Lensing of gravitational waves efficient wave-optics methods and validation with symmetric lenses Giovanni Tambalo1Miguel Zumalac arregui1Liang Dai2and Mark Ho-Yeuk Cheung3

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Lensing of gravitational waves: eﬃcient wave-optics methods and validation with

symmetric lenses

Giovanni Tambalo,1, ∗Miguel Zumalac´arregui,1, †Liang Dai,2, ‡and Mark Ho-Yeuk Cheung3, §

1Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

Am M¨uhlenberg 1, D-14476 Potsdam-Golm, Germany

2University of California at Berkeley, Berkeley, California 94720, USA

3William H. Miller III Department of Physics and Astronomy,

Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland, 21218, USA

(Dated: September 6, 2023)

Gravitational wave (GW) astronomy oﬀers the potential to probe the wave-optics regime of grav-

itational lensing. Wave optics (WO) eﬀects are relevant at low frequencies, when the wavelength is

comparable to the characteristic lensing time delay multiplied by the speed of light, and are thus

often negligible for electromagnetic signals. Accurate predictions require computing the condition-

ally convergent diﬀraction integral, which involves highly oscillatory integrands and is numerically

diﬃcult. We develop and implement several methods to compute lensing predictions in the WO

regime valid for general gravitational lenses. First, we derive approximations for high and low

frequencies, obtaining explicit expressions for several analytic lens models. Next, we discuss two

numerical methods suitable in the intermediate frequency range: 1) Regularized contour ﬂow yields

accurate answers in a fraction of a second for a broad range of frequencies. 2) Complex deformation

is slower, but requires no knowledge of solutions to the geometric lens equation. Both methods are

independent and complement each other. We verify sub-percent accuracy for several lens models,

which should be suﬃcient for applications to GW astronomy in the near future. Apart from mod-

elling lensed GWs, our method will also be applicable to the study of plasma lensing of radio waves

and tests of gravity.

CONTENTS

I. Introduction 1

II. Wave Optics Regime of Gravitational Lensing 2

III. Analytic Expansions 3

A. Geometric optics & beyond 3

1. Beyond geometric optics 3

2. Contribution from the cusp 4

B. Low-frequency expansion 5

1. Leading corrections for symmetric lenses 6

2. gSIS series expansion 7

IV. Regularized Contour Flow 8

A. Adaptive Sampling in the Time Domain 8

B. Time-domain Regularization and GO

Counterterms 10

V. Complex deformation 11

A. Flow of the integration domain 11

B. Extension to realistic lenses 13

VI. Accuracy and performance 14

A. Comparison to point lens 14

B. Comparison between methods 15

∗giovanni.tambalo@aei.mpg.de

†miguel.zumalacarregui@aei.mpg.de

‡liangdai@berkeley.edu

§hcheung5@jhu.edu

C. Performance 17

VII. Conclusions 17

Acknowledgments 19

A. Derivatives of F(w) via Regularized Contour

Flow 19

References 19

I. INTRODUCTION

Gravitational lensing, the deﬂection of waves by grav-

itational ﬁelds, has become an essential tool for explor-

ing the Universe’s structure and contents. The rich phe-

nomenology of gravitational lensing [1] has enabled many

applications, from inferring cosmological parameters to

testing dark matter models. Progress on these fronts has

relied exclusively on observations across the electromag-

netic spectrum. However, recent advances in GW astron-

omy [2,3] open up a new arena for gravitational lensing.

In time, searches of lensed GWs [4,5] are likely to turn

into conclusive detections and novel applications.

Diﬀerences between gravitational and electromagnetic

radiation from astrophysical sources make GW lens-

ing qualitatively distinct and complementary to electro-

magnetic observations. GWs emit coherently and at

much lower frequencies. GWs detectable by LIGO-Virgo-

Kagra (LVK) have wavelengths more than three orders of

magnitude longer than the lowest frequency radio waves

permitted by the Earth’s ionosphere. This diﬀerence may

arXiv:2210.05658v2 [gr-qc] 5 Sep 2023

allow the observation of WO eﬀects [6,7], which emerge

when the wavelength is comparable to the time delay

produced by the lens multiplied by the speed of light.

WO eﬀects are frequency dependent and their detection

would allow an accurate determination of the lens prop-

erties [8–12]. Moreover, WO lensing of GWs could serve

to identify stellar-scale microlenses [13–17] and test dark

matter scenarios [6,18–24].

Unfortunately, accurately computing lensed waveforms

in the WO regime is challenging. A closed-form analyti-

cal expression exists only for the simplest point-mass lens

[25], and series expansions have been developed for more

general lenses [26]. These solutions have been used widely

to study WO lensing. However, even these simple expres-

sions are costly to evaluate in practice, particularly at

high frequencies. General predictions require condition-

ally convergent integrals of rapidly oscillating functions

over the lens plane. Previous works used direct integra-

tion [6,27], Levin’s algorithm method [28,29], sampling

the Fermat potential over contours [16,30] (related to

our ﬁrst method) or discretely [14,15,17], discrete FFT-

convolution [31], and Picard-Lefschetz theory [32,33] (re-

lated to our second method). While these methods have

been used to study complex lenses (e.g. Refs [14–17]),

they have been validated (e.g. cross-validated with inde-

pendent calculations) only for simple examples.

Here we describe methods to obtain WO predictions

and cross-validate them on several lens models. In Sec-

tion II, we present the diﬀraction integral. Section III

presents expansions valid in the low- and high-frequency

limits before turning to general, numerical algorithms.

In Section IV we solve the Fourier transform of the in-

tegral by adaptively sampling contours of equal time de-

lay. Then, in Section Vwe analytically continue the inte-

gration variable to make the integral manifestly conver-

gent. Finally, in Section VI we validate the accuracy of

both methods and discuss their performance. We will ex-

plore the phenomenology of GW lensing separately [34].

Throughout this paper, we will work in a unit system

with c= 1.

II. WAVE OPTICS REGIME OF

GRAVITATIONAL LENSING

In this Section, we will review the equations govern-

ing gravitational lensing in the WO regime. In order to

focus on the mathematical problem we will not provide

a detailed derivation of the quantities involved. Read-

ers are referred to Refs. [8,34] for details. Our goal is

to evaluate the diﬀraction integral, which we will give in

dimensionless form:

F(w) = w

2πi Zd2xeiwϕ(x,y).(1)

See Ref. [35] for a derivation. The integration is over the

lens plane, with the coordinates rescaled by a dimension-

ful scale ξ0(e.g. a characteristic scale of the lens), so x

1.00

1.25

1.50

1.75

2.00

2.25

2.50

|F(w)|=|˜

hlens/˜

h0|

Method

bGO

O(w)

Perturbative

Geometric

Optics

10−310−210−1100101102103

w≡8πGMLz f

10−3

10−2

10−1

100

|F/FWO −1|

FIG. 1. Ampliﬁcation factor for an SIS lens (y= 0.3). Top

panel: The full WO solution (solid) is compared to diﬀerent

approximations: low wseries expansion (dotted), geometric

optics (dashed) and beyond geometric optics (dash-dotted).

The regions where systematic expansions are good descrip-

tions appear shaded. Bottom panel: Relative deviation

with respect to the WO solution.

is dimensionless. The impact parameter yis rescaled by

η0≡DSξ0/DL, where DS, DLare the angular diameter

distances to the lens and the source, respectively.

Here we introduced the dimensionless frequency

w≡8πGMLzf , (2)

which is given in terms of a redshifted eﬀective lens mass:

MLz ≡ξ2

2Gdeﬀ

.(3)

The factor deﬀ ≡DLDLS

(1+zL)DSalso depends on the angular

diameter distance between the lens and the source DLS .

For a point lens, MLz coincides with the total mass of

the lens (i.e. setting ξ0to be the Einstein radius), but

this is not true for extended lenses.

The integral depends on the Fermat potential :

ϕ(x,y) = 1

2|x−y|2−ψ(x)−ϕm(y).(4)

Here ψis the lensing potential, which depends on the

matter distribution projected on the lens plane and

whose derivative gives the deﬂection angle (Eq. (5) be-

low). We conventionally shift by ϕm(y), the global min-

imum value of the Fermat potential. From here on, we

will suppress ϕm(y) in our formulas and assume that it is

added to make the minimum arrival time equal to zero.

When necessary, we will introduce it back.

We will consider several lens models in this work, sum-

marized in Table I. All of them are spherically symmetric,

leading to an axi-symmetric projected mass and lens-

ing potential ψ(x) = ψ(x) (here and in the following

x≡ |x|). First, we consider the point lens, whose analytic

solution will help us test the accuracy of diﬀerent meth-

ods in Sec. VI. We will additionally consider three ex-

tended lenses: the Singular Isothermal Sphere (SIS), and

two one-parameter extensions. SIS lenses follow from a

matter proﬁle ρ∝1/r2and are often employed to model

lensing by galaxies. Our ﬁrst extension, the generalized-

SIS (gSIS), has an arbitrary slope ρ∝1/r1+k(0 < k < 2)

and can be used to model steeper or shallower lenses [35–

37]. Its central density diverges, but the enclosed mass

up to some radius remains ﬁnite if k < 2. Our second

extension, the Cored Isothermal Sphere (CIS), has a cen-

tral core of physical radius rc=xcξ0[38,39]. Therefore,

the matter density ρ∝1/(r2+r2

c) is ﬁnite at the centre.

Details about these lenses and their phenomenology will

be provided separately [34].

Figure 1shows the ampliﬁcation factor for an SIS lens

with impact parameter y= 0.3. The full WO solution

was obtained by regularized contour ﬂow, discussed in

Sec. IV, matched to an analytic expansion at high fre-

quencies. The remaining curves correspond to the ex-

pansions presented in Sec. III, each with limited range

of validity: Geometric Optics (GO) and its next-order

reﬁnement (bGO) (Sec. III A) are good descriptions at

high frequency, while the series expansion (Sec. III B)

is a good approximation only for w≪1. GO remains

bounded at all frequencies, while bGO and the series ex-

pansion diverge at low/high frequencies, respectively.

III. ANALYTIC EXPANSIONS

We now present analytic expansions valid in the high-

and low-frequency limits, Subsections III A,III B, respec-

tively.

A. Geometric optics & beyond

In the high-frequency limit, following the same argu-

ments leading to the stationary-phase approximation for

path integrals, only the neighbourhoods of extrema of

the Fermat potential (4) contribute to the ampliﬁcation

factor (1). Each extremum is associated with an image

J, located at a position xJin the image plane where the

lens equation

∇xϕ(xJ,y) = xJ−y−α(xJ)=0,(5)

is satisﬁed (here α(xJ)≡∇xψ(xJ) is the deﬂection an-

gle and ∇xis the gradient computed with respect to x).

The geometric optics regime emerges from a quadratic

expansion of the Fermat potential around each image so

that the diﬀraction integral can be performed analyti-

cally.

The GO ampliﬁcation factor (1) receives contributions

from each image J

F(w) = X

J|µJ|1/2eiwϕJe−iπnJ,(6)

where the magniﬁcation

µ−1≡det (ϕ,ij ) = 1−α(x)

x1−dα(x)

dx,(7)

is evaluated on the image position xJ(the second equal-

ity above applies to the speciﬁc case of axially-symmetric

lenses). In the above expressions, ϕJis the Fermat po-

tential of the J-th image, ϕ,ij ≡∂i∂jϕis its Hessian

matrix and α(x)≡ |α(x)|. As we are working in the two-

dimensional lens plane, i, j, ··· ∈ {1,2}, corresponding to

the x1and x2coordinates (the Cartesian components of

x). The Morse Phase [8,35,40] depends on the type of

image as

nJ=









0 if det (ϕ,ij ),tr (ϕ,ij )>0 (minima)

2if det (ϕ,ij )<0 (saddle)

1 if det (ϕ,ij )>0,tr (ϕ,ij )<0 (maxima)

(8)

Minima, saddle points and maxima of the time delay

function are also known as type I, II and III images,

respectively [41].

1. Beyond geometric optics

Beyond GO (bGO) corrections can be obtained as a

series expansion in 1/w. We now review the leading

bGO correction, following [27] and focusing on axially-

symmetric lenses (see also [42]). First of all, we expand

the lensing potential ϕaround each image xJup to quar-

tic order in ˜xi≡xi−xi

ϕ(x,y) = ϕJ+1

2(ϕJ),ij ˜xi˜xj+1

3!(ϕJ),ijk ˜xi˜xj˜xk

4!(ϕJ),ijkl ˜xi˜xj˜xk˜xl+O(˜x5).(9)

For a symmetric lens, the quadratic term in ˜xiis diag-

onal and can be written as (here yis taken to be along

the x1direction so that xi

J=xJδi

(ϕJ),ij ˜xi˜xj= (1 −ψ′′

J)˜x2

1+1−ψ′

xJ˜x2

= 2aJ˜x2

1+ 2bJ˜x2

2,(10)

where primes denote radial derivatives and we deﬁned

aJ≡(1 −ψ′′

J)/2 and bJ≡(1 −ψ′

J/xJ)/2. At this point,

in the diﬀraction integral Eq. (1) we shift and rescale xi

to zi≡√w˜xi. In terms of zi, the quadratic term at the

exponent is windependent. The cubic and quartic terms

of Eq. (9), multiplied by w, scale instead as O(w−1/2) and

Name ρ(r)ψ(x) Parameters

Point Lens δD(r) log(x) -

Singular Isothermal Sphere (SIS) 1

r2x-

Generalized SIS (gSIS) 1

r(k+1)

x(2−k)

(2−k)Slope k

Cored Isothermal Sphere (CIS) 1

r2+r2

√x2

c+x2+xclog 2xc

√x2

c+x2+xcCore size xc

TABLE I. Summary of lens models used in this work. For reference, we have included ρ(r), the density of a spherically

symmetric matter distribution leading to the lensing potential ψin each case.

O(w−1) respectively. For large w, we can then Taylor

expand the exponential and keep terms up to O(w−1).

At order O(w0) we recover the GO result (from the

quadratic part). The term of order O(w−1/2) van-

ishes since it leads to an odd integrand. The ﬁrst

correction comes instead at order O(w−1), where we

have two distinct contributions: one from squaring the

term (ϕJ),ijk ˜xi˜xj˜xkand another from (ϕJ),ijkl ˜xi˜xj˜xk˜xl.

Terms with higher powers of ˜xcontribute at order

O(w−3/2) or higher, and can thus be neglected at suf-

ﬁciently large frequencies.

After performing these two integrals, one is left with

the following simple result

F(w) = X

J|µJ|1/21 + i∆J

weiwϕJ−iπnJ+O(1/w2),

(11)

where the real number ∆Jcharacterizes the bGO correc-

tion, and is given by

∆J≡1

16 "ψ(4)

2a2

12a3

(ψ(3)

J)2+ψ(3)

JxJ

+aJ−bJ

aJbJx2

J#.

(12)

Here ψ(n)≡dn

dxnψ.

Equation (11) shows that the leading-order GO result

is a good approximation provided that ∆J/w ≪1 for

all images.1Note also that non-analytic features in the

Fermat potential (e.g. cusps) produce other O(w−1) con-

tributions without a corresponding GO image [27]. We

will now address the contribution of non-analytic features

in speciﬁc cases.

2. Contribution from the cusp

The leading terms in the GO expansion, Eq. (6), arise

from the stationary points of the Fermat potential and

1Another GO convergence criterion is that w(ϕI−ϕJ)≫1,∀I̸=

J. This can be understood from the contours framework (IV)

as the images being resolvable at ﬁnite frequency. Note that

this criterion is, in general, independent from bGO terms being

negligible, ∆/w ≪1.

capture the high-wcontributions to the ampliﬁcation fac-

tor. Nonetheless, other locations in the lens plane can

induce corrections at subleading order in the ∼1/wnex-

pansion and might be comparable to the bGO terms. In

particular, they can arise from singular points of the lens

equation (cusps in the lensing potential). See [27] for a

similar discussion on cusp contributions to F(w).

In this Subsection we are going to discuss these contri-

butions for the lens models featuring a central cusp (gSIS

and SIS lenses, Table I).2In particular, we focus on the

strong lensing regime, where ycan be taken as a small

number.

Let us consider the gSIS lens in the limit of large w. For

this lens, we distinguish two behaviours depending on the

value of the slope k. For 0 < k < 1 (broad proﬁles) the

lens equation is smooth at the lens’ centre and a central

image forms in the strong-lensing regime. In other terms,

the deﬂection angle αis bounded as the ray approaches

the centre of the lens. In the complementary interval 1 ≤

k < 2 (narrow proﬁles) the lens equation is singular at the

centre and no image forms. In both these cases we isolate

the contribution to F(w) from the centre by truncating

the integration range from x∈(0, Rc), for some radius

Rcsmall enough for the GO images not to be enclosed.

The range x > Rc, at high w, is then dominated by

the GO expansion around the minimum and/or saddle

(depending on y). In the lower integration interval, we

have instead

Fc(w) = w

2πi ZRc

dx x Zπ

dθ eiwϕ(x,y)

=−iweiwϕcZRc

dx xJ0(wyx)eiw(x2

2−ψ(x)) ,(13)

where ϕc≡y2/2−ϕmis the time delay associated to

the lens centre (here we re-introduced the minimum time

delay ϕm) and Jν(z) is the Bessel function of the ﬁrst kind

(obtained after performing the angular integral). The

2The centre is smooth for the CIS lens, so no new contribution

arises compared with bGO. The point lens is singular at the

centre, but the new contribution is highly suppressed in w(see

[27] for a discussion). Thus, the gSIS is the only relevant case

among the lenses we consider.

integrand, in the limit w≫1, peaks around x= 0 once

we rotate the integration line into the complex plane. To

see this, ﬁrst notice that J0(wyx)eiwx2/2≃1 for small

x(we will motivate better why xcan be taken small a-

posteriori ). By writing x e−iwψ(x)=elog x−iwψ(x)≡eΩ,

we can locate the peak as the stationary point xsof Ω:

dxΩ = 1

xs−iwx1/A−1

s= 0 .(14)

Here we deﬁned A≡1/(2 −k), which is a positive quan-

tity. Equation (14) is solved for xs= (iw)−A. Notice

that xsbecomes smaller for larger w, making our approx-

imation adequate in this limit (in particular the Gaussian

part at the exponent can be neglected since x2

s≪ψ(xs)).

Therefore, Fc(w) for w≫1 can be obtained using a

saddle-point approximation around xs. However, we pre-

fer to take a slightly diﬀerent approach that yields very

similar results: we evaluate J0(wyx)eiwx2/2in Eq. (13) at

the peak xs, while performing the exact integration over

for eΩ. Since the integral is highly localized for w≫1,

the calculation can be simpliﬁed by taking Rc→ ∞,

making exponentially small errors. We obtain

Fc(w)≃ −iweiwϕcJ0(wyxs)eiwx2

s/2Z∞

dx x e−iwψ(x)

=−eiwϕc(iwA)1−2AΓ(2A)J0(wyxs)eiwx2

s/2.

(15)

This formula is valid when the limits w≫1 and wyxs≪

1 are satisﬁed (therefore, for small enough y).3The

full F(w) is then given by the sum of Eq. (15) and the

usual GO expansion for the other images. In the fol-

lowing, we will refer to this expansion as resummed GO

(rGO). We can better understand the behaviour of Fc(w)

by ﬁrst looking at the SIS case k= 1. Here, neglect-

ing again the Gaussian and the Bessel function, we have

Fc(w)≃i/weiwϕc. This can be interpreted as an addi-

tional bGO contribution from the cusp x=xc= 0, with

time delay ϕcand with a vanishing GO term (i.e. not

accompanied by an image). More in general, for narrow

(broad) proﬁles, Fc(w) decays faster (slower) than 1/w.

For broad proﬁles, there is a caveat in the previous

derivation at very large w: the Gaussian part can start

contributing signiﬁcantly to the integral, thus leading to

the usual GO expansion for the central image. Therefore,

for 0 < k < 1, Fcin Eq. (15) is a better approximation

than bGO for the central image only in the range 1 ≲

w≲∆c, while for ∆c/w ≪1 bGO performs better (here

∆cis the bGO coeﬃcient of the central image, Eq. (12)).

This issue does not arise for k > 1, since here there is no

central image.

3Contribution from cusps are computed in [27] for SIS and gSIS

lenses. For the SIS, Eq. (15) matches with Eq. (21) of [27] for

small y. For the generic gSIS instead, the reference implicitly as-

sumes large y, so that our formulas cannot be directly compared.

From the discussion above we conclude that WO ef-

fects from the cusp are relevant even when no central

image forms. As we will study in [34], this has interest-

ing implications for parameter estimations with GWs.

B. Low-frequency expansion

We are now interested in understanding the behaviour

of the ampliﬁcation factor in the limit of small w. In this

limit, GO fails and one has to resort to other methods to

obtain good approximations.

For small w,F(w) approaches 1 since the wavelength

becomes much larger than the lens’ characteristic scale,

and the wave is unperturbed by the lens. Here we would

like to motivate that corrections to F(w)∼1 in this

limit correspond to an expansion in powers of the lensing

potential ψ(x). A physical motivation can be given as

follows. If the wavelength is much larger than the typi-

cal scale of the lens (i.e. Einstein radius), then the impact

parameter’s value cannot be precisely resolved. This im-

plies that the impact parameter should be irrelevant (at

least at leading order) in this low-frequency limit. Thus,

we could imagine performing the calculation for F(w)

with y≫1 (i.e. impact parameter much larger than the

scale of the lens, set by ξ0) but still much smaller than

the wavelength ∝1/w. In this case one can expand the

diﬀraction integral in powers of the lensing potential (see

[43] on the conditions for the applicability of this approx-

imation).

We can also see explicitly that this procedure gives a

sensible series expansion in w: higher powers on ψ(x)

lead to subleading terms in w. For simplicity, we show

this for axis-symmetric lenses.

First, we perform a rotation of the integration contour

to make the integrals manifestly convergent. Similarly

to Eq. (13), the ampliﬁcation factor can be written as

follows

F(w) = −iweiwy2/2Z∞

dx xJ0(wyx)eiw(x2/2−ψ(x))

=eiwy2/2Z∞

dz zJ0(eiπ/4√wyz)e−z2/2−iwψ(x(z))) ,

(16)

where in the second line we rescaled the radial vari-

able and rotated the integration contour by 45 degrees

in the complex plane: x=eiπ/4z/√w.4Note that the

Gaussian part dictates the leading behaviour at inﬁnity

of the integrand, since the Bessel function only grows

4Here we implicitly assumed ψ(x) to be analytic in the region 0 ≤

arg x < π/2 of the complex-xplane. For the lenses we consider in

Tab. Ithis is the case and it is possible to perform the 45-degrees

rotation in the complex plane without hitting singularities. For

more general situations Eq. (17) needs to be modiﬁed to include

the contribution of singularities.

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Lensingofgravitationalwaves:efficientwave-opticsmethodsandvalidationwithsymmetriclensesGiovanniTambalo,1,∗MiguelZumalac´arregui,1,†LiangDai,2,‡andMarkHo-YeukCheung3,§1MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute)AmM¨uhlenberg1,D-14476Potsdam-Golm,Germany2UniversityofCaliforniaat...

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Lensing of gravitational waves efficient wave-optics methods and validation with symmetric lenses Giovanni Tambalo1Miguel Zumalac arregui1Liang Dai2and Mark Ho-Yeuk Cheung3

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