Gravitational lensing in a universe with matter and a cosmological constant

TL;DR

This paper investigates gravitational lensing in a universe transitioning from matter dominance to dark energy dominance, comparing models to determine the significance of matter content and higher-order effects on lensing predictions.

Contribution

It extends previous lensing models to include the matter-dark energy transition and assesses the necessity of matter content in lensing calculations at observed redshifts.

Findings

No significant contribution from matter content beyond previous models.

Higher-order corrections do not measurably affect the bending angle.

The effect of the universe's accelerated expansion is consistent with earlier results.

Abstract

We extend the results obtained in \cite{Piattella_2016, mcvittie_2015} and \cite{Park_2008} for gravitational lensing in the McVittie metric by including the effect of the transition from the matter-dominated epoch of the Universe to the $\Lambda$-dominated era. We derive a formula that agrees with the previous results for the McVittie metric at lowest order, and compare the lensing angle predictions obtained from the Schwarzschild approximation, the McVittie model and higher order corrections to the McVittie model. In doing this, we test if, beyond the correction from the accelerated expansion of the Universe, there is a need for including the matter content of the Universe in modeling lens systems at the redshifts observed in lens systems. We investigate if there is a need for a modification of the lens equation from these corrections, and if so, to which order and whether it is measurable. We find that while the effect is of the same order as the one calculated previously, there is no significant contribution to the bending angle, as the 1st order effect is already of order $\mathcal{O}(\theta_O^4)$ in the observed angle.