The cusp-halo relation

Simulations have established that each halo of collisionless dark matter is expected to contain a $\rho = A r^{-1.5}$ density cusp at its center. This prompt cusp is a relic of the halo's earliest moments and has a mass comparable to the cutoff scale in the spectrum of initial density perturbations. In this work, we provide a framework to predict for each halo the coefficient $A$ of its central cusp. We also present a "cusp-NFW" functional form that accurately describes the density profile of a halo with a prompt cusp at its center. Accurate characterization of each halo's central cusp is of particular importance in the study of warm dark matter models, for which the spectral cutoff is on an astrophysically relevant mass scale. To facilitate easy incorporation of prompt cusps into any halo modeling approach, we provide a code package that implements the cusp-halo relation and the cusp-NFW density profile.