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PHYSICAL REVIEW D 102, 023029 (2020)

Effective J-factors for dwarf spheroidal with velocity-dependent annihilation

Kimberly K. Boddy ,1 Jason Kumar,2 Andrew B. Pace ,3,4 Jack Runburg ,2 and Louis E. Strigari3 1Department of Physics & Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 2Department of Physics & Astronomy, University of Hawai‘i, Honolulu, Hawaii 96822, USA 3Department of Physics and Astronomy, Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA 4McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, USA

(Received 21 October 2019; accepted 28 May 2020; published 24 July 2020)

We calculate the effective J-factors, which determine the strength of indirect detection signals from annihilation, for 25 dwarf spheroidal galaxies (dSphs). We consider several well-motivated assumptions for the relative velocity dependence of the dark matter annihilation cross section: σAv: 2 4 s-wave (velocity independent), p-wave (σAv ∝ v ), d-wave (σAv ∝ v ), and Sommerfeld-enhancement in the Coulomb limit (σAv ∝ 1=v). As a result we provide the largest and most updated sample of J-factors for velocity-dependent annihilation models. For each scenario, we use Fermi-LAT gamma-ray data to constrain the annihilation cross section. Due to the assumptions made in our gamma-ray data analysis, our bounds are comparable to previous bounds on both the p-wave and Sommerfeld-enhanced cross sections using dSphs. Our bounds on the d-wave cross section are the first such bounds using indirect detection data.

DOI: 10.1103/PhysRevD.102.023029

I. INTRODUCTION In particular, for Sagittarius II, we perform the first J-factor One of the most promising strategies for the indirect analysis for any annihilation model. Moreover, we are the detection of dark matter (DM) is the search for gamma rays first to our knowledge to calculate effective J-factors arising from DM annihilation in dwarf spheroidal galaxies of any dSph for d-wave annihilation, as well as for (dSphs). These targets are especially useful because they p-wave annihilation without assuming a Maxwell- have large dark-to-luminous mass ratios, large expected Boltzmann distribution. DM annihilation rates, and no standard astrophysical We use a Navarro-Frenk-White (NFW) density profile sources of gamma rays. for the dSph halos and assume that the DM velocity The flux of gamma rays arising from DM annihilation distribution is related to the density profile by the depends on the properties of the astrophysical source Eddington inversion formula [11]. Under the approxima- through the J-factor. Under the standard assumption of tion that a dSph spans a small angular size (which is well velocity-independent DM annihilation, the J-factor is deter- justified for all dSphs we consider), we employ previous mined by the DM density profile of the dSph. If, however, work that has determined the effective J-factor in terms of the annihilation cross section is velocity dependent, the the scale density, scale radius, and distance to the halo calculation of the J-factor must account for this velocity for all annihilation models we consider [6]. We then dependence by incorporating the full DM velocity distri- estimate these parameters by fitting the associated velocity bution [1–8]. Previous works have estimated these effective dispersion to stellar data and present results for the effective J-factors for some dSphs, using a variety of techniques, under the assumptions of Sommerfeld-enhanced DM anni- J-factors, integrated over various angular cones. Finally, we use the MADHAT code [12] to perform a hilation in the Coulomb limit (σAv ∝ 1=v) [3,5,7,9,10] and 2 stacked analysis of Fermi-LAT gamma-ray data [13] for p-wave annihilation (σAv ∝ v ) [4,9]. In this work, we calculate the effective J-factors for these targets. We obtain bounds on the DM annihilation 25 dSphs of the Milky Way (MW), under well-motivated cross section for each of the annihilation models we annihilation models: s-wave, p-wave, d-wave, and consider. Limits on Sommerfeld-enhanced annihilation Sommerfeld-enhancement in the Coulomb limit. We and p-wave annihilation have been previously obtained present the first effective J-factor analysis conducted for using a smaller set of dSphs, with effective J-factors many of these dSphs under certain annihilation scenarios. determined using different methodologies [9,14].

2470-0010=2020=102(2)=023029(14) 023029-1 © 2020 American Physical Society BODDY, KUMAR, PACE, RUNBURG, and STRIGARI PHYS. REV. D 102, 023029 (2020)

This paper is organized as follows. In Sec. II, we determine Majorana) that annihilates through a scalar current the effective J-factors for our set of 25 dSphs, describing coupling, regardless of the final state particles; our methodology in detail and comparing to previous results. in this case, the matrix element is only nonvanishing In Sec. III, we use these effective J-factors, along with if the DM initial state is p-wave (see, for example, Fermi-LAT data, to set bounds on the DM annihilation cross Ref. [17]). section. We conclude in Sec. IV. (iv) n ¼ 4: d-wave annihilation. This scenario is relevant if DM is instead a real scalar [18] that annihilates to Standard Model fermion/antifermion pairs II. EFFECTIVE J-FACTORS through an interaction that respects Minimal Flavor We express the DM annihilation cross section as σAv ¼ Violation. In this case, annihilation from an s-wave ðσAvÞ0Sðv=cÞ, where ðσAvÞ0 is a quantity independent of state is chirality-suppressed, and the p-wave initial the relative velocity v of the annihilating particles. The state is forbidden by symmetry of the wave func- differential photon flux arising from DM annihilation in tion [17,18]. any astrophysical target is Following Ref. [8], we assume that the DM velocity distribution is a function of only two dimensionful param- 2 eters: the scale radius rs and the scale density ρs. d Φ ðσAvÞ0 dN J Ω ; 1 Furthermore, we take the limit θ0 ≪ 1, where θ0 ≡ r =D Ω ¼ 8 2 Sð Þ ð Þ s d dEγ πmX dEγ is the characteristic angular scale of the target. Under these assumptions, the effective J-factor for a given annihilation where dN=dEγ is the photon spectrum produced per model parameter n may thus be written as annihilation and mX is the DM particle mass. We have assumed that the DM particle is its own antiparticle. The 4 2 n=2 ˜ 2 πGNρsrs ˜ Ω J ðθÞ¼2ρ r J˜ ðθÞ; ð3Þ effective J-factor, JSð Þ, encodes the information about the SðnÞ s s c2  SðnÞ DM distribution in the target. For a target with a central potential and DM particles on isotropic orbits, the DM ˜ ˜ ˜ θ ≡ θ=θ0 J θ velocity distribution fðr; v Þ is simply a function of the where and Sð Þ is the scale-free angular dis- p tribution that depends only on n and on the functional form distance from the center of the target and the velocity of of the velocity distribution, but not on the parameters ρ , r , the DM particle [11]. With this simplification, the effective s s O θ2 J-factor is or D. Deviations from this result scale as ð 0Þ, which is negligible for the dSphs we consider. ∞ Therefore, to determine the effective J-factor for any l 3 3 v − v JSðθÞ¼ d d v1 d v2Sðj 1 2j=cÞ dSph, it is only necessary to know the halo parameters (ρs, Z0 Z Z ˜ ˜ rs, and D) and scale-free angular distribution JSðnÞðθÞ. The × f½rðl; θÞ;v1Šf½rðl; θÞ;v2Š; ð2Þ latter has been previously calculated for an NFW density profile ρ r ρ r=r −1 1 r=r −2 and for all values of where l is the distance along the line of sight and θ is the ð Þ¼ sð sÞ ð þ sÞ n discussed above [8]. We make the same assumptions as angle between the light of sight and the line to the center of Ref. [8] about the DM velocity distribution, which is the target. The radial distance from the halo center can be related to the density profile through the Eddington recast via r2ðl; θÞ¼l2 þ D2 − 2lD cos θ, where D is the inversion formula. Using these results, we are able to distance to the center of the target. determine the various effective J-factors for individual We consider DM annihilation models of the form dSphs if we know ρ , r , and D. Sðv=cÞ¼ðv=cÞn, for integer n. In particular, we focus s s In the following subsections, we describe our procedure on the following possible scenarios: of determining these parameters and present the resulting (i) n ¼ −1: Sommerfeld-enhanced annihilation in the effective J-factors for specific dSphs. Coulomb limit [15,16]. If the annihilation proceeds −1 through a heavy mediator, then ðσvÞ0ð2πα Þðv=cÞ , X A. Halo parameters where αX is the DM self coupling. We fix αX ¼ 1=2π. (ii) n ¼ 0: s-wave velocity-independent annihilation. We use the halo parameter analysis, originally presented This scenario is the one that is usually considered. in Ref. [19], which calculated J-factors for 41 dSphs. We (iii) n ¼ 2: p-wave annihilation. This scenario is relevant consider the subset of 22 dSphs that have confidently if DM is a Majorana fermion, which annihilates to measured velocity dispersions and are MW satellites. The Standard Model fermion/antifermion pairs through general methodology for determining halo parameters in an interaction that respects Minimal Flavor Viola- dSphs is through a spherical Jeans analysis [20–22]. The tion. In this case, annihilation from an s-wave initial analysis involves solving the spherical Jeans equation state is chirality-suppressed. As another example, (which relates the velocity dispersion, stellar anisotropy, this scenario is relevant if DM is a fermion (Dirac or and gravitational potential) for a set of halo parameters,

023029-2 EFFECTIVE J-FACTORS FOR MILKY WAY DWARF … PHYS. REV. D 102, 023029 (2020)

FIG. 1. Median effective J-factors, integrated over cones with opening half-angles of 0.5° (hexagons) and 10.0° (diamonds), along with asymmetric 68% containment bands. We show results for Sommerfeld-enhanced (green), s-wave (blue), p-wave (red), and d-wave (orange) DM annihilation.

projecting it into the line-of-sight direction, and comparing This analysis includes a total of seven parameters: the the line-of-sight dispersion to observed stellar kinematics. three needed to find the effective J-factor (ρs, rs, and D) For our spherical Jeans analysis, we assume a Plummer and four others [average line-of-sight velocity, half- distribution for the stellar density [23],anNFW light radius (rp), ellipticity (ϵ), stellar anisotropy (β)]. profile for the DM distribution [24], and a constant stellar The half-light radius, ellipticity, and distance all contain anisotropy. Gaussian priors based on literature measurements. For

023029-3 BODDY, KUMAR, PACE, RUNBURG, and STRIGARI PHYS. REV. D 102, 023029 (2020)

FIG. 2. J-factors for s-wave DM annihilation integrated over a 0.5° cone from this analysis and others. the halo parameters, we assume Jeffreys priors: −2 < eliminate some unphysical points in the parameter space, −3 log10 ðrs=kpcÞ < 1 and 4 < log10ðρs=ðM⊙ kpc ÞÞ < 14. we use the global density slope-anisotropy inequality, The stellar anisotropy prior is uniform in a symmetrized γ⋆ðrÞ ≥ 2βðrÞ, where γ⋆ is the log stellar density slope – version: −0.95 < β˜ < 1, where β˜ ¼ β=ð2 − βÞ.1 To [25 28]. For a Plummer stellar profile and a constant stellar anisotropy, this constraint is β < 0. We use an unbinned likelihood [22] and determine the posterior 1 Generally, β ranges between −∞ and 1. Negative and positive distributions with a multimodal nested sampling algo- values correspond to tangential and radial orbits, respectively. rithm [29,30]. We refer the reader to Ref. [19] for more This alternate parameterization uniformly favors tangential and radial orbits. details.

023029-4 EFFECTIVE

2 −5 TABLE I. Integrated J-factors presented as log10ðJ=GeV cm Þ for s-wave, p-wave, d-wave, and Sommerfeld-enhanced annihilation for cones with opening half-angles of 0.1°, J

0.2°, 0.5°, and 10°. The posterior distributions are available at the following website: https://github.com/apace7/J-Factor-Scaling. DWARF WAY MILKY FOR -FACTORS

s-wave p-wave d-wave Sommerfeld 0.1° 0.2° 0.5° 10° 0.1° 0.2° 0.5° 10° 0.1° 0.2° 0.5° 10° 0.1° 0.2° 0.5° 10°

Aquarius II 18 12þ0.62 18 26þ0.63 18 39þ0.65 18 48þ0.72 9 99þ0.98 10 19þ1.03 10 37þ1.13 10 51þ1.27 2 11þ1.44 2 37þ1.52 2 58þ1.63 2 70þ1.86 22 45þ0.48 22 54þ0.48 22 62þ0.47 22 70þ0.48 . −0.56 . −0.57 . −0.58 . −0.62 . −0.87 . −0.90 . −0.99 . −1.08 . −1.25 . −1.34 . −1.47 . −1.56 . −0.43 . −0.42 . −0.42 . −0.44 Boötes I 17 79þ0.30 17 99þ0.29 18 19þ0.30 18 39þ0.45 9 38þ0.45 9 68þ0.49 9 98þ0.60 10 24þ0.93 1 30þ0.74 1 65þ0.83 2 01þ0.99 2 29þ1.42 22 28þ0.26 22 41þ0.24 22 54þ0.22 22 68þ0.24 . −0.28 . −0.27 . −0.28 . −0.38 . −0.42 . −0.46 . −0.58 . −0.76 . −0.68 . −0.78 . −0.96 . −1.17 . −0.25 . −0.23 . −0.21 . −0.23 I 17 19þ0.18 17 33þ0.15 17 43þ0.16 17 49þ0.24 9 01þ0.23 9 20þ0.30 9 33þ0.45 9 38þ0.63 1 07þ0.45 1 27þ0.60 1 41þ0.82 1 45þ1.04 21 53þ0.21 21 62þ0.17 21 69þ0.13 21 75þ0.12 . −0.17 . −0.14 . −0.15 . −0.17 . −0.21 . −0.25 . −0.31 . −0.34 . −0.35 . −0.43 . −0.51 . −0.54 . −0.18 . −0.15 . −0.12 . −0.12 Canes Venatici II 17 53þ0.42 17 68þ0.44 17 82þ0.48 17 92þ0.55 9 39þ0.71 9 62þ0.76 9 82þ0.87 9 95þ1.03 1 54þ1.08 1 81þ1.17 2 04þ1.31 2 16þ1.52 21 87þ0.33 21 96þ0.32 22 05þ0.32 22 12þ0.35 . −0.41 . −0.43 . −0.47 . −0.53 . −0.70 . −0.77 . −0.88 . −0.97 . −1.09 . −1.22 . −1.36 . −1.46 . −0.31 . −0.31 . −0.31 . −0.33 I 17 57þ0.17 17 72þ0.13 17 83þ0.09 17 88þ0.11 9 16þ0.13 9 37þ0.13 9 50þ0.21 9 54þ0.29 0 98þ0.18 1 20þ0.27 1 34þ0.40 1 37þ0.50 22 04þ0.18 22 13þ0.15 22 20þ0.12 22 25þ0.11 . −0.14 . −0.11 . −0.09 . −0.10 . −0.12 . −0.12 . −0.14 . −0.16 . −0.15 . −0.18 . −0.23 . −0.26 . −0.16 . −0.13 . −0.10 . −0.09 Carina II 18 02þ0.53 18 19þ0.52 18 37þ0.54 18 57þ0.66 9 29þ0.81 9 57þ0.83 9 83þ0.94 10 06þ1.29 0 91þ1.18 1 21þ1.30 1 49þ1.48 1 73þ1.93 22 66þ0.43 22 78þ0.41 22 89þ0.39 23 03þ0.41 . −0.51 . −0.50 . −0.52 . −0.59 . −0.77 . −0.82 . −0.90 . −1.05 . −1.15 . −1.23 . −1.34 . −1.51 . −0.41 . −0.39 . −0.38 . −0.39 I 18 61þ0.31 18 81þ0.32 19 01þ0.36 19 25þ0.57 10 32þ0.54 10 63þ0.61 10 96þ0.73 11 29þ1.15 2 40þ0.94 2 77þ1.03 3 16þ1.19 3 50þ1.75 23 03þ0.27 23 16þ0.25 23 30þ0.24 23 46þ0.29 . −0.32 . −0.32 . −0.36 . −0.50 . −0.53 . −0.61 . −0.78 . −1.05 . −0.92 . −1.07 . −1.32 . −1.62 . −0.26 . −0.25 . −0.24 . −0.28 14 91þ0.33 15 13þ0.30 15 35þ0.27 15 56þ0.25 5 48þ0.41 5 81þ0.39 6 14þ0.36 6 43þ0.42 −3 60þ0.49 −3 22þ0.48 2 84þ0.50 2 54þ0.63 19 92þ0.28 20 06þ0.26 20 20þ0.24 20 34þ0.20 Crater II . −0.29 . −0.28 . −0.25 . −0.23 . −0.38 . −0.35 . −0.34 . −0.37 . −0.46 . −0.46 − . −0.46 − . −0.54 . −0.25 . −0.24 . −0.21 . −0.18 18 43þ0.16 18 63þ0.13 18 82þ0.12 18 96þ0.20 10 41þ0.15 10 70þ0.17 10 97þ0.25 11 13þ0.47 2 68þ0.27 3 01þ0.35 3 32þ0.49 3 50þ0.75 22 72þ0.18 22 85þ0.15 22 97þ0.13 23 08þ0.13 I . −0.14 . −0.13 . −0.12 . −0.16 . −0.14 . −0.15 . −0.21 . −0.30 . −0.21 . −0.26 . −0.36 . −0.46 . −0.15 . −0.14 . −0.12 . −0.12 023029-5 þ0.12 þ0.11 þ0.10 þ0.10 þ0.12 þ0.10 þ0.09 þ0.09 þ0.12 þ0.10 þ0.10 þ0.10 þ0.12 þ0.11 þ0.10 þ0.10

17 91 18 02 18 09 18 11 9 78 9 93 10 02 10 05 1 86 2 04 2 14 2 16 22 22 22 30 22 34 22 36 … I . −0.13 . −0.11 . −0.10 . −0.09 . −0.13 . −0.10 . −0.09 . −0.09 . −0.12 . −0.10 . −0.09 . −0.10 . −0.13 . −0.12 . −0.11 . −0.10 I 17 09þ0.49 17 20þ0.49 17 29þ0.51 17 35þ0.58 8 46þ0.77 8 61þ0.84 8 72þ0.96 8 78þ1.10 0 05þ1.18 0 21þ1.32 0 34þ1.48 0 38þ1.68 21 65þ0.39 21 72þ0.38 21 79þ0.37 21 84þ0.38 . −0.52 . −0.51 . −0.52 . −0.54 . −0.78 . −0.82 . −0.88 . −0.91 . −1.12 . −1.20 . −1.28 . −1.31 . −0.41 . −0.39 . −0.39 . −0.39 Horologium I 18 92þ0.75 19 05þ0.75 19 17þ0.80 19 28þ0.87 10 93þ1.19 11 12þ1.25 11 29þ1.34 11 39þ1.59 3 23þ1.71 3 42þ1.84 3 60þ2.03 3 67þ2.39 23 20þ0.58 23 27þ0.58 23 34þ0.57 23 42þ0.59 . −0.62 . −0.65 . −0.70 . −0.80 . −1.07 . −1.18 . −1.30 . −1.39 . −1.67 . −1.80 . −1.96 . −2.03 . −0.48 . −0.47 . −0.48 . −0.51 I 18 21þ0.31 18 42þ0.30 18 65þ0.32 18 93þ0.57 9 51þ0.49 9 83þ0.55 10 19þ0.68 10 58þ1.21 1 20þ0.90 1 58þ1.00 2 00þ1.16 2 42þ1.85 22 86þ0.28 23 00þ0.25 23 14þ0.23 23 34þ0.28 . −0.29 . −0.28 . −0.31 . −0.47 . −0.46 . −0.53 . −0.71 . −1.03 . −0.84 . −0.99 . −1.24 . −1.61 . −0.26 . −0.24 . −0.21 . −0.25 17 38þ0.13 17 52þ0.10 17 61þ0.13 17 64þ0.19 9 41þ0.16 9 58þ0.25 9 69þ0.40 9 72þ0.53 1 65þ0.40 1 84þ0.54 1 97þ0.72 2 00þ0.85 21 62þ0.17 21 70þ0.13 21 77þ0.10 21 81þ0.10 I . −0.11 . −0.10 . −0.11 . −0.12 . −0.14 . −0.19 . −0.25 . −0.27 . −0.27 . −0.36 . −0.45 . −0.47 . −0.13 . −0.11 . −0.08 . −0.09 Leo II 17 63þ0.16 17 65þ0.15 17 66þ0.16 17 66þ0.16 9 30þ0.20 9 33þ0.20 9 33þ0.20 9 33þ0.20 1 16þ0.24 1 18þ0.26 1 19þ0.26 1 19þ0.26 22 02þ0.16 22 03þ0.16 22 03þ0.16 22 03þ0.16 . −0.16 . −0.15 . −0.15 . −0.15 . −0.18 . −0.18 . −0.18 . −0.18 . −0.23 . −0.23 . −0.23 . −0.23 . −0.17 . −0.15 . −0.15 . −0.15 II 18 54þ0.34 18 73þ0.34 18 94þ0.38 19 16þ0.64 9 99þ0.56 10 28þ0.65 10 59þ0.80 10 86þ1.38 1 78þ1.04 2 12þ1.17 2 47þ1.40 2 75þ2.13 23 10þ0.30 23 23þ0.28 23 36þ0.26 23 53þ0.32 . −0.32 . −0.32 . −0.38 . −0.53 . −0.56 . −0.66 . −0.85 . −1.08 . −0.99 . −1.16 . −1.40 . −1.66 . −0.28 . −0.26 . −0.25 . −0.30 17 07þ1.60 17 22þ1.47 17 35þ1.36 17 48þ1.23 8 20þ1.95 8 42þ1.77 8 62þ1.60 8 83þ1.42 0 39þ2.20 0 13þ2.02 0 10þ1.84 0 35þ1.66 21 77þ1.39 21 87þ1.31 21 95þ1.24 22 03þ1.16 Sagittarius II . −1.09 . −1.02 . −0.91 . −0.79 . −1.39 . −1.27 . −1.14 . −1.07 − . −1.61 − . −1.50 . −1.40 . −1.37 . −0.92 . −0.87 . −0.80 . −0.70 I 18 33þ0.11 18 47þ0.08 18 58þ0.05 18 63þ0.05 10 14þ0.08 10 36þ0.06 10 51þ0.10 10 55þ0.15 2 21þ0.09 2 45þ0.13 2 61þ0.21 2 65þ0.26 22 67þ0.12 22 77þ0.10 22 84þ0.08 22 88þ0.06 D REV. PHYS. . −0.10 . −0.07 . −0.05 . −0.05 . −0.07 . −0.06 . −0.08 . −0.09 . −0.08 . −0.10 . −0.14 . −0.16 . −0.11 . −0.09 . −0.07 . −0.05 18 73þ0.53 18 87þ0.48 19 00þ0.48 19 12þ0.63 9 89þ0.72 10 08þ0.78 10 21þ0.96 10 32þ1.40 1 29þ1.15 1 45þ1.36 1 57þ1.68 1 67þ2.30 23 40þ0.47 23 50þ0.43 23 59þ0.40 23 71þ0.39 . −0.68 . −0.67 . −0.68 . −0.68 . −0.99 . −1.01 . −1.02 . −1.08 . −1.36 . −1.38 . −1.44 . −1.50 . −0.58 . −0.54 . −0.53 . −0.53 I 17 40þ0.22 17 58þ0.18 17 75þ0.12 17 87þ0.12 8 93þ0.20 9 21þ0.15 9 45þ0.16 9 56þ0.33 0 78þ0.19 1 08þ0.21 1 32þ0.34 1 42þ0.61 21 92þ0.21 22 03þ0.19 22 13þ0.15 22 21þ0.10 . −0.22 . −0.17 . −0.11 . −0.10 . −0.18 . −0.14 . −0.13 . −0.17 . −0.18 . −0.18 . −0.22 . −0.26 . −0.24 . −0.20 . −0.15 . −0.09 II 18 56þ0.57 18 74þ0.55 18 93þ0.56 19 13þ0.65 10 40þ0.82 10 68þ0.85 10 96þ0.92 11 24þ1.13 2 57þ1.18 2 91þ1.24 3 23þ1.36 3 53þ1.66 22 91þ0.47 23 03þ0.45 23 16þ0.43 23 30þ0.44 . −0.53 . −0.51 . −0.50 . −0.56 . −0.74 . −0.76 . −0.82 . −0.99 . −1.04 . −1.11 . −1.23 . −1.44 . −0.44 . −0.42 . −0.41 . −0.39 I 18 07þ0.31 18 21þ0.28 18 33þ0.28 18 40þ0.37 9 85þ0.41 10 05þ0.45 10 20þ0.57 10 26þ0.82 1 88þ0.65 2 09þ0.78 2 22þ1.02 2 27þ1.36 22 44þ0.28 22 53þ0.25 22 61þ0.22 22 68þ0.22 . −0.29 . −0.28 . −0.28 . −0.32 . −0.41 . −0.45 . −0.51 . −0.55 . −0.63 . −0.69 . −0.73 . −0.76 . −0.28 . −0.25 . −0.22 . −0.23 102, 19 00þ0.45 19 21þ0.42 19 44þ0.41 19 72þ0.54 10 84þ0.61 11 17þ0.63 11 51þ0.71 11 90þ1.08 3 07þ0.93 3 44þ1.01 3 83þ1.15 4 25þ1.64 23 37þ0.39 23 51þ0.36 23 66þ0.33 23 85þ0.33 Ursa Major II . −0.40 . −0.39 . −0.39 . −0.49 . −0.58 . −0.61 . −0.69 . −0.93 . −0.87 . −0.94 . −1.10 . −1.39 . −0.33 . −0.32 . −0.30 . −0.32 18 56þ0.17 18 68þ0.14 18 76þ0.12 18 80þ0.11 10 35þ0.17 10 53þ0.14 10 65þ0.15 10 68þ0.18 2 39þ0.18 2 59þ0.19 2 71þ0.24 2 73þ0.29 22 92þ0.16 22 99þ0.15 23 05þ0.13 23 07þ0.12 I . −0.21 . −0.16 . −0.11 . −0.11 . −0.17 . −0.14 . −0.14 . −0.14 . −0.17 . −0.17 . −0.19 . −0.19 . −0.21 . −0.18 . −0.15 . −0.12 (2020) 023029 19 11þ0.44 19 24þ0.45 19 36þ0.52 19 46þ0.73 10 73þ0.74 10 90þ0.87 11 05þ1.11 11 14þ1.60 2 58þ1.33 2 78þ1.54 2 94þ1.86 3 01þ2.47 23 54þ0.43 23 63þ0.39 23 72þ0.38 23 82þ0.42 . −0.45 . −0.43 . −0.46 . −0.52 . −0.64 . −0.72 . −0.82 . −0.89 . −1.00 . −1.12 . −1.25 . −1.32 . −0.41 . −0.40 . −0.38 . −0.39 BODDY, KUMAR, PACE, RUNBURG, and STRIGARI PHYS. REV. D 102, 023029 (2020)

FIG. 3. J-factors for s-wave DM annihilation integrated over a 10° cone from this analysis and total J-factors from other analyses.

We also apply the same analysis to three additional follows: D¼117.5Æ1.1kpc, rp ¼ 31.1 Æ 2.5 arcmin [40]; 2 dSphs: Crater II [31], Hydrus I [32], and Sagittarius II. D ¼ 27.6 Æ 0.5 kpc, rp ¼ 7.42 Æ 0.58 arcmin, ϵ¼0.21Æ The literature properties we use for our modeling are as 0.11 [32]; and D ¼ 73.1 Æ 0.9 kpc, rp ¼1.7Æ0.05arcmin

2We note that the identification of Sagittarius II as a dSph versus a cluster is not yet definite. It has a very compact size and high luminosity compared to what is expected for a dSph [33,34]. The velocity dispersion is resolved, but the mass-to-light ratio is much lower than other ultrafaint dSphs, and the dispersion is possibly resolved [35]. Furthermore, there are other dwarf galaxies included in our sample whose nature is debated: Segue 1 [36,37] and Willman 1[38,39].

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FIG. 4. Effective J-factors for Sommerfeld-enhanced DM annihilation integrated over a 10° cone from this analysis and total J-factors from other analyses. The total effective J-factors from other analyses have been rescaled to αX ¼ 1=2π for comparison.

[35]. For Crater II and Hydrus I, we determine membership B. Results with a mixture model including a dSph and MW fore- Using the values of rs, ρs, and D from our nested ground component (see Ref. [41] for details on the mixture sampling runs, we are able to produce the posterior model). We include Gaia DR2 proper motions, which helps distributions for the effective J-factor [from Eq. (3)], identify dSph [42,43]. The J-factors of Hydrus I and integrated over a given angular region of each dSph for Crater II have been presented before [31,32], and our different annihilation models. In Fig. 1, we show the results are comparable to previous measurements. This is effective J-factors for our set of 25 dSphs, integrated the first J-factor analysis of Sagittarius II. over cones with opening half-angles of 0.5° and 10.0°,

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FIG. 5. Effective J-factors for p-wave DM annihilation, integrated over a cone of 0.5° (hexagons) or 10.0° (diamonds), or the total integrated effective J-factor (diamonds), as indicated in the legend.

for the scenarios of s-wave, p-wave, d-wave, and Table I lists the median integrated effective J-factors Sommerfeld-enhanced annihilation. In each case, we plot with their 1σ uncertainties for each of our annihilation the median J-factor, along with 68% containment bands. scenarios. We provide effective J-factors integrated over Note that, in general, there is very little difference between cones with half-angles of 0.1°, 0.2°, 0.5°, and 10°.3 the effective J-factor integrated over cones with half-angles of 0.5° and 10.0°, indicating that in most cases the 0.5° cone 3The posterior distributions are available at the following encompasses the dSph almost entirely. website: https://github.com/apace7/J-Factor-Scaling

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FIG. 6. Exclusion limits at 95% C.L. for s-wave (upper left), p-wave (upper right), d-wave (lower left) and Sommerfeld-enhanced (in the Coulomb limit, lower right) DM annihilation. We consider the following annihilation channels: bb¯ (orange), ττ¯ (blue), μμ¯ (red), and WþW− (green). Each central solid line is obtained using the median effective J-factors for all 25 dSphs considered, while the colored bands indicate the limits obtained by varying all effective J-factors either up or down by their 1σ uncertainty. The solid gray lines reproduce the exclusion limits for annihilation to the bb¯ channel at 95% C.L. found in Ref. [44] (s-wave), Ref. [9] [p-wave, expressed as a bound on ðσvÞ0], and Ref. [14] (Sommerfeld-enhanced, rescaled to αX ¼ 1=2π).

C. Comparison with other approaches Refs. [3,5,9,10].4 In Fig. 5, we compare our results for the We compare our results to other results found in the p-wave effective J-factor (integrated over a 0.5° cone) to literature. In Fig. 2, we compare our results for the s-wave those in Ref. [4] and for the p-wave effective J-factor J-factor (integrated over a 0.5° cone) to those found in (integrated over a 10° cone) to the total p-wave effective Refs. [19,31,32,45,46]. In Fig. 3, we compare our results J-factor found in Ref. [9]. for the s-wave J-factor (integrated over a 10° cone) to the total integrated J-factors found in Refs. [5,9,10]. In Fig. 4, we compare our results for the Sommerfeld-enhanced 4All of the effective J-factors in these other works have effective J-factor (integrated over a 10° cone) to the been rescaled to αX ¼ 1=2π for direct comparison with our total Sommerfeld-enhanced effective J-factors found in calculations.

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We note here one detail regarding the comparison of our For comparison, we also plot 95% C.L. bounds on results to those in Ref. [9] for the case of p-wave or annihilation to bb¯ from other analyses, represented by the Sommerfeld-enhanced annihilation. In Ref. [9], the DM solid gray lines in Fig. 6. For the s-wave scenario, we show velocity distribution is assumed to be Maxwell-Boltzmann, the bounds from the Fermi collaboration analysis [44]. with a velocity dispersion that is independent of position. These constraints are stronger than ours, but the two results In this case, the velocity and position integrals in Eq. (2) are in agreement within the level of their uncertainty bands factorize, and the total effective J-factor can be written as (the 95% containment bands from Ref. [44] are not shown) the product of the total s-wave J-factor and a velocity for mX ≲ 100 GeV. At higher DM masses, our s-wave integral that depends only on the assumed velocity bounds are considerably weaker than those found in dispersion. In Ref. [9], this integral is absorbed into the Ref. [44]. This discrepancy is likely due to our analysis definition of the thermally averaged annihilation cross using MADHAT, which limits the photon energy range to section. For the purposes of comparison, we have rescaled <100 GeV in order to achieve a model-independent the s-wave J-factors reported in Ref. [9] by the appropriate analysis framework, while the analysis in Ref. [44] uses integrals, in order to obtain their total effective J-factors. photons up to an energy of 500 GeV. For the case of p-wave [or Sommerfeld-enhanced] anni- We also show the bounds on p-wave annihilation found 2 −1=2 −1 hilation, the rescaling factor is 6ðv0=cÞ [or π ðv0=cÞ ], in Ref. [9], recast as a bound on ðσvÞ0. Finally, to provide a where the values of the velocity dispersion v0 are also taken comparison for Sommerfeld-enhanced annihilation in the from Ref. [9]. Coulomb limit, we obtain bounds using the effective J-factors found in Ref. [3], rescaled to αX ¼ 1=2π. In both cases, these bounds lie within our 1σ systematic uncertainty III. CONSTRAINTS ON DARK MATTER band for m < Oð100Þ GeV. ANNIHILATION χ As bounds on d-wave annihilation have not previously Having determined the effective J-factors for our set of been determined, we comment on the applicability of our dSphs under different DM annihilation models, we now d-wave constraints. If DM is a real scalar that annihilates to a constrain DM annihilation to a variety of Standard Model fermion/antifermion pair (ϕϕ → ff¯ ) through an interaction final states by performing a stacked dSph analysis with respecting minimal flavor violation, then annihilation from a Fermi-LAT gamma-ray data. We use the MADHAT 1.0 p-wave initial state is exactly forbidden, while annihilation software package [12], which is based on the model- from an s-waveinitial state is chirality-suppressed by a factor 2 independent analysis framework described in Ref. [14]. ∼ðmf=mϕÞ . For DM of mass mX ∼ Oð10Þ TeV annihilat- MADHAT uses Fermi-LAT Pass 8R3 data [47], collected ing to muons, the suppression factor is ∼Oð10−10Þ, which is over a time frame of nearly 11 years, and incorporates still larger than the v4-suppression factor associated with d- – gamma rays only in the energy range of 1 100 GeV, across wave DM annihilation in a dSph. As such, absent fine- which the Fermi-LAT effective area is treated as approx- tuning, we expect d-wave annihilation to dominate s-wave imately constant. This process makes it possible to apply a annihilation for DM much heavier than Oð10Þ TeV, in stacked analysis to any particle physics model in this which case the Fermi-LATwould not be the ideal instrument energy range without having to process Fermi-LAT data to set constraints. Note that these considerations are not for a given analysis to account for the energy dependence of necessarily relevant for constraints on p-wave annihilation, the detector. We simply need the gamma-ray spectrum since there are scenarios in which DM annihilation from an s- dN=dEγ for a specific annihilation channel to produce wave initial state is effectively forbidden, while DM anni- bounds with MADHAT, and we obtain relevant spectra from hilation from a p-wave state is necessarily dominant. the tools described in Ref. [48]. In Fig. 6, we plot constraints on s-wave, p-wave, IV. CONCLUSIONS d-wave, and Sommerfeld-enhanced DM annihilation to bb¯ , ττ¯ , μμ¯ , and WþW−. For each channel, the solid line is There are well-motivated DM models that produce the 95% C.L. bound derived from an analysis of all annihilation cross sections with power-law scalings of the 25 dSphs, setting the effective J-factors (integrated over relative velocity. For scenarios beyond velocity-independent a cone with an opening half-angle of 0.5°) to their median s-wave annihilation, standard calculations of the astrophysi- values, while the uncertainty band arises from adjusting cal J-factor for indirect detection must be modified to their values upward or downward by their 1σ systematic account for any velocity dependence. Under various sim- uncertainties.5 plifying assumptions, we can infer the DM halo velocity distribution from the density profile using the Eddington inversion formula. With the velocity distribution at hand, we 5We also obtained 95% C.L. bounds by stacking only dSphs with integrated effective J-factors that are at least 15% of the incorporate the velocity dependence of the annihilation largest integrated effective J-factor. Using this subset of dSphs cross section into the calculation of the J-factor to produce did not affect the limits in any significant way. an effective J-factor.

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We have determined these effective J-factors for 25 systems. Previous studies of p-wave DM annihilation in the dwarf spheroidal galaxies, with assumed NFW halo pro- considered the increase in density and files, for DM annihilation that is s-wave, p-wave, d-wave, characteristic velocities near the central black hole or Sommerfeld-enhanced in the Coulomb limit. We present [55–57] and used these to argue for stronger constraints the first analysis that we are aware of for several dSphs on p-wave models. There are also constraints on p-wave under certain annihilation models. In particular, we perform annihilation from the epoch of reionization, for which the the first analysis for Sagittarius II for any annihilation results depend on assumptions of the reionization history of model. Changing the assumed particle physics model for the intergalactic medium and the structure formation DM annihilation can change the effective J-factor by prescription used to determine annihilation boost factors several orders of magnitude. [58]. Future observatories, such as CTA [59], will target the We have used these effective J-factors and the MADHAT Galactic center in particular, and there are a variety of software package to determine bounds on DM annihilation upcoming and proposed instruments that will improve our in each of these scenarios with a stacked analysis of understanding of reionization. It is thus important to Fermi-LAT gamma-ray data from dSphs. The limits on compare the ultimate sensitivity that these types of obser- s-wave annihilation are consistent with those found pre- vations can achieve in comparison to dSphs. viously in the literature. We expect the discovery of many new dSphs from Although we have assumed the dSphs have an NFW current instruments, such as DECam [60,61] and Hyper- density profile, similar methods can be used for other Surprime Cam [62,63], as well as from future observato- profiles, such as the generalized NFW, Burkert, Einasto, ries, such as the LSST [64–66]. If nearby dSphs with large etc. It would be interesting to see how the choice of a different effective J-factors are found, observational sensitivity to profile affects the effective J-factors for non-s-wave models DM annihilation could improve significantly. It is standard of DM annihilation. Another avenue for improvement in our practice to estimate s-wave J-factors for dSphs and analysis is to lift the assumption of spherical symmetry. This dSph-candidates, but we have demonstrated that it is just has been explored in the s-wave J-factor case but has not yet as straightforward to estimate the effective J-factors rel- – been extended to velocity dependent models [49 52].We evant for velocity-dependent DM annihilation. Since the leave such a study to future work. fundamental nature of DM interactions is still mysterious, it Observations of systems other than dSphs may provide is important to use data to search for and constrain a variety competitive or stronger limits on velocity-dependent anni- of DM annihilation models. hilation. For instance, typical DM particle velocities can be quite small in the early , and thus strong con- straints on Sommerfeld-enhanced DM annihilation arise ACKNOWLEDGMENTS from observations of the cosmic microwave background We are grateful to Stephen Hill for useful discussions. and measurements of light element abundances [53,54]. J. K. is grateful to the organizers of the Mitchell Conference How competitive the cosmological constraints are com- on Collider, Dark Matter and Neutrino Physics 2019, where pared to dSphs, however, is model-dependent: the velocity work on this project began, for their hospitality. The work behavior of the Sommerfeld enhancement depends on of J. K. is supported in part by DOE Grant No. de- the mass of the particle mediating the dark matter self- sc0010504. The work of A. B. P. is supported by NSF interaction. Grant No. AST-1813881. The work of L. E. S. is supported While limits on p-wave and d-wave annihilation may be by DOE Grant No. de-sc0010813. This work has made use stronger from systems with larger characteristic velocities, of data from the European Space Agency (ESA) mission such as clusters or the Galactic center, the limits we have Gaia (https://www.cosmos.esa.int/gaia), processed by the derived are robust, because the DM distributions in dSphs Gaia Data Processing and Analysis Consortium (DPAC, are directly extracted from the data. Since baryons con- https://www.cosmos.esa.int/web/gaia/dpac/consortium). tribute a non-negligible amount to the potential of clusters Funding for the DPAC has been provided by national and MW-like galaxies in the regions where the DM institutions, in particular the institutions participating in annihilation signal arises, they represent an important the Gaia Multilateral Agreement. systematic uncertainty that must be dealt with in these

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