Inferring the Intermediate Mass Black Hole Number Density from Gravitational Wave Lensing Statistics
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Inferring the Intermediate Mass Black Hole Number Density from Gravitational Wave Lensing
Statistics
Joseph Gais,1, ∗ Ken Ng,2, 3 Eungwang Seo,1 Kaze W.K. Wong,4 and Tjonnie G. F. Li1, 5, 6
1
Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.
2
LIGO, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
3
Kavli Institute for Astrophysics and Space Research, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
4
Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
5
Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium.
6
Department of Electrical Engineering (ESAT), KU Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium
The population properties of intermediate mass black holes remain largely unknown, and understanding their
distribution could provide a missing link in the formation of supermassive black holes and galaxies. Gravi-
tational wave observations can help fill in the gap from stellar mass black holes to supermassive black holes.
In our work, we propose a new method for probing lens populations through lensing statistics of gravitational
arXiv:2201.01817v2 [gr-qc] 11 Jan 2022
waves, here focusing on inferring the number density of intermediate mass black holes. Using hierarchical
Bayesian inference of injected lensed gravitational waves, we find that existing gravitational wave observatories
at design sensitivity could either identify an injected number density of 106 Mpc−3 or place an upper bound of
. 104 Mpc−3 for an injected 103 Mpc−3 . More broadly, our method could be applied to probe other forms of
compact matter as well.
I. INTRODUCTION rates estimates suggest aLIGO could detect O(1)yr−1 lensed
events at design sensitivity [22–24].
To date, we have detected dozens of black holes within the Building off of [16], we consider the lensing of gravita-
stellar mass range O(1 − 100)M from binary black hole tional waves by IMBHs as a means of inferring the IMBH
merger gravitational wave emission [1–3] and X-ray binary number density nL . We develop an analytical model verified
observations [4, 5], as well as supermassive black holes of by simulation results for the distribution of the single-lensing
mass > O(106 )M , first identified from stellar orbits about event parameters, the normalized impact parameter y and red-
the center of the Milky Way [6] and now imaged by the Event shifted lens mass Mlz . We then use a hierarchical Bayesian
Horizon Telescope [7–12]. The least understood parameter model for constraining possible nL values from a population
space of black holes lies between these two ranges, the so- of recovered y’s alongside our simulated distributions of im-
called intermediate mass black holes (IMBH) in the mass pact parameter for different lens number density. Since a pri-
range [102 , 106 ]M . Understanding the formation channels ori we have no means of identifying a lensed gravitational
of supermassive black holes and galaxies themselves will re- wave, we conduct the parameter estimation on all gravita-
quire filling in the missing link of IMBHs. tional wave events, where the posterior of unlensed gravita-
IMBHs may soon be detected. Search methods include stel- tional waves should demonstrate significant support at large
lar and gas dynamical searches as well as accreting IMBHs y and little support at y . O(1). In contrast, lensed gravita-
within galactic nuclei suggest a number of tentative IMBH tional waves with y . 1 should be recovered from the param-
discoveries (see [13] for a recent review). Recently, the first eter estimation. For any gravitational wave event, we conduct
half of LIGO-Virgo’s third observing run has detected the parameter estimation of the redshifted lens mass, Mlz and y.
gravitational waves of a binary black hole merger with a rem- The set of lens parameter estimation allows us to build a dis-
nant mass of 142M [14], the first ever confirmed IMBH. In tribution for the full population of y values. In turn, we are
addition to measurements of IMBH remnants, another pos- able either to constrain the number density of IMBHs if no
sible method for detecting IMBHs lies in measuring gravita- IMBH mass range lenses are present within the full popula-
tional wave lensing effects. tion, or measure on the IMBH number density if IMBH lens
If a gravitational wave passes by an IMBH mass lens events are detected.
closely, the measured gravitational wave will have a frequency Injecting a catalog of ∼ 200 events drawn from nL =
dependent amplification factor altering the waveform [15]. {103 , 106 }Mpc−3 with a design sensitivity LIGO Hanford,
From careful study of detected gravitational waves, we may LIGO Livingston [25] and Virgo [26] observatory network,
determine the lens parameters, with recent work demonstrat- we can confidently detect the density of IMBH lenses at
ing the detection of mass of an IMBH lens [16] and how grav- 106 Mpc−3 or constrain to . 104 Mpc−3 for a number den-
itational wave lensing can constrain black hole populations sity of 103 Mpc−3 , on the scale of IMBH densities inferred
[17]. Although no gravitational wave event has yet been con- from gamma ray burst observations [27]. Combining mea-
clusively identified as being lensed [18–21], tentative lensing surements from lensing statistics as well as with parameter es-
timation of source masses in gravitational wave mergers could
then shed light on the largely unknown population of IMBH
lenses.
∗ 1155138494@link.cuhk.edu.hk We begin by describing the effect of a point mass lens on a2
gravitational wave in Sec. II. Then, in Sec. III, we derive a hi-
erarchical Bayesian model to infer the point mass lens popula-
tion from detected gravitational wave events. In Sec. IV A, we
detail an analytical population model for IMBH lenses, vali-
dating our model against simulated results. We then conduct
an injection campaign in the LIGO-Virgo detector network as
described in Sec. V. Finally, in Sec. VI, we present the recov-
ered lens number density from our injections, and discuss our
results and impact of improved detector networks on probing
the IMBH population in Sec. VII.
II. GRAVITATIONAL WAVE LENSING
When a gravitational wave passes by a massive object, it
is lensed in a manner similarly to electromagnetic waves. In
the geometric optics regime, i.e., when the dimensionless fre-
quency w = 8πMLz f 1, where MLz is the redshifted lens
mass with gravitational frequency f in the detector’s frame, FIG. 1. Basic lensing geometry for a gravitational wave lensed by
a point mass in the thin lens approximation. In the plane of the sky
the amplitude of the gravitational wave is either magnified
with the lens at the origin, the source is located at η , passes the lens
or demagnified while the phase content remains unchanged. plane with impact parameter ξ , and then deflected by the lens at the
However, in the wave optics regime where w . 1, both the lens plane, ultimately reaching the observer. DS signifies the angular
amplitude and phase of the gravitational wave are modulated diameter distance from the observer to the source, DL is the angular
in a frequency-dependent manner, yielding a rich structure diameter distance from observer to lens, and DLS is the angular di-
in the lensed gravitational wave. Lensed gravitational waves ameter distance from lens to source, which is not equal to DS − DL .
could soon be detected [18, 22, 23], with applications ranging
from improved sky localization [28], tests of the polarization
of gravitational waves [29], or probing dark matter [30]. parameter of the source-lens pair normalized by the lens’ Ein-
Here, we focus on the case of a gravitational wave lensed stein radius, are detectable from Bayesian parameter estima-
by a single point mass, illustrated in Fig. 1. The details of the tion of the lensed gravitational wave for IMBHs [16]. Follow-
analytical calculation for the lensing amplification factor are ing this example, we prepare a likelihood model for a lensed
outlined in App. A, resulting in an analytical solution for the gravitational wave, from which one can infer the posterior on
isolated point mass, the lensing parameters. When wy 2 /2 1, the amplification
factor is highly oscillatory in the frequency domain, the geo-
πw w w metric optics approximation can be used. Using a dynamical
F (w) = exp +i ln − 2φm (y)
4 2 2 lookup table in (w, wy 2 /2) for the evaluation of the hyper-
i
i i
geometric function in F (f ), we are able to rapidly evaluate
× Γ 1 − w 1 F1 w, 1; wy 2 , (1) the amplification factor such that lensing parameter estima-
2 2 2
tion is feasible, and use the geometric optics approximation
where w = 8πMLz f is the dimensionless frequency, y is for wy 2 > 1000 elsewhere.
the impact parameter normalized by the lens’ Einstein radius,
MLz is the redshifted lens mass, 1 F1 is the confluent hyper-
geometric function, and III. HIERARCHICAL BAYESIAN ANALYSIS
(xm − y)2 In this section, we list the mathematical details of the hier-
φm (y) = − ln xm , (2)
p 2 archical inference model for a generic lensing scenario. We
y + y2 + 4 seek to measure the properties of the lens population param-
xm = . (3)
2 eterized by Λ L . Given a dataset d = {di } of N detections
and the properties of source population parameterized by Λ S ,
The lensed waveform is then, we can compute the posterior of Λ L , pΛ (ΛΛL |dd, Λ S ), by com-
bining the measurement of waveform parameters x of each
ψ L (f ) = F (f )ψ0 (f ) (4) detection,
where ψ0 (f ) is the frequency-domain base waveform and N
ΛL |dd, Λ S ) Y
Z
F (f ) is the amplification factor. pΛ (Λ
Lgw di |x
xi πgw (x
xi |Λ xi ,
∝ ΛS , Λ L )dx
Previous studies demonstrate that the gravitational wave ΛS , Λ L )
πΛ (Λ i=1
event parameters and lens parameters, MLz and y, the impact (5)3
where Lgw di |x xi is the likelihood of the i-th gravitational are less likely to have a sufficiently high SNR. However, for
wave detection, πgw (x xi |Λ
ΛS , Λ L ) is the distribution of wave- the physically motivated regime of number densities we con-
form parameters given both the source and lens population sider, y 1 in most events, resulting in magnifications very
properties, and πΛ (Λ ΛS , Λ L ) is the prior of (Λ
ΛS , Λ L ). While close to unity, and so the SNR of any event is hardly affected
one can simultaneously infer (Λ ΛS , Λ L ), we expect that the by lensing (and by extension the lens number density). Thus,
population properties of sources and lenses are weakly cor- the SNR selection is unlikely to bias our results and we ignore
related and leave out ΛS for the rest of the paper for sim- it for simplicity.
plicity. We list our choice of source population properties,
such as BBH mass spectrum and redshift evolution, in App. B.
In the following, we separate the waveform parameters into IV. DISTRIBUTION OF THE NEAREST-EFFECTIVE
(y, x S , x L ), in which y can be thought of the parameter char- LENSES
acterizing the pairing of a source and a lens, x S = (zS , x̃ S ) is
the set of source parameters including source redshift zS and A. Notion of the nearest-effective lens
other parameters irrelevant to lensing, x̃S , and xL = (zL , x̃L )
is the set of lensing-relevant parameters including the lens red-
We observe the population of the source-lens systems rather
shift zL and the model-dependent parameters characterizing
than the population of isolated lenses. One needs to cau-
the internal properties of the lens, x̃ L . nt parameters given
tiously account for this subtle difference when modeling πL
hyperparameters Λ which we simulate directly.
in Eq. (7), which is no longer the intrinsic distribution of the
We expect that x̃ S and x L are independent of each other and
lenses. We assume that a source is solely diffracted by a single
hence their distributions are separable. We treat the constraint
lens, i.e., multiple lensing due to the next neighboring lenses
that a lens must be inside the volume within zS , zL < zS as
is negligible. Since the size of the Einstein ring also affects the
a condition imposed on the lens distribution in Bayes’ theo-
magnitude of y, the nearest-neighbor lens (i.e. with the small-
rem. One can further marginalize over other irrelevant source
est value of θS = η/DS ) does not necessarily give rise to the
parameters x̃ S . Putting these steps together, Eq. (5) becomes
strongest effect of diffraction. Instead, a source is the most
ΛL |dd) Y
pΛ (Λ
N ZZZZ diffracted by a lens whose parameters result in the smallest
Lgw di |x
xi πL y i , x iL |zSi , Λ L , P
∝ value of y. We call such lenses as the nearest-effective lenses.
ΛL )
πΛ (Λ In terms of the lensing statistics, the statement P is equiva-
i=1
× πS zSi , x̃ iS dzSi dy i dx
xiL dx̃ lent to the requirement of minimum y when pairing the lenses
i
x̃S , (6)
and sources. We can model the nearest-effective pairing by
where πL (y, x L |zS , Λ L , P) is the distribution of lens param- characterizing the distribution of neighboring lenses through
eters given a source at redshift zS , and πS is the prior of the a spatial Poisson process, which only depends on the spatial
source parameters. The conditional statement P denotes the distribution among the lenses but not on the internal properties
requirement of a source-lens pair having the strongest diffrac- of the lenses. This is achievable by considering y as an effec-
tion along the line of sight. We will explain the importance of tive distance between a source and its nearest effective lens on
this notion in Sec. IV A. the sky plane. Assuming the lenses are uniformly distributed
To evaluate Eq. (6), we can usei importance sampling
by on the sky plane, we can separate the joint distribution of y
i xi i i
recognizing
that L gw d |x Pr x = pgw x |d , where and x L into
Pr x i is the prior of waveform parameters used in the
parameter-estimation algorithm that estimates the posterior of πL (y, x L |zS , Λ L , P)
waveform parameters, pgw x i |di . We can reweigh the sam- = πy (y|zS , Λ L , P) πx L (zL , x̃ L |zS , Λ L , P) , (8)
ples drawn from the estimated posterior to evaluate the hierar-
chical likelihood, where πy and πx L are the distributions of y and x L condi-
N K i tioned on the nearest-effective pairing between sources and
ΛL |dd) Y πS (zSi,j , x̃ i,j
pΛ (Λ 1 X S ) lenses, respectively. In the following, we first derive πy and
∝
ΛL )
πΛ (Λ i=1
K j=1 Pr(y i,j , zS , x̃ S , x i,j
i i,j i,j
L )
πx L from the spatial Poisson process, then list out the math-
ematical details in the case of point-mass lenses, and validate
× πL (y i,j , x i,j |z i,j
, Λ L , P) , (7) the analytical model by comparing it to the direct simulation
L S
of the nearest-effective pairing of the source-lens systems.
where (·)i,j denote the j-th sample drawn from K i posterior
samples of the i-th event.
B. Spatial Poisson Process
Generically, in hierarchical Bayesian analysis of hyperpa-
rameters, the selection bias must be taken into account. For
y 1, the lensed waveform is greatly amplified [15, 31, 32], With a source centered at the origin, the probability that
resulting in higher SNR values. Selection of only those events there are k lenses within an effective distance y is
above a certain threshold will then bias the recovered hyper- k
parameter posterior towards higher lens number densities, as Σπy 2 2
events with higher y values (and thus, less of a lensing effect) Poisson(k|Σ) = e−Σπy , (9)
k!4
where Σπ is the effective density parameter of lenses within by its mass ML , i.e., x̃ L = ML . Throughout the study,
the volume of zS projected on the sky. The differential prob- we assume the intrinsic lens mass spectrum does not evolve
0 0
ability of finding the nearest-effective lens inside an infinites- with lens redshift, i.e., πL = πM L zL
0
π 0 , where πM L
and πz0 L
imal ring between y and y + dy is the product of the prob- are the one-dimensional intrinsic distribution of lens mass
ability that there is no lens within the circle of radius y, and lens redshift, respectively. We use a power-law mass
0
(ML |αL ) ∝ ML−αL , in
2
Poisson(0|Σ) = e−Σπy , and the probability of a lens lying spectrum with an index αL , πM L
inside the ring, 2Σπydy. Dividing this probability by dy, the the domain [ML,min = 100 M , ML,max = 20000 M ].
probability density function of the nearest-effective lens locat- For simplicity, we keep the lens number density constant in
ing at y is the comoving frame such that the prior of lens redshift is
πz0 L (zL |zS ) ∝ dVc (zL )/dzL for zL < zS . We note that one
2
p(y) = 2Σπye−Σπy . (10) can relax the assumption of constant density to infer the lens
redshift evolution. As such, we only have two hyperparame-
Since y is the dimensionless ratio of the angular separa- ters, Λ L = (nL,0 , αL ).
tion between the source and the lens to the angular size of the Now, we write down the expressions for πx L ≡ πzL πML
2
lens Einstein ring, the effective density parameter can be in- and Σ. Including the lensing bias factor, θE ∝ ML DLS /DL
terpreted as the mean fractional area of all lenses within zS at a fixed zS , we have
relative to the full sky plane (or, equivalently, the inverse of
the mean of y 2 ), i.e., DLS
πzL (zL |zS , ΛL , P) ∝ πz0 L (zL |zS , ΛL ) , (15)
DL
2 0
π θE ΛL πML (ML |zS , Λ L , P) ∝ πM (ML |zS , Λ L ) ML . (16)
L
Σ(zS , Λ L )π = NL (zS ) , (11)
4π
Rz Since F (f ) only depends on (y, MLz ) and zL is not directly
where NL (zS ) = 0 S nL (zL )dVc (zL ) is the total number of measured, we further marginalize πML πzL over zL to obtain
lenses within the comoving volume Vc (zS ) for an arbitrary the distribution of redshifted lens mass,
number density evolution of lenses nL (zL ), and
Z πMLz (MLz |zS , Λ L , P)
2 2 0 Z zS 1−αL
θE = θE (zL , x̃ L |zS ) πL (zL , x̃ L |zS , Λ L ) dzL dx̃
x̃L MLz DLS dVc dzL
Λ L ∝ , (17)
(12) 0 1 + zL DL dzL 1 + zL
is the mean area enclosed by the Einstein rings, with h·iΛ L for ML ∈ [ML,min , ML,max ], and is zero otherwise. The extra
being the mean quantity over the intrinsic lens distribution pa- factor of (1 + zL )−1 comes from the transformation of the
0
rameterized by Λ L , πL (zL , x̃ L |zS , Λ L ) is the joint distribu- differential dMLz = (1 + zL )dML . Finally, the expression of
tion of redshift and mass of the intrinsic lens population (i.e. Σ for πy is
regardless of the pairing with the sources). Thus, the term
4nL,0 χ3S
DLS
Σπy 2 in the exponent of Eq. (10) is equivalent to the mean Σ(zS , Λ L ) = hML iΛ L , (18)
number of lenses within the area πθS2 . The desired πy is then 3DS DL Λ L
πy (y|zS , Λ L , P) = 2πyΣ(zS , Λ L )e−Σ(zS ,Λ
ΛL )πy
.
2
(13) where χS is the comoving distance at zS , hML iΛ L is the mean
lens mass,
The pairing requirement, P, favors a source-lens system
L,max − ML,min
M
with the largest θE to minimize the value of y. One can think for αL = 1
ln (M L,max /ML,min )
of the pairing condition as choosing the lens with the largest
2
area, πθE . As a result, the final distribution of lens parame-
ln (ML,max /ML,min )
ters in the source-lens systems has an additional lensing bias for αL = 2
factor proportional to θE 2
for sources at the same zS . Math- hML iΛ L = −1
ML,min −1
− ML,max
ematically, the distribution of x L after the nearest-effective
pairing is 2−αL 2−αL
1 − αL ML,max − ML,min
otherwise,
1−αL
2 − αL ML,max 1−αL
2 0 − ML,min
πxL (zL , x̃ L |zS , Λ L , P) ∝ θE πL (zL , x̃ L |zS , Λ L ) , (14)
(19)
which is indeed the integrand of Eq. (12).
and hDLS /DL iΛ L is the mean distance factor given by
C. Lensing Statistics for Point-mass Lenses
Z zS
DLS DLS dVc
= dzL . (20)
DL ΛL 0 DL dzL
4
IMBHs with masses of ∼ O(100 − 10 ) M may serve
as point mass lenses to diffract gravitational waves. The We use Planck 18 cosmology [33] for the evaluation of cos-
mass profile of a point mass lens is entirely parameterized mological distances.5
D. Validation ×10−5
3.0 psim
Let us examine the behavior of πL . First, the inverse of πL
−1
the density parameter (Σπ) characterizes the scale of y. In 2.5
particular, the most probable value of y (or the peak of πy ) is
2.0
πΣ(zS )
−1/2
yp = (2Σπ) . This can be understood physically by in-
−1 1.5
terpreting (Σπ) as the ratio of the mean cross-section area,
π θS2 Λ ≡ 4π/NL , to the mean area of lenses, π θE 2
ΛL 1.0
L
(cf Eq. (11)). Second, in the limit of y → ∞, the Gaus-
2 0.5
sian term e−Σπy regulates the linear increase in πL with
2
ye−Σπy → 0. The impact parameter cannot be arbitrarily 0.0
large because the separation between adjacent lenses is char- 0.5 1.0 1.5 2.0 2.5 3.0
acterized by the scale of (Σπ)
−1/2
. Third, we consider the zS
limit of 0 < y < ymax , where ymax is the cut-off of y satisfy-
ing ymax (Σπ)−1/2 . In such limit, sources are distributed FIG. 2. πΣ(zS , nL = 1000 Mpc−3 ) as a function of source red-
uniformly around the vicinity of the nearest-effective lens, re- shift, comparing the analytical spatial Poisson model (orange) to
sulting in a linear distribution of y. Indeed, the spatial Pois- simulated, fitted values (blue). As Σ increases monotonically with
2 source redshift, π(y|zS , nL ) shifts towards smaller y. Thus, detec-
son piece, 2πyΣe−Σπy , is well approximated by 2y/ymax 2
tor networks with a larger detectable range are more likely to detect
2 −1
for y (Σπ) and independent of Σ. Together with lensed sources.
2
the lensing bias factor ∝ θE , the asymptotic form of πL for
−1/2
y (Σπ) is
psim (y|zS , Λ L , P) closely, validating the analytical model.
We note that the distributions πML (ML |zS , Λ L , P) and
0 −1/2
πL y, zL , x̃ L |zS , Λ L , P, y (Σπ)
πzL (zL |zS , Λ L , P) are altered from their pre-selection dis-
0
2y 2 0 tribution, πM (ML |zS , Λ L ) and πz0 L (zL |zS , Λ L ), with the
∝ θE πL (zL , x̃ L |zS , Λ L ) , (21) L
2
ymax bias-factored distributions matching the simulated distribu-
tions. After selecting, πML is now uniform, and so πMLz fol-
which, after the marginalization over zL , recovers the usual lows the approximate shape of πzL . The lens redshift distribu-
definition of the lensing optical depth (or the lensing proba- tion πzL is more skewed towards smaller redshifts, as the bias
bility) defined in the existing literature [34] for non-evolving factor DLS /DL is maximized at smaller lens redshifts. Addi-
point-mass lens distribution, tionally, drawing independent samples from the bias-factored
Z zS distributions, plotted in orange contours, we find that they
d2 τ 2 dVc 0 match the simulated contours, indicating that the lensing pa-
= 2yπθE nL (zL ) π (ML |zS , Λ L )dzL ,
dydML 0 dzL ML rameters (y, zL , ML ) are independent following selection of
(22) nearest effective lens-source pairs.
Fig. 2 shows the evolution of Σ with source redshift. In
up to some overall constants as nL (zL )dVc /dzL ∝ πz0 L and particular, note that the effective density increases monotoni-
0
πL is a normalized probability density function rather than a cally with source redshift, as more and more lenses are in the
probability function for the optical depth. plane of the sky. As a result, the y distribution shifts towards
To test that the spatial-Poisson process accurately models smaller values as zS increases, and Fig. 3 plots the decreasing
the lensing statistics described thus far, we directly simulate a peak value of p(y|zS , nL ) with zS .
population of lenses and sources for a fixed value of nL,0 =
1000 Mpc−3 . Lenses are placed uniformly in the plane, with a
redshift distribution uniform in comoving volume, and have a V. GRAVITATIONAL WAVE LENS PARAMETER
power-law mass distribution with αL = 1 between ML,min = ESTIMATION
100 M and ML,max = 20000 M . Source redshifts are
assumed to follow the Madau-Dickinson star formation rate. In order to effectively use lens parameter estimation to draw
We then compute the y value for each possible lens-mass pair, conclusions on the IMBH population, injected lens parameters
subject to the constraint that zS > zL . should be recoverable in the parameter estimation. To conduct
We can identify that our bias factor described in Sec. IV B parameter estimation, we use the Bilby library [35] with the
is correct with the aid of a corner plot of our simulation Dynesty sampler [36]. Fig. 5 and Fig. 6 demonstrate typical
in (y, zL , ML ). Figure 4 shows the corner plot with a results for the impact parameter of a lensed gravitational wave
fixed source redshift of zS = 3 after selecting source-lens injection, with an injected y < 1 and y 1 respectively.
pairs, with the simulated marginalized distributions (blue), In the case of y < 1 in Fig. 5, the injected y parameter is
bias-factored analytical model (orange), and model with- accurately recovered in the posterior of both y and MLz , and
out bias factoring (dashed black lines). The spatial Pois- the likelihood is only non-zero about the injected value. Thus,
son distribution πL (y) matches the simulated distribution injections with y < 1 for IMBHs are clearly detectable.6
104
ypk
103
y
102
15
0.
101
log10(y)
0 2 4 6 8 10 12 14
zS
00
0.
FIG. 3. Peak π(y|zS ) value as a function of zS . The y distribu-
15
0.
tion shifts towards smaller values as the effective lens surface density
−
grows.
30
0.
75
90
05
20
30
15
00
15
−
3.
3.
4.
4.
0.
0.
0.
0.
−
−
log10( M
M )
Lz log10(y)
psim
FIG. 5. Corner plot posterior for an injection with log10 (y) ∼ −0.05
πL and log10 (MLz ) ∼ 4, with the gold lines demarcating the injected
values of (MLz , y). While there is some degeneracy in (MLz , y)
πL0
as the parameter wy 2 determines the oscillatory behavior of the fre-
quency domain waveform, both the injected MLz and y values are
recovered with reasonable precision.
4 8 2 6 0 6 2 8 4 0
0. 0. 1. 1. 2. 0. 1. 1. 2. 3.
In constrast to the small y case, Fig. 6 illustrates the poste-
zL
rior for a large injected value, y 1. With a uniform in log
prior, the posterior remains relatively flat, and the posterior is
not localized about the injected value, as the effects of lensing
on the waveform are too small to be detected, and the MLz
ML(104M )
posterior is agnostic. However, the posterior has no support
for y . 1, ruling out the parameter space where lensing ef-
fects are significant. In this way, the diffraction effects of a
microlens can either be detected or ruled out.
At small nL,0 values the typical√y value is large, with the
y distribution peaking at yp ∼ 1/ Σ. This could present a
0
0
0
0
6
2
8
4
0
4
8
2
6
0
15
30
45
60
0.
1.
1.
2.
3.
0.
0.
1.
1.
2.
problem if multiple diffraction effects are combined, as the
y zL ML(104M ) lens with the smallest y value for the source could be large
enough that other lenses have a similar y value. However, as
FIG. 4. Corner plot of (y, zL , ML ) distributions simulated di- these parameter estimation results show, the diffraction effects
rectly (blue), without bias factor (dashed black line), and with are still minimal at large y, and so an arbitrarily large y value
bias factor (orange), for αL = 1. Because of the lens- can be injected without consideration of possible contami-
ing bias from selecting source-lens pairs with the smallest y nating effects from other source-lens pairings in a multiple-
value, the selected lens mass and lens redshift distributions, lensing scenario.
πzL (zL |zS , Λ L , P) πML (ML |zS , Λ L , P), are different from their
pre-selection distribution, πz0 L (zL |zS , Λ L ) πM
0
L
(ML |zS , Λ L ), and
direct simulations confirm our bias factor. After selection, A. Generating the Injection Bank
the lens redshift distribution now scales as DDLS dVc
L dzL
and
πML (ML |zS , Λ L , P) is uniform. Furthermore, independent sam-
pling of (y, zL , ML ) (orange contours) align with the direct simula- Finally, for a fixed lens number density and lens mass
tion contours (blue), and so the 1D distributions are indeed uncorre- power law, we create an injection set to test our ability to
lated following selection of minimum y. recover the lens number density hyperparameter. For the
lens parameters, the source position π(y|nL , zS ) is sampled
from Eq. (13), and the source parameters are sampled from7
0.20
0.15
p(nL,0|dd)
0.10
0.05
0
3.
log10(y)
0.00
0 1 2 3 4 5 6
5
log (nL,0/Mpc−3)
1.
0
0.
FIG. 7. Hierarchical likelihood for an injected nL = 103 Mpc−3
density, with ∼ 200 gravitational wave events. At 95% CI, the den-
5
sity is constrained to . 104.6 Mpc−3 . Further detections of gravi-
1.
−
3
6
9
2
5
0
5
0
tational waves unlensed by IMBHs could push this constraint further
3.
3.
3.
4.
1.
0.
1.
3.
−
down, as well as an expanded, more sensitive detector network.
log10( M
M )
Lz log10(y)
FIG. 6. Corner plot with an injected y 1 in {MLz , y}, with the
injected values marked by the gold lines. In constrast to the small y
1.2
case, at large y the lens mass posterior is completely agnostic, as is
the MLz posterior.
1.0
0.8
p(nL,0|dd)
the distributions discussed in Sec. B. For the base unlensed
waveform, we use the IMRPhenomD approximate [37, 38], 0.6
which encompasses the inspiral, merger, and ringdown. The
lensed waveform is then the product of the amplification fac- 0.4
tor and the base waveform. We threshold sampled injections
by signal-to-noise ratio (SNR), selecting only those injections 0.2
with network SNRs ρnet > 12 in a three detector network
0.0
consisting of the LIGO Livingston, LIGO Hanford, and Virgo 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
observatories at design sensitivity. log (nL,0/Mpc−3)
For the hyperparameters, we fix αL = 1, and generate in-
jection sets with IMBH densities nL = {103 , 106 }Mpc−3 . At FIG. 8. Hierarchical likelihood for an injected nL = 106 Mpc−3
nL = {103 , 106 }Mpc−3 the SNR gain due to strong lensing density, with ∼ 200 gravitational wave events. The likelihood cor-
is negligible, and so we neglect the selection effect. rectly recovers the injected hyperparameter 106 Mpc−3 , with 95% CI
intervals of 105.1 Mpc−3 and 106.5 Mpc−3 , and so is capable of not
only constraining the population properties of IMBHs but actually
VI. RESULTS OF HIERARCHICAL ANALYSIS detecting them.
Fig. 7 and Fig. 8 show the recovered hierarchical likeli-
hood for the cases of 103 Mpc−3 and 106 Mpc−3 respectively.
At 103 Mpc−3 , the recovered likelihood can constrain the hy-
perparameter to . 105 Mpc−3 at 90% confidence. This upper lens number densities.
constraint can improve with further unlensed detections.
For a density of 106 Mpc−3 , the injected hyperparameter is With a more sensitive network the volume of detectable
recoverable with this network, with the likelihood of Fig. 8 mergers grows, and since πy (y|zS , Λ L , P) increases mono-
ruling out both nL . 105 Mpc−3 and nL & 106.5 Mpc−3 at tonically with source redshift, the probability of encounter-
90% confidence. Thus, even with just a three detector net- ing a significantly lensed event increases. Thus, lensed events
work, the population properties of IMBH lenses are not only by IMBH lenses could be detectable even at these relatively
possible to constrain but even to detect. This is because O(1) small redshifts, and the recovered likelihood for an injected
events in our injection set are lensed with recoverable y injec- 103 Mpc−3 hyperparameter may resemble a true measure-
tion parameters in the parameter estimation, ruling out smaller ment, rather than just an upper bound.8
VII. DISCUSSION allows for a flexible extension to test other lens models, such
as the singular isothermal sphere or NFW profile [46–48], by
We present a novel method of probing population distri- considering the population as a mixture of different types of
butions for lenses of gravitational waves, using the statistics lenses. Notably, inclusion of galactic lenses could boost the
of gravitational wave lensing, assuming that multiple lens- detectability of y as shown in previous work [49]. For lenses
ing effects are negligible. Deriving population models for the that do not obtain circular symmetry, such as elliptical lenses,
lensing statistics of point-mass lenses be distributed uniformly the presented formalism still holds, with two modifications:
in comoving volume with a power-law mass distribution, we (1) including the dependence of the symmetry-breaking pa-
verify our models with direct simulations, and demonstrate a rameter (e.g., ellipticity or external shear) in x̃ L to calcu-
hierarchical Bayesian model for computing the likelihood of late F (f ), and (2) redefining the normalization of y that re-
the lens density from successive observations. We then con- spects the notion of the nearest-effective lens, i.e., the effect
duct an injection campaign with gravitational wave samples, of diffraction is stronger when y is smaller, to evaluate Σ and
generating catalogues of lensed injections with network SNR πL (y, x L |zS , Λ L , P). We will leave these extensions in the
ρnet > 12 for densities of {103 , 106 }Mpc−3 . Our results, future work.
shown in Figs. 7 and 8, show that we may either constrain or
directly detect the lens number density for {103 , 106 }Mpc−3
VIII. ACKNOWLEDGEMENTS
respectively.
In the specific case of IMBHs, our method can probe their
relatively unknown population properties with just a three- JG and ES are supported by grants from the Research
detector network of already existing gravitational wave ob- Grants Council of the Hong Kong (Project No. CUHK
servatories operating at design sensitivity. Since the effective 24304317), The Croucher Foundation of Hong Kong, and
lensing probability increases with source redshift, a more sen- the Research Committee of the Chinese University of Hong
sitive detector network could greatly improve our ability to Kong. KKYN is supported by the NSF through the award
probe the IMBH population, detecting or constraining lower PHY-1836814. KWKW is supported by the Simons Foun-
values of the lens number density. With the addition of a few dation. The authors are grateful for computational resources
more planned observatories, like LIGO-India or KAGRA, the provided by the LIGO Lab and supported by the National Sci-
IMBH number densities of ∼ 103 − 104 Mpc−3 could be di- ence Foundation Grants No. PHY-0757058 and No. PHY-
rectly detected. Additionally, third generation detectors like 0823459. This research has made use of data, software and/or
the Einstein Telescope [39] or Cosmic Explorer [40, 41] could web tools obtained from the Gravitational Wave Open Science
probe extremely high source redshifts of zS & 30, detect Center [50], a service of LIGO Laboratory, the LIGO Scien-
∼ 10000 binary black hole mergers per month [42], and be tific Collaboration, and the Virgo Collaboration.
sensitive to higher injected y values, so that smaller IMBH
densities would be detectable. Indeed, applying the third
Appendix A: Amplification function in wave optics
generation population forecast discussed in [43] with isolated
galactic field formation, dynamical globular cluster formation,
and Population III stars at high redshift subpopulations, we The background metric of a gravitational is given by
find that ∼ 1 event with y < 1 could be detected each month
ds2 = −(1 + 2U )dt2 + (1 − 2U )dr2 ≡ gµν
(B)
dxµ dxν , (A1)
for a density of nL = 103 Mpc−3 .
We end by noting that the common use of lensing optical with lens potential U (r) 1. For a gravitational wave prop-
depth in Eq. (22) carries the notion of a signal being lensed vs agating against the lens background, we consider a linear per-
unlensed, which is less well-defined in the wave-optics sce- turbation against the background metric, where
nario. The classification of the lensed signals relies on the (B)
choice of y ≤ ymax to down-select the data of the lensed- gµν = gµν + hµν . (A2)
only population for further analysis. One has to build up de- Under an appropriate gauge choice and applying the Eikonal
tection statistics, e.g. the Bayes factor statistics from a large approximation, we can express the gravitational wave hµν as
scale injection campaign [44] or the mismatch from the wave-
form [45], for identifying the events that belong to the lensed hµν = φeµν , (A3)
population. Besides being inflexible, this approach depends
with polarization tensor eµν and scalar φ. The change in the
on a number of artificial choices, such as the choice of prior
polarization tensor along the null geodesic is O(U ) 1 such
and the threshold of detection statistics for a lensed signal.
that we hold the polarization fixed. We then consider the prop-
As a result, such process can be fuzzy for weak signals and
agation of the scalar field as it interacts with the background
may misidentify the lensed population in the data. On the
lens potential, with propagation equation
other hand, our method makes full use of the parameteriza- p
tion of y and does not require the binary notion of “lensed ∂µ −g (B) g (B)µν ∂ν φ = 0. (A4)
vs unlensed”. With the hierarchical approach, we can treat the
data as a whole population to infer the lens properties robustly, In the frequency domain φ̃(f, r), Eq. (A4) satisfies,
given a detailed model of the source-lens systems.
∇2 + ω 2 φ̃ = 4ω 2 U φ̃,
The mathematical framework derived in Secs. III & IV also (A5)9
where ω = 2πf . We define the amplification function as the To improve computational efficiency at the limit of y 1
ratio of the lensed and unlensed (U = 0) gravitational-wave or w 1, we switch to the geometric approximation of the
amplitudes, such that magnification,
φ̃L (f )
p p
F (f ) = . (A6) Fgeo (w) = |µ+ | − i |µ− |eiw∆τ , (A11)
φ̃(f ) 1 y +2 2
µ± = ± p , (A12)
In the thin-lens approximation, we decompose the source’s 2 2y y 2 + 4
wave into wavelets of all possible paths and integrate their p p !
y y2 + 4 y2 + 2 + y
contribution by the Kirchhoff’s diffraction formula to obtain ∆τ = + ln p (A13)
the amplification function [15, 31, 32] 2 y2 + 2 − y
DS ξ02 (1 + zL ) f
Z
where µ+ and µ− are the magnifications of the two geometric
F (f ) = d2x exp[2πif td (x
x, y )], (A7)
DL DLS i images, and ∆τ is the normalized time delay between the two
images.
where DS and DL are the source’s and lens’ angular diameter
distances from the observer, respectively, zL is the lens red-
shift, DLS is the angular diameter distance between the source
and lens, ξ0 is the Einstein radius, x = ξ/ξ0 is the position of Appendix B: Source Distribution
the wavelet on the lens plane, y = (η/DS )/(ξ0 /DL ) is the
normalized impact parameter (or the normalized source posi- The parameters of the source distribution from which we
tion), and td is the arrival time of the wavelet at the observer. sample are as follows. For the mass distribution of the com-
In the case of a point-mass lens, Eq. (A7) may be analytically ponent source masses, we sample from the Power Law + Peak
integrated yielding the solution model from population studies of GWTC-2 [51]. The source
n πw w h w io redshift distribution is drawn from the phenomenological fit
F (w) = exp +i ln − 2φm (y) to the population synthesis rate [43, 52],
4 2 2
i i i 2
× Γ 1 − w 1 F1 w, 1; wy , (A8) dVC (1 + zS )1.57
2 2 2 p(zS ) ∝ . (B1)
dzS 1 + 1+zS 5.83
3.36
where w = 8πMLz f is the dimensionless frequency, MLz is
the redshifted lens mass, 1 F1 is the confluent hypergeometric The rest of the parameters, including the sky position,
function, and polarization angle, cosine of orbital inclination angle, and
aligned spins, are distributed uniformly. After sampling the
(xm − y)2
φm (y) = − ln xm , (A9) source parameters from the above distribution, we simulate
p 2 the gravitational-wave signals in the presence of detectors’
y + y2 + 4 noise, calculate the network SNR, and only select the signals
xm = . (A10)
2 with SNRs ≥ 12.
[1] B. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Acernese, black-hole binaries, Annu. Rev. Astron. Astrophys. 44, 49
K. Ackley, C. Adams, R. Adhikari, V. Adya, C. Affeldt, et al., (2006).
Gwtc-1: a gravitational-wave transient catalog of compact bi- [6] A. Ghez, S. Salim, S. D. Hornstein, A. Tanner, J. Lu, M. Morris,
nary mergers observed by ligo and virgo during the first and E. Becklin, and G. Duchêne, Stellar orbits around the galactic
second observing runs, Physical Review X 9, 031040 (2019). center black hole, The Astrophysical Journal 620, 744 (2005).
[2] R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley, [7] E. H. T. Collaboration et al., First m87 event horizon telescope
A. Adams, C. Adams, R. Adhikari, V. Adya, C. Affeldt, et al., results. i. the shadow of the supermassive black hole, arXiv
Gwtc-2: Compact binary coalescences observed by ligo and preprint arXiv:1906.11238 (2019).
virgo during the first half of the third observing run, arXiv [8] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K.
preprint arXiv:2010.14527 (2020). Baczko, D. Ball, M. Baloković, J. Barrett, D. Bintley, et al.,
[3] R. Abbott, T. Abbott, F. Acernese, K. Ackley, C. Adams, N. Ad- First m87 event horizon telescope results. ii. array and instru-
hikari, R. Adhikari, V. Adya, C. Affeldt, D. Agarwal, et al., mentation, The Astrophysical Journal Letters 875, L2 (2019).
Gwtc-3: Compact binary coalescences observed by ligo and [9] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K.
virgo during the second part of the third observing run, arXiv Baczko, D. Ball, M. Baloković, J. Barrett, D. Bintley, et al.,
preprint arXiv:2111.03606 (2021). First m87 event horizon telescope results. iii. data process-
[4] J. E. McClintock and R. A. Remillard, Black hole binaries, ing and calibration, The Astrophysical Journal Letters 875, L3
arXiv preprint astro-ph/0306213 (2003). (2019).
[5] R. A. Remillard and J. E. McClintock, X-ray properties of [10] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K.10
Baczko, D. Ball, M. Baloković, J. Barrett, D. Bintley, et al., 024001 (2014).
First m87 event horizon telescope results. iv. imaging the cen- [27] J. Paynter, R. Webster, and E. Thrane, Evidence for an
tral supermassive black hole, The Astrophysical Journal Letters intermediate-mass black hole from a gravitationally lensed
875, L4 (2019). gamma-ray burst, Nature Astronomy , 1 (2021).
[11] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. [28] O. A. Hannuksela, T. E. Collett, M. Çalışkan, and T. G. Li,
Baczko, D. Ball, M. Baloković, J. Barrett, D. Bintley, et al., Localizing merging black holes with sub-arcsecond precision
First m87 event horizon telescope results. v. physical origin of using gravitational-wave lensing, Monthly Notices of the Royal
the asymmetric ring, The Astrophysical Journal Letters 875, L5 Astronomical Society 498, 3395 (2020).
(2019). [29] S. Goyal, K. Haris, A. K. Mehta, and P. Ajith, Testing the na-
[12] K. Akiyama, A. Alberdi, W. Alef, K. Asada, R. Azulay, A.-K. ture of gravitational-wave polarizations using strongly lensed
Baczko, D. Ball, M. Baloković, J. Barrett, D. Bintley, et al., signals, Physical Review D 103, 024038 (2021).
First m87 event horizon telescope results. vi. the shadow and [30] J. Urrutia and V. Vaskonen, Lensing of gravitational waves as
mass of the central black hole, The Astrophysical Journal Let- a probe of compact dark matter, Monthly Notices of the Royal
ters 875, L6 (2019). Astronomical Society (2021).
[13] J. E. Greene, J. Strader, and L. C. Ho, Intermediate-mass black [31] T. T. Nakamura, Gravitational lensing of gravitational waves
holes, arXiv preprint arXiv:1911.09678 (2019). from inspiraling binaries by a point mass lens, Physical review
[14] R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley, letters 80, 1138 (1998).
C. Adams, R. Adhikari, V. Adya, C. Affeldt, M. Agathos, et al., [32] T. T. Nakamura and S. Deguchi, Wave optics in gravitational
Gw190521: A binary black hole merger with a total mass of lensing, Progress of Theoretical Physics Supplement 133, 137
150 m , Physical review letters 125, 101102 (2020). (1999).
[15] R. Takahashi and T. Nakamura, Wave effects in the gravitational [33] N. Aghanim, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont,
lensing of gravitational waves from chirping binaries, The As- C. Baccigalupi, M. Ballardini, A. J. Banday, R. Barreiro,
trophysical Journal 595, 1039 (2003). N. Bartolo, et al., Planck 2018 results-i. overview and the cos-
[16] K.-H. Lai, O. A. Hannuksela, A. Herrera-Martı́n, J. M. Diego, mological legacy of planck, Astronomy & Astrophysics 641,
T. Broadhurst, and T. G. Li, Discovering intermediate-mass A1 (2020).
black hole lenses through gravitational wave lensing, Physical [34] E. L. Turner, J. P. Ostriker, and J. R. Gott III, The statistics
Review D 98, 083005 (2018). of gravitational lenses-the distributions of image angular sep-
[17] J. M. Diego, Constraining the abundance of primordial black arations and lens redshifts, The Astrophysical Journal 284, 1
holes with gravitational lensing of gravitational waves at ligo (1984).
frequencies, Physical Review D 101, 123512 (2020). [35] G. Ashton, M. Hübner, P. D. Lasky, C. Talbot, K. Ackley,
[18] O. Hannuksela, K. Haris, K. Ng, S. Kumar, A. Mehta, D. Keitel, S. Biscoveanu, Q. Chu, A. Divakarla, P. J. Easter, B. Goncharov,
T. Li, and P. Ajith, Search for gravitational lensing signatures in et al., Bilby: A user-friendly bayesian inference library for
ligo-virgo binary black hole events, The Astrophysical Journal gravitational-wave astronomy, The Astrophysical Journal Sup-
Letters 874, L2 (2019). plement Series 241, 27 (2019).
[19] X. Liu, I. M. Hernandez, and J. Creighton, Identifying strong [36] J. S. Speagle, dynesty: a dynamic nested sampling package for
gravitational-wave lensing during the second observing run of estimating bayesian posteriors and evidences, Monthly Notices
advanced ligo and advanced virgo, The Astrophysical Journal of the Royal Astronomical Society 493, 3132 (2020).
908, 97 (2021). [37] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pürrer, X. J.
[20] C. McIsaac, D. Keitel, T. Collett, I. Harry, S. Mozzon, O. Edy, Forteza, and A. Bohé, Frequency-domain gravitational waves
and D. Bacon, Search for strongly lensed counterpart images of from nonprecessing black-hole binaries. ii. a phenomenologi-
binary black hole mergers in the first two ligo observing runs, cal model for the advanced detector era, Physical Review D 93,
Physical Review D 102, 084031 (2020). 044007 (2016).
[21] R. Abbott, T. D. Abbott, S. Abraham, F. Acernese, K. Ackley, [38] S. Husa, S. Khan, M. Hannam, M. Pürrer, F. Ohme, X. J.
A. Adams, C. Adams, R. Adhikari, V. Adya, C. Affeldt, et al., Forteza, and A. Bohé, Frequency-domain gravitational waves
Search for lensing signatures in the gravitational-wave observa- from nonprecessing black-hole binaries. i. new numerical wave-
tions from the first half of ligo–virgo’s third observing run, The forms and anatomy of the signal, Physical Review D 93, 044006
Astrophysical Journal 923, 14 (2021). (2016).
[22] M. Oguri, Effect of gravitational lensing on the distribution [39] M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Ander-
of gravitational waves from distant binary black hole mergers, sson, K. Arun, F. Barone, B. Barr, M. Barsuglia, M. Beker,
Monthly Notices of the Royal Astronomical Society 480, 3842 et al., The einstein telescope: a third-generation gravitational
(2018). wave observatory, Classical and Quantum Gravity 27, 194002
[23] K. K. Ng, K. W. Wong, T. Broadhurst, and T. G. Li, Precise ligo (2010).
lensing rate predictions for binary black holes, Physical Review [40] B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, K. Ack-
D 97, 023012 (2018). ley, C. Adams, P. Addesso, R. X. Adhikari, V. Adya, C. Af-
[24] S.-S. Li, S. Mao, Y. Zhao, and Y. Lu, Gravitational lensing of feldt, et al., Exploring the sensitivity of next generation grav-
gravitational waves: a statistical perspective, Monthly Notices itational wave detectors, Classical and Quantum Gravity 34,
of the Royal Astronomical Society 476, 2220 (2018). 044001 (2017).
[25] J. Aasi, B. Abbott, R. Abbott, T. Abbott, M. Abernathy, K. Ack- [41] D. Reitze, R. X. Adhikari, S. Ballmer, B. Barish, L. Barsotti,
ley, C. Adams, T. Adams, P. Addesso, R. Adhikari, et al., Ad- G. Billingsley, D. A. Brown, Y. Chen, D. Coyne, R. Eisenstein,
vanced ligo, Classical and quantum gravity 32, 074001 (2015). et al., Cosmic explorer: The us contribution to gravitational-
[26] F. a. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Alleman- wave astronomy beyond ligo, Bulletin of the American Astro-
dou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, nomical Society 51, 35 (2019).
et al., Advanced virgo: a second-generation interferometric [42] T. Regimbau, M. Evans, N. Christensen, E. Katsavounidis,
gravitational wave detector, Classical and Quantum Gravity 32, B. Sathyaprakash, and S. Vitale, Digging deeper: Observing11
primordial gravitational waves below the binary-black-hole- (1996).
produced stochastic background, Physical review letters 118, [48] J. F. Navarro, C. S. Frenk, and S. D. White, A universal density
151105 (2017). profile from hierarchical clustering, The Astrophysical Journal
[43] K. K. Ng, S. Vitale, W. M. Farr, and C. L. Rodriguez, Probing 490, 493 (1997).
multiple populations of compact binaries with third-generation [49] E. Seo, O. A. Hannuksela, and T. G. Li, Strong lensing: A mag-
gravitational-wave detectors, arXiv preprint arXiv:2012.09876 nifying glass to detect gravitational-wave microlensing, arXiv
(2020). preprint arXiv:2110.03308 (2021).
[44] S. Basak, A. Ganguly, K. Haris, S. Kapadia, A. Mehta, and [50] M. Vallisneri, J. Kanner, R. Williams, A. Weinstein, and
P. Ajith, Constraints on compact dark matter from gravitational B. Stephens, The ligo open science center, in Journal of
wave microlensing, arXiv preprint arXiv:2109.06456 (2021). Physics: Conference Series, Vol. 610 (IOP Publishing, 2015)
[45] Y. Wang, R. K. Lo, A. K. Li, and Y. Chen, Identifying type ii p. 012021.
strongly lensed gravitational-wave images in third-generation [51] R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ack-
gravitational-wave detectors, Physical Review D 103, 104055 ley, A. Adams, C. Adams, R. Adhikari, V. Adya, C. Affeldt,
(2021). et al., Population properties of compact objects from the second
[46] J. F. Navarro, C. S. Frenk, and S. D. White, Simulations of x-ray ligo-virgo gravitational-wave transient catalog, arXiv preprint
clusters, Monthly Notices of the Royal Astronomical Society arXiv:2010.14533 (2020).
275, 720 (1995). [52] K. Belczynski, D. E. Holz, T. Bulik, and R. O’Shaughnessy,
[47] J. F. Navarro, C. S. Frenk, and S. D. M. White, The structure The first gravitational-wave source from the isolated evolu-
of cold dark matter halos, The Astrophysical Journal 462, 563 tion of two 40-100 Msun stars, Nature 534, 512 (2016),
arXiv:1602.04531 [astro-ph.HE].You can also read
