Towards a New Proposal for the Time Delay in Gravitational Lensing

Abstract

One application of the Cosmological Gravitational Lensing in General Relativity is the measurement of the Hubble constant H 0 using the time delay ∆t between multiple images of lensed quasars. This method has already been applied, obtaining a value of H 0 compatible with that obtained from the SNe 1A, but non-compatible with that obtained studying the anisotropies of the CMB. This difference could be a statistical fluctuation or an indication of new physics beyond the Standard Model of Cosmology, so it desirable to improve the precision of the measurements. At the current technological capabilities it is possible to obtain H 0 to a percent level uncertainty, so a more accurate theoretical model could be necessary in order to increase the precision about the determination of H 0 . The actual formula which relates ∆t with H 0 is approximated; in this paper we expose a proposal to go beyond the previous analysis and, within the context of a new model, we obtain a more precise formula than that present in the literature.

Analysis of the Angular Dependence of Time Delay in Gravitational Lensing

“…The two formulas for time delay, the standard one [21] and the one we are proposing [20], at first sight look quite different. In particular, our proposal seems to depend on more degrees of freedom.…”

“…[20] we have obtained a new formula to determine the Hubble constant H 0 using time delay ∆t between multiple images of lensed objects (Throughout this paper we adopt the notations of [22]). …”

“…Finally, c k are the coefficients of the Taylor series, which can be easily computed and can be found in Ref. [20]. Equation (1) was built in the framework of a new theoretical model for GL whose geometry is described in Figure 1.…”

“…In the standard analysis [21], the time delay is calculated adding two contributions: the Shapiro delay from the gravitational potential of the lens and the geometric delay due to the deformations of the ray paths, approximated by straight lines [21]. The formula we are considering in this work derives from a different approach which allows to compute the time delay directly in a single shot, rather than two [20].…”

“…Our contribution towards an attempt to solve the disagreement [20] concerns the theoretical analysis of the time delay ∆t and its connection to H 0 . In the standard analysis [21], the time delay is calculated adding two contributions: the Shapiro delay from the gravitational potential of the lens and the geometric delay due to the deformations of the ray paths, approximated by straight lines [21].…”