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On the Phase-Magnitude Relation in Gravitational Lensing: Reformulation and Applications of the Kramers-Kronig relation

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The phase of a lensed gravitational-wave signal is fixed by its amplitude, with the low-frequency relation dictated by causality alone and independent of the lens model.

desk verdict The phase-magnitude KK reformulation is a useful and solid piece of lensing phenomenology, but the low-frequency "universal" claim is oversold and the alpha=1 branch of Eq. (3.7) contains a missing 2/pi factor. read the letter →

arxiv 2506.02430 v1 pith:MN5H7LDE submitted 2025-06-03 gr-qc astro-ph.CO

Teruaki Suyama , Shasvath J. Kapadia This is my paper
classification gr-qcastro-ph.CO PACS 98.62.Sb04.30.-w
keywords gravitationallensingwavesKramers-Kronigrelationamplificationfactorphase-magnitudeBlaschkeproductlow-frequencybehaviorwaveoptics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the phase of the gravitational-lensing amplification factor is not a free ingredient of any model: a Kramers-Kronig (KK) relation, which follows solely from the causal nature of lensing, ties the phase to the magnitude. The authors reformulate the standard KK relation between real and imaginary parts into one between magnitude and phase, and show the phase is fixed by the magnitude up to a Blaschke-product term that contributes only for certain lens geometries, such as an NFW (dark-matter halo) profile. Their central result is universal: in the low-frequency regime the leading phase is completely determined by the leading magnitude, with $|F(\omega)|=1+A\omega^\alpha$ implying $\theta(\omega)=-\tan(\pi\alpha/2)\ln|F(\omega)|$ for $0\le\alpha<1$ and $\theta(\omega)=\ln|F(\omega)|\ln\omega$ for $\alpha=1$, independent of any lens model. This matters for gravitational-wave data analysis because phenomenological searches that fit the magnitude of a candidate lensed signal must also respect the phase this relation dictates; the paper supplies examples, including a magnitude form whose KK phase is obtained in closed form.

What carries the argument

The argument is carried by three objects. (1) The logarithmic KK relation: given that $\ln F(\omega)$ is analytic in the upper half-plane (true when $F$ has no zeros there), Cauchy's theorem applied to $G(\omega)=\big(\frac{1}{\omega-\omega_1}-\frac{1}{\omega-\omega_2}\big)\ln F(\omega)$ yields the phase-from-magnitude integral and its reciprocal. (2) The Blaschke product $B(\omega)=\prod_n\frac{\omega-\mu_n}{\mu_n^*-\omega}$, in which the $\mu_n$ are the zeros of $F$ in the upper half-plane; it has unit magnitude on the real axis, so it shifts the phase while leaving the magnitude intact, and it accounts for the unbounded phase winding seen when the first-arrival image is not the brightest. (3) The low-frequency expansion $|F(\omega)|=1+A\omega^\alpha$ with $0<\alpha\le1$ together with the identity $\int_0^\infty\frac{s^\alpha-1}{s^2-1}\,ds=\frac{\pi}{2}\tan(\pi\alpha/2)$, which converts the KK integral into the universal leading-order phase formula. The same machinery yields the closed-form phase (4.5) from the magnitude $\ln|F|=\alpha\omega/(\omega+\beta)\cos(\gamma\omega+\phi)$ via sine and cosine integrals.

What would settle it

One concrete check: compute $F(\omega)$ in full wave optics for an NFW configuration like the one in Fig. 3 and test whether the exact phase minus the KK integral built from the exact magnitude equals the Blaschke phase $2\sum_n\tan^{-1}((\omega-\operatorname{Re}\mu_n)/\operatorname{Im}\mu_n)$ for the upper-half-plane zeros $\mu_n$ of $F$; if no zero set reproduces the difference, the reformulation is incomplete. A second check settles the $\alpha=1$ branch: for the point-mass lens at very small $w$, measure the slope of $\theta$ versus $w\ln w$ on a log-log plot and compare it with the prediction from $\ln|F|\approx A w$; the paper's own Eq. (3.5) gives $2A/\pi$, while Eq. (3.7) as printed implies $A$, so the data would fix the correct prefactor.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a pair of reciprocal relations connecting the magnitude and phase of the amplification factor $F(\omega)$, the ratio of the lensed to the unlensed waveform in the frequency domain. The phase is given by $$\$\theta$(\omega)=-\frac{2\omega}{\pi}\,\mathrm{P}\!\int_0^\infty\frac{\ln|F(\omega')|}{\omega'^2-\$omega^{2}$}\,d\omega' - i\ln B(\omega),$$ where $B(\omega)$ is a Blaschke product built from the zeros of $F$ in the upper half-plane; conversely, $F(0)=1$ forces $\ln|F(\omega)|$ to be uniquely determined by $\theta$ through a companion relation. Because the Blaschke factor has unit modulus on the real axis, it changes the phase without changing the magnitude, so the magnitude alone does not always fix the phase: the ambiguity is absent for the point-mass lens, whose $F$ has no upper-half-plane zeros, but present for an NFW profile, whose geometric-optics trajectory winds around the origin. In the low-frequency regime the Blaschke contribution is only of order $O(\omega)$, so the leading phase is set entirely by the magnitude: with $|F|=1+A\omega^\alpha$ ($0<\alpha\le1$), the leading term is $-\tan(\pi\alpha/2)\ln|F|$ for $\alpha<1$ and $\ln|F|\,\ln\omega$ for $\alpha=1$. This reproduces known results derived for specific lens profiles as a model-independent statement, and the paper demonstrates the construction of phases from two analytic magnitude forms, one yielding a closed form in terms of sine and cosine integrals that tracks the point-mass phase.

Load-bearing premise

The derivation assumes the amplification factor is analytic and bounded in the upper half of the complex frequency plane, a causal-lensing property taken from earlier work rather than re-derived here, and it assumes the low-frequency magnitude behaves as $1+A\omega^\alpha$ with $0<\alpha\le1$; if a real lens produced branch cuts, additional singularities, unbounded growth, or a different low-frequency scaling, the phase-magnitude relations would fail as stated.

Editorial extensions

If this is right

  • For any phenomenological magnitude of the form $|F(\omega)|=1+A\omega^\alpha$ with $0<\alpha\le1$, the leading low-frequency phase is no longer free: it must follow Eq. (3.7), on pain of violating the causality encoded in the KK relation.
  • Searches for lensed gravitational waves that use phenomenological or agnostic templates can predict the low-frequency phase from the measured magnitude alone, turning the KK relation into a causality-based consistency test that requires no lens model.
  • The phase is fixed by the magnitude only up to a Blaschke product: for configurations such as certain NFW alignments where the first-arrival image is not the brightest, $|F|$ alone cannot determine $\theta$, and the two may differ by a discrete phase shift; magnitude-only fits must therefore carry that caveat.
  • For the analytic magnitude $\ln|F(\omega)|=\alpha\omega/(\omega+\beta)\cos(\gamma\omega+\phi)$, the KK integral is evaluated in closed form, giving the phase through sine and cosine integrals; this yields phase-consistent phenomenological templates, demonstrated for a point-mass-like shape.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension the authors leave open: because the Blaschke term is real for NFW-like lenses, two lens models could in principle share the same $|F(\omega)|$ over all frequencies yet differ in phase; a matched-filter search sensitive to phase would then break a degeneracy invisible to magnitude-only analyses.
  • The $\alpha=1$ branch of Eq. (3.7) deserves a direct numerical check: substituting $\ln|F|\approx A\omega$ into the paper's own integral evaluation (3.5) gives a leading phase $(2A/\pi)\omega\ln\omega$, a factor $2/\pi$ away from the printed relation $\theta=\ln|F|\,\ln\omega$; fitting the exact point-mass phase at tiny $w$ would fix the prefactor.
  • The same phase-from-magnitude machinery should transfer to any causal wave-propagation problem whose response function is analytic and bounded in the upper half-plane with a $1+A\omega^\alpha$ low-frequency form, so the low-frequency locking is a general causality feature, not a lensing-specific accident.
  • A practical template strategy suggested by Eq. (4.5): fit the analytic magnitude form (4.3) to any numerically computed wave-optics template (not just the point-mass lens) and reuse the closed-form phase, giving fast, phase-consistent phenomenological templates whose KK consistency is exact by construction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper reformulates the Kramers-Kronig (KK) relation for the gravitational-lensing amplification factor F(ω) into a relation between |F(ω)| and its phase θ(ω), including the possible contribution of a Blaschke product when F has zeros in the upper half-plane. It then uses this reformulation to claim that, in the low-frequency regime, the leading behavior of θ(ω) is completely determined by the leading behavior of |F(ω)|, provided |F(ω)|=1+Aω^α with 0<α≤1. The paper also applies the magnitude-phase KK relation to construct phases for two phenomenological magnitude models, one of which yields an analytic closed-form phase. The central mathematical framework is standard complex analysis applied to the previously established analyticity and boundedness of F(ω) in the upper half-plane, and the point-mass numerical checks are consistent with the relation when the Blaschke product is absent.

Significance. If the low-frequency universality claim were established in the stated generality, it would be a useful model-independent constraint on lensed gravitational-wave signals: the phase could not be assigned freely in phenomenological templates, and the leading low-frequency phase would be fixed by the magnitude coefficient and exponent. The paper also makes a genuinely useful contribution by making the Blaschke-product ambiguity explicit for lensing, by checking the reformulated KK relation numerically for the point-mass lens, and by providing an analytic example whose phase is computed in closed form. However, the advertised universality is currently overstated: the derivation assumes 0<α≤1 without proving this for all lens models, and the α=1 branch of Eq. (3.7) is internally inconsistent with Eq. (3.5). These issues affect the central claim and require revision.

major comments (2)
  1. [Sec. 3, Eq. (3.7)] The α=1 branch of Eq. (3.7) is missing the factor 2/π. Equation (3.5) gives the leading phase as (2A/π)ω lnω, whereas for |F|=1+Aω one has ln|F|≈Aω, so the printed relation θ≈ln|F|lnω yields Aω lnω rather than (2A/π)ω lnω. This is not a harmless notational choice: for the point-mass lens in Eq. (3.8), A=π/2 and the actual leading phase is w lnw, so the printed formula would overstate the coefficient by π/2. The corrected statement is θ≈(2/π)ln|F|lnω, and the sentence claiming verification of Eq. (3.7) against the point-mass result should be revised accordingly.
  2. [Sec. 3, Eq. (3.1)] The claimed universality of the low-frequency phase-magnitude relation rests on the restriction 0<α≤1 in Eq. (3.1), but this restriction is neither derived from analyticity and boundedness in the upper half-plane nor proved for general lens profiles. The examples cited (point mass, SIS, generalized SIS) cover only a subclass. For example, F(ω)=(1−icω)^{-1} satisfies analyticity, boundedness, the reality condition, and F(0)=1, yet has |F|−1=O(ω²) and θ=O(ω), so no relation of the form of Eq. (3.7) holds. Cored lens models with finite ∫d²x ψ(x) may similarly give α=2. Either prove that all physical lensing amplification factors satisfy 0<α≤1, or explicitly narrow the universality claim in the abstract, Eq. (3.7), and the conclusions.
minor comments (5)
  1. [Sec. 3, Eq. (3.7)] The first branch of Eq. (3.7) is written as 0≤α≤1, which overlaps with the second branch at α=1; it should read 0≤α<1.
  2. [Sec. 2.1] There are typos in the text: 'comlex' should be 'complex', 'presense' should be 'presence', and in Sec. 3 'singular isothermal shere' should be 'singular isothermal sphere'.
  3. [Sec. 2.1, after Eq. (2.12)] The identity for the Blaschke-factor integral implicitly assumes the zeros μ_n lie strictly in the upper half-plane (Im μ_n>0). If zeros on the real axis were allowed, the factors would not be analytic on the integration contour and the principal-value treatment would need to be stated more carefully.
  4. [Sec. 4, Eq. (4.3)] The phenomenological form (4.3) is not even in ω, whereas a physical amplification factor satisfies |F(−ω)|=|F(ω)|. The KK phase formula (4.2) uses only positive frequencies, so the phase is constructed for that domain; please state the even extension or clarify explicitly that only positive frequencies are being modeled.
  5. [Fig. 3] The axis notation 'x/(y) axis' is unclear; it should state that the horizontal and vertical axes are Re F(ω) and Im F(ω), respectively.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the phase-magnitude relation is a genuine mathematical consequence of analyticity with the stated low-frequency ansatz.

full rationale

The paper's derivation chain is not circular. Equation (2.20) is a standard logarithmic reformulation of the Kramers-Kronig relation: given analyticity and boundedness of F in the upper half-plane, one relates ln|F| and arg F, with an explicit Blaschke-product term for possible zeros. This is mathematics, not a definition of the phase. Equation (3.7) follows by substituting the stated low-frequency expansion (3.1), |F|=1+Aω^α, into the KK integral; the later comparison with the literature results (3.8) is a consistency check, not an input used to fix parameters. The Section 4 examples fit a magnitude form and then compute the phase from the KK relation; they are explicitly presented as constructions or demonstrations, not as predictions from fitted phase information. The reliance on prior work [8] for analyticity and boundedness is a self-citation by one of the authors, but [8] is a parameter-free published result whose assumptions do not include the low-frequency phase claim, so it is independent evidence rather than circular support. The restriction 0<α≤1 in Eq. (3.1) and the apparent missing factor 2/π in the α=1 branch of Eq. (3.7) relative to Eq. (3.5) are correctness or limitation issues, not circularity. No fitted parameter is renamed as a prediction, and no result is assumed into existence by its own conclusion.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

Central results rest on the analyticity and boundedness of F (from [8]), the F(0)=1 normalization, the geometric-optics high-frequency form, and the assumed low-frequency power-law expansion. The examples introduce free fitting parameters but no new physical entities. The alpha=1 branch of Eq. (3.7) contains a coefficient error, noted separately in red flags.

free parameters (4)
  • α (low-frequency exponent) = not fitted; lens-model dependent (1 for point mass, 1/2 for SIS)
    Introduced in Eq. (3.1) as the exponent of the leading correction to |F|; the universal low-frequency formula depends on α.
  • A (low-frequency amplitude) = not fitted; lens-model dependent
    Constant in Eq. (3.1); cancels in the α<1 relation but appears in the α=1 branch.
  • (α, β, γ, ϕ) of Eq. (4.3) = (0.4, 0.1, 2.1, -1.6)
    Hand chosen to fit the point-mass ln|F| in Fig. 8; these are fitting parameters in the worked example.
  • (b, k, φ) of Eq. (4.1), with a fixed by F(0)=1 = (b, k, φ) = (0.5, 0.1, π); a = 1.5
    Fitting parameters of the phenomenological magnitude used in Fig. 7; the physical normalization F(0)=1 fixes one parameter.
assumptions (6)
  • domain assumption F(ω) is analytic and bounded in the upper-half complex frequency plane, due to the causal nature of gravitational lensing.
    Invoked in Sec. 2.1 and Sec. 2.2 as the premise for all KK relations; established in the authors' prior work [8], not re-derived here.
  • domain assumption The amplification factor is normalized so that F(0) = 1.
    Used to derive Eq. (2.21) and implicitly in the low-frequency expansion; this is the standard definition of the lensed-to-unlensed ratio at zero frequency.
  • domain assumption The geometric-optics form Eq. (2.13) is valid at high frequency and may be used over the whole integration range in Eq. (2.14), with an error of order O(1/ω).
    Invoked after Eq. (2.14) to argue that the Blaschke product can contribute when the first-arrival image is not the brightest; the approximation is not proven for arbitrary lens profiles.
  • domain assumption The low-frequency expansion |F| = 1 + Aω^α + ... with 0 < α ≤ 1 is valid, and for α=1 the integral must be cut off at a scale Λ still in the low-frequency regime.
    Used in Sec. 3 to derive the universal relation; not every lens model necessarily exhibits a single power-law leading correction.
  • standard math Cauchy's integral theorem applies to G(ω) = [1/(ω-ω_1) - 1/(ω-ω_2)] ln A(ω), and the Blaschke product decomposition is valid for bounded analytic functions.
    Foundation of Eqs. (2.4) through (2.12); this is background mathematics from Ref. [10].
  • domain assumption For the examples in Sec. 4, the Blaschke product is absent (B=1).
    Verified numerically for the point-mass lens in Fig. 4, but assumed for the phenomenological magnitude forms in Sec. 4.

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Cite this review

Pith. "Pith review of On the Phase-Magnitude Relation in Gravitational Lensing: Reformulation and Applications of the Kramers-Kronig relation." pith.science (2026). https://pith.science/paper/MN5H7LDE

@misc{pith2026250602430,
  author       = {Pith},
  title        = {Pith review of: On the Phase-Magnitude Relation in Gravitational Lensing: Reformulation and Applications of the Kramers-Kronig relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MN5H7LDE}},
  note         = {Machine review of arXiv:2506.02430}
}
read the original abstract

It is known that the amplification factor, defined as the ratio of the lensed to the unlensed waveform in the frequency domain, satisfies the Kramers-Kronig (KK) relation, which connects the real and imaginary parts of the amplification factor for any lensing signal. In this work, we reformulate the KK relation in terms of the magnitude and phase of the amplification factor. Unlike the original formulation, the phase cannot be uniquely determined from the magnitude alone due to the possible presence of a Blaschke product. While this ambiguity does not arise in the case of a point-mass lens, it can appear in more complex lens models, such as those with an NFW lens profile. As an application of our formulation, we demonstrate that the leading-order behavior of the phase in the low-frequency regime is completely determined by the leading-order behavior of the magnitude in the same regime. This reproduces known results from the literature, derived via low-frequency expansions for specific lens models. Importantly, our result does not rely on any particular lens model, highlighting a universal feature that the low-frequency behavior of the amplification factor is tightly constrained by the KK relation. As a further application, we present two examples in which the phase is constructed from a given analytic form of the magnitude using the newly derived KK relation. In particular, the second example allows for an analytic evaluation of the KK integral, yielding an explicit expression for the phase. This study offers a potentially powerful method for applying the KK relation in model-agnostic searches for lensing signals.

Figures

Figures reproduced from arXiv: 2506.02430 by the authors.

Figure 1
Figure 1. Contour Γ of integral used to derive the KK relation for magnitude and phase. as A(ω) does. To write down its explicit form, let us define a new function G(ω) by G(ω) =  1 ω − ω1 − 1 ω − ω2  ln A(ω), (2.3) where ω1, ω2 ∈ R. This function is analytic and decays as O(ω −2 ) for |ω| → ∞ in I+. Then, from Cauchy’s theorem, the integral of G(ω) along the contour Γ shown in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Schematic picture showing the configuration of the GW source, lens, and the observer. is the light travel time and ψ(x) is the lens potential. The last term ϕm(y) has been introduced so that minx td(x, y) = 0. For the NFW profile, the lens potential is given by [15]: ψ(x) =    κs 2 h ln x 2 2 − (arctanh√ 1 − x 2 ) 2 i x ≤ 1 κs 2 h ln x 2 2 + (arctan√ x 2 − 1)2 i x > 1 (2.18) where κs ≡ 16πGρsrs DLDLS DS and: w … view at source ↗
Figure 3
Figure 3. Trajectory of F(ω) as ω is varied for the case where the lens profile is given by the NFW profile. x/(y) axis is ReF(ω)/(ImF(ω)) and the range of ω in terms of w is 2 × 103 ≤ w ≤ 1.2 × 104 . from the Blaschke product: θ(ω) = − 2ω π − Z ∞ 0 ln |F(ω ′ )| ω′2 − ω2 dω′ − i X n ln  ω − µn µ∗ n − ω  . (2.20) This means that measurement of the magnitude of F(ω) alone does not fix the phase of F(ω) and additional assumpti… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Plot of |F(ω)| in the upper-half complex ω-plane for the point-mass lens with y = 1. point-mass lens as an example and numerically compare the left-hand side and the right￾hand side of the relation (2.20). There exits an analytic form of the amplification factor for th…
Figure 5
Figure 5. Figure 5: The left panel shows a plot of the left-hand side and the right-hand side of the KK relation (2.20) as a function of w = 4GMω (y is fixed to y = 1). The right panel shows the relative error between the two sides. 0 1 2 3 4 5 -0.6 -0.4 -0.2 0.0 0.2 y Point mass lens (w=…
Figure 6
Figure 6. Figure 6: The left panel shows a plot of the left-hand side and the right-hand side of the KK relation (2.20) as a function of y (w is fixed to w = 1). The right panel shows the relative error between the two sides. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Left panel: plot of |F(ω)| given by Eq. (4.1) as a function of ω/ω0 for a parameter choice (b, k, φ) = (0.5, 0.1, π). Right panel: plot of the phase θ(ω) computed by the KK relation (4.2) where the magnitude |F(ω)| is given by the left panel. the case of weak lensing f…
Figure 8
Figure 8. Figure 8: The left panel shows a comparison between ln |F(w)| for the point-mass lens with y = 1 (Exact) and Eq. (4.3) with (α, β, γ, ϕ) = (0.4, 0.1, 2.1, −1.6) (Approximate). The right panel shows a comparison between arg F(ω) for the point mass and θ(ω) given by Eq. (4.5) for …

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Works this paper leans on

17 extracted references · 5 canonical work pages

  1. [8]

    Abbott et al., GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run , Phys

    KAGRA, VIRGO, LIGO Scientific Collaboration, R. Abbott et al., GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run , Phys. Rev. X 13 (2023), no. 4 041039, [ http://arxiv.org/abs/2111.03606 arXiv:2111.03606 ]

    arXiv 2023

  2. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

  3. [2]

    Aasi et al., Advanced LIGO , Class

    LIGO Scientific Collaboration, J. Aasi et al., Advanced LIGO , Class. Quant. Grav. 32 (2015) 074001, [ http://arxiv.org/abs/1411.4547 arXiv:1411.4547 ]

    arXiv 2015

  4. [3]

    Acernese et al., Advanced Virgo: a second-generation interferometric gravitational wave detector , Class

    Virgo Collaboration, F. Acernese et al., Advanced Virgo: a second-generation interferometric gravitational wave detector , Class. Quant. Grav. 32 (2015), no. 2 024001, [ http://arxiv.org/abs/1408.3978 arXiv:1408.3978 ]

    arXiv 2015

  5. [4]

    Akutsu et al., Overview of KAGRA: Detector design and construction history , PTEP 2021 (2021), no

    KAGRA Collaboration, T. Akutsu et al., Overview of KAGRA: Detector design and construction history , PTEP 2021 (2021), no. 5 05A101, [ http://arxiv.org/abs/2005.05574 arXiv:2005.05574 ]

    arXiv 2021

  6. [5]

    LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs , http://arxiv.org/abs/1811.12907 arXiv:1811.12907

    arXiv

  7. [6]

    Abbott et al., GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run , Phys

    LIGO Scientific, Virgo Collaboration, R. Abbott et al., GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run , Phys. Rev. X 11 (2021) 021053, [ http://arxiv.org/abs/2010.14527 arXiv:2010.14527 ]

    arXiv 2021

  8. [7]

    Abbott et al., GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run , Phys

    LIGO Scientific, VIRGO Collaboration, R. Abbott et al., GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run , Phys. Rev. D 109 (2024), no. 2 022001, [ http://arxiv.org/abs/2108.01045 arXiv:2108.01045 ]

    arXiv 2024

Show all 17 references
  1. [9]

    Tanaka and T

    S. Tanaka and T. Suyama, Kramers-Kronig relation in gravitational lensing , Phys. Rev. D 108 (2023), no. 4 044015, [ http://arxiv.org/abs/2303.05650 arXiv:2303.05650 ]

    2023 arXiv

  2. [10]

    Tanaka, G

    S. Tanaka, G. Prabhu, S. J. Kapadia, and T. Suyama, Towards model-independent identification of lensed gravitational waves using Kramers-Kronig relation , http://arxiv.org/abs/2504.21320 arXiv:2504.21320

  3. [11]

    Stern, Elementary theory of the optical properties of solids , vol

    F. Stern, Elementary theory of the optical properties of solids , vol. 15 of Solid State Physics , pp. 299--408. Academic Press, 1963

    1963

  4. [12]

    H. G. Choi, C. Park, and S. Jung, Small-scale shear: Peeling off diffuse subhalos with gravitational waves , Phys. Rev. D 104 (2021), no. 6 063001, [ http://arxiv.org/abs/2103.08618 arXiv:2103.08618 ]

    2021 arXiv

  5. [13]

    Tambalo, M

    G. Tambalo, M. Zumalac\'arregui, L. Dai, and M. H.-Y. Cheung, Lensing of gravitational waves: Efficient wave-optics methods and validation with symmetric lenses , Phys. Rev. D 108 (2023), no. 4 043527, [ http://arxiv.org/abs/2210.05658 arXiv:2210.05658 ]

    2023 arXiv

  6. [14]

    Chakraborty and S

    A. Chakraborty and S. Mukherjee, -GLANCE: A Novel Technique to Detect Chromatically and Achromatically Lensed Gravitational Wave Signals , http://arxiv.org/abs/2410.06995 arXiv:2410.06995

    arXiv

  7. [15]

    Suyama, On arrival time difference between lensed gravitational waves and light , Astrophys

    T. Suyama, On arrival time difference between lensed gravitational waves and light , Astrophys. J. 896 (2020), no. 1 46, [ http://arxiv.org/abs/2003.11748 arXiv:2003.11748 ]

    2020 arXiv

  8. [16]

    Takahashi, Wave effects in the gravitational lensing of gravitational waves from chirping binaries , 2004

    R. Takahashi, Wave effects in the gravitational lensing of gravitational waves from chirping binaries , 2004. http://cosmo.phys.hirosaki-u.ac.jp/takahasi/dt.pdf

    2004

  9. [17]

    Schneider, J

    P. Schneider, J. Ehlers, and E. Falco, Gravitational Lenses . Astronomy and Astrophysics Library. Springer New York, 2012

    2012

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