On the possibility to detect gravitational waves from post-merger super-massive neutron stars with a kilohertz detector

The detection of a secular post-merger gravitational wave (GW) signal in a binary neutron star (BNS) merger serves as strong evidence for the formation of a long-lived post-merger neutron star (NS), which can help constrain the maximum mass of NSs and differentiate NS equation of states. We specifically focus on the detection of GW emissions from rigidly rotating NSs formed through BNS mergers, using several kilohertz GW detectors that have been designed. We simulate the BNS mergers within the detecting limit of LIGO-Virgo-KARGA O4 and attempt to find out on what fraction the simulated sources may have a detectable secular post-merger GW signal. For kilohertz detectors designed in the same configuration of LIGO A+, we find that the design with peak sensitivity at approximately $2{\rm kHz}$ is most appropriate for such signals. The fraction of sources that have a detectable secular post-merger GW signal would be approximately $0.94\% - 11\%$ when the spindowns of the post-merger rigidly rotating NSs are dominated by GW radiation, while be approximately $0.46\% - 1.6\%$ when the contribution of electromagnetic (EM) radiation to the spin-down processes is non-negligible. We also estimate this fraction based on other well-known proposed kilohertz GW detectors and find that, with advanced design, it can reach approximately $12\% - 45\%$ for the GW-dominated spindown case and $4.7\% - 16\%$ when both the GW and EM radiations are considered.


INTRODUCTION
Since the initial detection of the gravitational wave (GW) event GW150914, which resulted from a binary-black-hole (BBH) merger (Abbott et al. 2016, for summaries), a significant number of GW events arising from binary-compact-object mergers have been detected during the first three operational periods of the LIGO-Virgo-KAGRA collaboration (Abbott et al. 2019(Abbott et al. , 2021;; The LIGO Scientific Collaboration et al. 2021).However, it is worth noting that only two binary-neutron-star (BNS) mergers (GW170817 (Abbott et al. 2017a) and GW190425 (Abbott et al. 2020b)) have been identified thus far.Additionally, in the case of these BNS merger events, only GWs from the inspiral phase were successfully detected.
The post-merger GW emission from a BNS-merger system depends on the type of central remnant.In theory, there are four possible evolving routes for the central remnant: (Rosswog et al. 2000;Rezzolla et al. 2010Rezzolla et al. , 2011;;Howell et al. 2011;Lasky et al. 2014;Rosswog et al. 2014;Ravi & Lasky 2014;Gao et al. 2016;Radice et al. 2018;Ai et al. 2020): 1)it directly collapses into a black hole (BH); 2) it produces a differential-rotating hyper-massive NS (HMNS) that ★ E-mail: shunke.ai@whu.edu.cn† E-mail: gaohe@bnu.edu.cn‡ E-mail: zhuzh@bnu.edu.cncollapses into a BH within seconds; 3) it produces a differentially rotating HMNS that transitions to a rigidly rotating super-massive neutron star (SMNS) after the differential rotation vanishes and eventually collapses into a BH after the SMNS spins down; 4) it produces a stable neutron star (SNS) after the HMNS phase.The timescale of the post-merger GW signal is determined by the lifetime of the post-merger NS.In BNS-merger simulations, if a HMNS is formed, it is expected to emit a significant amount of post-merger GWs dominated by the quadrupolar  -mode, with a typical frequency range of approximately 2 kHz to 4 kHz (Xing 1994;Ruffert et al. 1996;Shibata & Uryū 2000;Hotokezaka et al. 2013).In fact, depending on the equation of state (EoS) of neutron stars, the post-merger GW emission can start at around 1 kHz (Maione et al. 2017).However, this type of GW emission rapidly damps within ∼ 100ms due to either the collapse of the HMNS into a BH (route 2) or the formation of a rigidly rotating NS (route 3 and 4) (Baiotti et al. 2008;Kiuchi et al. 2009;Rezzolla et al. 2011;Hotokezaka et al. 2013;Kiuchi et al. 2018).A post-merger rigidly rotating NS can be approximated as a triaxial ellipsoid, and its ellipticity, defined as where  3 is the moment of inertia about the rotational axis and  1 and  2 are the other two moments of inertia, quantifies its de-formation (Bonazzola & Gourgoulhon 1996;Palomba 2001;Cutler 2002).This deformation can be induced through various mechanisms, including magnetic-field-induced deformation (Bonazzola & Gourgoulhon 1996;Palomba 2001;Cutler 2002), bar-mode instability (Lai & Shapiro 1995;Corsi & Mészáros 2009), and r-mode instability (Andersson 1998;Lindblom et al. 1998).Consequently, the long-lived post-merger NS could emit secular GWs with a frequency twice that of the NS's rotation frequency.Efforts have been made to search for the post-merger GW signal associated with GW170817, but no signal related to it has been detected yet.Unfortunately, due to the limited sensitivity of advanced LIGO in the kilohertz (kHz) frequency range, meaningful constraints on the properties of the central remnant cannot be derived (Abbott et al. 2017b).
The nature of the remnant can potentially be inferred from electromagnetic (EM) observations.In the case of a long-lived post-merger remnant with a significant magnetic field, such as a magnetar, several characteristic features may arise (see Zhang 2018, for a review).One commonly accepted feature indicating the formation of long-lived NS is the so called internal X-ray plateau.This plateau phase is characterized by a nearly-flat light curve that typically lasts for a few hundred seconds and followed by a steep power-law decay with a slope index greater than 3.This feature is difficult to explain in the framework of an external-shock-driven afterglow but can be naturally powered by a long-lived NS (magnetar) that eventually collapses into a BH during the steep decay phase (Rowlinson et al. 2010(Rowlinson et al. , 2013;;Lü & Zhang 2014;Lü et al. 2015).For GW170817, although EM counterparts have been observed across a wide range from gamma-ray to radio frequencies, the lack of distinct EM signals and the absence of early X-ray observations make it challenging to determine the formation of a long-lived neutron star.Even if it was believed that a long-lived neutron star was formed in GW170817, its surface magnetic field is expected to be relatively low (Ai et al. 2018).Considering the large uncertainties in EM radiation models, confirming the existence of a long-lived neutron star after a binary neutron star merger requires the search for secular post-merger signals using upgraded GW detectors, particularly those with peak sensitivity in the kilohertz range.
In this study, our focus is on the GW signals emitted by long-lived SMNSs or SNSs formed through BNS mergers.We do not delve into specific mechanisms for deformation, but instead treat  as a general parameter.In section 2, we introduce the basic formula for GW emission from a rigidly rotating NS.In Section 3, we present several designs to enhance the sensitivity of GW detectors in kilohertz.In Section 4, we investigate the population properties related to BNS mergers and their central remnants.In Section 5, we discuss the feasibility of detecting this type of GWs using kilohertz GW detectors.Specifically, we aim to find out that, among all the BNS mergers identified by the currently operating GW detector LIGO A+, what fraction would have a detectable secular post-merger GW signal, with different kilohertz detector used.Finally, our conclusions and discussions are presented in Section 5.

BASIC FORMULA (SINGLE EMITTER)
Consider a newly formed post-merger rigidly-rotating NS (Hereafter we call it as post-merger NS) with radius , poloidal magnetic field   , momentum of inertia  and initial spin period  0 .Also, suppose the post-merger NS is slightly deformed as an triaxial ellipsoid with ellipticity .It would spin down via both GW and magnetic dipole radiation, where the spin-down luminosity can be expressed as (Shapiro et al. 1983;Zhang & Mészáros 2001) (2) Here Ω = 2/ represents the angular velocity of the spinning NS, whose evolution can be calculated as Ω/ =  sd /(Ω).Using the convention  = 10    for cgs units hereafter, the spin-down timescale for the EM dominated and GW dominated cases can be estimated as and respectively, where  rot = (1/2)Ω 2 0 ≈ 2 × 10 52  45  −2 i,−3 erg is the initial total rotational kinetic energy.
In our expression,  determines the strength of the GW radiation, from which the strain of GWs can be derived as where  is the distance from the source to the observer.In the frequency domain, the characteristic strain could be expressed as where  = Ω/ = 2/P stands for the frequency of the GWs so that  / = (1/)Ω/.From ℎ  one can calculated the signal-noiseratio (SNR) for the GW signal with a specific detector as1 with   representing the sensitivity of the detector.In practical,  max and  min represent the initial and final frequency related to the initial and final spin period of the post-merger NS during the detection.If post-merger NS collapses into a BH during the detection,  min should be derived from the spin period when the collapsing happens.
In the GW-dominated case, the EM radiation term in Equation 2 can be ignored, thus the characteristic strain of the GWs in the frequency domain can be further written as Considering a rigidly rotating NS formed right after the differential rotating phase, it is reasonable to assume its initial spin period to be close to the breaking limit (i.e.Keplerian period), which is a function of the NS mass.During the merger, the total baryonic mass of the system is conserved, which can be calculated as where  ,1 ,  ,2 and  b,rem are the baryonic masses for the two pre-merger NSs and the post-merger central remnant.Once the gravitational masses ( 1 and  2 ) for the pre-merger NSs are determined, the baryonic masses can be calculated from a universal relation   =   +0.802  (e.g.Gao et al. 2020), for the slow-spinning NSs. ej is the ejected mass during the merger, which can be obtained from the EM follow-up observation or the numerical simulations for BNS mergers.We adopt  ej = 0.01 ⊙ in the whole paper 2 .Considering that the post-merger NS are rapidly rotating, the relation of  and   for the slow-spinning NSs is no longer applicable.For a NS spins at the Keplerian period, a new relation   =  + 0.064 2  would apply, from which the initial gravitational mass for the central remnant ( rem,0 ) can be estimated.Further, one can estimate the initial spin period from another universal relation (Ai et al. 2020) where P 0 =  0 / k,min is the normalized initial spin period.Considering a NS with arbitrary spin period, a more general relation for the baryonic and gravitational masses, which reads as can be used to convert  b,rem back to the gravitational mass of the central remnant ( rem ). 1.4 is the radius in km for the non-rotating NS when  NS = 1.4 ⊙ .P = / k,min is defined as the normalized NS spin period, where  k,min represents the minimum period of a rigid rotating NS before breaking up.In this work,  1.4 = 12.38km and  k,min = 5.85 × 10 −4 s are adopted3 .Following the spin-down history of the post-merger NS, together with Equation 11, one can calculate the its spin period as well as the gravitational mass at any time.When the criterion (Ai et al. 2020) is satisfied, the massive NS should collapse into a BH.Therefore, the collapsing time ( col ) and the collapsing spin period ( col ) can be read out.If the collapsing time is smaller than the length of effective GW data ( d ), we use  col to calculate  min .Otherwise, we should follow the spin-down history until  d , and use the spin period at that time to calculate  min .
Examples for the evolution of ℎ  are shown in Figure 1, with  d ∼ 5 × 10 3 s assumed as the length of effective GW data for our aimed signal.When magnetic dipole radiation has contribution to the spin-down process, even though with the same  value, the characteristic strain (ℎ  ) of the GW signal would be smaller than the GW-dominated case.If a longer   is assumed, the trajectory of ℎ  would extend to lower frequency, thus a higher SNR is expected.

KILOHERTZ GW DETECTORS
Previous calculations indicate that, the GWs from the post-merger NSs are mainly in kilohertz.In this section, targeting on this type of GW sources, we propose several kHz GW detectors, with central frequency at 1 kHz, 2 kHz, and 3 kHz, respectively, for comparison.
Usually, extending the length of the signal recycling cavity is the most convenient and efficient way to shift the detection frequency band from low to high, which is the main approach used in our designs.
In this work, we design kHz detectors based on LIGO A+.Considering that the target detecting frequency band is far away from the free spectrum range of the configuration of the kHz detector, the influence of antenna response can be ignored.The arm length are short enough that the gravitational wave strain can be approximately treated as static (Essick et al. 2017).The signal recycling cavity (SRC) and arm cavity can be considered as coupled cavities.By decreasing the transmission of the signal recycling mirror, the wide band can be changed to narrow band.Meanwhile, the sensitivity around the resonant frequency will become better.In this case, the sum of the phase shift reflected from the arm cavity  arm and the phase shift of light through SRC  src need to achieve resonance conditions, which can be described as (Miao et al. 2018;Martynov et al. 2019) arm and  src are calculated as and with where  src is the length of SRC.Ω = 2  is the sideband frequency.
arm is the arm cavity bandwidth. arm is the length of arm cavity. arm is the transmission of the arm cavity.We adopt the same arm cavity configuration of LIGO A+ (Miller et al. 2015). src and sideband frequency  have a corresponding relationship.The transmission of SRM  src determines the width of the sideband of the detector's power spectral density.The parameters of the detectors designed in this paper is shown in Table 1.Meanwhile, the sensitivity curves for our designed detectors, with power spectrum density as the vertical axis, are shown in Figure 1.Moreover, by increasing the power within the arm cavity and implementing an opto-mechanical filter, one can significantly amplify the signal, resulting in an approximately five-fold increase in the SNR (Miao et al. 2018).To achieve the same goal by simply increasing the light intensity, it needs to be increased by about 25 times, which is extremely hard due to the difficulties in controlling the input laser power, dealing with the mirror absorption thermal effect and the dynamic instability.Although the opto-mechanical filter on GW detectors is also a raw technique right now, designs have been made in the literature (e.g.Miao et al. 2018), which can increase the sensitivity of the detector in a much wider frequency range, totally covering the peak frequencies for our designed kHz detector.We further consider the design of the Neutron Star Extreme Matter Observatory (NEMO) (Ackley et al. 2020), which is a proposed 2.5-generation GW detector to be constructed in Australia with peak frequency in kilohertz band.NEMO has higher arm circulating power (4.5 MW), lower longer laser wavelength (2 m) and advanced cooling technique compared to our designs.Their sensitivity curves are also shown in Figure 1.  and Miao et al. (2018), respectively.Other solid lines are generated by using the python module pygwinc (Rollins et al. 2020).We adopt the parameters for the post-merger NS as  0 = 0.5ms,  = 1.5 × 10 45 g cm 2 ,  = 1.2 × 10 5 cm.Assume the post-merger NS will never collapse and locate at 40Mpc from the observer.For the black dashed line,  = 10 −2 and   = 0 are assumed.

distribution of redshift
The red-shift distribution of BNS mergers is determined by their merger rate at each red-shift.Usually, the BNS merger rate in unit volume follows the star formation rate (SFR) convoluting with the a time delay from the formation of the BNS system to the merger (Wanderman & Piran 2015;Sun et al. 2015), which reads as where the  () = SFR(z) / SFR(z=0) is the normalized SFR, with the SFR fitted as (Yüksel et al. 2008) where  = −10 is adopted. stands for the time delay and  () describes its probability density function.Assume the time delay follows a log-normal distribution (Wanderman & Piran 2015;Sun et al. 2015), which reads as where   = 2.9 Gyr and   = 0.2 ln(Gyr).
The BNS merger rate in a certain rad-shift bin can finally be calcualted as where  is the comoving volume of the universe and with  0 = 68.7kms −1 Mpc −1 and Ω  = 0.308 adopted (Planck Collaboration et al. 2016).

distribution of mass
The mass of the merger remnant can be calculated from the masses of the two pre-merger NSs.Before the detection of GW190425, the mass distribution for BNS mergers are supposed to follow that for the Galactic double NS, which can be described by a Gaussian distribution with central value  = 1.33 and standard deviation  = 0.11 (Kiziltan et al. 2013).However, the total mass of the BNSmerger system GW190425 reaches  tot = 3.4 ⊙ (Abbott et al. 2020b), which exceeds the 5 limit of the Gaussian distribution presented above, indicating the BNS merger systems may divert from that for the Galactic double NSs.
In this work, since the correct distribution of BNSs are still under debate, we still use the mass distribution for Galactic double NSs to simulate all the BNS merger systems.This treatment will make our estimate on the possibility to detect the secular post-merger GW signals optimal, due to more rigidly rotating NSs would be formed.To simulate the merger remnants, another crucial physical quantity is  TOV .Recently, a millisecond pulsar with mass  = 2.35 ± 0.17 has been found (Romani et al. 2022), indicating that very likely  TOV ≳ 2.35 ⊙ .Combined with the previous constraints on  TOV based on the generally believed scenario that the merger remnant of GW170817 collapsed into a BH, e.g. TOV ≲ 2.3 ⊙ (Shibata et al. 2019), we adopt  TOV = 2.35 ⊙ .However, the constraints on  TOV from GW170817 is not necessarily correct because the type of merger remnant are not determined, thus the value of  TOV could be higher.The value adopt here may offset some overestimation of the production of rigidly rotating NSs as the merger remnants introduced by the mass distribution assumed above.

distributions of 𝜖 and 𝐵 𝑝
The distribution of  and   can be inferred from the breaking times of the X-ray internal plateaus.The SGRBs followed by a X-ray internal plateau as well as the breaking times are listed in Table A1 (See Appendix A1).
First of all, we assume the magnetic field on all of the post-merger NSs are efficiently small so that the spin-down processes are dominated by GW radiation.In this scenario,  becomes the only free parameter.According to the Bayesian theory, the posterior distribution of  can be expressed as where () represent the prior distribution of  and L ( b,obs | ) represents the likelihood of a certain .Instead of an analytical formula, the likelihood is obtained by conducting Monte Carlo simulation.For each value of , we simulate 10 3 BNS-merger sources according to the methods introduced in Section 2 and Section 4.3.For the sources where a long-lived super-massive neutron star is produced, the corresponding collapsing time can also be calculated.We compare the simulated collapsing time samples with the observed breaking times of X-ray internal plateaus, by conducting a Kolmogorov-Smirnov (KS) test using the python module scipy.stats.kstest, to check if they follow the same distribution.The p value of the KS test represents the probability for the two datasets coming from the same distribution.
And we assume the likelihood in Equation 22is proportional to this p value.Here, we adopt a uniform prior distribution for  from 10 −5 to 10 −1 in logarithmic space.Then, we run the Markov Chain Monte Carlo (MCMC) simulation based on Equation 22, using the python module emcee (Foreman-Mackey et al. 2013), to obtain a sample of .The distribution of collapsing times of SMNSs and the resultant distribution of  from MCMC simulation are shown in Figure 2.
Here lg() = −2.66+0.04 −0.03 with the uncertainty at 1 confident level is obtained.Using the inferred  distribution, based on the mass distribution of BNS mergers and  TOV , we generate a set of collapsing times, whose distribution is shown with the black line and histogram in the upper panel of Figure 2. The reproduced collapsing time distribution is generally consistent with the distribution of the observed ones, which is shown with the blue line and histogram.
Next, as the intermediate step, we assume the spin-down processes for all the post-merger rigidly rotating NSs are dominated by magnetic dipole radiation.In this scenario,   becomes the only free parameter.We adopt a uniform prior distribution for   from 10 13 G to 10 17 G in logarithmic space and follow the procedure above to find the posterior distribution of   .The result is is shown in the upper panel of Figure 3, while The resultant collapsing time distribution for simulated SMNSs, and the observed distribution of the breaking times of X-ray internal plateaus, are shown in the lower panel.
For a rigidly rotating NS, if its triaxial shape is induced by the magnetic field, some theories suggest a relation  ∝  2  (Bonazzola & Gourgoulhon 1996;Haskell et al. 2008).However, since the origin of  is unclear, to be conservative, we treat  and   as independent.When both GW and EM radiation have contributions to the spindown process, the range of  and   should in the range constrained in Figure 2 and 3, otherwise, due to they are independent, collapsing times outside the range of the observed ones would be generated.Here, we use the 3 limits (99.73%) of the two parameters as their boundaries (lg() ∈ [−2.79, −2.52] and lg(  ) ∈ [14.81, 15.07]).The joint posterior distribution of  and   can be written as while again the likelihood is calucated by Monte Carlo simulation.The only difference compared to the procedure describe above is that we simulate the BNS mergers to find their corresponding the collapsing times under each set of  and   .The joint constraint on  and   is shown in the upper panel of Figure 4. From the lower panel, it can be seen that the generated collapsing times have a distribution being consistent with that for the observed ones.

DETECTABILITY
Knowing basic formula for a single emitter and the population properties for the sources, we can estimate their the detectability with a specific GW detector.Here we focus on the BNS mergers that can  be detected during LIGO-Virgo-KAGRA O4, which has a detecting limit as 170Mpc (Abbott et al. 2020a), and try to find out on what fraction the identified BNS mergers may followed by a detectable secular post-merger GW emission.We randomly generate 5 × 10 5 BNS merger systems within 170Mpc, according to the red-shift distribution introduced in Section 4.1.In principle, our simulated sources are all at the near universe where the event rate density,  merger , can be treated as constant over red-shift.Although the method to calculate the red-shift distribution introduced above is overqualified, we still use it to be general.For each system, the masses of pre-merger NSs are randomly set, according to the mass distribution presented in 4.2.Then we could judge whether a rigidly rotating NS is formed after the merger, based on the criterion  rem,0 < 1.2 TOV (Cook et al. 1994;Lasota et al. 1996;Breu & Rezzolla 2016;Ai et al. 2020).Once a post-merger rigidly rotating NS is produced, we randomly generate  and   according to their inferred distributions presented in Section 4.3.Next, we follow its spin-down history and calculate the characteristic strain for the GW radiation and its SNR with a specific GW detector.Setting a threshold SNR, for each detector, we can count the number of sources with a detectable post-merger GW signal and record the fraction.

GW-dominated spindown
For different kilohertz GW detectors, the cumulative distribution for SNRs of secular post-merger GWs are shown in the left panels of Figure 5.We also show the distribution of SNR when LIGO A+ is used to search for the secular post-merger GWs.Assuming SNR > 5 as the criterion, the fraction of the simulated sources that have a detectable secular post-merger GW signal can be read.The values are listed in Table 2.
When the length of valid data (  ) is relatively short, the GW detectors with peak sensitivity at a higher frequency has better performance.This is because, with a short   , at the end of detection, most of the newly formed rigidly rotating NSs have not spun down yet, in which case the evolving trajectory of ℎ  would only span a small range of frequency so that the integral SNR remains relatively small.As the length of valid data (  ) increases, for all the detectors, the fraction of sources with detectable post-merger GW signals would increase.This is also reasonable, because the evolving trajectory of ℎ  for some long-lived NSs would extend to lower frequencies along with the spin-down processes, resulting in the increases of the integral SNRs, especially when GW detectors peaks at lower frequencies are used.This explains why the detector with peak sensitivity at 3000Hz is much better than that peaks at 1000Hz when   = 5 × 10 2 s, while it becomes opposite when   = 5 × 10 4 s.Although for most of the cases, increasing the data length   can effectively lead to a higher fraction of sources that have detectable post-merger GW signals, a saturated fraction might exist.For example, using the detector proposed by Miao et al. (2018), this fraction no longer increases when   increases from 5×10 3 s to ×10 4 s.This is because, for any source within 170Mpc, if it can survive for 5 × 10  and has a  value within the inferred range, it must have reached threshold SNR at an earlier time.The histogram of the distances of all the sources with a detectable secular post-merger GW signal are shown in the right panels of Figure 5.The vertical axes are normalized with the total number of simulated sources.As can be seen, for the secular post-merger GW signals, the detecting rate might not necessarily be proportional to the cube of the detecting limit of a detector, because the detecting limit is determine by the GWs from SNSs, while some high-frequency detectors might also sensitive to the sources collapsing at earlier times.The performances of the kilohertz GW detectors designed basically with the same technique used in LIGO A+ are not satisfactory.Even when   = 5 × 10 4 s, only 3% − 10% BNS sources identified in LIGO-Virgo-KAGRA O4 can have a detectable post-merger GW signal.When   = 5 × 10 2 s is assumed, this fraction decreases to less than 1%.However, they have been much better than when LIGO A+ is used to search for the secular post-merger GW signals.Among them, the performance of the detector peaks at 2000Hz is the best and the most stable.
Using NEMO, the fraction of BNS-merger events with a secular  post-merger signal would be much higher.For example, the fraction increases from ∼ 5% when   = 5 × 10 2 s to ∼ 42% when   = 5 × 10 4 s.If more quantum technologies were applied, i.e. using the design in Miao et al. (2018), this fraction could reach ∼ 11% when   = 5 × 10 2 s and approaches a saturated value as ∼ 45% when   > 5 × 10 3 s.

The effect of dipole magnetic field
In the case that both the contributions of GW and EM radiation to the spin-down process are not negligible, the fractions of the simulated sources that have a detectable post-merger GW signal are list in Table 3 , while the cumulative distribution of SNRs and the distribution of distances for the detectable post-merger sources are shown in Figure 6.
The general trend in this case remains the same as that in the case of GW-dominated spindown, but the fraction with the kilohertz detector designed by Miao et al. (2018) have not reached a saturated value.In this case, it is even more unlikely for the secular post-merger GW signals to be detected.For the kilohertz detectors designed with the same techniques as LIGO A+, the fraction can only reach 1.6% even with   = 5×10 4 s while be as low as 0.5% when   = 5×10 2 s.

CONCLUSION AND DISCUSSION
In this paper, we studied the secular post-merger GW emissions from the newly-formed rigidly-rotating NSs after the BNS mergers.We discussed several kilohertz GW detectors targeting on this type of signals.Then, we simulated the BNS mergers within the detecting limit of LIGO-Virgo-KAGRA O4 (170Mpc), and infer the fraction of the sources that have a detectable secular post-merger GW signal for each designed kilohertz GW detector, to test their performance.
For the kilohertz GW detector proposed in this paper with only the length of SRC adjusted while other structures remain the same as LIGO A+, the fraction of the sources that have a detectable secular post-merger GW signal is at most ∼ 11% when the length of the effective GW data is   = 5 × 10 4 s, and ∼ 1% when   = 5 × 10 2 s, in the GW-dominated spindown case.If the effect of magnetic dipole radiation on the spin-down process is not negligible, this fraction would decrease to 0.5% − 1.6%.These results are based on the detector with peak frequency at 2000Hz, which is better than that with peak frequency at 1000Hz and 3000Hz, and much better than the case when just searching for secular post-merger GW signals in the data of LIGO A+ itself.According to this fraction, the detecting rate should not be satisfactory.Therefore, new technique needs to be applied to upgrade the detector.
The proposed 2.5-generation detector NEMO has much wider sensitive frequency range in kilohertz band, from 1000Hz to 3000Hz, which covers almost all the possible frequencies of the GW emission from a post-merger rigidly rotating NS.It it was used, the fraction of the sources that followed by a detectable secular post-merger GW signal would reach ∼ 42% if the spin-down processes for the post-merger rigidly rotating NSs are all dominated by GW radiation and   > 5 × 10 4 s, while the fraction decreases to ∼ 5% when   = 5×10 2 .If magnetic dipole radiation also contributes to the spindown of the post-merger rigidly rotating NSs, this fraction decrease to ∼ 3% when   = 5 × 10 3 − 5 × 10 4 s and to ∼ 1.5% when   = 5 × 10 2 s.Furthermore, if more quantum technologies were applied to GW detectors as proposed by Miao et al. (2018), the sensitivity of the GW detector and the results obtained will be even better.The fraction of the sources that followed by a detectable secular postmerger GW signal would reach ∼ 45% if the spin-down processes for the post-merger rigidly rotating NSs are all dominated by GW radiation and   > 5 × 10 3 s, while the fraction decreases to ∼ 11% when   = 5 × 10 2 .If magnetic dipole radiation also contributes to the spindown of the post-merger rigidly rotating NSs, this fraction decrease to ∼ 15% when   = 5 × 10 3 − 5 × 10 4 s and to ∼ 5% when   = 5 × 10 2 s.
There are also other design schemes to obtain high-frequency gravitational wave detectors at the cost of decreasing the sensitivity.For example, detuning the phase of SRC directly without changing the length of SRC can easily change the peak frequency of detector by adjusting the phase of signal recycling mirror (Ganapathy et al. 2021).However, the power spectrum density of this detector in peak frequency will be twice than the detector proposed in this paper in theoretical because of losing half of the sideband.For another example, we can using an auxiliary resonant cavity with maintaining the same length of SRC.This auxiliary resonant cavity is used to compensate for phase to get the peak frequency we want.Of course, using an auxiliary resonant cavity will definitely bring new losses and noises and lead to the decreasing of sensitivity.The advantage of this design is that we can mainly adjust the transmission of the resonant cavity, thereby controlling its scale within a certain space.Besides, another way to improve the sensitivity of the GW detector is to use longer arm length.longer length of the arm cavity means the higher power in the main interferometer.It can improve the sensitivity of the detector in all frequency bands, not just in the kilohertz band.In this case, the antenna response is needed to be considered.It will decrease the SNR of the GW detector.Of course, it is also a very interesting subject to know the ultimate light intensity that the interferometer can withstand.
In addtion, we compared the kilohertz detectors discussed in this paper with future ground-base GW detectors.We find that, targeting on this specific kind of sources, the performance of the kilohertz detectors designed with the same technique used for LIGO A+ could be comparable to LIGO Voyager, while the kilohertz detector with more quantum technologies (i.e.Miao et al. (2018)) could be comparable to the third-generation GW detectors (i.e.Einstein Telescope and Cosmic Explorer).With Einstein Telescope and Cosmic Explorer, we will definitely detect more of such signals, but the fraction of the sources that have a detectable secular post-merger GW signal may not necessarily increase due to the highly enhanced detecting limit for the inspiral signal of BNS mergers.
In this work, we did not consider the inclination angle of the postmerger NSs, which might make the characteristic strains of the GW radiation slightly overestimated.Therefore, the estimated fraction of the sources that have a detectable secular post-merger GW signal could be even lower.We treat the  value as a constant during the whole spin-down process because its evolution over time is highly model-dependent thus unclear.Therefore, the  value in this work should be an effective value.Our results could be refined if the evolution of  is well determined in the future.

APPENDIX A: X-RAY INTERNAL PLATEAU
We conducted an analysis of 144 short-duration gamma-ray bursts (SGRBs) that were observed with the Swift satellite between January 2005 and January 2023.Our aim was to investigate the presence of plateau features in the light curves of their X-ray afterglows, which could indicate the collapse of a supra-massive neutron star (NS) into a black hole (BH).We extrapolated the available BAT-band data (15-150 keV) to XRT-band (0.3-10 keV) using a single power-law spectrum.We then applied a temporal fit to the combined light curve, utilizing a smooth broken power-law function, in order to identify potential plateau features.These features are characterized by shallow decay slopes smaller than 0.5, followed by a steep decay index greater than 3. Our analysis led to the identification of 35 SGRBs displaying internal plateau characteristics, constituting approximately 24% of the total sGRB sample.A detailed list of the 35 SGRBs with internal plateau characteristics, and their corresponding breaking times, can be found in Table A1.
This paper has been typeset from a T E X/L A T E X file prepared by the author.

Figure 1 .
Figure1.The sensitivity curves for the designed kilohertz GW detectors (solid lines) and the trajectories for ℎ  evolution (dashed lines) in the frequency domain.As comparison, the sensitivity curve for LIGO A+ is also shown.The red and purple solid lines are extracted fromAckley et al. (2020) andMiao et al. (2018), respectively.Other solid lines are generated by using the python module pygwinc(Rollins et al. 2020).We adopt the parameters for the post-merger NS as  0 = 0.5ms,  = 1.5 × 10 45 g cm 2 ,  = 1.2 × 10 5 cm.Assume the post-merger NS will never collapse and locate at 40Mpc from the observer.For the black dashed line,  = 10 −2 and   = 0 are assumed.For the grey dashed line,  = 10 −2 and   = 10 15 G are adopted.

Figure 2 .
Figure 2. The distributions of collapsing time of SMNSs are shown in the upper panel while the distribution of  is shown in the lower panel.In the upper panel, the solid lines represent the fitted probability density function with kernal density estimation using the python module scipy.stats.gaussiankde.The black line and histogram represent the collapsing times reproduced from the  distribution in the lower panel, considering the mass distribution of BNS mergers and  TOV .

Figure 4 .
Figure 4. Similar as Figure 2 and 3, the distribution of collapsing time of SMNSs are shown in the upper panel while the joint distribution of  and   is shown in the lower panel.

Figure 5 .
Figure5.The left panels show the cumulative SNR distribution for the simulated post-merger GW signals, while the right panels show the distribution of the distances from the GW sources to the observer for those with SNR > 5.The upper, middle and lower panels show the cases that   = 5 × 10 2 , 5 × 10 3 , and 5 × 10 4 , respectively.Different colors stand for different GW detectors applied.All of these panels is in the case that the spin-down process is dominated by the GW radiation.

Table 1 .
Parameters of different kHz detector designed based on LIGO A+

Table 3 .
The fraction of BNS mergers, inside the detecting limit of LIGO-Virgo-KAGRA O4 (170 Mpc), with a detectable post-merger GW signal.The spin-down process is considered both in GW radiation and EM radiation.

Table A1 .
breaking times of X-ray internal plateaus