OBSERVATIONS OF MILKY WAY DWARF SPHEROIDAL GALAXIES WITH THE FERMI-LARGE AREA TELESCOPE DETECTOR AND CONSTRAINTS ON DARK MATTER MODELS

For a power-law spectrum, the differential flux is written as

$$\begin{equation} \frac{dN}{dEdAdt} = N_0\left(\frac{E}{E_0}\right)^{-\Gamma }\,, \end{equation} \tag{ 1 }$$

where E is the photon reconstructed energy in the restricted energy range 100 MeV–50 GeV, and E₀ is an arbitrary energy scale set to 100 MeV. Such a model has two unknown parameters, the photon index Γ and the normalization parameter N₀. In this analysis, we fix Γ to five possible values, Γ =  1, 1.8, 2.0, 2.2, and 2.4. While the last four indices provide constraints on standard astrophysical source spectra, the very hard index of Γ =  1 is motivated by the dark matter annihilation models in Essig et al. (2009). N₀ is fitted to the data in each ROI separately, together with the isotropic and Galactic diffuse normalizations. The best-fit values and corresponding errors of these three parameters, for the case Γ = 2, are gathered in Table 2. In the third column, the errors on N₀ are several orders of magnitude larger than the fitted values, which means that the latter are compatible with zero. We thus conclude that no significant signals from the direction of the selected dSphs are detected. The same conclusion is reached for the other values of the photon index Γ.

We expect variations across the ROIs around the ideal value of 1 for the normalizations of the two diffuse components, due to possibly slightly different exposure, statistical fluctuations, and inaccuracy of the underlying spatial model and spectral shape. Nevertheless, the fit results for the isotropic component remain close to 1, while the deviations for the Galactic diffuse component are somewhat larger, which is expected as spatial inaccuracies of the model are expected to vary from one ROI to another. Figure 1 shows examples of the spectral fits to the data and the residuals of these fits for Willman 1, which has the largest fit residuals, and Draco, which has residuals typical of most of the fits.

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Flux upper limits are then derived, based on a profile likelihood technique (Bartlett 1953; Rolke et al. 2005). In this method, the normalization of the power-law source representing the dwarf galaxy is scanned away from the fitted minimum while the remaining two free parameters (the normalizations of the two diffuse backgrounds) are fit at each step. The scanning proceeds until the difference of the logarithm of the likelihood function reaches 1.35, which corresponds to a one-sided 95% confidence level (C.L.). Extensive work to test this method, using Monte Carlo simulations as well as bootstrapping the real data, indicate that profile likelihood method as implemented in the Science Tools is slightly overcovering the expected confidence level.⁵⁸ As a result, we believe the ULs presented in this paper are conservative.

In Table 3, we report flux upper limits in two different energy ranges (from 100 MeV to 50 GeV and from 1 GeV to 50 GeV), for the five different values of Γ. As expected, the Fermi-LAT limits are stronger for a hard spectrum which predicts relatively more photons at the dwarf location at high energy. At higher energies, the LAT PSF is significantly reduced while the effective area is significantly higher; in addition, the diffuse backgrounds have relatively soft spectra compared to all but the softest models considered. For example, for a power law index of 1 when analyzing the full energy range, the upper limits are roughly 10 times lower than for an index of 2.2.

As can be seen from Table 3, the different dwarfs give roughly similar gamma-ray flux upper limits, as expected given the fairly uniform Fermi exposure across the sky. However, the limits do vary from dwarf to dwarf due to, for example, the direction dependence of the diffuse background intensity and the proximity of bright gamma-ray sources. In general, Ursa Minor gives the lowest flux limits while Sculptor, which is within a couple of degrees of a bright background point source, gives the highest flux limits.

While power-law spectra can be justified on astrophysical grounds, a proper search for a dark matter signal should take account of the specific spectrum resulting from WIMP annihilations. At a given photon energy E, the γ-ray flux originating from WIMP particle annihilations with a mass m_WIMP can be factorized into two contributions (Baltz et al. 2008): the "astrophysical factor" J(ψ) related to the density distribution in the emission region and the "particle physics factor" Φ^PP which depends on the candidate particle characteristics:

$$\begin{equation} \phi _{\rm WIMP}(E,\psi)=J(\psi)\times \Phi ^{\rm PP}(E),\, \end{equation} \tag{ 2 }$$

where ψ is the angle between the direction of observation and the dSph center (as given in Table 1). Following notations of Baltz et al. (2008), J(ψ), and Φ^PP are defined as

$$\begin{equation} \Phi ^{\rm PP}(E) =\frac{1}{2}\frac{\langle \sigma v\rangle }{4\pi \ m_{\rm WIMP}^2} \sum _f \frac{dN_f}{dE} B_f \end{equation} \tag{ 3 }$$

and

$$\begin{equation} J(\psi)= \int _{\rm {l.o.s}} dl(\psi) \rho ^2(l(\psi)), \end{equation} \tag{ 4 }$$

where 〈σv〉 is the relative velocity times the annihilation cross section of the two dark matter particles, averaged over their velocity distribution, and the sum runs over all possible pair-annihilation final states f, with dN_f/dE and B_f the corresponding photon spectrum and branching ratio, respectively. The integral in Equation (4) is computed along the line of sight (l.o.s.) in the direction ψ, and the integrand ρ(l) is the assumed mass density of dark matter in the dSph.

For each galaxy, we model the dark matter distribution with a NFW (Navarro et al. 1997) density profile within the tidal radius, as is reasonable for cold dark matter subhalos in Milky Way-type host halos (Diemand et al. 2007; Springel et al. 2008):

$$\begin{equation} \rho (r) = \left\lbrace \begin{array}{@{}ll@{}} {{\rho _s r_s^{3}} \over {r}(r_s+r)^2} & \mbox{for} \quad r < r_t \\[6pt] 0 & \mbox{for} \quad r \ge r_t \end{array} \right., \end{equation} \tag{ 5 }$$

where ρ_s is the characteristic density, r_s is the scale radius, and r_t is the tidal radius. The l.o.s. integral in Equation (4) may be computed once ρ_s, r_s, and r_t are known. The tidal radius of the dwarf's dark matter halo in the gravitational potential of the Milky Way is self-consistently computed from the Jacobi limit (Binney & Tremaine 1987) for each set of ρ_s and r_s values assuming a mass profile for the Milky Way given by Catena & Ullio (2009). The sharp cut-off in the density profile is rather extreme, but it is conservative in the sense that it truncates the probability distribution of expected γ-ray fluxes at the high end.

The halo parameters (ρ_s, r_s), and the resulting J factor from Equation (4), are estimated following the methodology outlined in Martinez et al. (2009). The observed l.o.s. stellar velocities are well described by a Gaussian distribution (Muñoz et al. 2005, 2006; Walker et al. 2007, 2009; Geha et al. 2009), and we include the dispersion arising from both the motion of the stars and the measurement errors as Strigari et al. (2007):

$$\begin{eqnarray} {\mathcal {L}}(\mathscr{A}) &\equiv & P(\lbrace v_i\rbrace | {\mathscr{A}}) \nonumber\\ &=& \prod _{i=1}^{n} \frac{1}{\sqrt{2\pi \big(\sigma _{{\rm los}, i}^2 + \sigma _{m, i}^2\big)}} \exp \left[-\frac{1}{2} \frac{(v_i -u)^2}{\sigma _{{\rm los}, i}^2 + \sigma _{m, i}^2} \right]\,,\qquad \end{eqnarray} \tag{ 6 }$$

where {v_i} are the individual l.o.s. stellar velocity measurements and σ_m,i are the measurement errors on these velocities. The mean l.o.s. velocity of the dwarf galaxy is denoted by u. The full set of astrophysical parameters is $\mathscr{A}={\rho _s,r_s,\Upsilon _\star,\beta,u}$, and we discuss the two new parameters ϒ_⋆ and β below. The theoretical l.o.s. dispersion, σ_los, is the projection of the three-dimensional velocity dispersion on the plane of the sky and this is determined using the Jeans equation (see Binney & Tremaine 1987) once $\mathscr{A}$ is specified. ϒ_⋆ is the stellar mass to light ratio and it sets the mass of the baryons in these dwarf galaxies given the stellar luminosity. The velocity dispersion anisotropy is β ≡ 1 − σ²_t/σ²_r, where σ_t and σ_r are the tangential and radial velocity dispersion of the stars (measured with respect to the center of the dwarf galaxy). We assume that β is constant for this analysis. The probability of the astrophysical parameters, $\mathscr{A}$ given a data set {v_i} is obtained via Bayes' theorem: $\mathcal {P} (\mathscr{A}|\lbrace v_i\rbrace) \propto \mathcal {P}(\lbrace v_i\rbrace |\mathscr{A}) \mathcal {P}(\mathscr{A})$. The prior probability, $\mathcal {P}(\mathscr{A})$, for the halo parameters, {r_s, ρ_s} is based on ΛCDM simulations (Diemand et al. 2007; Springel et al. 2008) and described in detail in Martinez et al. (2009). For ϒ_⋆ we take the prior to be uniform between 0.5 and 5, and for β the prior is uniform between −1 and 1.

The astrophysical factor J after marginalization over all the parameters in $\mathscr{A}$ for each dwarf galaxy within an angular region of diameter 1° is given in Table 4. The chosen 1° region for the calculation of J is a good match to the LAT PSF at energies of 1–2 GeV, where most of the models under consideration are best constrained. At lower energies, the PSF is significantly larger, but beyond 1° the dwarf dark matter density has a negligible impact on the overall J computation, and at higher energies, the statistics with the current data are rather limited. Note that, due to their uncertain nature as true dark matter dominated dSphs or large uncertainties in their dark matter content, the Segue 2, Willman 1, and Bootes II dSphs have not been considered in this analysis. In addition, new stellar data on Segue 1 and Bootes II are being currently reduced and will be used in a forthcoming publication. We also exclude Ursa Major I, Hercules, and Leo IV, because their J values are smaller than those of the rest of the sample, yielding a final sample of 8 dSphs used for the dark matter constraints.

Table 4. Properties of the Dark Matter Halos of Dwarf Spheroidals Used in this Study

Name	[〈R〉, 〈P〉]	[〈R2〉 − 〈R〉2, 〈P2〉 − 〈P〉2, 〈RP〉 − 〈R〉〈P〉]	JNFW
		R ≡ log10(rs/kpc), P ≡ log10(ρs/M☉ kpc−3)	($10^{19} \frac{{\rm GeV}^2}{{\rm cm}^{5}}$)
Ursa Major II	[−0.78, 8.54]	[0.0417, 0.0986, −0.0554]	0.58+0.91−0.35
Coma Berenices	[−0.79, 8.41]	[0.0603, 0.132, −0.0820]	0.16+0.22−0.08
Bootes I	[−0.57, 8.31]	[0.0684, 0.165, −0.0931]	0.16+0.35−0.13
Usra Minor	[−0.19, 7.99]	[0.0430, 0.116, −0.0697]	0.64+0.25−0.18
Sculptor	[−0.021, 7.57]	[0.0357, 0.0798, −0.0528]	0.24+0.06−0.06
Draco	[0.32, 7.41]	[0.0236, 0.0364, −0.0286]	1.20+0.31−0.25
Sextans	[−0.43, 7.93]	[0.0302, 0.109, −0.0570]	0.06+0.03−0.02
Fornax	[−0.24, 7.82]	[0.0474, 0.140, −0.0798]	0.06+0.03−0.03

Notes. These parameters are obtained from measured stellar (l.o.s.) velocities. ρ_s and r_s are the density and scale radius for the dark matter halo distribution. The first column, [log₁₀(ρ_s), log₁₀(r_s)], is the average in the joint log₁₀(r_s) − log₁₀(ρ_s) parameter space, whose posterior is well described by a Gaussian distribution centered on the average value given. The second column gives the diagonal and off-diagonal components of the covariance matrix that may be used to approximate the joint probability distribution of ρ_s and r_s as a Gaussian in log₁₀(r_s) and log₁₀(ρ_s). The last column provides J^NFW (see Equation (4)), which is proportional to the pair-annihilation flux coming from a cone of solid angle 2.4 × 10⁻⁴ sr centered on the dwarf. The errors on J^NFW are obtained from the full MCMC probability distribution and bracket the range which contains 68% of the total area under the probability distribution.

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In principle, annihilations in cold and dense substructure in the dwarf galaxy halo can increase J. However, previous studies have shown that this boost due to annihilations in substructure is unlikely to be larger than a factor of few (see, e.g., Martinez et al. 2009). Similarly, a boost in the annihilation cross section in dwarfs due to a Sommerfeld enhancement (e.g., Arkani-Hamed et al. 2009), where the annihilation cross section depends on the relative velocity of the particles, would increase the expected gamma-ray signal and improve our constraints. In order to be conservative, we have not included either of these effects.

Using Equation (2), the results given in Table 4, and the DMFit package (Jeltema & Profumo 2008a) as implemented in the Science Tools, 95% C.L. upper limits on photon fluxes (above 100 MeV) and on 〈σv〉 have been derived as a function of the WIMP mass, for each dSph and for specific annihilation channels. Our choices for the pair-annihilation final states are driven by theoretical as well as phenomenological considerations: a prototypical annihilation final state is into a quark–antiquark pair. The resulting γ rays stem dominantly from the decay of neutral pions produced in the quark and antiquark hadronization chains, and do not crucially depend upon the specific quark flavor or mass; in fact, a very similar γ-ray spectrum is produced by the (typically loop-suppressed) gluon-gluon final state. Here, for illustration we consider the specific case of a $b\bar{b}$ final state: this choice is motivated by the case of supersymmetric dark matter (see Jungman et al. 1996, for a review). In supersymmetry with R-parity conservation, the prototypical WIMP dark matter candidate is the lightest neutralino, the mass eigenstate resulting from the superposition of the fermionic partners of the hypercharge and SU(2) neutral gauge bosons and of the two neutral Higgs bosons. At moderate to large values of tan β, if the lightest neutralino is bino-like (i.e., if the U(1) hypercharge gauge eigenstate is almost aligned with the lightest neutralino mass eigenstate), dark matter dominantly pair annihilates into $b\bar{b}$. Another final state that is motivated by supersymmetric dark matter is into τ⁺τ⁻, that dominates in the case of a low-mass scalar superpartner of the τ lepton, as is the case e.g., in the so-called co-annihilation region of minimal supergravity (mSUGRA). An intermediate case with a mixed $b\bar{b}$ and τ⁺τ⁻ final state is also ubiquitous in supersymmetry, since those are the two heaviest down-type matter fermions in the Standard Model. The additional color factor and a larger value for the mass typically favor a relatively larger $b\bar{b}$ branching fraction. While the choice of the final states we consider is motivated here by supersymmetry, the results we find apply to generic WIMP models and not only to neutralinos.

Figure 2 shows the derived upper limits on the photon fluxes for all selected dwarfs and for various annihilation final states (respectively, 100% $b\bar{b}$ in the upper left panel, 100% τ⁺τ⁻ in the upper right panel, and a mixed 80$\%\ b\bar{b}$ + 20%τ⁺τ⁻ final state in the lower left panel). The lower right plot of Figure 2 illustrates, for the case of Ursa Minor, how upper limits on the γ-ray fluxes change as a function of mass depending on the selected final state. Final states producing a hard γ-ray spectrum, such as μ⁺μ⁻ and τ⁺τ⁻ result in the best upper limits, since they predict abundant photons fluxes at larger energies, where the diffuse background is lower. With increasing mass, the spectrum from WIMP annihilation is rigidly shifted to higher energies, and the advantage of having a harder spectrum is less critical: for m_DM ∼ 1 TeV the upper limits we obtain for all the final states we consider vary within a factor 3, versus more than one order of magnitude at lower masses.

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The results presented in Figure 2 bracket realistic cases of theoretically motivated particle dark matter models, but since they only refer to the WIMP annihilation final state they do not depend on the particular assumed particle theory model. We consider next a few motivated specific WIMP dark matter scenarios, and study how the corresponding parameter space is constrained. We consider first the well-known case of mSUGRA (Chamseddine et al. 1982; Barbieri et al. 1982; see also Nilles 1984 for an early review), where the supersymmetry breaking parameters are typically specified at a high-energy scale (typically taken to be the grand unification scale M_GUT ≃ 2 × 10¹⁶ GeV) and assumed to be universal at that scale. Those parameters are the universal high-scale supersymmetry-breaking sfermion mass m₀, gaugino mass M_1/2 and trilinear scalar coupling A₀; an additional (low-scale) parameter is the ratio of the vacuum expectation values of the two Higgs doublets, tan β, and the sign of the higgsino mass parameter μ. Here, we adopt the following ranges for the various parameters (linear scan): 80 < m₀ < 10⁵, 80 < M_1/2 < 4500, A₀ = 0, 1.5 < tan β < 60 and with sign(μ) being not constrained.

A less constrained alternative is to consider a "phenomenological" Minimal Supersymetric Standard Model (MSSM) setup (see e.g., Chung et al. 2005, for a review of the most general MSSM Lagrangian), where all soft supersymmetry breaking parameters (i.e., the positive mass-dimension coefficients of Lagrangian terms that explicitly break supersymmetry) are defined at the electro-weak (low-energy) scale, possibly with a few simplifying assumptions (see e.g., Profumo & Yaguna 2004, for an early attempt at a scan of the MSSM parameter space). Here, we consider the reduced set of parameters considered in Gondolo et al. (2004), and perform a logarithmic scan over the following parameters: |μ| < 10⁴, |M₂| < 10⁴, 100 < m_A < 1000, 1.001 < tan β < 60, $100 < m_{\tilde{q}} < 2 \times 10^{4}$, while the scan is linear in $-5 < A_{t}/m_{\tilde{q}} < 5$ and $-5 < A_{b}/m_{\tilde{q}} < 5$.

In addition to the two scenarios considered above, we also entertain two additional specific WIMP dark matter models. The first one is the lightest Kaluza–Klein particle of universal extra dimensions (UED; see e.g., Cheng et al. 2002; Servant & Tait 2003; for a review, see Hooper & Profumo 2007), where, in the minimal setup, the dark matter candidate corresponds to the first Kaluza–Klein excitation of the U(1) hypercharge gauge boson, also known as B⁽¹⁾. In this case, there is an almost fixed relationship between the dark matter mass and its pair-annihilation cross section, and a thermal relic abundance in accord with the dark matter density is obtained for masses around 700 GeV (Servant & Tait 2003).

The second model, which was recently considered in Kane et al. (2009) as a natural and well-motivated scenario in connection with the anomalous positron fraction reported by PAMELA (Adriani et al. 2009), is that of wino-like neutralino dark matter (Moroi & Randall 2000). Wino-like neutralinos (which for brevity we will refer to as "winos"), i.e., neutralino mass eigenstates dominated by the component corresponding to the supersymmetric fermionic partners of the SU(2) gauge bosons of the Standard Model, pair annihilate very efficiently into pairs of W⁺W⁻ (if their mass is larger than the W mass) and the pair-annihilation cross section is fixed by gauge invariance once the wino mass is given. Winos arise in various supersymmetry breaking scenarios and in several string motivated setups, where e.g., the anomaly mediation contribution to the gaugino masses dominates over other contributions, setting M₂ < M₁. Typical such scenarios of anomaly mediated supersymmetry breaking (AMSB) were considered e.g., in Randall & Sundrum (1999); Giudice et al. (1998). Although winos produce a thermal relic density matching the universal dark matter density for masses around 2 TeV, several non-thermal production mechanisms have been envisioned that could explain a wino dark matter scenario with lighter dark matter candidates than a TeV.

Figure 3 compares the resulting LAT sensitivity in the (m_wimp,〈σv〉) plane with predictions from mSUGRA, MSSM, Kaluza–Klein dark matter in UED, and wino-like dark matter in AMSB. All mSUGRA and MSSM plotted models are consistent with accelerator constraints. Red points are compatible with the 3σ WMAP constraint on the universal matter density under thermal production while blue points would have a lower thermal relic density. For the blue points, we assume that the production mechanism is non-thermal in order to produce the observed universal matter density, and we therefore do not rescale the neutralino density by the factor (Ω_thermal/Ω_DM)², which would result assuming exclusive thermal production. This is very natural in the context of several string-theory motivated frameworks, where moduli generically decay into both Standard Model particles and their supersymmetric partners, which in turn eventually decay into the lightest neutralinos Moroi & Randall (2000). Topological objects such as Q-balls can also decay and produce neutralinos out of equilibrium, as envisioned e.g., by Fujii & Hamaguchi (2002) and Fujii & Ibe (2004). Another possible scenario is one where the expansion history of the universe is more rapid than in a radiation-dominated setup, for instance because of a dynamical "quintessence" field in a kinetic-dominated phase (Salati 2003; Profumo & Ullio 2003).

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Figure 3 clearly shows that, after less than a year of Fermi data survey, the upper limits on the γ-ray flux from dSphs are already starting to be competitive for MSSM models, provided that these models correspond to low thermal relic density. Draco and Ursa Minor dSphs set the best limits so far. Pending more data, they may also start to constrain mSUGRA models with low thermal relic density as well. Furthermore, these flux upper limits already disfavor AMSB models with masses <300 GeV. Interestingly, our results strongly constrain the models considered in Kane et al. (2009), invoking a 200 GeV mass wino.

The recent detection by the PAMELA experiment of a positron fraction that increases with energy above 10 GeV (Adriani et al. 2009) and the possibility that dark matter annihilation in the Galaxy could produce this "positron excess" (among more mundane explanations such as pulsars) have spurred great interest in the particle physics community. The pair annihilation of galactic WIMP dark matter can, in principle, produce an anomalous excess in the positron fraction at energies between a few GeV and ∼100 GeV. The spectrum of high-energy e⁺ + e⁻, although compatible with a purely canonical cosmic-ray origin (Abdo et al. 2009; Grasso et al. 2009), can also accommodate an additional component due to galactic dark matter annihilation.

A dark matter annihilation interpretation of the positron excess implies preferentially a leptonic final state, to avoid the overproduction of antiprotons. This is very hard to achieve in the minimal supersymmetric extension of the Standard Model (see however Kane et al. 2009, for the case of AMSB, which we will not consider further here). In addition, the spectral shape of high-energy e⁺e⁻ points toward rather large masses, and the level of the needed local positron flux indicates either a very large pair-annihilation rate, or a strong enhancement in the local dark matter density. Using a canonical primary electron injection spectrum, the analysis of Bergström et al. (2009) further indicates that a preferred annihilation final state is μ⁺μ⁻, or the somewhat softer (in the produced e⁺e⁻ spectrum) but essentially very similar four body μ⁺μ⁻μ⁺μ⁻ final state. Theoretical arguments that could explain this peculiar annihilation final state, possibly involving mechanisms to enhance the low-velocity annihilation rate, have been proposed (Arkani-Hamed et al. 2009; Nomura & Thaler 2009). With standard assumptions on the dark matter density profile, and assuming a μ⁺μ⁻ final state, the regions in the pair-annihilation cross section versus mass plane preferred by the Fermi-LAT e⁺e⁻ data are shown in orange in Figure 4, while those favored by the PAMELA positron fraction data are highlighted in light blue (for details on the computation of these regions see Profumo & Jeltema 2009).

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In a pair-annihilation event producing a μ⁺μ⁻ pair, γ rays result from both the internal bremsstrahlung of the muons (final state radiation), with the well-known hard power-law spectrum dn_γ/dE_γ ∼ E⁻¹_γ, and from the IC up-scattering of cosmic microwave background (CMB) light by the e⁺e⁻ resulting from muon decay. The dark matter interpretation of the "cosmic-ray lepton anomalies" implies significant γ-ray emission from a variety of sources; predictions and constraints on these models have been discussed extensively in the recent literature (see e.g., Profumo & Jeltema 2009, for a discussion of the constraints from the expected IC emission from annihilation at all redshifts and in all halos).

The left panel of Figure 4 illustrates the constraints we derive from 11 months of Fermi data on local dSph on a generic WIMP dark matter pair annihilating into a μ⁺μ⁻ final state (we do not specify here any particular particle physics scenario, although as stated above several examples have been considered in the literature), considering the final state radiation emission only. Here the spectral modeling was done using the DMFit package as in Section 3.2. The hierarchy among the constraints derived from the various dSph is very similar to what we find for the softer pair-annihilation final states considered in Figure 3. Neglecting any low-velocity enhancement of the annihilation rate (Arkani-Hamed et al. 2009), which would boost the γ-ray signal from dSph and hence the constraints we show, the final state radiation alone does not yet exclude portions of the parameter space favored by the dark matter annihilation interpretation of the cosmic-ray lepton data.

The calculation of the γ-ray yield from IC is complicated, in systems as small as dSphs, by the fact that e⁺e⁻ are not confined (i.e., their diffusion lengths are typically larger than the physical size of the system). In fact, the typical energy-loss length-scale for TeV e⁺e⁻ loosing energy dominantly via IC off of photons in the CMB is of the order of hundreds of kpc, much larger than the size of dSph. Assumptions on cosmic-ray diffusion are therefore critical, as discussed, e.g., in Colafrancesco et al. (2007) and in Jeltema & Profumo (2008b). In the absence of any direct cosmic-ray data for external galaxies such as dSph (the only piece of information being that dSph are gas-poor environments with typically low magnetic fields), we consider the usual diffusion-loss equation and solve it in a spherically symmetric diffusive region with free-escape boundary conditions. We employ a diffusion coefficient at the level of what is usually inferred for cosmic rays in our own Milky Way (e.g., Strong et al. 2007). For a thorough discussion of the diffusion model we adopt here, we refer the reader to Colafrancesco et al. (2007) and Jeltema & Profumo (2008b). Specifically, we consider two values for the diffusion coefficient, D₀ = 10²⁸ cm² s⁻¹ and D₀ = 10²⁹ cm² s⁻¹ bracketing the values typically inferred for the Galaxy: the larger the diffusion coefficient, the larger the cosmic-ray mean free path, and the larger the leakage of cosmic-ray e⁺e⁻ out of the dwarf, leading to a suppression of the IC signal. We also assume a power-law dependence of the diffusion coefficient on energy given by $D(E)=D_0\ \left(\frac{E}{\rm 1\ GeV}\right)^{1/3}$. In the energy-loss term, IC emission off of CMB photons by far dominate over both synchrotron and starlight IC losses.

We show our results, including both final state radiation (FSR) and IC emission off of CMB photons, in Figure 4, right panel, for the case of the Ursa Minor dSph. Here the spectrum was modeled self-consistently with custom spectra which include the expected γ-ray emission from both IC and FSR for a given assumed particle mass and diffusion coefficient for a grid of particle mass values ranging from 100 MeV to 10 TeV. The Ursa Minor dSph was chosen as an example of one of the best cases for this particular study. In the smaller ultrafaint dwarfs, diffusion is expected to have a much larger effect due to the much smaller diffusive region (modeled based on the stellar extent), and IC emission for the diffusion coefficients assumed here will not add significant flux above what is expected from FSR alone.

The lower and upper lines in Figure 4 indicate the range of uncertainty in the determination of the dark matter density profile of the Ursa Minor dSph. With the smaller diffusion coefficient choice, and for large enough masses (producing higher energy e⁺e⁻ and subsequent IC photons), the IC emission dominates, and it exceeds our γ-ray upper limits for models that fit the PAMELA data and have masses larger than 1 TeV. Excessive emission is predicted also in the more conservative case with D₀ = 10²⁹ cm² s⁻¹ for some models with masses in the 2–5 TeV range.