Repeated gravitational lensing of gravitational waves in hierarchical black hole triples

Abstract

We consider binary black holes (BBHs) in a hierarchical triple system where a more compact, less-massive binary is emitting detectable gravitational waves (GWs), and the tertiary is a supermassive BH at the center of a nuclear star cluster. As previous works have shown, the orbital motion of the outer binary can generate a detectable relativistic Doppler boost of the GWs emitted by the orbiting inner binary. We show here that for outer-binary orbits with a period of order one year, there can be a non-negligible probability for repeated gravitational lensing of the GWs emitted by the inner binary. Repeating gravitational lensing events could be detected by the LISA observatory as periodic GW amplitude spikes before the BBH enters the LIGO band. Such a detection would confirm the origin of some BBH mergers in nuclear star clusters while non-detection could constrain such formation channels. GW lensing also offers new testing grounds for strong gravity.

Detecting gravitational lensing in hierarchical triples in galactic nuclei with space-borne gravitational-wave observatories

“…Approximating the lens as a point mass (as we typically have a o ∼ 100-1000M 3 ≫ M 3 for systems of interest), we then have a time-dependent Fðt; fÞ as [57] (see also Refs. [46,61]; note that this is applied only when x o > 0 or the source is behind the lens) ηðtÞ is defined as ηðtÞ ¼ βðtÞ=θ Ein ðtÞ, where θ Ein ðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4M 3 D LS ðtÞ=ðD OS D OL Þ p is the angular Einstein radius. Under the SPA, the time t can be further treated as a function of f via Eq.…”

“…Recently, Ref. [46] further considered the strong gravitational lensing of the BBH's GW caused by the SMBH and showed that there is a high geometrical probability (∼10%) for lensing to happen if the BBH is in an outer orbit with a period of less than a year (shorter than the duration of the observation). Reference [46] referred to this as "repeated lensing."…”

“…[46] further considered the strong gravitational lensing of the BBH's GW caused by the SMBH and showed that there is a high geometrical probability (∼10%) for lensing to happen if the BBH is in an outer orbit with a period of less than a year (shorter than the duration of the observation). Reference [46] referred to this as "repeated lensing." Indeed, if the outer period is long (much longer than the observation duration), then in order for the BBH to be lensed it needs to have both the right azimuthal and polar angles such that the angular separation between the source and the lens is sufficiently small to be comparable with the Einstein ring.…”

“…Reference [46] focused on the standard strong lensing, which treats the SMBH as a Newtonian point particle and lensing happens when the source is behind the lens (i.e., the source and the observer are on opposite sides of the lens). Meanwhile, the strong gravity field of the SMBH can further lead to relativistic lensing signatures [47][48][49][50][51].…”

“…Furthermore, the parameter space where repeated lensing happens [46] largely overlaps with the parameter space where the de Sitter (dS) precession is detectable [41]. It is thus critical to incorporate both effects in the waveform modeling.…”