# Improvement of the parameter measurement accuracy by the third-generation
gravitational wave detector Einstein Telescope

Hee-Suk Cho<EMAIL_ADDRESS>Department of Physics, Pusan National
University
Busan, 46241, Korea
Extreme Physics Institute, Pusan National University
Busan, 46241, Korea

###### Abstract

The Einstein Telescope (ET) has been proposed as one of the third-generation
gravitational wave (GW) detectors. The sensitivity of ET would be a factor of
10 better than the second-generation GW detector, Advanced LIGO (aLIGO); thus,
the GW source parameters could be measured with much better accuracy. In this
work, we show how the precision in parameter estimation can be improved
between aLIGO and ET by comparing the measurement errors. We apply the
TaylorF2 waveform model defined in the frequency domain to the Fisher matrix
method which is a semi-analytic approach for estimating GW parameter
measurement errors. We adopt as our sources low-mass binary black holes with
the total masses of $M\leq 16M_{\odot}$ and the effective spins of
$-0.9\leq\chi_{\rm eff}\leq 0.9$ and calculate the measurement errors of the
mass and the spin parameters using $10^{4}$ Monte-Carlo samples randomly
distributed in our mass and spin parameter space. We find that for the same
sources ET can achieve $\sim 14$ times better signal-to-noise ratio than aLIGO
and the error ratios ($\sigma_{\lambda,\rm ET}/\sigma_{\lambda,\rm aLIGO}$)
for the chirp-mass, symmetric mass ratio, and effective spin parameters can be
lower than $7\%$ for all binaries. We also consider the equal-mass binary
neutron stars with the component masses of 1, 1.4, and 2 $M_{\odot}$ and find
that the error ratios for the mass and the spin parameters can be lower than
$1.5\%$. In particular, the measurement error of the tidal deformability
$\tilde{\Lambda}$ can also be significantly reduced by ET, with the error
ratio of $3.6-6.1\%$. We investigate the effect of prior information by
applying the Gaussian prior on the coalescence phase $\phi_{c}$ to the Fisher
matrix and find that the error of the intrinsic parameters can be reduced to
$\sim 70\%$ of the original priorless error ($\sigma_{\lambda}^{\rm
priorless}$) if the standard deviation of the prior is similar to
$\sigma_{\phi_{c}}^{\rm priorless}$.

Keywords: gravitational waves; parameter estimation; binary black holes;
binary neutron stars; Einstein Telescope

##  1 Introduction

In the past few years, a new window to see the universe has been opened
through the successful observations of the gravitational waves (GWs) by the
ground-based GW detectors, Advanced LIGO (aLIGO) [1] and Advanced Virgo [2].
So far, LIGO and Virgo have conducted a few times science observations
successfully and they are currently preparing the next operation with upgraded
detector performance. The source of the first detected GW signal, named
GW150914, was a merging binary black hole (BBH) with the masses of $\sim
36M_{\odot}$ and $\sim 29M_{\odot}$ [3]. GW150914 was not only the first
direct detection of GWs but also the first observation of a BBH merger. This
demonstrated the existence of stellar-mass black holes more massive than $\sim
25M_{\odot}$. On the other hand, a GW signal whose origin was a merging binary
neutron star (BNS) was detected on 17 August 2017; thus, named GW170817. [4].
At the same time, the detection of the gamma-ray burst GRB170817A and the
electromagnetic follow-up were made successfully with respect to the same
source of GW170817 [5, 6]. Consequently, this event could begin a new era of
multi-messenger astronomy.

The efficiency of GW searches mainly depends on the detector sensitivity, so
the GW observatories have continuously upgraded the instrument performance.
The first-generation GW detectors, initial LIGO and initial Virgo, were able
to observe the GWs only in the frequency range from $40$ Hz to a few hundred
Hz, and did not detect any GW signals. The second-generation GW detectors,
aLIGO and Advanced Virgo, could lower the low-frequency limit down to
$10$–$20$ Hz and consequently have detected GW signals successfully. About 50
GW signals have been observed so far, all of them were originated from merging
compact binaries consisting of a black hole and/or a neutron star [7, 8].

After detection of a GW signal, the parameters of the source, such as the
mass, the spin, the distance, the sky position, etc., can be measured by the
parameter estimation analysis. The measurement accuracy of each parameter
depends on how the parameter contributes to the description of the binary
motion and consequently, to the shape of the waveforms. The leading and the
next-to-leading contributions to the waveforms are the mass and the spin
parameters, respectively; thus, those can be mainly measurable in GW parameter
estimation. For the case GW150914, the $90\%$ symmetric credible intervals are
10–20 % for the component masses and 0.14–0.15 for the effective spin
$\chi_{\rm eff}$ [9, 10]. On the other hand, the neutron star tidal effect can
be incorporated in the post-Newtonian (PN) waveforms. However, since the tidal
correction terms appear from the 5PN order, its effect is relatively very
small compared to that of the mass or the spin. For the case GW170817,
assuming the low-spin prior ($|\chi_{i}|\leq 0.05$), the one-sided $90\%$
lower or upper limits for the component masses are $(1.36,1.60)M_{\odot}$ and
$(1.16,1.36)M_{\odot}$, the effective spin is $0.00^{+0.02}_{-0.01}$ (where
$x^{+z}_{-y}$ represent the median, $5\%$ lower limit, and $95\%$ upper
limit), and the $90\%$ symmetric credible intervals for the tidal
deformability $\tilde{\Lambda}$ is $300^{+500}_{-190}$ [11]. Although the
tidal deformability was poorly measured, that result could roughly constrain
various theoretical models on the neutron star equation of state [11, 12].

The Einstein Telescope (ET) is conceived to be one of the third-generation GW
detectors. ET will achieve a greatly improved sensitivity by increasing the
size of the interferometer from the 3km arm length of the Virgo detector to
10km, and by implementing a series of new technologies [13]. The sensitivity
of ET would be a factor of 10 better than the second-generation detectors and
it could also extend the low-frequency limit down to 1–3 Hz. Thus, it is
expected that the measurement accuracy of GW source parameters could be much
improved. In particular, due to the wide frequency band, ET could observe the
BNS inspiral phase and the onset of tidal effects with a high signal-to-noise
ratio (SNR). Figure 1 shows the detector sensitivity curves for ET and aLIGO.
We consider the zero-detuned, high-power noise power-spectral-density (PSD)
for aLIGO [14] and one of the design noise PSD curves for ET [15]. Here, the
ET curve is obtained by using the analytical fit to the ET-B noise PSD given
by [16]

$S_{n}(f)=10^{-50}(2.39\times
10^{-27}x^{-15.64}+0.349x^{-2.145}+1.76x^{-0.12}+0.409x^{1.10}),$ (1)

where $x=f/100$ Hz.

Figure 1: Detector sensitivity curves for aLIGO and ET and amplitudes $|h|$ of
the inspiral-merger-ringdown waveforms for BBHs with the total masses of
$6,15$ and $30M_{\odot}$ calculated by using the inspiral-merger-ringdown
waveform model (IMRPhenomD) [17]. The effective distance is 400 Mpc. The
vertical line of each waveform indicates the frequency of the innermost stable
circular orbit.

We use the TaylorF2 waveform model which is an analytic function expressed in
the frequency domain. This model is the most simple waveform model and has
been widely used in GW data analysis studies. Typically, the TaylorF2 waveform
terminates at the frequency of the innermost stable circular orbit (ISCO) and
describes only the inspiral part of the inspiral-merger-ringdown waveforms for
BBHs. As the binary mass decreases, the ISCO frequency becomes higher,
consequently, the merger-ringdown contribution to the wave phasing becomes
smaller. Therefore, TaylorF2 is suitable only for a sufficiently low-mass
binary system in which the merger-ringdown contribution is negligible compared
to the inspiral contribution. In figure 1, for example, we depict the
amplitude spectra of the waveforms as well as the ISCO frequencies for BBHs
with masses of $6,15,$ and $30M_{\odot}$. Several authors have investigated
the effectualness of the inspiral-only template models with respect to the
inspiral-merger-ringdown signals [18, 19, 20, 21, 22, 23]. They pointed out
that the total mass should be lower than $\sim 15M_{\odot}$ to achieve the
fitting factors over $\sim 0.97$, which corresponds to a loss of detection
rates of $\sim 10\%$.

We calculate the measurement errors of the source parameters using both the
aLIGO and the ET detector sensitivities and show how the measurement accuracy
can be improved with ET. We adopt the Fisher matrix method which is a semi-
analytic approach for estimating the measurement errors of the GW source
parameters. The Fisher matrix approximation may not represent the results of a
full parameter estimation analysis due to several well-known limitations. For
example, since the Fisher matrix is constructed assuming a high SNR, it is
only valid at high SNR limits. In addition, there is no simple way to
incorporate non-Gaussian priors into the Fisher matrix posterior function,
especially if the parameter being considered has physical boundaries, such as
the spin parameter ($|\chi_{i}|\leq 1$). The authors of [24] have also found
that the disagreement between the Fisher Matrix method and the Markov-chain
Monte Carlo (MCMC) simulations increases with total mass. Later, the authours
of [25] and [26] have found that such disagreement is more likely to occur
with higher-mass binaries, where the inspiral-only waveform abruptly
terminates within the detector frequency band. This is also a major cause of
the loss of fitting factor in search analysis as discussed above. In this
work, therefore, we only consider the BNSs and the low-mass BBHs with the
total masses of $M\leq 16M_{\odot}$. We use the geometrized unit $G=c=1$.

##  2 Waveform model : TaylorF2

In the GW data analysis, TaylorF2 is the most commonly used waveform model
because this model is expressed simply by the post-Newtonian (pN) series
expansions in closed form [27, 28, 29]. The TaylorF2 waveform can be obtained
from the time-domain waveform via the stationary phase approximation and the
wave function can be expressed in the frequency domain as

$h(f)={M_{\rm c}^{5/6}\over\pi^{2/3}D_{\rm eff}}\sqrt{{5\over
24}}f^{-7/6}e^{i\Psi(f)},$ (2)

where $M_{\rm c}=\eta^{3/5}M$ (where $M=m_{1}+m_{2}$ is the total mass of the
binary and $\eta=m_{1}m_{2}/M^{2}$ is the symmetric mass ratio) is the chirp
mass and $D_{\rm eff}$ is the effective distance of the source. The effective
distance is determined by the extrinsic parameters, those are the true
distance of the source and several geometrical factors that relate the source
orientation to the detector orientation [30]. In a single-detector analysis,
the only well-constrained extrinsic factor is $D_{\rm eff}$, if we include
$D_{\rm eff}$ in the Fisher matrix, we can also calculate the error of $D_{\rm
eff}$. However, the correlations between $D_{\rm eff}$ and the intrinsic
parameters are negligible, so the inclusion of $D_{\rm eff}$ does not change
the measurement errors of the intrinsic parameters (see, table 3). Therefore,
we do not take into account the extrinsic parameters in our analysis by
choosing a fixed effective distance. We show the effect of the choice of
various extrinsic parameters on the measurement errors of the intrinsic
parameters in A. In addition, we only consider an aligned-spin binary system,
in which the binary does not precess.

The wave phase of TaylorF2 is given as

$\Psi(f)=2\pi ft_{c}-2\phi_{c}-{\pi\over 4}+{3\over 128\eta v^{5}}[\phi_{\rm
3.5pN}(f)+\phi^{\rm Tidal}(f)],$ (3)

where $v=(\pi fM)^{1/3}$ is the post-Newtonian expansion parameter, $t_{c}$
and $\phi_{c}$ are the coalescence time and the phase at coalescence instant.
The 3.5PN order phase equation is expressed as [31]

$\displaystyle\phi_{\rm 3.5pN}=1$ $\displaystyle+$
$\displaystyle\left(\frac{3715}{756}+\frac{55}{9}\eta\right)v^{2}+(4\beta-16\pi)v^{3}$
(4) $\displaystyle+$
$\displaystyle\left(\frac{15293365}{508032}+\frac{27145}{504}\eta+\frac{3085}{72}\eta^{2}-10\sigma\right)v^{4}$
$\displaystyle+$
$\displaystyle\left(\frac{38645\pi}{756}-\frac{65\pi}{9}\eta-\gamma\right)(1+3\log
v)v^{5}$ $\displaystyle+$
$\displaystyle\bigg{\\{}\frac{11583231236531}{4694215680}-\frac{640}{3}\pi^{2}-\frac{6848\gamma_{\rm
E}}{21}-\frac{6848\log\left(4{v}\right)}{21}$
$\displaystyle+\left(\frac{2255{\pi}^{2}}{12}-\frac{15737765635}{3048192}\right)\eta+\frac{76055}{1728}\eta^{2}-\frac{127825}{1296}\eta^{3}\bigg{\\}}v^{6}$
$\displaystyle+$
$\displaystyle\left(\frac{77096675\pi}{254016}+\frac{378515\pi}{1512}\eta-\frac{74045\pi}{756}\eta^{2}\right)v^{7},$

where $\gamma_{\rm E}=0.577216...$ is the Euler constant, and the terms
$\beta,\sigma,$ and $\gamma$ denote the leading-order spin-orbit coupling,
leading-order spin-spin coupling, and next-to-leading-order spin-orbit
coupling, respectively [32]:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{113}{12}\left(\chi_{s}+\delta\chi_{a}-\frac{76\eta}{113}\chi_{s}\right),$
$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\chi_{a}^{2}\left(\frac{81}{16}-20\eta\right)+\frac{81\chi_{a}\chi_{s}\delta}{8}+\chi_{s}\left(\frac{81}{16}-\frac{\eta}{4}\right),$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\chi_{a}\delta\left(\frac{140\eta}{9}+\frac{732985}{2268}\right)+\chi_{s}\left(\frac{732985}{2268}-\frac{24260\eta}{81}-\frac{340\eta^{2}}{9}\right)$
(5)

where $\delta\equiv(m_{1}-m_{2})/(m_{1}+m_{2})$,
$\chi_{s}\equiv(\chi_{1}+\chi_{2})/2$ and
$\chi_{a}\equiv(\chi_{1}-\chi_{2})/2$ with $\chi_{i}\equiv S_{i}/m_{i}^{2}$,
$S_{i}$ being the spin angular momentum of the $i{\rm th}$ compact object. On
the other hand, it is more efficient to treat the two spin parameters with a
single effective spin parameter $\chi_{\rm eff}$ because the two spins are
strongly correlated in an aligned-spin system [32, 33, 34, 35, 36]. The
effective spin is defined by

$\chi_{\rm eff}={m_{1}\chi_{1}+m_{2}\chi_{2}\over M}.$ (6)

If we choose the same component spin values in the wave function, the value of
$\chi_{\rm eff}$ can be given as $\chi_{\rm eff}=\chi_{1}=\chi_{2}$. We also
use $\\{M_{c},\eta\\}$ as our mass parameters rather than $\\{m_{1},m_{2}\\}$
since those parameters are generally more efficient in the GW data analysis.

In equation (3), $\phi^{\rm Tidal}$ represents the tidal effect induced by the
tidal deformation of neutron stars; thus, this term is not applicable to the
BBH waveforms. The tidal correction term first appears from 5PN order but its
impact on the wave phase is comparable to the 3.5PN order phase term [37]. We
here consider the tidal correction up to 6PN order as [38, 39]

$\phi^{\rm
Tidal}=-\bigg{[}\frac{39\tilde{\Lambda}}{2}v^{10}+\bigg{(}\frac{3115\tilde{\Lambda}}{64}-\frac{6595\sqrt{1-4\eta}\delta\tilde{\Lambda}}{364}\bigg{)}v^{12}\bigg{]},$
(7)

where $\tilde{\Lambda}$ and $\delta\tilde{\Lambda}$ are defined by [38, 39,
40]

$\displaystyle\tilde{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{8}{13}[(\rm
1+7\eta-31\eta^{2})(\Lambda_{1}+\Lambda_{2})+\sqrt{1-4\eta}(1+9\eta-11\eta^{2})(\Lambda_{1}-\Lambda_{2})],$
$\displaystyle\delta\tilde{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\sqrt{1-4\eta}\bigg{(}1-\frac{13272\eta}{1319}+\frac{8944\eta^{2}}{1319}\bigg{)}(\Lambda_{1}+\Lambda_{2})$
(8) $\displaystyle+$
$\displaystyle\bigg{(}1-\frac{15910\eta}{1319}+\frac{32850\eta^{2}}{1319}+\frac{3380\eta^{3}}{1319}\bigg{)}(\Lambda_{1}-\Lambda_{2})\bigg{]},$

where $\rm\Lambda_{i}$ is the dimensionless tidal deformability. For
simplicity, we only consider equal-mass BNSs, then
$\tilde{\Lambda}=\Lambda_{1}=\Lambda_{2}$ and $\delta\tilde{\Lambda}$ term
vanishes in equation (7).

##  3 Parameter measurement error

The waveforms emitted from compact binary systems can be modeled reasonably
well, thus the matched filter technique is likely to be used in the GW data
analysis. The matched filter output is given by using the inner product
$\langle...|...\rangle$ as

$\langle x|h\rangle=4{\rm Re}\int_{f_{\rm low}}^{f_{\rm
max}}\frac{\tilde{x}(f)\tilde{h}^{*}(f)}{S_{n}(f)}df,$ (9)

where the tilde denotes the Fourier transform of the time-domain waveform,
$x(t)$ is the detector data stream, $h(t)$ is the filter template, $S_{n}(f)$
is the noise PSD for the detector, $f_{\rm low}$ and $f_{\rm max}$ are the low
and maximum frequency limits, respectively. Once a detection is made in the GW
search pipeline, a more detailed analysis is conducted in the parameter
estimation pipeline to find the source properties, such as the distance, mass,
spin, etc. [41]. By means of Bayesian statistics, the result of the parameter
estimation analysis is given by the marginalized posterior probability density
functions (PDFs) for the parameters. The posterior probability that the GW
signal in the data stream $x$ is characterized by the parameters $\lambda$,
can be expressed by

$p(\lambda|x)\propto p(\lambda)L(x|\lambda),$ (10)

where $p(\lambda)$ is the prior probability, $L(s|\lambda)$ is the likelihood
given by [28, 42]

$L(x|\lambda)\propto{\rm exp}\bigg{[}-\frac{1}{2}\langle
x-h(\lambda)|x-h(\lambda)\rangle\bigg{]},$ (11)

where $h(\lambda)$ is a model waveform. If the SNR is high enough, i.e., the
detector noise is negligible with respect to the incident signal strength, the
likelihood can be given by

$L\propto{\rm exp}\bigg{[}-\frac{1}{2}\\{\langle s|s\rangle+\langle
h|h\rangle-2\langle s|h\rangle\\}\bigg{]},$ (12)

where $s$ is the GW signal. With the assumption of flat priors in equation
(10), the Bayesian posterior PDF is determined only by the likelihood.

Since the parameter estimation algorithm explores the entire parameter space,
the procedure usually takes a very long time. On the other hand, the
measurement errors of the parameters can be determined by using the Fisher
matrix method which is a semi-analytic approach [43, 44]. Fisher matrix can
estimate the $1\sigma$ statistical errors of parameters for a given signal
with known true values. Using the definition of the inner product given in
equation (9), the SNR ($\rho$) for the signal $s$ can be given by [30]

$\rho=\sqrt{\langle s|s\rangle}.$ (13)

If we choose a template whose waveform is similar to the signal waveform, we
have $\rho^{2}=\langle s|s\rangle\simeq\langle h|h\rangle$. Then, we can
rewrite equation (12) as [45]

$L(\lambda)\propto{\rm
exp}[-\rho^{2}\\{1-\langle\hat{s}|\hat{h}(\lambda)\rangle\\}],$ (14)

where $\hat{h}$ denotes the normalized waveform defined by $\hat{h}=h/\rho$.
Generally, the likelihood has the form of a Gaussian distribution in the high
SNR limit [28, 42].

If one describes a Gaussian function centred around $x_{0}$ as

$p(x)\propto e^{-\Gamma\Delta x^{2}/2},$ (15)

where $\Delta x=x-x_{0}$, then the variance $\sigma^{2}$ corresponds to
$\Gamma^{-1}$, and $\Gamma$ can be calculated by

$\Gamma=-{d^{2}\ln p(x)\over d^{2}x}\bigg{|}_{x=x_{0}}.$ (16)

Similarly, for a N-dimensional case, $\Gamma_{ij}$ is a $N\times N$ matrix and
the Gaussian function can be expressed as

$p(x_{i})=e^{-\Gamma_{ij}\Delta x_{i}\Delta x_{j}/2}.$ (17)

In this case, the matrix $\Gamma_{ij}$ is determined by

$\Gamma_{ij}=-{\partial^{2}\ln p(x)\over\partial x_{i}\partial
x_{j}}\bigg{|}_{x=x_{0}}.$ (18)

If we assume flat priors in equation (10), we have $p(\lambda|x)\propto
L(x|\lambda)$. By substituting $L(x|\lambda)$ for $p(x)$ in equation (18), and
using equation (14) and the relation $\hat{s}=\hat{h}_{0}$, we can get

$\Gamma_{ij}=-{\partial^{2}\ln
L\over\partial\lambda_{i}\partial\lambda_{j}}\bigg{|}_{\lambda=\lambda_{0}}=-\rho^{2}{\partial^{2}\langle\hat{h}_{0}|\hat{h}(\lambda)\rangle\over\partial\lambda_{i}\partial\lambda_{j}}\bigg{|}_{\lambda=\lambda_{0}}.$
(19)

Using this equation and the relation $h=\rho\hat{h}_{0}$, the matrix
$\Gamma_{ij}$ can be obtained directly from the wave function $h(\lambda)$ as
follows [43, 46]

$\Gamma_{ij}=\bigg{\langle}{\partial
h\over\partial\lambda_{i}}\bigg{|}{\partial
h\over\partial\lambda_{j}}\bigg{\rangle}\bigg{|}_{\lambda=\lambda_{0}}.$ (20)

This is the familiar expression of the standard Fisher matrix introduced in
[28, 29]. The above procedure implies that the Fisher matrix can be obtained
by taking only the leading-order terms of the Taylor expansion in all
parameters of the Likelihood about the maximum likelihood point
($\lambda=\lambda_{0}$) and this approximation is only valid in the high SNR
limit. The inverse of Fisher matrix represents the covariance matrix
($\Sigma_{ij}=(\Gamma^{-1})_{ij}$) of the parameter errors, and the
measurement error $\sigma_{i}$ (standard deviation of a normal distribution)
of each parameter and the correlation coefficient (${\cal C}_{ij}$) between
two parameters are given by

$\sigma_{i}=\sqrt{\Sigma_{ii}},\ \ \ {\cal
C}_{ij}={\Sigma_{ij}\over\sqrt{\Sigma_{ii}\Sigma_{jj}}}.$ (21)

From equations (19) and (21), one can see that the measurement error is
inversely proportional to the SNR for a given signal $\hat{s}$.

In Bayesian parameter estimation simulation, any form of prior distribution
can be imposed on the analysis. However, in the Fisher matrix formalism, only
the Gaussian prior function can be applied analytically. If the parameter
$\lambda_{i}$ has a Gaussian prior distribution centered at the true value
$\lambda_{0}$ with variance $P_{\lambda}^{2}$, the (prior-incorporated)
covariance matrix can be determined simply by adding the component
$\Gamma_{ii}^{0}=P_{\lambda}^{-2}$ as [28, 29]

$\Sigma^{P}_{ij}=(\Gamma_{ij}+\Gamma_{ii}^{0})^{-1},$ (22)

where $\Gamma_{ij}$ is the original Fisher matrix in which no prior
information is imposed.

One element of the Fisher matrix can be determined independently of the
computation of the other elements. Although an N-dimensional Fisher matrix
rises to N+n dimension by adding n parameters to the original Fisher matrix,
the original elements remain unchanged. However, all elements of the original
N-dimensional covariance matrix can be changed by the addition of the n
parameters. Typically, the measurement errors of the intrinsic parameters can
significantly increase by adding another intrinsic parameter because the
intrinsic parameters are usually strongly correlated with each other. In
contrast, adding extrinsic parameters to the Fisher matrix does not affect the
result of the intrinsic parameters due to the weak correlation between the
extrinsic and intrinsic parameters.

One should note that the error calculated by
$\sigma_{i}=\sqrt{(\Gamma_{ii})^{-1}}$ is different from the error calculated
by $\sigma_{i}=\sqrt{(\Gamma^{-1})_{ii}})$. The former represents the error of
the parameter $\lambda_{i}$ obtained by using only $i$th diagonal element of
the N-dimensional Fisher matrix, in this case, the true values of the other
parameters are assumed to be exactly known. Meanwhile, the latter represents
the error of $\lambda_{i}$ obtained by using the inversion of the
N-dimensional Fisher matrix as described in equation (22), and in that case,
the error $\sigma_{\lambda_{i}}$ corresponds to the marginalized one-
dimensional error that incorporates the correlations with the other
parameters. All errors given in this work are of the latter case.

##  4 Result : comparison between aLIGO and ET

###  4.1 Binary black holes

We apply the TaylorF2 wave function to the Fisher matrix formalism given in
equation (20) to calculate the measurement errors. For the BBH system, the
Fisher matrix is given by a $5\times 5$ matrix with the three intrinsic
parameters $\\{M_{c},\eta,\chi_{\rm eff}\\}$ and $\\{t_{c},\phi_{c}\\}$. The
two parameters $t_{c}$ and $\phi_{c}$ are arbitrary constants, so the choice
of their true values does not affect the errors of the intrinsic parameters.
However, since those parameters are strongly correlated with the intrinsic
parameters, omitting them significantly affects the errors of the intrinsic
parameters (see, table 3). Although $t_{c}$ and $\phi_{c}$ are physically
uninteresting parameters, they must be included in the construction of the
Fisher matrix. In this work, thus, we consider $t_{c}$ and $\phi_{c}$ together
with the intrinsic parameters in the Fisher matrix but present only the
results for the intrinsic parameters. We consider an L-shaped single-detector
configuration and use the aLIGO and the ET detector sensitivity curves given
in figure 1. According to the shape of the sensitivity curves, the low-
frequency limit is given as $f_{\rm low}=10\ (1)$ Hz for aLIGO (ET). The
maximum-frequency limit is given as $f_{\rm max}=f_{\rm ISCO}$ for both the
cases, where the ISCO frequency is given by

$f_{\rm ISCO}={1\over 6^{3/2}\pi M}.$ (23)

Although the masses of the BBHs detected by aLIGO and Advance Virgo are
broadly distributed between $10M_{\odot}$ and $160M_{\odot}$ [7, 8], we only
consider the low-mass BBHs with $M\leq 16M_{\odot}$ due to the validity of the
TaylorF2 model as discussed above. We also consider the component masses of
$m_{1},m_{2}\geq 3M_{\odot}$ and the effective spin of $-0.9\leq\chi_{\rm
eff}\leq 0.9$. We assume the effective distance to be $D_{\rm eff}=400$ Mpc
for all BBH sources. For the errors of the chirp mass in our result, we use
the dimensionless relative error $\sigma_{M_{c}}/M_{c}$, while we use the
absolute error $\sigma_{\lambda}$ for the other parameters. Finally, for
completeness, it is worth mentioning that using the same waveform, parameters,
and PSD as [29] we are able to reproduce their result with greater than $99\%$
accuracy.

We generate $10^{4}$ Monte-Carlo BBH samples, which are randomly distributed
in the $m_{1}$–$m_{2}$–$\chi_{\rm eff}$ parameter space in the range of
$m_{1},m_{2}\geq 3M_{\odot}(m_{1}\geq m_{2}),M\leq 16M_{\odot},$ and
$-0.9\leq\chi_{\rm eff}\leq 0.9$. Using equation (13), we calculate the SNRs
for all binaries for the aLIGO and the ET cases, respectively. In figure 2,
the upper panel shows the SNRs in the $m_{1}$–$m_{2}$ plane. Note that in this
result, we use all of the $10^{4}$ samples, so the SNR dependence on
$\chi_{\rm eff}$ is projected onto the $m_{1}$–$m_{2}$ plane. We have the SNRs
of $17\lesssim\rho\lesssim 37$ for aLIGO and $240\lesssim\rho\lesssim 540$ for
ET, and we find that ET can achieve $\sim 14$ times larger SNRs than aLIGO for
all binaries. Most of the signal strength is determined by its amplitude, so
the SNR is only a function of $M_{c}$ (i.e., $\rho\propto M_{c}^{5/6}$) for a
given $D_{\rm eff}$. This relation is clearly described in the lower panel. In
this panel, we also use all of the $10^{4}$ samples, so the small width of the
curves indicates a negligible dependence of $\eta$ and $\chi_{\rm eff}$.

Figure 2:  The SNRs calculated with $10^{4}$ Monte-Carlo BBH samples. The
samples are randomly generated in the range of $m_{2},m_{2}\geq
3M_{\odot},M\leq 16M_{\odot},-0.9\leq\chi_{\rm eff}\leq 0.9$ assuming $D_{\rm
eff}=400{\rm Mpc}$. The lower panels show the strong dependence of SNR on
$M_{c}$ ($\rho\propto M_{c}^{5/6})$ and negligible dependence on $\eta$ and
$\chi_{\rm eff}$. For all binaries, we have ${\rm SNR_{ET}/SNR_{aLIGO}}\sim
14$. Figure 3:  Measurement errors of the chirp mass (top), the symmetric mass
ratio (middle), and the effective spin (bottom) for the binaries with
$\chi_{\rm eff}=0$. We assmue $D_{\rm eff}=400$ Mpc.

Figure 3 shows the measurement errors of the chirp mass, the symmetric mass
ratio, and the effective spin obtained with aLIGO and ET for the sources with
$\chi_{\rm eff}=0$. Among the three parameters, the most accurately measured
one is the chirp mass, of which the measurement errors are
$\sigma_{M_{c}}/M_{c}\sim{\cal O}(10^{-3})$ for aLIGO and $\sim{\cal
O}(10^{-5})$ for ET. The measurement errors of $\eta$ and $\chi_{\rm eff}$ are
$\sigma_{\eta,\chi_{\rm eff}}\sim{\cal O}(10^{-1})$ for aLIGO and $\sim{\cal
O}(10^{-3})$ for ET, but the accuracy for $\eta$ is 2–4 times better. One can
see that the parameter measurement between the two detectors is greatly
improved. Overall, the measurement errors for ET can be lower than 1% of those
for aLIGO. The trends of the contours for aLIGO and ET are similar for
$\sigma_{M_{c}}$ but quite different for $\sigma_{\eta}$ and
$\sigma_{\chi_{\rm eff}}$. For aLIGO, $\sigma_{\eta}$ and $\sigma_{\chi_{\rm
eff}}$ have the smallest values when the binary mass is smallest, i.e.
$(3M_{\odot},3M_{\odot})$, but for ET, they have the smallest values when the
component masses are most asymmetric even though the binary is the heaviest in
our mass range, i.e. $(13M_{\odot},3M_{\odot})$. On the other hand, the
results of the symmetric binaries ($\eta\sim 1$) show that the measurement
error of the intrinsic parameter increases as $M$ increases even though the
signal strength becomes stronger. This can be understood by the fact that the
GW signal of a less massive system (for a given value of $\eta$) contains more
wave cycles as the frequency sweeps through the detector bandwidth, and the
phasing of the signal plays the largest role in measuring the mass and the
spin parameters. (Clearly, this is only true for the intrinsic parameters,
which affect the wave phasing. The measurement of the extrinsic parameters
mainly depends on the SNR because they are integrated into the wave
amplitude.)

Table 1: Parameter measurement errors and SNRs for BBHs with aLIGO and ET. The
effective distance is 400 Mpc. The ratios are defined by ${{\cal
R}_{\lambda}}=\sigma_{\lambda,\rm{ET}}/\sigma_{\lambda,\rm{aLIGO}}$ and ${\cal
R}_{\rm SNR}={\rm SNR_{ET}/SNR_{aLIGO}}$.

  * aLIGO |
---|---
$m_{1}[M_{\odot}]$ | $m_{2}[M_{\odot}]$ | $\chi_{\rm eff}$ | $\sigma_{M_{c}}/M_{c}{\times}{10^{3}}$ | $\sigma_{\eta}$ | $\sigma_{\chi_{\rm eff}}$ | SNR
3.0 | 3.0 | 0.0 | 0.61 | 0.061 | 0.175 | 16.8
5.0 | 5.0 | 0.0 | 1.29 | 0.100 | 0.295 | 25.5
8.0 | 8.0 | 0.0 | 3.20 | 0.208 | 0.622 | 37.0
13.0 | 3.0 | 0.0 | 2.83 | 0.086 | 0.353 | 28.9
ET |
$m_{1}[M_{\odot}]$ | $m_{2}[M_{\odot}]$ | $\chi_{\rm eff}$ | $\sigma_{M_{c}}/M_{c}{\times}{10^{5}}$ | $\sigma_{\eta}{\times}{10^{3}}$ | $\sigma_{\chi_{\rm eff}}{\times}{10^{2}}$ | SNR
3.0 | 3.0 | 0.0 | 0.38 | 0.78 | 0.203 | 241.5
5.0 | 5.0 | 0.0 | 0.63 | 0.96 | 0.265 | 368.2
8.0 | 8.0 | 0.0 | 1.07 | 1.26 | 0.361 | 537.5
13.0 | 3.0 | 0.0 | 0.85 | 0.46 | 0.177 | 419.6
Comparison |
$m_{1}[M_{\odot}]$ | $m_{2}[M_{\odot}]$ | $\chi_{\rm eff}$ | ${\cal R}_{M_{c}}{\times}{10^{2}}$ | ${\cal R}_{\eta}{\times}{10^{2}}$ | ${\cal R}_{\chi_{\rm eff}}{\times}{10^{2}}$ | ${\cal R}_{\rm{SNR}}$
3.0 | 3.0 | 0.0 | 0.63 | 1.27 | 1.16 | 14.3
5.0 | 5.0 | 0.0 | 0.49 | 0.96 | 0.90 | 14.4
8.0 | 8.0 | 0.0 | 0.33 | 0.61 | 0.58 | 14.5
13.0 | 3.0 | 0.0 | 0.30 | 0.54 | 0.50 | 14.5

Figure 4:  Measurement errors obtained from the $10^{4}$ Monte-Carlo
simulations. For comparison between aLIGO and ET, we define the error ratio as
${{\cal R}_{\lambda}}=\sigma_{\lambda,\rm{ET}}/\sigma_{\lambda,\rm{aLIGO}}$
Figure 5:  One-dimensional error curves given as a function of $\chi_{\rm
eff}$ for the selected binaries with different masses.

To see the spin dependence of the error, we display the errors as a function
of $\chi_{\rm eff}$ in figure 4. The distribution of errors differs
significantly between aLIGO and ET, showing no symmetry about $\chi_{\rm
eff}=0$ in both cases. The result shows sharp peaks at $\chi_{\rm eff}\sim
0.1$ for aLIGO but smooth peaks around $\chi_{\rm eff}=0.4$ for ET. At
$\chi_{\rm eff}=-0.9$, the error ratios are ${\cal R}_{M_{c}}\leq 1\%$ and
${\cal R}_{\eta},{\cal R}_{\chi_{\rm eff}}\leq 2.5\%$, and they slowly
decrease with $\chi_{\rm eff}$ and from $\chi_{\rm eff}\sim 0.2$, they are
widely distributed up to $5-7\%$. Although not shown here, for all of the
samples we have $\sigma_{t_{c}}<0.02\ (0.0035)$ s for aLIGO (ET). For easy
understanding of the distribution of the errors, we select four BBH sources
with different masses and display the one-dimensional error curves in figure
5. We find that the curves coincide with the error distributions. We also find
that the upper part of the error distribution can be obtained from the
symmetric and higher-mass binaries ($m_{1},m_{2}\sim 8M_{\odot}$), except for
the high-spin region. In table 1, we present exact values of the SNR,
$\sigma_{\lambda}$, and ${\cal R}_{\lambda}$ for the selected binaries with
$\chi_{\rm eff}=0$. On the other hand, in the case of aLIGO, we figure out
that the sharp peaks are caused by the strong correlations between the
intrinsic parameters and $\phi_{c}$. In figure 6, one can see that for aLIGO
the errors of $\phi_{c}$ also have a sharp peak at $\chi_{\rm ef}\sim 0.1$,
and all correlations with the intrinsic parameters can have very high values
at the same region around $\chi_{\rm ef}=0.1$. Consequently, the binary
samples in the high-correlation area constitute these sharp peaks. Note that
none of the high-correlation samples encounters the matrix inversion problem
in our Fisher matrix computations (see E). For ET, the errors
$\sigma_{\phi_{c}}$ for all samples are less than 0.4.

Figure 6:  Measurement error of $\phi_{c}$ (upper left) and the correlations
with the intrinsic parameters for aLIGO. All correlations have very high
values at the same region around $\chi_{\rm ef}=0.1$ (see, the inset). Note
that the correlation ${\cal C}_{\eta\phi_{c}}$ is negative.

###  4.2 Effectt of prior

As discussed in section 3, the measurement error can be reduced by imposing
the prior information if the prior distribution is narrow enough. To evaluate
the impact of prior, using equation (22), we calculate the prior-imposed
measurement error ($\sigma^{\rm prior}$) varying the prior variance $P^{2}$
and compare that with the original priorless error ($\sigma^{\rm priorless}$).
In the left panel of figure 7, we display the ratio $\sigma^{\rm
prior}/\sigma^{\rm priorless}$ as a function of $P_{\phi_{c}}$ for the
intrinsic parameters assuming a binary with the true values of
$(m_{1},m_{2},\chi_{\rm eff})=(6M_{\odot},3M_{\odot},-0.5)$. This result
describes how the measurement error can be reduced as the prior variance
decreases. In equation (22), the original Fisher matrix depends on the SNR,
while the prior variance ($P^{2}$) is independent of the SNR. Then, it can be
shown that the prior contribution is stronger for lower SNRs, which is
described in the left panel of figure 7. In this panel, the upper and lower
curves are obtained by assuming $\rho=60$ and 20, respectively. If we rescale
the x-axis to $P_{\phi_{c}}/\sigma^{\rm priorless}_{\Phi_{c}}$, the two curves
are exactly the same (see, the inset). Therefore, the prior contribution can
be generalized independently of the SNR by defining the quantity $\sigma^{\rm
prior}_{M_{c}}/\sigma^{\rm priorless}_{M_{c}}$ as a function of
$P_{\phi_{c}}/\sigma^{\rm priorless}_{\Phi_{c}}$. For example, if
$P_{\phi_{c}}/\sigma^{\rm priorless}_{\Phi_{c}}\sim 1$, then $\sigma^{\rm
prior}_{\lambda}\sim 70\%$ of $\sigma^{\rm priorless}_{\lambda}$. As seen in
equation (22), adding the component $\Gamma_{ii}^{0}=P_{\theta}^{-2}$ to the
original Fisher matrix changes every element of the covariance matrix, so all
correlations change. The right panel of figure 7 shows the correltions between
the intrinsic parameters and $\phi_{c}$ as a function of
$P_{\phi_{c}}/\sigma^{\rm priorless}_{\Phi_{c}}$. This result shows that the
strong correlations rapidly weaken as the prior variance decreases.

Figure 7:  Left: Ratio between the prior-imposed measurement error
($\sigma^{\rm prior}$) and the priorless error ($\sigma^{\rm priorless}$) for
the intrinsic parameters $\lambda_{i}=(M_{c},\eta,\chi_{\rm eff})$.
$P_{\phi_{c}}$ represents the standard deviation of the Gaussian prior
function. The virtical dotted lines indicate $\sigma_{\phi_{c}}^{\rm
priorless}$ for the SNRs of 60 (black) and 20 (red). If we rescale the x-axis
to $P_{\phi_{c}}/\sigma^{\rm priorless}_{\Phi_{c}}$, the two curves are
exactly the same and independent of the SNR (see, the inset). Right:
Correlations between $\lambda_{i}$ and $\phi_{c}$ given as a function of
$P_{\phi_{c}}/\sigma^{\rm priorless}_{\Phi_{c}}$. Here, we use a binary with
the true values of $(m_{1},m_{2},\chi_{\rm
eff})=(6M_{\odot},3M_{\odot},-0.5)$.  Figure 8:  Measurement errors obtained
by imposing the Gaussian prior on $\phi_{c}$ ($P_{\phi_{c}}=2\pi,4\pi$) using
$10^{3}$ Monte-Carlo samples.

One can see in figure 6 that for aLIGO a fairly large number of samples can
produce the errors $\sigma_{\phi_{c}}$ much larger than $2\pi$, which is a
physically reasonable range for $\phi_{c}$. We apply the Gaussian prior on
$\phi_{c}$ ($P_{\phi_{c}}=2\pi,4\pi$) to our analysis and recalculate the
measurement errors of the intrinsic parameters using $10^{3}$ Monte-Carlo
samples. The result is given in figure 8. For the case of $P_{\phi_{c}}=4\pi$,
the errors are greatly reduced, and the sharp peaks are relieved, compared to
the result in figure 4. The errors for $P_{\phi_{c}}=2\pi$ are more reduced by
about a factor of 2 compared to those for $P_{\phi_{c}}=4\pi$, as expected. We
also perform the same analysis for ET but the result is indistinguishable from
that in figure 4. This is because the errors $\sigma_{\phi_{c}}$ for ET are
much smaller than $2\pi$ and the SNRs are high enough. In this case, the
posteriors are almost unaffected by the prior $P_{\phi_{c}}\sim 2\pi$ as
described in figure 7. Concrete examples for the effect of priors are also
given in table 5 of [47].

###  4.3 BNSs

For a BNS system, our Fisher matrix is given by a $6\times 6$ matrix with the
parameters $\\{M_{c},\eta,\chi_{\rm eff},\tilde{\Lambda},t_{c},\phi_{c}\\}$
assuming a fixed effective distance. The choice of $f_{\rm low}$ and $f_{\rm
max}$ is the same as in the case of BBHs but the effective distance is assumed
to be 40 Mpc in order to obtain sufficiently high SNRs. We consider three
equal-mass binaries ($1M_{\odot},1M_{\odot}$), ($1.4M_{\odot},1.4M_{\odot}$),
and ($2M_{\odot},2M_{\odot}$). We choose the true value of the tidal parameter
$\tilde{\Lambda}$ using the soft APR4 equation-of-state (EOS) model and the
stiffer EOS model MPA1 given in [48].

Table 2: Parameter measurement errors and SNRs for BNSs with aLIGO and ET. The
effective distance is 40 Mpc. The ratios are defined by ${{\cal
R}_{\lambda}}=\sigma_{\lambda,\rm{ET}}/\sigma_{\lambda,\rm{aLIGO}}$ and ${\cal
R}_{\rm SNR}={\rm SNR_{ET}/SNR_{aLIGO}}$. For the errors of $M_{c},\eta,$ and
$\chi_{\rm eff}$ we only present the result for the MPA1 EOS model but the
difference from those for ARP4 is within $1\%$. For convenience, we list the
result for $\tilde{\Lambda}$ separately, the error of $\tilde{\Lambda}$
depends on both the mass and the EOS.

  * aLIGO
---
$m_{i}[M_{\odot}]$ | $\chi_{\rm eff}$ | $\sigma_{M_{c}}/M_{c}{\times}{10^{5}}$ | $\sigma_{\eta}{\times}{10^{3}}$ | $\sigma_{\chi_{\rm eff}}{\times}{10^{2}}$ | SNR
2.0 | 0.0 | 5.44 | 7.60 | 2.04 | 120.0
1.4 | 0.0 | 3.64 | 6.26 | 1.62 | 89.2
1.0 | 0.0 | 2.58 | 5.42 | 1.34 | 67.4
ET
$m_{i}[M_{\odot}]$ | $\chi_{\rm eff}$ | $\sigma_{M_{c}}/M_{c}{\times}{10^{7}}$ | $\sigma_{\eta}{\times}{10^{5}}$ | $\sigma_{\chi_{\rm eff}}{\times}{10^{4}}$ | SNR
2.0 | 0.0 | 3.19 | 9.33 | 2.19 | 1724.8
1.4 | 0.0 | 2.29 | 8.29 | 1.82 | 1281.6
1.0 | 0.0 | 1.68 | 7.50 | 1.53 | 968.3
Comparison
$m_{i}[M_{\odot}]$ | $\chi_{\rm eff}$ | ${\cal R}_{M_{c}}{\times}{10^{2}}$ | ${\cal R}_{\eta}{\times}{10^{2}}$ | ${\cal R}_{\chi_{\rm eff}}{\times}{10^{2}}$ | ${\cal R}_{\rm{SNR}}$
2.0 | 0.0 | 0.587 | 1.23 | 1.07 | 14.4
1.4 | 0.0 | 0.627 | 1.32 | 1.13 | 14.4
1.0 | 0.0 | 0.653 | 1.39 | 1.14 | 14.4
Result for $\tilde{\Lambda}$
$m_{i}[M_{\odot}]$ | EOS | $\tilde{\Lambda}$ |  $\sigma_{\tilde{\Lambda}}\ (\sigma_{\tilde{\Lambda}}/\tilde{\Lambda})$ [aLIGO] |  $\sigma_{\tilde{\Lambda}}\ (\sigma_{\tilde{\Lambda}}/\tilde{\Lambda})$ [ET] | ${\cal R}_{\tilde{\Lambda}}{\times}{10^{2}}$
2.0 | MPA1 | 48 | 35.8 (0.746) | 2.09 (0.0435) | 5.84
2.0 | APR4 | 15 | 34.1 (2.273) | 2.07 (0.138) | 6.08
1.4 | MPA1 | 513 | 75.9 (0.148) | 3.89 (0.00758) | 5.13
1.4 | APR4 | 250 | 63.5 (0.254) | 3.77 (0.0151) | 5.93
1.0 | MPA1 | 3150 | 218.3 (0.0693) | 7.94 (0.00252) | 3.64
1.0 | APR4 | 1734 | 154.9 (0.0893) | 7.27 (0.00419) | 4.69

In table 2, we summarize the SNRs and the measurement errors of the intrinsic
parameters for BNSs with aLIGO and ET and show a comparison between aLIGO and
ET. We have the SNRs of $70\lesssim\rho\lesssim 120$ for aLIGO and
$970\lesssim\rho\lesssim 1720$ for ET. As in the case of BBHs, we find the SNR
ratio of ${\cal R}_{\rm{SNR}}\sim 14$. The measurement error of the chirp mass
is $\sigma_{M_{c}}/M_{c}\sim{\cal O}(10^{-5})$ for aLIGO and $\sim{\cal
O}(10^{-7})$ for ET. The measurement errors of the symmetric mass ratio and
the effective spin are $\sigma_{\eta}\sim{\cal O}(10^{-3}),\ \sigma_{\chi_{\rm
eff}}\sim{\cal O}(10^{-2})$ for aLIGO and $\sigma_{\eta}\sim{\cal
O}(10^{-5}),\ \sigma_{\chi_{\rm eff}}\sim{\cal O}(10^{-4})$ for ET. As in the
case of BBHs, as $M$ increases the measurement accuracy for $M_{c},\eta,$ and
$\chi_{\rm eff}$ decreases independently of the increasing SNR. Overall, the
error ratios are ${\cal R}_{M_{c}}\leq 0.7\%,{\cal R}_{\eta}\leq 1.5\%,$ and
${\cal R}_{\chi_{\rm eff}}\leq 1.2\%$. Here, we only present the errors for
the MPA1 EOS model but the difference from those for ARP4 is within $1\%$.

For convenience, we list the result for $\tilde{\Lambda}$ at the bottom of
table 2, separately. For aLIGO, the tidal deformability measurement is much
worse than the measurements of the other parameters. The errors are widely
distributed as $35\lesssim\sigma_{\tilde{\Lambda}}\lesssim 155$ depending on
the mass for aLIGO, and they can be reduced to
$2\lesssim\sigma_{\tilde{\Lambda}}\lesssim 8$ for ET, resulting in the error
ratios $3.6\%\lesssim{\cal R}_{\tilde{\Lambda}}\lesssim 6.1\%$. The error
difference between the two EOSs is less than $5\ (1)\%$ for aLIGO (ET) for the
binary $m_{i}=2M_{\odot}$, but it becomes larger for lower masses, giving
$\sim 30\ (9)\%$ for the binary $m_{i}=1M_{\odot}$. Unlike the intrinsic
parameters, the measurement accuracy of the tidal parameter has a strong
dependence on the SNR (the SNRs are given in the last column in the upper two
subtables). This is because, most tidal effects occur above several hundred
Hz, so the measurement is almost unaffected by the entire wave cycles. Thus,
the measurement of the tidal parameter depends on the SNR rather than the
binary mass that governs the entire wave cycles.

Figure 9: Comparison of parameter measurement errors of the chirp mass (top),
the symmetric mass ratio (middle), and the effective spin (bottom) between
aLIGO and ET for BNSs. The error ratio is defined by ${{\cal
R}_{\lambda}}=\sigma_{\lambda,\rm{ET}}/\sigma_{\lambda,\rm{aLIGO}}$. We only
present the results for the MPA1 EOS model, but the results for ARP4 are
indistinguishable from them.  Figure 10: Measurement errors of the tidal
deformability. The error ratio is defined by ${{\cal
R}_{\lambda}}=\sigma_{\lambda,\rm{ET}}/\sigma_{\lambda,\rm{aLIGO}}$. For the
true values of $\tilde{\Lambda}$, we use the soft APR4 EOS model (solid line)
and the stiffer EOS model MPA1 (dotted line) given in [48].

We display the errors as a function of $\chi_{\rm eff}$ in figure 9 for the
parameters $M_{c},\eta,$ and $\chi_{\rm eff}$. Contrary to the case of BBHs,
the error curves show similar trends between aLIGO and ET and relatively
simple behavior. The errors $\sigma_{M}/M_{c}$ and $\sigma_{\eta}$
monotonically increase with increasing $\chi_{\rm eff}$, while
$\sigma_{\chi_{\rm eff}}$ decreases and increases again from $\chi_{\rm
eff}\sim 0.7$. The comparison results also show a simple behavior. We have the
error ratios of ${\cal R}_{M_{c}}\leq 0.8\%,\ {\cal R}_{\eta}\leq 1.6\%$, and
${\cal R}_{\chi_{\rm eff}}\leq 1.5\%$ exept for the high spin region. Again,
we only present the results for the MPA1 EOS model, but the results for ARP4
are indistinguishable from them.

We also show the result for $\tilde{\Lambda}$ in figure 10, separately.
Overall, the variations in the errors are very small, so the true value of
$\chi_{\rm eff}$ has little effect on the measurement of $\tilde{\Lambda}$. In
our result, the maximum value of ${\cal R}_{\tilde{\Lambda}}$ is $\sim 9\%$ at
$\chi_{\rm eff}\sim 0.65$ for the binary $m_{i}=2M_{\odot}$. For all BNSs, we
have $\sigma_{\phi_{c}}\ll 2\pi$, so we do not consider the prior effect.

##  5 Summary and future work

In this work, we demonstrated the improvement of accuracy in measuring the
source parameters between aLIGO and ET for low-mass BBHs with $M\leq
16M_{\odot}$ and BNSs. We calculated the Fisher matrices using $10^{4}$ BBH
Monte-Carlo samples randomly distributed in our mass and spin parameter space
and located at $D_{\rm eff}=400{\rm Mpc}$. We found that ET can detect the GW
signals with $\sim 14$ times better SNRs than aLIGO and the measurement errors
of the intrinsic parameters $M_{c},\eta,$ and $\chi_{\rm eff}$ for ET can be
below $\sim 3\ (7)\%$ of those for aLIGO in the range $\chi_{\rm eff}<0.3\
(>0.3)$. By applying the Gaussian prior with a variance $P^{2}$ to the Fisher
matrix, we investigated the effect of prior on the measurement accuracy. We
found that the error of the intrinsic parameters can be reduced to $\sim 70\%$
of the original priorless error $\sigma_{\phi_{c}}^{\rm priorless}$ if the
prior size is $P_{\phi_{c}}\sim\sigma_{\phi_{c}}^{\rm priorless}$. We also
found that for aLIGO, the inclusion of prior information about the parameter
$\phi_{c}$ as $P_{\phi_{c}}=2\pi$ can greatly reduce the error of the
intrinsic parameters, but the results for ET are not affected by this prior.

We also performed the same analysis for the BNSs using equal-mass binaries
with masses of $m_{i}=1,1.4,2M_{\odot}$ located at $D_{\rm eff}=40$ Mpc and
obtained the error ratios between aLIGO and ET for the intrinsic paremeters as
${\cal R}_{M_{c}}\leq 0.7\%,{\cal R}_{\eta}\leq 1.5\%,$ and ${\cal
R}_{\chi_{\rm eff}}\leq 1.2\%$. These results are almost independent of the
EOS. Assuming a soft and a stiffer EOS models, we calculated the errors of
tidal deformability $\tilde{\Lambda}$ and found that the measurement error of
$\tilde{\Lambda}$ can also be significantly reduced by ET. We obtained the
errors of $35\lesssim\sigma_{\tilde{\Lambda}}\lesssim 155$ for aLIGO and
$2\lesssim\sigma_{\tilde{\Lambda}}\lesssim 8$ for ET and the error ratio of
$3.6\%\lesssim{\cal R}_{\tilde{\Lambda}}\lesssim 6.1\%$. These precise
measurements of the NS tidal parameter will play an important role in
constraining various theoretical models for the NS EOS.

Although only low-mass BBHs were considered here, the masses of the observed
BBHs are broadly distributed up to $\sim 160M_{\odot}$ [7, 8]. Since TaylorF2
is only valid for low-mass systems, one has to use the inspiral-merger-
ringdown waveform models, such as the phenomenological models [17, 32, 33], to
obtain correct measurement errors for high-mass BBHs. However, the
phenomenological models are much more complicated than TaylorF2; thus,
applying those models to the Fisher matrix method would be very difficult.
Nevertheless, we will extend this work to the high-mass BBH system by using
one of the recent phenomenological models and cross-check the results in this
work. We will also use the python package GWBench which was recently developed
by [49] designed specifically for GW Fisher Matrix analyses. GWBench supports
the IMRPhenom models via numerical differentiation methods, we will check the
consistency between GWBench and the analytical method. Some results for high-
mass BBHs are given in F obtained by using GWBench.

This work was supported by the National Research Foundation of Korea (NRF)
grants funded by the Korea government (No. 2018R1D1A1B07048599, No.
2019R1I1A2A01041244, and No. 2020R1C1C1003250

##  Appendix A Effect of the extrinsic parameters on the measurement errors of
the intrinsic parameters

The effective distance is given by [30]

$D_{\rm
eff}=D/\sqrt{F^{2}_{+}\bigg{(}\frac{1+\cos^{2}\iota}{2}\bigg{)}^{2}+F^{2}_{\times}\cos^{2}\iota},$
(24)

where $D$ is the true distance of the source, $F_{+}$ and $F_{\times}$ are the
antenna response functions for the incident signal; these functions can be
given by the sky position (RA, Dec) of the source with respect to the
reference frame of the detector and the polirization angle $\psi$. If the
detector is placed at the x–y plane and the detector arms are aligned with the
axes, the sky position is given by $(\theta,\phi)$ in the spherical coordinate
system, and the antenna response functions are given by

$\displaystyle F_{+}=\frac{1}{2}\cos 2\phi\cos
2\psi\left(\cos^{2}\theta+1\right)-\sin 2\phi\sin 2\psi\cos\theta,$ (25)
$\displaystyle F_{\times}=\frac{1}{2}\cos 2\phi\sin
2\psi\left(\cos^{2}\theta+1\right)+\sin 2\phi\cos 2\psi\cos(\theta).$ (26)

For a fixed $D$, the maximum SNR (or minimum $D_{\rm eff}$) can be obtained
when $\theta=\phi=\iota=0$. In this case, $D_{\rm eff}$ is the same as $D$ and
independent of $\psi$.

On the other hand, different true values for $\theta,\phi$, and $\iota$ give
different SNRs, so all measurement errors of the intrinsic parameters are
adjusted simultaneously by the new SNR. In order to see the effect of the
extrinsic parameters on the measurement of the intrinsic parameters, we
investigate the variation of SNR by changing the extrinsic parameters. It is
straightforward that the SNR is inversely proportional to $D$ for a given sky
position and inclination. At the optimal sky position ($\theta=\phi=0$) with
$\psi=0$, the effective distance can be only a function of inclination:

$D_{\rm eff}=\frac{2D}{\cos^{2}{\iota}+1},$ (27)

which gives, max($D_{\rm eff})=2D$ at $\iota=\pi/2$, and min($D_{\rm eff})=D$
at $\iota=0$ or $\pi$. The strength of the incident signal is independent of
the polarization angle $\phi$ because that parameter is only determined by the
polarization basis for a fixed source. On the other hand, assuming $\iota=0$
in equation (24), we have $D_{\rm eff}=D/\sqrt{F^{2}_{+}+F^{2}_{\times}}$, so
$\rho^{2}\propto D_{\rm eff}^{-2}\propto F^{2}_{+}+F^{2}_{\times}$, and the
function

$F^{2}_{+}+F^{2}_{\times}=\frac{1}{4}\cos^{2}2\phi\left(\cos^{2}\theta+1\right)^{2}+\sin^{2}2\phi\cos^{2}\theta$
(28)

corresponds to the antenna power pattern of a single detector, which is
independent of $\psi$. This is often referred to as the “peanut diagram”. If
the signal comes from the sky position $\theta=\pi/2$ and $\phi=(1+2n)\pi/4$,
$D_{\rm eff}\rightarrow\infty$. Therefore, there can be four hidden positions
for a single detector observation.

The measurement error of $D_{\rm eff}$ can also be calculated by including
that parameter in the Fisher matrix, although $D_{\rm eff}$ is mostly
uncorrelated with the intrinsic parameters. A comparison between a
5-dimensional Fisher matrix neglecting $D_{\rm eff}$ and a 6-dimensional
Fisher marix including $D_{\rm eff}$ is shown in table 3. On the other hand,
one single extrinsic parameter incorporated in the function of $D_{\rm eff}$
can also be used in the Fisher matrix, where the true values of the other four
parameters should be set to be fixed. However, if we choose $\iota$ as the
variable, not all angles are amenable to the Fisher Matrix approximation. For
example, if $\iota=n\pi/2$, then $\partial{h}/\partial{\iota}=0$, and the
Fisher matrix becomes degenerate.

Table 3: Comparison of the errors ($\sigma$) between 3, 5, and 6-dimensional
Fisher matrices. The true values are assmumed to be ($m_{1},m_{2},\chi_{\rm
eff},t_{c},\phi_{c},D_{\rm eff})=(10M_{\odot},5M_{\odot},-0.5,0,0,400{\rm
Mpc}$). In the 6-dimensional Fisher matrix result, the correlations between
$D_{\rm eff}$ and the intrinsic parameters are given as ($\cal{C}_{\it
M_{c}D_{\rm eff}},\cal{C}_{\it\eta D_{\rm eff}},\cal{C}_{\it\chi_{\rm
eff}D_{\rm eff}}$)=(0.0295, -0.0288, 0.0289). We use the aLIGO PSD and $f_{\rm
low}=10$Hz.

  * Dimension | $\sigma_{M_{c}}/M_{c}{\times}{10^{3}}$ | $\sigma_{\eta}$ | $\sigma_{\chi_{\rm eff}}$ | $t_{c}[{\rm s}]$ | $\phi_{c}$ | $\sigma_{D_{\rm eff}}$
---|---|---|---|---|---|---
3-D | 0.1093 | 0.005416 | 0.01896 | - | - | -
5-D | 1.0712 | 0.036853 | 0.18528 | 0.0015430 | 8.648 | -
6-D | 1.0716 | 0.036868 | 0.18539 | 0.0015436 | 8.652 | 12.1

##  Appendix B Comparison between the Fisher matrix and the Bayesian parameter
estimation posteriors

We perform Monte-Carlo parameter estimation simulations using the Bayesian
parameter estimation package Bilby [50], which is a user-friendly Bayesian
inference library. We use one of the packaged samplers,
Dynesty111https://github.com/joshspeagle/dynesty. As a fiducail binary, we
adopt a BBH with the true parameter values of
$(m_{1},m_{2},\chi_{1},\chi_{2},t_{c},\phi_{c})=(6M_{\odot},3M_{\odot},-0.5,-0.5,0,0)$,
and we use the aLIGO PSD assuming $f_{\rm low}=20$Hz. Our Fisher matrix is
5-dimensional since we use $\chi_{\rm eff}$ instead of $\chi_{1},\chi_{2}$,
while the Monte-Carlo simulations run in 6-dimensional space with fixed
extrinsic parameters. Since we choose sufficiently high SNR ($\rho=100$), we
do not need to consider the effect of priors on our Fisher matrix result. In
the Bayesian parameter estimation, the priors are given as flat distributions
within physical boundaries, e.g. $[0,2\pi]$ for $\phi_{c}$. In figure 11, we
show a comparison between the Fisher matrix and the Bayesian parameter
estimation posteriors. One can see that our Fisher matrix result agrees
remarkably well with the Bayesian posteriors. In Bilby, one can choose the
parameters when displaying the Bayesian posterior PDFs, so we use $\chi_{\rm
eff}$ for direct comparison with the Fisher matrix result. On the other hand,
the Bayesian posterior of $\phi_{c}$ cannot be given because we have selected
the phase marginalization option for efficiency.

Figure 11:  Comparison between the Fisher matrix and the Bayesian parameter
estimation posteriors for a BBH. We assume the true values to be
$(m_{1},m_{2},\chi_{1},\chi_{2},t_{c},\phi_{c})=(6M_{\odot},3M_{\odot},-0.5,-0.5,0,0)$
and use the aLIGO PSD with $f_{\rm low}=20$. The SNR is $100$.

##  Appendix C Comparison between the optimal SNR and the matched-filter SNR

We perform the injection studies to have more realistic SNRs using Bilby. We
inject $10^{4}$ BBH signals into Gaussian noise using the aLIGO and ET PSDs,
respectively, and compute the matched-filter SNRs. The true parameter values
are randomly distributed in our 3-dimensional mass and spin parameter space
with a fixed $D_{\rm eff}$. The matched-filter SNR is given by $<s+n|s>/\rho$,
where $n$ is the Gaussian noise and $\rho$ is the optimal SNR defined in
equation (13). Thus, if we remove the Gaussian noise in this computation, the
matched-filter SNR can be the same as the optimal SNR. We compute both the
optimal SNR and the matched-filter SNR for aLIGO and ET and calculate the SNR
ratio between the two detectors. Figure 12 shows a comparison between the
optimal SNR ratio (${\cal R}_{\rm SNR}^{\rm optimal}$) and the matched-filter
SNR ratio (${\cal R}_{\rm SNR}^{\rm search}$). One can see that the
distribution of ${\cal R}_{\rm SNR}^{\rm optimal}$ is very narrow
$\sim[14.7,14.9]$ and shows a peak near the highest region, while ${\cal
R}_{\rm SNR}^{\rm search}$ shows a much wider and Gaussian-like distribution,
centered at the same position as the peak position of ${\cal R}_{\rm SNR}^{\rm
optimal}$. Here, we assume $f_{\rm low}=20\ (10)$ Hz for aLIGO (ET) due to
difficulties in setting the initial frequency below 20 (10) Hz in Bilby, while
we used $f_{\rm low}=10\ (1)$ for aLIGO (ET) in our analyses in section 4.
However, we found that the difference between the SNRs for $f_{\rm low}=10\
(1)$ Hz and $f_{\rm low}=20\ (10)$ Hz is within $\sim 2\ (0.5)\%$ for aLIGO
(ET). Because of these small differences, ${\cal R}_{\rm SNR}^{\rm optimal}$
in this figure is slightly larger than ${\cal R}_{\rm SNR}$ in table 1.

Figure 12:  The optimal SNR ratio (left) and the matched-filter SNR ratio
(right) between aLIGO and ET. We assume $f_{\rm low}=20\ (10)$ Hz for aLIGO
(ET).

##  Appendix D Multi-detector SNR analysis for the second and third-generation
detectors

The advantage of multi-detector networks in the measurement accuracy of
intrinsic parameters can be achieved mainly by an increase in the SNR. Here,
we investigate the single and the network SNRs for the second and third-
generation detectors. For the second-generation detectors we consider aLIGO
Hanford (H) and Livingstone (L) [51], advanced Virgo (V) [2], and KAGRA (K)
[52]. For third-generation detectors, we consider the Einstein Telescope (E)
[53] and the Cosmic Explorer (C) [54]. For ET, we use the triangular detector
configuration consisting of the three sub-detectors E1, E2, and E3. Bilby
includes geographic coordinates, orientations, and PSDs for all of the
detectors above. We inject $10^{4}$ BBH sources distributed over the sky
position and calculate the matched-filter SNRs for the detectors. In this
calculations, we randomly generate the angle RA in the range $[0,2\pi]$ and
$\cos({\rm Dec})$ in the range $[-1,1]$. We do not take into account the
binary orientation by choosing $\iota=0,\psi=0,$ and $D=400$ Mpc for all
samples. As a fiducial binary, we choose a nonspinning BBH with the masses
$(6M_{\odot},3M_{\odot})$. We assume $f_{\rm low}=20$ H and $10$ Hz for the
second and the third-generation detectors, respectively.

The result is given in figure 13. The upper-left panel shows the single SNRs
for H and the network SNRs for the second-generation detectors, and the upper-
right panel shows the single and the network SNRs for the third-generation
detectors. Note that, since ET consists of the three sub-detectors, a single
SNR for ET corresponds to the network SNR for the three sub-detectors.
Finally, the bottom panel shows the ratios between the second and the third-
generation network SNRs assuming various combinations.

Figure 13:  PDFs for the single and network SNRs for the second-generation
(upper left panel) and the third-generation (upper right panel) detectors and
their SNR ratios (lower panel). As a ficucial binary, we assume a nonspinning
BBH with the masses $(6M_{\odot},3M_{\odot})$. We assume $f_{\rm low}=20$ H
and $10$ Hz for the second-generation detectors and the third-generation
detectors, respectively.

##  Appendix E Subtleties in the inversion of the Fisher matrix

In this work, we evaluate the Fisher matrix and its inversion using a
Mathematica code. Following the methodology of [55], we check the reliability
of the result simply by multiplying the inversion by the original matrix. We
define the numerical identity matrix as $I^{\rm num}$ and measure the
deviation from the true indentity matrix:

$\epsilon_{\rm inv}\equiv{\rm max}_{ij}|I^{\rm num}_{ij}-\delta_{ij}|,$ (29)

where $\delta_{ij}$ is the Kronecker symbol. We find that all of the BBH
samples used in our 5-dimensional Fisher matrices could achieve $\epsilon_{\rm
inv}<10^{-6}$, and $\epsilon_{\rm inv}<10^{-5}$ for the 6-dimensional Fisher
matrices for BNSs (cf. Appendix B of [55]).

##  Appendix F High-mass BBHs

Here, we show some results for high-mass BBHs obtained by using GWBench. We
employ IMRPhenomHM [56] which is a higher-multipole model based on IMRPhenomD,
this model is more accurate than dominant-multipole-only models, especially
for highly-asymmetric binaries. Figure 14 shows the errors as a function of
the total mass assuming symmetric and asymmetric-mass binaries with
$m_{1}/m_{2}=1$ and 4. Since we fix the spin as $\chi_{1}=\chi_{2}=-0.1$, this
result does not show any spin dependence. We will perform a more comprehensive
study for high-mass BBHs in a broad parameter space in future work. Since only
the component spins can be used in GWbench, we only use the spin parameter
$\chi_{1}$ so the Fisher matrix can be given by $5\times 5$ matrix with the
parameters $(M_{c},\eta,\chi_{1},t_{c},\phi_{c})$. The choice of the spin
parameter between $\chi_{1}$ or $\chi_{\rm eff}$ has a little impact on the
measurement errors of the parameters considered. Note that, using both
component spins in the Fisher matrix can significantly increase the errors due
to increasing parameter dimension. Figure 15 shows the SNRs as a function of
the chirp mass. Unlike the result for small-mass BBHs in figure 2, the two
curves are clearly distinct. That means, the dependence of the SNR on the mass
ratio becomes stronger for higher-mass systems due to the non-negligible
contribution of the merger-ringdown. The SNR ratio between aLIGO and ET is
given as $13\lesssim{\cal R}_{\rm SNR}^{\rm optimal}\lesssim 16$.

Figure 14:  One-dimensional error curves for high-mass BBHs given as a
function of the total mass with fixed mass ratios of $m_{1}/m_{2}=1$ and 4. We
assume $\chi_{1}=\chi_{2}=-0.1,D_{\rm eff}=400$ Mpc, and $f_{\rm low}=10$ and
$1$ Hz for aLIGO and ET, respectively. Figure 15:  The optimal SNRs for high-
mass BBHs. The SNR ratio is $13\lesssim{\cal R}_{\rm SNR}^{\rm
optimal}\lesssim 16$.

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