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

## TL;DR

This paper derives new analytic expressions for weak gravitational lensing deflection angles caused by two-power-law density profiles using the Gauss-Bonnet theorem, connecting topological properties of the optical manifold to lensing behavior.

## Contribution

It introduces a novel application of the Gauss-Bonnet theorem to compute deflection angles for a broad class of density profiles, extending previous topological insights.

## Key findings

- Derived analytic deflection angles for various two-power-law models.
- Confirmed agreement with known results for Hernquist and NFW profiles.
- Linked deflection behavior to topological properties of the optical manifold.

## Abstract

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. The limiting values of these deflection angles (at zero or infinite impact parameter) are either vanishing or similar to the deflection due to a singular isothermal sphere. We show that these behaviors can be attributed to the topological properties of the optical manifold, thus extending the pioneering insight of Werner and Gibbons to a broader class of mass densities.