4. Testing the Equivalence Principle with strong lensing time delay variations

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Context: In an expanding universe, the redshift is subject to time dependence, with an anticipated variation of roughly \(10^-10\) per year. However, this variation has yet to be measured. I demonstrated that the non-detection of this variation can be used productively. Specifically, redshift-dependent quantities are also time-dependent, indicating that any observable object serves as a probe for the expansion of the universe. One crucial quantity is the angular diameter distance, which provides the redshift of an object. Together with Leonardo Giani, I developed a test of the equivalence principle using strong lensing observations. Through this test, modified gravity scenarios can be constrained through the effective gravitational coupling, \(G\), thereby providing a new probe at intermediate scales in an almost model-independent manner, where only homogeneity and isotropy of the background is assumed.

Method: The analysis is similar when examining strong lensing deviations from any observed object. In addition to the angular diameter distance, the time delay between two multiple images has two components: the geometric time delay, which is a result of differences in the length of light ray trajectories, and the potential time delay, which is due to the gravitational potential of the source. Assuming that only the potential term is impacted by changes in time when \(G\) varies, and assuming that the light path is unperturbed, we calculate the relative variation in time delay. To do this, I simulated light curves produced in the Time Delay Challenge 1 (TDC1) along with real light curves from the quasar DES J0408-5354. I used the Python Curve Shifting software PyCS3 from the COSMOGRAIL collaboration to calculate time delays with their associated uncertainties over both the entire observation period and half of the observation period. By examining whether there was a variation in time or not, I provided an upper bound on the temporal variation of \(G\).

Results:

• Redshift drifts determined from strong lensing put upper bounds on \(\dot{G}/G\) ranging from \(10^{-1}\) to \(10^{-2}\) per year.