Measurability of neutron star tidal deformability from merging neutron star-black hole binaries Hee-Suk Cho Department of Physics Pusan National University Busan 46241 Korea and

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Measurability of neutron star tidal deformability from merging neutron star-black hole binaries

Hee-Suk Cho∗

Department of Physics, Pusan National University, Busan, 46241, Korea and

Extreme Physics Institute, Pusan National University Busan, 46241, Korea

(Dated: October 7, 2022)

The neutron star-black hole binary (NSBH) system has been considered one of the promising detection candi-

dates for ground-based gravitational-wave (GW) detectors such as LIGO and Virgo. The tidal effects of neutron

stars (NSs) are imprinted on the GW signals emitted from NSBHs as well as binary neutron stars. The NS

tidal deformability (λNS) was successfully measured by the binary neutron star signal GW170817 but could

not be constrained in the analysis of the two NSBH signals GW200105 and GW200115 due to the low signal-

to-noise ratio. In this work, we study how accurately the parameter λNS can be measured in GW parameter

estimation for NSBH signals. We set the parameter range for the NSBH sources to [4M,10M]for the black

hole mass, [1M,2M]for the NS mass, and [−0.9,0.9] for the dimensionless black hole spin. For realistic

populations of sources distributed in different parameter spaces, we calculate the measurement errors of λNS

(σλNS ) using the Fisher matrix method. In particular, we perform a single-detector analysis using the advanced

LIGO and the Cosmic Explorer detectors and a multi-detector analysis using the 2G (advanced LIGO-Hanford,

advanced LIGO-Livingstone, advanced Virgo, and KAGRA) and the 3G (Einstein Telescope and Cosmic Ex-

plorer) networks. We show the distribution of σλNS for the population of sources as a one-dimensional probabil-

ity density function. Our result shows that the probability density function curves are similar in shape between

advanced LIGO and Cosmic Explorer, but Cosmic Explorer can achieve ∼15 times better accuracy overall in

the measurement of λNS. In the case of the network detectors, the probability density functions are maximum

at σλNS ∼130 and ∼4for the 2G and the 3G networks, respectively, and the 3G network can achieve ∼10

times better accuracy overall. Speciﬁcally, we investigate the distribution of σλNS for 103Monte Carlo sources

in our parameter range with the NS mass ﬁxed to m2= 1.4M, and the result shows that if the sources are

located at dL'100Mpc, the parameter estimation results for ∼80% of the sources can distinguish between

the theoretical EOS models at the 1–σlevel, using the 3G network. Additionally, we demonstrate that our PDF

results are almost unaffected by different choices of the true value of λNS.

I. INTRODUCTION

Since the ﬁrst gravitational-wave (GW) signal was detected

in 2015 [1], the network of the two advanced LIGO (aLIGO)

[2] and advanced Virgo [3] detectors has observed 90 GW

candidates [4–7] through three observing runs. All GW sig-

nals were emitted from a compact binary coalescence (CBC)

system such as binary black hole (BBH), binary neutron star

(BNS), and neutron star-black hole binary (NSBH). Most GW

sources originated from BBHs, and various BH masses and

spins were measured from these signals. The sources of the

two signals GW170817 [8, 9] and GW190425 [10] were iden-

tiﬁed as BNSs, and GW170817 enabled us to directly measure

the NS tidal deformability for the ﬁrst time through observa-

tion means. In particular, since GW170817 had a high signal-

to-noise ratio (SNR) ∼32, it could be inferred that the soft

equation-of-state (EOS) model is preferred over the stiff EOS

model [9]. In parameter estimation for BNS signals, a well-

constrained tidal parameter is the effective tidal deformabil-

ity rather than the component tidal deformability. The effec-

tive tidal deformability is deﬁned by the combination of the

masses (mi) and the component tidal parameters (λi). Since

λ1and λ2are generally strongly correlated, their measure-

ment errors can be large even though the effective tidal param-

eter is well constrained as shown in the result of GW170817

[9].

∗chohs1439@pusan.ac.kr

On the other hand, the two NSBH signals GW200105 and

GW200115 were also captured by the LIGO-Virgo network

during the third observing run [11] (for a brief overview of

NSBH mergers, refer to [12]). Since the GWs from NSBHs

also contain an NS tidal effect, information on the tidal param-

eter can be extracted from those signals. However, the con-

tribution of tidal deformability to the waveform of the NSBH

system is relatively small compared to that of the BNS system,

especially when the BH mass is much larger than the NS mass.

Therefore, a sufﬁciently high SNR is required to measure the

tidal deformability from the NSBH signals. Unfortunately, the

observed NSBH signals were not able to constrain the tidal

deformability of the NSs well due to their low SNRs. Mean-

while, a large advantage of the NSBH signals when measuring

the tidal parameter is that the individual NS tidal deforma-

bility rather than the effective tidal deformability can be ob-

tained directly through parameter estimation because the tidal

deformability of BH is zero.

The purpose of this work is to investigate how accurately

the NS tidal deformability (λNS) can be measured from NSBH

signals. To this end, we utilize the Fisher matrix method im-

plemented in the python package GWBench [13] and calcu-

late the measurement errors of λNS for realistic populations of

NSBH sources. Several parameter estimation studies on the

measurability of the NS tidal deformability have been done

using Bayesian analysis with stochastic sampling on NSBH

systems as well as Fisher Matrix studies. Lackey et al.[14]

estimated tidal deformability from NSBH systems with non-

spinning BHs by applying the Fisher matrix method to the

hybrid waveforms based on BBH waveforms calibrated to

arXiv:2210.02610v1 [gr-qc] 6 Oct 2022

NSBH numerical simulations, and they extended it to the spin-

ning NSBH systems in subsequent work [15]. In the former,

they showed that for a single aLIGO detector, the tidal param-

eter λNS can be extracted to 10–40% accuracy from single

events for mass ratios of q= 2 and 3at a distance of 100Mpc,

and in the latter, for q= 2–5, BH spins χBH =−0.5–

0.75, NS masses mNS = 1.2M–1.45M, and a distance of

100Mpc, a single aLIGO detector can measure λNS to a 1–σ

uncertainty of ∼10–100%. For both works, they showed that

the uncertainty in λNS is an order of magnitude smaller for the

3G detector Einstein Telescope. On the other hand, Kumar et

al.[16] performed a Bayesian analysis of NSBH systems with

spinning BHs to study the measurability of λNS with aLIGO

and found that 20–35 events can constrain λNS within 25–

50%, depending on the EOS.

This paper is organized as follows. In Sec. II, we intro-

duce various waveform models recently developed for use in

GW data analysis for BNS or NSBH systems based on the

Effective-One-Body (EOB) formalism and the phenomeno-

logical ﬁt approach. Next, we give a brief overview of the

Bayesian parameter estimation and the Fisher matrix ap-

proach in terms of parameter measurement accuracy and list

the 2G and the 3G detectors used in our analysis. The re-

sults are given in Sec. III. First, we compare the parame-

ter measurement errors obtained from the Fisher matrix ap-

proach with those obtained from the Bayesian parameter esti-

mation simulations and verify the reliability of the Fisher ma-

trix method in our analysis. Next, we investigate the suitability

of four recent waveform models for applying the Fisher matrix

method to NSBH sources in our parameter range and adopt

a representative waveform model (denoted by SEOBNR T

in this work). Then, we apply the Fisher matrix method to

SEOBNR T and calculate the measurement errors of λNS for

our NSBH sources. We perform a single-detector analysis as

well as a multi-detector analysis and provide comparisons be-

tween the 2G and the 3G detectors. In particular, using the

results for the 3G network, we present a speciﬁc example de-

scribing how well the theoretical EOS models can be con-

strained by the NSBH signals. In Sec. IV, we summarize our

results and provide some discussion.

II. METHOD

A. Waveform models

To simulate an NSBH signal, the waveform model requires

the full Inspiral-Merger-Ringdown (IMR) expression, includ-

ing the NS tidal effect. To date, various IMR waveform mod-

els have been developed and implemented in LAL (LIGO Al-

gorithm Library [17]) for use in GW data analysis. Those

models are roughly classiﬁed into two waveform families ac-

cording to the construction formalism. One is based on the

Effective-One-Body (EOB) formalism and the other is based

on the phenomenological ﬁt approach. In both approaches, the

waveform from the late inspiral to the merger-ringdown is cal-

ibrated against the aligned-spin BBH NR waveforms. So they

are represented by the “SEOBNR” and “IMRPhenom” mod-

Full name (implemented in LAL) References

Short label (used in this work) Base model Corrections

SEOBNRv4 ROM NRTidalv2 [18–20] [21, 22]

SEOBNR T

SEOBNRv4 ROM NRTidalv2 NSBH [23] [18–20] [21, 22, 24]

SEOBNR NSBH

IMRPhenomPv2 NRTidalv2 [25–27] [21, 22]

IMRPhenomP T

IMRPhenomNSBH [28] [26, 27, 29] [22, 24]

IMRPhenom NSBH

TABLE I: Waveform models used in our analysis for NSBH

systems.

els.

The EOB formalism is basically constructed in the time

domain. For computational efﬁciency, the frequency-domain

model SEOBNRv4 ROM [18, 19] has been developed based

on the time-domain model SEOBNRv4 using a reduced-

order-quadrature rule [20]. SEOBNRv4 ROM NRTidalv2

builds on SEOBNRv4 ROM by adding tidal correction terms

that are constructed from high-resolution BNS NR simu-

lations [21, 22]. SEOBNRv4 ROM NRTidalv2 NSBH [23]

was built from SEOBNRv4 ROM NRTidalv2 to generate

aligned-spin NSBH waveforms by adding corrections to the

wave amplitude [24].

Meanwhile, the IMRPhenom models are deﬁned in the fre-

quency domain. Early versions of the IMRPhenom models

were developed for the BBH system. IMRPhenomPv2 has

been mainly used for CBC analyses in recent years. This

model is based on the precessing-spin model IMRPhenomP

[25] and the aligned-spin model IMRPhenomD [26, 27]. IMR-

PhenomPv2 NRTidalv2 is based on IMRPhenomPv2 and in-

cludes the same tidal correction terms [21, 22] as in SEOB-

NRv4 ROM NRTidalv2. The IMRPhenom family also has

an NSBH model IMRPhenomNSBH [28]. This model is

based on the amplitude of IMRPhenomC [29] and the phase

of IMRPhenomD [26, 27], and incorporates NS tidal ef-

fects [22] and amplitude corrections [24] similar to SEOB-

NRv4 ROM NRTidalv2 NSBH.

In this work, four recent IMR waveform models contain-

ing NS tidal effects are considered, which are listed in Table

I. Finally, we note that the TaylorF2 model also includes NS

tidal effects but generates inspiral-only waveforms. TaylorF2

is reliable in parameter estimation for BNS systems such as

GW170817 [9] and GW100425 [10]. However, after some

consistency tests, we have veriﬁed that TaylorF2 is not suit-

able for the analysis of our NSBH sources.

B. Bayesian parameter estimation

The physical properties of the GW source can be mea-

sured in the parameter estimation procedure [30]. This process

is based on Bayesian inference statistics, and the algorithm

explores the entire parameter space, computing the overlaps

between model waveforms and detector data. The result of

Bayesian parameter estimation can be given as the posterior

probability density functions (PDFs) of the parameters con-

sidered. Given the detector data xcontaining the GW signal s

and noise n, the overlap between xand the model waveform

his deﬁned as

hx|hi= 4Re Z∞

˜x(f)˜

h∗(f)

Sn(f)df, (1)

where the tilde denotes the Fourier transform of the time-

domain waveform, Sn(f)is the detector’s noise power spec-

tral density (PSD). For efﬁciency, the integration is performed

in the frequency range [fmin, fmax], and the choice of these

frequencies depends on the PSD curve.

The Bayesian posterior probability that the GW signal s

contained in the data xis characterized by the parameters θ,

where θis the set of parameters considered in the analysis, can

be given by the prior p(θ)and the likelihood L(x|θ)as

p(θ|x)∝p(θ)L(x|θ).(2)

The likelihood is given as [31, 32]

L(x|θ)∝exp−1

2hx−h(θ)|x−h(θ)i(3)

= exp−1

2hs+n−h(θ)|s+n−h(θ)i.(4)

(5)

If the signal is strong enough (i.e., high SNR limit), the noise

can be removed from the above equation, giving the likelihood

function as

L(θ)∝exp−1

2{hs|si+hh(θ)|h(θ)i − 2hs|h(θ)i}.(6)

Employing the deﬁnition of SNR ρ=phs|si[33], the above

equation can be re-written as

L(θ)∝exp[−ρ2{1− hˆs|ˆ

h(θ)i}],(7)

where ˆ

h≡h/ρ, and we assume that the waveform model can

describe the signal waveform almost exactly, i.e., h(θ0)'

s(where θ0represents the true parameter values), hence

hh(θ0)|h(θ0)i ' hs|si=ρ2. In the above equation, the term

hˆs|ˆ

h(θ)irepresents the distribution of the normalized overlaps

between the signal and the model waveforms, and this over-

lap distribution is maximum at θ=θ0. Therefore, the shape

of the likelihood surface can be given by the overlap distri-

bution, and its scale of interest depends on the SNR [34]. At

small scales (i.e., high SNRs), the shape of the overlap dis-

tribution is nearly quadratic around the maximum position.

If we assume a ﬂat prior in Eq. 2, the posterior distribution is

equivalent to the likelihood distribution. Therefore, in the high

SNR limit, the posterior PDF follows a multivariate Gaussian

distribution centered around the position of the true parameter

values.

C. Fisher matrix

If a multivariate Gaussian function is represented by

f(x) = exp(−Σ−1

ij xixj/2),(8)

Σij corresponds to the covariance matrix, and its inverse ma-

trix represents the Fisher matrix (Γij ). Therefore, given the

Gaussian function above, the Fisher matrix can be obtained

Γij =−∂2ln f(x)

∂xi∂xj

.(9)

Analogously, since the likelihood in Eq. 7 follows a multivari-

ate Gaussian distribution, the corresponding Fisher matrix can

be given by

Γij =−∂2lnL(θ)

∂θi∂θj



θ=θ0

=−ρ2∂2hˆs|ˆ

h(θ)i

∂θi∂θj



θ=θ0

.(10)

Thus, the Fisher matrix describes the curvature of the log-

likelihood or the overlap surface at the position of the true pa-

rameter values, Furthermore, the second equality means that a

speciﬁc iso-match contour in the overlap surface corresponds

to a speciﬁc conﬁdence region of the likelihood distribution

for a given SNR [35–37]. Using the relation ˆ

h≡h/ρ, the

above formula can equivalently be written as [38–40]

Γij =−ρ2∂2hˆs|ˆ

h(θ)i

∂θi∂θj



θ=θ0

=∂h(θ)

∂θi



∂h(θ)

∂θj



θ=θ0

(11)

The last term is the familiar expression of the Fisher matrix.

In a multivariate Gaussian distribution, the measurement error

(σi) and the correlation coefﬁcient (Cij ) can be obtained from

the Fisher matrix as

σi=p(Γ−1)ii, Cij =(Γ−1)ij

p(Γ−1)ii(Γ−1)jj

.(12)

Since the Fisher matrix has a simple functional form and is

easy to apply to analytical waveform models, this approach

has been mainly used in many past works since it was in-

troduced in [32, 41]. However, the Fisher matrix has some

well-known limitations (for detailed reviews, refer to [39]).

First of all, the Fisher matrix is only reliable at high SNRs be-

cause it is derived with a high SNR assumption as described

above. In addition, since the computation of the Fisher matrix

is entirely dependent on the waveform model h(θ)as in Eq.

11, the results highly rely on the accuracy of the model used.

Some issues induced by using the inspiral-only waveform

model TaylorF2 have been thoroughly studied in past works

[36, 42, 43]. Recently, Harry and Lundgren [44] pointed out

that the TaylorF2-applied Fisher matrix is unsuitable to pre-

dict the match between two BNS waveforms when including

tidal terms. Another well-known limitation of the Fisher ma-

trix is the poor applicability of prior information. Bayesian pa-

rameter estimation allows for all forms of prior information,

while only Gaussian prior functions can be applied analyti-

cally to the Fisher matrix method [32, 41]. Cho [45] showed

that the measurement error of the intrinsic parameters can be

reduced to ∼70% of the original priorless error (σpriorless

θ)

if the standard deviation of the Gaussian prior is similar to

σpriorless

θ, and thus the prior effect can be ignored at sufﬁ-

ciently high SNRs. Therefore, we choose the IMR waveform

model and the high SNR in our analysis to avoid the above

limitations.

To describe the waveforms of an aligned-spin NSBH sys-

tem, ﬁve extrinsic parameters (true distance dL, orbital in-

clination θJN , polarization angle Ψ, and sky position angles

RA, DEC), ﬁve intrinsic parameters (two masses m1, m2, di-

mensionless spins χBH, χNS, and dimensionless NS tidal de-

formability λNS), and two arbitrary constants (coalescence

time tcand coalescence phase φc) are required. Since the NS

mass is much smaller than the BH mass and the NS spins

observed from BNS systems are very small (χNS <

∼0.05)

[46, 47], the NS spin has a negligible contribution to the wave

phase and thus has no effect on our analysis. Therefore, for

simplicity, we assume the NS spin to be zero and consider

only the BH spin (χBH) in this work. In addition, we adopt the

soft EOS model APR4 [48], which is one of the most preferred

models in the parameter estimation results for GW170817 [9],

to choose the true value of λNS.

A GW waveform can be described as

h(f) = A(f)eiψ(f).(13)

The signal strength (i.e., SNR) is entirely governed by the

wave amplitude (A), and the amplitude is given by the ex-

trinsic parameters and the chirp mass (Mc≡(m1+m2)η3/5,

where η≡(m1m2)/(m1+m2)2is the symmetric mass ra-

tio). The wave phase ψ(f)is only a function of the intrinsic

parameters and tcand φc. The true values of tcand φccan

be arbitrarily selected, and their choice does not affect the

measurement accuracy of other parameters. However, since

these two parameters are strongly correlated with the intrin-

sic parameters, they must be considered variables when con-

structing the Fisher matrix (e.g., see Table A1 of [45]). On the

other hand, the extrinsic parameters are strongly correlated

with each other but weakly correlated with the intrinsic pa-

rameters. Thus, when focusing on the intrinsic parameters, it

is very efﬁcient to use a single effective parameter that rep-

resents the ﬁve extrinsic parameters, and we use the param-

eter deﬀ (effective distance [33]) in this work. At leading or-

der, the wave amplitude can be given by A∝M5/6

c/deﬀ . For

ﬁxed deﬀ , the measurement errors of the intrinsic parameters

are independent of the choice of the individual extrinsic pa-

rameters, so the extrinsic parameters are not considered vari-

ables in our Fisher matrix. Therefore, in this work, the Fisher

matrix can be given by a 6×6matrix with the components

{Mc, η, χBH, λNS, tc, φc}.

D. Detectors

We consider the four 2G GW detectors, aLIGO-Hanford

(H) and Livingstone (L) [2], advanced Virgo (V) [3], and KA-

1 5 10 50 100 500 1000

10-25

10-24

10-23

10-22

10-21

10-20

frequency

[

]

ASD [Hz-1/2]

1 5 10 50 100 500 1000

10-25

10-24

10-23

10-22

10-21

10-20

frequency [Hz]

ASD [Hz-1/2]

aLIGO

aVirgo

KAGRA

(H & L)

(V)

(K)

Cosmic Explorer (CE)

Einstein Telescope (ET)

1 5 10 50 100 500 1000

10-25

10-24

10-23

10-22

10-21

10-20

frequency [Hz]

ASD [Hz-1/2]

FIG. 1: Amplitude spectral densities (ASDs) pSn(f)of the

2G and 3G detectors used in this work. The frequency range

is set to fmin = 10(5) Hz for the 2(3)G detectors and

fmax = 2048 Hz for all detectors.

GRA (K) [49], and the two 3G detectors, Cosmic Explorer

(CE) [50] and Einstein Telescope (ET) [51]. The sensitivity

curves of the detectors are shown in Fig. 1. These PSDs are

available in GWBench [13], labeled aLIGO (H & L), V+

(V), K+ (K), ET (ET), and CE1-40-CBO (CE). We assume

fmin = 10 and 5 Hz for the 2G and the 3G detectors, respec-

tively, and fmax = 2048 Hz for all detectors. The locations

(longitude and latitude) and the orientations (orientation of the

y-arm with respect to due East) of the detectors are summa-

rized in Table III and Fig. 4 of [13]. Note that H and L have

the same PSD curve but their locations and orientations are

different. CE (Idaho, USA) is located at a site similar to H

(Washington, USA) but has a different orientation. ET is set

to the same coordinates as V (Cascina, Italy) and consists of

three V-shaped detectors, ET1, ET2, and ET3, that form an

equilateral triangle, and one of them has the same orientation

as V.

III. RESULT

A. Comparison between Bayesian parameter estimation and

Fisher matrix

We assume our ﬁducial NSBH source with the true values

{m1, m2, χBH, λNS, tc, φc}={5M,1.4M,0,251,0,0}.

Here, the true value of λNS is determined by the NS mass

(m2) according to the APR4 EOS model. We inject the ﬁdu-

cial NSBH signal into the aLIGO PSD1and perform Bayesian

1For our ﬁducial binary system, the time to merger from fmin = 5Hz

is about 2400 seconds, which is about 6.5 times longer than that from

fmin = 10Hz, so a much longer time is required to run parameter estima-

tion for 3G detectors. Moreover, the reliability of the Fisher matrix method

is almost independent of PSD. Therefore, in this work, for comparison with

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Measurabilityofneutronstartidaldeformabilityfrommergingneutronstar-blackholebinariesHee-SukChoDepartmentofPhysics,PusanNationalUniversity,Busan,46241,KoreaandExtremePhysicsInstitute,PusanNationalUniversityBusan,46241,Korea(Dated:October7,2022)Theneutronstar-blackholebinary(NSBH)systemhasbeenconside...

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Measurability of neutron star tidal deformability from merging neutron star-black hole binaries Hee-Suk Cho Department of Physics Pusan National University Busan 46241 Korea and

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