From Matter Density to Deflection Angle and Gravitational Lensing Using a Perturbative Method

TL;DR

This paper introduces a perturbative approach to calculate deflection angles and gravitational lensing effects in static, spherically symmetric spacetimes with arbitrary density profiles, validated by numerical comparisons.

Contribution

The work develops a novel perturbative method linking matter density distributions to gravitational lensing observables in SSS spacetimes.

Findings

Analytical formulas match numerical results well.

Method applicable to various density profiles.

Provides insights into density influence on lensing.

Abstract

In this work, we develop a perturbative method to compute the deflection angle of null or timelike signals in spacetimes filled with a static and spherically symmetric (SSS) perfect fluid with fairly arbitrary density distributions. After solving the Tolman-Oppenheimer-Volkoff equations, the metric functions of the spacetime are obtained either as asymptotic series or as expansions around a finite boundary. The deflection angles of null or timelike signals in the weak-field limit in such spacetimes can then be expressed as series expansions in terms of the impact parameter, with coefficients determined by the metric expansions and, in turn, the density distribution function. Gravitational lensing equations are also solved perturbatively to derive the apparent angles of the lensed images. Comparing our analytical formulas with numerical results demonstrates the validity and efficiency of our method and results. This procedure establishes a direct connection between the mass density, the deflection angle, and the apparent angles of gravitationally lensed images. We apply these methods and results to the generalized Navarro-Frenk-White model and some other density profiles to analyze the influence of the density parameters.