Time delay in the Einstein-Straus solution

Abstract

The time delay of strong lensing is computed in the framework of the Einstein-Straus solution. The theory is compared to the observational bound on the time delay of the lens SDSS J1004+4112.Comment: 20 pages, 4 tables, 1 figur

Time delay in Swiss cheese gravitational lensing

“…To incorporate the lenses into the mean homogeneous mass density and to enable us to evaluate time delays beyond the linear term we use a simple Swiss cheese cosmology [3][4][5]. Estimations of these time delays using a similar lensing model have recently appeared in [6,7] and earlier estimates, some using less precise models, appeared in [8][9][10][11][12]. These attempts primarily focus on determining the cosmological constant's effect on lensing, including time delays, whereas this analytic work emphasizes the significantly larger effect caused by embedding the lens into the cosmology rather than the conventional approach of simply superimposing the lensing mass on top of the homogeneous mean density.…”

Deflection of light and time delay in closed Einstein-Straus solution

“…In this region where prevails Kottler metric, we shall not go deeper in the details leading to the same results already discussed in the flat case [11,18], and what we would have to do is to calculate first the scale factors a ′ SchüS and a SchüS and then their corresponding times t ′ SchüS and t SchüS , at which the two photons enter inside the vacuole, by integrating the Friedman equation with final conditions at the exit points from the vacuole.…”

“…Within the same framework, Ishak, Rindler, Schücker, Kantowski et al [10,11,12,13], have proved that the effect of the cosmological constant on light bending is only diminished without however being dropped, contrary to what has been argued in [14,15,16]. In references [17,18,19], the authors have gone a step further and investigated the cosmological constant's effect on time delay in which case the computation of the photon's travel time outside the vacuole is particularly simple using some properties of Euclidean geometry.…”

“…However, as in the flat case [18], one can proceed otherwise, without further numerical computation to obtain the time t SchüE for lower trajectory photon. The method consists in determining t SchüE by difference with t ′ SchüE through an approximate analytical expression.…”

“…However, one can proceed, as in the flat case [18], in a different manner computing directly the time delay through an approximate analytical expression. Combining (99) and (102), one gets, after evaluating the integrals with respect to time,…”