# Towards a New Proposal for the Time Delay in Gravitational Lensing

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## Abstract

**:**

## 1. Introduction

## 2. Standard Analysis

#### 2.1. Basics of Gravitational Lensing

- The Shapiro delay, or potential time delay, caused directly by the motion of the light through the gravitational potential of the lens
- The geometric delay, caused by the increased length of the total light path from the source to the earth.

#### 2.2. The Shapiro Time Delay in Minkowski Metric

#### 2.3. Geometric Time Delay

## 3. An Easy Extension

#### 3.1. The Extension of $\Delta {t}_{G}$

#### 3.2. The Shapiro Time Delay in RW Metric

## 4. Cosmological Born-Oppenheimer Approximation for Time Delay

#### The Idea

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Christoffel Symbols

#### Appendix A.2. Coefficients of the Expansion

## References

- Blasi, A.; Maggiore, N. Massive deformations of rank-2 symmetric tensor theory (a.k.a. BRS characterization of Fierz–Pauli massive gravity). Class. Quantum Gravity
**2017**, 34, 015005. [Google Scholar] [CrossRef] - Blasi, A.; Maggiore, N. Massive gravity and Fierz—Pauli theory. Eur. Phys. J. C
**2017**, 77, 614. [Google Scholar] [CrossRef] - Blasi, A.; Braggio, A.; Carrega, M.; Ferraro, D.; Maggiore, N.; Magnoli, N. Non-Abelian BF theory for 2+1 dimensional topological states of matter. New J. Phys.
**2012**, 14, 013060. [Google Scholar] [CrossRef] - Blasi, A.; Ferraro, D.; Maggiore, N.; Magnoli, N.; Sassetti, M. Symanzik’s method applied to the fractional quantum hall edge states. Ann. Phys.
**2008**, 17, 885. [Google Scholar] [CrossRef] - Amoretti, A.; Blasi, A.; Caruso, G.; Maggiore, N.; Magnoli, N. Duality and dimensional reduction of 5D BF theory. Eur. Phys. J. C
**2013**, 73, 2461. [Google Scholar] [CrossRef] - Amoretti, A.; Braggio, A.; Maggiore, N.; Magnoli, N. Thermo-electric transport in gauge/gravity models. Adv. Phys. X
**2017**, 2, 409. [Google Scholar] [CrossRef] - Amoretti, A.; Braggio, A.; Caruso, G.; Maggiore, N.; Magnoli, N. Holography in flat spacetime: 4D theories and electromagnetic duality on the border. JHEP
**2014**, 1404, 142. [Google Scholar] [CrossRef] - Carroll, S.M. Spacetime and Geometry: An Introduction to General Relativity; Addison-Wesley: San Francisco, CA, USA, 2004; p. 513. [Google Scholar]
- Efstathiou, G.; Bond, J.R. Cosmic confusion: Degeneracies among cosmological parameters derived from measurements of microwave background anisotropies. Mon. Not. R. Astron. Soc.
**1999**, 304, 75. [Google Scholar] [CrossRef] - Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartlett, J.G.; Bartolo, N.; et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys.
**2016**, 594, A13. [Google Scholar] [CrossRef] - Suyu, S.H.; Bonvin, V.; Courbin, F.; Fassnacht, C.D.; Rusu, C.E.; Sluse, D.; Treu, T.; Wong, K.C.; Auger, M.W.; Ding, X.; et al. H0LiCOW—I. H0 Lenses in COSMOGRAIL’s Wellspring: Program overview. Mon. Not. R. Astron. Soc.
**2017**, 468, 2590. [Google Scholar] [CrossRef] - Sluse, D.; Sonnenfeld, A.; Rumbaugh, N.; Rusu, C.E.; Fassnacht, C.D.; Treu, T.; Suyu, S.H.; Wong, K.C.; Auger, M.W.; Bonvin, V.; et al. H0LiCOW—II. Spectroscopic survey and galaxy-group identification of the strong gravitational lens system HE0435-1223. Mon. Not. R. Astron. Soc.
**2017**, 470, 4838. [Google Scholar] [CrossRef] - Rusu, C.E.; Fassnacht, C.D.; Sluse, D.; Hilbert, S.; Wong, K.C.; Huang, K.H.; Suyu, S.H.; Collett, T.E.; Marshall, P.J.; Treu, T.; et al. H0LiCOW—III. Quantifying the effect of mass along the line of sight to the gravitational lens HE 0435-1223 through weighted galaxy counts. Mon. Not. R. Astron. Soc.
**2017**, 467. [Google Scholar] [CrossRef] - Wong, K.C.; Suyu, S.H.; Auger, M.W.; Bonvin, V.; Courbin, F.; Fassnacht, C.D.; Halkola, A.; Rusu, C.E.; Sluse, D.; Sonnenfeld, A.; et al. H0LiCOW—IV. Lens mass model of HE 0435-1223 and blind measurement of its time-delay distance for cosmology. Mon. Not. R. Astron. Soc.
**2017**, 465, 4895. [Google Scholar] [CrossRef] - Bonvin, V.; Courbin, F.; Suyu, S.H.; Marshall, P.J.; Rusu, C.E.; Sluse, D.; Tewes, M.; Wong, K.C.; Collett, T.; Fassnacht, C.D.; et al. H0LiCOW—V. New COSMOGRAIL time delays of HE 0435-1223: H
_{0}to 3.8 per cent precision from strong lensing in a flat ΛCDM model. Mon. Not. R. Astron. Soc.**2017**, 465, 4914. [Google Scholar] [CrossRef] - Refsdal, S. On the possibility of determining Hubble’s parameter and the masses of galaxies from the gravitational lens effect. Mon. Not. R. Astron. Soc.
**1964**, 128, 307. [Google Scholar] [CrossRef] - Riess, A.G.; Macri, L.M.; Hoffmann, S.L.; Scolnic, D.; Casertano, S.; Filippenko, A.V.; Tucker, B.E.; Reid, M.J.; Jones, D.O.; Silverman, J.M.; et al. A 2.4% Determination of the Local Value of the Hubble Constant. Astrophys. J.
**2016**, 826, 56. [Google Scholar] [CrossRef] - Di Valentino, E.; Melchiorri, A.; Mena, O. Can interacting dark energy solve the H
_{0}tension? Phys. Rev. D**2017**, 96, 043503. [Google Scholar] [CrossRef] - Schneider, P.; Ehlers, J.; Falco, E.E. Gravitational Lenses; Springer: Berlin, Germany, 1992. [Google Scholar]
- CASTLES Catalogue. Available online: https://www.cfa.harvard.edu/castles (accessed on 24 September 2017).
- Weinberg, S. Cosmology; Oxford University Press: Oxford, UK, 2008; p. 593. [Google Scholar]
- Alchera, N.; Bonici, M.; Maggiore, N.; Panizzi, L.; Università di Genova, Genova, Italy. In progress. 2017.

**Figure 1.**${S}_{1}$ and ${S}_{2}$ are the images of the source S. The points ${P}_{1}$ and ${P}_{2}$ are the deflection points of the light rays deflected by the lens L and observed in E. $S{P}_{1}E$ and $S{P}_{2}E$ approximate the deflected photon geodesics.

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**MDPI and ACS Style**

Alchera, N.; Bonici, M.; Maggiore, N.
Towards a New Proposal for the Time Delay in Gravitational Lensing. *Symmetry* **2017**, *9*, 202.
https://doi.org/10.3390/sym9100202

**AMA Style**

Alchera N, Bonici M, Maggiore N.
Towards a New Proposal for the Time Delay in Gravitational Lensing. *Symmetry*. 2017; 9(10):202.
https://doi.org/10.3390/sym9100202

**Chicago/Turabian Style**

Alchera, Nicola, Marco Bonici, and Nicola Maggiore.
2017. "Towards a New Proposal for the Time Delay in Gravitational Lensing" *Symmetry* 9, no. 10: 202.
https://doi.org/10.3390/sym9100202