Consistency relations of amplitude and phase fluctuations of gravitational waves magnified by strong gravitational lensing

TL;DR

This paper investigates how strong and weak gravitational lensing affect the amplitude and phase fluctuations of gravitational waves, establishing that certain consistency relations remain valid even with a strong lens present.

Contribution

It introduces a systematic method to evaluate weak lensing effects on gravitational waves in the presence of a strong lens, extending previous relations to more complex lensing scenarios.

Findings

Consistency relations hold with a strong lens present.

Weak lensing causes both magnification and Fresnel scale shifts.

The amplification factor is derived up to second order in weak lensing potential.

Abstract

We discuss the amplitude and phase fluctuations of gravitational waves due to wave optics lensing in the presence of both a strong lens and cosmological weak lenses. By applying the geometric optics approximation to the strong lens and treating the weak lensing potential perturbatively, we obtain the amplification factor up to the second order in the weak lensing potential. Additionally, we establish a methodology to systematically evaluate the weak lensing effects based on diagrammatic rules. Based on the derived amplification factor, we evaluate the statistics of the fluctuations and demonstrate that the consistency relations originally established in the absence of a strong lens still hold in exactly the same form when a strong lens is present. The physical origin of these relations is also discussed. Furthermore, we demonstrate that for the mean of the weak lensing signal, both the magnification of the signal and the shift of the Fresnel scale to larger scales occur, consistent with the behavior observed in the variance.