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

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.