8 able analytic calculations of the survival fraction are sim- ilarly difficult. In earlier work [15], we used the nearly self-similar behavior of hierarchical clustering to develop an analytic approach to estimate the substructure sur- vival. The central result of that work was a prediction that the probability P ( ρ ) for a given point in the Galac- tic halo to have a local density ρ will have a high-density power-law tail. Here, we have fit the parameters of that analytic model to new N-body simulations that can resolve the high- density tail in P ( ρ ). The analytic model then allows us to extrapolate the behavior of the simulations to mass scales far below the simulation’s resolution scale. As a result, we have a simple analytic expression for the dis- tribution of dark-matter densities within the Milky Way halo, as a function of Galactocentric radius. The key qualitative result is that the fraction of the Milky Way volume occupied by substructures is small (1 − f$_{s}$ ≲ 10 − 3 in the central regions, ∼ 10 − 2 in the outskirts). In par- ticular, when we calculate the boost factor for WIMPs with the canonical (i.e., no) dependence of 〈 σv 〉 on the velocity, we find that it is small: only about 50% at the solar radius, and only ∼ 17 for the total boost within the virial radius. The PDF, supplemented with a scal- ing for the subhalo velocity dispersion with subhalo den- sity, allows us to also analytically estimate the boost factor from substructure in the presence of a Sommer- feld enhancement. The central result here is given in Eqs. (10, 11), which provide a boost factor in terms of the power-law index β for the velocity scaling of 〈 σv 〉 , the saturation velocity v$_{sat}$ , the maximum substructure density ρ$_{max}$ ≃ 80 GeV cm − $^{3}$, and the velocity dispersion v$_{0}$ of the smallest halos. We estimate numerically that local (8 kpc) boost factors ≳ 10 3 can be obtained with β ≳ 0 . 69, or for β = 1 with v$_{sat}$/c < 3 . 3 × 10 − $^{7}$. We also discussed the finite width of the density dis- tribution smoothly-distributed component of halo dark matter, but find that the effects of this finite width on the boost factor are small. There are several caveats to our results and several improvements that can be pursued in future work. (1) Hierarchical clustering is not a precisely self-similar pro- cess. In particular, given that the primordial mass power spectrum P ( k ) → k n has a power-law index that be- comes smaller at higher k (smaller distance/mass scales, earlier collapse times), the power-law index α in our work will probably have some scale dependence, becoming, if anything, larger at higher densities. Taking this into ac- count, our estimates for the boost factors due to sub- structure are probably on the high side. (2) The small- scale mass cutoff M$_{c}$ for canonical WIMPs [11] may not apply if the new interactions required for a Sommerfeld enhancement are taken into account. The cutoff mass may therefore be very different. If it is much larger, than the boost factors will be reduced (substructures will not extend to such small scales). If the cutoff mass is much smaller, the boost factor may be increased relative to our estimates, but, given the weak dependence of z$_{c}$ and It has long been recognized that there may be a hierar- chy of structure in the Milky Way halo, with substructure existing all the way down to the mass scale of a fraction of an Earth mass, more than 22 orders of magnitude from the ∼ 10 12 M$_{⊙}$ Milky Way halo mass. It has also been noted that this substructure may have serious implica- tions for the dark-matter–annihilation rate in the halo. If the WIMP has the canonical velocity-independent 〈 σv 〉 , then the boost factor could be as high as ( ρ$_{max}$/ ¯ ρ ) ∼ 200, if all of the substructure was preserved. If the WIMP has a Sommerfeld enhancement to the annihilation rate, the boost factor could be even higher, and perhaps dramati- cally so. Unfortunately, the 22 or more orders of magnitude be- tween the substructure cutoff mass and the Milky Way mass prevent the survival of the smallest-scale substruc- ture to be addressed directly with simulations, and reli- Most generally, there may be annihilation in both the smooth component and in the clumped component, in which case the angular dependence of the intensity will fall somewhere in between [21], as illustrated in Fig. 7. Measurement of this angular variation may help shed em- pirical light on the existence of a boost factor. FIG. 7. Intensity of gamma-ray radiation from WIMP an- nihilation in the Milky Way as a function of the (cosine of the) angle ψ that the line of sight makes with the Galactic center. All three curves are normalized to have the same in- tensity at cos ψ = 0. The top (solid) curve is the intensity due to annihilation in a smooth halo; i.e., for I ∝ ∫ ρ 2 dl . The bottom (dotted) curve is that for annihilation entirely in subhalos; i.e., I ∝ ∫ ρ dl . The middle (dashed) curve is for a Sommerfeld-enhanced annihilation [i.e., Eq. (11)] with β = 1, v$_{sat}$/c = 5 . 0 × 10 − $^{5}$. VII. DISCUSSION