Amplitude and phase fluctuations of gravitational waves magnified by strong gravitational lensing

We discuss how small-scale density perturbations on the Fresnel scale affect amplitudes and phases of gravitational waves that are magnified by gravitational lensing in geometric optics. We derive equations that connect the small-scale density perturbations with the amplitude and phase fluctuations to show that such perturbative wave optics effects are significantly boosted in the presence of macro model magnifications such that amplitude and phase fluctuations can easily be observed for highly magnified gravitational waves. We discuss expected signals due to microlensing by stars, dark matter substructure, fuzzy dark matter, and primordial black holes.

Figure 1

The kernel function $I_S^j$ defined in Eq. (37) for different macro model magnifications, ${\mu }_0={\mu }_{j,1}{\mu }_{j,2}=1$ (solid), 10 (dotted), and 100 (dashed), which are computed changing ${\mu }_{j,1}$ while fixing ${\mu }_{j,2}=1$. They are computed assuming the lens redshift $z=0.3$, the source redshift $z_{\mathrm{s}}=2$, and the frequency of gravitational waves $f=10\mathrm{Hz}$, for which the inverse of the Fresnel scale is $1/r_{\mathrm{F}}\simeq 3\times {10}^{6}h{\mathrm{Mpc}}^{-1}$.

Figure 2

The small-scale dimensionless power spectra ${\mathrm{\Delta }}^2(k)=k^{2}P(k)/(2\pi )$ at around the Einstein radius due to dark matter substructure $P_{\kappa }(k)$, Eq. (46), and microlensing by stars $P_{\kappa ,\mathrm{shot}}(k)$, Eq. (50). See the text for parameters used for the calculation. For microlensing, we consider two cases with the stellar mass fraction of $f_{*}=0.5$ and 0.005.

Figure 3

Phase dispersions as a function of the macro model magnification ${\mu }_0$ computed by Eq. (36) with convergence power spectra of dark matter substructure $P_{\kappa }(k)$, Eq. (46), and microlensing by stars $P_{\kappa ,\mathrm{shot}}(k)$, Eq. (50), as shown in Fig. 2. The frequency of gravitational waves of $f=10\mathrm{Hz}$ is adopted.

Figure 4

Similar to Fig. 3, but for $f=0.1\mathrm{Hz}$.

Figure 5

Phase dispersions as a function of the macro model magnification ${\mu }_0$ in the FDM (solid) and PBH (dashed) models for the frequency of $f=10\mathrm{Hz}$. The calculation in the FDM model is done using the convergence power spectrum shown in Eq. (51) with the FDM mass of ${10}^{-20}\mathrm{eV}$, while that in the PBH model is based on the convergence power spectrum shown in Eq. (53) with $f_{\mathrm{PBH}}=0.1$ and $M_{\mathrm{PBH}}=30M_{\odot }$. Here we assume the stellar mass fraction of $f_{*}=0.5$, and the other parameters are same as those used in Sec. 3b.