Model-independent constraints on dark matter annihilation in dwarf spheroidal galaxies

We present a general, model-independent formalism for determining bounds on the production of photons in dwarf spheroidal galaxies via dark matter annihilation, applicable to any set of assumptions about dark matter particle physics or astrophysics. As an illustration, we analyze gamma-ray data from the Fermi Large Area Telescope to constrain a variety of nonstandard dark matter models, several of which have not previously been studied in the context of dwarf galaxy searches.

The plan of this work is as follows. In Section II, we describe the general analysis framework. In Section III, we use this framework to present bounds on several DM particle and astrophysics scenarios, including several scenarios for which no previous bounds have been exhibited. We conclude with a discussion of our results in Section IV.

II. ANALYSIS FRAMEWORK
The expected number of photons within a given energy range that arise from DM annihilation in a particular dSph is where J(∆Ω) is the astrophysical J-factor, T obs is the observation time,Ā eff is the average effective area of the detector, and Φ PP , a quantity determined only by the DM particle physics model, is given by where m X is the DM mass and dN γ /dE γ is the photon energy spectrum per annihilation. The effective area of the Fermi-LAT is energy dependent; however, we work in an energy range in which the effective area is approximately constant [A eff (E γ ) =Ā eff ] at the few percent level. The annihilation cross section times relative velocity is often assumed to be constant: (σv) = (σv) 0 . To account for a nontrivial dependence on v on the calculation of the J-factor (which is determined by the DM velocity distribution [10]), we write the annihilation cross section as where S(v) is some function of the relative velocity. The J-factor is then where is the distance along the line-of-sight and f (r, v) is the DM velocity distribution. In the limit of s-wave annihilation [S(v) = 1], we recover the standard result for σv = (σv) 0 : J = ∆Ω dΩ d ρ 2 . Note that although dSphs are ideal systems for searches of DM annihilation, this formalism is also applicable for DM decay by substituting (σv) 0 /2m X → Γ and J → J D ≡ ∆Ω dΩ d ρ, where Γ is the decay width. The factors that go into N DM can thus be categorized in the following way: 1. T obsĀeff depends on the specifications of the detector.
2. Φ PP depends only the particle physics model for DM annihilation. 3. J contains information about the DM distribution, as well as information about the velocity-dependence of the particle physics model for DM annihilation.
In particular, Φ PP is completely independent of the choice of target dSph, while T obsĀeff depends on the region of the sky being observed (i.e., the location of the dSph). On the other hand, J relies on the detailed properties of the target dSph and is subject to significant systematic uncertainty. The expected total number of photons arising from DM annihilation in a set of dSphs is Our aim is to place a bound on this quantity using Fermi-LAT data. The data provide T obsĀeff for each dSph, and we use values of J from a variety of previous works. The bound on N tot DM then translates into a bound on Φ PP . In order to place constraints on N tot DM , we first need to find the background distributions for the dSphs.
A. Estimating the astrophysical background One of the major advantages of using dSphs to search for DM annihilation is their low baryonic content and clean environment. Well above the Galactic disk, the expected astrophysical contribution to the observed gamma-ray spectrum is from diffuse emission and point sources. We can choose a region of interest (ROI) around a particular dSph and quantify how likely it is that the number of counts coming from the location of the dSph is or is not consistent with a DM source, given the number of counts in the ROI slightly away from the dSph. Following Ref. [6], we find the empirical background distribution for each dSph with the following procedure: 1. Choose an ROI, labeled by i, that is centered on the dSph with a radius of 10 • on the sky.
2. The number of observed photons, N i obs , from the dSph are all those within a radius of 0.5 • (∆Ω = 2.4 × 10 −4 ) of the dSph's central location.
3. Randomly choose 10 5 sample regions within the ROI of the same size as the target dSph (0.5 • ).
4. Reject any sample region whose boundary intersects the border of the ROI or the boundary of a known source region, defined to be within 0.8 • of a known point source.
5. Histogram the number of counts for the surviving sample regions.
The resulting histogram is the probability mass function P i bgd (N i bgd ) for the ROI to contain N i bgd counts in an arbitrary region of 0.5 • .
Increasing the number of sample regions or increasing the size of the source masks has negligible effects on our overall results. We chose the size of the target and sample regions to be 0.5 • because many J-factor calculations are performed over a cone of radius 0.5 • . We note that there are certain dSphs for which a known point source is within 1.3 • , which violates the above criteria for distinguishing the target, background, and known point source regions. Previous studies [6,7] have included these "contaminated" dSphs in their counting analyses, possibly weakening their results. While we acknowledge this issue, the gain from including more dSphs outweighs the disadvantage of incorporating additional photons whose origin is likely a nearby point source. Using the contaminated dSphs is acceptable for placing upper limits on DM, but we note that they cannot be used to make a claim for a DM signal.

B. Constraining dark matter
Once we have determined the background distributions for individual dSphs, we convolve these distributions to find the total probability mass function for a set of stacked dSphs: The total number of observed photons is N tot obs = i N i obs . For a given expected number of photons arising from DM annihilation, we assume that the actual number of such photons is drawn from a Poisson distribution, The expected total distribution is the convolution of the DM signal and the background. For an input value of N tot DM , the probability of producing more than the total observed number of photons N tot obs from the dSphs is Then, the upper bound on the expected number of photons arising from DM annihilation (at confidence level β), N bound (β), is given by Any model for which N tot DM > N bound (β) may be rejected at the β confidence level. Note that this upper bound on the expected number of total photons arising from DM annihilation is derived entirely from Fermi-LAT data, with no dependence on either the particle physics model or any astrophysical assumptions about the DM velocity or density distribution.
The corresponding upper bound on Φ PP at β confidence level is We treat the systematic uncertainties in the J-factors following the approach of Ref. [6].

III. RESULTS
We apply this formalism to the Fermi-LAT Pass 8 data set in the mission elapsed time range of 239557417 to 533867602 seconds. We incorporate photons in the energy range 1-100 GeV, with evclass=128 and evtype=3. We set zmax=100 and use the filter '(DATA_QUAL>0)&&(LAT_CONFIG==1)'. To process the Fermi-LAT data, we use the Fermi Science Tools, v10r0p5.
In the following subsections, we verify our methodology, determine N bound (β) for several different sets of dSphs, and present our constraints on Φ PP . Finally, we apply this analysis to constrain model parameters in several particle physics scenarios.

A. Comparison to prior results
In order to verify that our stacking procedure gives reasonable bounds on Φ PP relative to more complicated analyses, we first reproduce the analysis of Ref. [6]. They weight events by the signal-to-noise ratio expected from each individual dSph. In their analysis, an excess event from a dSph with a larger J-factor has a greater probability of being a signal event and thus has more constraining power. We mimic their Pass 7 analysis as closely as possible, with the exception of using a more recent version of Fermi Science Tools, and find Φ PP using the same J-factors from Ref. [2]. We find In the following analysis we opt for the simplest weighting scheme for stacking dSphs, i.e., all events are equally weighted, as described at the beginning of this section. This stacking procedure yields Φ PP = 6.62 +9.38 −4.27 × 10 −30 cm 3 s −1 GeV −2 at 95% C.L. Although the stacking bound is weaker, it is consistent with the signal-to-noise bound, given the uncertainties in the J-factors.

B. Determination of N bound (β) from Fermi-LAT data
We consider five sets of dSphs as detailed in Table I in the Appendix: 1. Set 1: The set of 45 objects considered in Ref. [5], which includes 28 confirmed dSphs, 13 likely galaxies, and 4 ambiguous systems.
(a) Set 1a: The subset that includes only the 28 confirmed dSphs.
(b) Set 1b: The subset that includes the 28 confirmed dSphs and the 13 likely galaxies.
(c) Set 1c: The subset that includes 27 dSphs for which the 0.5 • radius around the central location of each dSph does not intersect the 0.8 • mask around nearby point sources.

Set 2:
The set of 27 dSphs for which s-wave J-factors have been calculated in Ref. [11].
3. Set 3: The set of 24 dSphs for which J-factors for non-spherical halos have been calculated in Ref. [12].

Set 4:
The set of 7 dSphs for which J-factors modified for foreground effects have been calculated in Refs. [13,14].

Set 5:
The set of 5 dSphs considered in Ref. [10], for which Sommerfeld-enhanced J-factors have been calculated in the Coulomb limit.
Each of these is a different set of objects, although many dwarfs appear in multiple sets. The differences between these sets lie in one's assessment of which objects are actually dwarf spheroidal galaxies (and thus should be used in a search for DM annihilation), in the possibility of background contamination from point sources, and in assumptions about how one computes the J-factor (including how to treat systematic uncertainties, assumptions about the DM mass distribution, and the effect of the velocity-distribution on DM annihilation). These assumptions thus determine which of the above sets of objects are appropriate for a DM search. Given that choice of the set of objects, the quantity that is relevant to a search for DM annihilation is N bound (β), the upper bound (at confidence level β) on the total expected number of photons arising from DM annihilation in that set of objects. This quantity encapsulates everything that one needs to know from Fermi-LAT photon data. In Fig. 1, we plot the total background distribution for each set of dSphs, as well as the number of photons observed. For each set, the expected total number of background counts (N tot bgd ) and the actual number of photons observed (N tot obs ) are given in the figure legend. The background photon distributions for each of the individual dSphs is provided in Fig. 10 in the Appendix and in the supplementary file. In Fig. 2, we plot N bound (β) for each set of dSphs. Note that the normalizations of the background distributions do depend implicitly on the Fermi-LAT exposures on each dwarf, which are listed in Table I in the Appendix. Although the choice of the appropriate set of dSphs may be motivated by assumptions about astrophysics, N bound (β) itself is entirely independent of any assumptions about DM physics. For example, to constrain a model of Sommerfeld-enhanced DM annihilation, the dSphs given in Set 5 should be used, because Sommerfeld-enhanced J-factors have been computed for those objects. If Sommerfeld-enhanced J-factors are eventually determined for all of the objects in the larger Set 1, then one may instead use that set of objects; the only input needed from Fermi-LAT photon data would be N Set 1 bound (β) already presented above. Indeed, Sommerfeld-enhanced J-factors have recently been computed in Ref. [15] for a set of 20 dSphs, though using a different methodology than that used in Ref. [10]. Although this set of 20 dSphs is not one of those for which we have plotted N bound (β), it is possible to compute N bound (β) for any set of the 47 objects of Fig. 10, using the background distributions and the numbers of observed counts found therein, as well as the formulae in Section II B.

C. Constraints on Φ PP
To transform N bound (β) into a constraint on Φ PP , we must now plug in specific J-factors. We consider the data sets of the previous subsection with their associated J-factors. In Fig. 3, we plot Φ bound PP (β) as a function of β for each of these sets. In each case, the width of the band arises from varying all J-factors through their 1σ uncertainties.
At 95% C.L.  Using Eq. (10) and the exposures given in Table I in the Appendix, Φ bound PP (β) can be rescaled appropriately for any determination of the relevant J-factors.
Note that for Set 5, the Sommerfeld-enhanced J-factors were computed assuming that the dark fine structure constant is α X = 0.01, and in the limit of a Coulomb-like interaction, the Sommerfeld-enhanced J-factor is proportional to α X [10]. For a different choice of α X , Φ bound,Set 5 PP (β) should be rescaled by a factor 0.01/α X .

D. Constraints on particle physics parameters
Finally, we translate constraints on Φ PP into constraints on DM parameters, for several choices of interaction models. For the purpose of illustration in this subsection, we focus on obtaining constraints on different particle physics models, while making nominal assumptions about DM astrophysics. We consider the following particle physics scenarios: 1. Particle Physics Scenario 1 : DM with mass m X annihilates with a total s-wave cross section (σv) 0 to a twobody final state. We considerτ τ ,bb, W + W − andμµ final states, each with 100% branching fraction, and the dSphs and associated J-factors of Set 1. strength α X = 10 −2 and annihilates through a contact interaction with cross section (σv) 0 . DM annihilation is thus Sommerfeld-enhanced. We considerτ τ ,bb, W + W − andμµ final states, each with 100% branching fraction, and the dSphs and associated J-factors of Set 5.

Particle Physics
3. Particle Physics Scenario 3 : DM with mass m X annihilates with a total s-wave cross section (σv) 0 to a threebody final stateμµγ via internal bremsstrahlung. This situation occurs if DM annihilation toμµ is p-wave suppressed and internal bremsstrahlung is the dominant annihilation channel. We consider the model presented in Ref. [16], with two charged mediators of masses m 1 and m 2 and left-right mixing angle θ LR , and the dSphs and associated J-factors of Set 1.

4.
Particle Physics Scenario 4 : DM with mass m X annihilates with total s-wave cross section (σv) 0 to a pair of intermediate particles φ (of mass m φ ), each of which decays to two photons (XX → φφ → 4γ). We consider the dSphs and associated J-factors of Set 1. [17], the lightest component of which has mass m 0 and annihilates with cross section (σv) 0 to a pair of intermediate particles φ, each of which decays to two photons (X i X i → φφ → 4γ) [18]. The heavier components of the ensemble (with mass m n ) annihilate to the same final state, but with a rate which scales as ∝ (m n /m 0 ) ξ , where ξ is a parameter. We consider the dSphs and associated J-factors of Set 1.

Particle Physics Scenario 5 : DM consists of a Dynamical Dark Matter (DDM) ensemble
We plot representative photon spectra for these scenarios in Fig. 4 assuming that the primary contributions to the photon flux are final state radiation and secondary decays, and that propagation effects in dSphs are negligible. Each spectrum is normalized to the average number of photons produced per annihilation, and in all cases of singlecomponent DM, we take m X = 100 GeV. Particle Physics Scenario 1 is widely studied, and for this case we plot spectra obtained from the tools provided in Ref. [19], for the final statesτ τ (blue solid),bb (red solid),μµ (green solid), and W + W − (black solid). Particle Physics Scenarios 2 yields the same photon spectra as Particle Physics Scenario 1, but with different J-factors. The spectrum for Particle Physics Scenario 3 is plotted as a green dashed curve assuming the masses of the charged mediators are m 1 = 120 GeV and m 2 = 450 GeV and the mediator left-right mixing angle is θ LR = 0. The dotted black curve shows an example photon spectrum for Particle Physics Scenario 4, under the assumption that the mediator mass is m φ = 60 GeV. Finally, the spectra for the DDM Particle Physics on the 95% C.L. limits due to the 1σ variation in J-factors for all objects considered (as presented in Fig. 3), with the red shading. For reference, the grey dashed line in each panel indicates a cross section of (σv) 0 = 3×10 −26 cm 3 s −1 . The limits are expectedly stronger for the Sommerfeld-enhanced scenario than in the absence of a Sommerfeld enhancement. Focusing on Particle Physics Scenario 1 (left), one can compare the limits presented here to those presented in Ref. [4]; for m X = 10 GeV, the limits presented here are weaker by a factor of ∼ 2 − 5, which is less than the systematic uncertainty of this analysis. Of course, a direct comparison of these methodologies is not readily possible, since the set of targets for the two analyses are different. However, it is unsurprising that a dedicated study of a particular particle physics model proves more constraining than a generic search. The more interesting application of this method is to models for which current constraints are inapplicable, as shown in the right panel.
For Particle Physics Scenario 3, we first consider the case in the absence of left-right mixing (i.e., θ LR = 0), which exhibits a substantial bump in the photon spectrum near the DM mass due to virtual internal bremsstrahlung (VIB). In the limit in which the lightest charged mediator is nearly degenerate with the DM (m 1 ∼ m X ), the photon is very hard and the spectrum is not very different from that of a line. On the other hand, if m 1 m X , the effects of VIB are largely irrelevant. We focus on the intermediate case, m 1 m X , for which the spectral shape is not well approximated by typical spectra utilized in dSph searches. In Fig. 6, we plot the bounds on (σv) 0 for Particle Physics Scenario 3, assuming θ LR = 0 and for mediator masses m 1 = 1.2m X and m 2 = 4.5m X .
In Fig. 7, we again consider Particle Physics Scenario 3 with mediator masses m 1 = 1.2m X and m 2 = 4.5m X , but now with fixed m X = 100 GeV. We plot the bounds on (σv) 0 as a function of θ LR [16]. In the left panel, we show the full range of θ LR between 0 and π/2; while in the right panel, we consider small θ LR where the effect of left-right mixing is substantial. For θ LR near 0 or π/2, the VIB bump is substantial and dependent on the value of θ LR , leading to a θ LR -dependent limit. For moderate values of θ LR , where the limit is flat in the left panel, the photon spectrum does not exhibit a substantial VIB bump and is therefore independent of θ LR .
For Particle Physics Scenario 5, we plot in Fig. 9. This quantity, multiplied by the J-factor, is the 95% C.L. bound on the total photon flux at the Fermi-LAT arising from DM annihilation. If ∆m is a constant mass splitting between successive DM components, and if m 0 , Ω 0 We also show the effect on the variation in the 95% C.L. limits due to the 1σ variation in J-factors for all objects considered, as presented in Figure 3, with the shaded region. The grey dashed line indicates a cross section of (σv)0 = 3 × 10 −26 cm 3 s −1 . where we take the DM mass to be mX = 100 GeV and the scalar mediator masses to be m1 = 120 GeV and m2 = 450 GeV. In the left panel, we show the range of θLR between 0 and π/2, while in the right panel we focus on small θLR. We also show the effect on the variation in the 95% C.L. limits due to the 1σ variation in J-factors for all objects considered, as presented in Figure 3, with the shaded region in each panel.
and (σv) 0 are the mass, abundance, and annihilation cross section of the lightest component, respectively, then we findΦ where m max is the mass of the heaviest DM component. We set ξ = −3, and determine a 95% C.L. bound onΦ for a model parameterized by (m φ , m 0 , m max ) = (10 GeV, m, 10000 GeV) (cyan), (10 GeV, 100 GeV, m) (magenta), We also show the effect on the variation in the 95% C.L. limits due to the 1σ variation in J-factors for all objects considered, as presented in Fig. 3, with the shaded regions surrounding the grey and cyan curves.

IV. CONCLUSIONS
We have described a formalism for deriving model-independent constraints on the number of photons produced by DM annihilation in a set of dwarf spheroidal galaxies. Our approach differs from previous attempts in that our constraints are independent of both the DM particle physics model and the DM astrophysics. Essentially, once the number of background photons is estimated by using data taken slightly off-axis, the number of photons originating from DM annihilation can be statistically constrained, independent of any assumptions about how the DM actually produces those photons. Although such a general search is indeed less powerful than a targeted search strategy for any particular model, the loss in constraining power is not dramatic.
With increasingly diverse models of DM being considered, the utility of a model-independent constraint on DM annihilation in dSphs is clear. Since models with multibody annihilation final states, with final-state cascades, with multi-component DM, etc., have gained popularity, dSph analyses targeted towards particular sets of photon spectra are not generally applicable to a specific model of interest. Similarly, not only is there significant uncertainty in the standard J-factors applicable for s-wave annihilation, but also uncertainty as to whether this is even a correct type of J-factor to apply. If DM decays, or if DM annihilation has nontrivial velocity-dependence, then the modified J-factors can be very different from the standard J-factors. In such cases, an analysis which weights the statistical power of photons based on a putative set of J-factors would again be inapplicable.  TABLE I. Properties of each dSph. The columns give the name of the dSph, the average Fermi-LAT exposure, the average number of expected background events, the number of observed events in the dSph region, and the J-factors used in the various sets described in the text. Set 1a, Set 1b, and Set 1c (labeled simply as "a", "b", and "c") are subsets of Set 1, so we do not rewrite the value of the J-factor; instead, we indicate whether or not this dSph is included in the subset by a check mark.