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Time delay in matched exterior and interior Kottler solutions

Mourad Guenouche

TL;DR

This work addresses how interior mass structure affects gravitational-lensing time delays by modeling the lens as a uniform-density fluid sphere described by the interior Kottler solution and smoothly matching it to exterior Schwarzschild–de Sitter spacetime at $r_B$. Using conserved quantities for null geodesics, the authors derive analytic expressions for the bending angle and travel-time, and decompose the total time delay into an exterior SdS part plus an interior correction $4GM\mathcal{E}_{in}$. Numerical results for galaxy- and cluster-scale lenses show interior corrections can amount to roughly 60–70% of the exterior contribution, producing delays that differ by tens of days (galaxies) to years (clusters). The findings emphasize the importance of incorporating interior lens structure in strong-lensing analyses and offer a relativistic framework that can be extended to more realistic mass profiles and dynamical cosmological backgrounds.

Abstract

We extend the study of gravitational lensing to photons traversing the lens mass, modeled as a uniform-density fluid described by the interior Kottler (Schwarzschild--de Sitter) solution smoothly matched to the exterior Kottler region. An analytic expression for the time delay is derived, allowing the interior contribution to be explicitly isolated relative to the vacuum Kottler case. This correction is found to systematically enhance the total time delay, an effect corroborated by numerical evaluations for astrophysically relevant lenses at both galaxy and cluster scales. These results underscore the importance of accounting for the interior lens structure in accurate modeling of strong lensing time delays.

Time delay in matched exterior and interior Kottler solutions

TL;DR

This work addresses how interior mass structure affects gravitational-lensing time delays by modeling the lens as a uniform-density fluid sphere described by the interior Kottler solution and smoothly matching it to exterior Schwarzschild–de Sitter spacetime at . Using conserved quantities for null geodesics, the authors derive analytic expressions for the bending angle and travel-time, and decompose the total time delay into an exterior SdS part plus an interior correction . Numerical results for galaxy- and cluster-scale lenses show interior corrections can amount to roughly 60–70% of the exterior contribution, producing delays that differ by tens of days (galaxies) to years (clusters). The findings emphasize the importance of incorporating interior lens structure in strong-lensing analyses and offer a relativistic framework that can be extended to more realistic mass profiles and dynamical cosmological backgrounds.

Abstract

We extend the study of gravitational lensing to photons traversing the lens mass, modeled as a uniform-density fluid described by the interior Kottler (Schwarzschild--de Sitter) solution smoothly matched to the exterior Kottler region. An analytic expression for the time delay is derived, allowing the interior contribution to be explicitly isolated relative to the vacuum Kottler case. This correction is found to systematically enhance the total time delay, an effect corroborated by numerical evaluations for astrophysically relevant lenses at both galaxy and cluster scales. These results underscore the importance of accounting for the interior lens structure in accurate modeling of strong lensing time delays.

Paper Structure

This paper contains 7 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Matching interior and exterior Kottler metrics
  3. Integrating null geodesic equations
  4. Bending of light
  5. Time delay
  6. Numerical Results and Discussion
  7. Conclusion

Figures (3)

  • Figure 1: Two light rays emitted by a source S, bent outside and inside the SdS fluid lens L, and received at the blue Earth E.
  • Figure 2: Evolution of time delay versus reception angle $\alpha\in[1.18^{\prime\prime},1.31^{\prime\prime}]$ with $\alpha^\prime=\alpha/0.9$ for a galaxy lens in matched Kottler spacetime.
  • Figure 3: Evolution of time delay versus reception angle $\alpha\in[11.76^{\prime\prime},13.07^{\prime\prime}]$ with $\alpha^\prime=\alpha/0.9$ for a cluster lens in matched Kottler spacetime.