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

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.

💡 Research Summary

The paper tackles the long‑standing problem of identifying microlensed gravitational‑wave (GW) signals without relying on specific source‑waveform or lens‑model templates. Traditional searches for GW microlensing have used matched‑filter pipelines that require a priori knowledge of both the binary‑coalescence waveform and the lens amplification factor (e.g., point‑mass or singular isothermal sphere). This approach suffers from two major drawbacks: (i) the true source waveform and lens mass distribution are generally unknown, and (ii) certain unlensed waveforms can masquerade as microlensed ones, leading to false‑alarm events.

The authors propose a fundamentally different strategy based on the Kramers‑Kronig (KK) relations, which are a consequence of causality in any linear response system. They argue that the gravitational‑lensing (GL) of GWs can be treated as a linear, causal system where the unlensed waveform ϕ₀(ω) is the input, the lensed waveform ϕL(ω) is the output, and the complex amplification factor F(ω) = ϕL/ϕ₀ acts as the response function. Causality guarantees that F(ω)−1 is a Herglotz function, and therefore its real and imaginary parts obey the Hilbert‑transform‑type KK integral: \