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

TL;DR

This work investigates how negative spatial curvature affects strong gravitational lensing in a Swiss-cheese cosmology built from an open FLRW background ($k=-1$) with Einstein–Straus–de Sitter embedded vacuoles. By exactly integrating null geodesics across the vacuole and surrounding spacetime, it shows a curvature threshold at $a_0 \approx 2.6\times10^{27}$ m (|$Ω_{k0}$| ≈ 0.0025) beyond which lensing observables converge to flat-space values, while below this threshold curvature enhances deflection and time delays. For a representative lensed quasar SDSS J1004+4112 with $Ω_{k0}=-0.15$, deflection increases by about 0.9% and the time delay by about 10% relative to the flat case, underscoring a stronger sensitivity of time delay to curvature. Across all configurations, the cosmological constant $Λ$ continues to reduce light bending, indicating that $Λ$ effects persist even in negatively curved spacetimes and must be included in precise lensing analyses.

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 $|Ω_{k0}| \gtrsim 0.0025$. In particular, for $Ω_{k0} = -0.15$, 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.

Figures (8)

• Figure 1: Two photons emitted by a source S, bent inside the SdS vacuole by a lens L and finally received at Earth E. Outside the vacuole the trajectories diverge due to the pseudospherical geometry of negative curvature.
• Figure 2: Evolution of galaxy cluster mass $M$ with current cosmic scale factor $a_0$ in hyperbolic Einstein-Straus--de Sitter model. Cosmological constant $\Lambda$, angles $\alpha$ and $\alpha^{\prime}$ are fixed at their central values.
• Figure 3: Evolution of deflection angle $-\varphi_\text{S}$ with current cosmic scale factor $a_0$ in hyperbolic Einstein-Straus--de Sitter model. Cosmological constant $\Lambda$, angles $\alpha$ and $\alpha^{\prime}$ are fixed at their central values.
• Figure 4: Evolution of time delay $\Delta t$ with current cosmic scale factor $a_0$ in hyperbolic Einstein-Straus--de Sitter model. Cosmological constant $\Lambda$, angles $\alpha$ and $\alpha^{\prime}$ are fixed at their central values.
• Figure 5: Evolution of galaxy cluster mass $M$ with current curvature density $\Omega_{k0}$ in hyperbolic Einstein-Straus--de Sitter model. Cosmological constant $\Lambda$, angles $\alpha$ and $\alpha^{\prime}$ are fixed at their central values.