Weak gravitational lensing by two-power-law densities using the Gauss-Bonnet theorem

Understand

We study the weak-field deflection of light by mass distributions described by two-power-law densities $\rho(R)=\rho_0 R^{-\alpha}(R+1)^{\beta-\alpha}$, where $\alpha$ and $\beta$ are non-negative integers.

• New analytic expressions of deflection angles are obtained via the application of the Gauss-Bonnet theorem to a chosen surface on the optical manifold.
• Some of the well-known models of this two-power law form are the Navarro-Frenk-White (NFW) model $(\alpha,\beta)=(1,3)$, Hernquist $(1,4)$, Jaffe $(2,4)$, and the singular isothermal sphere $(2,2)$.
• The calculated deflection angles for Hernquist and NFW agrees with that of Keeton and Bartelmann, respectively.