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 \geq 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 ( ) and exemplified by .
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 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 .