Deflection and gravitational lensing of null and timelike signals in general asymptotically (anti-)de Sitter spacetimes

TL;DR

This paper investigates how the cosmological constant affects gravitational lensing and signal deflection in asymptotically (anti-)de Sitter spacetimes, revealing that null deflection is unaffected while other lensing properties are influenced.

Contribution

It introduces a perturbative method to compute lensing effects in (anti-)de Sitter spacetimes, accounting for finite source/detector distances and analyzing the impact of the cosmological constant.

Findings

Null ray deflection is independent of $\Lambda$ under certain conditions.

Small positive $\Lambda$ decreases apparent image angles.

Time delays between signals are affected by $\Lambda$, decreasing for signals from the same side with different energies.

Abstract

The deflection and gravitational lensing of light and massive particles in arbitrary static, spherically symmetric and asymptotically (anti-)de Sitter spacetimes are considered in this work. We first proved that for spacetimes whose metric satisfying certain conditions, the deflection of null rays with fixed closest distance will not depend on the cosmological constant $\Lambda$, while that of timelike signals and the apparent angle in gravitational lensing still depend on $\Lambda$. A two-step perturbative method is then developed to compute the change of the angular coordinate and total travel time in the weak field limit. The results are quasi-series of two small quantities, with the finite distance effect of the source/detector naturally taken into account. These results are verified by applying to some known de Sitter spacetimes. Using an exact gravitational lensing equation, we solved the apparent angles of the images and time delays between them and studied the effect of $\Lambda$ on them. It is found that generally, a small positive $\Lambda$ will decrease the apparent angle of images from both sides of the lens and increase the time delay between them. The time delay between signals from the same side of the lens but with different energy however, will be decreased by $\Lambda$.