4 There is r dependence in this boost factor via Eq. (4) in f$_{s}$ , via Eq. (1) in ρ$_{h}$ ≈ ¯ ρ , and in principle also in Δ although it turns out to be negligible. There are two contributions to this boost factor: The first comes from the finite width Δ of the smooth component—i.e., B$_{s}$ = f$_{s}$e Δ $^{2}$—and it depends very strongly on Δ. Given that we find Δ ≲ 0 . 2 in our sim- ulations, we infer that the boost factor due to the finite width of the smooth component is no more than a few percent. We now turn to the second term, that due to substruc- ture, the central focus of this work. Given that ρ$_{max}$ ≫ ¯ ρ in the Milky Way halo, the boost factor would be essen- tially independent of ρ$_{max}$ if α were α > 1. However, we find in the simulations that α < 1 in a Milky-Way–like halo, in which case the boost-factor results for the canon- ical WIMP do indeed depend on ρ$_{max}$ . The fact that the integral in Eq. (6) is dominated by the high-density end implies that the boost factor due to clumping is deter- mined primarily by the early-collapsing highest-density (and lowest-mass) substructures. where 〈 σv 〉 is the annihilation cross section (times rel- ative velocity v , averaged over the velocity distribution of the halo, and m$_{χ}$ is the WIMP mass; i.e. 〈 σv 〉 = ∫ ∞ 0 dv f ( v ) σv , where f ( v ) (normalized to ∫ ∞ 0 f ( v ) dv = 1) is the WIMP pairwise velocity distribution at that point in the Galactic halo. For the canonical WIMP, σv is approximately velocity independent at galactic-halo ve- locities, and so 〈 σv 〉 = ( σv ), a constant. If σv depends on velocity, then 〈 σv 〉 may vary from one point in the halo to another due to the possible variation of f ( v ) with po- sition in the halo. We now consider the boost of the annihilation rate in a halo with substructure relative to the rate in the canonical smooth-halo model. The annihilation rate (per unit volume) at any point in the Galactic halo is (not shown here) it remains unresolved, and we conclude only that Δ ≲ 0 . 2 at radii less than 100 kpc. Let us first consider the canonical WIMP where the cross-section factor 〈 σv 〉 is velocity-independent. In this case, the change in the velocity dispersion that may ac- company clumping is irrelevant. The total annihilation rate in some volume V is then proportional to a volume integral of the density squared—i.e., ∝ ∫ ρ 2 dV . The annihilation rate for a fixed total mass of dark matter enclosed within a given volume will thus be enhanced by a “boost factor” if the matter is not uniformly distributed within that volume. For the canonical WIMP, this boost factor B ( r ) is the ratio What is an appropriate value for ρ$_{max}$ ? A first guess might be the virial density of the earliest-collapsing ha- los, ρ$_{vir}$ = 178 ρ$_{crit}$ ( z$_{c}$ ) ≃ 16 GeV cm − 3 ( z$_{c}$/ 40) 3 (for a collapse redshift z$_{c}$ ≫ 0). However, depending on its density profile, most of a halo’s volume might have considerably higher densities. In Fig. 3 we show the density probability distribution P ( ρ ) ≡ 1 /V dV/dρ for an isolated NFW halo. At low densities, in the out- skirts of the halo where ρ ∼ r − $^{3}$, the density proba- bility falls as ρ − $^{2}$, matching the power law tail of the local density probability function (Eq. (2)). In the in- where the second equality follows since P ( ρ, r ) = (1 /V )( dV/dρ ) (with V the volume in the halo). The quantity ρ$_{max}$ is the maximum density. We take that to be ∼ 80 GeV cm − $^{3}$, which is about five times the mean virial density of the rare first collapsed structures in the Universe (178 times the mean density of the Universe at z ≈ 40); we will say more later about ρ$_{max}$ . In the standard WIMP model, where 〈 σv 〉 =constant, the boost factor B ( r ) is given by (noting that ¯ ρ ≃ ρ$_{h}$ ) B ( r ) = ∫ ρ 2 dV ∫ [¯ ρ ( r )] 2 dV = ∫ ρ$_{max}$ 0 P ( ρ, r ) ρ 2 [¯ ρ ( r )] 2 dρ, (6) B ( r ) = f$_{s}$e Δ 2 +(1 − f$_{s}$ ) 1 + α 1 − α [ $^{(}$ρ$_{max}$ ρ$_{h}$ ) 1 − α − 1$^{]}$ . (7) Γ = 〈 σv 〉 ρ 2 2 m 2 χ , (5) IV. ANNIHILATION BOOST FACTOR A. Standard velocity dependence FIG. 3. The density probability distribution function P ( ρ ) for an isolated NFW halo with concentration c = 10, truncated at r$_{vir}$ . The P ( ρ ) ∼ ρ − 2 behavior continues beyond ρ$_{vir}$ (vertical dashed line) to ∼ ρ ( r$_{s}$ ) = 1 / 12 c $^{3}$/f ( c ) ρ$_{vir}$ .