Analysis of the angular dependence of time delay in gravitational lensing

TL;DR

This paper proposes a generalized formula for gravitational lensing time delay that applies to arbitrary angles, extending the standard small-angle approximation, with potential applications in cosmology and astrophysics.

Contribution

It introduces a new, more general formula for gravitational lensing time delay that reduces to the standard formula at small angles and offers new insights into lens mass and potential constraints.

Findings

The new formula matches the standard at small angles.

It provides a constraint on the gravitational potential.

It offers a method to estimate lens mass in central potential cases.

Abstract

We consider an alternative formula for time delay in gravitational lensing. Imposing a smoothness condition on the gravitationally deformed paths followed by the photons from the source to the observer, we show that our formula displays the same degrees of freedom as the standard one. In addition to this, it is shown that the standard expression for time delay is recovered when small angles are involved. These two features strongly support the claim that the formula for time delay studied in this paper is the generalization to the arbitrary angles of the standard one, which is valid at small angles. This could therefore result in a useful tool in Astrophysics and Cosmology which may be applied to investigate the discrepancy between the various estimates of the Hubble constant. As an aside, two interesting consequences of our proposal for time delay are discussed: the existence of a constraint on the gravitational potential generated by the lens and a formula for the mass of the lens in the case of central potential.