# Event Rate Density and Luminosity Function of Newborn-Magnetar-Driven X-Ray Transients from Neutron Star Binary Mergers

Le Zou, Ji-Gui Cheng, Rui-Chong Hu, Wen-Jin Xie, En-Wei Liang

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Le Zou, Ji-Gui Cheng, Rui-Chong Hu, Wen-Jin Xie, En-Wei Liang. Event Rate Density and Luminosity Function of Newborn-Magnetar-Driven X-Ray Transients from Neutron Star Binary Mergers. The Astrophysical Journal, 2025, 986 (1), pp.28. 10.3847/1538-4357/add2f7. hal-05077794

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# Event Rate Density and Luminosity Function of Newborn-magnetar-driven X-Ray Transients from Neutron Star Binary Mergers

Le Zou<sup>1,2</sup>①, Ji-Gui Cheng<sup>3</sup>①, Rui-Chong Hu<sup>4</sup>①, Wen-Jin Xie<sup>5</sup>, and En-Wei Liang<sup>4,5</sup>

¹Department of Physics, Xiangtan University, Xiangtan 411105, People's Republic of China

²Key Laboratory of Stars and Interstellar Medium, Xiangtan University, Xiangtan 411105, People's Republic of China

³School of Physics and Electronics, Hunan University of Science and Technology, Xiangtan 411201, People's Republic of China

⁴Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, People's Republic of China; lew@gxu.edu.cn

⁵Université Paris Cité, CEA Paris-Saclay, IRFU/DAP-AIM, 91191 Gif-sur-Yvette, France

Received 2022 December 29; revised 2025 April 13; accepted 2025 April 29; published 2025 June 4

## Abstract

X-ray transients (XTs) driven by newborn magnetars from mergers of neutron star binaries (NSBs) have occasionally been detected in the narrow-field Chandra Deep Field-South (CDF-S) survey and the Swift/X-Ray Telescope observations of short gamma-ray bursts. Quantifying their event rate density (ERD) and luminosity function (LF) is critical for understanding NSB coalescence and magnetar formation. Utilizing population synthesis calculations incorporating various equations of state (EoSs), we derive a local ERD of ~300 Gpc⁻³ yr⁻¹ and a redshift-dependent ERD profile peaking at z = 1.81 followed by a rapid decline beyond z ~ 4. Constructing an XT sample based on CDF-S and Swift observations, we characterize the LF by a single power-law function at L ≤ 4.75 × 10⁴⁶ erg s⁻¹ with a slope of -1.03, followed by a broken power-law function in which the break luminosity is L_b = 4.38 × 10⁴⁷ erg s⁻¹ and the slopes are -0.28 and -1.66. Based on the ERD and the LF, we estimate that the Einstein Probe (EP) detection rate is ~31 yr⁻¹, adopting a conservative threshold flux of 10⁻⁹ erg s⁻¹, a luminosity range of L ∈ [2 × 10⁴⁴, 2 × 10⁴⁹] erg s⁻¹, and a correction for the jet opening angle of ~16°. This detection rate is consistent with the EP observations during its first year of operation. It is important to note that our estimation is subject to uncertainties arising from the LF derivation. Future EP observations of these XT events will be crucial in reducing these uncertainties.

Unified Astronomy Thesaurus concepts: Stellar mergers (2157); Magnetars (992); Gamma-ray bursts (629)

## 1. Introduction

The discovery of the association between gravitational wave (GW) burst, gamma-ray burst (GRB), and kilonova on 2017 August 17, i.e., GW170817/sGRB 170817A/AT2017gfo, from a neutron star binary (NSB) merger has opened the multimessenger astronomical era (B. P. Abbott et al. 2017a, 2017b). It is highly prospective that NSB mergers are the sites for the synergic observations of the multiwavelength electromagnetic (EM) and GW signals. Theoretical studies and simulations suggest that an NSB merger can give birth to either a black hole (BH) or a neutron star (NS), depending on the total mass of the binary system and the NS equation of state (EoS; A. Rowlinson et al. 2010; B. Giacomazzo & R. Perna 2013; P. D. Lasky et al. 2014; A. Li et al. 2016). The newborn NSs could be classified into three different types, i.e., a hypermassive NS (HMNS), which survives for only about 100 ms before collapsing into a BH (T. W. Baumgarte et al. 2000; M. Shibata & K. Taniguchi 2006), a supramassive NS (SMNS), which has a longer lifetime, ranging from a few seconds to several hours before collapse (A. Rowlinson et al. 2010; H.-J. Lü et al. 2015; H. Gao et al. 2016), and a stable NS (Z. G. Dai & T. Lu 1998; B. Zhang & P. Mészáros 2001; Y.-W. Yu et al. 2010; B. D. Metzger et al. 2011). It is suggested that all three types of NS possess a strong magnetic field (i.e., 10¹⁴~10¹⁵ G) (B. Giacomazzo & R. Perna 2013;

B. Giacomazzo et al. 2015) and can be recognized as magnetars (R. C. Duncan & C. Thompson 1992; C. Thompson & R. C. Duncan 1995, 1996).

It has been suggested that newborn magnetars are powerful central engines of GRBs (E. Troja et al. 2007; A. Rowlinson et al. 2013; H.-J. Lü et al. 2015). These magnetars dissipate their spin energy via the GW emission and magnetic dipole radiation (DR). The continuous energy injection from the DR can power X-ray transients (XTs), in addition to the GRBs and their afterglows, as well as the kilonova. Their light curves are typically characterized as an X-ray plateau or a shallow decay segment, i.e., L ∝ (1 + t/τ)⁻², where τ is the characteristic spin-down timescale of a magnetar (V. V. Usov 1992; Z. G. Dai & T. Lu 1998; B. Zhang & P. Mészáros 2001; B. D. Metzger et al. 2011; H.-J. Lü & B. Zhang 2014). Such a signature is not only present in early X-ray afterglow light curves of long GRBs (E. Troja et al. 2007) but also is observed in the X-ray light curves of some short GRBs (sGRBs; A. Rowlinson et al. 2013; H.-J. Lü et al. 2015). Thus, magnetar-driven XTs make the diversity of the observed early X-ray afterglow light curves of GRBs, depending on the relative flux levels between the fireball afterglows and the XTs (H.-J. Lü et al. 2018). The XTs may be identified through X-ray follow-up observations (e.g., in GRBs 070110A and 180620A; E. Troja et al. 2007; L. Zou & E.-W. Liang 2022) and are occasionally detected as orphan X-ray or soft gamma-ray transients. As proposed by L. Zou et al. (2021b), the peculiar GRB 101225A detected with Swift/Burst Alert Telescope (BAT) is a bright orphan XT powered by the DR of a newborn magnetar. More interestingly, Y. Q. Xue et al. (2019) found that the X-ray light curve of the known-redshift

<figure><img src="image_3.png" /></figure>

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CDF-S XT2 is consistent with predictions of XTs driven by newborn magnetars. Several orphan redshift-known XTs have already been discovered in the CDF-S X-ray survey data (J. Quirola-Vásquez et al. 2022, 2023).

Magnetar-driven XTs are crucial probes for investigating NS physics. Based on the formation history of magnetars, these XTs can be further divided into two groups, i.e., ones powered by magnetars formed in NSB mergers (NSB magnetars) and ones powered by magnetars born from the collapse of massive stars (MS magnetars). Over the years, many studies have been conducted to determine the luminosity function (LF) and the event rate density (ERD) of the two XT populations. W.-J. Xie et al. (2020) constrained the LF of the MS-magnetar-driven XTs as a smooth broken power-law, breaking at $L_b = 3.2 \times 10^{48}$ erg s⁻¹ with the index changing from $\beta_1 = -0.79$ to $\beta_2 = -2.22$. H. Sun et al. (2017) constrained the ERD and the LF of the NSB-magnetar-driven XTs by adopting a redshift distribution of NSB mergers derived from the empirical merger delay model (F. J. Virgili et al. 2011; D. Wanderman & T. Piran 2015). Their results showed that the LF can be fitted with a two-log-normal distribution peaking around $2.5 \times 10^{46}$ erg s⁻¹ and $4.0 \times 10^{49}$ erg s⁻¹, and the ERD is around a few tens of Gpc⁻³ yr⁻¹ for luminosity above $10^{45}$ erg s⁻¹. Despite these studies, the ERD and LF of NSB-magnetar-driven XTs remain poorly understood, due to the absence of wide field-of-view (FoV) X-ray surveys.

The launch of the soft X-ray Einstein Probe (EP) telescope on 2024 January 9, presents an excellent opportunity to investigate NSB-magnetar-driven XTs. In its first year of operation, EP has detected 69 XTs, including EP240315a (W. J. Zhang et al. 2024), EP240408a (J. W. Hu et al. 2024), EP240413a (T. Y. Lian et al. 2024b), and EP240414a (T. Y. Lian et al. 2024a). Most of these detections exhibit peak fluxes around $1 \times 10^{-9}$ erg cm⁻² s⁻¹, with a few reaching up to $1 \times 10^{-8}$ erg cm⁻² s⁻¹. Some of the detections may relate to NSB magnetars. GW observations are also providing valuable insights, as NSB-driven XTs are among the most promising EM counterparts of GW sources. For instance, B. P. Abbott et al. (2019) infer a local NSB merger rate density of 110–3840 Gpc⁻³ yr⁻¹ at a 90% confidence interval, using the O1/O2 runs of Advanced LIGO and Virgo. This merger rate can be used to constrain the ERD of NSB-driven XTs. In this paper, we aim to estimate the EP detection rate of NSB-magnetar-driven XTs and compare it with the EP observations. In Section 2, we use population synthesis simulations, combined with current GW observations, to obtain the ERD. In Section 3, we derive the LF via empirical function fitting of the Swift and Chandra sample of NSB-magnetar-driven XTs candidates. The final EP detection rate is presented in Section 4. Discussions and conclusions are given in Sections 5 and 6, respectively. Throughout this paper, we adopt the standard ΛCDM cosmology with parameters $H_0 = 70.0$ km s⁻¹, Mpc⁻¹, $\Omega_M = 0.3$, and $\Omega_\Lambda = 0.7$.

## 2. ERD

We investigate the ERD of NSB magnetars based on available results of population synthesis simulations implemented by the StarTrack code, which is used to generate populations of NSB systems (K. Belczynski et al. 2002, 2008). This code has been improved by K. Belczynski et al. (2020), including updated treatments of a common envelope (CE)

<figure><img src="image_4.png" /><figcaption>**Figure 1.** The distribution of the remnant masses of NSB mergers derived from the population synthesis simulations implemented by the StarTrack code. According to the different EoSs with the $M_{\text{TOV}}$, we classify the remnant masses into the different types, i.e., an HMNS with ~1.2$M_{\text{TOV}} \lesssim M_{\text{tot}} \lesssim 1.3–1.6M_{\text{TOV}}$, an SMNS with $M_{\text{TOV}} \lesssim M_{\text{tot}} \lesssim 1.2M_{\text{TOV}}$, and a stable NS with $M_{\text{tot}} < M_{\text{TOV}}$ (B. Margalit & B. D. Metzger 2017). The different EoSs are marked with color bars. The magenta, green, yellow, orange, and cyan represent the EoSs GM1, DD2, AB-N, AB-L, and APR, respectively.</figcaption></figure>

evolution, compact object mass calculations, the new stellar remnant natal kick, and the angular momentum transport. K. Belczynski et al. (2020) generated a broad set of submodels of compact binary systems using the upgraded code. We adopt the results of the submodel M33.A for our analysis since predictions of this model are consistent with the observational limits for generating the population of NSBs. In this model, the rapid supernova engine model for NS mass is adopted from C. L. Fryer et al. (2012) with a high natal kick ($\sigma = 265$ km s⁻¹). 50% nonconservative Roche lobe overflow and 5% Bondi–Hoyle accretion are set for the CE evolution modeling. The cosmic star formation density is taken from P. Madau & T. Fragos (2017). The average metallicity is $Z = 0.014Z_\odot$, where $Z_\odot$ is the solar metallicity. For more details of the simulations, please refer to K. Belczynski et al. (2020).

Based on the simulation results, we are able to track every NSB event. Their merger information, including the total mass of the binary system $M_{\text{tot}}$ and the merger delay timescale, is recorded. The typical mass of material outflow from an NSB system is ~$10^{-4}–10^{-2}M_\odot$ (e.g., L. Rezzolla et al. 2011; K. Hotokezaka et al. 2013; S. Rosswog et al. 2013), which could be negligible in comparison with the total mass of the NSB system ($M_{\text{tot}}$). Therefore, the merger remnant mass can be approximated as $M_{\text{tot}}$. $M_{\text{tot}}$ of an NSB system places a strong constraint on the outcome of the remnant, which could be classified into four types concerning the NS maximum mass $M_{\text{TOV}}$ (the Tolman–Oppenheimer–Volkov mass), i.e., a BH with $M_{\text{tot}}/M_{\text{TOV}} \gtrsim 1.3–1.6$ (G. B. Cook et al. 1994), an HMNS with ~1.2 $\lesssim M_{\text{tot}}/M_{\text{TOV}} \lesssim 1.3–1.6$, an SMNS with $1 \lesssim M_{\text{tot}}/M_{\text{TOV}} \lesssim 1.2$, and a stable NS with $M_{\text{tot}}/M_{\text{TOV}} < 1$ (B. Margalit & B. D. Metzger 2017). Since we consider only the remnant as an NS, we exclude NSB events with $M_{\text{tot}} \gtrsim 1.3 M_{\text{TOV}}$. The $M_{\text{TOV}}$ depends on the EoS of an NS.

The distribution of the $M_{\text{tot}}$ for the selected remnants of NSB is illustrated in Figure 1. The range of $M_{\text{tot}}$ spans from 2.22 $M_\odot$ to 3.15 $M_\odot$. We exclude EoSs that result in an $M_{\text{TOV}}$ lower than 2.22 $M_\odot$. For our analysis, we adopt four EoSs with the following corresponding $M_{\text{TOV}}$ values: EoS DD2 with
<figure><img src="image_5.png" /><figcaption>Figure 2. Cosmic event rate history of the NSB-magnetar-XTs derived from a StarTrack M33.A population synthesis simulation with different EoSs. The black dots are the total simulation data, green, magenta, and orange dots represent the redshift distributions of different remnants, respectively. The black, green, magenta, and orange dashed lines are the best-fit empirical model given in Equation (1). The black dot line presents the peak R at z = 1.81. The blue dashed dot line marks the cosmic evolution R ∝ z −3.85 at z > 4.</figcaption></figure>

$M_{\text{TOV}} = 2.42 M_{\odot}$ (S. Typel et al. 2010), EoS AB-L with $M_{\text{TOV}} = 2.71 M_{\odot}$, EoS AB-N with $M_{\text{TOV}} = 2.67 M_{\odot}$ (W. D. Arnett & R. L. Bowers 1977), and EoS GM1 with $M_{\text{TOV}} = 2.37 M_{\odot}$ (N. K. Glendenning & S. A. Moszkowski 1991). It should be noted that EoS Akmal, Pandharipande, and Ravenhall (APR) with $M_{\text{TOV}} = 2.20 M_{\odot}$ (A. Akmal et al. 1998), which is only slightly smaller than $2.22 M_{\odot}$. Hence, the EoS APR is also considered in our analysis. The mass ranges of the different remnant types, determined by the $M_{\text{TOV}}$ corresponding to each EoS, are indicated by color bars in Figure 1.

The NSB ERDs ($R(z)$) as a function of z for the global sample, the stable NS sample, the SMNS sample, and the

HMNS sample are shown in Figure 2. We fit these curves with an empirical model, i.e.,

$$R(z) = R_0 \left[ (1+z)^{a\eta} + \left( \frac{1+z}{B} \right)^{b\eta} \right]^{\frac{1}{\eta}}, \quad (1)$$

where $R_0$ approximates the local ERD at z = 0 and η measures the curvedness of the curve. This model is composed of two smooth joint broken power-law functions. The parameter B is related to the slopes of the two components as $B = (1 + z_b)^{1-a/b}$, where $z_b$ is the transition redshift of the two components. It has an advantage for the derivation of the
<table><caption>**Table 1**
The Fitting Results for the Different Merger Remnants with Different EoSs</caption><thead><tr><th></th><th>R<sub>0</sub><br>(Gpc<sup>-3</sup> yr<sup>-1</sup>)</th><th>a</th><th>b</th><th>B</th><th>η</th></tr></thead><tbody><tr><td colspan="6"><strong>DD2(M<sub>TOV</sub> = 2.42 M<sub>☉</sub>)</strong></td></tr><tr><td>Total</td><td>298.14 ± 4.66</td><td>2.19 ± 0.03</td><td>-4.45 ± 0.07</td><td>5.74 ± 0.03</td><td>-0.81 ± 0.03</td></tr><tr><td>Stable NSs</td><td>214.15 ± 4.03</td><td>2.13 ± 0.04</td><td>-4.65 ± 0.09</td><td>5.65 ± 0.03</td><td>-0.77 ± 0.04</td></tr><tr><td>SMNSs</td><td>81.67 ± 1.18</td><td>2.33 ± 0.03</td><td>-4.07 ± 0.04</td><td>6.01 ± 0.03</td><td>-0.88 ± 0.03</td></tr><tr><td>HMNSs</td><td>2.50 ± 0.07</td><td>2.15 ± 0.07</td><td>-5.45 ± 0.23</td><td>5.78 ± 0.07</td><td>-0.57 ± 0.05</td></tr><tr><td colspan="6"><strong>AB-L(M<sub>TOV</sub> = 2.71 M<sub>☉</sub>)</strong></td></tr><tr><td>Total</td><td>298.95 ± 4.68</td><td>2.19 ± 0.03</td><td>-4.45 ± 0.07</td><td>5.74 ± 0.03</td><td>-0.81 ± 0.03</td></tr><tr><td>Stable NSs</td><td>291.63 ± 4.59</td><td>2.19 ± 0.03</td><td>-4.43 ± 0.06</td><td>5.74 ± 0.03</td><td>-0.82 ± 0.03</td></tr><tr><td>SMNSs</td><td>6.82 ± 0.19</td><td>2.23 ± 0.07</td><td>-5.41 ± 0.20</td><td>5.89 ± 0.06</td><td>-0.59 ± 0.05</td></tr><tr><td>HMNSs</td><td>0.57 ± 0.03</td><td>2.29 ± 0.15</td><td>-5.07 ± 0.36</td><td>6.21 ± 0.13</td><td>-0.57 ± 0.09</td></tr><tr><td colspan="6"><strong>AB-N(M<sub>TOV</sub> = 2.67 M<sub>☉</sub>)</strong></td></tr><tr><td>Total</td><td>298.84 ± 4.68</td><td>2.19 ± 0.03</td><td>-4.45 ± 0.07</td><td>5.74 ± 0.03</td><td>-0.81 ± 0.03</td></tr><tr><td>Stable NSs</td><td>290.84 ± 4.58</td><td>2.19 ± 0.03</td><td>-4.43 ± 0.06</td><td>5.74 ± 0.03</td><td>-0.82 ± 0.03</td></tr><tr><td>SMNSs</td><td>7.51 ± 0.22</td><td>2.23 ± 0.07</td><td>-5.40 ± 0.20</td><td>5.89 ± 0.06</td><td>-0.60 ± 0.05</td></tr><tr><td>HMNSs</td><td>0.57 ± 0.03</td><td>2.25 ± 0.13</td><td>-5.01 ± 0.33</td><td>6.22 ± 0.12</td><td>-0.61 ± 0.09</td></tr><tr><td colspan="6"><strong>GM1(M<sub>TOV</sub> = 2.37 M<sub>☉</sub>)</strong></td></tr><tr><td>Total</td><td>297.98 ± 4.67</td><td>2.19 ± 0.03</td><td>-4.45 ± 0.07</td><td>5.74 ± 0.03</td><td>-0.81 ± 0.03</td></tr><tr><td>Stable NSs</td><td>17.54 ± 0.25</td><td>2.25 ± 0.03</td><td>-5.61 ± 0.10</td><td>5.02 ± 0.02</td><td>-0.68 ± 0.03</td></tr><tr><td>SMNSs</td><td>276.35 ± 4.62</td><td>2.19 ± 0.04</td><td>-4.42 ± 0.07</td><td>5.79 ± 0.03</td><td>-0.81 ± 0.03</td></tr><tr><td>HMNSs</td><td>3.71 ± 0.10</td><td>2.09 ± 0.07</td><td>-5.43 ± 0.21</td><td>5.67 ± 0.06</td><td>-0.59 ± 0.05</td></tr><tr><td colspan="6"><strong>APR(M<sub>TOV</sub> = 2.20 M<sub>☉</sub>)</strong></td></tr><tr><td>Total</td><td>294.65 ± 4.62</td><td>2.19 ± 0.03</td><td>-4.44 ± 0.06</td><td>5.74 ± 0.03</td><td>-0.81 ± 0.03</td></tr><tr><td>SMNSs</td><td>290.30 ± 4.58</td><td>2.19 ± 0.03</td><td>-4.42 ± 0.06</td><td>5.74 ± 0.03</td><td>-0.82 ± 0.03</td></tr><tr><td>HMNSs</td><td>4.39 ± 0.17</td><td>2.35 ± 0.09</td><td>-5.45 ± 0.24</td><td>6.03 ± 0.07</td><td>-0.60 ± 0.06</td></tr></tbody></table>

local ERD. We employ a least-squares algorithm to fit the curves to the data and the results of fitting are summarized in Table 1 and depicted in Figure 2. Our results demonstrate that the model can well fit the data at $z < 5$. However, at $z > 5$, the ERDs of SMNS and HMNS samples deviate from the fitting curves for certain EoSs. The derived parameter values, namely, *a*, *b*, and *B*, are roughly consistent among the NS samples in different EoSs. Globally, the local total ERD is $R_0 \sim 300 \text{ Gpc}^{-3} \text{ yr}^{-1}$, with a peak ERD $R_{\text{peak}}$ occurring at $z_{\text{peak}} = 1.81$. Beyond $z = 4$, $R(z)$ drops rapidly as $R \propto z^{-3.85}$. Q. Chu et al. (2022) investigated the properties of NSBs and their mergers by combining population synthesis models for binary stellar evolution with cosmological galaxy formation and evolution models. Their methodology is consistent with that in K. Belczynski et al. (2020), albeit with different initial and boundary conditions. By incorporating observed galactic NSBs and the local NSB merger rate density inferred from GW observations, they constrained the NSB merger rate density as $R(z) \sim R_0(1 + z)^\zeta$ at $z \lesssim 0.5$, where $R_0 \sim 316-784 \text{ Gpc}^{-3} \text{ yr}^{-1}$ and $\zeta \sim 1.34-2.04$.

The values of $R_0$ differ significantly among the remnant types for various EoSs. In the scenarios of EoSs DD2, AB-L, and AB-N, the event rate of the stable NSs dominates the total event rate, especially in the latter two scenarios. Conversely, in the EoS GM1 and APR scenarios, the event rate of SMNSs dominates the global sample due to the relatively low $M_{\text{TOV}}$ values. Even no stable NS can be produced in the EoS APR scenario.

### 3. LF

At present, no wide FoV survey sensitive in the soft X-ray band is available for constraining the LF of the XTs. Some of the XTs are occasionally detected with the Swift and Chandra missions. Early X-ray follow-up observations with the Swift/X-Ray Telescope (XRT) reveal a steady X-ray emission component embedded in the jet afterglow fireball of about half of the GRBs triggered by the Swift/BAT. We attribute this component to an XT powered by the DRs of a newborn magnetar. Considerable samples of the XTs are accumulated with the XRT data over ~20 yr operation of the Swift/XRT emission. Moreover, the extended soft gamma-rays observed in some GRBs with the Swift/BAT may also be from the XTs (B. Zhang 2013; P. Kumar & B. Zhang 2015).

We conduct a comprehensive analysis of XTs and sGRBs reported in the literature, which are believed to originate from NSB magnetars. L. Zou et al. (2021a) analyzed the Swift GRB catalog up to 2021, identifying 28 sGRBs with X-ray plateaus (XPs), of which 13 had measured redshifts. This subset accounted for ~26% of all sGRBs recorded at that time. By analyzing the Swift GRB catalog after 2021, we find only one additional sGRB with an XP but without a redshift measurement, indicating that sGRBs with XPs currently represent ~21% of the Swift sGRB catalog, with only ~10% having both XPs and redshift measurements. We also select 11 extended emissions (EEs) with redshift measurements in sGRBs as reported by A. Lien et al. (2014), H.-J. Lü et al. (2015), N. Anand et al. (2018), and X. J. Li et al. (2021), and collect 10 plausible XTs with a plateau phase from F. E. Bauer
<table><caption>**Table 2**
XRTs Discovered by Observations with the Chandra/CDF-S, Swift/BAT, and Swift/XRT in Our Sample</caption><thead><tr><th>XT</th><th>z</th><th>Γ</th><th>log L<sub>b</sub> (erg s<sup>-1</sup>)</th><th>α<sub>1</sub></th><th>α<sub>2</sub></th><th>t<sub>b</sub>/10<sup>3</sup> s</th></tr></thead><tbody><tr><td>XT2</td><td>0.738</td><td>1.57 ± 0.55</td><td>44.98 ± 0.05</td><td>0.09 ± 0.11</td><td>2.43 ± 0.19</td><td>2.53 ± 0.24</td></tr><tr><td>XT1</td><td>2.23</td><td>1.43 ± 0.26</td><td>47.21 ± 0.10</td><td>-0.04 ± 0.10</td><td>1.60 ± 0.10</td><td>0.20 ± 0.10</td></tr><tr><td>080819</td><td>0.7</td><td>3.00 ± 1.10</td><td>44.07 ± 0.17</td><td>-0.20 ± 0.10</td><td>2.80 ± 1.90</td><td>5.30 ± 0.20</td></tr><tr><td>170901</td><td>1.44</td><td>2.20 ± 0.60</td><td>45.86 ± 0.06</td><td>0.10 ± 0.10</td><td>1.90 ± 0.50</td><td>2.10 ± 0.30</td></tr><tr><td>210423</td><td>1.51</td><td>2.20 ± 0.70</td><td>45.68 ± 0.05</td><td>0.20 ± 0.10</td><td>3.80 ± 1.20</td><td>4.40 ± 0.40</td></tr><tr><th>sGRB-XP</th><th>z</th><th>Γ</th><th>log L<sub>b</sub> (erg s<sup>-1</sup>)</th><th>α<sub>1</sub></th><th>α<sub>2</sub></th><th>t<sub>b</sub>/10<sup>3</sup> s</th></tr><tr><td>051221A</td><td>0.5465</td><td>1.83 ± 0.21</td><td>45.25 ± 0.10</td><td>0.31 ± 0.11</td><td>1.34 ± 0.07</td><td>28.53 ± 6.92</td></tr><tr><td>060614</td><td>0.1254</td><td>1.75 ± 0.01</td><td>44.43 ± 0.03</td><td>0.05 ± 0.05</td><td>1.82 ± 0.04</td><td>45.73 ± 2.62</td></tr><tr><td>060801</td><td>1.31</td><td>1.64 ± 0.37</td><td>48.34 ± 0.09</td><td>-0.20 ± 1.16</td><td>1.16 ± 0.27</td><td>0.15 ± 0.07</td></tr><tr><td>061201</td><td>0.111</td><td>1.30 ± 0.18</td><td>45.20 ± 0.17</td><td>0.62 ± 0.12</td><td>2.12 ± 0.13</td><td>2.88 ± 0.64</td></tr><tr><td>070724A</td><td>0.457</td><td>1.70 ± 0.78</td><td>44.24 ± 1.24</td><td>0.20 ± 0.00</td><td>2.53 ± 0.00</td><td>41.22 ± 4.28</td></tr><tr><td>070809</td><td>0.219</td><td>1.48 ± 0.32</td><td>44.55 ± 0.15</td><td>0.04 ± 0.18</td><td>1.17 ± 0.28</td><td>7.11 ± 3.74</td></tr><tr><td>090426</td><td>2.6</td><td>1.81 ± 0.34</td><td>48.40 ± 0.12</td><td>-0.40 ± 0.88</td><td>1.05 ± 0.11</td><td>0.22 ± 0.09</td></tr><tr><td>100117A</td><td>0.92</td><td>1.45 ± 0.12</td><td>48.35 ± 0.07</td><td>0.67 ± 0.16</td><td>4.53 ± 0.17</td><td>0.23 ± 0.02</td></tr><tr><td>100625A</td><td>0.425</td><td>2.03 ± 0.32</td><td>46.26 ± 0.24</td><td>0.46 ± 0.59</td><td>3.16 ± 1.02</td><td>0.24 ± 0.07</td></tr><tr><td>101219A</td><td>0.718</td><td>1.34 ± 0.32</td><td>47.94 ± 0.15</td><td>0.23 ± 0.69</td><td>6.24 ± 1.91</td><td>0.17 ± 0.02</td></tr><tr><td>130603B</td><td>0.356</td><td>1.77 ± 0.20</td><td>46.21 ± 0.11</td><td>0.42 ± 0.07</td><td>1.68 ± 0.07</td><td>3.13 ± 0.60</td></tr><tr><td>140903A</td><td>0.351</td><td>1.60 ± 0.21</td><td>45.72 ± 0.06</td><td>0.07 ± 0.07</td><td>1.22 ± 0.08</td><td>8.52 ± 1.52</td></tr><tr><td>190627A</td><td>1.942</td><td>1.99 ± 0.16</td><td>47.57 ± 0.09</td><td>0.09 ± 0.11</td><td>1.51 ± 0.09</td><td>21.36 ± 3.88</td></tr><tr><th>sGRB-EE</th><th>z</th><th>Γ</th><th>log L<sub>b</sub> (erg s<sup>-1</sup>)</th><th>α<sub>1</sub></th><th>α<sub>2</sub></th><th>t<sub>b</sub>/10<sup>3</sup> s</th></tr><tr><td>050724</td><td>0.257</td><td>1.65 ± 0.17</td><td>48.41 ± 0.15</td><td>0.15 ± 0.13</td><td>3.09 ± 0.07</td><td>0.11 ± 0.00</td></tr><tr><td>061006</td><td>0.4377</td><td>1.74 ± 0.17</td><td>48.66 ± 1.28</td><td>0.34 ± 0.40</td><td>4.90 ± 0.70</td><td>0.06 ± 0.01</td></tr><tr><td>061210</td><td>0.41</td><td>1.56 ± 0.28</td><td>48.04 ± 2.79</td><td>0.18 ± 0.27</td><td>1.60 ± 0.81</td><td>1.20 ± 3.84</td></tr><tr><td>070714B</td><td>0.92</td><td>1.36 ± 0.19</td><td>49.79 ± 0.03</td><td>-0.16 ± 0.19</td><td>2.41 ± 0.08</td><td>0.09 ± 0.00</td></tr><tr><td>071227</td><td>0.383</td><td>1.91 ± 0.53</td><td>47.42 ± 0.21</td><td>0.58 ± 0.13</td><td>6.69 ± 0.79</td><td>0.18 ± 0.01</td></tr><tr><td>080123</td><td>0.495</td><td>1.50 ± 0.13</td><td>48.83 ± 0.04</td><td>0.25 ± 0.32</td><td>1.92 ± 0.11</td><td>0.07 ± 0.02</td></tr><tr><td>090510</td><td>0.903</td><td>1.60 ± 0.12</td><td>47.58 ± 0.08</td><td>0.70 ± 0.04</td><td>2.26 ± 0.11</td><td>1.65 ± 0.25</td></tr><tr><td>100724A</td><td>1.288</td><td>1.95 ± 0.27</td><td>47.58 ± 0.10</td><td>0.26 ± 0.44</td><td>2.35 ± 0.40</td><td>0.69 ± 0.16</td></tr><tr><td>111117A</td><td>2.211</td><td>1.50 ± 0.40</td><td>48.31 ± 0.10</td><td>0.10 ± 0.40</td><td>3.10 ± 1.50</td><td>0.18 ± 0.05</td></tr><tr><td>131004A</td><td>0.717</td><td>1.90 ± 0.50</td><td>47.34 ± 0.32</td><td>-0.20 ± 0.57</td><td>1.61 ± 0.20</td><td>0.84 ± 0.50</td></tr><tr><td>150423A</td><td>1.394</td><td>1.26 ± 0.23</td><td>46.90 ± 0.09</td><td>0.20 ± 0.44</td><td>2.10 ± 0.50</td><td>0.77 ± 0.13</td></tr></tbody></table>

et al. (2017), Y. Q. Xue et al. (2019), J. Quirola-Vásquez et al. (2022), and J. Quirola-Vásquez et al. (2023). Considering the luminosity of some XTs falls below $10^{44}$ erg s$^{-1}$, which is unlikely to be produced by an NSB magnetar (B. D. Metzger & A. L. Piro 2014; H. Sun et al. 2017; B. D. Metzger et al. 2018), we exclude five XTs from our sample. Finally, these data allowed us to compile a sample of 29 events.

For these events, we model their light curves with an empirical broken power-law function, i.e., $F = F_0[(t/t_b)^{\omega_{01}} + (t/t_b)^{\omega_{02}}]^{-1/\omega}$, in which $t_b$ presents the break time, $\omega$ describes the sharpness of the break, $\alpha_1$ and $\alpha_2$ are the decay indices before and after $t_b$, respectively. To determine the photon spectral index ($\Gamma$), we fit the spectra of the magnetic dipole emission-dominated epoch using an absorbed single power-law function. The parameters of our sample are detailed in Table 2, where $F_b$ corresponds to the break flux at $t_b$ and $L_b = 4\pi D_L^2 F_b$ represents the corresponding luminosity. We adopt the break luminosity as the characteristic luminosity for our sample.

In Figure 3, we categorize the XTs observed by different instruments into nine equal-interval luminosity bins in the logarithmic space. We initially applied a single power-law function to fit the data. The result shows that the function does not fit well after the fifth data point. Consequently, we consider employing a combination of a single power-law function and a broken power-law function, i.e., $\Phi(L) \propto L^a$ for data points before the fifth and $\Phi(L) \propto [(L/L_b)^{\omega_{\beta1}} + (L/L_b)^{\omega_{\beta2}}]^{1/\omega}$ for the

subsequent points, to better constrain the LF. The best fit yields $a = -1.03$, $\beta_1 = -0.28$, $\beta_2 = -1.66$, and $L_b = 4.38 \times 10^{47}$ erg s$^{-1}$. Our results show that the derived LF aligns well with the observed data, as illustrated in Figure 3. Note that the derived LF is expected to have uncertainties arising from two factors. One is the instrument selection effect in constructing the XT sample. Chandra just occasionally captures some XTs in the deep survey and the Swift/XRT only tracks the rapid follow-up of GRB afterglows, neither of which is an all-sky survey project. And the other is the relatively small sample size. These factors may cause the derived LF to deviate significantly from the true one. However, such uncertainties are currently unavoidable, as the derivation of the LF relies on the available observational sample. In the future, as the number of detected XTs increases, these uncertainties will be significantly mitigated.

## 4. Prospects for the EP XT Survey

For a detector with a flux threshold of $F_{th}$ and an FoV of $\Omega$, the number of detectable events within an operational time $T$ can be estimated through

$$N = \frac{\Omega T}{4\pi} \int_{L_{\min}}^{L_{\max}} \Phi(L) dL \int_{0}^{z_{\max}(L)} \frac{R(z)}{1+z} \frac{dV(z)}{dz} dz, \quad (2)$$
<figure><img src="image_6.png" /><figcaption>**Figure 3.** The constructed LF of the XTs driven by NSB magnetars. The blue dashed dot line and red dashed line are the best-fit single power-law function and a combination of a single power-law function and a broken power-law function, respectively. The data with error bars are calculated with the XTs reported in Table 2.</figcaption></figure>

where $\Phi(L)$ is normalized to the unit in luminosity range [$L_{min}, L_{max}$], $z_{max}$ is the maximum detectable redshift for an XT with luminosity $L$, and $dV/dz$ is the comoving volume at redshift $z$, which is calculated under the standard $\Lambda$CMD cosmology as

$$ \frac{dV}{dz} = \frac{c}{H_0 (1+z)^2} \frac{4\pi D_L^2}{[\Omega_M (1+z)^3 + \Omega_\Lambda]} \quad (3) $$

The $z_{max}$ is determined by the threshold flux through

$$ F_{\text{th}} = \frac{L}{4\pi D_L^2(z_{\text{max}})k(z_{\text{max}}, \Gamma)}, \quad (4) $$

where $k(z, \Gamma)$ is the $k$-correction factor depending on redshift and $\Gamma$ (J. S. Bloom et al. 2001).

The EP will provide a great opportunity to investigate the LF and the ERD of the NSB-magnetar-driven XTs. It aims to monitor the XTs and variable objects in the soft X-ray band (0.5–4 keV) (W. Yuan et al. 2015, 2017). A wide-field X-ray telescope (WXRT) with a large FoV of 3600 deg² and a narrow-field follow-up XRT are on board EP. Using the novel lobster-eye optics, a WXRT may offer high sensitivity for X-ray all-sky monitors. Its sensitivity is $F_{EP} \sim 5 \times 10^{-11} \text{ erg cm}^{-2} \text{ s}^{-1}$, corresponding to an exposure time of $10^3$ s. Considering the current observations, we adopt the $F_{EP}$ as $1 \times 10^{-9} \text{ erg cm}^{-2} \text{ s}^{-1}$ and the derived $R(z)$ and LF to estimate the detectable XTs with EP/WXRT. We consider a luminosity range of $[10^{44}, 10^{51}] \text{ erg s}^{-1}$ (B. D. Metzger & A. L. Piro 2014; H. Sun et al. 2017; B. D. Metzger et al. 2018). Table 3 presents the estimated event rates of detectable XTs powered by the remnants for different EoSs. The type of remnant primarily affects the shape of the observed XT light curves. The HMNSs would rapidly evolve and collapse into a BH within $\sim 100$ ms. This short survival duration makes it challenging to detect the XTs they generate. Therefore, XTs detectable with EP/WXRT are mainly driven by stable NSs or SMNSs, which can survive for an extended period as long as the NS is rotating rapidly. Once SMNSs lose significant angular momentum, they will collapse into a BH. The light curves of XTs produced by SMNSs typically exhibit a plateau followed by a very sharp drop, often referred to as an internal plateau. Our results indicate that $\sim 30\%$ of XTs would show an internal plateau in the EoS DD2, while $\sim 98\%$ of the XTs generated by the EoSs AB-L and AB-N exhibit a plateau

<caption>**Table 3**</caption>

The Event Rates of XTs Powered by the Remnant Types for Different EoSs

<table><thead><tr><th></th><th>Stable NSs<br>(yr<sup>-1</sup>)</th><th>SMNSs<br>(yr<sup>-1</sup>)</th><th>HMNSs<br>(yr<sup>-1</sup>)</th></tr></thead><tbody><tr><td>DD2</td><td>1226.28</td><td>539.75</td><td>13.84</td></tr><tr><td>AB-L</td><td>1750.56</td><td>40.84</td><td>3.48</td></tr><tr><td>AB-N</td><td>1745.82</td><td>45.35</td><td>3.52</td></tr><tr><td>GM1</td><td>100.57</td><td>1660.44</td><td>19.87</td></tr><tr><td>APR</td><td>...</td><td>1740.51</td><td>29.21</td></tr></tbody></table>

followed by a declining phase of approximately 1–2, known as the external plateau. In the EoS GM1 scenario, 94% of light curves display an internal plateau. All light curves of XTs present a sharp drop in the EoS APR scenario. The total estimated detectable event rate is $\sim 1.8 \times 10^3 \text{ yr}^{-1}$ without considering any beaming effect. If the XT emission is collimated within a jet, the detection rate with EP/WXRT would be significantly reduced. Considering a median jet opening angle of $\theta_j \sim 16^\circ$, as observed in sGRBs, a median beaming factor of $f_b \approx 0.04$ can be calculated (W. Fong et al. 2015). Taking this into account, we estimate the detection event of XTs to be $\sim 70 \text{ yr}^{-1}$ with EP/WXRT.

In our estimation, we adopt a broad luminosity range, extending the upper boundary to $10^{51} \text{ erg s}^{-1}$. This corresponds to the theoretical maximum power of a newborn magnetar central engine. However, XTs with such extreme luminosities are exceedingly rare in observations. As shown in Figure 3, the brightest XT in our sample has a luminosity of only $\sim 2 \times 10^{49} \text{ erg s}^{-1}$. Therefore, lowering the upper boundary to this value, i.e., $L \in [2 \times 10^{44}, 2 \times 10^{49}] \text{ erg s}^{-1}$, might provide a more realistic estimation of the detection rate. The reduced upper boundary is expected to lower the detection rate since the $z_{max}$ is also reduced. Applying the new luminosity range, we obtain a detection rate of $\sim 780 \text{ yr}^{-1}$, half of the previous estimate. Accounting for the jet collimation effect, the rate further decreases to $\sim 31 \text{ yr}^{-1}$, which is consistent with the EP observations during its first year of operation.

We perform Monte Carlo simulations to create a mock sample of NSB-magnetar-driven XTs for visualizing the probability distribution of detectable such XTs with the EP/WXRT. We generate $2 \times 10^5$ XTs from different remnant types corresponding to different EoSs. Each XT in this sample is characterized by a set of parameters ($z_{sim}$, $L_{sim}$, $\Gamma_{sim}$), randomly chosen from the normalized probabilities distribution of the $R(z)$ where $z \in [0, 10]$, $\Phi(L)$ where $L_{iso} \in [10^{44}, 10^{51}] \text{ erg s}^{-1}$ and the Gaussian distribution of $\Gamma$ as presented in L. Zou et al. (2021a). We then screen the generated XTs using the WXRT threshold$^6 F_{sim} \ge F_{EP} = 1 \times 10^{-9} \text{ erg cm}^{-2} \text{ s}^{-1}$, where $F_{sim} = L_{sim} / 4\pi D_L^2(z_{sim}) k(z_{sim}, \Gamma_{sim})$. Figure 4 illustrates the distributions and the density contours of the mock XT sample among the remnant types for different EoSs in the $\log(1+z) - \log L$ plane. For the EoS DD2, we obtain 2089 detectable XTs with an internal plateau and 5968 detectable XTs with an external plateau, respectively. In the case of the EoSs AB-L and AB-N, we find 196 detectable XTs with an internal plateau and 7840 detectable XTs with an external plateau, and 221

<footnote>$^6$ The designed trigger threshold of the EP/WXRT is 10 mCrab ($\sim 5 \times 10^{-16} \text{ erg cm}^{-2} \text{ s}^{-1}$ assuming a photon index of 2.1 in 0.5–4 keV band) with 600 s exposure time; https://www.nssdc.ac.cn/). Most XTs triggered by WXRT in 2024 are brighter than $10^{-9} \text{ erg cm}^{-2} \text{ s}^{-1}$. Thus, we adopt a robust threshold of $10^{-9} \text{ erg cm}^{-2} \text{ s}^{-1}$ in our analysis.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Simulated probability distribution contours of observable XTs with the EP/WXRT among the remnant types for different EoSs. The solid red, cyan, and black lines represent the contours of 68.3% and 95.5% of the mock XTs distribution, respectively. The observed data (solid dots) are also shown. The dashed magenta, blue, and olive lines stand for the roughly estimated sensitivity of EP/WXRT (1 × 10⁻⁹ erg cm⁻² s⁻¹), Swift/XRT (1 × 10⁻¹³ erg cm⁻² s⁻¹), and Chandra/CDF-S (4 × 10⁻¹⁵ erg cm⁻² s⁻¹), respectively. The data and the sensitivities of Swift/XRT and Chandra/CDF-S have been converted to the 0.5–4 keV band.</figcaption></figure>

detectable XTs with an internal plateau and 7898 detectable XTs with an external plateau, respectively. For the EoS GM1, we find 7336 detectable XTs with an internal plateau and 611 detectable XTs with an external plateau. Finally, for the EoS APR, we obtain 8028 detectable XTs with an internal plateau. In summary, we estimate that ~8 × 10³ XTs would be detectable with EP/WXRT out of the 2 × 10⁵ XTs generated, which is comparable to the distribution of the XTs detected by EP/WXRT over 3 yr of mission operation.

We also compare the XTs listed in Table 2 with the detection thresholds of Chandra, EP/WXRT, and plot them in

Figure 4 for comparison. It should be noted that all data points and threshold lines in this figure are rescaled to the EP/WXRT energy band of 0.5–4 keV. From the plot, we can observe that the sGRB-EEs represent the high-luminosity end of the XT population and about half of the XTs extracted from the sGRB-EEs would be detectable with EP/WXRT. However, EP/WXRT may not be able to capture XTs similar to those found in the X-ray afterglow plateau data and the CDS-S data. Nevertheless, EP/WXRT still possesses an excellent capability to establish a significant sample of these types of XTs, which could enrich our understanding of this XT population.
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Distributions of $z_{sim}$ and $L_{sim}$ for the mock sample for NSB magnetars (black lines) and MS magnetars (blue lines), respectively, and the $z$ and $L_b$ distributions of our sample.</figcaption></figure>

Without considering the specific light-curve shapes of XTs, we examine the distributions of $z_{sim}$ and $L_{sim}$ in our mock sample, using the results obtained with EoS DD2 as an example. The distributions are shown in Figure 5. In the same figure, we also compare the distributions of $z$ and $L_b$ in our sample with those of the XTs driven by MS magnetars taken from W.-J. Xie et al. (2020). We find that the distribution of $z_{sim}$ in our mock sample is roughly consistent with that taken from W.-J. Xie et al. (2020). The distribution of $L_b$ is broad, and the higher luminosity end aligns well with the distribution of $L_{sim}$ for XTs driven by MS magnetars. This is mainly due to the high-luminosity XTs in our sample corresponding to the extended emissions (EEs) observed in sGRBs. According to the distributions in Figure 5, we find that there is no way to tell from $z$ and $L_b$ alone whether the XTs are driven by an MS-magnetar or an NSB magnetar. Additionally, it is intriguing to note that some nearby XTs with $z < 0.01$ could potentially be discovered in the EP/WXRT survey.

# 5. Discussions

## 5.1. Uncertainties in Deriving the Detection Rate

As shown in Equation (2), our detection rate estimation relies on several assumptions, which introduce potential uncertainties. First, we assume that every newborn magnetar successfully produces an XT, as we are unable to acquire the failure rate. This assumption may lead to an overestimation of the detection rate. Second, we impose a luminosity range of [$L_{min}, L_{max}$] for NSB-magnetar-driven XTs. So that the choices of $L_{min}$ and $L_{max}$ are critical in deriving the detection rate. A larger $L_{max}$ is expected to increase the detection rate, as more XTs surpass the instrument threshold flux, whereas a lower $L_{min}$ reduces the detection rate by increasing the number of XTs that fall below the threshold. Finally, additional factors may prevent some XTs from being detected, such as the jet collimation effect mentioned earlier and various absorption effects in the propagation X-ray emission. Additionally, as emphasized in Section 3, the derived LF for NSB-magnetar-driven XTs is induced with uncertainties from the instrument selection effect and small sample size.

Comparison of our results with those from previous studies may provide more valuable insights. J. Quirola-Vásquez et al. (2022) conduct a comprehensive investigation on Chandra-detected XTs, in discriminating their physical origin. Their

work adopts a different approach from ours to estimate the detection rate. By extrapolating the Chandra detection of XTs with instrument parameters like the effective area and the exposure time accounted for, they estimate that the EP detection rate of the XTs is ~35 – 69 yr⁻¹. Adopting the same limit flux<fnref n="7" /> of the EP/WXRT used in their estimation, we estimate that the detection rate is ~1638 yr⁻¹ without considering the jet collimation effect, which is significantly higher than theirs. There are several expectable reasons for the discrepancy. The first is that our method tends to overestimate the detection rate, as already discussed. If taking the jet collimation factor $f_b \approx 0.04$ into consideration, our detection rate is reduced to ~66 yr⁻¹, which becomes comparable to their prediction. The second is that their prediction depends solely on the number of Chandra-detected XTs and instrument parameters and ignores the physical origins of the XTs, so that their results would also be influenced by the selection effect. Considering that we are focusing on a specific subclass of XTs, our detection rate remains slightly higher after incorporating the jet collimation effect. This discrepancy may indicate that not every newborn magnetar can produce an XT.

## 5.2. EoSs and the Type of XTs

The EoSs of NSs remain an open question in astrophysics, and numerous studies have been conducted to constrain the EoS using various observational data. For example, P. D. Lasky et al. (2014) analyzed eight individual sGRBs and compared the observed data with five different EoSs. Their findings suggested that the EoS GM1 was most consistent with the observed data. In a more extensive analysis, H. Gao et al. (2016) employed Monte Carlo simulations to test these five EoSs against observational constraints from a broader sGRB sample. They concluded that only the EoS GM1 could satisfy the constraints on the fraction of SMNSs, while the other four EoSs could not. Further investigations by A. Li et al. (2016) expanded the analysis to include additional NS EoSs and quark star (QS) EoSs. They found that all except one (EoS BCPM) could satisfy the observational constraints on SMNSs/QSs. Regarding specific observations, A. Li et al. (2021) and H.-J. Lü et al. (2021) proposed that the EoS DD2 is favored by

<footnote n="7">⁷ $F_{\text{lim,EP/WXT}} \approx 5 \times 10^{-10} \text{ erg cm}^{-2} \text{ s}^{-1}$. This limit flux was approximated as 10 times the WXRT threshold flux $F_{\text{th,EP/WXRT}} = 5 \times 10^{-11} \text{ erg cm}^{-2} \text{ s}^{-1}$ at a $10^3$ s exposure, to avoid the influence of the Poisson noise effects (J. Quirola-Vásquez et al. 2022).</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Observable event number as a function of instrument flux threshold based on the cosmic ERD and the LF derived from our analysis. The red dashed line presents the best fit. The blue and green dashed lines mark the sensitivities for Chandra/CDF-S and EP/WXRT, respectively.</figcaption></figure>

the observations of GW170817 and CDF-S XT2. However, it should be noted that there are different possibilities for the remnant of GW170817, and this may lead to very different constraints on NS EoSs. The population synthesis results could provide a novel and valuable method for constraining the EoS of NSs.

Considering the remnant types for different EoSs, our population synthesis results indicate that ~30% of detectable XTs with an internal plateau are expected for the EoS DD2. For the EoSs AB-L and AB-N, ~98% of the detectable XTs with an external plateau, while ~94% of the detectable XTs in the EoS GM1 and all detectable XTs in the EoS APR are expected to display an internal plateau. In a systematic analysis conducted by L. Zou et al. (2021a), they investigated the magnetar model using sGRB X-ray afterglow data observed by Swift/XRT. Combining their results with our sample, we obtained 29 sGRBs with XPs, in which eight sGRBs presented an internal plateau. Assuming these XPs are the candidate XTs powered by NSs, it suggests that about 28% of the XPs observed by Swift/XRT show an internal plateau, consistent with the results obtained for the EoS DD2. These results may indicate that the EoS DD2 could be more satisfied with the observations without considering the selection effect of the instrument. However, it is important to note that the bursts analyzed in our study represent only a small fraction of the observations with Swift/XRT. To further study the EoSs of NSs and derive more robust conclusions, a larger sample of XTs is needed for comprehensive analysis.

### 5.3. N as a Function of $F_{th}$

To further explore the differences in detectable event rates, we examine the relationship between the observable event number *N* and the instrumental threshold $F_{th}$. In Figure 6, we illustrate *N* as a function of $F_{th}$, incorporating the FoV of

Chandra ($\Omega_{cha}$), along with the derived $R(z)$ and the LF. One can observe that the relationship between *N* and $F_{th}$ is well described by a broken power-law function, with best-fit parameters $\alpha_1 = 0.02 \pm 0.01$, $\alpha_2 = 1.55 \pm 3.55$, and a break at $F_{th,b} = (3.00 \pm 0.25) \times 10^{-9}$ erg cm$^{-2}$ s$^{-1}$. Based on this result, we find that the expected event rate of XTs observed by Chandra ($N_{Cha}$) is about one order of magnitude larger than $N_{EP}$ within the same energy band and FoV, owing to the lower $F_{th}$ of Chandra compared to EP/WXRT. However, when considering the FoVs of both instruments and defining the ratio ($\mathcal{R}$) of the event rate observable with the EP/WXRT to that with Chandra as $\mathcal{R} = (N_{EP}\Omega_{EP})/(N_{Cha}\Omega_{Cha})$, we derive $\mathcal{R} \simeq 50$ in the 0.5–7 keV band, which indicates that the number of events detectable by EP/WXRT is approximately 50 times larger than those detected by Chandra, primarily due to the significantly wider FoV of EP/WXRT.

## 6. Conclusions

Based on the results of the population synthesis numerical simulation, we have investigated the event rate densities $R(z)$ of NSB mergers as a function of $z$ for different EoSs. By considering the M33.A model simulation results and assuming that a newborn magnetar is the outcome of an NSB event with a total mass $M_{tot} \le 1.3 M_{TOV}$, we estimate the global local ERD of NSB magnetars to be $R_0 \sim 300 \text{ Gpc}^{-3} \text{ yr}^{-1}$. The cosmic history of $R(z)$ peaks at $z_{peak} = 1.81$ and exhibits a rapid decrease as $R \propto z^{-3.85}$ for $z > 4$. Regardless of the instrumental selection effects and based on the plausible XTs observed in the Chandra/CDF-S data and the early X-ray afterglow data of sGRBs, we have parameterized the LF of NSB-magnetar XTs as a combination of a single power-law function and a broken power-law function with indices of $a = -1.03$, $\beta_1 = -0.28$, and $\beta_2 = -1.66$, as well as a break luminosity $L_b = 4.38 \times 10^{47} \text{ erg s}^{-1}$. Our analysis predicts that
the detectable rate of the XTs with EP/WXRT is ~780 yr⁻¹ in the luminosity range of [2 × 10⁴⁴, 2 × 10⁴⁹] erg s⁻¹ without considering any beaming effect. When considering an ejecta opening angle of 16°, similar to that observed in sGRBs, this number significantly drops to ~31 yr⁻¹. According to the results reported by the EP/WXRT in the past year, our result, considering the jet opening angle, does not violate the current observations. However, it is noted that our estimation is subject to uncertainties arising from the LF derivation, due to the instrumental selection effect and the limited sample size. Currently, we are unable to quantify their impact on the estimated detection rate. We expect that the EP observations will be crucial in reducing these uncertainties. Furthermore, according to the population synthesis results, we can also determine the fraction of the types of XTs among the remnants for different EoSs. Our results suggest that the EoS DD2 more closely aligns with observations when not considering the selection effect of the instrument.

These XTs as NSB post-merger products are the most prospective observable EM counterparts of the GW sources. Assuming the mass of the NSB merger is below $2.5 M_⊙$, K. Belczynski et al. (2020) give a local merger rate density of $R_0 = 321 \text{ Gpc}^{-3} \text{ yr}^{-1}$. The GW detection rate for LIGO/Virgo's mid-high sensitivity curve is $R_{GW} = 1.329 \text{ yr}^{-1}$, with the capability to detect sources up to a redshift of $z_{dis} = 0.082$. These findings highlight that, despite a high NSB merger rate, current instruments are likely to detect only a limited number of GW events. When adopting $R_0 = 321 \text{ Gpc}^{-3} \text{ yr}^{-1}$, the detectable event rate for NSB-magnetar XTs is $3.80 \text{ yr}^{-1}$ at $z_{dis}$. If considering $R_0 = 300 \text{ Gpc}^{-3} \text{ yr}^{-1}$, the detectable event rate is $3.53 \text{ yr}^{-1}$.

H. Sun et al. (2017) investigated the peak LF and ERD of the XTs powered by the NSB magnetars associated with sGRBs via Monte Carlo simulations. They proposed a bimodal LF, peaking at $2.5 \times 10^{46} \text{ erg s}^{-1}$ and $4 \times 10^{49} \text{ erg s}^{-1}$, corresponding to the line of sight to the trapped (the direction where X-rays are initially trapped by the dynamical ejecta) and the free (the direction where no sGRB is observed) zones, respectively. They inferred an ERD of a few tens Gpc⁻³ yr⁻¹ at $L > 10^{45} \text{ erg s}^{-1}$. Note that the GRB jet emission could be choked by extended material and stalled far below the photosphere (N. Senno et al. 2016) or the misalignment of their jet axis to the light of sight (L. Zou et al. 2019, 2021b), the rate estimated by H. Sun et al. (2017) should be only a rough lower limit of the XT rate.

## Acknowledgments

We very much appreciate the helpful comments and suggestions from the referee. We acknowledge the use of the public data from the Swift data archive and the UK Swift Science Data Center. This work is supported by the National Key R&D Program of China (grant No. 2024YFA1611700) and the National Natural Science Foundation of China (grant Nos. 12403056 and 12133003).

## ORCID iDs

Le Zou https://orcid.org/0000-0003-4639-5397

Ji-Gui Cheng https://orcid.org/0000-0002-2585-442X

Rui-Chong Hu https://orcid.org/0000-0002-6442-7850

En-Wei Liang https://orcid.org/0000-0002-7044-733X

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