PHYSICAL REVIEW D 102, 023029 (2020) Effective J-factors for Milky Way dwarf spheroidal galaxies 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 dark matter 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;v1f½rðl; θÞ;v2; 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.