral index has a statistical uncertainty, we interpolated to the largest value of TS⁹⁹_ext that was consistent with the ±1σ error for each source for a conservative estimate of TS⁹⁹_ext.

To select sources that were spectrally consistent with WIMP dark matter annihilation, we designed a test for spectral curvature. The continuum γ-ray emission from WIMP annihilation has two different contributions: secondary photons from tree-level annihilation (Baltz et al. 2008; Cesarini et al. 2004) and additional photons from QED corrections—i.e., final state radiation (FSR; Beacom et al. 2005). For tree-level annihilations, the leading channels among the kinematically allowed final states are predicted to be $b \bar{b}$, $t\overline{t}$, W⁺W⁻, Z⁰Z⁰, and τ⁺τ⁻. The γ-ray spectra from these channels are quite similar, except for the τ-channel which is considerably harder (Cesarini et al. 2004). We chose the $b \bar{b}$ channel as a representative proxy for the tree-level annihilation spectrum.

We defined a test statistic to evaluate the consistency of the data with dark matter and power-law spectra. This spectral test statistic,

$$\begin{eqnarray} \mathrm{TS}_\mathrm{spec}^{} &= - 2 \ln \left(\frac{\mathcal {L}(H_\mathrm{pwl})}{\mathcal {L}(H_{b \bar{b}})} \right)\!, \nonumber \\ &= \mathrm{TS} _{b \bar{b}} - \mathrm{TS} _\mathrm{pwl} \end{eqnarray} \tag{ 14 }$$

was the difference in source TS calculated with an unbinned analysis using gtlike assuming a $b \bar{b}$ dark matter spectral model ($\mathrm{TS} _{b \bar{b}}$) and a power-law (TS_pwl) spectral model. These two hypotheses are not nested, and thus the significance of this test was evaluated with simulations. When performing our fits, we modeled candidate sources as point-like⁵⁸ and left their fluxes and the fluxes of the diffuse backgrounds free. Additionally, the power-law and dark matter spectral models contain a spectral free parameter (the dark matter mass or power-law index).

Using the same representative simulations described in Section 3.3, we defined TS⁹⁹_spec to be the value of TS_spec which was larger than that calculated for 99% of simulated power-law sources (Table 1). When calculating TS⁹⁹_spec for a particular source, we chose the largest value from a bilinear interpolation to the ±1σ errors on fitted flux and spectral index (as discussed at the end of Section 3.3). These tests of spatial extension and spectral character allowed us to select non-point-like and non-power-law sources with a contamination of 1 in 10⁴ assuming they were independent.

We applied the cuts on spatial extension and spectral character to select dark matter satellite candidates from the 385 unassociated high-latitude LAT sources and source candidates. Two of the 385 unassociated sources, 1FGL J1302.3−3255 and 1FGL J2325.8−4043, passed the cut on spatial extension. One of these, 1FGL J1302.3−3255, also passed our spectral test, preferring a $b \bar{b}$ spectrum to a power-law spectrum. However, we do not believe that either of these sources is a viable dark matter satellite candidate for reasons discussed below.

As their names imply, both 1FGL J1302.3−3255 and 1FGL J2325.8−4043 are present in the 1FGL (Abdo et al. 2010b); we summarize information about each source in Table 2. While 1FGL J1302.3−3255 was unassociated when the 1FGL was published, and has previously been considered as a promising dark matter satellite candidate (Buckley & Hooper 2010), it has since been associated with a millisecond pulsar by radio follow-up observation (Hessels et al. 2011). The other possibly extended source, 1FGL J2325.8−4043, has a high probability of association with two active galactic nuclei (AGNs) in the first LAT AGN Catalog (Abdo et al. 2010f). 1FGL J2325.8−4043 is assigned a 70% probability of association to 1ES 2322−409 and a 55% probability of association with PKS 2322−411. Additionally, each of these sources was checked for consistency with an E⁻¹ power-law spectrum, as would be expected from some FSR models (Essig et al. 2009), and both can be excluded with 99% confidence (Wang 2011). We further discuss sources with hard spectral indices in the Appendix.

Since AGNs are not expected to be spatially extended at an angular scale resolvable by the LAT, we cross checked 1FGL J2325.8−4043 against the Second LAT Source Catalog (Abdo et al. 2011). In two years of data, two sources were found within 05 of the location of 1FGL J2325.8−4043. In one year of data, these two sources could not be spatially resolved, but their existence was enough to favor an extended source hypothesis. Spurious measurements of a finite extent for point sources are not unexpected. Testing extension with a purity of 99%, the Poisson probability of finding at least one spurious source in our 385 tests is 98%. Since 1FGL J1302.3−3255 is associated with a pulsar and 1FGL J2325.8−4043 does not appear to be truly extended, we conclude that there were no unassociated, high-latitude spatially extended γ-ray sources in the first year of LAT data. Thus, according to the criteria defined in Section 3, no viable dark matter satellite candidates were found.

To better understand possible misidentification of pulsars (such as 1FGL J1302.3−3255) as candidate dark matter satellites, we applied our test of spectral shape to 25 high-latitude LAT-detected pulsars. Of these 25 pulsars, 14 were identified when the 1FGL was published and 11 were subsequently identified through follow-up observations by radio telescope (Hessels et al. 2011).

Interestingly, 24 of these pulsars passed our spectral cut, preferring a $b \bar{b}$ spectrum to a power-law spectrum. This can be understood by comparing the exponentially cutoff power-law model commonly used to fit pulsars with a $b \bar{b}$ WIMP annihilation spectrum. The exponentially cutoff power law has the form (Abdo et al. 2010g):

$$\begin{equation} \frac{dN}{dE}=KE^{-\Gamma }_{\mathrm{\,GeV}}{\exp \left(-\frac{E}{E_\mathrm{cut}}\right),\ } \end{equation} \tag{ 15 }$$

where Γ is the photon index at low energy, E_cut is the cutoff energy, and K is a normalization factor (in units of  photons cm⁻² s⁻¹ MeV⁻¹). In Figure 3, we plot both the exponentially cutoff power-law model and a low-mass (M_WIMP ∼ 25 GeV) $b \bar{b}$ spectrum and show that for E > 200 MeV the two curves are very similar.

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By fitting $b \bar{b}$ spectra to the 25 LAT-detected pulsars, we found that they tend to be best fit by low dark matter masses (Figure 4). Although our statistics are limited, the distribution peaks around 30 GeV, with most pulsars having a best-fit dark matter mass M_WIMP < 60 GeV. This suggests that unidentified, high-latitude pulsars can present a source of confusion in spectral searches for dark matter satellites. In general, many unassociated LAT sources have spectra that are inconsistent with a power-law model (Bonamente 2010; Abdo et al. 2011). The fact that these sources passed our spectral test does not imply that they are best fit by $b \bar{b}$ spectra, merely that $b \bar{b}$ spectra fit better than a simple power law. These unassociated, non-power-law sources were not found to share a consistent spectrum, as would be expected from dark matter annihilation.

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The abundance of non-power-law γ-ray sources emphasizes the importance of testing for spatial extension when attempting to identify dark matter satellites at high latitudes. Some concerns remain due to the fact that the LAT detects spatially extended pulsar wind nebulae located around some pulsars (Ackermann et al. 2011b). However, we do not expect the older pulsars at high Galactic latitudes to have nebulae that are spatially extended on a scale detectable by the LAT. Of course, there is always a risk that a chance coincidence with a low-flux neighboring source will cause apparent source extension.

The detection efficiency of our selection was defined as the fraction of dark matter satellites that pass the cuts in Section 3 and was calculated from Monte Carlo simulations. The efficiency for detecting a dark matter satellite depended on spectral shape (i.e., dark matter mass and annihilation channel), flux, and spatial extension. For a 100 GeV WIMP annihilating through the $b \bar{b}$ channel, we examined the efficiency for satellites with characteristic fluxes ranging from 5.0 × 10⁻¹¹ photons cm⁻² s⁻¹ to 5.0 × 10⁻⁸ photons cm⁻² s⁻¹ and characteristic spatial extension (as described in Section 3.3) from 05 to 20. These ranges were chosen to reflect the fluxes of the unassociated high-latitude LAT sources and angular extents to which the LAT is sensitive.⁵⁹

For each set of characteristics listed in Table 3, we utilized gtobssim to simulate 200 dark matter satellites with NFW profiles and $b \bar{b}$ spectra from a 100 GeV WIMP. These simulations were embedded in LAT data at random high-latitude locations, and Sourcelike was used to compute TS_ext, TS_spec, and the detection TS for each. The satellite detection efficiency was computed as the fraction of satellites with Sourcelike TS > 24, TS_ext > TS_ext⁹⁹, and TS_spec > TS_spec⁹⁹. The first requirement was included as a proxy for the efficiency of the source-finding algorithm. The creation of this efficiency table (Table 3) was computationally intensive and the result is likely model dependent, which limited this analysis to the examination of only the 100 GeV $b \bar{b}$ model. To expedite the generation of this table, we found the flux value with efficiency <0.05 and conservatively set the efficiency for sources with less flux to 0.

Table 3. Detection Efficiency

Fluxa	Extension
( photons cm−2 s−1)	05	10	20
0.2 × 10−8	<0.05	<0.05	<0.05
0.5 × 10−8	0.16	0.28	0.31
1.0 × 10−8	0.74	0.76	0.83
2.0 × 10−8	0.99	1.0	0.99
5.0 × 10−8	1.0	1.0	1.0

Notes. Satellite detection efficiency for 100 GeV WIMP annihilating through the $b \bar{b}$ channel. ^aIntegral flux from 200 MeV to 300 GeV.

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The VL-II and Aquarius simulations (described in Section 2) were used to predict the Galactic dark matter satellite population. Picking a vantage point 8.5 kpc from the center of each simulation (the solar radius), we calculated the spatial extension and integrated J-factor for each satellite. To account for variation in the local satellite population, we repeated this procedure, creating six realizations from maximally separated vantage points at the solar radius. It is important to note that while the VL-II and six Aquarius simulations are statistically independent, these different realizations are not—i.e., the same satellites appear in multiple realizations. Thus, we have seven independent simulations, each with six not-independent realizations (collectively referred to as 42 "visualizations").

After excluding undetectably faint satellites with J-factors an order of magnitude less than the lower bound the J-factor of Draco (Section 2.3), we compared the distributions of J-factors and spatial extensions across the visualizations. We found reasonable agreement in the detectable satellite distributions between the VL-II and Aquarius simulations. The variation in satellite number in bins of flux and spatial extension was much larger between different simulations than between realizations of the same simulation (as is expected, because the realizations are not independent). On average, satellites with α₀ > 05 make up ∼30% of the total integrated J-factor from satellites in these simulations.

For each of the 42 visualizations of VL-II and Aquarius, we calculated the γ-ray fluxes of all satellites for a given 〈σv〉 using Equation (1). With these fluxes and the true spatial extension for each satellite, we performed a bilinear interpolation on Table 3 to determine the detection efficiency for each satellite. The probability that the LAT would observe none of the satellites in visualization i is

$$\begin{equation} P_i(\langle \sigma v \rangle) = \prod _{j} (1 - \epsilon _{i,j}(\langle \sigma v \rangle)), \end{equation} \tag{ 16 }$$

where _{i, j} is the detection efficiency for satellite j in visualization i. Because there is no reason to favor any one visualization, we calculated the average null detection probability over the N = 42 visualizations as

$$\begin{equation} \bar{P} (\langle \sigma v \rangle) = \frac{1}{N} \sum _i^N P_i(\langle \sigma v \rangle). \end{equation} \tag{ 17 }$$

To set an upper limit on the dark matter annihilation cross section, we increased 〈σv〉 until the probability of a null observation was <5%, i.e., $\bar{P} < 0.05$. This corresponds to 95% probability that, for this 〈σv〉, at least one satellite would have passed our selection criteria. Using this methodology, the LAT null detection constrains 〈σv〉 to be less than 1.95 × 10⁻²⁴ cm³ s⁻¹ for a 100 GeV WIMP annihilating through the $b \bar{b}$ channel.