Deflection of light and time delay in hyperbolic Einstein-Straus--de Sitter solution

TL;DR

This paper investigates how negative spatial curvature and the cosmological constant influence gravitational lensing and time delay in a specific cosmological model, revealing thresholds where curvature effects become significant.

Contribution

It introduces a detailed analysis of gravitational lensing in the Einstein-Straus--de Sitter model within an open universe, highlighting the impact of spatial curvature and cosmological constant on observables.

Findings

Negative curvature increases light bending by about 1% for certain parameters.

Time delay can increase by approximately 10% due to curvature effects.

Lensing observables become insensitive to scale factor beyond a critical threshold.

Abstract

We analyze strong gravitational lensing by a spherically symmetric mass distribution within the Einstein-Straus-de Sitter framework in a spatially open Universe with negative curvature ($k = -1$). Applying the theory to the lensed quasar SDSS J1004+4112, we identify a critical threshold for the current scale factor $a_0$ of approximately $2.6 \times 10^{27}$ m, below which the effects of negative spatial curvature on lensing observables become significant, corresponding to a current curvature density of $|\Omega_{k0}| \gtrsim 0.0025$. In particular, for $\Omega_{k0} = -0.15$, the light bending increases slightly by about 1%, while the time delay exhibits a more substantial increase of about 10%. Beyond this threshold, however, the lensing observables are found to be insensitive to the current scale factor and converge to those characteristic of a spatially flat Universe. Importantly, our results indicate that, even where spatial curvature would otherwise enhance lensing observables, the effect of the cosmological constant remains present and acts to reduce light bending, corroborating the claim of Rindler and Ishak.