Kilonovae and Optical Afterglows from Binary Neutron Star Mergers. II. Optimal Search Strategy for Serendipitous Observations and Target-of-opportunity Observations of Gravitational Wave Triggers

The total number of BNS mergers in the universe can be estimated as (e.g., Sun et al. 2015)

$$\begin{eqnarray}&&{\dot{N}}_{\mathrm{BNS}}\approx {\int }_{0}^{{z}_{\max }}\displaystyle \frac{{\dot{\rho }}_{0,\mathrm{BNS}}f(z)}{1+z}\displaystyle \frac{{dV}(z)}{{dz}}{dz},\end{eqnarray} \tag{ 1 }$$

where ${\dot{\rho }}_{0,\mathrm{BNS}}$ is the local BNS event rate density, f(z) is the dimensionless redshift distribution factor, and ${z}_{\max }$ is the maximum redshift for BNS mergers. The comoving volume element dV(z)/dz in Equation (1) is

$$\begin{eqnarray}&&\displaystyle \frac{{dV}}{{dz}}=\displaystyle \frac{c}{{H}_{0}}\displaystyle \frac{4\pi {D}_{{\rm{L}}}^{2}}{{\left(1+z\right)}^{2}\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}}},\end{eqnarray} \tag{ 2 }$$

where c is the speed of light and D_L is the luminosity distance, which is expressed as

$$\begin{eqnarray}&&{D}_{{\rm{L}}}=(1+z)\displaystyle \frac{c}{{H}_{0}}{\int }_{0}^{z}\displaystyle \frac{{dz}}{\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{{\rm{m}}}{\left(1+z\right)}^{3}}}.\end{eqnarray} \tag{ 3 }$$

Recently, Abbott et al. (2021b) estimated the local BNS event rate density as ${\dot{\rho }}_{0,\mathrm{BNS}}={320}_{-240}^{+490}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ based on the GW observations during the first half of the third observing (O3) run (see Mandel & Broekgaarden 2022, for a review of ${\dot{\rho }}_{0,\mathrm{BNS}}$). Hereafter, if not otherwise specified, ${\dot{\rho }}_{0,\mathrm{BNS}}$ used in our calculations is simply set as the median value of the GW constraint by the LIGO/Virgo Collaboration (LVC), i.e., ${\dot{\rho }}_{0,\mathrm{BNS}}\simeq 320\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$.

BNS mergers can be thought as occurring with a delay timescale with respect to the star formation history. The Gaussian delay model (Virgili et al. 2011), log-normal delay model (Nakar et al. 2006; Wanderman & Piran 2015), and power-law delay model (Virgili et al. 2011; Hao & Yuan 2013; D'Avanzo et al. 2014) are the main types of delay time distributions. Sun et al. (2015) suggested that the power-law delay model leads to a wider redshift distribution of BNS mergers than the other two models, while recent observations of sGRBs by Zevin et al. (2022), Fong et al. (2022), O'Connor et al. (2022), and Nugent et al. (2022) supported the power-law delay. Although much debate remains, for simplicity, we only adopt the log-normal delay model as our merger delay model, and the analytical fitting expression of f(z) is presented as Equation (A8) in Zhu et al. (2021d). With a known redshift distribution f(z), we randomly simulate a group of ${n}_{\mathrm{sim}}=5\times {10}^{6}$ BNS events in the universe based on Equation (1). For each BNS event, we then generate the EM emission. We briefly assume that all BNS events in the universe would only power three main types of EM signals, i.e., the jet afterglow, the kilonova, and the sGRB. We assume that cosmological kilonovae are AT2017gfo-like with the consideration of the viewing-angle effect, while the properties of afterglows are consistent with those of cosmological sGRB afterglows. The modeling details of the redshift distribution, jet afterglow, and kilonova emission of BNS mergers have been presented in Paper I. Our viewing-angle-dependent semi-analytical model of sGRB emission follows Song et al. (2019) and Yu et al. (2021). The signature of sGRBs depends on the on-axis equivalent isotropic energy E₀, the core half-opening angle θ_c, and the latitudinal viewing angle θ_view, while the afterglow emission has a dependence on four additional parameters, i.e., the number density of interstellar medium n, the power-law index of the electron distribution p, and the fractions of shock energy distributed in electrons, ε_e , and in magnetic fields, ε_B . Furthermore, the kilonova emission is only determined by θ_view. According to the distributions of the above parameters, as described in Paper I in detail, one can randomly generate the EM emission components for each simulated BNS event.

We divide the detectable events into two main groups based on the relative brightness of the detected kilonova and afterglow. If the peak kilonova flux is larger than 5 times the afterglow flux, i.e., F_ν,KN(t_KN,p) > 5F_ν,AG(t_KN,p), where t_KN,p is the peak time of the kilonova, we classify these events into the "kilonova-dominated sample." For such events, kilonova emission at the peak time would be at least 2 magnitudes brighter than that of the associated afterglow emission, so that this requirement can guarantee a clear kilonova signal for observers. For on-axis or near-on-axis afterglows, some bright kilonovae can appear detectable as an excess flux compared to the afterglow power-law decay, which are also defined as kilonova-dominated events. Other events are classified in the "afterglow-dominated sample," as the observed kilonova signals of these events may be ambiguous. In Paper I, we have shown that ∼50% on-axis and nearly on-axis afterglows are brighter than the associated kilonovae at the peak time. Thus, most of them would be afterglow dominated. Only at large viewing angles with $\sin {\theta }_{{\rm{v}}}\gtrsim 0.20$, would the EM signals of most BNS mergers be kilonova dominated, and some off-axis afterglows may emerge ∼5–10 days after the mergers.

Some optically discovered EM counterparts of BNS mergers could be associated with sGRB observations. For the GW-triggered ToO searches, the observations of sGRBs can cooperate on the constraint on the sky location for BNS GW alerts, which would help us find the EM counterparts. On the basis of whether or not an sGRB is detected, we can further divide each sample into two subsamples, i.e., (1) kilonova-dominated sample: kilonova with (w/) an sGRB and kilonova without (w/o) an sGRB; (2) afterglow-dominated sample: afterglow w/ an sGRB and afterglow w/o an sGRB. Furthermore, a fraction of the BNS mergers may only be detected in the γ-ray band without any detection of an associated optical afterglow or kilonova. Thus, we totally define five subsamples for the EM counterparts of BNS mergers (see Table 1).

SGRBs are believed to be triggered if F_γ > F_γ,limit, where F_γ is the γ-band flux for each BNS GW event (see Song et al. 2019; Yu et al. 2021, for details on the sGRB model) and F_γ,limit is the effective sensitivity limit for various γ-ray detectors. Many GRB detectors with quick response and wide FoVs, e.g., Swift (Gehrels et al. 2004), AstroSAT (Singh et al. 2014), Fermi (Meegan et al. 2009), and the Gravitational wave high-energy Electromagnetic Counterpart All-sky Monitor (GECAM; Zhang et al. 2019; Song et al. 2019), will work during O4. In our calculations, we simply set F_γ,limit ∼ 2 ×10⁻⁷ erg s⁻¹ in 50–300 keV, which is the effective sensitivity limit of the Fermi Gamma-ray Burst Monitor (GBM; Meegan et al. 2009) and GECAM (Zhang et al. 2019; Song et al. 2019), in view of the fact that Fermi-GBM and GECAM can nearly achieve an all-sky coverage to detect GRB events. ¹³

Following Zhu et al. (2021d), we adopt a method of probabilistic statistical analysis to estimate the EM detection rate for BNS mergers. The probability that a single simulated event can be detected could be considered as the ratio of the survey area within the time duration (Δt) that the brightness of the associated EM signal is above the limiting magnitude (m_limit) to the area of the celestial sphere (Ω_sph = 41252.96 deg²). The maximum probability for a source to be detected is ${{\rm{\Omega }}}_{\mathrm{FoV}}{\dot{t}}_{\mathrm{ope}}{\rm{\Delta }}t/{{\rm{\Omega }}}_{\mathrm{sph}}({n}_{\exp }{t}_{\exp }+{t}_{\mathrm{oth}})$, where Ω_FoV is the field of view (FoV) for the specific survey project, ${\dot{t}}_{\mathrm{ope}}$ is the average operation time per day, ${n}_{\exp }$ is defined as the exposure number for each visit, ${t}_{\exp }$ is the exposure time, and t_other is the rest of the time spent for each visit. However, high-cadence observations would restrict the survey area, which means that the probability of a source being detected by the high-cadence search would be a constant, i.e., ${{\rm{\Omega }}}_{\mathrm{FoV}}{\dot{t}}_{\mathrm{ope}}{t}_{\mathrm{cad}}/{{\rm{\Omega }}}_{\mathrm{sph}}({n}_{\exp }{t}_{\exp }+{t}_{\mathrm{oth}})$, where the cadence time t_cad is defined as the interval between consecutive observations of the same sky area by a telescope. Furthermore, the event should appear in the sky coverage of the survey telescope so one can have a chance to discover it. Thus, we simply set an upper limit on the probability of a source to be detected, which is expressed as Ω_cov/Ω_sph, with Ω_cov being the detectable sky coverage for a specific survey project. By counting the detection probabilities of all simulated events, one can write the EM detection rate for serendipitous observations as

$$\begin{eqnarray}&&{\dot{N}}_{\mathrm{EM}}\approx \displaystyle \frac{{\dot{N}}_{\mathrm{BNS}}}{{n}_{\mathrm{sim}}}\displaystyle \sum _{i=1}^{{n}_{\mathrm{sim}}}\min \left[\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{cov}}}{{{\rm{\Omega }}}_{\mathrm{sph}}},\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{FoV}}{\dot{t}}_{\mathrm{ope}}\min ({t}_{\mathrm{cad}},{\rm{\Delta }}{t}_{i})}{{{\rm{\Omega }}}_{\mathrm{sph}}({n}_{\exp }{t}_{\exp }+{t}_{\mathrm{oth}})}\right].\end{eqnarray} \tag{ 4 }$$

We roughly assume that the average operation time per day is ${\dot{t}}_{\mathrm{ope}}\approx 6\,\mathrm{hr}\,{\mathrm{day}}^{-1}$ for all survey projects except for SiTian. The time spent for each visit t_oth is dependent on the technical performance of the specific survey project and the different search strategy. Because t_oth is uncertain, we set it as a constant for each survey project, i.e., t_oth = 15 s.

In order to reject the supernova background and other rapid-evolving transients, in Paper I, we showed that one can use the unique color evolution of kilonovae and afterglows to identify them among the observed transients. We require that the judgment condition for the detection of the kilonova and/or afterglow by a serendipitous search is that "two different exposure filters have at least two detection epochs." It would be ${n}_{\exp }=1$ for Mephisto and SiTian as these two survey projects can achieve simultaneous imaging in three bands, ¹⁴ while ${n}_{\exp }=2$ for ZTF, WFST, and LSST.

The values of some technical parameters, including the expected limiting magnitude (which is a logarithmic function of the exposure time in each band), the FoV, and the detectable sky coverage, for the survey telescopes we considered are presented in Table 2. As examples, we also list the g-band limiting magnitudes m_g,limit for different exposure times of ${t}_{\exp }=30,180,300\,{\rm{s}}$ for the survey telescopes in Table 2. Thus, survey telescopes with apertures smaller than those of ZTF and SiTian can have a limiting magnitude of m_limit ≲ 20 mag. The detection depths of ZTF and SiTian lie in the range 20 mag ≲ m_limit ≲ 22 mag. A range of 22 mag ≲ m_limit ≲ 24 mag applies to Mephisto and WFST, and 24 ≲ m_limit ≲ 26 mag can be only achieved by LSST.

Table 2. Summary Technical Information for Each Survey

Telescope	${m}_{\mathrm{limit}}=a\times {t}_{\exp }^{b}$						${t}_{\exp }/{\rm{s}}$	mg,limit/mag	FoV/deg2	Sky Coverage/deg2	References
ZTF	g		r		i		30	20.3	47.7	30,000	(1)
	18.62		18.37		17.91		180	21.3
	0.026		0.026		0.027		300	21.6
Mephisto	u	v	g	r	i	z	30	22.4	3.14	26,000	(2)
	18.45	18.54	19.91	19.91	19.68	18.71	180	23.8
	0.043	0.042	0.034	0.032	0.030	0.033	300	24.2
WFST	u	g	r	i	z	w	30	23.0	6.55	20,000	(3)
	20.70	21.33	21.13	20.46	19.41	21.33	180	23.9
	0.022	0.022	0.022	0.023	0.024	0.022	300	24.2
LSST	u	g	r	i	z	y	30	25.1	9.6	20,000	(4)
	22.03	23.60	22.54	21.73	21.83	21.68	180	25.9
	0.025	0.018	0.023	0.030	0.031	0.032	300	26.2
SiTian a	g		r		i		30	20.3	600	30,000	(5)
	18.62		18.37		17.91		180	21.3
	0.026		0.026		0.027		300	21.6

Notes. The columns are: [1] the survey project; [2] the search limiting magnitude m_limit as a logarithmic function of the exposure time ${t}_{\exp }$ in different bands for specific survey projects (parameters a and b are, respectively, the values at the second and third subrows of each row); [3] the exposure time ${t}_{\exp };$ [4] the g-band limiting magnitude m_g,limit corresponding to different exposure times; [5] the FoV Ω_FoV; [6] the detectable sky coverage Ω_cov; [7] References: (1) Bellm et al. (2019), Masci et al. (2019), (2) X.‐Z. Er et al. (2023, in preparation), Lei et al. (2021), (3) X. Kong et al. (2023, in preparation), Shi et al. (2018), (4) LSST Science Collaboration et al. (2009), (5) Liu et al. (2021).

^a The technical specification of the limiting magnitude in the g-band stacked images for SiTian is similar to that for ZTF (Liu et al. 2021). SiTian would simultaneously observe the same visit in three different filters (u, g, i). Due to the lack of the technical information in u and i band of SiTian, we simply use the technical information of ZTF in the gri bands to calculate the EM detection rates by SiTian.

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As listed in Table 3, we show the 90% credible regions of two timescales, i.e., Δt_KN and Δt_AG, with different filters and different limiting magnitudes. These two parameters are, respectively, defined as the timescales during which the brightness of the associated kilonova and afterglow is above the limiting magnitude in different bands. Because the gri bands are the common filters used by various survey projects, we only show the probability density functions of Δt_KN and Δt_AG with different searching magnitudes in these three bands in Figure 1.

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Table 3. Time During which the Brightness of the EM Counterpart is above the Limiting Magnitude

Filter	Parameter	mlimit = 18 mag	19 mag	20 mag	21 mag	22 mag	23 mag	24 mag	25 mag	26 mag
u	ΔtKN	⋯	${0.67}_{-0.67}^{+0}$	${0.65}_{-0.65}^{+0.77}$	${0.82}_{-0.72}^{+0.63}$	${0.8}_{-0.6}^{+1.1}$	${0.7}_{-0.6}^{+1.4}$	${0.7}_{-0.5}^{+1.3}$	${0.7}_{-0.6}^{+1.2}$	${0.8}_{-0.6}^{+1.2}$
	ΔtAG	${0.12}_{-0.11}^{+0.55}$	${0.14}_{-0.12}^{+0.68}$	${0.18}_{-0.16}^{+0.91}$	${0.2}_{-0.2}^{+1.3}$	${0.3}_{-0.3}^{+1.9}$	${0.4}_{-0.4}^{+3.0}$	${0.6}_{-0.5}^{+4.7}$	${0.9}_{-0.8}^{+7.5}$	${2}_{-1}^{+12}$
g	ΔtKN	${0.83}_{-0.83}^{+0}$	${0.7}_{-0.7}^{+1.1}$	${1.3}_{-1.2}^{+0.8}$	${1.2}_{-1.0}^{+1.4}$	${1.2}_{-0.9}^{+1.9}$	${1.1}_{-0.8}^{+1.7}$	${1.0}_{-0.7}^{+1.7}$	${0.9}_{-0.7}^{+1.8}$	${0.9}_{-0.7}^{+1.9}$
	ΔtAG	${0.12}_{-0.11}^{+0.58}$	${0.15}_{-0.13}^{+0.73}$	${0.19}_{-0.17}^{+0.97}$	${0.2}_{-0.2}^{+1.4}$	${0.3}_{-0.3}^{+2.1}$	${0.4}_{-0.4}^{+3.3}$	${0.6}_{-0.5}^{+5.2}$	${1.0}_{-0.9}^{+8.1}$	${2}_{-2}^{+13}$
v	ΔtKN	${1.4}_{-1.4}^{+0}$	${0.7}_{-0.6}^{+1.1}$	${1.1}_{-0.8}^{+1.5}$	${1.0}_{-0.8}^{+2.4}$	${1.1}_{-0.7}^{+2.3}$	${1.0}_{-0.8}^{+2.0}$	${1.1}_{-0.8}^{+1.9}$	${1.2}_{-0.9}^{+2.0}$	${1.3}_{-0.7}^{+1.9}$
	ΔtAG	${0.13}_{-0.12}^{+0.60}$	${0.15}_{-0.13}^{+0.77}$	${0.2}_{-0.2}^{+1.0}$	${0.2}_{-0.2}^{+1.5}$	${0.3}_{-0.3}^{+2.2}$	${0.4}_{-0.4}^{+3.4}$	${0.6}_{-0.6}^{+5.4}$	${1.0}_{-1.0}^{+8.6}$	${2}_{-2}^{+13}$
w	ΔtKN	${0.5}_{-0.5}^{+1.1}$	${0.89}_{-0.60}^{+0.88}$	${1.2}_{-0.9}^{+1.6}$	${1.2}_{-0.8}^{+2.3}$	${1.2}_{-0.8}^{+2.3}$	${1.1}_{-0.8}^{+2.3}$	${1.1}_{-0.8}^{+2.2}$	${1.2}_{-0.8}^{+2.1}$	${1.3}_{-1.0}^{+2.0}$
	ΔtAG	${0.13}_{-0.12}^{+0.61}$	${0.15}_{-0.13}^{+0.78}$	${0.2}_{-0.2}^{+1.0}$	${0.2}_{-0.2}^{+1.5}$	${0.3}_{-0.3}^{+2.2}$	${0.4}_{-0.4}^{+3.5}$	${0.6}_{-0.6}^{+5.5}$	${1.1}_{-1.0}^{+8.7}$	${2}_{-2}^{+14}$
r	ΔtKN	${0.7}_{-0.7}^{+1.0}$	${1.07}_{-0.69}^{+0.88}$	${1.3}_{-1.0}^{+1.7}$	${1.2}_{-0.8}^{+2.5}$	${1.1}_{-0.8}^{+2.4}$	${1.3}_{-0.9}^{+2.1}$	${1.3}_{-0.9}^{+2.2}$	${1.2}_{-0.9}^{+2.4}$	${1.3}_{-0.9}^{+2.4}$
	ΔtAG	${0.13}_{-0.12}^{+0.63}$	${0.15}_{-0.13}^{+0.79}$	${0.2}_{-0.2}^{+1.1}$	${0.3}_{-0.2}^{+1.5}$	${0.3}_{-0.3}^{+2.3}$	${0.4}_{-0.4}^{+3.6}$	${0.7}_{-0.6}^{+5.6}$	${1.1}_{-1.0}^{+8.9}$	${2}_{-2}^{+14}$
i	ΔtKN	${0.7}_{-0.7}^{+1.9}$	${1.3}_{-0.7}^{+1.3}$	${1.5}_{-1.3}^{+2.2}$	${1.4}_{-0.9}^{+3.1}$	${1.3}_{-1.0}^{+2.6}$	${1.3}_{-1.0}^{+2.4}$	${1.4}_{-0.9}^{+2.4}$	${1.4}_{-1.0}^{+2.5}$	${1.7}_{-1.2}^{+2.6}$
	ΔtAG	${0.13}_{-0.12}^{+0.67}$	${0.16}_{-0.14}^{+0.84}$	${0.2}_{-0.2}^{+1.1}$	${0.3}_{-0.2}^{+1.6}$	${0.3}_{-0.3}^{+2.4}$	${0.5}_{-0.4}^{+3.8}$	${0.7}_{-0.6}^{+6.0}$	${1.1}_{-1.1}^{+9.4}$	${2}_{-2}^{+15}$
z	ΔtKN	${0.8}_{-0.8}^{+2.3}$	${1.6}_{-1.1}^{+1.6}$	${2.0}_{-1.4}^{+2.3}$	${1.8}_{-1.3}^{+3.0}$	${1.8}_{-1.3}^{+2.7}$	${1.8}_{-1.3}^{+2.6}$	${1.6}_{-1.1}^{+2.9}$	${1.6}_{-1.2}^{+3.1}$	${1.7}_{-1.3}^{+2.8}$
	ΔtAG	${0.14}_{-0.13}^{+0.69}$	${0.16}_{-0.14}^{+0.89}$	${0.2}_{-0.2}^{+1.2}$	${0.3}_{-0.2}^{+1.7}$	${0.4}_{-0.3}^{+2.6}$	${0.5}_{-0.4}^{+4.0}$	${0.7}_{-0.7}^{+6.3}$	${1}_{-1}^{+10}$	${2}_{-2}^{+15}$
y	ΔtKN	${1.0}_{-1.0}^{+2.2}$	${1.8}_{-1.7}^{+1.5}$	${2.1}_{-1.7}^{+2.2}$	${1.8}_{-1.4}^{+3.6}$	${1.9}_{-1.4}^{+2.9}$	${1.9}_{-1.4}^{+2.7}$	${1.9}_{-1.4}^{+2.9}$	${1.9}_{-1.5}^{+2.8}$	${1.7}_{-1.4}^{+3.2}$
	ΔtAG	${0.14}_{-0.12}^{+0.71}$	${0.17}_{-0.15}^{+0.90}$	${0.2}_{-0.2}^{+1.2}$	${0.3}_{-0.2}^{+1.8}$	${0.4}_{-0.3}^{+2.6}$	${0.5}_{-0.5}^{+4.1}$	${0.8}_{-0.7}^{+6.5}$	${1}_{-1}^{+10}$	${2}_{-2}^{+16}$

Note. The values are the timescales during which the brightness of the associated kilonova (Δt_KN) and afterglow (Δt_AG) is above the 5σ limiting magnitude in different bands with a 90% interval.

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As shown in Table 3, for a limiting magnitude of m_limit ≤ 19 mag, the values of Δt_KN may be imprecise, due to the limited amount of available data. For m_limit ≥ 20 mag, one can see that the median value of Δt_KN is ∼0.6–1.4 days in the optical and ∼1.4–2.1 days in the infrared, which may be uncorrelated with the limiting magnitude m_limit. If the observer wants to achieve at least two detection epochs for at least 50% of the observable kilonova signals, the cadence time t_cad should be less than half of the median value of Δt_KN. This means t_cad should be t_cad ≲ 0.3–0.7 days if one uses the optical band to search for kilonovae and t_cad ≲ 0.7–1.0 days in an infrared-band search for all survey projects.

Unlike Δt_KN, there exists a positive correlation between Δt_AG and m_limit, as shown in Table 3. The median value of Δt_KN would always be larger than Δt_AG if m_limit ≲ 24 mag. Thus, LSST, which has a limiting magnitude of m_limit ≳ 24 mag, can find ≳50% detectable afterglows brighter than the searching limiting magnitude by adopting a cadence to search for kilonovae. As shown in Figure 1, the probability density function of Δt_AG is significantly higher than that of Δt_KN, especially for searching with a relatively shallow limiting magnitude in a bluer filter band. Thus, it may be easier to discover optical afterglows by adopting the cadence of searching for kilonovae.

We show the detection rates of the kilonova-dominated and afterglow-dominated samples for ZTF, Mephisto, WFST, and LSST in Figures 2 and 3, by considering exposure times from 30 to 300 s and five different cadence timescales, t_cad = 0.5 hr, 1 hr, 3 hr, 1 day, and 2 days. The results shown in Figures 2 and 3 are obtained considering only the gri bands as these three bands are commonly used for these survey telescopes. Because SiTian is an integrated network of dozens of survey and follow-up telescopes, its survey strategy should largely differ from those of other survey projects. We give a separate calculation of the EM detection rates for SiTian in Section 3.4.

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As shown in Figure 2, for each survey project, the difference in the detection rates of the kilonova-dominated events between different bands seems very small, of the order of unity. For the cadence choice, we find that a one-day cadence strategy can always discover the highest number of kilonovae. Regarding the choice of the exposure time, the kilonova detection rates by ZTF, WFST, and LSST would decline as the exposure time increases. On the contrary, a longer exposure time can lead to the discovery of more kilonova events for Mephisto, although the increase in the amount of discovered kilonovae for longer exposure times is not significant. The simultaneous imaging in three bands by Mephisto is the reason for the difference in the detection rates between Mephisto and other survey projects. To sum up, a one-day cadence strategy with a ∼30 s exposure time is recommended to achieve optimal search for kilonovae. Based on Figure 2, the maximum kilonova detection rates for ZTF, Mephisto, WFST, and LSST are ∼0.3 yr⁻¹, ∼0.6 yr⁻¹, ∼1 yr⁻¹, and 20 yr⁻¹, respectively.

For afterglow-dominated events, there is no significant difference between searching in the optical and in the infrared bands for each survey project. The detection rates would drop with the increase in the exposure time. By adopting the optimal search strategy for kilonovae, one can also discover many afterglows from BNS mergers whose detection rate is much higher than that of kilonovae. For this case, the afterglow detection rates for ZTF, Mephisto, WFST, and LSST are ∼50 yr⁻¹, ∼60 yr⁻¹, ∼100 yr⁻¹, and ∼800 yr⁻¹, respectively.

SiTian (Liu et al. 2021; Yang et al. 2022a) is composed of a number of "units" deployed partly in China and partly at various sites around the world. Each unit includes three 1 m class Schmidt telescopes with a FoV of Ω_FoV = 25 deg², which will simultaneously observe the same visit in three different optical filters. There will be also three or four 4 m class telescopes for spectral identification and follow-up studies within the project.

SiTian at full design will scan at least 10,000 deg² of sky every 30 minutes, down to a detection limit of g ≈ 21 mag with an exposure time of ∼1 minute using at least 14 units in China. Furthermore, at least 10 units outside China can survey an additional ∼20,000 deg² with a slightly lower cadence (a few hours). Based on this fiducial search plan of SiTian, we change the cadence time and exposure time for all of units to explore the optical search strategy of SiTian, while preserving the same sky coverage.

The results of the EM detection rates for SiTian are shown in Table 4. We show that SiTian can detect ∼2 × 10³ yr⁻¹ afterglow-dominated events. The detection rate of kilonova-dominated events is ∼(2–4) yr⁻¹ by adopting the fiducial search plan of SiTian. As kilonovae are very faint, a better search strategy would be to increase the exposure time of the telescopes with the expense of losing the cadence. The detection rate of kilonova-dominated events would slightly rise to ∼(3–7) yr⁻¹ if an exposure time of 165 s is used.

Table 4. EM Detection Rates for SiTian

${t}_{\exp }/{\rm{s}}$	ΩFoV,1/deg2	tcad,1/minutes	ΩFoV,2/deg2	tcad,2/minutes	${\dot{N}}_{\mathrm{KN}}/{\mathrm{yr}}^{-1}$			${\dot{N}}_{\mathrm{AG}}\times {10}^{3}/{\mathrm{yr}}^{-1}$
					g	r	i	g	r	i
45 (fiducial)	350	30	250	80	2.0	3.4	3.6	1.5	1.7	1.9
75		45		120	2.4	4.3	4.7	1.7	1.9	2.0
105		60		160	2.7	5.0	5.6	1.7	1.9	2.1
165		90		240	3.0	5.8	7.1	1.7	1.9	2.0

Note. We assume that the time between two visits is t_oth = 15 s. The operation times for units in China and outside China are assumed to be ${\dot{t}}_{\mathrm{ope}}=8\,\mathrm{hr}\,{\mathrm{day}}^{-1}$ and 16 hr day⁻¹, respectively. The columns are: [1] the exposure time; [2] the total FoV of SiTian units in China; [3] the corresponding cadence time for SiTian units in China; [4] the total FoV of SiTian units outside China; [5] the corresponding cadence time for SiTian units outside China; [6]–[8] the detection rates of kilonova-dominated events in the gri bands; [9]–[11] the detection rates of afterglow-dominated events in the gri bands.

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By adopting an optimal serendipitous search strategy, i.e., a one-day cadence strategy, we show the redshift distributions of the detectable EM signals for a g-band limiting magnitude of m_g,limit = 20, 22, 24, and 26 mag in Figure 4. As for each detectable EM signal, we randomly simulate the detection epochs 1000 times and calculate the median difference value between these detection epochs as the fading rate. Figure 5 shows the distributions of the fading rate for detectable EM signals. For the same search depth of each filter, there is no significant difference between searching in different bands. It is important to note that we collect all EM events with t_cad ≥ 1 day when we calculate the redshift distributions of the detectable EM signals; so the distributions shown in Figure 4 do not consider their detection probabilities.

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For a limiting magnitude of m_limit = 20 mag, the most likely EM counterpart of BNS mergers to be detected is individual sGRB emission. Due to this relatively shallow search depth, afterglow emissions associated with these individual sGRBs could have at most one recorded epoch. In this case, it may be hard to establish the link between the sGRBs and the associated afterglows by optically serendipitous searches. Detectable afterglow emissions should be much easier to be discovered than kilonova emissions. However, these optical afterglows should always be associated with sGRB emissions. The most probable detectable redshift for these individual sGRBs and GRB-associated afterglows is z ∼ 0.5, which is consistent with the observations (e.g., Fong et al. 2015). Some detectable orphan afterglows with much lower detection rates could take place at a range of z ≳ 0.75. In our simulations, the largest distance of detectable kilonovae is ${z}_{\max }\sim 0.02$ $({D}_{{\rm{L}},\max }\sim 80\,\mathrm{Mpc})$. Most (∼80%–90%) of these detectable kilonovae are expected to be discovered individually without the detection of accompanied sGRB emissions.

The improvement in the search depth would lead to a proportionate decrease in the number of individual detectable sGRB events. If m_limit ≳ 22 mag, more near-on-axis orphan afterglows and nearby off-axis orphan afterglows can be discovered, which would become the primary detectable EM counterparts of BNS mergers. For a limiting magnitude of m_limit ≳ 24 mag, one can always find the associated afterglow and kilonova emissions after sGRB triggers by optically serendipitous searches. Due to the limited instrument sensitivity of γ-ray telescopes, the largest distance of sGRB-associated afterglows and kilonovae is ${z}_{\max }\sim 1.75$. We note that this simulated largest distance of sGRB triggers is obtained by adopting the effective sensitivity limit of Fermi-GBM and GECAM. A few Swift sGRBs were found to have photometric redshifts of z ≳ 2, as presented by Nugent et al. (2022), because the Swift Burst Alert Telescope (BAT) has a lower sensitivity compared with Fermi-GBM and GECAM, and hence a deeper detection depth. With the increase in the search depth, kilonovae would play a leading role in nearby detectable EM counterparts. For limiting magnitudes of m_g,limit = 22, 24, and 26 mag, the median (largest) distances of these detectable kilonovae are z = 0.04, 0.1, and 0.25 (${z}_{\max }=0.06,0.21,\mathrm{and}\ 0.55$), respectively. The search depth has little effect on the ratio between detectable kilonovae w/sGRBs and kilonovae w/o sGRBs.

The fading rates of detectable afterglows always peak at ∼1.3 mag day⁻¹ and have a wide distribution between ∼−0.5 mag day⁻¹ and ∼4.5 mag day⁻¹. Comparing with afterglows, kilonova-dominated events have more slow-evolving light curves. Their fading rates peak at ∼0–0.1 mag day⁻¹. For a limiting magnitude of m_limit ≲ 22 mag, the fading rates of kilonovae are in the range from −0.25 day⁻¹ to ∼1 mag day⁻¹. As shown in Figure 5, by adopting a limiting magnitude of m_limit ≳ 24 mag (m_limit ≳ 26 mag), some fast-evolving sGRB-associated kilonovae (kilonovae w/sGRBs and kilonovae w/o sGRBs) with a fading rate of ≳1 mag day⁻¹ can be discovered. For these fast-evolving kilonova events, their early-stage observations would be contributed by the associated afterglows while the kilonova emissions would lead to late-stage observations.

It is expected that two Advanced LIGO detectors (H1 and L1) in the USA (Harry & LIGO Scientific Collaboration 2010; LIGO Scientific Collaboration et al. 2015), the Advanced Virgo detector (V1) in Europe (Acernese et al. 2015), and the KAGRA detector (K1) in Japan (Aso et al. 2013; Kagra Collaboration et al. 2019) will start the fourth observation run (O4) together in 2023. Hereafter, the network composed of these second-generation detectors is referred to as the "HLV era". Here, the sensitivities of H1, L1, and V1 in the HLV era are adopted as their respective design sensitivities (Abbott et al.2020a) because their sensitivities are dynamic and change over time, ¹⁵ while K1 is ignored in our simulations as it will work toward improving most of time during O4. ¹⁶ The second-generation detectors would finish their upgrade to 2.5th-generation detectors in ∼2025. The subsequent upgrades of Advanced LIGO, Advanced Virgo, and KAGRA are called Advanced LIGO Plus (A+; Miller et al. 2015), Advanced Virgo Plus (AdV+; Abbott et al. 2020a), and KAGRA+ (Michimura et al. 2020). Hereafter, we refer to the era during which these four detectors will upgrade to 2.5th-generation detectors as the "PlusNetwork era." After ∼2030, the third-generation GW detectors are expected to start their observation. The currently proposed third-generation detector plans include LIGO Voyager (Adhikari et al. 2020) as a possible upgrade upon LIGO A+ (strictly speaking, it is more like quasi-third generation, but as its sensitivity is much higher than that of the 2.5th-generation detectors, we classify it as 3G for convenience), the Einstein Telescope (ET) in Europe (Punturo et al. 2010a, 2010b; Maggiore et al. 2020), and the Cosmic Explorer (CE) in the USA (Reitze et al. 2019). Due to the as-yet undetermined locations of ET and CE, we directly place ET at the current Virgo detector position and two CE detectors at the current H1 and L1 positions, according to convention (Vitale & Evans 2017; Vitale & Whittle 2018).

For each BNS system, we randomly simulate the masses of individual NSs based on the observationally derived mass distribution of Galactic BNS systems, i.e., a normal distribution ${M}_{\mathrm{NS}}/{M}_{\odot }\sim { \mathcal N }(1.32,{0.11}^{2})$ (Lattimer 2012; Kiziltan et al. 2013). The NS equation of state (EoS) DD2 (Typel et al. 2010), which is one of the stiffest EoS allowed by present constraints (e.g., Gao et al. 2016; Abbott et al. 2019), is adopted. With known M_NS, z, and EoS, we use the IMRPhenomPv2_NRTidalv2 (Dietrich et al. 2019) waveform model to simulate the GW waveform in the geocentric coordinate system, and then project it to different detectors to obtain the detector-frame strain signal. The optimal signal-to-noise ratio (S/N) can be obtained by

$$\begin{eqnarray}&&{\rho }_{\mathrm{opt}}^{2}=4{\int }_{{f}_{\min }}^{{f}_{\max }}\displaystyle \frac{| \tilde{h}(f){| }^{2}}{{S}_{n}(f)}{df}={\int }_{{f}_{\min }}^{{f}_{\max }}\displaystyle \frac{{\left(2| \tilde{h}(f)| \sqrt{f}\right)}^{2}}{{S}_{n}(f)}d\,\mathrm{ln}(f),\end{eqnarray} \tag{ 5 }$$

where f is the frequency, $\tilde{h}(f)$ is the strain signal in the frequency domain, and S_n (f) is the one-sided power spectral density of the GW detector, which is the square of the amplitude spectral density (ASD). The ASD for each detector is shown in the Appendix. We set the maximum frequency ${f}_{\max }$ as 2048 Hz. The low-frequency cutoff ${f}_{\min }$ is set to 20 Hz for O3; 10 Hz for the second- and 2.5th-generation detectors (Miller et al. 2015), and LIGO Voyager (Adhikari et al. 2020); 5 Hz for CE (Reitze et al. 2019) and 1 Hz for ET (Punturo et al. 2010a). We use the optimal S/N to approximate the matched filtering S/N of the GW signal detected by each detector, and then calculate the network S/N of the entire detector network, i.e., the root sum squared of the S/N of all detectors. In each GW era, when the S/N for a single detector is greater than the threshold of 8 and the network S/N is greater than 12, we expect that the corresponding GW signal is detected.

We consider the exact duty cycle of O3 (see Table 5), calculated following the timeline released from LVC, ¹⁷ to simulate the GW observations of BNS mergers in O3 and O4. In view of the significant technology upgrades in GW detections and the fact that more detectors will join GW campaigns, the duty cycle in the 2.5th- and third-generation detector networks could be highly uncertain. Thus, we only calculate their best cases, i.e., "all detectors in the corresponding era have reached the design sensitivity and work normally" as the optimal situation. During the 2.5th-generation detector network, the best case is that A+, AdV+, and KAGRA+ all work normally, which we abbreviate as "PlusNetwork." LIGO Voyager is separately discussed. Furthermore, "ET&CE" represents the best case of the third-generation era.

We need to localize the BNS through GW signals for EM follow-ups. As we need to calculate a large number of simulated signals, we use the Fisher information matrix (FIM; Cutler & Flanagan 1994) to approximate the localization area estimated by the more computationally expensive Bayesian method (Thrane & Talbot 2019). The FIM is based on linear signal approximation (LSA; Cutler & Flanagan 1994) and uses a Gaussian distribution to approximate the posterior distribution of the parameters. This assumption requires the signal to have a sufficiently high S/N, so we only calculate the GW localization for the signal whose network S/N meets the detection threshold. The FIM of the detector network is a linear summation of the FIM of the individual detector in that network

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{ij}}=\displaystyle \sum _{k}{\unicode{x03008}{\partial }_{i}h| {\partial }_{j}h\unicode{x03009}}_{k},\end{eqnarray} \tag{ 6 }$$

where the bracket means the inner product

$$\begin{eqnarray}&&\unicode{x03008}a| b\unicode{x03009}=4{\mathfrak{R}}{\int }_{{f}_{\min }}^{{f}_{\max }}\displaystyle \frac{\tilde{a}(f){\tilde{b}}^{* }(f)}{{S}_{n}(f)}\,{df},\end{eqnarray} \tag{ 7 }$$

where k is the index of the detector in that network, and ∂_i h or ∂_j h refers to the partial differentiation of the detector-frame signal in the frequency domain with respect to a certain parameter. In our FIM calculation, the parameters are chosen from the detector-frame chirp mass ${ \mathcal M }$, the symmetric mass ratio η, the luminosity distance D_L , the coalescence time t_c , the coalescence phase ϕ_c , the inclination angle l, the polarization angle ψ, the R.A. θ, the decl. ϕ, and the tidal deformation parameters $\tilde{{\rm{\Lambda }}}$ and $\delta \tilde{{\rm{\Lambda }}}$. Note that, for the third-generation detector network, we also take the Earth's rotation into account (Liu & Shao 2022). In order to reduce the matrix singularity issue, we do not take partial differentiation of the tidal parameters for the cases before the third-generation.

For high-S/N signals, the inverse of the FIM is less or equal to the covariance matrix of parameters, the so-called "Cramer–Rao lower bound"

$$\begin{eqnarray}&&\mathrm{cov}\left(i,j\right)\geqslant {\left({{\boldsymbol{\Gamma }}}^{-1}\right)}_{{ij}};\end{eqnarray} \tag{ 8 }$$

for a specific parameter, we can use the square root of the corresponding diagonal element in the inverse of the FIM as the bias. In our case, we care about ${\rm{\Delta }}\cos \theta$ and Δϕ. Then we can derive the sky localization area (Barack & Cutler 2004)

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{GW}}=2\pi \sqrt{{\left({\rm{\Delta }}\cos \theta {\rm{\Delta }}\phi \right)}^{2}-\langle {\rm{\Delta }}\cos \theta {\rm{\Delta }}\phi {\rangle }^{2}};\end{eqnarray} \tag{ 9 }$$

we consider a 90% confidence of this area hereafter.

4.2.1. GW Detection Rate, Detectable Distance, and Sky Localization

We consider the exact duty cycle shown in Table 5, and the simulated GW detection results of O3 and O4 are summarized in Table 6. We check that the GW detection rate in O3 should be $\sim {2.4}_{-1.8}^{+3.6}\,{\mathrm{yr}}^{-1}$, which is consistent with the observations of LVC (Abbott et al. 2020a, 2021c). The median detectable luminosity distance is ∼110 Mpc, nearly approximate to the observed distance of GW190425 (Abbott et al. 2020b). In the HLV (O4) era, we predict that one can detect ∼11 yr⁻¹ BNS GW events with the median detectable distance at z ∼ 0.040 and the horizon at ${z}_{\max }\sim 0.084$.

Table 6. GW Detection Results

Case	Era	${\dot{N}}_{\mathrm{GW}}/{\mathrm{yr}}^{-1}$	z	${z}_{\max }$	${\mathrm{log}}_{10}({{\rm{\Omega }}}_{\mathrm{GW}}/{\deg }^{2})=a\times {\mathrm{log}}_{10}(z)+b$
		$({\dot{N}}_{\mathrm{GW}}/{\dot{N}}_{\mathrm{BNS}})$	(DL/100 Mpc)	$({D}_{{\rm{L}},\max }/100\,\mathrm{Mpc})$	a	b
HLV (O3)	2nd	${2.4}_{-1.8}^{+3.6}$	${0.025}_{-0.013}^{+0.025}$	0.062	⋯	⋯
		(0.001%)	$({1.1}_{-0.5}^{+1.2})$	(2.9)
HLV (O4)	2nd	${11}_{-8}^{+17}$	${0.040}_{-0.025}^{+0.025}$	0.084	1.93	${4.02}_{-0.42}^{+0.85}$
		(0.004%)	$({1.8}_{1.1}^{1.2})$	(4.0)
PlusNetwork	2.5th	${210}_{-160}^{+320}$	${0.099}_{-0.055}^{+0.050}$	0.190	1.87	${2.85}_{-0.55}^{+0.45}$
		(0.08%)	$({4.7}_{-2.7}^{+2.6})$	(9.5)
LIGO Voyager	3rd	${1.8}_{-1.4}^{+2.8}\times {10}^{3}$	${0.22}_{-0.13}^{+0.12}$	0.43	⋯	⋯
		(0.73%)	$({11.0}_{-6.7}^{+7.4})$	(24.3)
ET&CE	3rd	${2.4}_{-1.8}^{+3.6}\times {10}^{5}$	${0.97}_{-0.57}^{+0.71}$	3.77	2.00	${0.85}_{-0.46}^{+0.69}$
		(90.7%)	$({65}_{-43}^{+64})$	(343)

Note. The GW detection rates and luminosity distance distributions for detectable GWs in the second-generation era are simulated by adopting the exact duty cycle labeled in Table 5, while GW detection results in the 2.5th- and third-generation eras are obtained under ideal operation conditions. The columns are: [1] the case of different generation eras; [2] the generation of GW detectors; [3] the median GW detection rates adopting a 90% interval and a local event rate density of ${\dot{\rho }}_{0,\mathrm{BNS}}={320}_{-240}^{+490}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ (Abbott et al. 2021b), while the numbers in brackets are the corresponding detectable proportions of the number of BNS mergers per year in the universe (${\dot{N}}_{\mathrm{BNS}}$); [4] the median detectable redshifts and detectable luminosity distances at 90% intervals; [5] the maximum detectable redshifts and detectable luminosity distances; [6] the GW sky localizations as function of z, where a and b are fitting parameters.

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We simulate the sky localization area (Ω_GW) for detectable BNS GW events when two or three detectors are online simultaneously during O4. As the localization for GW mainly relies on the time delay between different detectors, one second-generation GW detector cannot localize GW signals. We find that the relationship between the sky localization and redshift for BNS mergers detected in O4 with the H1, L1, and VIRGO network can be well explained by a log-linear trend (Figure 6), while the log-linear relationships with only two GW detectors network are not obvious due to their limited detections. In Figure 7, the median sky localization area is expected to be ∼10 deg² for detectable BNS GW events. We also collected the cumulative fraction of the sky localization from Abbott et al. (2020a). Comparing with our simulation, they considered the operation of K1 in O4 and used the BAYESTAR (Singer & Price 2016) code to perform sky localization for BNS GW events. A duty cycle of 70% for each detector uncorrelated with the other detectors, which is slightly different from our simulations, was adopted by Abbott et al. (2020a). However, our simulation result for the sky localization in O4 is nearly consistent with that shown in Abbott et al. (2020a).

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4.2.2. EM Detectability in O4

Based on the BNS GW detection results during O4, we now estimate the EM detection rates of detectable BNS GW events for ZTF, SiTian, Mephisto, WFST, and LSST. For serendipitous observations, the event would appear in an arbitrary position of the celestial sphere due to the lack of ToO alerts. For the GW-triggered ToO observations, the survey project would just need to cover the sky localization of GW events in search of their associated EM counterparts. Thus, one can replace Ω_sph with Ω_GW in Equation (4) to estimate the EM detection rate for the ToO observations, i.e.,

$$\begin{eqnarray}\begin{array}{rcl}{\dot{N}}_{\mathrm{EM}} & \approx & \displaystyle \frac{{\dot{N}}_{\mathrm{BNS}}}{{n}_{\mathrm{sim}}}\cdot \displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{cov}}}{{{\rm{\Omega }}}_{\mathrm{sph}}}\\ & & \times \,\displaystyle \sum _{i=1}^{{n}_{\mathrm{GW}}}\displaystyle \sum _{j}\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{FoV}}{P}_{\mathrm{duty},j}\min ({t}_{\mathrm{cad},{ij}},{\rm{\Delta }}{t}_{{ij}})}{\max ({{\rm{\Omega }}}_{\mathrm{FoV}},{{\rm{\Omega }}}_{\mathrm{GW},j})({n}_{\exp ,{ij}}{t}_{\exp ,{ij}}+{t}_{\mathrm{oth}})},\end{array}\end{eqnarray} \tag{ 10 }$$

where j = {HL, HV, LV, HLV} and n_GW represents the number of detectable BNS GW events. Here, we adopt ${t}_{\exp }=300\,{\rm{s}}$ to make GW-triggered follow-up observations. Thus, the cadence time t_cad for each event is related to ${t}_{\exp }$ and Ω_GW for each event. The judgment condition for the detection of the kilonova and/or afterglow by a follow-up search after GW triggers is required to be two different exposure filters have at least two detection epochs.

As SiTian will not operate during O4, we only show our simulated detection rates for ZTF, Mephisto, WFST, and LSST at this era. Based on the technical information of the survey projects listed in Table 2, our simulation results show that ZTF/SiTian/Mephisto/WFST/LSST can detect ∼5/4/3/3 kilonovae (∼1/1/1/1 afterglows) in O4, respectively. ∼5% kilonovae and ∼90% afterglows after GW triggers are expected to be associated with the detection of sGRBs.

4.3.1. GW Detection Rate, Detectable Distance, and Sky Localization

We summarize all our simulated GW detection results of the 2.5th- and third-generation eras in Table 6. The total mass, S/N, and redshift for the detectable GW signals in different eras are shown in Figure 8. For the 2.5th-generation GW detector network, the optimal detection rate is ∼210 yr⁻¹. The GW detection distance would be doubled compared with the detection distance in O4, i.e., the horizon can reach ${z}_{\max }\sim 0.2$. For the LIGO Voyager in the third-generation era, the optimal detection rate can be increased to ∼1800 yr⁻¹ and the detection distance would be twice compared with that in the last era, i.e., a horizon of ${z}_{\max }\sim 0.4$. However, these numbers are much smaller than those for the newly designed third-generation detectors. For the ET&CE network, the optimal detection rate would be ∼2.4 × 10⁵ yr⁻¹, which would account for ∼91% of the total BNS GW events in the universe. The events detected by ET&CE are mainly dominated by BNS mergers at z ∼ 1, which is near the most probable redshift where BNS mergers occurred in the universe. The most remote detectable events by ET&CE would be at ${z}_{\max }\sim 3.8$. Except for the ET&CE era, the median distance of detectable GW events is always set to half of the horizon in each GW era.

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In Figure 9, the relationships between the 90% credible area of GW sky localization and the redshift for BNS mergers detected at the PlusNetwork, LIGO Voyager, and ET&CE eras can be represented by log-linear trends, similar to that of the second-generation network. The fitting results of these log-linear trends are listed in Table 6. In the PlusNetwork era, most of detectable BNS mergers will be localized to ≲10 deg². The network comprising one ET detector and two CE detectors can have a more remarkable capability to observe and localize BNS GW events. For BNS mergers occurring at z ≲ 0.2, the GW sky localizations constrained in the ET&CE era will be about 2 orders of magnitude lower than those constrained in the PlusNetwork era. The median localization for BNS mergers at z ∼ 0.5 (z ∼ 1) is shown to be ∼1 deg² (∼10 deg²). In these regimes, the present and future wide-FoV survey projects will be able to cover the sky localizations given by GW detections in a few pointings and achieve deep detection depths with relatively short exposure integration times. As the current GW operation plan during the LIGO Voyager era only includes two GW detectors, we find that the sky localizations will span from a few hundred square degrees to tens of thousands of square degrees. Thus, EM follow-ups might be very difficult if there are no more GW detectors to join the campaign at the LIGO Voyager era.

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4.3.2. EM Detectability

The BNS GW detectabilities for the networks of 2.5th- and third-generation GW detectors have been studied in detail in Section 4.3.1. Based on these results, we now discuss the EM detection probabilities and optimistic EM detection rates for GW-triggered ToO observations.

As the luminosity distributions in different bands are consistent, we only show g-band luminosity distributions for the EM signals of detectable GW events at different GW eras in Figure 10. For the future GW eras of PlusNetwork, LIGO Voyager, and ET&CE, the critical magnitudes for the detection of EM emissions from all BNS GW events would be ∼23.5 mag, ∼25 mag, and ≳26 mag, respectively. Present and foreseeable-future survey projects can hardly find all EM signals of BNS GW events detected during the ET&CE era. Comparing with the results of adjacent GW eras, one can see that there appears little difference in the number of detectable kilonovae if adopting the detection depth as the critical magnitude of earlier GW eras. However, one can find much more remote sGRBs and afterglows in later GW eras. As the search limiting magnitude increases, the amount of detectable kilonovae would increase exponentially. There is also an exponential increase with a slower rising slope for the amount of afterglows. At the critical magnitude of each era, ∼80% BNS GW events can observe clear kilonova signals, while afterglows would account for the other ∼20% BNS GW events. Most of the detectable kilonova-dominated BNS GW events would be not accompanied by observations of sGRBs. Kilonova events associated with sGRBs like the observations of GW170817/GRB170817A/AT2017gfo would be scarce.

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Based on the GW detections in different eras, we then estimate the optimistic EM detection rates for specific survey projects listed in Table 2. Under ideal operation conditions in the PlusNetwork and ET&CE eras, due to the precise sky localizations of GW events, these survey projects will be able to cover the sky localizations given by GW detections in a few pointings. Thus, we define Ω_cov/Ω_GW ∼ 1 in Equation (10) to estimate the EM detection rates. We note that, although the current plan in the LIGO Voyager era shows relatively poor sky localization for GW events, we still estimate the EM detection rate of this era by defining Ω_cov/Ω_GW ∼ 1 under the assumption that more GW detectors joining the campaign might significantly improve the sky localizations. The optimistic detection rates of EM signals at different GW eras for specific survey projects are displayed in Figure 11 and labeled in Table 7. By comparing the limiting magnitudes of specific survey projects and the critical magnitudes in the different GW eras (Figure 10), we find that these wide-field surveys (ZTF, SiTian, Mephisto, and WFST) are unlikely to detect a larger number of kilonovae despite the upgraded GW detectors improving the BNS detection rates. Optimistically, ZTF/SiTian/Mephisto/WFST can detect ∼5/5/150/120 kilonovae per year at the 2.5th- and third-generation eras, while ∼100/300/1200 yr⁻¹ kilonovae per year can be discovered by LSST during the PlusNetwork/LIGO Voyager/ET&CE eras, respectively. At later GW eras, ToO observations of BNS GW events can always discover more afterglows, almost all of which are associated with sGRBs. However, we find a special case in which LSST follow-up in the ET&CE era will detect as much as ∼75% of detectable afterglows, which will be largely orphans unaccompanied by an sGRB.

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Table 7.  g-band Optimistic EM Detection Rates in Each GW Era

Sample	Era	ZTF	SiTian	Mephisto	WFST	LSST
KNe w/sGRBs	PlusNetwork	0.8	0.9	6.3	4.9	4.9
	Voyager	0.8	0.9	16.6	12.8	22.1
	ET&CE	0.8	0.9	23.6	18.1	75.4
KNe w/o sGRBs	PlusNetwork	4.0	4.4	82.4	63.4	74.0
	Voyager	4.0	4.4	118	90.9	438
	ET&CE	4.0	4.4	130	100	1.11 × 103
AFs w/sGRBs	PlusNetwork	16.8	16.8	27.1	20.9	20.9
	Voyager	36.6	36.6	154	118	131
	ET&CE	101	101	905	696	1.37 × 103
AFs w/o sGRBs	PlusNetwork	0.4	0.5	3.1	2.4	2.4
	Voyager	0.7	0.7	36.2	27.9	47.4
	ET&CE	3.3	3.3	392	302	2.93 × 103
Total EM Signals	PlusNetwork	22.1	22.6	119	91.5	102
	Voyager	42.2	42.7	325	250	639
	ET&CE	109	110	1.45 × 103	1.12 × 103	5.49 × 103

Note. The values represent the simulated BNS merger detection rates (in unit of yr⁻¹) in different GW eras.

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