Detectability of strongly lensed gravitational waves using model-independent image parameters_2

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Detectability of strongly lensed gravitational waves using model-independent image
parameters
Saif Ali ,
1, ∗
Evangelos Stoikos ,
1, †
Evan Meade ,
1
Michael Kesden ,
1, ‡
and Lindsay King
1, §
1
Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
(Dated: October 6, 2022)
Strong gravitational lensing of gravitational waves (GWs) occurs when the GWs from a compact
binary system travel near a massive object. The mismatch between a lensed signal and unlensed
templates determines whether lensing can be identified in a particular GW event. For axisymmetric
lens models, the lensed signal is traditionally calculated in terms of model-dependent lens parameters
such as the lens mass
ML
and source position
y
. We propose that it is useful to parameterize this
signal instead in terms of model-independent image parameters: the flux ratio
I
and time delay ∆
td
between images. The functional dependence of the lensed signal on these image parameters is far
simpler, facilitating data analysis for events with modest signal-to-noise ratios. In the geometrical-
optics approximation, constraints on
I
and ∆
td
can be inverted to constrain
ML
and
y
for any lens
model including the point mass (PM) and singular isothermal sphere (SIS) that we consider. We use
our model-independent image parameters to determine the detectability of gravitational lensing in
GW signals and find that for GW events with signal-to-noise ratios
ρ
and total mass
M
, lensing
should in principle be identifiable for flux ratios I&2ρ−2and time delays ∆td&M−1.
I. INTRODUCTION
The first direct detection of gravitational waves (GWs)
from merging compact objects was reported by the LIGO
and Virgo collaborations in 2016 [
1
]. To date, Advanced
LIGO [
2
] and Advanced Virgo [
3
] have reported about 90
events, most of which are mergers between stellar-mass
black holes, during their first three observing runs [
4
].
Kamioka Gravitational Wave Detector (KAGRA) [
5
–
7
]
has joined the preexisting ground-based GW detectors to
form the Advanced LIGO-Virgo-Kagra (LVK) network.
The increased sensitivity of detectors such as LVK has
allowed us to detect an increasing number of GW events
and to perform various general relativistic and cosmolog-
ical tests [
8
,
9
]. With the increasing sensitivity of the
current ground-based detector network and future detec-
tors such as the Cosmic Explorer (CE) [
10
], the Einstein
Telescope (ET) [
11
], the Deci-Hertz Interferometer Gravi-
tational Wave Observatory (DECIGO) [
12
], and the Laser
Interferometer Space Antenna (LISA) [
13
], the number
of observed GW events will increase dramatically, as will
the probability of observing new propagation effects such
as gravitational lensing that have yet to be detected [
14
].
When GWs travelling through the Universe encounter
a massive object, such as a compact object, galaxy or
galaxy cluster, that can act as a lens, deflection of these
GWs, i.e. gravitational lensing, will occur [
15
–
21
]. Strong
lensing of GWs will arise when a lens is very close to the
line of sight. This will result in the GWs splitting into
different lensed images, each with its own magnification
and phase [
19
,
22
]. There will also be an associated time
∗sxa180025@utdallas.edu
†Evangelos.Stoikos@utdallas.edu
‡kesden@utdallas.edu
§lindsay.king@utdallas.edu
delay between the lensed images which could range from
seconds to years depending on the mass of the lens and
geometry of the lens system [23,24].
GW lensing, if detected, could facilitate several exciting
scientific studies. It could be used to extract information
about the existence of intermediate-mass (mass ranging
from
∼
10
2
-10
5M
) [
25
] or primordial black holes [
26
,
27
]
and test general relativity [
28
–
30
], including through con-
straints from GW polarization content [
31
]. In addition,
if a lensed electromagnetic (EM) counterpart of the lensed
GW event is observed, it could help to locate the host
galaxy at sub-arcsecond precision [
32
]. Combining the
information from the two messengers, i.e. GW and EM
lensing, could enable high-precision cosmography [
33
–
38
].
There are two major differences between the gravita-
tional lensing of EM waves and GWs from the point of
view of wave-optics effects. The first difference is in the
applicability of the geometrical-optics approximation. In
the case of EM waves, this approximation, typically valid
when the wavelength
λ
of the waves is much smaller than
the Schwarzchild radius
Rs
of the lens, applies to the
vast majority of observations. This is not always the
case for GWs, since ground-based detectors such as the
LVK network observe at frequencies (10 —10
4
)Hz, lower
than even the lowest-frequency radio telescopes. These
GWs have wavelengths longer than the Schwarzschild
radii of lenses with masses
ML.
10
4M
, leading to non-
negligible wave-optics effects. The second difference is
that the GWs emitted by compact binaries, unlike most
EM sources, are coherent, causing interference between
lensed images when the signals overlap at the observer.
In this paper, we focus on strong gravitational lensing
by stellar-mass objects and GW sources consistent with
those seen by the LVK network. However, our treatment is
also applicable to more massive lenses, and to sources such
as supermassive binary black holes that will be detectable
by LISA. In the frequency domain, the modulation of
GWs due to gravitational lensing is characterized by a
arXiv:2210.01873v1 [gr-qc] 4 Oct 2022
2
multiplicative factor known as the amplification factor.
Typically, this factor is parameterized in terms of model-
dependent lens parameters such as the source position
y
and lens mass
ML
[
17
,
19
]. In the limit where the
geometrical-optics approximation is valid, for a particular
axisymmetric lens model, analytical equations relate these
lens parameters to a set of model-independent image pa-
rameters, the flux ratio
I
and time delay ∆
td
between the
images. Motivated by current GW search pipelines which
use unlensed GW templates, we explore the detectability
of lensing signatures using a match-filtering analysis be-
tween lensed GW source and unlensed GW templates. For
the axisymmetric lens models, we explore the mismatch
between the lensed and unlensed GW waveforms in both
the lens and image parameter spaces.
This paper is organized as follows. In Sec. II, we be-
gin with a pedagogical outline of gravitational lensing
of GWs, discussing the time delay and amplification fac-
tor due to the lens. We then present the prescription
used to generate the GWs in the inspiral phase of binary
compact objects using the post-Newtonian approximation
following [
39
]. In Sec. III, we present a detailed analysis
of the point mass (PM) and singular isothermal sphere
(SIS) axisymmetric lens-mass profiles and introduce the
model-independent image parameters. In Sec. IV, we
perform a match-filtering analysis in which we calculate
the mismatch between lensed and unlensed GWs. Appen-
dices Aand Binvestigate the mismatch between lensed
GW source and unlensed templates. Throughout the
paper, we assume c=G= 1.
II. BASIC FORMALISM
In this section, we briefly review the basic theory of the
gravitational lensing of GWs.
A. Gravitational lensing
In the strong gravitational-lensing regime, we observe
multiple images (or a single very distorted image) of
a distant background source due to the presence of an
intervening massive astrophysical object known as a lens.
Lensing occurs when the GWs from a compact binary
system travel near a lens as shown for the general lensing
geometry in Fig. 1[
24
]. The extents of the lens and
the source are taken to be much less than the observer-
lens and lens-source distances, in which case they can be
localized to the lens and source planes. An optic axis
connects the observer and the center of the lens. The
lens and source planes are at angular-diameter distances
DLand DSrespectively. The angular-diameter distance
between the lens and source planes is
DLS
. A GW source
is located on the source plane at displacement
η
with
respect to the optic axis. After being emitted by the
source, the GWs travel to the lens plane, with an impact
parameter
ξ
, and are deflected through an angle
ˆ
α
by
DS
DLS
DL
observer
source plane
lens plane
Figure 1. A typical gravitational lens system consisting of a
compact binary system in the source plane, a lens in the lens
plane, and an observer.
DL
,
DLS
and
DS
are the angular-
diameter distances from observer to lens, lens to source, and
observer to source respectively. The vector
ξ
is the impact
parameter in the lens plane, and the vector
η
is the location
of the source with respect to the optic axis in the source plane.
ˆ
αis the deflection angle measured on the lens plane.
the gravitational potential of the lens.
x≡ξ/ξ0
and
y≡
η/
(
ξ0DS/DL
)are dimensionless vectors on the lens and
source plane respectively, where
ξ0
is a model-dependent
characteristic length scale on the lens plane called the
Einstein radius. GWs that reach the observer satisfy the
lens equation
y=x−α(x),(1)
where
α(x) = DLDLS
ξ0DS
ˆα(ξ0x) = ∇xψ(x),(2)
is the scaled deflection angle at the observer. The lensing
potential
ψ
(
x
)is given by the two-dimensional Poisson
equation
∇2
xψ(x) = 2Σ(x)
Σcr
,(3)
where Σis the surface mass density of the lens and Σ
cr ≡
DS/
4
πDLDLS
is the critical surface mass density. For
the formation of multiple images, Σ
/
Σ
cr >
1is a sufficient,
but not necessary condition [40].
Gravitational lensing causes a time delay between the
lensed images at the observer. The arrival time has two
components, one arising from the geometry of the path
traveled, and the other due to the gravitational potential
of the lens known as the Shapiro time delay. The time
3
delay at the observer due to a lens at redshift zLis
td(x,y) = DSξ2
0(1 + zL)
DLDLS 1
2|x−y|2−ψ(x) + φm(y),
(4)
where
φm
(
y
)is chosen such that the minimum value of
the time delay is 0.
The lensing amplification factor
F
(
f
) =
˜
hL
(
f
)
/˜
h
(
f
)re-
lates the lensed waveform
˜
hL
(
f
)to the unlensed waveform
˜
h
(
f
)for GWs of frequency
f
. It is given by Kirchhoff’s
diffraction integral [19,24]
F(f) = DSξ2
0(1 + zL)
DLDLS
f
iZd2xexp[2πiftd(x,y)] .(5)
This integral over the lens plane accounts for all the
trajectories in which the wave can propagate; it is unity
in the absence of a lens.
1. Geometrical-optics approximation
In the geometrical-optics approximation, generally valid
for GW frequencies
f
and lens masses
ML
for which
f
M−1
L
[
41
], discrete images form at the stationary points
xj
of the time-delay function at which
∇xtd
(
x,y
) = 0.
Only these points contribute to the lensing amplification
factor
F(f) = X
j|µj|1/2exp (2πiftd(xj,y)−iπnj),(6)
where
µj
= 1
/det
(
∂y/∂xj
)is the magnification of the
jth
image and the Morse index
nj
has values of 0, 1/2, or 1
depending on whether
xj
is a minimum, saddle point, or
maximum respectively of the time-delay surface,
td
(
x,y
).
B. Gravitational waveform
We restrict our analysis to the inspiral phase of the
GW evolution from binary black hole (BBH) mergers and
use the post-Newtonian (PN) approximation to model
our unlensed waveform [39]
˜
h(f) = (A
DM5/6f−7/6eiΨ(f),0< f < fcut
0, fcut < f , (7)
where
D
is the the luminosity distance to the source, Ψ(
f
)
is the GW phase, and the GW amplitude
A
is a function
of sky localization and source geometry of order unity as
discussed in [
42
]. For a BBH system with masses
m1
and
m2
,
M
=
m1
+
m2
is the total mass,
η
=
m1m2/M2
is the
symmetric mass ratio,
Mz
= (1 +
z
)
M
is the redshifted
total mass, and
M
=
η3/5Mz
is the redshifted chirp mass.
The cutoff frequency
fcut
= 1
/
(6
3/2πMz
)is chosen to
be twice the orbital frequency at the innermost stable
circular orbit of a BH of mass Mz.
To 1.5PN order [39], the GW phase is
Ψ(f) = 2πftc−φc−π
4
+3
4(8πMf)−5
31 + 20
9743
336 +11η
4x−16πx3
2,
(8)
where
tc
and
φc
are the coalescence time and phase and
x≡(πMzf)2/3is the PN expansion parameter.
III. AXISYMMETRIC LENS MODELS
In this section, we discuss two axisymmetric lens models,
the singular isothermal sphere (SIS) and the point mass
(PM), that produce at most two images in the geometrical-
optics approximation. We introduce model-independent
image parameters that describe the amplification factor
F
in this approximation, and assess the validity of these
new parameters as the geometrical-optics approximation
breaks down at low frequencies.
A. Singular isothermal sphere (SIS)
The SIS density profile Σ(
ξ
) =
σ2
v/
2
ξ
, where
σv
is the
velocity dispersion, is the most simple profile that can
effectively describe the flat rotation curves of galaxies [
43
].
It leads to the lensing potential
ψ
(
x
) =
x
by Eq. (3) and
the amplification factor [19,44]
F(f) = −iweiwy2/2Z∞
0
dx xJ0(wxy)
×exp iw 1
2x2−x+φm(y)
=ei
2w(y2+2φm(y))
∞
X
n=0
Γ(1 + n
2)
n!
×(2wei3π
2)n
21F11 + n
2,1; −i
2wy2,(9)
by Eq. (5), where
w
= 8
πMLf
,
φm
(
y
) =
y
+1
/
2,
J0
is the
Bessel function of zeroth order,
ξ0
= 4
π2σ2
vDLDLS /DS
is the Einstein radius, and
ML
=
σ2
v
(1 +
zL
)
ξ0
is the lens
mass inside the Einstein radius.
B. Point mass (PM)
The PM is the simplest mass distribution for a gravi-
tational lens. It leads to a lensing potential
ψ
(
x
) =
ln x
and amplification factor [19,45]
F(f) = exp nπw
4+iw
2hln w
2−2φm(y)io
×Γ1−i
2w1F1i
2w, 1; i
2wy2,(10)
摘要：

Detectabilityofstronglylensedgravitationalwavesusingmodel-independentimageparametersSaifAli,1,EvangelosStoikos,1,yEvanMeade,1MichaelKesden,1,zandLindsayKing1,x1DepartmentofPhysics,TheUniversityofTexasatDallas,Richardson,Texas75080,USA(Dated:October6,2022)Stronggravitationallensingofgravitationalwav...

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