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

TL;DR

This paper introduces a new family of biorthogonal potential-density pairs for modeling galaxy haloes, offering more realistic outer slopes and better accuracy for NFW-like profiles compared to previous expansions.

Contribution

The authors develop a novel biorthogonal potential-density family using Hankel transforms of Laguerre polynomials, improving the representation of galaxy haloes with realistic density slopes.

Findings

The new family spans a range of inner density cusps from simulations.

It provides more accurate modeling of NFW haloes than previous expansions.

The expansion coefficients distribution for simulated haloes is analyzed.

Abstract

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 potential-density pairs constructed using the Hankel transform of the Laguerre polynomials. The lowest-order density basis functions are double-power-law profiles cusped like $\rho \sim r^{-2 + 1/\alpha}$ at small radii with asymptotic density fall-off like $\rho \sim r^{-3 -1/(2\alpha)}$. Here, $\alpha$ is a parameter satisfying $\alpha \ge 1/2$. 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 (1996) and exemplified by Hernquist & Ostriker (1992). When $\alpha =1$, the lowest-order density profile has an inner density cusp of $\rho \sim r^{-1}$ and an outer density slope of $\rho \sim r^{-3.5}$, similar to the famous Navarro, Frenk & White (1997) 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.