Galactic Substructure and Dark Matter Annihilation in the Milky Way Halo arXiv:1001.3144v2 [astro-ph.GA] 20 Jan 2010 Marc Kamionkowski, 1 Savvas M. Koushiappas, 2 and Michael Kuhlen 3 $^{1}$California Institute of Technology, Mail Code 350-17, Pasadena, CA 91125 ∗ $^{2}$Department of Physics, Brown University, 182 Hope Street, Providence, RI 02912 † $^{3}$Theoretical Astrophysics Center, University of California Berkeley, Berkeley, CA 94720 ‡ We study the effects of substructure on the rate of dark-matter annihilation in the Galactic halo. We use an analytic model for substructure that can extend numerical simulation results to scales too small to be resolved by the simulations. We first calibrate the analytic model to numerical simulations, and then determine the annihilation boost factor, for standard WIMP models as well as those with Sommerfeld (or other) enhancements, as a function of Galactocentric radius in the Milky Way. We provide an estimate of the dependence of the gamma-ray intensity of WIMP annihilation as a function of angular distance from the Galactic center. This methodology, coupled with future numerical simulation results can be a powerful tool that can be used to constrain WIMP properties using Fermi all-sky data. PACS numbers: 95.35.+d,98.35.Gi, 98.35.Pr, 98.62.Gq Weakly-interacting massive particles (WIMPs) provide perhaps the most promising class of dark-matter candi- dates. These are particles that arise in theories of new electroweak-scale physics, such as low-energy supersym- metry (SUSY) [1] or models with universal extra dimen- sions (UEDs) [2]. Experiments that seek to directly de- tect these particles in low-background experiments, or to indirectly detect them through observation of energetic neutrinos from WIMP annihilation in the Sun, are now beginning to dig into the favored WIMP parameter space. However, there are also prospects for indirectly detect- ing WIMPs through observation of gamma rays and/or cosmic-ray positrons, antiprotons, or antideuterons from WIMP annihilation in the Galactic halo. These annihi- lation products have received considerable attention in the recent literature with the attribution of some re- ported (and still controversial) cosmic-ray anomalies [3] to WIMP-annihilation products. The total rate at which WIMPs annihilate in the Galactic halo is proportional to the volume integral of the square of the dark-matter density. In the canonical (and simplest) model, the dark matter is smoothly dis- tributed in the halo in a spherically-symmetric way with a dark-matter density ρ ( r ) that is a monotonically de- creasing function of r ; for example, it is often modeled as an isothermal or a Navarro-Frenk-White (NFW) profile [4]. However, analytic arguments and numerical simula- tions show that there should be substructure in the dark- matter distribution in the Galactic halo [5–7]. The dark matter may be clumped; some of it may be bound in higher-density self-bound subhalos; and some may be in tidal streams [8]. This is an outcome of hierarchical clus- tering, in which small dense halos form first and then merge to form progressively larger structures. If dark matter in the halo is clumped, then the total annihila- tion rate will be enhanced by some boost factor [9], the increased rate per unit volume in dense regions outweigh- ing the decreased rate per unit volume in lower-density regions. In the canonical WIMP scenario, the substructure may have a roughly scale-invariant distribution in substruc- ture mass/size extending all the way down to substruc- tures on mass scales ∼ 10 − 10 M$_{⊙}$ [7, 10, 11], roughly 22 orders of magnitude smaller than the ∼ 10 12 M$_{⊙}$ Milky Way halo. While state-of-the-art numerical calculations now have the resolution to simulate several decades in this hierarchy, they are very far from being able to follow the survival and evolution of the smallest substructures through all of the generations in the structure-formation hierarchy that result in a Milky Way halo. Analytic calculations of these survival probabilities are difficult [12, 13]. However, given that the smallest subhalos are likely to be the densest, the boost factor may depend significantly on the existence of these substructures. This is partic- ularly true in models with a Sommerfeld enhancement in the annihilation rate [14]—those where the annihila- tion rate increases with lower WIMP velocities—since the smallest subhalos are also likely to have the smallest ve- locity dispersion. In a previous paper [15], we presented an analytic model to describe the self-similar substructure expected from hierarchical clustering. The model predicted a high- density power-law tail for the probability for a given point in the halo to be in a clump of density ρ . Sub- sequent to that paper, another appeared [16] presented N-body simulations that showed this power-law tail. Af- ter calibration to numerical simulations, the model can be used to extrapolate the results of numerical simula- tions to substructure-mass scales far smaller than those currently accessible to the simulations. We investigated $^{∗}$kamion@tapir.caltech.edu $^{†}$koushiappas@brown.edu $^{‡}$mqk@astro.berkeley.edu I. INTRODUCTION