thus ρ$_{max}$ on M$_{c}$ , not by much. (3) While identification of the high-density power-law tail in P ( ρ ) and measure- ment of its small amplitude is a big step forward, our N-body measurement of the parameters required to de- scribe the distribution P ( ρ, r ) can certainly be improved upon. It will be important in future work to measure f$_{s}$ ( r ) more precisely, and to measure and determine the r dependence of α and Δ. (4) Fig. 3 indicates that at densities above ρ$_{max}$ , the α = 0 power law for P ( ρ ) will steepen to something closer to α = 2. This steep high- density tail is due to the small-r ρ ∼ 1 /r dependence in the earliest NFW halos. For a canonical WIMP with a velocity-independent 〈 σv 〉 , this steepening will not affect the results, since the integrand in Eq. (6) is proportional to ρ − α and hence dominated by the low density (i.e., near ρ$_{max}$ ) end of this steep tail. With a Sommerfeld enhancement, however, the integrand is proportional to ρ 1 . 75 β − α (Eq. (9)), which for α = 2 is dominated by the high-density end if β > 8 / 7. In other words, if β > 8 / 7, the substructure annihilation enhancement will be dom- inated by the 1 /r cusps in the earliest NFW subhalos, 

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