Galaxy halo expansions: A new biorthogonal family of potential-density pairs

Efficient expansions of the gravitational field of (dark) haloes have two main uses in the modelling of galaxies: first, they provide a compact representation of numerically constructed (or real) cosmological haloes, incorporating the effects of triaxiality, lopsidedness or other distortion. Secondly, they provide the basis functions for self-consistent field expansion algorithms used in the evolution of N-body systems. We present a new family of biorthogonal potentialdensity pairs constructed using the Hankel transform of the Laguerre polynomials. The lowest order density basis functions are double-power-law profiles cusped like ρ ~ r-2+1/α at small radii with asymptotic density fall-off like ρ ~ r-3-1/(2α). Here, α is a parameter satisfying α ≥ 1/2. The family therefore spans the range of inner density cusps found in numerical simulations, but has much shallower - and hence more realistic - outer slopes than the corresponding members of the only previously known family deduced by Zhao and exemplified by Hernquist & Ostriker. When α = 1, the lowest order density profile has an inner density cusp of ρ ~ r-1 and an outer density slope of ρ ~ r-3.5, similar to the famous Navarro, Frenk & White (NFW) model. For this reason, we demonstrate that our new expansion provides a more accurate representation of flattened NFW haloes than the competing Hernquist-Ostriker expansion.We utilize our new expansion by analysing a suite of numerically constructed haloes and providing the distributions of the expansion coefficients. © 2018 The Author(s).