11institutetext: Korea Astronomy and Space Science Institute, 776 Daedeokdae-
ro, Yuseong-gu, Daejeon 34055, Republic of Korea
11email<EMAIL_ADDRESS>22institutetext: Department of Physics, Chung-Ang
University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Republic of Korea
22email<EMAIL_ADDRESS>
# Binary microlensing by high eccentric stellar-mass black hole binaries

Kyungmin Kim 11 Yeong-Bok Bae, Corresponding author22 Yoon-Hyun Ryu 11

(Received ; accepted )

###### Abstract

Context. Microlensing is one of the most promising tools for discovering
stellar-mass black holes (BHs) in the Milky Way because it allows us to probe
dark or faint celestial compact objects.

Aims. While the existence of stellar-mass BHs has been confirmed through
observation of X-ray binaries within our galaxy and gravitational waves from
extragalactic BH binaries, a conclusive observation of microlensing events
caused by Galactic BH binaries has yet to be achieved. In this study, we focus
on those with high eccentricity, including unbound orbits, which can
dynamically form in star clusters and could potentially increase the
observation rate.

Methods. We demonstrate parameter estimation for simulated light curves
supposing various orbital configurations of BH binary lenses. We employ a
model-based fitting using the Nelder-Mead method and Bayesian inference based
on the Markov chain Monte Carlo method for the demonstration.

Results. The results show that we can retrieve true values of the parameters
of high eccentric BH binary lenses within the $1\sigma$ uncertainty of
inferred values.

Conclusions. We conclude it is feasible to find high eccentric Galactic BH
binaries from the observation of binary microlensing events.

###### Key Words.:

Gravitational lensing:micro – Stars:black holes – Methods: statistical

##  1 Introduction

Microlensing is a powerful tool that allows us to probe dark or faint
celestial objects, such as planets, dwarfs, neutron stars, and black holes
(BHs), as well as massive compact halo objects by magnifying the luminosity of
background sources (Paczynski, 1986; Mao & Paczynski, 1991; Gould & Loeb,
1992). Thanks to several photometric surveys conducted hitherto—such as
Expérience pour la Recherche d’Objets Sombres (EROS), MAssive Compact Halo
Objects (MACHO) project, Microlensing Observations in Astrophysics (MOA),
Optical Gravitational Lensing Experiment (OGLE), and Korea Microlensing
Telescope Network (KMTNet)—tens of thousands of microlensing events within our
own and nearby galaxies have been observed (Aubourg et al., 1993; Alcock et
al., 2000; Sumi et al., 2011; Udalski et al., 2015; Yock & Muraki, 2018; Byun
et al., 2022).

Stellar-mass BHs are one of the possible lens objects of microlensing events.
They have been identified through X-ray observations (Liu et al., 2007;
Tetarenko et al., 2016), but microlensing provides a unique opportunity to
detect isolated stellar-mass BHs as well. Based on the stellar mass function,
stellar-mass BHs are expected to account for approximately 1% of microlensing
events (Gould, 2000). However, only about a dozen stellar-mass BH candidates
have been found from microlensing events to date, even though they are
anticipated to be overrepresented in the observation due to the longer event
duration resulting from their substantial masses (Agol et al., 2002; Bennett
et al., 2002; Mao et al., 2002; Poindexter et al., 2005; Dong et al., 2007;
Shvartzvald et al., 2015; Wyrzykowski et al., 2016; Mróz & Wyrzykowski, 2021;
Sahu et al., 2022).

As for stellar-mass BH binaries, they have not been conclusively discovered
through microlensing events, which contrasts sharply with the fact that nearly
100 BH binaries have been identified through gravitational-wave (GW)
observations (Abbott et al., 2019, 2021a, 2021b, 2021c). However, we should
note that this situation arises from differences in the observable spatial
volume and time, rather than suggesting a scarcity of wide-orbit BH binaries
that produce microlensing events. This lack of detection could be attributed
to the small microlensing parallax of the BH binary due to their large mass as
well as the lens’s orbital motion that complicates the evaluation of the
parallax, which makes it difficult to accurately determine the lens mass
(Karolinski & Zhu, 2020; Ma et al., 2022).

It is known that BH binaries can form through the evolution of massive stellar
binaries (Marchant & Bodensteiner, 2023; Langer et al., 2020). Also, they can
form via dynamical interactions at the center of star clusters (O’Leary et
al., 2009; Banerjee et al., 2010; Bae et al., 2014; Hong & Lee, 2015; Park et
al., 2017). In particular, in highly dense regions like the center of nuclear
star clusters, many BHs can be gathered in a small volume through the
dynamical friction and mass segregation (Hailey et al., 2018), so the number
of interactions of BHs increases, which triggers the dynamical formation of BH
binaries.

In this study, we consider BH binary lenses with a range of eccentricities,
from low to high, including those in unbound orbits with eccentricities equal
to or exceeding unity. Generally, the formation of BH binaries from massive
stellar binaries tends to favor low eccentricity, while dynamical formation
processes markedly favor the production of BH binaries with high
eccentricities (Bae et al., 2014; Hong & Lee, 2015). Furthermore,
gravitational perturbations from the supermassive BH at the center of the
galaxy can amplify the eccentricity of BH binaries, known as the Kozai-Lidov
mechanism (Hoang et al., 2018). Additionally, in multi-body interactions—such
as binary-single or binary-binary encounters—highly eccentric encounters occur
repeatedly (Samsing, 2018; Zevin et al., 2019), commonly resulting in unbound
orbits. Thus, the occurrence rate of high-eccentricity encounters, especially
in cluster centers, is significant and cannot be overlooked.

In the determination of the characteristics of binary microlensing from a
light curve of a luminous celestial source, well-aligned configurations
between the source, binary lens, and observer ought to be satisfied, and
several fundamental parameters, e.g., the distances to the source and lens,
the mass of the binary lens, pericenter distance or separation between the
components of a binary, the trajectory and relative velocity of the source,
and so on, should be taken into account. Assuming negligible parallax for BH
binary lens due to its small value as previously mentioned, in this study, we
focus on the effects of orbital properties of BH binary lenses, specifically
including unbound BH trajectories, that can appear on the light curve of
microlensing events. We explore the signature of binary lensing depending on
the eccentricity of a BH binary and compare the cases of elliptic and
hyperbolic orbits with varying other parameters characterising a binary lens
system, such as the pericentre distance and mass ratio between two component
BHs, and the inclination angle between the orbital axis and the line of sight.
Additionally, we test whether we can restore the true value of the parameters
of a BH binary lens by using a model-based fitting method, the Nelder-Mead
fitting (Nelder & Mead, 1965), and Bayesian inference using the Markov chain
Monte Carlo sampling method (Goodman & Weare, 2010).

We find from this study that the employed parameter estimation methods can
retrieve true values of the binary microlensing parameters within the
$1\sigma$ uncertainty of inferred values. For some samples of which the
Nelder-Mead fitting failed in the uncertainty estimation, Bayesian inference
based on the Markov chain Monte Carlo method can succeed in estimating the
uncertainty. On the other hand, if the median value of the posterior
distribution obtained by the Bayesian inference deviates from the true value
more than that from the initial Nelder-Mead fitting, it turns out that using
the uncertainty estimated from the Bayesian inference helps reduce overall
deviations of all parameters.

This paper is organized as follows: In Sec. 2, we summarize how to compute the
orbits of two black holes and how to simulate binary microlensing phenomena
that can occur by the two black holes acting like a binary lens. We
investigate the parameter dependency in the binary microlensing signature and
show the resulting caustics and light curve in Sec. 3. We provide the result
of parameter estimation for the light curve of a simulated binary microlensing
event in Sec. 4. In Sec. 5, we discuss our result and the future prospect of
observing binary microlensing events by stellar-mass binary black holes within
our and/or nearby galaxies.

Figure 1: Schematic of BH binary lens system. We suppose the line of sight
towards the lens system is perpendicular to the lens plane which may not be
identical to the orbital plane of the orbiting two BHs. We define the angle
between the orbital axis and the line of sight as the inclination angle
$\iota$. For a source constantly moving along the grey dotted line, the angle
between the real axis $z_{x}$ on the lens plane and the source trajectory
projected onto the lens plane is defined as $\theta$.

##  2 Methods

###  2.1 Lens equation and magnification for binary lens system

We consider the light of a luminous source being lensed by two point-like
stellar-mass black holes (BHs) acting like a binary lens. As depicted in Fig.
1, we also take into account the lens plane may not be identical to the
orbital plane of the BH binary.

To simulate the binary microlensing by two BHs projected onto the lens plane,
we define the complex coordinate axes $z_{x}$ and $z_{y}$ on the lens plane
corresponding to the real and imaginary parts of a complex coordinate $z$,
respectively. Based on the complex coordinate, we then adopt the complex
notation of the lens equation (Witt, 1990; Witt & Mao, 1995; Meneghetti,
2021):

$z_{s}=z-\frac{m_{1}}{z^{*}-z_{1}^{*}}-\frac{m_{2}}{z^{*}-z_{2}^{*}}\leavevmode\nobreak\
,$ (1)

where $z$ is a position on the lens plane, $z_{s}$ is the position of the
source on the source plane, and $z_{1,2}$ are the positions of two lenses on
the lens plane. The asterisk (∗) denotes the complex conjugate. With the
complex conjugate form of Eq. (1), the lens equation then can be rewritten as
a complex polynomial equation of degree 5:

$c_{5}z^{5}+c_{4}z^{4}+c_{3}z^{3}+c_{2}z^{2}+c_{1}z+c_{0}=0\leavevmode\nobreak\
,$ (2)

where $c_{i}(i\\!=\\!0,1,\ldots,5)$ are the polynomial coefficients and their
exact forms are given in Eq. (14).

Similar to the lens equation, for a given phase angle $\phi\in[0,2\pi)$ with
respect to the real axis of the lens plane, we can obtain the critical curve
of the binary lens via

$\frac{m_{1}}{(z^{*}-z_{1}^{*})^{2}}+\frac{m_{2}}{(z^{*}-z_{2}^{*})^{2}}=e^{i\phi}\leavevmode\nobreak\
,$ (3)

and it can be turned into another complex polynomial equation of degree 4:

$d_{4}z^{4}+d_{3}z^{3}+d_{2}z^{2}+d_{1}z+d_{0}=0\leavevmode\nobreak\ ,$ (4)

with $d_{j}(j\\!=\\!0,1,\ldots,4)$ given in Eq. (16). Also, we can compute the
caustics on the source plane by inserting the solution of Eq. (4) into Eq.
(1).

The total magnification $\mu$ occurring by the binary lens at a certain
observation time $t$ can be obtained as follows

$\mu(t)=\sum^{N_{\textrm{images}}}_{j}|\mu_{j}(t)|\leavevmode\nobreak\ ,$ (5)

where $\mu_{j}$ is the individual magnification of each image $j$ of a source
at the given $t$: For a given binary lens system, we can calculate
$\mu_{j}(t)$ by

$\mu_{j}(t)=\left(1-\left|\frac{m_{1}}{z^{*}_{j}(t)-z^{*}_{1}}+\frac{m_{2}}{z^{*}_{j}(t)-z^{*}_{2}}\right|^{2}\right)^{-1}\leavevmode\nobreak\
.$ (6)

###  2.2 Orbital motion of black hole binary

As we suppose two BHs causing binary lensing orbit each other with a non-zero
eccentricity ($e$), we have to calculate the time-dependent trajectory of two
BHs, i.e., $z_{1}(t)$ and $z_{2}(t)$, in order to solve Eqs. (2) and (4). For
the orbital motion of the two BHs in the orbital plane, we solve the Newtonian
equation of motion using the Verlet integration method (Verlet, 1967), which
is a symplectic method that preserves the energy of the system. The time step
is fixed at 0.1 day. The resulting solution is a Kepler orbit as is well
known. Defining the angle between the orbital axis and the line of sight
perpendicular to the lens plane as the inclination angle ($\iota$), the
initial condition is adjusted with the given eccentricity and the pericentre
distance projected onto the lens plane.

##  3 Caustics and light curves

In this work, we assume observing a binary lensing event is made with a
constant time interval for a total observation time. Following the assumption,
we interpolate the trajectory of orbit obtained by solving the equation of
motion for the given observation time so that the time steps of the trajectory
are equally spaced. Consequently, we can simulate a binary lensing light curve
with $\sim 1\leavevmode\nobreak\ \textrm{day}$ cadence. In addition, in order
to ensure a sufficient observation time for identifying the signature of
binary lensing, i.e., two sharp spikes connected with magnified ‘U’-shape
light curve, we take twice of the Einstein cross time, calculable for the
Einstein radius of the total mass of a BH binary, as the total observation
time.

Regarding the consideration discussed thus far and denoting a model parameter
of this work as $\lambda$, we configure a set ${\Lambda}$ as follows

${\Lambda}=\\{\lambda\\}=\\{D_{S},D_{L},v_{r},\theta,m_{1},m_{2},s,e,\iota\\}\leavevmode\nobreak\
,$ (7)

where $D_{S}$ and $D_{L}$ are the distance to source and BH binary lens,
$v_{r}$ is the relative velocity of source against to the lens, $\theta$ is
the angle between the real axis $z_{x}$ on the lens plane and the source
trajectory projected onto the lens plane, $m_{1}$ and $m_{2}$ are the mass of
component BHs of a binary lens, $s$ is the pericenter distance between two
BHs, $e$ is the eccentricity of the orbital motion of BH binary, and $\iota$
is the inclination angle defined by the angle between the orbital axis and the
line of sight. In order to focus on the original and projected orbital motion
of binary BHs, we fix $D_{S}=8\leavevmode\nobreak\ \textrm{kpc}$,
$D_{L}=5\leavevmode\nobreak\ \textrm{kpc}$, $v_{r}=200\leavevmode\nobreak\
\textrm{km/s}$, $\theta=120^{\circ}$, and alter $m_{1}$, $m_{2}$, $s$, $e$,
and $\iota$.

Figure 2: The lens orbits and the magnitude color maps for the selected points
on the orbit. The left two columns are for the mass ratio $q=1$ and the
pericenter distance $s=10$ AU in the case of elliptic ($e=0.3$) and hyperbolic
($e=2.0$) lens orbits. The middle and right two columns show the cases of
$q=1$, $s=20$ AU, and $q=0.3$ ($m_{1}=30M_{\odot}$, $m_{2}=10M_{\odot}$),
$s=20$ AU, respectively. The points on the orbits in the left, middle, and
right two columns are all at the same time, and the color maps are arranged in
the order in which the lens moves away from the pericenter. The positions are
normalized to the Einstein radius. Figure 3: Caustics and corresponding light
curve for each case of Fig. 2. We suppose observing microlensed light of an
arbitrary moving luminous source is made for the twice of the Einstein cross
time. For the same given source, we can see that the signature of binary
microlensing appearing in the light curve changes by different values of $q$,
$s$, and $e$ as expected.

###  3.1 Zero inclination angle

To examine the $e$-dependent variations in microlensing signature, e.g.,
caustics and magnification, we draw color maps depicting the magnification of
the source’s brightness for five specific lens positions along the lenses’
orbits (Fig. 2). Here, we choose two $e$ values, $e\\!=\\!0.3$ and $2.0$, as
representing values of elliptic and hyperbolic orbits, respectively. Despite
of many other parameters contribute to determine the binary microlensing
signature as discussed in the previous section, for simplicity, we first
consider $\iota=0^{\circ}$ and assume a source of an arbitrary magnitude in
the direction of the Galactic center, with a distance of 8 kpc to the source
($D_{S}=8$ kpc) and 5 kpc to the lens ($D_{L}=5$ kpc). Then, we investigate
three example cases of {$q,\leavevmode\nobreak\ s$}, i.e., {$q\\!=\\!1.0$,
$s\\!=\\!10\leavevmode\nobreak\ \textrm{AU}$}, {$q\\!=\\!1.0$,
$s\\!=\\!20\leavevmode\nobreak\ \textrm{AU}$}, and {$q\\!=\\!0.3$,
$s\\!=\\!20\leavevmode\nobreak\ \textrm{AU}$}. Hereafter, we label each pair
of {$q,\leavevmode\nobreak\ s$} as Case I, II, and III, respectively, for
convenience.

As is well known, caustics are formed in the direction perpendicular to the
connecting direction of the two lenses when the distance between two lenses is
close, but as the two lenses move away from each other, the shape of the
caustics is formed in the direction connecting the two lenses. As shown in
Fig. 2, in the case of a binary lens with an elliptical orbit, the distance
between the two lenses is kept within a certain range, so a caustic of a
certain size is also maintained. On the other hand, in the case of a binary
lens with a hyperbolic orbit, the caustic becomes elongated as it moves away
from the pericenter, and in this case, the magnification of the source
brightness is likely to occur for a very short period as the source traverses
the caustic. After that, the effect of each point mass lens will occur as the
distance becomes larger.

If the position of lens does not change significantly during the observation
period, the brightness change of the source due to its movement can be
inferred from this color map. However, in reality, both the lens and the
source are moving, so it is difficult to grasp the actual light curve with a
magnitude color map at a specific time such as Fig. 2. In addition, the
parameter space that determines the lens effect is very wide, so it is not
possible to examine all situations. Thus, we have examined a few examples
focused on the parameters that determine the lens dynamics.

Figure 4: Comparing caustics and corresponding light curves between two
$\iota$ values, $\iota=0^{\circ}$ and $\iota=15^{\circ}$ for each case in Fig.
2. One can see that non-zero $\iota$ results in slightly different light
curves from those of $\iota=0^{\circ}$ for the given $q$, $s$, and $e$.

We explore how the $e$-dependant shapes, sizes, and orientations of caustics
shown in Fig. 2 affects the profile of light curves. To this end, we simulate
light curves of the same source and BH binary lenses considered in Fig. 2. We
further suppose the source moves along a straight path corresponding to
$\theta=120^{\circ}$ with a constant relative velocity
$v_{r}=200\leavevmode\nobreak\ \textrm{km/s}$ to the BH binary. We set the
observation period as the twice of the Einstein cross time calculated by the
given binary microlensing configuration. For simplicity, we assume the
background source passes the barycentre of the BH binary, i.e., zero impact
parameter, at the middle of the observation period.

Fig. 3 shows the locations and orientations of caustics for given $q$, $s$,
and $e$ (top panels) and corresponding light curves (bottom panels). We can
see that the $e$-dependent differences in caustics determine not only the
width and height of ‘U’-shape double peaks and the presence of another double
peak or single peak on the light curve. It is easy to understand that such
differences originate from the changes in the shape, size, and orientation of
caustics for the given observation period and the location of the background
source at each moment of observation.

We have compared the $e$-dependent characteristics of caustics between $e=0.3$
and $e=2.0$ corresponding to an elliptic and hyperbolic orbits, respectively,
and resultant profiles of binary microlensing light curves thus far. We have
indeed seen that a BH binary of a hyperbolic orbit results in different light
curves compared to that of an elliptic orbit: Amongst tested cases, light
curves of Case I or Case III of $e=2.0$ show wider ‘U’-shape double peaks and
another peak(s) after the first appearance of ‘U’-shape peaks. However, Case
II of $e=2.0$ only shows small difference in the width and height of the
‘U’-shape peaks compared to that of $e=0.3$ because of relatively smaller
change in the size and orientation of the caustics.

###  3.2 Non-zero inclincation angle

Next, we consider $\iota\neq 0^{\circ}$ because it is more realistic than
$\iota=0^{\circ}$. We test how much non-zero $\iota$ changes the caustics and
light curve because, if $\iota\neq 0^{\circ}$, $s$ and $e$ of two BHs
projected on the source plane will be obviously adjusted by $\cos\iota$ from
their actual values. In Fig. 4, we present two cases of $\iota=0^{\circ}$ and
$15^{\circ}$ for the configurations studied in Figs. 2 and 3, but focusing on
$e=2.0$.

One can see that, if $\iota=15^{\circ}$, not only the projected $s$ of two BHs
becomes shorter than that of $\iota=0^{\circ}$ but also the shape and
orientation of the caustics varies for both $e=0.3$ and $e=2.0$ as expected.
For instance, the caustics of $\iota=15^{\circ}$ is slightly narrower than
$\iota=0^{\circ}$ for all cases because of the shorten $s$ and increased $e$
on the source plane. Also, we can observe from the light curve for each
$\iota$ that the position and number of spikes shown when $e=2.0$ are
different from the spikes of $e=0.3$ because of the $e$-dependent variations
in the shape, size, and orientation of corresponding caustics.

Thus far, we have investigated the difference between $e=0.3$ and $e=2.0$ via
example caustics and light curves. We have seen from the examples presented in
the previous section that the shape and orientation of the caustic of two
stellar-mass BHs indeed depends on the tested parameters and they consequently
affect the profile of the light curve. In specific, we could see that $e=2.0$
gives different characteristics of a BH binary lensing compared to the case of
$e=0.3$. However, we understand from this investigation that it is hard to
find a general pattern in considering $e=2.0$ instead of $e=0.3$ because
caustics and resultant light curve are highly degenerated. For example, when
we compare $e=0.3$ and $2.0$ with $\iota=0^{\circ}$, we could see additional
peak(s) from Case I and III, but, from Case II, we only saw small changes in
the width and height of the ‘U’-shape double peaks. Similarly, if we consider
$\iota\neq 0^{\circ}$, it is also nontrivial to find a common behaviour for
$e=2.0$ because, for example, the secondary ‘U’-shape double peaks of Case I
and a single bump around $-100$ day of Case II that are shown in
$\iota=0^{\circ}$ disappear when $\iota=15^{\circ}$. On top of that, there is
quite small shift in the location and three peaks of Case III for
$\iota=15^{\circ}$ that makes us be confused in recognising the signature of
$e=2.0$ from a simple visual inspection.

This can be a limitation on finding a general feature about the effect of
different eccentricities of a BH binary lens system. Moreover, if we consider
other degrees of freedom such as the source trajectory, distance to the binary
lens, and distance to the source, the degeneracy will be more complicated. At
this point, the discussed limitation raises a question whether we can
correctly recover those highly degenerated parameters of a BH binary lens
system if a light curve represents the signature of binary microlensing. To
answer the question, in the following section, we demonstrate parameter
estimations in order to investigate whether the unknown characteristic
parameters of a BH binary lens system can be correctly recovered from a binary
microlensing light curve.

##  4 Parameter Estimation

We demonstrate a fitting-based parameter estimation of binary microlensing
events. In practice, a successful fitting of a given light curve data with a
binary microlensing model can be a primary approach for estimating the most
promising values of microlensing parameters from the value of model parameters
used in the fitting. We adopt the Nelder-Mead (NM) method (Nelder & Mead,
1965) as the fitting method of this work.

Table 1: Prior ranges for the Nelder-Mead fitting. We assume uniform distribution for all priors for simplicity. Parameter (unit) | Prior range | Distribution
---|---|---
$D_{S}$ (kpc) | $[0,30]$ | Uniform
$D_{L}$ (kpc) | $[0,15]$ | Uniform
$v_{r}$ (km/s) | $[150,250]$ | Uniform
$\theta$ (∘) | $[0,360]$ | Uniform
$m_{1}$ ($M_{\odot}$) | $[5,100]$ | Uniform
$m_{2}$ ($M_{\odot}$) | $[5,100]$ | Uniform
$s$ (AU) | $[1,50]$ | Uniform
$e$ (–) | $[0,10]$ | Uniform
$\iota$ (∘) | $[-90,+90]$ | Uniform
Table 2: Estimated parameters from the Nelder-Mead fitting method for samples of test parameters, $m_{1}$, $m_{2}$, $s$, and $e$, including parameters Case I, II, and III of Figs. 3 and 4. Samples 1–15, 16–30, and 31–45 fundamentally correspond to Case I, II, and III, respectively, but more values of $e$ and $\iota$ are tested. The number in parentheses indicates estimated uncertainty $\sigma_{\lambda}$ of each parameter $\lambda$, if available. We summarize $\chi^{2}_{r}$ to see the fitting quality and $\left<\delta_{\lambda}\right>$ to evaluate the performance of the fitting-based parameter estimation, respectively, in the last two columns. Note that we have discarded samples of $\iota=0^{\circ}$ because not only it is less realistic but also $\iota_{T}=0^{\circ}$ makes $\delta_{\iota}$ diverge.  Sample no. | Fixed parameters |  | Altered parameters |  | Fitting results | Remarks
---|---|---|---|---|---|---
$D_{S}$ | $D_{L}$ | $v_{r}$ | $\theta$ |  | $m_{1}$ | $m_{2}$ | $s$ | $e$ | $\iota$ |  | $\chi^{2}_{r}$ | $\left<\delta_{\lambda}\right>$
| (kpc) | (kpc) | (km/s) |  (∘) |  |  ($M_{\odot}$) |  ($M_{\odot}$) | (AU) | (-) |  (∘) |  | (-) | (-) |
1 | $8.01(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.01(\textrm{N/A})$ | $119.92(\textrm{N/A})$ |  | $9.99(\textrm{N/A})$ | $10.00(\textrm{N/A})$ | $10.00(\textrm{N/A})$ | $0.98(\textrm{N/A})$ | $5.03(\textrm{N/A})$ |  | $1.14$ | $2.75\\!\times\\!10^{-3}$ |
2 | $7.72(0.26)$ | $5.01(0.49)$ | $199.99(17.68)$ | $119.94(0.08)$ |  | $10.59(1.49)$ | $10.58(1.49)$ | $9.94(0.62)$ | $0.90(0.18)$ | $14.58(9.34)$ |  | $0.99$ | $3.26\\!\times\\!10^{-2}$ |
3 | $8.22(0.01)$ | $5.01(0.02)$ | $201.08(7.44)$ | $120.03(0.09)$ |  | $9.63(0.03)$ | $9.63(0.03)$ | $10.03(0.01)$ | $1.10(0.14)$ | $29.99(0.17)$ |  | $1.08$ | $2.24\\!\times\\!10^{-2}$ |
4 | $7.96(0.70)$ | $5.03(0.97)$ | $200.00(6.67)$ | $120.00(0.08)$ |  | $10.08(1.48)$ | $10.07(1.48)$ | $9.98(0.82)$ | $1.48(0.08)$ | $4.81(9.54)$ |  | $1.05$ | $8.50\\!\times\\!10^{-3}$ |
5 | $8.07(0.54)$ | $4.99(0.80)$ | $200.00(14.15)$ | $120.06(0.08)$ |  | $9.91(1.36)$ | $9.91(1.36)$ | $10.03(0.66)$ | $1.55(0.18)$ | $14.99(7.58)$ |  | $1.07$ | $7.67\\!\times\\!10^{-3}$ |
6 | $8.03(0.02)$ | $4.95(0.05)$ | $200.00(5.57)$ | $120.07(0.08)$ |  | $10.10(0.11)$ | $10.10(0.11)$ | $9.99(0.03)$ | $1.48(0.12)$ | $28.72(0.50)$ |  | $1.03$ | $9.85\\!\times\\!10^{-3}$ |
7 | $8.04(0.34)$ | $5.11(0.77)$ | $200.32(25.33)$ | $120.01(0.08)$ |  | $9.97(2.00)$ | $9.97(2.00)$ | $9.99(0.88)$ | $2.02(0.37)$ | $6.89(30.88)$ |  | $1.05$ | $4.69\\!\times\\!10^{-2}$ |  2nd largest $\left<\delta_{\lambda}\right>$
8 | $8.00(0.25)$ | $4.99(0.64)$ | $200.01(25.29)$ | $119.94(0.07)$ |  | $10.01(1.78)$ | $9.99(1.78)$ | $10.00(0.70)$ | $1.99(0.37)$ | $15.01(13.90)$ |  | $0.90$ | $1.22\\!\times\\!10^{-3}$ |
9 | $8.30(0.04)$ | $4.98(0.10)$ | $200.68(6.77)$ | $119.99(0.07)$ |  | $9.91(0.20)$ | $9.94(0.20)$ | $10.34(0.08)$ | $2.16(0.19)$ | $30.68(0.81)$ |  | $0.99$ | $2.20\\!\times\\!10^{-2}$ |
10 | $7.95(0.34)$ | $5.09(0.78)$ | $199.93(29.25)$ | $119.89(0.08)$ |  | $9.94(2.19)$ | $9.94(2.19)$ | $9.97(0.92)$ | $2.48(0.49)$ | $10.00(24.39)$ |  | $1.09$ | $1.17\\!\times\\!10^{-1}$ |  Largest $\left<\delta_{\lambda}\right>$
11 | $7.77(0.37)$ | $4.96(0.68)$ | $200.04(21.53)$ | $120.07(0.08)$ |  | $10.48(1.89)$ | $10.50(1.89)$ | $10.05(0.81)$ | $2.37(0.36)$ | $15.54(11.19)$ |  | $1.08$ | $2.55\\!\times\\!10^{-2}$ |
12 | $7.95(0.06)$ | $5.00(0.13)$ | $199.99(8.32)$ | $119.98(0.08)$ |  | $10.14(0.36)$ | $10.15(0.36)$ | $9.99(0.14)$ | $2.44(0.23)$ | $29.68(1.35)$ |  | $1.16$ | $7.84\\!\times\\!10^{-3}$ |
13 | $8.01(10^{-3})$ | $5.00(0.01)$ | $200.00(4.32)$ | $119.93(0.10)$ |  | $9.98(0.02)$ | $9.99(0.02)$ | $10.00(10^{-3})$ | $2.97(0.14)$ | $4.99(0.55)$ |  | $1.04$ | $1.74\\!\times\\!10^{-3}$ |
14 | $8.00(0.60)$ | $5.00(0.89)$ | $200.00(16.60)$ | $120.03(0.08)$ |  | $9.98(1.60)$ | $10.01(1.60)$ | $10.00(0.78)$ | $3.00(0.34)$ | $15.04(8.79)$ |  | $1.06$ | $6.86\\!\times\\!10^{-4}$ |
15 | $7.88(0.07)$ | $5.00(0.15)$ | $199.96(8.52)$ | $120.07(0.07)$ |  | $10.28(0.45)$ | $10.27(0.45)$ | $9.95(0.17)$ | $2.89(0.25)$ | $29.40(1.63)$ |  | $0.85$ | $1.50\\!\times\\!10^{-2}$ |
16 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $10.00(\textrm{N/A})$ | $10.00(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $1.00(\textrm{N/A})$ | $5.02(\textrm{N/A})$ |  | $0.93$ | $7.64\\!\times\\!10^{-4}$ |
17 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $9.99(\textrm{N/A})$ | $10.02(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $1.01(\textrm{N/A})$ | $15.00(\textrm{N/A})$ |  | $0.81$ | $9.25\\!\times\\!10^{-4}$ |
18 | $8.04(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $119.86(\textrm{N/A})$ |  | $9.96(\textrm{N/A})$ | $9.95(\textrm{N/A})$ | $20.02(\textrm{N/A})$ | $0.98(\textrm{N/A})$ | $29.97(\textrm{N/A})$ |  | $0.96$ | $3.89\\!\times\\!10^{-3}$ |
19 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.03(\textrm{N/A})$ |  | $10.01(\textrm{N/A})$ | $9.97(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $1.48(\textrm{N/A})$ | $5.03(\textrm{N/A})$ |  | $1.06$ | $2.64\\!\times\\!10^{-3}$ |
20 | $7.98(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $10.02(\textrm{N/A})$ | $10.07(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $1.51(\textrm{N/A})$ | $14.97(\textrm{N/A})$ |  | $0.90$ | $2.32\\!\times\\!10^{-3}$ |
21 | $8.00(0.03)$ | $5.00(0.06)$ | $199.99(19.61)$ | $120.03(0.07)$ |  | $9.99(0.20)$ | $9.99(0.20)$ | $20.00(0.17)$ | $1.50(0.45)$ | $29.97(0.59)$ |  | $1.00$ | $7.71\\!\times\\!10^{-4}$ |
22 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.01(\textrm{N/A})$ | $119.99(\textrm{N/A})$ |  | $10.00(\textrm{N/A})$ | $9.99(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.00(\textrm{N/A})$ | $5.01(\textrm{N/A})$ |  | $0.93$ | $4.20\\!\times\\!10^{-4}$ |
23 | $8.01(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $119.98(\textrm{N/A})$ |  | $10.00(\textrm{N/A})$ | $9.99(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.01(\textrm{N/A})$ | $15.00(\textrm{N/A})$ |  | $1.05$ | $6.85\\!\times\\!10^{-4}$ |
24 | $8.00(10^{-3})$ | $5.00(0.01)$ | $200.00(9.02)$ | $119.99(0.04)$ |  | $10.00(0.02)$ | $9.99(0.02)$ | $20.00(0.02)$ | $2.00(0.25)$ | $30.00(0.06)$ |  | $0.90$ | $2.49\\!\times\\!10^{-4}$ |
25 | $8.00(\textrm{N/A})$ | $5.01(\textrm{N/A})$ | $200.08(\textrm{N/A})$ | $119.92(\textrm{N/A})$ |  | $10.02(\textrm{N/A})$ | $10.03(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.51(\textrm{N/A})$ | $4.98(\textrm{N/A})$ |  | $1.03$ | $1.98\\!\times\\!10^{-3}$ |
26 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.07(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $10.00(\textrm{N/A})$ | $10.03(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.52(\textrm{N/A})$ | $14.93(\textrm{N/A})$ |  | $0.82$ | $1.80\\!\times\\!10^{-3}$ |
27 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.01(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $10.00(\textrm{N/A})$ | $10.00(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.50(\textrm{N/A})$ | $30.00(\textrm{N/A})$ |  | $1.04$ | $7.68\\!\times\\!10^{-5}$ |  Smallest $\left<\delta_{\lambda}\right>$
28 | $8.05(\textrm{N/A})$ | $5.01(\textrm{N/A})$ | $199.99(\textrm{N/A})$ | $120.01(\textrm{N/A})$ |  | $9.93(\textrm{N/A})$ | $9.90(\textrm{N/A})$ | $20.01(\textrm{N/A})$ | $3.01(\textrm{N/A})$ | $4.89(\textrm{N/A})$ |  | $0.94$ | $5.76\\!\times\\!10^{-3}$ |
29 | $8.01(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.01(\textrm{N/A})$ | $119.94(\textrm{N/A})$ |  | $9.99(\textrm{N/A})$ | $9.99(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $3.01(\textrm{N/A})$ | $15.00(\textrm{N/A})$ |  | $0.94$ | $7.12\\!\times\\!10^{-4}$ |
30 | $8.00(\textrm{N/A})$ | $4.99(\textrm{N/A})$ | $199.99(\textrm{N/A})$ | $119.97(\textrm{N/A})$ |  | $9.96(\textrm{N/A})$ | $9.96(\textrm{N/A})$ | $20.01(\textrm{N/A})$ | $3.01(\textrm{N/A})$ | $30.20(\textrm{N/A})$ |  | $0.94$ | $2.26\\!\times\\!10^{-3}$ |
31 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.02(\textrm{N/A})$ | $120.03(\textrm{N/A})$ |  | $30.00(\textrm{N/A})$ | $10.05(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $1.06(\textrm{N/A})$ | $5.01(\textrm{N/A})$ |  | $1.02$ | $7.18\\!\times\\!10^{-3}$ |
32 | $8.00(0.01)$ | $4.99(0.01)$ | $200.01(133.94)$ | $119.94(0.08)$ |  | $30.02(0.20)$ | $10.02(0.12)$ | $20.00(0.03)$ | $1.04(2.71)$ | $14.97(0.33)$ |  | $1.13$ | $5.77\\!\times\\!10^{-3}$ |
33 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.02(\textrm{N/A})$ | $119.90(\textrm{N/A})$ |  | $30.03(\textrm{N/A})$ | $10.04(\textrm{N/A})$ | $19.99(\textrm{N/A})$ | $1.13(\textrm{N/A})$ | $30.01(\textrm{N/A})$ |  | $1.02$ | $1.54\\!\times\\!10^{-2}$ |
34 | $7.99(0.01)$ | $4.99(0.03)$ | $199.99(47.42)$ | $119.95(0.08)$ |  | $30.02(0.41)$ | $10.00(0.13)$ | $20.00(0.15)$ | $1.53(1.17)$ | $5.16(0.94)$ |  | $1.19$ | $5.96\\!\times\\!10^{-3}$ |
35 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.02(\textrm{N/A})$ | $119.97(\textrm{N/A})$ |  | $30.02(\textrm{N/A})$ | $10.04(\textrm{N/A})$ | $19.99(\textrm{N/A})$ | $1.59(\textrm{N/A})$ | $14.98(\textrm{N/A})$ |  | $1.02$ | $7.69\\!\times\\!10^{-3}$ |
36 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $30.01(\textrm{N/A})$ | $10.02(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $1.55(\textrm{N/A})$ | $29.99(\textrm{N/A})$ |  | $1.02$ | $4.22\\!\times\\!10^{-3}$ |
37 | $8.06(0.08)$ | $5.04(0.11)$ | $200.01(75.29)$ | $120.07(0.07)$ |  | $29.76(2.83)$ | $9.95(0.95)$ | $19.99(0.92)$ | $2.10(2.24)$ | $5.10(6.68)$ |  | $1.06$ | $1.13\\!\times\\!10^{-2}$ |
38 | $7.98(\textrm{N/A})$ | $5.01(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $119.97(\textrm{N/A})$ |  | $30.12(\textrm{N/A})$ | $10.00(\textrm{N/A})$ | $19.99(\textrm{N/A})$ | $1.92(\textrm{N/A})$ | $15.07(\textrm{N/A})$ |  | $0.94$ | $5.87\\!\times\\!10^{-3}$ |
39 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.00(\textrm{N/A})$ |  | $30.00(\textrm{N/A})$ | $10.00(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.00(\textrm{N/A})$ | $30.00(\textrm{N/A})$ |  | $1.03$ | $2.25\\!\times\\!10^{-4}$ |  2nd smallest $\left<\delta_{\lambda}\right>$
40 | $7.99(0.15)$ | $5.00(0.20)$ | $200.01(37.97)$ | $119.94(0.06)$ |  | $30.07(6.30)$ | $9.99(2.09)$ | $20.01(2.05)$ | $2.44(1.18)$ | $4.92(13.44)$ |  | $1.09$ | $5.10\\!\times\\!10^{-3}$ |
41 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $120.02(\textrm{N/A})$ |  | $29.97(\textrm{N/A})$ | $10.01(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.54(\textrm{N/A})$ | $14.99(\textrm{N/A})$ |  | $1.11$ | $2.00\\!\times\\!10^{-3}$ |
42 | $7.96(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $199.98(\textrm{N/A})$ | $119.91(\textrm{N/A})$ |  | $30.27(\textrm{N/A})$ | $10.06(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.45(\textrm{N/A})$ | $29.99(\textrm{N/A})$ |  | $0.90$ | $4.74\\!\times\\!10^{-3}$ |
43 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $119.99(\textrm{N/A})$ |  | $30.00(\textrm{N/A})$ | $9.98(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.99(\textrm{N/A})$ | $5.01(\textrm{N/A})$ |  | $0.99$ | $8.90\\!\times\\!10^{-4}$ |
44 | $7.98(\textrm{N/A})$ | $4.99(\textrm{N/A})$ | $200.00(\textrm{N/A})$ | $119.96(\textrm{N/A})$ |  | $30.06(\textrm{N/A})$ | $9.99(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.89(\textrm{N/A})$ | $15.15(\textrm{N/A})$ |  | $0.98$ | $6.14\\!\times\\!10^{-3}$ |
45 | $8.00(\textrm{N/A})$ | $5.00(\textrm{N/A})$ | $199.99(\textrm{N/A})$ | $120.05(\textrm{N/A})$ |  | $29.98(\textrm{N/A})$ | $9.99(\textrm{N/A})$ | $20.00(\textrm{N/A})$ | $2.95(\textrm{N/A})$ | $30.00(\textrm{N/A})$ |  | $1.06$ | $1.96\\!\times\\!10^{-3}$ |

We perform the fitting with the prior range tabulated in Table 1 and summarize
the result obtained using the method in a Python package, lmfit (Newville et
al., 2016), in Table 2 for simulated samples prepared with the combination of
different values of selected parameters $q$, $s$, $e$, and $\iota$ including
the samples studied in the previous section. Here, we test following parameter
values, $q=\\{0.3,1.0\\}$, $s=\\{10\leavevmode\nobreak\
\textrm{AU},20\leavevmode\nobreak\ \textrm{AU}\\}$,
$e=\\{1.0,1.5,2.0,2.5,3.0\\}$, and
$\iota=\\{5^{\circ},15^{\circ},30^{\circ}\\}$.

Figure 5: True value of eccentricity $e$ versus averaged relative difference
$\left<\delta_{\lambda}\right>$ of Case I (left), II (middle), and III
(right). We collect the result of five different values of $\iota$, i.e.,
$\iota=5^{\circ}$, $10^{\circ}$, $15^{\circ}$, $20^{\circ}$, and $30^{\circ}$,
into one figure of each case for comparison. One can see that there is no
correlation between $e$ and $\left<\delta_{\lambda}\right>$. Also, this result
shows that the value of $\iota$ is unrelated to
$\left<\delta_{\lambda}\right>$ with given $e$.

(a) Sample 27 (Smallest $\left<\delta_{\lambda}\right>$) (b) Sample 10
(Largest $\left<\delta_{\lambda}\right>$)

Figure 6: Input and best fit light curves of Sample 27 (left) and 10 (right)
corresponding to the smallest and the largest $\left<\delta_{\lambda}\right>$,
respectively.

As tabulated in Table 2, in order to measure the goodness-of-fitting, we
compute the reduced $\chi^{2}$, i.e., $\chi^{2}_{r}$, which is defined as

$\chi^{2}_{r}=\sum^{N}_{i}\rho_{i}^{2}/(N-N_{\textrm{vars}})\leavevmode\nobreak\
,$ (8)

where $\rho_{i}$ is the residual of magnitude subtracting the inferred
magnitude $d({\Lambda})_{i}$ from the true magnitude $d_{T,i}$ and scaled by
the data uncertainty $\sigma_{i}$, i.e.,

$\rho_{i}=\frac{d_{T,i}-d({\Lambda})_{i}}{\sigma_{i}}$ (9)

at each observation point $i$. In this work, we adopt the same form of
$\sigma_{i}$ used in Ma et al. (2022) which is given by

$\sigma_{i}=\sqrt{\sigma_{\textrm{sys}}^{2}+\sigma_{0}^{2}10^{0.8(d_{T,i}-d_{0})}},$
(10)

where $\sigma_{\textrm{sys}}=0.004$ is the systematic floor and
$\sigma_{0}=0.04$ is the uncertainty at the reference magnitude $d_{0}$ set as
18. $N$ and $N_{\textrm{vars}}$ denote the number of observation data points
and the number of variables that are fixed as $400$ and $9$ in this work,
respectively. Note that we take $\sum_{i}\rho_{i}$ as the cost function of the
NM method for seeking the best fitting parameters. Along with $\chi_{r}^{2}$,
for the figure-of-merit of the fitting-based parameter estimation, we
calculate the averaged relative difference $\left<\delta_{\lambda}\right>$ as
follows

$\left<\delta_{\lambda}\right>=\sum_{\lambda}\delta_{\lambda}/N_{\textrm{vars}}\leavevmode\nobreak\
,$ (11)

where $\delta_{\lambda}$ is the relative difference between the true value
$\lambda_{T}$ and inferred value $\lambda_{I}$ of each parameter $\lambda$,
i.e., $\delta_{\lambda}=|\lambda_{T}-\lambda_{I}|/\lambda_{T}$, and summarize
the value for each sample in Table 2.

Looking at Table 2, we can see that the 9 parameters of binary microlensing
are mostly well recovered by the NM-based fitting. Also, the goodness-of-fit,
represented by the value of $\chi^{2}_{r}$, of all tested samples results in
around $1$ which means the extent of residuals is in accord with given degrees
of freedom, i.e., $(N-N_{\textrm{var}})$, in the fitting. When we compare the
range of $\left<\delta_{\lambda}\right>$ of samples in each case, the range of
$\left<\delta_{\lambda}\right>$ of Case I, resulted in
$\sim\\!10^{-3}$–$10^{-2}$, are relatively larger than the ranges of Case II
and III, $\sim\\!10^{-4}$–$10^{-3}$ and $\sim\\!10^{-4}$–$10^{-2}$,
respectively. This result tells us that relatively smaller/narrower caustics
of shorter $s$ such as $s=10\leavevmode\nobreak\ \textrm{AU}$ hinders precise
estimations than longer $s$ such as $s=20\leavevmode\nobreak\ \textrm{AU}$.

Next, we examine the relation between the value of $e$ and
$\left<\delta_{\lambda}\right>$ because our main interest is whether the
precise parameter estimation is possible or not for a BH binary lens of high
eccentric orbit such as $e\geq 1$. In order to examine the correlation, we
draw the plot of $e$ versus $\left<\delta_{\lambda}\right>$ in Fig. 5. Also,
for comparison, we collect the result of five different values of $\iota$,
i.e., $\iota=\\{5^{\circ}$, $10^{\circ}$, $15^{\circ}$, $20^{\circ}$,
$30^{\circ}\\}$, into the figure of each case. One can see from the subfigures
of Fig. 5 that there is no $e$-dependency in $\left<\delta_{\lambda}\right>$.
In other words, it is possible to obtain low $\left<\delta_{\lambda}\right>$
even for a BH binary of higher $e$. The lesson we can learn from this result
is that there is no obvious hindrance in the parameter estimation of a BH
binary of $e\geq 1$ from binary microlensing light curves, if a well-prepared
model is applied to the fitting. Beside, we can see that
$\left<\delta_{\lambda}\right>$ is also independent from the value of $\iota$.
Hence, the NM-based fitting with the considered binary lens model allows us to
recognise the $\iota$-dependent changes in $s$ and $e$.

From this fitting-based study, it turns out that the values of
$\left<\delta_{\lambda}\right>$ of Sample 27 and 39 are the smallest and
second smallest $\left<\delta_{\lambda}\right>$ while Sample 10 and 7 are the
largest and second largest $\left<\delta_{\lambda}\right>$, respectively,
amongst all tested samples. In specific, we can see that the estimated value
of $\iota$ of Sample 10 and 7 are more different from the true value,
$+5.00^{\circ}$ and $+1.89^{\circ}$, respectively, than other samples which
mostly resulted in less than $\pm 1^{\circ}$ deviations from the true $\iota$.
Despite of it, the difference between the true value and estimated value of
other parameters are less than $10\%$ of the true value. Indeed, when we plot
the light curve with the best fit parameters, as presented in Fig. 6 for
Sample 27 and 10, we see that the mismatch between the input and best fit
light curves mainly happens at peaks/bumps and the best fit light curves can
more or less perfectly reconstruct input light curves.

On the other hand, the NM fitting for most samples of Case II and III failed
in estimating uncertainties of all 9 parameters. This is because, for example,
(i) if a variable has no practical contribution on the fit or (ii) if the fit
estimate the value of a variable being near the maximum or minimum of given
bounds, the variable likely causes the covariance matrix singular that making
standard errors impossible to estimate (Newville et al., 2016). In reality,
the uncertainty of estimated parameter is quite important because it not only
gives us confidence in the estimation but also it is helpful to conduct
follow-up analyzes with the given range of uncertainty. Therefore, we need an
alternative of parameter estimation that allows us to estimate uncertainties.

Table 3: True value $\lambda_{T}$ and inferred value $\lambda_{I}$ for the 9 model parameters of Sample 27 and 10 corresponding to the smallest and the largest $\left<\delta_{\lambda}\right>$ values in Table 2. For comparison, we tabulate $\lambda_{I,\textrm{NM}}$ and $\delta_{\lambda,\textrm{NM}}$ together with $\lambda_{I,\textrm{MCMC}}$ and $\delta_{\lambda,\textrm{MCMC}}$. The uncertainty given for $\lambda_{I}$ represents $1\sigma$ error, except $\lambda_{I,\textrm{NM}}$ that were failed in the uncertainty estimation from the NM fitting. We also summarize $\delta_{\lambda}$ of each parameter calculated with the true value and the median value. $\lambda$ (unit) | Sample 27 |  | Sample 10
---|---|---|---
| $\lambda_{T}$ | $\lambda_{I,\textrm{NM}}$ | $\delta_{\lambda,\textrm{NM}}$ | $\lambda_{I,\textrm{MCMC}}$ | $\delta_{\lambda,\textrm{MCMC}}$ |  | $\lambda_{T}$ | $\lambda_{I,\textrm{NM}}$ | $\delta_{\lambda,\textrm{NM}}$ | $\lambda_{I,\textrm{MCMC}}$ | $\delta_{\lambda,\textrm{MCMC}}$
$D_{S}$ (kpc) | $8$ | $8.00\\!\pm\\!\textrm{N/A}$ | $2.42\\!\times\\!10^{-5}$ | $8.01_{-0.29}^{+0.57}$ | $7.71\\!\times\\!10^{-4}$ |  | $8$ | $7.95\\!\pm\\!0.34$ | $6.18\\!\times\\!10^{-3}$ | $8.66_{-1.48}^{+1.62}$ | $8.28\\!\times\\!10^{-2}$
$D_{L}$ (kpc) | $5$ | $5.00\\!\pm\\!\textrm{N/A}$ | $6.09\\!\times\\!10^{-5}$ | $5.00_{-0.19}^{+0.28}$ | $4.90\\!\times\\!10^{-5}$ |  | $5$ | $5.09\\!\pm\\!0.78$ | $1.78\\!\times\\!10^{-2}$ | $5.45_{-0.88}^{+0.85}$ | $8.95\\!\times\\!10^{-2}$
$v_{r}$ (km/s) | $200$ | $200.01\\!\pm\\!\textrm{N/A}$ | $3.32\\!\times\\!10^{-5}$ | $201.94_{-7.33}^{+5.44}$ | $9.72\\!\times\\!10^{-3}$ |  | $200$ | $199.93\\!\pm\\!29.25$ | $3.64\\!\times\\!10^{-4}$ | $201.00_{-5.49}^{+5.38}$ | $4.99\\!\times\\!10^{-3}$
$\theta$ (∘) | $120$ | $120.00\\!\pm\\!\textrm{N/A}$ | $1.68\\!\times\\!10^{-5}$ | $120.00_{-0.02}^{+0.03}$ | $6.07\\!\times\\!10^{-6}$ |  | $120$ | $119.89\\!\pm\\!0.08$ | $9.07\\!\times\\!10^{-4}$ | $119.93_{-0.08}^{+0.08}$ | $5.92\\!\times\\!10^{-4}$
$m_{1}$ ($M_{\odot}$) | $10$ | $10.00\\!\pm\\!\textrm{N/A}$ | $5.07\\!\times\\!10^{-5}$ | $10.16_{-0.46}^{+0.40}$ | $1.58\\!\times\\!10^{-2}$ |  | $10$ | $9.94\\!\pm\\!2.19$ | $5.78\\!\times\\!10^{-3}$ | $11.08_{-1.96}^{+2.00}$ | $1.08\\!\times\\!10^{-1}$
$m_{2}$ ($M_{\odot}$) | $10$ | $10.00\\!\pm\\!\textrm{N/A}$ | $8.16\\!\times\\!10^{-5}$ | $10.16_{-0.46}^{+0.40}$ | $1.56\\!\times\\!10^{-2}$ |  | $10$ | $9.94\\!\pm\\!2.19$ | $5.53\\!\times\\!10^{-3}$ | $11.09_{-1.95}^{+2.02}$ | $1.09\\!\times\\!10^{-1}$
$s$ (AU) | $20$ | $20.00\\!\pm\\!\textrm{N/A}$ | $7.95\\!\times\\!10^{-5}$ | $20.30_{-0.99}^{+1.16}$ | $2.79\\!\times\\!10^{-2}$ |  | $10$ | $9.97\\!\pm\\!0.92$ | $3.44\\!\times\\!10^{-3}$ | $10.83_{-1.73}^{+2.38}$ | $8.34\\!\times\\!10^{-2}$
$e$ (–) | $2.5$ | $2.50\\!\pm\\!\textrm{N/A}$ | $3.20\\!\times\\!10^{-4}$ | $2.57_{-0.25}^{+0.17}$ | $5.44\\!\times\\!10^{-3}$ |  | $2.5$ | $2.48\\!\pm\\!0.49$ | $9.24\\!\times\\!10^{-3}$ | $2.50_{-0.11}^{+0.10}$ | $1.38\\!\times\\!10^{-3}$
$\iota$ (∘) | $30$ | $30.00\\!\pm\\!\textrm{N/A}$ | $2.46\\!\times\\!10^{-5}$ | $30.16_{-0.83}^{+0.93}$ | $7.16\\!\times\\!10^{-4}$ |  | $5$ | $10.00\\!\pm\\!24.39$ | $1.00\\!\times\\!10^{0}$ | $4.99_{-4.78}^{+5.39}$ | $1.18\\!\times\\!10^{-3}$
Figure 7: Results of MCMC-based parameter estimation for Sample 27. The corner
plot represents posterior distributions of 9 parameters inferred by the MCMC-
based analysis. We mark the true value as black solid line and the median
value as red solid line with the $1\sigma$ uncertainty depicted as black
dashed line. The upper-right inset figure shows the reconstructed light curve
using the median values of 9 parameters and the residuals after the
subtraction of reconstructed light curve from the input.  Figure 8: Results of
MCMC-based parameter estimation for Sample 10. Similar to Fig. 7, we present
corner plot and reconstructed light curve.

In this work, we consider the Markov chain Monte Carlo (MCMC)-based Bayesian
inference (Goodman & Weare, 2010) as an alternative of parameter estimation.
For the Bayesian inference, we conduct sampling of the posterior probability
density of parameters of interest using the ensemble sampler implemented in
the emcee (Foreman-Mackey et al., 2013) package. For consistency, we take the
same prior range used in the NM fitting. Finding the global minimum is tested
with the log-likelihood function of a $\Lambda$-dependent light curve data,
given the input data $d$, which is defined as

$\ln
P(d|{\Lambda})=-\frac{1}{2}\sum_{i}\left[\frac{(d({\Lambda})_{i}-d_{i})^{2}}{\sigma_{i}^{2}}+\ln
2\pi\sigma_{i}^{2}\right],$ (12)

where $d({\Lambda})$ is the light curve drawn by the model with a given
parameter set ${\Lambda}$ and $\sigma_{i}$ is the same data uncertainty taken
for the calculation of $\rho_{i}$ in the NM fitting. Then, we let each MCMC
process take 20 walkers and each walker draws 5,000 samples from the posterior
distribution.

In Table 3, we summarize the result of the MCMC-based parameter estimation for
Sample 27 that was resulted in the smallest $\left<\delta_{\lambda}\right>$
but the uncertainty estimation was failed in the NM fitting. Also, we examine
Sample 10 resulted in the largest $\left<\delta_{\lambda}\right>$ although its
uncertainty estimation is done even with the NM fitting. One can see from
$\lambda_{I,\textrm{MCMC}}$ of Sample 27 that the uncertainty estimation for
each inferred value is available with the MCMC-based parameter estimation as
desired. On top of that, the estimated uncertainty successfully covers each
$\lambda_{T}$ although the median value itself is slightly deviated from
$\lambda_{T}$. However, when we compare $\delta_{\lambda,\textrm{MCMC}}$ with
$\delta_{\lambda,\textrm{NM}}$ for both Sample 27 and 10, it is shown that the
deviation of $\lambda_{I,\textrm{MCMC}}$ from $\lambda_{T}$ is mostly bigger
than the deviation of $\lambda_{I,\textrm{NM}}$ from the same $\lambda_{T}$.
In consequence, for the two samples, $\left<\delta_{\lambda}\right>$ is
calculated as $1.00\times 10^{-2}$ and $5.35\times 10^{-2}$, respectively,
that were $7.68\times 10^{-5}$ and $1.17\times 10^{-1}$ from the NM fitting.
Despite this, the uncertainty estimated from the MCMC-based parameter
estimation results in better precision on several parameters, e.g., $v_{r}$,
$m_{1}$, $m_{2}$, $e$, and $\iota$, than the NM-based parameter estimation. In
specific, the parameter estimation of $e$ and $\iota$ is improved by the MCMC-
based parameter estimation in terms of not only the precision but also the
accuracy.

Table 4: Inferred parameter value $\lambda_{I}$ for Sample 10, obtained from the initial NM-based fitting ($\lambda_{I,\textrm{NM}}$) and from refitting with NM method based on the $1\sigma$ error bar of MCMC result ($\lambda_{I,\textrm{NM}_{\textrm{refit}}}$) for the 9 model parameters of Sample 10.  Parameter ($\lambda_{T}$) | $\lambda_{I,\textrm{NM}}$ | $\delta_{\lambda,\textrm{NM}}$ | $\lambda_{I,\textrm{NM}_{\textrm{refit}}}$ | $\delta_{\lambda,\textrm{NM}_{\textrm{refit}}}$
---|---|---|---|---
$D_{S}$ ($8$ kpc) | $7.95\\!\pm\\!0.34$ | $6.18\\!\times\\!10^{-3}$ | $7.99\\!\pm\\!\textrm{N/A}$ | $6.73\\!\times\\!10^{-4}$
$D_{L}$ ($5$ kpc) | $5.09\\!\pm\\!0.78$ | $1.78\\!\times\\!10^{-2}$ | $4.89\\!\pm\\!\textrm{N/A}$ | $2.15\\!\times\\!10^{-2}$
$v_{r}$ ($200$ km/s) | $199.93\\!\pm\\!29.25$ | $3.64\\!\times\\!10^{-4}$ | $200.01\\!\pm\\!\textrm{N/A}$ | $4.81\\!\times\\!10^{-5}$
$\theta$ ($120^{\circ}$) | $119.89\\!\pm\\!0.08$ | $9.07\\!\times\\!10^{-4}$ | $119.93\\!\pm\\!\textrm{N/A}$ | $5.45\\!\times\\!10^{-4}$
$m_{1}$ ($10\leavevmode\nobreak\ M_{\odot}$) | $9.94\\!\pm\\!2.19$ | $5.78\\!\times\\!10^{-3}$ | $10.02\\!\pm\\!\textrm{N/A}$ | $1.64\\!\times\\!10^{-3}$
$m_{2}$ ($10\leavevmode\nobreak\ M_{\odot}$) | $9.94\\!\pm\\!2.19$ | $5.53\\!\times\\!10^{-3}$ | $10.03\\!\pm\\!\textrm{N/A}$ | $2.95\\!\times\\!10^{-3}$
$s$ ($10$ AU) | $9.97\\!\pm\\!0.92$ | $3.44\\!\times\\!10^{-3}$ | $10.07\\!\pm\\!\textrm{N/A}$ | $7.36\\!\times\\!10^{-3}$
$e$ (2.5) | $2.48\\!\pm\\!0.49$ | $9.24\\!\times\\!10^{-3}$ | $2.50\\!\pm\\!\textrm{N/A}$ | $1.59\\!\times\\!10^{-3}$
$\iota$ ($5^{\circ}$) | $10.00\\!\pm\\!24.39$ | $1.00\\!\times\\!10^{0}$ | $5.17\\!\pm\\!\textrm{N/A}$ | $3.36\\!\times\\!10^{-2}$

(a) Best fit light curve of Sample 10 from the initial NM (b) Best fit light
curve of Sample 10 from NM${}_{\textrm{refit}}$

Figure 9: Input and best fit light curves of Sample 10 resulted from the
initial NM fitting (top) and the refitting (bottom), respectively.

We present the posterior distributions of the 9 selected parameters obtained
from the MCMC-based parameter estimation. To this end, as shown in Figs. 7
(Sample 27) and 8 (Sample 10), we draw the corner plot which consists of 1D
histograms and 2D contour plots. We also depict the noisy input light curve
and reconstructed light curve based on the median value of the posterior
distribution of each parameter. In the 1D histogram and 2D contour plot, we
mark the true value and the median of inferred values as the black-solid line
and red-solid line, respectively. The black-dashed vertical lines indicates
the lower and upper bounds of $1\sigma$ error for the median value. Looking at
1D histograms and 2D contour plots in Fig. 7, we find that all posterior
distributions peak around the true value and all true values place within the
$1\sigma$ error bar. Similarly, we can see from Fig. 8 that the lower and
upper bounds of $1\sigma$ error of all posterior distributions for Sample 10
contains true values successfully although there are some deviations between
the true value and the inferred value that seen in Table 3. However, when we
look at the reconstructed light curves and the residuals presented in both
Figs. 7 and 8, the mismatch between the input and reconstructed ones become
much bigger than the best fit result shown in Fig. 6 because of the more
bigger difference in the inferred values of the MCMC-based result than that of
the NM-based result.

To address the issue on bigger difference in the MCMC-based parameter
estimation, we examine whether utilising the uncertainty information obtained
from MCMC can be helpful in precise parameter estimation. As a demonstration,
we provide the result from refitting Sample 10 with the NM method based on the
$1\sigma$ error bar of the MCMC result in Table 4 and Fig. 9. From Table 4,
one can see improvements in $\delta_{\lambda}$ of each parameter, except
$D_{L}$ and $s$, although $\delta_{D_{L},\textrm{NM}_{\textrm{refit}}}$ and
$\delta_{s,\textrm{NM}_{\textrm{refit}}}$ are still enhanced compared to
$\delta_{D_{L},\textrm{MCMC}}$ and $\delta_{s,\textrm{MCMC}}$, respectively.
Also, the improvement in the inferred value of most parameters concludes to
$\left<\delta_{\lambda}\right>$ too, from
$\left<\delta_{\lambda}\right>_{\textrm{NM}}=1.17\times 10^{-1}$ to
$\left<\delta_{\lambda}\right>_{\textrm{NM}_{\textrm{refit}}}=7.76\times
10^{-3}$, even though the refitting failed in uncertainty estimation for all
parameters unlike the initial fitting. Therefore, we can confirm that
utilising MCMC-based parameter estimation is indeed helpful in obtaining
better estimation with a fitting-based parameter estimation.

Despite of such improvement, when we compare the best fit light curves
presented in Fig. 9, we observe that the residual around the right-hand-side
bump is still larger than that of the initial NM fitting result. This result
addresses not only that a relatively good fitting does not always guarantee
the best parameter estimation but also that attaining the best result in both
fitting and parameter estimation is quite nontrivial because of the degeneracy
in the parameter-dependent light curve.

##  5 Discussions

In this study, we have explored the characteristics of microlensing events
caused by the stellar-mass BH binary, focusing on high eccentric lens orbits.
We have compared the shapes of caustics and light curves in several selected
cases, depending on the orbital properties of BH binary lens, and attempted to
estimate the parameters that describe the lens and source system from
simulated light curves. We have demonstrated parameter estimation by utilizing
two different methods—the Nelder-Mead fitting and a MCMC-based Bayesian
inference—and confirmed that both approaches successfully recover the true
parameter values within a reasonable uncertainty. The lesson we could learn
from this study is that microlensing by BHs of hyperbolic orbits can even be
distinguished from the microlensing by elliptic BH binaries if adequate
estimation methods are employed for given binary microlensing light curves.

However, we should note that we primarily examined the dynamical properties of
the BH binary lens system, rather than exploring other parameters associated
with the source and observer. In particular, as previously mentioned,
microlensing parallax was disregarded due to its predicted small magnitude and
the complexity of its evaluation for the BH binary lens. However, if we are
able to appropriately account for the parallax effect through additional
future work, more reliable parameter estimation will be achievable in actual
observations.

To date, BH binaries have only been observed through GW detections. Due to the
frequency range of the current GW detectors, the observed BH binaries have
been limited to those with very small separations, just before merging.
However, the BH system with an orbital size of about 10 AU, considered in this
work, emits GWs at a frequency corresponding to the frequency band of Pulsar
Timing Array (PTA). Moreover, if they are located in our Galaxy, the GW
signals from these systems could be observed by the PTA using Square Kilometre
Array. If the stellar-mass BH binaries are also observed through the
microlensing effect by forthcoming telescopes such as Nancy Grace Roman Space
Telescope, it would mark a significant milestone in multi-messenger astronomy,
enriching our understanding of the BH population.

###### Acknowledgements.

We thank Jeongcho Kim and Changsu Choi for fruitful discussions. This work is
supported by the National Research Foundation of Korea (NRF) grants funded by
the Ministry of Science and ICT (MSIT) of the Korea Government
(NRF-2020R1C1C1005863, NRF-2021R1F1A1051269). The work of K.K. is partially
supported by the Korea Astronomy and Space Science Institute under the R&D
program (Project No. 2024-1-810-02) supervised by the MSIT. K.K. also
acknowledges that the computational work reported in this paper was partly
performed on the KASI Science Cloud platform supported by Korea Astronomy and
Space Science Institute. We acknowledge the hospitality at the Asia Pacific
Center for Theoretical Physics where a part of this work was done.

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##  Appendix A Coefficients of Complex Polynomial Lens Equation and Critical
Lines for Binary Lenses

The lens equation for the binary lens can be represented as a complex
polynomial equation of degree 5:

$c_{5}z^{5}+c_{4}z^{4}+c_{3}z^{3}+c_{2}z^{2}+c_{1}z+c_{0}=0\leavevmode\nobreak\
,$ (13)

for two point-like lenses, $m_{1}$ and $m_{2}$, at $z_{1}$ and $z_{2}$,
respectively, in the lens plane. The coefficients $c_{i}$ of the polynomial
equation are given as

$\displaystyle c_{5}$ $\displaystyle=$ $\displaystyle-
z_{1}^{*}z_{2}^{*}\\!+\\!(z_{1}^{*}\\!+\\!z_{2}^{*})z_{s}^{*}\\!-\\!z_{s}^{*2}\leavevmode\nobreak\
,$ $\displaystyle c_{4}$ $\displaystyle=$ $\displaystyle
m_{1}z_{1}^{*}\\!+\\!m_{2}z_{2}^{*}\\!+\\!(z_{1}^{*}z_{2}^{*}\\!+\\!z_{s}^{*2})(2z_{1}\\!+\\!2z_{2}\\!+\\!z_{s})\\!-\\!z_{s}^{*}[2\mu$
$\displaystyle+(z_{1}^{*}\\!+\\!z_{2}^{*})(2z_{1}\\!+\\!2z_{2}\\!+\\!z_{s})]\leavevmode\nobreak\
,$ $\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle-
m_{1}(z_{1}z_{1}^{*}\\!-\\!z_{1}z_{2}^{*}\\!+\\!2z_{2}z_{1}^{*})\\!-m_{2}(z_{2}z_{2}^{*}\\!-\\!z_{2}z_{1}^{*}\\!+\\!2z_{1}z_{2}^{*})$
$\displaystyle-2\mu(z_{1}^{*}\\!+\\!z_{2}^{*})z_{s}\\!+\\!2z_{s}^{*}(m_{1}z_{2}\\!+\\!m_{2}z_{1}\\!+\\!2\mu
z_{s})\\!-\\![z_{1}^{*}z_{2}^{*}$
$\displaystyle+z_{s}^{*2}-(z_{1}^{*}\\!+\\!z_{2}^{*})z_{s}^{*}][(z_{1}\\!+\\!z_{2})^{2}\\!+\\!2z_{1}z_{2}\\!+\\!2(z_{1}\\!+\\!z_{2})z_{s}]\leavevmode\nobreak\
,$ $\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle
4\mu^{2}z_{s}\\!-\\!2\mu(m_{1}z_{1}\\!+\\!m_{2}z_{2})\\!+\\!2\nu(z_{1}^{2}z_{2}^{*}\\!-\\!z_{2}^{2}z_{1}^{*})\\!-\\!4\nu
z_{1}z_{2}$
$\displaystyle\times(z_{1}^{*}\\!-\\!z_{2}^{*})\\!+\\!(m_{1}z_{1}\\!+\\!m_{2}z_{2}\\!+\\!2m_{1}z_{2}\\!+\\!2m_{2}z_{1})(z_{1}^{*}\\!+\\!z_{2}^{*})$
$\displaystyle\times
z_{s}\\!+\\!2z_{1}z_{2}(z_{1}\\!+\\!z_{2})z_{1}^{*}z_{2}^{*}\\!+\\!(z_{1}\\!+\\!z_{2})^{2}z_{1}^{*}z_{2}^{*}z_{s}\\!+\\!2z_{1}z_{2}z_{1}^{*}$
$\displaystyle\times
z_{2}^{*}z_{s}\\!+\\!z_{s}^{*2}[2z_{1}z_{2}(z_{1}\\!+\\!z_{2})\\!+\\!(z_{1}\\!+\\!z_{2})^{2}z_{s}\\!+\\!2z_{1}z_{2}z_{s}]$
$\displaystyle-
z_{s}^{*}[2\nu(z_{1}^{2}\\!-\\!z_{2}^{2})\\!+\\!2m_{1}(z_{1}\\!+\\!2z_{2})z_{s}\\!+\\!2m_{2}(2z_{1}\\!+\\!z_{2})z_{s}$
$\displaystyle+(z_{1}^{*}\\!+\\!z_{2}^{*})\\{2z_{1}z_{2}(z_{1}+z_{2})\\!+\\!(z_{1}+z_{2})^{2}z_{s}\\!+\\!2z_{1}z_{2}z_{s}\\}]$
$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
2z_{1}z_{2}[m_{1}^{2}\\!+\\!m_{2}^{2}\\!+\\!m_{1}(z_{2}z_{2}^{*}\\!+\\!2z_{1}z_{2}^{*}\\!-\\!z_{2}z_{1}^{*})\\!+\\!m_{2}(z_{1}z_{1}^{*}$
$\displaystyle+2z_{2}z_{1}^{*}\\!-\\!z_{1}z_{2}^{*})]\\!+\\!m_{1}m_{2}(z_{1}\\!+\\!z_{2})(z_{1}\\!+\\!z_{2}\\!-\\!2z_{s})$
$\displaystyle-2[m_{1}^{2}z_{2}\\!+\\!m_{2}^{2}z_{1}\\!-\\!(z_{1}^{*}\\!+\\!z_{2}^{*})\\{m_{1}^{2}z_{1}^{2}\\!+\\!m_{2}^{2}z_{2}^{2}\\!-\\!2\mu$
$\displaystyle\times(z_{1}\\!+\\!z_{2})^{2}\\}]z_{s}\\!-\\!z_{1}z_{2}z_{1}^{*}z_{2}^{*}[z_{1}z_{2}\\!-\\!2(z_{1}\\!+\\!z_{2})]\\!-\\!z_{1}z_{2}z_{s}^{*2}$
$\displaystyle\times[z_{1}z_{2}\\!+\\!2(z_{1}\\!+\\!z_{2})z_{s}]\\!-\\!z_{s}^{*}[2(m_{1}z_{1}^{2}z_{2}\\!+\\!m_{2}z_{1}z_{2}^{2})\\!-\\!2$
$\displaystyle\times(m_{1}z_{2}^{2}\\!+\\!m_{2}z_{1}^{2}\\!+\\!4\mu
z_{1}z_{2})z_{s}\\!-\\!z_{1}^{2}z_{2}^{2}(z_{1}^{*}\\!+\\!z_{2}^{*})\\!-\\!2z_{1}z_{2}$
$\displaystyle\times(z_{1}\\!+\\!z_{2})(z_{1}^{*}\\!+\\!z_{2}^{*})z_{s}]\leavevmode\nobreak\
,$ $\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle(m_{1}z_{2}\\!+\\!m_{2}z_{1})^{2}z_{s}\\!-z_{1}z_{2}(m_{1}^{2}z_{2}\\!+\\!m_{2}^{2}z_{1})\\!-\\!m_{1}m_{2}z_{1}z_{2}$
(14)
$\displaystyle\times(z_{1}\\!+\\!z_{2})\\!-\\!z_{1}^{2}z_{2}^{2}(m_{1}z_{2}^{*}\\!+\\!m_{2}z_{1}^{*})\\!+\\!z_{1}z_{2}(m_{1}z_{2}\\!+\\!m_{2}z_{1})$
$\displaystyle\times(z_{1}^{*}\\!+\\!z_{2}^{*})z_{s}\\!+\\!z_{1}^{2}z_{2}^{2}z_{1}^{*}z_{2}^{*}z_{s}\\!+\\!z_{1}^{2}z_{2}^{2}z_{s}z_{s}^{*2}\\!+\\!z_{s}^{*}[2\mu
z_{1}^{2}z_{2}^{2}$
$\displaystyle-2z_{1}z_{2}(m_{1}z_{2}\\!+\\!m_{2}z_{1})z_{s}\\!-\\!z_{1}^{2}z_{2}^{2}(z_{1}^{*}\\!+\\!z_{2}^{*})z_{s}]\leavevmode\nobreak\
,$

where $\mu\\!=\\!(m_{1}\\!+\\!m_{2})/2$ and $\nu\\!=\\!(m_{2}\\!-\\!m_{1})/2$.
If we assume the masses of two lenses are the same, i.e.,
$z_{2}\\!=\\!-z_{1}$, and they are on the real axis, i.e., $z_{1}=z_{1}^{*}$,
the coefficients in Eq. (14) are reduced to the ones in Equation (2) of Witt &
Mao (1995).

For the given binary lens system, the equation of the critical curves in Eq.
(3) is also rewritten as a complex polynomial equation of degree 4:

$d_{4}z^{4}+d_{3}z^{3}+d_{2}z^{2}+d_{1}z+d_{0}=0\leavevmode\nobreak\ ,$ (15)

for $\phi\in[0,2\pi)$, where

$\displaystyle d_{4}$ $\displaystyle=$ $\displaystyle
e^{-i\phi}\leavevmode\nobreak\ ,$ $\displaystyle d_{3}$ $\displaystyle=$
$\displaystyle-2e^{-i\phi}(z_{1}\\!+\\!z_{2})\leavevmode\nobreak\ ,$
$\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle
e^{-i\phi}(z_{1}^{2}\\!+\\!4z_{1}z_{2}\\!+\\!z_{2}^{2})\\!-\\!(m_{1}\\!+\\!m_{2})\leavevmode\nobreak\
,$ $\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle-2e^{-i\phi}z_{1}z_{2}(z_{1}\\!+\\!z_{2})\\!+\\!2(m_{1}z_{2}\\!+\\!m_{2}z_{1})\leavevmode\nobreak\
,$ $\displaystyle d_{0}$ $\displaystyle=$ $\displaystyle
e^{-i\phi}z_{1}^{2}z_{2}^{2}\\!-\\!m_{1}z_{2}^{2}\\!-\\!m_{2}z_{1}^{2}\leavevmode\nobreak\
.$ (16)

Then, using the lens equation, we can map the solutions of Eq. (15) to the
source plane, and, eventually, get the caustics.


# Text to Image Generation: Leaving no Language Behind

Pedro Reviriego
Universidad Politécnica de Madrid
28040 Madrid, Spain
<EMAIL_ADDRESS>
Elena Merino-Gómez
Universidad de Valladolid
47011 Valladolid, Spain
<EMAIL_ADDRESS>

###### Abstract

One of the latest applications of Artificial Intelligence (AI) is to generate
images from natural language descriptions. These generators are now becoming
available and achieve impressive results that have been used for example in
the front cover of magazines. As the input to the generators is in the form of
a natural language text, a question that arises immediately is how these
models behave when the input is written in different languages. In this paper
we perform an initial exploration of how the performance of three popular
text-to-image generators depends on the language. The results show that there
is a significant performance degradation when using languages other than
English, especially for languages that are not widely used. This observation
leads us to discuss different alternatives on how text-to-image generators can
be improved so that performance is consistent across different languages. This
is fundamental to ensure that this new technology can be used by non-native
English speakers and to preserve linguistic diversity.

##  1 Introduction

In the last decade, Artificial Intelligence (AI) has made significant
breakthroughs and today outperforms humans in a number of tasks. As large
datasets for training become available for many applications, and more
sophisticated models are developed, AI is expected to overtake humans in more
tasks in the coming years [1]. One application that has experience a dramatic
improvement in the last years are text-to-image AI generators. These tools
take as input a natural language description of an image in the form of a text
prompt and generate an image that corresponds to what is described in the
text. A good example are the DALL-E and Glide tools developed by OpenAI
[2],[3],[4] or Imagen [5] and Parti [6] from Google. There are many other
tools such as Cogview [7] and some are available as open-source projects like
Craiyon (formerly DALL-E Mini) [8] or offer publicly available interfaces to
generate images like MidJourney111Available at www.midjourney.com or DALL-E2
[3]. These text-to-image generators are attracting a lot of interest in many
different communities ranging from computer science and graphic designers to
artists, and even the general public. A good example of this interest is the
use of DALL-E2 to create the front cover of a popular magazine222Available at
www.cosmopolitan.com/lifestyle/a40314356/dall-e-2-artificial-intelligence-
cover/. This interest is expected to drive the development of this technology
with new tools or new versions of existing tools being released in the coming
months. These state of the art text-to-image generators are trained with huge
datasets and have billions of parameters. For example, the latest Google
generator, Parti can have up to 20 billion parameters and is trained on
several billions of text/image pairs [6]. This means that the training of the
generator requires a large computational cost and time.

Another area in which AI has also improved significantly is machine
translation of texts [9]. Machine translation is now being used in many
applications and is expected to be adopted in many others in the coming years
[10],[11]. Machine translation is indeed a key technology to ensure that the
latest technologies are available to humans regardless of their language thus
ensuring diversity and inclusiveness [12]. Many tools have been implemented
and are available for end-users or developers. There are indeed large efforts
to make natural language processing in general and machine translation in
particular, available in many languages. For example, BLOOM333Available at
https://huggingface.co/blog/bloom an advanced large language model with close
to 200 billion parameters and that can generate text in 46 languages has been
recently released as an open source project. As for text-to-image generators,
training requires a huge computational effort, indeed more than 100 days
running on a supercomputer were needed to train BLOOM. Another recent
development is the No Language Left Behind (NLLB) initiative by Meta that
provides translation among 200 different languages and for which the code has
been released as open source [9].

As text-to-image generator technology emerges and consolidates, supporting
language diversity and inclusiveness will become a priority. In this paper we
take an initial step in this direction and analyze how current generators
performance depends on the language used to input the text. We consider
several generators and languages using simple texts to get a preliminary
understanding on the state of the technology. These initial results show that
for widely used languages, like Spanish, some of the generators have a similar
performance to English, but others suffer a significant performance
degradation. Instead, for languages used by smaller communities or no longer
used, all the generators have a large performance degradation in most cases.
Based on these results, we briefly discuss how language support can be
achieved in text-to-image generators exploring different options. All of them
seem to lead to an interaction between text-to-image generators and machine
translation, opening an interesting topic for further research.

The rest of the paper is organized as follows. In section 2 we describe how we
have designed our initial evaluation experiments and report and discuss the
results. Then in section 3 we briefly discuss different options to make the
performance of text-to-image generators consistent across many languages. The
paper ends with the conclusion in section 4.

##  2 Initial Evaluation of Language Support in Text to Image Generators

In order to have a preliminary idea of how different languages are supported
on text-to-image generators, an initial evaluation has been done using four
languages:

  1. 1.

English.

  2. 2.

Spanish.

  3. 3.

Basque.

  4. 4.

Latin.

English is used as the benchmark for comparison; Spanish is included as a
broadly used language that has also similarities with other Roman languages
such as Italian, French or Portuguese; Basque444A ”language isolate” used in
parts of northern Spain that has no demonstrated links with any major
language. as an example of a language used by a small community that has no
similarity with any major language; and Latin as an example of a language that
is no longer used but has strongly influenced many widely used languages. The
translation of the prompts has been done using Google translate and then
checked and corrected when needed by native speakers and an expert in Latin.

In terms of generators, three commonly used and publicly available tools have
been used:

  1. 1.

DALL-E2.

  2. 2.

MidJourney.

  3. 3.

Craiyon.

DALL-E2 is a state of the art text-to-image generator developed by OpenAI.
DALL-E2 has been trained using several hundreds of millions of text/image
pairs and uses an AI model with billions of parameters. The algorithms,
datasets and performance are described in some detail in [3]. MidJourney does
not disclose details about the implementation, but it is also publicly
available and has issued several releases. In our test we used the
implementation available during August 2022. Finally, Craiyon (formerly known
as DALL-E Mini) is an open-source project that aims to reproduce the results
of OpenAI’s DALL-E with a simpler design. In more detail, Craiyon uses only
0.4 billion parameters in the model compared to the several billion of DALL-E2
which makes the training process much faster, and also reduces the storage
requirements and the time needed to generate images. Craiyon was trained with
three image/text datasets using in total approximately 15 million image/text
pairs.

Finally, we have selected five random text prompts555The prompts with the
translations used for each language are listed in an appendix at the end of
the paper. from the MS-COCO dataset [13] that have been used to illustrate the
performance of DALL-E2 (see Figure 12 in [3]). These prompts are short and
simple sentences and contain only common words and thus one would expect a
text-to-image generator to be able to create images that relate to the text.
This gives us a total of 60 different sets of images that are sufficient to
illustrate how performance varies significantly across languages in current
text-to-image generators.

The results obtained for MidJourney are summarized in Figure 1. In this case,
four images are shown per text prompt. It can be observed that for English,
the images capture the meaning of the text input reasonably well. For Spanish
part of the information in the text is lost and does not appear in the images.
Finally, for Basque and Latin, the tool is unable to interpret the text and
produces images that are completely unrelated to the meaning of the text.
Therefore, it seems that this tool has a performance that depends strongly on
the input language even for widely used languages.

The results for Craiyon are presented in Figure 2. This tool produces nine
images per run with the default settings, the best image obtained is shown in
the figure. In this case, the images for Spanish text prompts have a similar
quality as those of English. Instead, results for Basque and Latin are bad,
with Craiyon identifying part of the elements described in the text in the
best cases and producing completely unrelated images in the rest. Finally, the
results for DALL-E2 are shown in Figure 3. The tool generates four images by
default and the best one is shown. It can be observed that the quality of the
images is similar for English and Spanish. Instead, there is a significant
performance loss for both Basque and Latin.

Therefore, this simple experiment confirms that current text-to-image
generation technology does not provide consistent performance across languages
even when using very simple text prompts. Indeed, some of the tools suffer a
significant performance degradation even when using widely used languages such
as Spanish. The degradation is dramatic for Basque and Latin. In the first
case, probably few image/text pairs have been used to train the tools as it is
not a widely used language. The second shows that current AI generators cannot
infer the meaning of Latin texts even when they have been trained with
multiple languages derived from it such as Spanish, Italian or French.

Text prompt 1: “a green train is coming down the tracks” (English, Spanish,
Basque, Latin)

Text prompt 2: “a group of skiers are preparing to ski down a mountain”
(English, Spanish, Basque, Latin)

Text prompt 3: “a small kitchen with a low ceiling” (English, Spanish, Basque,
Latin)

Text prompt 4: “a group of elephants walking in muddy water” (English,
Spanish, Basque, Latin)

Text prompt 5: “a living area with a television and a table” (English,
Spanish, Basque, Latin)

Figure 1: Results for MidJourney

Text prompt 1: “a green train is coming down the tracks” (English, Spanish,
Basque, Latin)

Text prompt 2: “a group of skiers are preparing to ski down a mountain”
(English, Spanish, Basque, Latin)

Text prompt 3: “a small kitchen with a low ceiling” (English, Spanish, Basque,
Latin)

Text prompt 4: “a group of elephants walking in muddy water” (English,
Spanish, Basque, Latin)

Text prompt 5: “a living area with a television and a table” (English,
Spanish, Basque, Latin)

Figure 2: Results for Craiyon

Text prompt 1: “a green train is coming down the tracks” (English, Spanish,
Basque, Latin)

Text prompt 2: “a group of skiers are preparing to ski down a mountain”
(English, Spanish, Basque, Latin)

Text prompt 3: “a small kitchen with a low ceiling” (English, Spanish, Basque,
Latin)

Text prompt 4: “a group of elephants walking in muddy water” (English,
Spanish, Basque, Latin)

Text prompt 5: “a living area with a television and a table” (English,
Spanish, Basque, Latin)

Figure 3: Results for DALL-E2

##  3 Leaving no Language Behind in Text to Image Generation

In this section we briefly discuss possible approaches to achieve similar
performance across many languages in text-to-image generators.

The first solution could be to include a relevant number of text/image pairs
in all the target languages during the training of the generators. However,
this has a number of issues: 1) getting those pairs does not seem to be
straightforward and 2) training the model would be significantly more complex
as the size of the dataset would grow by two orders of magnitude to cover a
relevant number of languages. In more detail, there are some efforts to
develop datasets of annotated images on several languages (see [14] and
references within) but they are limited in the number of languages and number
of images, very far from the hundreds of millions or billions used in state of
the art text-to-image generators. A potential solution to the lack of
annotated images for some languages could be to use automatic machine
translation of the English annotations. Indeed, this has shown good
performance in previous works such as [15]. However, even if we can generate
many text/image pairs for all the target languages, we would still have to
face the second issue, training the system with a much larger dataset.

A second approach would be to add a natural language processing module to the
text-to-image generators that detects the language of the input text and when
it corresponds to a language for which the system has poor performance applies
machine translation to English and uses the generated text as input. This has
a number of advantages compared to the previous solution: 1) the text-to-image
generator can be trained on a much smaller dataset and 2) the machine
translation is independent of the text-to-image generator so that advances in
translation can be used as they become available or different implementations
of the translation can be used with the same text-to-image generator.
Therefore, it seems this second solution is more flexible and scalable.

A final consideration is that the lack of human annotated images for many
languages would in any case limit the ability of text-to-image generators to
what can be achieved in translation (that has to be used either in the
generation of the training dataset or to translate the input text) and thus
minority languages would be somewhat lost in translation.

##  4 Conclusion

In this paper we have performed an initial exploration of the performance of
several recently developed text-to-image generators that are publicly
available when using different languages for the text input. The results show
that all generators suffer a significant degradation on the quality of their
results when using languages that are not widely used and the degradation is
even worse for classical languages such as Latin. Therefore, current
generators do not seem to support language diversity and inclusiveness which
are fundamental goals to make AI fair and responsible. Based on that
observation, we have also discussed several alternatives to make the
performance of text-to-image generators consistent across different languages.
The analysis suggests that for low resources languages an attractive option is
to use machine translation to preprocess the text and convert it to English.
In any case, as text-to-image technology is still in its infancy, further
evaluation and analysis is needed to fully understand how consistent
performance across languages can be best achieved.

## References

  * [1] K. Grace, J- Salvatier, A. Dafoe, B. Zhang, and O. Evans, Viewpoint: When will AI exceed human performance? Evidence from AI experts. Journal of Artificial Intelligence Research, 62, 729–754, 2019.
  * [2] A. Ramesh, M. Pavlov, G. Goh, S. Gray, C. Voss, A. Radford, M. Chen, and I. Sutskever. Zero-shot text-to-image generation. In International Conference on Machine Learning, pages 8821–8831. PMLR, 2021.
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  * [4] A. Nichol, P. Dhariwal, A. Ramesh, P. Shyam, P. Mishkin, B. McGrew, I. Sutskever, and M. Chen. Glide: Towards photorealistic image generation and editing with text-guided diffusion models. arXiv preprint arXiv:2112.10741, 2021.
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  * [8] B. Dayma, S. Patil, P. Cuenca, K. Saifullah, T. Abraham, P. Le Khac, L. Melas, R. Ghosh. DALL·E Mini, https://github.com/borisdayma/dalle-mini, 2021.
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APPENDIX: Text Prompts

The text prompts used in the different languages (English, Spanish, Basque,
Latin666For Latin, new words were taken from “Lexicon Recentis Latinitatis”. 2
vols. Vaticano: Libraria Editoria Vaticana, vol I: 1982; vol II: 1997 and
Egger, Carolus. “Sermo Latinus Hodiernus”, Roma: Opus Fundatum “Latinitas”,
1986. ) are:

  1. 1.

"a green train is coming down the tracks”, "un tren verde viene por las vías",
"tren berde bat dator trenbideetatik behera", "viridis hamaxósticus ferrivia
venit"

  2. 2.

“a group of skiers are preparing to ski down a mountain”, "un grupo de
esquiadores se prepara para esquiar montaña abajo", "eskiatzaile talde bat
menditik behera eskiatzeko prestatzen ari da", "nartatorum turma parat nartis
prolabi a monte"

  3. 3.

“a small kitchen with a low ceiling”, “una pequeña cocina con un techo bajo”,
"sukalde txiki bat sabai baxuarekin", "parva coquina humili tecto"

  4. 4.

“a group of elephants walking in muddy water”, "un grupo de elefantes
caminando en agua fangosas", "elefante talde bat ur zingiratsuetan ibiltzen",
"multitudo elephantorum ambulans in turbida aqua”

  5. 5.

“a living area with a television and a table”, "una sala de estar con un
televisor y una mesa", "egongela bat telebista con mahai batekin" Ban eta
"egongela bat telebista eta mahai batekin", "oecus televisione et mensa"