Gravitational Lensing as an Optical Framework for Modified Gravity Theories

TL;DR

This paper introduces an optical framework for gravitational lensing that simplifies analysis of modified gravity theories using undergraduate mathematics, deriving analytical expressions and validating them with simulations.

Contribution

It develops a unified optical approach to analyze various modified gravity models in gravitational lensing, providing analytical formulas and educational tools.

Findings

Derived general deflection angle for spherically symmetric fields.

Established observational baseline using general relativity.

Validated analytical results with numerical ray-tracing simulations.

Abstract

We present a framework that reformulates gravitational lensing as an optical phenomenon governed by an effective refractive index, enabling exploration of modified gravity theories using undergraduate-level mathematics and optics. After deriving the general deflection angle for arbitrary spherically symmetric fields, we establish the observational baseline using standard general relativity, including the lens equation and Einstein ring properties. Assuming the optical relation holds for modified effective potentials, we apply the formalism to deep-MOND, Yukawa-type, and power-law ($f(R)$) models, providing closed-form analytical expressions for the deflection angle and Einstein radius. Numerical ray-tracing simulations validate these analytical results. This framework serves as a conceptual bridge to contemporary research, offering students computational experience and critical awareness of gravitational lensing foundations.