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

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.

Figures (5)

• Figure 1: Gravitational lensing geometry. We adopt a coordinate system $(\chi,\bm{\theta})$ with the origin at the source, where $\chi$ is the comoving distance from the source and $\bm{\theta}$ is the two-dimensional vector perpendicular to the line of sight. The observer's position is denoted by $(\chi_{s},\bm{\theta}_{s})$. The strong lens component is treated using the thin-lens and geometric optics approximations, while the weak lens potential is treated perturbatively within the framework of wave optics.
• Figure 2: Summary of diagrammatic rules and their corresponding mathematical expressions. The table defines the propagator and the two distinct types of vertices representing the strong and weak lensing potentials, respectively. The explicit mathematical form of the propagator is given by Eq. \ref{['propag']}. The amplification factor is constructed by arranging these diagrammatic elements.
• Figure 3: The kernel functions of $\langle S^{2}_{j}\rangle_{W}$ (left) and $\langle S_{j}\rangle_{W}$ (right) are plotted for three different values of the magnification factor $\mu_{j}=\mu_{j,1}\mu_{j,2}=1\ (\text{solid}), 10\ (\text{dotted}), 100\ (\text{dashed})$. The variation in $\mu_{j}$ is achieved by changing $\mu_{j,1}$ while holding $\mu_{j,2}=1$ fixed. We assume a weak lens redshift of $z=0.1$, a strong lens redshift of $z_{l}=0.3$, a source redshift of $z_{s}=2$, and a gravitational wave frequency of $10\ \text{Hz}$, for which the inverse of the Fresnel scale is $1/ r_{F}\sim 6\times10^{6}h \text{Mpc}^{-1}$.
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