Strong Field Gravitational Lensing in Full Wave Optics

Introduction to the show: ident: General Relativity and Quantum Cosmology. The day in general relativity and quantum cosmology.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Strong Field Gravitational Lensing in Full Wave Optics".

Mira: Here's the gist of the paper. Compact binaries evolving in the vicinity of massive black holes may be commonplace in the Universe, and their gravitational waves carry signatures of their environment The paper studies the strong lensing of these signals by the massive companion, a configuration that violates the standard assumptions of the lensing literature Framework and Setup The study investigates the strong gravitational lensing of a stellar-mass binary (SB) in the spacetime of a Kerr black hole (MBH) with mass M and spin parameter a The setup involves an equal mass binary (SB), stationary, on the x axis, with coordinates (rS, π/2, 0), and with its spin S⃗ either pointing in the −xˆ (SB I) or in the zˆ direction (SB II) The inner and outer horizons are located at the roots of ∆, r± = M ± p/M squared - a squared The center of mass (CoM) of the SB is stationary, in the equatorial plane θ = π/2, and at r = rS > 2M, but fully in the strong field region Wave Optics Modeling The full wave optics model is constructed by exploiting Dixon’s formalism

56–59: to describe the SB in the MBH spacetime The gravitational perturbation produced by the SB is solved using the Teukolsky equation in the frequency domain The asymptotic expression for the Weyl scalar is given by Ψ4 ∼ Xl,m,q Z∞lmq −2S almωq (θ)e imϕ e −iωqu r The key quantity in any lensing analysis is the amplification function F(ω; Θ) = hL(ω; Θ)/hU(ω; Θ) Results for Non-Rotating Lenses For a Schwarzschild lens and the SBI configuration, the frequency dependence of the amplification function approaches the same universal curve after the expected caustic rescaling, even for Mω ∼ O(1) This demonstrates that the nullgeodesic arrival-time structure retains predictive power well beyond the strict geometric-optics limit The toy model is able to capture the deviation of the peak from alignment, and it also accurately predicts the direction of the shift Implications for Point Lenses The paper quantifies that a point lens (PL) model can reproduce fringe spacings or maximum amplification near alignment, but not both, and using this model leads to significantly biased lens parameters The corrected PL does not reproduce both in these comparisons The results provide a direct link between black-hole scattering and the traditional language of gravitational-wave lensing The distinction is relevant for the astrophysical systems motivating this work The effects of the strong field geometry, the effects of the spin-2 nature of the source, and future improvements of the RT model will be important Appendix Details The toy model shows that when ∆φS ≠ 0 there will not be maximum constructive interference when t1 − t2 = 0, as one usual expects The caustic crossing happens at the antipodal point when a → 0 The angular size of the caustic is given by the term in the second line of Eq.

Kai: (B1), and for the fiducial system is ∆ϕS ∼ 10−3 (a/M)2 The maximum amplification does not coincide with perfect alignment due to the breaking of axisymmetry Appendix B Details The angular offset of the Kerr caustics ϕc used in Sec.

Mira: III B is given by Eq.

Kai: (B2) The agreement between the Kerr WO and Schwarzschild RT results stems from the fact that we simply do not have the frequency resolution to resolve the small-scale structure of the caustic The true astroidal caustic structure of Kerr is left unresolved and smears into a point-like caustic, reminiscent of the Schwarzchild case Final Summary The results we have presented here have three clear messages about strong field lensing of a stationary SB source near a MBH (i) Even at Mω ∼ 1, the dependence of the amplification factor on the frequency follows a universality relation, predicted by catastrophe theory (ii) A RT model can reproduce the amplification in this region to a very good degree (iii) In the strong field, the quadrupole emission of the SB significantly impacts the angular shape of the diffraction pattern The PL is recovered asymptotically in the weak field, but its image positions and arrival-time differences deviate from those of the Schwarzschild lens as the source approaches the MBH Higher order corrections to the deflection angle can be absorbed into an effective PL mass and substantially improve the fringe spacing They cannot, however, simultaneously reproduce the strong-field magnifications and arrival-time differences Deep in the strong field regime, the prescription ceases to have a meaningful geometrical interpretation altogether The approach to the point-lens limit is slow, and O(1 − 10)% corrections to the effective lensing scale can remain Even relatively small errors in the image arrival times accumulate into appreciable dephasing at high frequency Consequently, applying a standard PL template to these systems can bias the inferred lens parameters and, in less symmetric source configurations, can miss structure associated with the quadrupolar emission and polarization Together, these results provide a direct link between black-hole scattering and the traditional language of gravitational-wave lensing They show both why a raybased diffraction construction can remain useful surprisingly deep into the wave-optics regime and where the standard Newtonian lensing description begins to fail The RT model can describe these results, while a PL model proves inadequate for these sources Acknowledgments We are grateful to Vitor Cardoso, Jose Maria Ezquiaga, Nicolas Aimar, Conor Dyson, Giulia Cusin, and Brian Seymour for their helpful comments and suggestions This work was supported by Research Grants No.

Paper summary: Kai: We talked about how this paper, "Strong Field Gravitational Lensing in Full Wave Optics," looks at compact binaries near massive black holes and how their gravitational waves are lensed. The main idea is that they are challenging the standard assumptions of geometric optics and thin lenses in this strong field environment.

Mira: They use a perturbative framework to calculate the full wave optics amplification factors for face-on and edge-on binaries, including spinning lenses. Their main claim is that in the edge-on configuration, peak amplification doesn't occur at perfect source– lens–observer alignment, even when the lens isn't spinning.

Lev: And they connect this to the anisotropic, coherent quadrupolar emission of the stellar-mass binary breaking the axisymmetry of the system. This is because they restrict their analysis to a plane near the optical axis.

Kai: They also found that despite these complexities, restricting to that equatorial plane near the optical axis, they found a frequency dependence in their amplification factor that stays universal across every configuration studied.

Mira: They checked this by comparing it against a strong-field ray tracing calculation for a face-on binary lensed by a Schwarzschild black hole and got agreement at about ten percent.

Lev: For the edge-on case, they only matched the angular peak shift and fringe spacing to that accuracy. And for Kerr lenses, once frame dragging is accounted for, their face-on amplification factor matches the Schwarzschild one to that same ten percent level.

Kai: But they did flag a big issue with point mass lens models because they said those models fail to reproduce the features of the system, which could really affect how we forecast lensing by massive black holes in strong fields.

Conclusion: Mira: So to wrap up this paper, "Strong Field Gravitational Lensing in Full Wave Optics," it shows a direct link between black-hole scattering and the traditional language of gravitational-wave lensing.

Lev: They demonstrate that even at M omega equal to one the dependence of the amplification factor on frequency follows a universality relation predicted by catastrophe theory <ref:2610.18958#pg1>.

Kai: And they show that ray tracing models can reproduce this amplification in this region to a very good degree, which is something we need to keep in mind.

Mira: But they also showed that in strong field, the quadrupole emission of the stellar-mass binary really impacts the angular shape of the diffraction pattern.

Lev: And they pointed out that while point lens models look like them in weak field, their image positions and arrival-time differences deviate from those of a Schwarzschild lens as we get closer to the massive black hole.

Kai: The paper suggests that higher order corrections to the deflection angle can be absorbed into an effective point mass and improve fringe spacing, but they can't reproduce both the strong-field magnifications and arrival-time differences simultaneously.

Mira: This means that deep in the strong field regime, the prescription stops having a meaningful geometrical interpretation altogether, and applying a standard point lens template to these systems can bias inferred lens parameters.

Lev: So what we have here is that ray-based diffraction construction can remain useful surprisingly deep into the wave optics regime, but where the standard Newtonian lensing description starts to fail.

Kai: It really highlights that these results show both why a ray-based diffraction construction can be useful surprisingly deep into the wave optics regime and where the standard Newtonian lensing description begins to fail.