Towards model-independent identification of lensed gravitational waves using Kramers-Kronig relation

TL;DR

This paper proposes a model-independent approach to identify microlensed gravitational waves by leveraging the Kramers-Kronig (KK) relation, which ties the real and imaginary parts of the amplification factor $F( u)$ in a causal, linear gravitational-lensing system. By treating GL as a causality-driven response, the method tests whether an inferred amplification factor—computed without assuming a lens model—is KK-consistent; violations arise if the assumed unlensed waveform or the signal class is incorrect. The authors develop a procedure to dismiss false amplification factors that fail KK consistency within a finite observational band, using quantities like $K( u)$, $S( u)$, and a measurable $r$, and they validate the framework with simulations for point-mass and SIS lenses as well as non-lensed scenarios including eccentric and spinning binaries. In idealized, noiseless bandwidths, the KK method can restrict the GW-template parameter space for true GL signals and dismiss non-GL mimics, though extension to realistic detector noise remains for future work and is essential for practical deployment.

Abstract

Observations of microlensed gravitational waves (GWs) emanated by compact binary coalescences (CBCs) are essential for studying the mass density distribution in the universe, including black holes and dark matter halos. However, no confident detection of microlensed GWs have been reported to date. There are two important challenges in the identification of microlensed GWs. The first is that the source waveform and lens structure models are not known a-priori. The second is that certain classes of unlensed GWs could mimic microlensed GWs, resulting in undesirable false alarms. In this work, we propose to use the Kramers-Kronig relation for gravitational lensing systems. We argue that such systems are essentially linear response systems obeying causality, where KK relation must hold. The power of this method lies in the fact that microlensed GWs, regardless of the lens structure, must obey KK relation, while unlensed GW events are not in general expected to obey it. This, in principle, allows us to identify microlensed GWs while dismissing microlensing mimickers. We provide the first important steps towards a methodology that exploits KK relation, and test its usefulness under idealized conditions.

Figures (14)

• Figure 1: Schematic picture of GL. $\bm{\xi}$ and $\bm{\eta}$ are the position vectors in the lens and the source plane, respectively. $D_L$ is the angular diameter distance between the observer and the lens, $D_{LS}$ is between the lens and the source, and $D_S$ is between the observer and the source.
• Figure 2: The amplification factors in the PML (left panel) and SIS models (right panel). The solid lines are the real parts and the dotted lines are the imaginary parts. In both cases, the lens parameters are $M_{Lz}=300M_{\odot},~y=0.1$.
• Figure 3: Plot of the $|\Delta_{\rm{tr}}(f_{\rm{mid}})/K(f_{\rm{min}})|$ in Eq. (\ref{['estimation']}). The solid line is for PML and the dotted line is for SIS. The parameters are $y=0.1,~f_{\rm{min}}=1\text{ [Hz]},~f_{\rm{max}}=1024\text{ [Hz]}$. In this parameter region, we can confirm that the $|\Delta_{\rm{tr}}(f_{\rm{mid}})/K(f_{\rm{min}})|$ is indeed $\mathcal{O}(1)$ as expected in Eq. (\ref{['estimation']}). The reason we stopped $M_{Lz}$ at $10M_{\odot}$ is that below this value, the amplification factor is in the WO regime in $[f_{\rm{min}}, f_{\rm{max}}]$, so the condition (\ref{['hierarchy']}) is not satisfied and is not our target as explained in the last paragraph of Sec. \ref{['Sec: Dismiss parameters']}. See Appendix for an discussion of why $|\Delta_{\rm{tr}}(f_{\rm{mid}})/K(f_{\rm{min}})|$ increases as $M_{Lz}$ decreases.
• Figure 4: Comparison of the left-hand side (solid line) and right-hand side (dashed line) of Eq. (\ref{['KK in GL KS truncated']}) with frequency range $[1~\text{Hz},~1024~\text{Hz}]$. The left panel is for the true amplification factor in the PML model with $M_{Lz}=300M_{\odot},~y=0.1$. In this case, the KK relation is slightly violated only due to the limited frequency range. The right panel is for the false amplification factor obtained from GWs with the eccentric orbit (see Sec. \ref{['Sec: ECC signal']}). The parameters are $e = 0.01,~\bm{\theta}_0=\{30M_{\odot},~1000~\text{Mpc},~0~\text{s},~0~\text{rad}\},~\bm{\theta}=\{30.005M_{\odot},~1000~\text{Mpc},~0.01~\text{s},~-0.01~\text{rad}\}$. For this false amplification factor, the KK relation is obviously violated, indicating that the violation is not only due to the limited frequency range.
• Figure 5: Schematic illustration of the hierarchy. This is the amplification factor of the SIS model with $M_{Lz}=500M_{\odot},~ y=0.1$. The shaded areas on the left and right are the WO and GO regimes, respectively. For the KK relation to be valid, there must be a sufficient hierarchy between the values of the amplification factor at $f_{\rm{mid}}$ and $f_{\rm{min}}$, as shown in the figure.