Astro 321: Halo Model 3

Halo Profile

The Navarro-Frenk-White (NFW profile) is given by

ρ(r, M) ∝ \frac{1}{rc/r_{h}(1 + rc/r_{h})^{2}}  (1)

where r_{h} = [3M/4πρ_{m}(z = 0)Δ_{M}]^{1/3} is the comoving radius associated with a halo of mass M at an overdensity of Δ_{M} = 180. c is the concentration and for the purposes of this project take it to be c = 10 independent of mass (we also ignore the distiction between the halo mass and virial mass). The normalization is set so that the volume integral over the halo profile out to R returns the mass M.

Compute the Fourier transform of the NFW profile for use in the 1-halo term of the halo model

y(k, M) = \frac{1}{M} ∫_{0}^{r_{h}} dr4πr^{2}ρ(r, M)\frac{sin(kr)}{kr}  (2)

normalized such that y(0, M) = 1.