# Prospects for detecting neutron star–white dwarf mergers with decihertz
gravitational-wave observatories

Yacheng Kang,1,2 Chang Liu,1,2,3 Jin-Ping Zhu,4,5 Yong Gao,1,2,6 Lijing
Shao,2,7 Bing Zhang,8,9Hui Sun,7 Yi-Han Iris Yin${}^{10},$ and Bin-Bin
Zhang10,11,12
1Department of Astronomy, School of Physics, Peking University, Beijing
100871, China
2Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
3Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3), Université de Toulouse,
CNRS, UPS, F-31062 Toulouse Cedex 9, France
4School of Physics and Astronomy, Monash University, Clayton Victoria 3800,
Australia
5Monash Centre for Astrophysics, School of Physics and Astronomy, Monash
University, Clayton, VIC 3800, Australia
6Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
Am Mühlenberg 1, Potsdam 14476, Germany
7National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China
8Nevada Center for Astrophysics, University of Nevada, Las Vegas, NV 89154,
USA
9Department of Physics and Astronomy, University of Nevada, Las Vegas, NV
89154, USA
10School of Astronomy and Space Science, Nanjing University, Nanjing 210093,
China
11Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, China
12Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023,
China E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Based on different neutron star–white dwarf (NS-WD) population models, we
investigate the prospects of gravitational-wave (GW) detections for NS-WD
mergers, with the help of early warnings from two space-borne decihertz GW
observatories, DO-Optimal and DECIGO. We not only give quick assessments of
the GW detection rates for NS-WD mergers with the two decihertz GW detectors,
but also report systematic analyses on the characteristics of GW-detectable
merger events using the method of Fisher matrix. With a sufficient one-day
early-warning time, the yearly GW detection number for DO-Optimal is in the
range of $(1.5$–$1.9)\times 10^{3}$, while it is $(3.3$–$4.6)\times 10^{4}$
for DECIGO. More importantly, our results show that most NS-WD mergers can be
localized with an uncertainty of $\mathcal{O}(10^{-2})\,\mathrm{deg}^{2}$.
Given the NS-WD merger as a possible origin for a peculiar long-duration
gamma-ray burst, GRB 211211A, we further suggest that the GW early-warning
detection would allow future electromagnetic telescopes to get prepared to
follow-up transients after some special NS-WD mergers. Based on our analyses,
we emphasize that such a feasible “wait-for” pattern can help to firmly
identify the origin of GRB 211211A-like events in the future and bring
excellent opportunities for the multimessenger astronomy.

###### keywords:

gravitational waves – gamma-ray bursts – binaries: general

††pubyear: 2023††pagerange: Prospects for detecting neutron star–white dwarf
mergers with decihertz gravitational-wave observatories–B

##  1 Introduction

The mergers of double-compact-object (DCO) systems have been explored
extensively, both as progenitors of various electromagnetic (EM) transients
and as gravitational wave (GW) sources. For example, binary white dwarfs
(BWDs) are not only one of the most promising candidates of type Ia supernova
(SN) progenitors (Livio & Mazzali, 2018; Soker, 2018; Wang, 2018; Ruiter,
2020), but are also expected to be the dominant GW sources by numbers in our
Milky Way (MW) for the Laser Interferometer Space Antenna (LISA)
mission111LISA is an ESA-led space-borne GW observatory, with NASA as a junior
partner. It is due to be launched in 2034 with the aim to record and study GWs
in the millihertz frequency band. More information about the LISA mission can
be found at https://lisa.nasa.gov. (Lamberts et al., 2018; Bayle et al., 2022;
Amaro-Seoane et al., 2023). For compact binary mergers containing at least one
neutron star (NS), it has long been proposed that the mergers of binary
neutron stars (BNSs) and neutron star–black holes (NS-BHs) are the progenitors
of short-duration gamma-ray bursts (sGRBs; Paczynski, 1986; Eichler et al.,
1989; Narayan et al., 1992; Zhang, 2018). The milestone co-detection of GW and
EM signals during GW170817/GRB 170817A/AT 2017gfo events (Abbott et al.,
2017a, b; Goldstein et al., 2017; Savchenko et al., 2017; Coulter et al.,
2017; Evans et al., 2017; Pian et al., 2017; Kilpatrick et al., 2017; Zhang et
al., 2018) has not only provided us a smoking-gun evidence for the BNS merger
origin of sGRBs and kilonovae (Li & Paczynski, 1998), but also an
unprecedented opportunity to explore gravity theories and fundamental physics
in extreme environments (Abbott et al., 2017c, 2018b, 2019; Shao et al., 2017;
Sathyaprakash et al., 2019; Arun et al., 2022).

Nevertheless, the mergers of other kinds of DCO systems, such as NS-WD
binaries, have not been explored to the same depth in literature. Until
recently, Yang et al. (2022) adopted a NS-WD merger scenario to explain the
kilonova-like emission following a peculiar long-duration gamma-ray burst
(lGRB), GRB 211211A (Rastinejad et al., 2022; Troja et al., 2022; Mei et al.,
2022; Gompertz et al., 2023), attracting significant attention on NS-WD
mergers and their detection prospects (Zhong et al., 2023; Yin et al., 2023).
Compared with other types of DCO systems (e.g., BWDs, BNSs, NS-BHs, etc.), the
different component properties of NS-WD binaries could have significant
impacts on the early-time dynamics of mass-transfer phase, initial conditions
of accretion disk, and final merger remnants, etc. (King et al., 2007;
Chattopadhyay et al., 2007; Paschalidis et al., 2011; Margalit & Metzger,
2016, 2017). In particular, Zhong et al. (2023) showed that GRB 211211A-like
events could arise from a NS-WD merger if the central engine leaves a magnetar
behind. They proposed that the magnetic bubble eruptions from the toroidal
magnetic field amplification of the pre-merger NS could successfully produce
the main burst of GRB 211211A. NS-WD systems are likely the most common type
of DCO systems besides BWDs (Nelemans et al., 2001b; Toonen et al., 2018), and
may have various observable explosive transients (Metzger, 2012; Zenati et
al., 2019; Fernández et al., 2019; Gillanders et al., 2020; Bobrick et al.,
2022; Kaltenborn et al., 2022), which will bring excellent opportunities for
future multimessenger astronomy. Overall, it is worth analyzing the
consequences of NS-WD mergers in more detail.

Based on the above arguments, in this work we discuss the realistic prospects
of detecting NS-WD mergers with GW early warnings using space-borne decihertz
GW detectors. Given the lower sensitive frequency range
($0.01$–$1\,\mathrm{Hz}$) of decihertz detectors, they can offer alerts to NS-
WD mergers much earlier than the current and future ground-based
detectors222Ground-based GW detectors include the current Advanced Laser
Interferometer Gravitational-wave Observatory (LIGO), the Advanced Virgo, the
Kamioka Gravitational Wave Detector (KAGRA), and the future next-generation
detectors, such as the Einstein Telescope (ET) and the Cosmic Explorer (CE).
More detailed descriptions of these ground-based GW detectors can be found in
Abbott et al. (2018a), Maggiore et al. (2020) and Reitze et al. (2019).
(usually sensitive in the $10$–$10^{4}\,\mathrm{Hz}$ band), or even earlier
than the EM facilities (Liu et al., 2022b; Kang et al., 2022). In addition,
another motivation to consider decihertz GW early warnings is that the upper
cutoff GW frequency is expected to be $\lesssim 1$ Hz for NS-WD mergers in
their inspiral phase. This is because the runaway mass-transfer phase and
tidal disruption outcome occur earlier for NS-WD mergers (Verbunt & Rappaport,
1988; Paschalidis et al., 2009; Margalit & Metzger, 2016; Kaltenborn et al.,
2022), especially when compared with BNS and NS-BH merger events.333Note that
for BNS and NS-BH mergers in the inspiral phase, the upper cutoff GW frequency
can be approximated by the GW frequency at the innermost stable circular
orbit. In view of this, it would be difficult for the NS-WD inspiral GW
signals to enter the ground-based GW detectors, unlike the situation for more
massive DCO systems. Nevertheless, it allows us to propose a feasible “wait-
for” GW detecting mode for NS-WD mergers with decihertz GW detectors, and
provide helpful inputs for future multimessenger astrophysics.

Before quantitatively analysing the early warnings from two representative
decihertz GW observatories, DO-Optimal (DO-OPT; Sedda et al., 2020, 2021) and
DECIGO (DEC; Kawamura et al., 2011, 2021), we first use the method of
convolution with different star formation rates (SFRs) and delay-time
distributions (DTDs) to obtain four kinds of NS-WD merger population models.
Given the short tidal-disruption timescale (see, e.g., Kaltenborn et al.,
2022), $\lesssim{\cal O}(\mathrm{min})$, and possibly observable EM transients
(King et al., 2007; Chattopadhyay et al., 2007; Metzger, 2012; Margalit &
Metzger, 2016, 2017; Yang et al., 2022) for NS-WD mergers, we follow Kang et
al. (2022) and set the early-warning time to be $t_{\mathrm{e}}=1\,\mathrm{d}$
and signal-to-noise ratio (SNR) threshold value to be 8 for our GW detection
strategy. Differently from BNS mergers or NS-BH mergers, we define the time
when the WD starts to experience the Roche-lobe overflow as the merger time
for NS-WD binaries, considering that there is no well-modeled GW waveform
during the runaway mass-transfer phase. With the aim to extend early studies
on decihertz GW alerts for NS-WD mergers, we not only give quick assessments
of yearly detection numbers and percentages with DO-OPT and DEC, but also
report more detailed analyses on the characteristics of GW-detectable merger
events using the Fisher matrix. We find that DEC has better performance than
DO-OPT as a whole on GW early-warning detections and localization abilities,
especially for high-redshift (high-$z$) events. For those mergers that would
yield the best estimation results of distance and angular resolution, we
collect them into a Golden Sample Set, and present the detection rates of NS-
WD mergers in the Golden Sample Set for different population models with DO-
OPT and DEC.

Taking the recent peculiar lGRB, GRB 211211A, as an example, we further
suggest that the GW early-warning detection would allow future EM telescopes
to get prepared for possible follow-up transients after some special NS-WD
mergers. Although there are a few groups suggesting different origins for GRB
211211A,444For examples, Waxman et al. (2022) have shown that the thermal
emission from dust could explain the observed near-infrared data; Zhu et al.
(2022c) have suggested that a NS-BH merger could roughly reproduce the
multiwavelength observations; Gompertz et al. (2023) concluded that the
spectral evolution could be explained by a transition from a fast-cooling mode
to a slow-cooling regime, favoring a BNS merger scenario rather than a NS-BH
origin; Barnes & Metzger (2023) also found that the afterglow-subtracted
emission of GRB 211211A is in best agreement for collapsar models with high
kinetic energies. Unfortunately, GRB 211211A was detected prior to the fourth
observing run of the LIGO-Virgo-KAGRA Collaboration. Overall, the origin of
GRB 211211A is still under debate. at least most studies have supported or
directly considered compact star merger scenarios for this event, namely
compact-binary lGRBs (cb-lGRBs), given the features of kilonova-like emissions
and host-galaxy properties including the offset, etc.. Regardless of the exact
composition of this binary system, with a sufficient early-warning time
($t_{\mathrm{e}}=1\,\mathrm{d}$) and localization accuracies
($\Delta\Omega\lesssim 1\,\mathrm{deg}^{2}$), we point out that one can
prepare well in advance for future EM transients of GRB 211211A-like events,
and no longer needs to consider the field of view (FoV) discounts and complex
searching strategies.555We refer readers to Kang et al. (2022) for more
descriptions. Such a feasible wait-for pattern, if realized in future, can
help to firmly identify the origin of GRB 211211A-like events and enhance the
comprehension of GRB’s physical type.

The organization of this paper is as follows. We first overview the
construction of the NS-WD merger population models in Section 2. In Section 3,
we introduce the GW detecting strategy with two space-borne decihertz GW
observatories. Using the above ingredients, we report our results and detailed
analyses on GW early-warning detections of NS-WD mergers in Section 4.
Finally, Section 5 concludes the paper. Throughout this paper, we adopt a
standard $\rm\Lambda CDM$ model with the matter density parameter
$\Omega_{\mathrm{m}}=0.315$, the dark-energy density parameter
$\Omega_{\Lambda}=0.685$, and the Hubble-Lemaître constant
$H_{0}=67.4\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ (Aghanim et al.,
2020).

Table 1: Simulated NS-WD merger numbers per year for different population
models with $\dot{\rho}_{0}=390\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$. In the
last row, we list the lower and upper values in brackets, by rescaling our
results with
$\dot{\rho}_{0}\approx[90,5800]\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$. More
descriptions of the four population models are given in Section 2.1.
Population Model
---
A |  B |  C |  D
$5.6\times 10^{5}$ |  $4.1\times 10^{5}$ |  $4.9\times 10^{5}$ |  $3.8\times 10^{5}$
($1.3\times 10^{5}$ –$8.3\times 10^{6}$) |  ($9.5\times 10^{4}$ – $6.1\times 10^{6}$) |  ($1.1\times 10^{5}$ – $7.4\times 10^{6}$) |  ($8.7\times 10^{4}$ – $5.6\times 10^{6}$)

##  2 NS-WD Populations

In this section, we briefly describe how we obtain the NS-WD populations
prepared for later analyses. Following Sun et al. (2015), we also ignore the
possible redshift evolution of intrinsic system parameters for NS-WD mergers.
Within this framework, one can separately discuss the redshift distribution of
NS-WD systems in Section 2.1. We show more details about mass distribution of
our NS-WD populations in Section 2.2.

###  2.1 Event rate and redshift distribution

The number density per unit time for NS-WD mergers at a given redshift $z$ can
be estimated as,

$\frac{d\dot{N}}{dz}=\frac{\dot{\rho}_{0}f(z)}{1+z}\frac{dV(z)}{dz}\,,$ (1)

where $\dot{\rho}_{0}$ is the local NS-WD merger rate density, $f(z)$ is the
dimensionless redshift distribution factor, and $\frac{dV(z)}{dz}$ is the
comoving volume element,

$\frac{dV(z)}{dz}=\frac{c}{H_{0}}\frac{4\pi
D_{\mathrm{L}}^{2}}{(1+z)^{2}\sqrt{\Omega_{\Lambda}+\Omega_{\mathrm{m}}(1+z)^{3}}}\,,$
(2)

where $c$ is the speed of light, and $D_{\mathrm{L}}$ is the luminosity
distance,

$D_{\mathrm{L}}=(1+z)\frac{c}{H_{0}}\int_{0}^{z}\frac{dz}{\sqrt{\Omega_{\Lambda}+\Omega_{\mathrm{m}}(1+z)^{3}}}\,.$
(3)

The function $f(z)$ in Equation (1) depends on the DTD of NS-WD mergers
superposed on the SFR. Considering different combinations of SFRs and DTDs, we
illustrate in Appendix A the redshift distributions of $f(z)$ with more
details. We consider two kinds of analytical models from Yuksel et al. (2008)
and Madau & Dickinson (2014) for SFRs, abbreviated as ‘Y08’ and ‘MD14’,
respectively; as for DTDs, two models—abbreviated as
‘$\gamma\alpha$-$\mathrm{H}$’ and ‘$\gamma\alpha$-$\mathrm{V}$’—are adopted in
this work (see Appendix A for more descriptions). The population models are
briefly summarized as follows:

  1. (1)

Model A: Y08 + $\gamma\alpha$-$\mathrm{H}$;

  2. (2)

Model B: Y08 + $\gamma\alpha$-$\mathrm{V}$;

  3. (3)

Model C: MD14 + $\gamma\alpha$-$\mathrm{H}$;

  4. (4)

Model D: MD14 + $\gamma\alpha$-$\mathrm{V}$.

For the local NS-WD merger rate density, we adopt
$\dot{\rho}_{0}~{}{=390\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}}$ from Zhao et
al. (2021) as the fiducial value, which is in agreement with many studies
(Nelemans et al., 2001a; Kim et al., 2004; O’Shaughnessy & Kim, 2010; Bobrick
et al., 2017; Toonen et al., 2018). Moreover, in a similar way to that in Liu
et al. (2022b), we will rescale our results in later sections by adjusting
$\dot{\rho}_{0}$ in the range of $[8,500]\,\mathrm{Myr}^{-1}$ per MW-like
Galaxy, corresponding to
$\dot{\rho}_{0}\approx[90,5800]\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$
hereafter (Kaltenborn et al., 2022).666In most recent studies, a few authors
obtained the conversion factor with the galactic blue luminosity of $1.7\times
10^{10}\,\mathrm{~{}L}_{\mathrm{B},\odot}$ (Kalogera et al., 2001) and the
blue luminosity for the local Universe of $1.98\times
10^{8}\,\mathrm{~{}L}_{\mathrm{B},\odot}\,\mathrm{Mpc}^{-3}$ (Kopparapu et
al., 2008), where $\mathrm{~{}L}_{\mathrm{B},\odot}=2.16\times
10^{33}\,\mathrm{erg}\,\mathrm{s}^{-1}$ is the Solar luminosity in the
$B$-band. We regard this as a crude but reasonable treatment to assess the
systematic uncertainties in our predictions. The simulated NS-WD merger rates
in the Universe for different population models are listed in Table 1.

###  2.2 Mass distribution of NS-WD binaries

The mass distribution of NS-WD systems is a key ingredient in our calculation
of GW detection rates. However, it is difficult to construct an analytical
model for the probability density function of the component masses in NS-WD
binaries. Therefore, for simplicity, we make use of the population-synthesis
simulation results obtained by Kaltenborn et al. (2022). Given that there are
two strong peaks in the mass distribution of NSs at
$M_{\mathrm{NS}}=1.11\,\mathrm{M}_{\odot}$ and
$M_{\mathrm{NS}}=1.26\,\mathrm{M}_{\odot}$ (see Figure 1 in Kaltenborn et al.,
2022), we assume that each NS-WD merger may have
$M_{\mathrm{NS}}=1.11\,\mathrm{M}_{\odot}$ or
$M_{\mathrm{NS}}=~{}{1.26\,\mathrm{M}_{\odot}}$ with an equal probability. The
former is referred to as ‘Case I’, and the latter corresponds to ‘Case II’. In
each case, without considering the detailed properties of NS-WD systems (e.g.,
the WD composition, the DTD models, etc.),777This treatment is acceptable
because Toonen et al. (2018) have suggested that there is no significant
evolution of the average WD mass with the delay time for NS-WD binaries. We
also find that the fractions of NS-WD mergers with different WD components
(i.e., CO, ONe or He WDs) remain almost unchanged when different DTD models
are adopted (see Table 2 in Toonen et al., 2018). Moreover, fractions shown in
Kaltenborn et al. (2022) are also remarkably consistent with the results in
Toonen et al. (2018). we choose to fit the mass distribution of WD components
using a model composed of multi-Gaussian components,

$\displaystyle f^{\mathrm{I}}_{\mathrm{M}}\propto$
$\displaystyle\left\\{a_{1}\exp\left[-\frac{\left(M_{\mathrm{WD}}-\mu_{1}\right)^{2}}{2\sigma_{1}^{2}}\right]+a_{2}\exp\left[-\frac{\left(M_{\mathrm{WD}}-\mu_{2}\right)^{2}}{2\sigma_{2}^{2}}\right]\right.$
(4)
$\displaystyle\left.+a_{3}\exp\left[-\frac{\left(M_{\mathrm{WD}}-\mu_{3}\right)^{2}}{2\sigma_{3}^{2}}\right]\right\\}\,,$

and

$\displaystyle f^{\mathrm{II}}_{\mathrm{M}}\propto$
$\displaystyle\left\\{a_{4}\exp\left[-\frac{\left(M_{\mathrm{WD}}-\mu_{4}\right)^{2}}{2\sigma_{4}^{2}}\right]+a_{5}\exp\left[-\frac{\left(M_{\mathrm{WD}}-\mu_{5}\right)^{2}}{2\sigma_{5}^{2}}\right]\right\\}\,.$
(5)

Note that in the above equations, the WD component mass $M_{\mathrm{WD}}$ is
in units of $\mathrm{M}_{\odot}$. We plot the distribution of
$M_{\mathrm{WD}}$ for Case I and Case II in Figure 1. The best-fit values of
each parameter in Equation (4) and Equation (5) are

$\displaystyle a_{1}=0.048\,,\quad\mu_{1}=0.73\,,\quad\sigma_{1}=0.027\,,$ (6)
$\displaystyle
a_{2}=~{}{0.061}\,,\quad\mu_{2}=1.18\,,\quad\sigma_{2}=0.051\,,$ (7)
$\displaystyle a_{3}=0.094\,,\quad\mu_{3}=1.14\,,\quad\sigma_{3}=0.17\,,$ (8)

and

$\displaystyle a_{4}=0.25\,,\quad\mu_{4}=0.71\,,\quad\sigma_{4}=0.020\,,$ (9)
$\displaystyle a_{5}={0.15}\,,\quad\mu_{5}=0.79\,,\quad\sigma_{5}=0.028\,.$
(10)

Figure 1: Distributions of the WD component mass obtained by Kaltenborn et al.
(2022) for Case I (red) and Case II (blue). With the parameters given in
Section 2.2, we also plot the best-fit lines (black) for each case. Note that
here we have considered $M_{\mathrm{WD}}\lesssim M_{\mathrm{ch}}$, where
$M_{\mathrm{ch}}\approx 1.45\,\mathrm{M}_{\odot}$ is the Chandrasekhar mass
limit.

Finally, let us elaborate on the meaning of ‘merger’ for NS-WD systems in this
work. Regardless of the evolution history, once a NS-WD binary is formed, it
keeps losing its orbital energy through GW emission, causing the orbit to
shrink. When GW emission drives the NS-WD system to the Roche-lobe overflow
(RLOF) orbital separation, the mass from the WD begins to transfer to the NS
companion. A well-known fit of the RLOF separation is given by (Eggleton,
1983),

$a_{\mathrm{RLOF}}\approx
R_{\mathrm{WD}}\frac{0.6q^{2/3}+\ln\left(1+q^{1/3}\right)}{0.49q^{2/3}}\,,$
(11)

where $q=M_{\mathrm{WD}}/M_{\mathrm{NS}}$ and $R_{\mathrm{WD}}$ is the WD
radius. The latter can be well approximated by (Nauenberg, 1972),

$R_{\mathrm{WD}}\approx
10^{9}\mathrm{~{}cm}\left(\frac{M_{\mathrm{WD}}}{0.7\,\mathrm{M}_{\odot}}\right)^{-1/3}\left[1-\left(\frac{M_{\mathrm{WD}}}{M_{\mathrm{ch}}}\right)^{4/3}\right]^{1/2}\,,$
(12)

where $M_{\mathrm{ch}}\approx 1.45\,\mathrm{M}_{\odot}$ is the Chandrasekhar
mass limit assuming the mean molecular weight per electron,
$\mu_{\mathrm{e}}=2$.

During the mass-transfer phase, the binary separation can increase due to the
conservation of angular momentum; on the other hand, the WD radius increases
due to its mass loss [see Equation (12)], which can increase the critical RLOF
separation [i.e., $a_{\mathrm{RLOF}}$ in Equation (11)]. The competition
between the two effects can result in different evolution processes. If
$a_{\mathrm{RLOF}}$ grows much faster, the NS-WD system will quickly progress
into runaway mass transfer, tidally disrupting the WD on a dynamical time-
scale. Otherwise, the NS-WD system can maintain its stable mass transfer for a
longer time. Many studies have found that the stability of mass transfer
depends heavily on the mass ratio $q$ (Verbunt & Rappaport, 1988; Paschalidis
et al., 2009; Margalit & Metzger, 2016). They suggest that the mass transfer
is unstable for binaries with $q_{\text{crit }}\gtrsim 0.43$–$0.53$.888Bobrick
et al. (2017) have recently suggested that winds from the accreting stream are
far more important to the stability, which brings a much smaller
$q_{\text{crit }}\gtrsim 0.20$. Note that such a lower $q_{\text{crit }}$
value is also in line with the results in Kaltenborn et al. (2022). Combined
with the distribution of $M_{\mathrm{WD}}$ in our consideration (see Figure
1), it is safe to say that most NS-WD systems would finally have a runaway
mass-transfer phase, rather than the stable mass transfer. Given that there is
no well-modeled GW waveform for the NS-WD system during the runaway mass-
transfer phase, we define the time when the WD starts to experience the RLOF
as the merger time. We regard this as a reasonable treatment, especially
considering the short tidal-disruption timescale, $\lesssim{\cal
O}(\mathrm{min})$ (see, e.g., Kaltenborn et al., 2022) and possibly observable
explosive outcomes for NS-WD mergers (King et al., 2007; Chattopadhyay et al.,
2007; Metzger, 2012; Margalit & Metzger, 2016, 2017; Yang et al., 2022; Zhong
et al., 2023). Note that such definitions would be very different from those
in BNS and NS-BH mergers in many studies (Kyutoku et al., 2011; Shibata &
Taniguchi, 2011; Zhu et al., 2021; Liu & Shao, 2022; Liu et al., 2022b).
Essentially, for the NS-WD mergers in this work, the GW early-warning
detections are actually to offer alerts on the RLOF time. For this reason, we
will only consider the GW signal detectability in the inspiral phase, when the
NS and WD can be regarded as well-separated bodies that gradually spiral
towards one another. More descriptions of GW early-warning detections are
presented in Section 3.

##  3 GW detection strategy

As mentioned in the Introduction, due to the higher sensitive frequency range
of the ground-based GW detectors, they cannot offer alerts as early as the GW
detectors in the decihertz band. Moreover, for space-borne decihertz GW
detectors, NS-WD signals can exist from the start of the mission to their
mergers or even to the end of the mission. Therefore, such GW signals will
exist in decihertz detectors long enough to guarantee a relatively stable
parameter estimation precision, especially for the localization. This means
that the decihertz GW detectors can provide early-warning alerts to other GW
and EM detectors for follow-ups. Following Liu et al. (2022b), our realistic
detection strategy is performed as follows. Note that we use geometric units
where $G=c=1$ in this section.

Throughout this paper, we mainly compare the performance on GW early warnings
between two space-borne decihertz GW detectors, DO-OPT and DEC. DO was
envisaged in the ESA’s Voyage 2050 call (Sedda et al., 2020, 2021), where DO-
OPT is the one with more ambitious LISA-like designs. DEC is a future Japanese
space-borne decihertz GW mission with four independent LISA-like detectors
(Kawamura et al., 2011, 2021). Parameters of the two detectors are available
in, e.g., Liu & Shao (2022). In the community, there are more proposed
decihertz detectors, including moon-based ones (see e.g., Li et al., 2023;
Shao, 2023).

We define $t_{\mathrm{c}_{0}}$ as the NS-WD’s time to merge since the start of
the observation with decihertz GW detectors. To obtain the yearly detection
rates, we only use early-warning NS-WD samples that will merge in
1–$2\,\mathrm{yr}$ since the mission begins (e.g., $1\,\mathrm{yr}\leq
t_{\mathrm{c}_{0}}\leq 2\,\mathrm{yr}$). The sources that merge within 1 yr
($t_{\mathrm{c}_{0}}<1\,\mathrm{yr}$) are discarded. This is because their GW
signals only stay shortly in the decihertz detectors, and not enough
information is accumulated to obtain precise parameter estimations. Liu et al.
(2022b) have shown that decihertz detectors have poor performances for BNSs
that merge within 1 yr, which also applies to our NS-WD mergers. Considering
that NS-WD mergers typically produce signals at around 0.1 Hz, there will only
be a slight frequency chirp effect in the decihertz band (see Figure 2). This
means that the choice of the maximum value of $t_{\mathrm{c}_{0}}$ will have
minor effects on the yearly detection results as long as it is not too
long.999Note that people usually set a 4-yr mission time for DO-OPT and DEC.
For sources that only inspiral within the whole 4-yr observational span (e.g.,
$t_{\mathrm{c}_{0}}>4\,\mathrm{yr}$), Liu et al. (2022b) have shown that their
timing accuracies have a sharp decreasing trend, which is unfavorable for GW
early-warning detections. In contrast, sources with $1\,\mathrm{yr}\leq
t_{\mathrm{c}_{0}}\leq 4\,\mathrm{yr}$ are in general distributed uniformly in
time (see Table 1 in Liu et al., 2022b). The above discussions explain why we
only consider NS-WD mergers with $1\,\mathrm{yr}\leq t_{\mathrm{c}_{0}}\leq
2\,\mathrm{yr}$.

Figure 2: Sensitivity curves of DO-OPT (green) and DEC (red). The simulated
NS-WD inspiral signals at different redshifts are plotted with blue dashed-
dotted lines. Taking $M_{\mathrm{NS}}=M_{\mathrm{WD}}=1.1\,\mathrm{M}_{\odot}$
as an example, source signals are plotted with a duration of $\sim
100\,\mathrm{yr}$. For each source, the short dashed vertical lines mark the
time before RLOF occurs (i.e., the rightmost endpoint). More descriptions of
the sky-averaged effective noise $\sqrt{S_{\mathrm{n}}}$ for various detectors
can be found in Liu et al. (2020).

Following Liu et al. (2022b), we generate NS-WD mergers weekly according to
the population models. Then we calculate their SNRs. If SNR > 8, we claim the
detection and calculate parameter precisions using the Fisher matrix (Finn,
1992; Cutler & Flanagan, 1994). As a fast way to estimate parameter
statistical errors, a Fisher information matrix (FIM) shows the Cramer–Rao
bound of parameters. In other words, a FIM tells us how precisely we can
determine the model parameters for events with a high SNR and Gaussian noises
(see Wang et al. (2022) for extensions of the FIM). We follow the settings in
Liu & Shao (2022) with slight revisions. In total nine system parameters are
used in the FIM, collectively denoted as

$\bm{\Xi}=\left\\{\mathcal{M}_{z},\eta,t_{\mathrm{c}},\phi_{\mathrm{c}},D_{\mathrm{L}},\bar{\theta}_{\mathrm{S}},\bar{\phi}_{\mathrm{S}},\bar{\theta}_{\mathrm{L}},\bar{\phi}_{\mathrm{L}}\right\\}\,.$
(13)

In $\bm{\Xi}$, $\eta\equiv
M_{\mathrm{NS}}M_{\mathrm{WD}}/\left(M_{\mathrm{NS}}+M_{\mathrm{WD}}\right)^{2}$
is the symmetric mass ratio for each NS-WD merger system;
$\mathcal{M}_{z}\equiv(1+z)\left(M_{\mathrm{NS}}+M_{\mathrm{WD}}\right)\eta^{3/5}$
is the detector-frame chirp mass; $t_{\mathrm{c}}$ and $\phi_{\mathrm{c}}$ are
the time and orbital phase at coalescence, respectively; and
$\left\\{\bar{\theta}_{\mathrm{S}},\bar{\phi}_{\mathrm{S}},\bar{\theta}_{\mathrm{L}},\bar{\phi}_{\mathrm{L}}\right\\}$
are the source direction and angular momentum direction in the Solar system
barycentric frame (see Figure 1 in Liu et al., 2020). Note that in generating
sources, we adopt $\cos\bar{\theta}_{\mathrm{S}}\in\mathcal{U}(-1,1)$,
$\cos\bar{\theta}_{\mathrm{L}}\in\mathcal{U}(-1,1)$,
$\bar{\phi}_{\mathrm{S}}\in\mathcal{U}(0,2\pi)$, and
$\bar{\phi}_{\mathrm{L}}\in\mathcal{U}(0,2\pi)$, where
$\mathcal{U}(\cdot\,,\cdot)$ denotes a uniform distribution.

After performing the FIM calculation, we obtain the angular resolution
$\Delta\Omega$ via (Cutler, 1998; Barack & Cutler, 2004),

$\Delta\Omega=2\pi\sqrt{\left(\Delta\bar{\mu}_{\mathrm{S}}\Delta\bar{\phi}_{\mathrm{S}}\right)^{2}-\left\langle\delta\bar{\mu}_{\mathrm{S}}\delta\bar{\phi}_{\mathrm{S}}\right\rangle^{2}}\,,$
(14)

where $\Delta\bar{\mu}_{\mathrm{S}}$ and $\Delta\bar{\phi}_{\mathrm{S}}$ are
the root-mean-square errors of $\bar{\mu}_{\mathrm{S}}$ and
$\bar{\phi}_{\mathrm{S}}$ with
$\bar{\mu}_{\mathrm{S}}\equiv\cos\bar{\theta}_{\mathrm{S}}$, and
$\left\langle\delta\bar{\mu}_{\mathrm{S}}\delta\bar{\phi}_{\mathrm{S}}\right\rangle$
is the covariance of $\bar{\mu}_{\mathrm{S}}$ and $\bar{\phi}_{\mathrm{S}}$.
In this work, our attention focuses on the estimations of $\Delta\Omega$ and
$\Delta D_{\mathrm{L}}$.

For the FIM in the frequency domain, we follow Kang et al. (2022) with slight
revisions. We set the integration limit to be $f_{\text{in}}$ and
$f_{\text{out}}$. In an idealized setting of two point particles, assuming a
circular Keplerian orbit with quadrupolar GW damping, the GW frequency will
formally diverge at a finite value of time, $t_{\text{div}}$. However, for a
real NS-WD system, we have noted previously in Section 2.2 that the merger
time $t_{\mathrm{c}_{0}}$ corresponds to the time when the WD starts to
experience the RLOF. Thus the upper cutoff GW frequency in this work is
actually at the critical RLOF separation, which can be calculated, to a
sufficient precision, by Kepler’s third law,

$f_{\max}\simeq\frac{1}{\pi(1+z)}\sqrt{\frac{M_{\mathrm{NS}}+M_{\mathrm{WD}}}{a_{\mathrm{RLOF}}^{3}}}\,.$
(15)

Note that the $(1+z)$ foctor is included in the denominator due to the
cosmological time dilation. With a known $f_{\max}$ value for each NS-WD
system, we can then calculate the remaining time to divergence (Maggiore,
2008),

$\displaystyle
t_{\mathrm{RLOF}}=t_{\text{div}}-t_{\mathrm{c}_{0}}=5\mathcal{M}_{z}^{-5/3}(8\pi
f_{\max})^{-8/3}\,.$ (16)

For a NS-WD system to merge in time $t_{\mathrm{c}_{0}}$, we thus have

$\displaystyle f_{\text{in}}$
$\displaystyle=\left[(t_{\mathrm{c}_{0}}+t_{\mathrm{RLOF}})/5\right]^{-3/8}\mathcal{M}_{z}^{-5/8}/8\pi\,,$
(17) $\displaystyle f_{\text{out}}$
$\displaystyle=~{}{\left[(t_{\mathrm{e}}+t_{\mathrm{RLOF}})/5\right]^{-3/8}\mathcal{M}_{z}^{-5/8}/8\pi}\,,$
(18)

with $t_{\text{e}}$ the early-warning time before the merger. The early-
warning time $t_{\text{e}}$ is remarkably significant to the GW early-warning
detections of NS-WD mergers, given the short tidal-disruption timescale and
possibly observable EM transients during the runaway mass-transfer phase. We
will present the results with $t_{\mathrm{e}}=1\,\mathrm{d}$ in detail. More
descriptions of the GW waveform construction and parameter estimation method
were discussed in Liu & Shao (2022).

Table 2: Yearly detection numbers and percentages (in brackets) of NS-WD mergers for different population models with DO-OPT and DEC. We assume an early-warning time $t_{\mathrm{e}}=1\,\mathrm{d}$. The SNR threshold is set to be 8. Note that we only list results with $\dot{\rho}_{0}=390\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$. Readers can rescale in the way as for the total simulated numbers in Table 1. GW Detector |  Population Model
---|---
|  A |  B |  C |  D
DO-OPT |  1915 |  1637 |  1722 |  1489
(0.34 %) |  (0.40 %) |  (0.35 %) |  (0.40 %)
DEC |  46049 |  36134 |  41272 |  33314
(8.25 %) |  (8.75 %) |  (8.34 %) |  (8.86 %)

Finally, a word on the tidal deformability included in our GW waveform. At the
late stages of the inspiral, the quadrupolar tidal field $\mathcal{E}_{ij}$ of
one compact component would induce a quadrupole moment $Q_{ij}$ to the other.
To the leading order in the adiabatic approximation, $\mathcal{E}_{ij}$ and
$Q_{ij}$ are related by a linear response function,
$Q_{ij}=-\lambda\mathcal{E}_{ij}$, where $\lambda$ is the tidal Love number
(Hinderer, 2008). The static tidal effect enters the GW phase at the 5-th
post-Newtonian order101010Post-Newtonian order represents the power of
velocity squared relative to the leading Newtonian order for point-particle
binary motion. through the dimensionless binary tidal deformability
$\tilde{\Lambda}$ (Flanagan & Hinderer, 2008; Hinderer, 2008; Favata, 2014),

$\tilde{\Lambda}=\frac{16}{13}\frac{\left(M_{\mathrm{NS}}+12M_{\mathrm{WD}}\right)M_{\mathrm{NS}}^{4}\Lambda_{\mathrm{NS}}+\left(M_{\mathrm{WD}}+12M_{\mathrm{NS}}\right)M_{\mathrm{WD}}^{4}\Lambda_{\mathrm{WD}}}{\left(M_{\mathrm{NS}}+M_{\mathrm{WD}}\right)^{5}}\,,$
(19)

where $\Lambda_{\mathrm{NS}}$ and $\Lambda_{\mathrm{WD}}$ correspond
respectively to the dimensionless tidal deformability of the NS and the WD in
a NS-WD binary system. With
$\Lambda_{\mathrm{WD}}\gg\Lambda_{\mathrm{NS}}$111111For cold, slowly rotating
WDs, we can see in Figure 2 of Wolz et al. (2020) that $\Lambda_{\mathrm{WD}}$
is of the order of $10^{13}$–$10^{21}$. However, in comparison,
$\Lambda_{\mathrm{NS}}$ is $\lesssim\mathcal{O}(10^{3})$ in GW170817 (Abbott
et al., 2017a). we can simplify Equation (19) to

$\tilde{\Lambda}\simeq\frac{\left(M_{\mathrm{WD}}+12M_{\mathrm{NS}}\right)M_{\mathrm{WD}}^{4}\Lambda_{\mathrm{WD}}}{\left(M_{\mathrm{NS}}+M_{\mathrm{WD}}\right)^{5}}\,.$
(20)

We follow Wolz et al. (2020) to obtain $\Lambda_{\mathrm{WD}}$ via the
universal relation,

$\ln\Lambda_{\mathrm{WD}}=2.02942+2.48377\ln\bar{I}_{\mathrm{WD}}\,,$ (21)

where $\bar{I}_{\mathrm{WD}}=I_{\mathrm{WD}}/M_{\mathrm{WD}}^{3}$ is the
dimensionless moment of inertia of the WD, which can be calculated with
another fitting universal relation,

$\displaystyle\ln\bar{I}_{\mathrm{WD}}=$ $\displaystyle\
24.7995-39.0476\left(\frac{M_{\mathrm{WD}}}{1\,\mathrm{M}_{\odot}}\right)+95.9545\left(\frac{M_{\mathrm{WD}}}{1\,\mathrm{M}_{\odot}}\right)^{2}$
(22)
$\displaystyle-138.625\left(\frac{M_{\mathrm{WD}}}{1\,\mathrm{M}_{\odot}}\right)^{3}+98.8597\left(\frac{M_{\mathrm{WD}}}{1\,\mathrm{M}_{\odot}}\right)^{4}-27.4\left(\frac{M_{\mathrm{WD}}}{1\,\mathrm{M}_{\odot}}\right)^{5}\,.$

Note that although many theoretical studies have discussed the effects of mass
transfer and dynamical tides for WDs (Lai, 2012; Kremer et al., 2017; Tauris,
2018; McNeill et al., 2020; Kuns et al., 2020; Lau et al., 2022), there are
still many uncertainties and unsettled problems, especially on the GW waveform
construction with mass transfer. In addition, many studies have focused on the
low-mass WD donors (i.e., He WDs) or BWD systems, which can be different from
NS-WD binaries in our consideration. In the following sections, we mainly
focus on the estimation of the accuracy of distance $\Delta D_{\mathrm{L}}$
and angular resolution $\Delta\Omega$. These two quantities can be extracted
from the inspiral chirp signal alone. We leave other complex effects involving
WDs for future studies.

##  4 Results

Figure 3: SNR as a function of redshift for yearly detectable NS-WD mergers
for DO-OPT (left) and DEC (right). We only show the population Model A as an
example. Dashed horizontal red lines correspond to the detection threshold
$\mathrm{SNR}=8$. In each plot, the purple histograms show the redshift and
SNR distributions in upper and right panels, respectively.

In Table 2 we first summarize yearly detection numbers of NS-WD mergers for
the four population models with DO-OPT and DEC. We also list corresponding
percentages in brackets, showing the ratio to the total number of yearly
simulated NS-WD mergers (cf Table 1). The SNR threshold is set to 8. Combining
all the results shown in Table 2, we conclude that DEC performs much better
than DO-OPT on GW early-warning detections as a whole, with the yearly
detection number in the range of $\sim(3.3\mbox{--}4.6)\times 10^{4}$. The
detection number for DO-OPT is in the range of $\sim(1.5\mbox{--}1.9)\times
10^{3}$, which is approximately 20 times fewer than DEC. Given that DEC has a
lower noise level (see Figure 2), it is reasonable for DEC to show better
detection abilities, especially for detection at higher redshift, which is
clearly shown in Figure 3. Taking the population Model A as an example, Figure
3 shows that DO-OPT can detect early-warning sources up to $z\lesssim 1.7$,
while DEC could reach $z\lesssim 7.6$. Moreover, the detectable NS-WD mergers
for DEC are clustered at $z\simeq 0.8$, consistent with the peak of the
redshift factor $f(z)$ for the population Model A (see Figure 7). In
comparison, due to its limited detection abilities, most detectable NS-WD
mergers for DO-OPT are clustered around $z\simeq 0.25$, which is much closer
than those of DEC. Similarly, for the maximum SNR value, Figure 3 shows that
DEC could reach $\mathrm{SNR}\simeq 650$, while DO-OPT only reaches
$\mathrm{SNR}\simeq 75$.

Figure 4: Distributions of $f_{\text{out}}$ (defined in Section 3) for
detectable NS-WD mergers in DO-OPT (left) and DEC (right). The total numbers
of detectable merger events are listed in Table 2. We set the early-warning
time to be $t_{\mathrm{e}}=1\,\mathrm{d}$. Different colours correspond to
different population models.

As mentioned in Section 3, GW early-warning detections of NS-WD mergers are
essentially to offer alerts on the RLOF time. Assuming an early-warning time
$t_{\mathrm{e}}=1\,\mathrm{d}$ for each NS-WD merger event, we calculate the
integration upper limit $f_{\text{out}}$ with the known $f_{\max}$ at the
critical RLOF separation. We plot in Figure 4 the distributions of
$f_{\text{out}}$ for detectable NS-WD mergers observed by DO-OPT and DEC.
Different colours correspond to different population models. As clearly shown
in Figure 4, the majority of total detectable NS-WD mergers for DEC would
cluster at $f_{\text{out}}\simeq 0.1\,\mathrm{Hz}$. In comparison, the peak of
$f_{\text{out}}$ distribution for DO-OPT would be a little higher in a wider
rage ($0.16\mbox{--}0.25\,\mathrm{Hz}$). These could be explained by the
cosmological redshift effect, given that DEC detect more high-$z$ events.

Furthermore, based on the GW early-warning NS-WD mergers, we now present
estimations for the accuracy of distance ($\Delta D_{\mathrm{L}}$) and angular
resolution ($\Delta\Omega$), which will be of great use to synergy
observations with EM facilities. As described in Section 3, our analyses are
performed using the FIM method. We first compare ${\Delta
D_{\mathrm{L}}}/{D_{\mathrm{L}}}$-$\Delta\Omega$ distributions of our yearly
GW early-warning samples with DO-OPT for the four population models in Figure
5; similar distributions for DEC are plotted in Figure 6. We mark all
detectable NS-WD mergers with circles in different colours based on their
SNRs. Our results seem to suggest that the mergers with higher SNRs tend to
have better localization accuracies (i.e., smaller $\Delta\Omega$ and $\Delta
D_{\mathrm{L}}/D_{\mathrm{L}}$), which are mainly distributed in the lower
left of these plots in Figure 5 and Figure 6. In each plot, the purple
histograms show the $\Delta\Omega$ and $\Delta D_{\mathrm{L}}/D_{\mathrm{L}}$
distributions in upper and right panels, respectively. Both Figure 5 and
Figure 6 show that the peak of the $\Delta\Omega$ distribution is at
$\simeq\mathcal{O}(0.01)\,\mathrm{deg}^{2}$ level, regardless of the
population models in our considerations. For the $\Delta
D_{\mathrm{L}}/D_{\mathrm{L}}$ distributions, most early-warning samples of
DO-OPT are clustered in the range of $0.3\lesssim\Delta
D_{\mathrm{L}}/D_{\mathrm{L}}\lesssim 1$. In comparison, DEC provides more
accurate estimations of $\Delta D_{\mathrm{L}}$, given that the peak of the
$\Delta D_{\mathrm{L}}/D_{\mathrm{L}}$ distribution is mostly at
$\simeq\mathcal{O}(0.1)$ level.

To go a step further, we are particularly interested in those mergers that
would yield the best estimation results of the distance and angular
resolution. This is because these events can provide helpful inputs for future
multimessenger astronomy with decihertz GW early warnings and EM follow-ups.
In view of this, we further define a Golden Sample (denoted as ‘GS’ hereafter)
set for our yearly GW early-warning samples with the two criteria that

  1. (i)

$\Delta\Omega<1\,\mathrm{deg}^{2}$, and

  2. (ii)

$\Delta D_{\mathrm{L}}/D_{\mathrm{L}}<0.3$.

Given that many current and planned wide-field optical survey projects have
FoV $\gtrsim 1\,\mathrm{deg}^{2}$ (e.g., see in Table 5 the summary of the
technical information for some optical survey telescopes), we follow the
discussions in Kang et al. (2022) to set $1\,\mathrm{deg}^{2}$ to be the
threshold in criterion (i). For criterion (ii), we arbitrarily set 30% to be
the selection criterion on $\Delta D_{\mathrm{L}}/D_{\mathrm{L}}$. Note that
similar crude treatments in fact are commonly adopted in many studies (e.g.,
Tamanini & Danielski, 2019; Kang et al., 2021). Based on the above analyses,
the NS-WD mergers in GS would matter the most to future multimessenger early-
warning detections. In Figure 5 and Figure 6 we plot black dashed lines to
denote the selection functions corresponding to the criteria (i) and (ii)
above. Therefore, the shaded blue regions in Figure 5 and Figure 6 delimit the
parameter space of GS.

Figure 5: ${\Delta D_{\mathrm{L}}}/{D_{\mathrm{L}}}$ versus $\Delta\Omega$ for
yearly GW early-warning samples with DO-OPT for different population models.
The colour represents the SNR of each merger event. In each plot, the purple
histograms show the $\Delta\Omega$ and $\Delta D_{\mathrm{L}}/D_{\mathrm{L}}$
distributions in the upper and right panels, respectively. The black dashed
lines denote the dividing lines used to select the GS set in the shaded blue
region (see Section 4).

We list in Table 3 yearly detection numbers of NS-WD mergers in GS for the
four population models with DO-OPT and DEC. The percentages in brackets show
the ratio to the total number of yearly detectable NS-WD mergers (see Table
2). Combining all the results shown in Table 3, we conclude that DEC performs
much better on GS detections as a whole, with its yearly detection number in
the range of $\sim(2.1\mbox{--}2.8)\times 10^{4}$. In comparison, the GS
detection number for DO-OPT is in the range of $\sim(3.8\mbox{--}4.5)\times
10^{2}$, which is approximately 50 times fewer than DEC. In particular, we
note that the percentage of GS detections with DEC to its total yearly
detectable mergers is $\gtrsim 60\%$, while it is much smaller for DO-OPT,
with the ratio to be $\lesssim 25\%$.

Figure 6: Same as Figure 5, but for DEC. Note that all detectable NS-WD merger
events are color-coded by the logarithmic SNRs in each plot.

Finally, let us move to the potential for future multimessenger early-warning
detections of NS-WD mergers. Taking the peculiar lGRB, GRB 211211A, as an
example,121212Recently, there is another extremely bright cb-lGRB, GRB
230307A, at a redshift of $z=0.065$ ($\approx 300\,\mathrm{Mpc}$; Fausnaugh et
al., 2023; Levan et al., 2023; Sun et al., 2023; Mereghetti et al., 2023;
Gillanders et al., 2023). However, given the different features of the
afterglow/kilonova-like emissions and precursor properties from GRB 211211A,
most investigation was not in favor of the NS-WD merger origin for GRB 230307A
(Wang et al., 2023b; Levan et al., 2023; Dichiara et al., 2023; Song, 2023;
Yang et al., 2023a). It is more likely to be a BNS merger. the observational
facts show that it is a nearby bright GRB at $z=0.0763$ ($\approx
350\,\mathrm{Mpc}$) with a possible NS-WD merger origin (Rastinejad et al.,
2022; Troja et al., 2022; Yang et al., 2022; Mei et al., 2022; Gompertz et
al., 2023). Using a set of pseudo-GRBs, Yin et al. (2023) have shown that a
GRB 211211A-like event could be detectable up to a maximum redshift $z=0.52$
($\approx 3000\,\mathrm{Mpc}$). On the other hand, our results suggest that
the number of yearly detectable NS-WD mergers with $z\lesssim 0.52$ is in the
range of $\sim 990\mbox{--}1300$ for DO-OPT, and the corresponding percentage
to the total number of yearly detectable merger events (see Table 2) is $\sim
65\%$. They are $\sim 5200\mbox{--}6700$ and $\sim 15\%$ for DEC. In view of
this, if there are $\gtrsim 1$‰ of yearly NS–WD mergers with $z\lesssim 0.52$
that can successfully produce GRB 211211A-like signals (Yang et al., 2022; Yin
et al., 2023; Zhong et al., 2023), the possibility of multimessenger early-
warning detections indeed exists, with the help of the two decihertz GW
detectors and different $\gamma$-ray missions.

Besides the peculiar GRB signal, optical/infrared kilonova emissions
associated with GRB 211211A were also observed. Many recent studies have
proposed various kilonova models and suggest that the peak AB absolute
magnitude of GRB 211211A-like kilonovae should be $\approx{-17}$ in $\it{g}$,
$\it{r}$, and $\it{i}$ bands (Yang et al., 2022; Barnes & Metzger, 2023;
Kunert et al., 2023; Zhong et al., 2023). Based on the above finding, if we
set $z=0.52$ as the redshift upper limit for detections of GRB 211211A-like
events, the limiting magnitude $m^{*}$ for wide-field optical survey projects
should satisfy $m^{*}\gtrsim 25$ to ensure the kilonova detections. From Table
5 we find that LSST and CSST meet the above requirements. Therefore, more
cooperative observations with different optical surveys and decihertz GW
detectors would be very promising in the future. Given that the true nature of
GRB 211211A-like events remains unknown, we emphasize that the multimessenger
observations with GW early warnings and the above multiband EM follow-ups
would shed light on the origin of such peculiar lGRBs. Note that even for the
NS-WD merger scenario, the properties of their EM follow-up transients could
vary widely, depending on the component masses, the magnetic field strength
and configuration, specific afterglow and kilonova models, etc..131313For
example, Yang et al. (2022) pointed out that a massive WD component near the
Chandrasekhar mass limit should be involved to trigger the accretion-induced
collapse of the WD during the GRB 211211A event. We leave the impacts of those
uncertainties on the realistic multimessenger observations for future studies.

##  5 Conclusion

Table 3: Yearly detection numbers and percentages (in brackets) of NS-WD mergers in the GS for different population models with DO-OPT and DEC. Note that the percentages in brackets show the ratio to the total number of yearly detectable NS-WD mergers (see Table 2). GW Detector |  Population Model
---|---
|  A |  B |  C |  D
DO-OPT |  451 |  396 |  428 |  378
(23.55 %) |  (24.19 %) |  (24.85 %) |  (25.39 %)
DEC |  28229 |  22320 |  25301 |  20677
(61.30 %) |  (61.77 %) |  (61.30 %) |  (62.07 %)

In this work, we follow Liu et al. (2022b) to discuss GW detections and early-
warning predictions of NS-WD mergers with two space-borne decihertz GW
detectors, DO-OPT and DEC. Based on different SFR and DTD models, we use the
method of convolution with a series of Monte Carlo simulations to obtain four
NS-WD merger population models. For all NS-WD mergers with the merger time
$1\,\mathrm{yr}\leq t_{\mathrm{c}_{0}}\leq 2\,\mathrm{yr}$, we set the early-
warning time to $t_{\mathrm{e}}=1\,\mathrm{d}$ and SNR threshold to 8 to
perform GW detection strategy. Based on the NS-WD merger populations, we not
only give quick assessments of GW detection rates with the two decihertz GW
detectors, but also report systematic analyses on the characteristics of GW-
detectable merger events using the FIM. We find that DEC has better
performance than DO-OPT as a whole, especially for high-$z$ events. The yearly
GW detection number for DEC is in the range of $\sim(3.3\mbox{--}4.6)\times
10^{4}$, while it is only $\sim(1.5\mbox{--}1.9)\times 10^{3}$ for DO-OPT,
approximately 20 times fewer. Taking the population Model A as an example, DO-
OPT can detect early-warning sources up to $z\lesssim 1.7$, while DEC could
reach $z\lesssim 7.6$. Moreover, detectable NS-WD mergers are clustered at
$z\simeq 0.8$ for DEC. In comparison, due to the limited detection abilities,
most detectable NS-WD mergers are clustered around $z\simeq 0.25$ for DO-OPT.
Similarly, for the maximum SNR, we show that DEC reaches $\mathrm{SNR}\simeq
650$, while DO-OPT reaches $\mathrm{SNR}\simeq 75$.

Differently from BNS mergers or NS-BH mergers, we note that GW early-warning
detections of NS-WD mergers are essentially to offer alerts on the RLOF time,
especially considering that there is no well-modeled GW waveform for NS-WD
systems during the runaway mass-transfer phase. In view of this, for the FIM
in the frequency domain, the integration upper limit $f_{\text{out}}$ should
be related to the GW cutoff frequency at the critical RLOF separation. We plot
the distributions of $f_{\text{out}}$ for detectable NS-WD mergers observed by
DO-OPT and DEC. The majority of total detectable NS-WD mergers for DEC cluster
at $f_{\text{out}}\simeq 0.1\,\mathrm{Hz}$, while in comparison, the peak of
$f_{\text{out}}$ distribution for DO-OPT is a little higher in a wider range
($0.16\mbox{--}0.25\,\mathrm{Hz}$). These could be explained by the
cosmological redshift effect, given that DEC will detect more high-$z$ events.

As for the estimations of the accuracy of distance and angular resolution, we
compare ${\Delta D_{\mathrm{L}}}/{D_{\mathrm{L}}}$-$\Delta\Omega$
distributions of yearly GW early-warning samples for DO-OPT and DEC with
different population models. Our results show that the peak of the
$\Delta\Omega$ distribution is expected to be at
$\simeq\mathcal{O}(0.01)\,\mathrm{deg}^{2}$ level, regardless of the changes
in decihertz GW detectors and population models. For the $\Delta
D_{\mathrm{L}}/D_{\mathrm{L}}$ distributions, most early-warning samples of
DO-OPT are clustered in the range of $0.3\lesssim\Delta
D_{\mathrm{L}}/D_{\mathrm{L}}\lesssim 1$, while in comparison, DEC can provide
more accurate estimations of $\Delta D_{\mathrm{L}}$, given that the peak of
the $\Delta D_{\mathrm{L}}/D_{\mathrm{L}}$ distribution is at
$\simeq\mathcal{O}(0.1)$ level. Furthermore, for those mergers that would
yield the best estimation results of the distance and angular resolution, we
define them as a GS. We present the yearly detection numbers of NS-WD mergers
in GS for different population models with DO-OPT and DEC. We conclude that
DEC performs much better on GS detections as a whole, with its yearly
detection number in the range of $\sim(2.1\mbox{--}2.8)\times 10^{4}$, while
it is $\sim(3.8\mbox{--}4.5)\times 10^{2}$ for DO-OPT, approximately 50 times
fewer than DEC. The percentage of GS detections with DEC to its total yearly
detectable NS-WD mergers could achieve $\gtrsim 60\%$, while it is much
smaller for DO-OPT with the ratio to be $\lesssim 25\%$.

Finally, taking the recent peculiar lGRB, GRB 211211A, as an example, we
further discuss the potential for future multimessenger early-warning
detections of NS-WD mergers. With a sufficient early-warning time (e.g.,
$t_{\mathrm{e}}=1\,\mathrm{d}$) and the localization accuracies, we suggest
that the GW early-warning detection will allow future EM telescopes to prepare
well in advance for the possible follow-up transients after some special NS-WD
mergers. Given that the nature of cb-lGRBs remains an open question, the
multimessenger observations with the GW early warnings and the multiband EM
follow-ups are expected to shed new light on their properties. For example,
compared with BNS and NS-BH scenarios, no inspiral GW signals of NS-WD mergers
are expected to enter the ground-based detectors following decihertz GW early
warnings. The EM follow-up transients with different origins may also vary
widely. In view of this, our results can provide meaningful references and
helpful inputs for upcoming EM follow-up projects.

There are, of course, many ways in which this study can be extended. At the
end of Section 3, we have emphasized that precise GW-waveform constructions
should take into account the effects of mass transfer and dynamical tides for
WDs, especially in the late inspiral stages of NS-WD mergers. On the other
hand, even if there are no observable EM transients after NS-WD mergers, we
suggest that the GW-detectable NS-WD populations in GS could still be of great
use for the dark-siren cosmology (see, e.g., Schutz, 1986; Fishbach et al.,
2019; Soares-Santos et al., 2019; Zhu et al., 2022a; Zhu et al., 2022b; Liu et
al., 2022a; Yang et al., 2023b; Seymour et al., 2023). We hope that more
studies can be carried out in the future.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(11975027, 11991053, U2038105, U1831135, 12121003, 12103065), the National SKA
Program of China (2020SKA0120300, 2022SKA0130102), the National Key Research
and Development Programs of China (2022YFF0711404), the science research
grants from the China Manned Space Project with No. CMS-CSST-2021-B11, the Max
Planck Partner Group Program funded by the Max Planck Society, the Fundamental
Research Funds for the Central Universities, the Program for Innovative
Talents and Entrepreneur in Jiangsu, and the High-Performance Computing
Platform of Peking University. C.L. was supported by the China Scholarship
Council (CSC).

## Data Availability

The data underlying this paper will be shared on a reasonable request to the
corresponding authors.

## References

  * Abbott et al. (2017a) Abbott B. P., et al., 2017a, Phys. Rev. Lett., 119, 161101
  * Abbott et al. (2017b) Abbott B. P., et al., 2017b, ApJ, 848, L13
  * Abbott et al. (2017c) Abbott B. P., et al., 2017c, ApJ, 850, L39
  * Abbott et al. (2018a) Abbott B. P., et al., 2018a, Living Rev. Relativ., 21, 3
  * Abbott et al. (2018b) Abbott B. P., et al., 2018b, Phys. Rev. Lett., 121, 161101
  * Abbott et al. (2019) Abbott B. P., et al., 2019, Phys. Rev. Lett., 123, 011102
  * Abell et al. (2009) Abell P. A., et al., 2009, preprint, p. (arXiv:0912.0201)
  * Aghanim et al. (2020) Aghanim N., et al., 2020, A&A, 641, A6
  * Amaro-Seoane et al. (2023) Amaro-Seoane P., et al., 2023, Living Rev. Relativ., 26, 2
  * Arun et al. (2022) Arun K. G., et al., 2022, preprint, p. (arXiv:2205.01597)
  * Barack & Cutler (2004) Barack L., Cutler C., 2004, Phys. Rev. D, 69, 082005
  * Barnes & Metzger (2023) Barnes J., Metzger B. D., 2023, ApJ, 947, 55
  * Bayle et al. (2022) Bayle J.-B., Bonga B., Caprini C., Doneva D., Muratore M., Petiteau A., Rossi E., Shao L., 2022, Nature Astron., 6, 1334
  * Becker et al. (2012) Becker W., Prinz T., Winkler P. F., Petre R., 2012, ApJ, 755, 141
  * Bellm et al. (2019) Bellm E. C., et al., 2019, PASP, 131, 018002
  * Bobrick et al. (2017) Bobrick A., Davies M. B., Church R. P., 2017, MNRAS, 467, 3556
  * Bobrick et al. (2022) Bobrick A., Zenati Y., Perets H. B., Davies M. B., Church R., 2022, MNRAS, 510, 3758
  * Chatterjee et al. (2005) Chatterjee S., et al., 2005, ApJ, 630, L61
  * Chattopadhyay et al. (2007) Chattopadhyay T., Misra R., Chattopadhyay A. K., Naskar M., 2007, ApJ, 667, 1017
  * Cordes et al. (1993) Cordes J. M., Romani R. W., Lundgren S. C., 1993, Nature, 362, 133
  * Coulter et al. (2017) Coulter D. A., et al., 2017, Science, 358, 1556
  * Cutler (1998) Cutler C., 1998, Phys. Rev. D, 57, 7089
  * Cutler & Flanagan (1994) Cutler C., Flanagan E. E., 1994, Phys. Rev. D, 49, 2658
  * Dichiara et al. (2023) Dichiara S., Tsang D., Troja E., Neill D., Norris J. P., Yang Y. H., 2023, ApJ, 954, L29
  * Eggleton (1983) Eggleton P. P., 1983, ApJ, 268, 368
  * Eichler et al. (1989) Eichler D., Livio M., Piran T., Schramm D. N., 1989, Nature, 340, 126
  * Evans et al. (2017) Evans P. A., et al., 2017, Science, 358, 1565
  * Fausnaugh et al. (2023) Fausnaugh M. M., et al., 2023, Res. Notes AAS, 7, 56
  * Favata (2014) Favata M., 2014, Phys. Rev. Lett., 112, 101101
  * Fernández et al. (2019) Fernández R., Margalit B., Metzger B. D., 2019, MNRAS, 488, 259
  * Finn (1992) Finn L. S., 1992, Phys. Rev. D, 46, 5236
  * Fishbach et al. (2019) Fishbach M., et al., 2019, ApJ, 871, L13
  * Flanagan & Hinderer (2008) Flanagan E. E., Hinderer T., 2008, Phys. Rev. D, 77, 021502
  * Gillanders et al. (2020) Gillanders J. H., Sim S. A., Smartt S. J., 2020, MNRAS, 497, 246
  * Gillanders et al. (2023) Gillanders J. H., et al., 2023, preprint, p. (arXiv:2308.00633)
  * Goldstein et al. (2017) Goldstein A., et al., 2017, ApJ, 848, L14
  * Gompertz et al. (2023) Gompertz B. P., et al., 2023, Nat. Astron., 7, 67
  * Gong et al. (2019) Gong Y., et al., 2019, ApJ, 883, 203
  * Gunn & Ostriker (1970) Gunn J. E., Ostriker J. P., 1970, ApJ, 160, 979
  * Hinderer (2008) Hinderer T., 2008, ApJ, 677, 1216
  * Hobbs et al. (2005) Hobbs G., Lorimer D. R., Lyne A. G., Kramer M., 2005, MNRAS, 360, 974
  * Ivanova et al. (2013) Ivanova N., et al., 2013, A&A Rev., 21, 59
  * Ivezić et al. (2019) Ivezić v., et al., 2019, ApJ, 873, 111
  * Kalogera et al. (2001) Kalogera V., Narayan R., Spergel D. N., Taylor J. H., 2001, ApJ, 556, 340
  * Kaltenborn et al. (2022) Kaltenborn M. A., Fryer C. L., Wollaeger R. T., Belczynski K., Even W., Kouveliotou C., 2022, preprint, p. (arXiv:2209.13061)
  * Kang et al. (2021) Kang Y., Liu C., Shao L., 2021, AJ, 162, 247
  * Kang et al. (2022) Kang Y., Liu C., Shao L., 2022, MNRAS, 515, 739
  * Kawamura et al. (2011) Kawamura S., et al., 2011, Class. Quantum Gravity, 28, 094011
  * Kawamura et al. (2021) Kawamura S., et al., 2021, Prog. Theor. Exp. Phys., 2021, 05A105
  * Khokhriakova & Popov (2019) Khokhriakova A. D., Popov S. B., 2019, J. High Energy Astrophys., 24, 1
  * Kilpatrick et al. (2017) Kilpatrick C. D., et al., 2017, Science, 358, 1583
  * Kim et al. (2004) Kim C.-L., Kalogera V., Lorimer D. R., White T., 2004, ApJ, 616, 1109
  * King et al. (2007) King A., Olsson E., Davies M. B., 2007, MNRAS, 374, L34
  * Kopparapu et al. (2008) Kopparapu R. K., Hanna C., Kalogera V., O’Shaughnessy R., González G., Brady P. R., Fairhurst S., 2008, ApJ, 675, 1459
  * Kremer et al. (2017) Kremer K., Breivik K., Larson S. L., Kalogera V., 2017, ApJ, 846, 95
  * Kunert et al. (2023) Kunert N., et al., 2023, preprint, p. (arXiv:2301.02049)
  * Kuns et al. (2020) Kuns K. A., Yu H., Chen Y., Adhikari R. X., 2020, Phys. Rev. D, 102, 043001
  * Kyutoku et al. (2011) Kyutoku K., Okawa H., Shibata M., Taniguchi K., 2011, Phys. Rev. D, 84, 064018
  * Lai (2012) Lai J. F. D., 2012, MNRAS, 421, 426
  * Lamberts et al. (2018) Lamberts A., et al., 2018, MNRAS, 480, 2704
  * Lau et al. (2022) Lau S. Y., Yagi K., Arras P., 2022, Phys. Rev. D, 106, 103038
  * Levan et al. (2023) Levan A., et al., 2023, preprint, p. (arXiv:2307.02098)
  * Li & Paczynski (1998) Li L.-X., Paczynski B., 1998, ApJ, 507, L59
  * Li et al. (2023) Li J., et al., 2023, Sci. China Phys. Mech. Astron., 66, 109513
  * Liu & Shao (2022) Liu C., Shao L., 2022, ApJ, 926, 158
  * Liu et al. (2020) Liu C., Shao L., Zhao J., Gao Y., 2020, MNRAS, 496, 182
  * Liu et al. (2022a) Liu M., Liu C., Hu Y.-M., Shao L., Kang Y., 2022a, Phys. Dark Univ., 38, 101136
  * Liu et al. (2022b) Liu C., Kang Y., Shao L., 2022b, ApJ, 934, 84
  * Livio & Mazzali (2018) Livio M., Mazzali P., 2018, Phys. Rept., 736, 1
  * Lyne & Lorimer (1994) Lyne A. G., Lorimer D. R., 1994, Nature, 369, 127
  * Madau & Dickinson (2014) Madau P., Dickinson M., 2014, ARA&A, 52, 415
  * Maggiore (2008) Maggiore M., 2008, Gravitational Waves. Vol. 1: Theory and Experiments. Oxford Univ. Press, Oxford, Oxford Master Series in Physics
  * Maggiore et al. (2020) Maggiore M., et al., 2020, J. Cosmology Astropart. Phys., 03, 050
  * Margalit & Metzger (2016) Margalit B., Metzger B. D., 2016, MNRAS, 461, 1154
  * Margalit & Metzger (2017) Margalit B., Metzger B. D., 2017, MNRAS, 465, 2790
  * Masci et al. (2019) Masci F. J., et al., 2019, PASP, 131, 018003
  * McNeill et al. (2020) McNeill L. O., Mardling R. A., Müller B., 2020, MNRAS, 491, 3000
  * Mei et al. (2022) Mei A., et al., 2022, Nature, 612, 236
  * Mereghetti et al. (2023) Mereghetti S., Rigoselli M., Salvaterra R., Tiengo A., Pacholski D., 2023, preprint, p. (arXiv:2307.13514)
  * Metzger (2012) Metzger B. D., 2012, MNRAS, 419, 827
  * Narayan et al. (1992) Narayan R., Paczynski B., Piran T., 1992, ApJ, 395, L83
  * Nauenberg (1972) Nauenberg M., 1972, ApJ, 175, 417
  * Nelemans et al. (2001a) Nelemans G., Yungelson L. R., Portegies Zwart S. F., Verbunt F., 2001a, A&A, 365, 491
  * Nelemans et al. (2001b) Nelemans G., Yungelson L. R., Portegies Zwart S. F., 2001b, A&A, 375, 890
  * O’Shaughnessy & Kim (2010) O’Shaughnessy R., Kim C., 2010, ApJ, 715, 230
  * Paczynski (1986) Paczynski B., 1986, ApJ, 308, L43
  * Paschalidis et al. (2009) Paschalidis V., MacLeod M., Baumgarte T. W., Shapiro S. L., 2009, Phys. Rev. D, 80, 024006
  * Paschalidis et al. (2011) Paschalidis V., Liu Y. T., Etienne Z., Shapiro S. L., 2011, Phys. Rev. D, 84, 104032
  * Pian et al. (2017) Pian E., et al., 2017, Nature, 551, 67
  * Rastinejad et al. (2022) Rastinejad J. C., et al., 2022, Nature, 612, 223
  * Reitze et al. (2019) Reitze D., et al., 2019, preprint, p. (arXiv:1907.04833)
  * Ruiter (2020) Ruiter A. J., 2020, Int. Astron. Union Symp., 357, 1
  * Sathyaprakash et al. (2019) Sathyaprakash B., et al., 2019, BAAS, 51, 251
  * Savchenko et al. (2017) Savchenko V., et al., 2017, ApJ, 848, L15
  * Schutz (1986) Schutz B. F., 1986, Nature, 323, 310
  * Sedda et al. (2020) Sedda M. A., et al., 2020, Class. Quantum Gravity, 37, 215011
  * Sedda et al. (2021) Sedda M. A., et al., 2021, Exper. Astron., 51, 1427
  * Seymour et al. (2023) Seymour B. C., Yu H., Chen Y., 2023, Phys. Rev. D, 108, 044038
  * Shao (2023) Shao L., 2023, Sci. China Phys. Mech. Astron., 66, 119531
  * Shao et al. (2017) Shao L., Sennett N., Buonanno A., Kramer M., Wex N., 2017, Phys. Rev. X, 7, 041025
  * Shi et al. (2018) Shi D. D., Zheng X. Z., Zhao H. B., Lou Z., Wang H. R., Qian Y., Liu W., Yao D. Z., 2018, Acta Astron. Sin., 59, 22
  * Shibata & Taniguchi (2011) Shibata M., Taniguchi K., 2011, Living Rev. Relativ., 14, 6
  * Soares-Santos et al. (2019) Soares-Santos M., et al., 2019, ApJ, 876, L7
  * Soker (2018) Soker N., 2018, SCPMA, 61, 049502
  * Song (2023) Song X.-Y., 2023, preprint, p. (arXiv:2307.16547)
  * Sun et al. (2015) Sun H., Zhang B., Li Z., 2015, ApJ, 812, 33
  * Sun et al. (2023) Sun H., et al., 2023, preprint, p. (arXiv:2307.05689)
  * Tamanini & Danielski (2019) Tamanini N., Danielski C., 2019, Nat. Astron., 3, 858
  * Tauris (2018) Tauris T. M., 2018, Phys. Rev. Lett., 121, 131105
  * Toonen et al. (2018) Toonen S., Perets H. B., Igoshev A. P., Michaely E., Zenati Y., 2018, A&A, 619, A53
  * Troja et al. (2022) Troja E., et al., 2022, Nature, 612, 228
  * Verbunt & Rappaport (1988) Verbunt F., Rappaport S., 1988, ApJ, 332, 193
  * Verbunt et al. (2017) Verbunt F., Igoshev A., Cator E., 2017, A&A, 608, A57
  * Virgili et al. (2011) Virgili F. J., Zhang B., O’Brien P., Troja E., 2011, ApJ, 727, 109
  * Wanderman & Piran (2015) Wanderman D., Piran T., 2015, MNRAS, 448, 3026
  * Wang (2018) Wang B., 2018, Res. Astron. Astrophys., 18, 049
  * Wang et al. (2022) Wang Z., Liu C., Zhao J., Shao L., 2022, Astrophys. J., 932, 102
  * Wang et al. (2023a) Wang T., et al., 2023a, preprint, p. (arXiv:2306.07590)
  * Wang et al. (2023b) Wang Y., Xia Z.-Q., Zheng T.-C., Ren J., Fan Y.-Z., 2023b, ApJ, 953, L8
  * Waxman et al. (2022) Waxman E., Ofek E. O., Kushnir D., 2022, preprint, p. (arXiv:2206.10710)
  * Wolz et al. (2020) Wolz A., Yagi K., Anderson N., Taylor A. J., 2020, MNRAS, 500, L52
  * Yang et al. (2022) Yang J., et al., 2022, Nature, 612, 232
  * Yang et al. (2023a) Yang Y.-H., et al., 2023a, preprint, p. (arXiv:2308.00638)
  * Yang et al. (2023b) Yang T., Cai R.-G., Cao Z., Lee H. M., 2023b, Phys. Rev. D, 107, 043539
  * Yin et al. (2023) Yin Y.-H. I., Zhang B.-B., Sun H., Yang J., Kang Y., Shao L., Yang Y.-H., Zhang B., 2023, ApJ, 954, L17
  * Yuan et al. (2020) Yuan X., et al., 2020, Proc. SPIE, 11445, 114457M
  * Yuksel et al. (2008) Yuksel H., Kistler M. D., Beacom J. F., Hopkins A. M., 2008, ApJ, 683, L5
  * Zenati et al. (2019) Zenati Y., Perets H. B., Toonen S., 2019, MNRAS, 486, 1805
  * Zhang (2018) Zhang B., 2018, The Physics of Gamma-Ray Bursts. Cambridge University Press, Cambridge
  * Zhang et al. (2018) Zhang B. B., et al., 2018, Nat. Commun., 9, 447
  * Zhao et al. (2021) Zhao Z. Y., Zhang G. Q., Wang Y. Y., Tu Z. L., Wang F. Y., 2021, ApJ, 907, 111
  * Zhong et al. (2023) Zhong S.-Q., Li L., Dai Z.-G., 2023, ApJ, 947, L21
  * Zhu et al. (2021) Zhu J.-P., et al., 2021, ApJ, 917, 24
  * Zhu et al. (2022a) Zhu L.-G., Hu Y.-M., Wang H.-T., Zhang J.-d., Li X.-D., Hendry M., Mei J., 2022a, Phys. Rev. Res., 4, 013247
  * Zhu et al. (2022b) Zhu L.-G., et al., 2022b, Sci. China Phys. Mech. Astron., 65, 259811
  * Zhu et al. (2022c) Zhu J.-P., Wang X. I., Sun H., Yang Y.-P., Li Z., Hu R.-C., Qin Y., Wu S., 2022c, ApJ, 936, L10
  * Zhu et al. (2023) Zhu J.-P., et al., 2023, ApJ, 942, 88

##  Appendix A The redshift distribution factor

As mentioned in Section 2, when we ignore the possible redshift evolution of
intrinsic system parameters for NS-WD mergers, the redshift-dependent merger
event rate density $\dot{\rho}(z)$ is,

$\dot{\rho}(z)=\dot{\rho}_{0}f(z)\,,$ (23)

where $z$ is the redshift, $\dot{\rho}_{0}$ is the local NS-WD event rate
density, and $f(z)$ is the dimensionless redshift distribution factor. On the
other hand, $\dot{\rho}(z)$ can also connect to the cosmological SFR density
$\dot{\rho}_{*}(z)$ by accounting for the probability density function of
delay time $P(\tau)$ of NS-WD mergers via

$\displaystyle\dot{\rho}(z)$
$\displaystyle\propto\int_{\tau_{\min}}^{\tau_{\max}}\dot{\rho}_{*}\left[z^{\prime}(\tau)\right]P(\tau)d\tau$
(24)
$\displaystyle\propto\int_{z^{\prime}_{\min}}^{z^{\prime}_{\max}}\dot{\rho}_{*}\left(z^{\prime}\right)P\left[\tau\left(z^{\prime}\right)\right]\left[-\frac{dt\left(z^{\prime}\right)}{dz^{\prime}}\right]dz^{\prime}\,,$

where $\tau=t(z)-t\left(z^{\prime}\right)$ is the delay time141414As defined
in Section 3.2.1 of Toonen et al. (2018), the delay time is the evolutionary
time between the merger and formation of the binary with two zero-age main
sequence stars. and $t\left(z^{\prime}\right)$ is the time when the binaries
are formed with two zero-age main sequence stars, $t(z)$ is the merger time
for such binaries, and $\tau_{\min}$ and $\tau_{\max}$ are the minimum and
maximum delay times, respectively.

With Equations (23) and (24), we can obtain $f(z)$ by normalizing it to unity
in the local Universe (i.e., $z=0$). Note that here $t$ is the age of the
Universe and we have,

$\frac{dt(z)}{dz}=-\frac{1}{H_{0}(1+z)\sqrt{\Omega_{\Lambda}+(1+z)^{3}\Omega_{\mathrm{m}}}}\,.$
(25)

Therefore, for NS-WD mergers at a fixed $z$, we can calculate the
corresponding $z^{\prime}_{\min}$ by the definition of $\tau_{\min}$ and
Equation (25). Based on the population-synthesis results presented in Toonen
et al. (2018), we set $\tau_{\min}=0.1\,\mathrm{Gyr}$ in our calculations.
Following Khokhriakova & Popov (2019), we also set the upper integration
redshift limit $z^{\prime}_{\max}$ to be 11.247 (corresponding to the time
when the age of our Universe was $400\,\mathrm{Myr}$) when the intensive star
formation began.

For the SFR density $\dot{\rho}_{*}(z)$, we consider two kinds of analytical
models from Yuksel et al. (2008) and Madau & Dickinson (2014) (abbreviated as
‘Y08’ and ‘MD14’, respectively) in our work. They read,

$\displaystyle\dot{\rho}_{*}^{\mathrm{Y08}}(z)$
$\displaystyle\propto{\left[(1+z)^{3.4\eta}+\left(\frac{1+z}{5000}\right)^{-0.3\eta}\right.}\left.+\left(\frac{1+z}{9}\right)^{-3.5\eta}\right]^{1/\eta}\,,$
(26) $\displaystyle\dot{\rho}_{*}^{\mathrm{MD14}}(z)$
$\displaystyle\propto\frac{(1+z)^{2.7}}{1+[(1+z)/2.9]^{5.6}}\,.$ (27)

Note that $\eta=-10$ is adopted in Equation (26).

Table 4: The best-fit $\alpha_{\mathrm{t}}$ values derived from different DTDs shown in Section 3.2.1 of Toonen et al. (2018) by Equation (28). We consider four combinations of two CE evolution models (model ‘$\alpha\alpha$’ and model ‘$\gamma\alpha$’) and two SN-kick velocity distributions (abbreviated as ‘$\mathrm{H}$’ and ‘$\mathrm{V}$’). More details about these models are given in Appendix A. Model |  $\alpha_{\mathrm{t}}$
---|---
$\alpha\alpha$-$\mathrm{H}$ |  $-1.12$
$\alpha\alpha$-$\mathrm{V}$ |  $-0.98$
$\gamma\alpha$-$\mathrm{H}$ |  $-1.11$
$\gamma\alpha$-$\mathrm{V}$ |  $-0.95$

Figure 7: Redshift distributions of $f(z)$ derived from the method of convolution by Equation (24). The Y08 SFR model is adopted in the left plot, while the MD14 model is applied in the right plot. The coloured solid lines and dashed lines denote different DTD models. The redshift cutoff $z=8$ is adopted here. Table 5: Technical information for some wide-field optical survey projects from Zhu et al. (2021, 2023). Note that a 300-s exposure time is adopted for the search limiting magnitude $m^{*}$ in the three most common filters ($\it{g},\it{r},\it{i}$). The references are: (1) Masci et al. (2019), Bellm et al. (2019); (2) Abell et al. (2009), Ivezić et al. (2019); (3) Shi et al. (2018), Wang et al. (2023a); (4) Yuan et al. (2020); (5) Gong et al. (2019). Survey |  $m^{*}$/$\mathrm{mag}$ |  FoV/$\mathrm{deg}^{2}$ |  Sky Coverage/$\mathrm{deg}^{2}$ |  Reference
---|---|---|---|---
|  $\it{g}$ |  $\it{r}$ |  $\it{i}$ |  |  |
ZTF |  21.6 |  21.3 |  20.9 |  47.7 |  30000 |  (1)
LSST |  26.2 |  25.7 |  25.8 |  9.6 |  20000 |  (2)
WFST |  24.2 |  24.0 |  23.3 |  6.55 |  20000 |  (3)
Mephisto |  24.2 |  23.9 |  23.4 |  3.14 |  26000 |  (4)
CSST |  26.3 |  26.0 |  25.9 |  1.1 |  17500 |  (5)

As for the probability density function of delay time $P(\tau)$, many studies
have assumed some empirical forms and derived the best-fit parameters with
simulated data. Such a treatment has been widely applied to other DCO systems
(e.g., BNS and NS-BH systems; Virgili et al., 2011; Wanderman & Piran, 2015;
Sun et al., 2015; Zhu et al., 2021). Therefore, for simplicity, we assume a
power-law dependence for the DTD of NS-WD mergers,

$P(\tau)\propto\tau^{-\alpha_{\mathrm{t}}}\,,$ (28)

where $\alpha_{\mathrm{t}}$ is a phenomenological parameter. Considering
different common-envelope (CE) evolutions151515As a mass-loss phase in the
formation of DCO systems, CE evolution can lead to a severe shrinkage of the
binary orbit. We refer readers to see Ivanova et al. (2013) for a
comprehensive review. and different NS natal-kick161616NS natal-kicks can also
be called SN-kicks. Many studies of pulsar scale heights (Gunn & Ostriker,
1970), proper motions of pulsars (Cordes et al., 1993; Lyne & Lorimer, 1994;
Hobbs et al., 2005; Verbunt et al., 2017), and high velocities of some single
NSs (Chatterjee et al., 2005; Becker et al., 2012) have suggested that a kick
should be imparted to the NS during core-collapse SNs. distributions, Toonen
et al. (2018) have recently presented different DTDs of NS-WD mergers. Here,
we fit four kinds of DTDs taken from Toonen et al. (2018)—abbreviated as
‘$\alpha\alpha$-$\mathrm{H}$’, ‘$\alpha\alpha$-$\mathrm{V}$’,
‘$\gamma\alpha$-$\mathrm{H}$’, and ‘$\gamma\alpha$-$\mathrm{V}$’—by Equation
(28) and derive the best-fit $\alpha_{\mathrm{t}}$ values, which are listed in
Table 4. Note that ‘$\alpha$’ and ‘$\gamma$’ are commonly used to denote
different treatments for the CE phase where the ‘$\alpha$’-formalism is based
on the energy conservation while the ‘$\gamma$’-formalism is based on the
balance of angular momentum instead of the energy (Ivanova et al., 2013). More
specifically, following Toonen et al. (2018), we use ‘$\alpha\alpha$’ and
‘$\gamma\alpha$’ to denote two kinds of CE evolutionary models for NS-WD
mergers where the ‘$\alpha$’-prescription is used to determine the outcome of
every CE in model ‘$\alpha\alpha$’. In model ‘$\gamma\alpha$’, the
‘$\gamma$’-prescription is introduced to describe the first CE phase, while
the ‘$\alpha$’-prescription is applied in the second CE phase. For SN-kick
velocity distributions, we adopt two kinds of models from Hobbs et al. (2005)
and Verbunt et al. (2017), respectively (abbreviated as ‘$\mathrm{H}$’ and
‘$\mathrm{V}$’ when combined with model ‘$\alpha\alpha$’ and model
‘$\gamma\alpha$’). The former (i.e., ‘$\mathrm{H}$’) is a Maxwellian
distribution with a one-dimensional root mean square of
$\sigma_{\mathrm{H}}=265\,\mathrm{km}\,\mathrm{s}^{-1}$, while the latter
(i.e., ‘$\mathrm{V}$’) includes two Maxwellian components with
$\sigma_{\mathrm{V}}^{1}=75\,\mathrm{km}\,\mathrm{s}^{-1}$ and
$\sigma_{\mathrm{V}}^{2}=316\,\mathrm{km}\,\mathrm{s}^{-1}$ from a direct
comparison of pulsar parallaxes and proper motions. More detailed descriptions
of the above DTDs can be found in Toonen et al. (2018).

Based on the above SFR and DTD models, we follow Zhu et al. (2021) to use the
method of convolution by Equation (24) and obtain the dimensionless redshift
distribution factor $f(z)$, which are shown in Figure 7. Given a specific DTD
model, Figure 7 shows that $f(z)$ has a dramatic dependence on the SFR model
in the relatively lower redshift regime ($z\lesssim 4.5$), while in the
high-$z$ regime ($z>4.5$), there is little difference between the Y08 SFR
model and the MD14 model. On the other hand, when we adopt the same SFR model,
both Table 4 and Figure 7 suggest that different CE evolution models will not
influence $f(z)$ significantly, while $f(z)$ can be indeed influenced by the
choice of the SN-kick velocity distribution. For this reason, we decide to
report detailed analyses only between the model $\gamma\alpha$-$\mathrm{H}$
and the model $\gamma\alpha$-$\mathrm{V}$ in the main text for more in-depth
comparisons.

##  Appendix B Information for optical surveys

Technical parameters for five current and planned optical time-domain surveys
are listed in Table 5, which are: the Zwicky Transient Facility (ZTF; Masci et
al., 2019; Bellm et al., 2019); the Large Synoptic Survey Telescope (LSST,
newly named as the Vera Rubin Observatory; Abell et al., 2009; Ivezić et al.,
2019); the Wide Field Survey Telescope (WFST; Shi et al., 2018; Wang et al.,
2023a); the Multi-channel Photometric Survey Telescope (Mephisto; Yuan et al.,
2020); and the Chinese Space Station Telescope (CSST; Gong et al., 2019).
Given the search limiting magnitude $m^{*}$, one can obtain the effective
limiting flux $F_{\nu}^{*}$ for each survey telescope by $F_{\nu}^{*}\simeq
3631\,\mathrm{Jy}\times 10^{-m^{*}/2.5}$. Note that Table 5 only lists $m^{*}$
values with the three most common bands ($\it{g},\it{r},\it{i}$), assuming a
300-s exposure time for each survey.

Recently, Kang et al. (2022) have investigated the prospects of multimessenger
early-warning detections for BNS mergers, with the help of early warnings from
different decihertz GW observatories and optical survey missions. We leave
similar analyses on NS-WD mergers for future studies.