Issue 
A&A
Volume 622, February 2019



Article Number  A194  
Number of page(s)  5  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201834552  
Published online  18 February 2019 
Gravitational waves from postmerger radially oscillating millisecond pulsars
^{1}
School of Astronomy and Space Science, Nanjing University, Nanjing 210093, PR China
email: dzg@nju.edu.cn
^{2}
Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, PR China
Received:
31
October
2018
Accepted:
14
January
2019
Observations of shortduration gammaray bursts and their afterglows show that a good fraction (perhaps ≳50%) of binary neutron star mergers lead to strongly magnetized, rapidly rotating pulsars (including millisecond magnetars), no matter whether the pulsar remnants are short or longlived. Such compact objects are very likely to have significant radial oscillations and high interior temperatures, as indicated in recent numerical simulations. In this paper, we have investigated rotationinduced gravitational radiation from possibly existing, radially oscillating pulsars after binary neutron star mergers, and find that this mechanism can efficiently damp the radial oscillations. The resulting gravitational waves (GWs) could have a nonnegligible contribution to the highfrequency spectrum. We provide an orderofmagnitude estimate of the event rate and suggest that such GW events would be detectable with the advanced LIGO/Virgo or nextgeneration detectors. Our discussion can also be applied to newborn, radially oscillating, millisecond pulsars formed through the other astrophysical processes.
Key words: gravitational waves / pulsars: general / stars: neutron / stars: oscillations
© ESO 2019
1. Introduction
Plenty of indirect evidence shows that binary neutron star mergers are the origin of short gammaray bursts (GRBs) (for reviews see Nakar 2007; Berger 2014). Direct evidence comes from the recentlydiscovered gravitational wave (GW) event GW170817 and its electromagnetic counterparts (Abbott et al. 2017a,b). Postmerger central objects include two possibilities: black holes and stronglymagnetized millisecond pulsars, which are dependent on the total mass of a binary neutron star system and the equation of state for neutronstar matter.
The central millisecond pulsar engine (including millisecond magnetars) was proposed as the predicted temporal plateaus of GRB afterglows by taking rotational energy injection into account for postburst relativistic blast waves (Dai & Lu 1998a,b; Zhang & Mészáros 2001; Dai 2004; Dall’Osso et al. 2011), and to explain latetime Xray flares in short GRBs by considering magnetic reconnection events taking place over the surfaces of postmerger millisecond pulsars (Dai et al. 2006). This kind of engine is not only consistent with general relativistic hydrodynamic simulations of binary neutron star mergers (Giacomazzo & Perna 2013; Piro et al. 2017) but also with a large sample of short GRB afterglows with temporal plateaus (Rowlinson et al. 2010, 2013). A statistical analysis shows that such an engine occurs in a good fraction (perhaps ≳ 50%) of short GRB afterglows (Gao et al. 2016; Piro et al. 2017). For GW170817, a neutron star remnant is suggested to interpret an observed multiwavelength kilonova (Yu et al. 2018; Ai et al. 2018; Li et al. 2018; Metzger et al. 2018), a broadband longevous afterglow (Geng et al. 2018), and a possible Xray flare appearing at ∼155 days after this merger event (Piro et al. 2019).
Recent numerical simulations indicate radial oscillations in postmerger neutron star remnants (e.g., Bauswein et al. 2018). The remnants have a radialpulsation amplitude α ∼ 0.03 − 0.1 and a high temperature T ≳ 10 − 20 MeV/k (where k is the Boltzmann constant). For these rapidly rotating neutron stars, dynamical barmode and secular instabilities are unlikely to take place because the ratio of the stellar rotational energy to gravitational binding energy is below 0.14 for many realistic neutronmatter equations of state (Piro et al. 2017). The remaining spinrelated instabilities such as unstable rmodes (Andersson 1998; Owen et al. 1998; Ho & Lai 2000; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Dai et al. 2016) and higher multipole (including quadrupolar) fmodes (Passamonti et al. 2013) are attributed to l = m = 2 and l = m > 2 nonradial pulsations respectively. However, their pulsation amplitudes necessarily spend a very long time – at least a few hundreds of seconds – increasing from extremely small to saturation values. Therefore, these instabilities cannot affect the rotational evolution of newborn millisecond pulsars with age ≲1 s.
In this paper, we investigate GWs from postmerger rapidlyrotating radially oscillating neutron stars and show that such GW events would be detectable with the advanced LIGO/Virgo or nextgeneration detectors. Although GWs from binary neutron star mergers have been presented through numerical simulations (e.g., Bernuzzi et al. 2015; Bauswein & Stergioulas 2015; Takami et al. 2016; Chatziioannou et al. 2017; Shibata & Kiuchi 2017; Zappa 2018), GWs from postmerger radially oscillating neutron stars remain unexplored^{1}. Interestingly, Weinberg et al. (2013) and Andersson & Ho (2018) explored dynamical tidal effects on GWs from inspiraling neutron star binaries before the mergers, and Dall’Osso et al. (2015) studied GWs from postmerger neutron stars by considering mass quadrupole moments that are induced by magnetic field amplification in the interiors during the mergers. In order to assess the importance of GWs from postmerger radially oscillating neutron stars, we propose a physical model and find that they could have a nonnegligible contribution to the postmerger spectrum at a frequency ≳3 kHz. We also suggest that this model can be used to discuss GWs from newborn, radially oscillating, millisecond pulsars formed through other astrophysical processes.
This paper is organized as follows. We first analyze three damping mechanisms of the radial oscillations, described in Sect. 2, and then we discuss the detectability of the resulting GWs in Sect. 3. We apply this analysis to GW170817 in Sect. 4, and discuss some implications of our model and give an orderofmagnitude estimate of the event rate in Sect. 5. Finally, we summarize our conclusions in Sect. 6.
2. Damping mechanisms
We consider a model in which a radially oscillating, strongly magnetized, rapidly rotating pulsar occurs just after the merger of two neutron stars. At this moment, any nonradial hydrodynamical effect is insignificant so that the pulsar remnant has a radial oscillation alone, as shown in recent numerical simulations (Bauswein et al. 2018). We also neglect the effect of any nonradial instability (e.g., unstable rmode and fmode) because its initial pulsation amplitude is extremely small so that the stellar rotation remains unchanged for t ≲ 1 s (Andersson 1998; Owen et al. 1998; Ho & Lai 2000; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Passamonti et al. 2013; Dai et al. 2016). The radial oscillation energy of this neutron star is calculated by (Chau 1967; Sawyer 1989)
where M and R are the stellar mass and radius respectively, α ≡ ΔR/R is the radial pulsation amplitude, ω is the angular pulsation frequency, and the convention Q_{x} = Q/10^{x} is adopted in cgs units. By linearizing the differential hydrodynamic equations describing the stellar nonequilibrium mechanical behavior and considering only smallamplitude adiabatic oscillations, the angular pulsation frequency turns out to be (Cox 1980)
where γ is the adiabatic index of neutronstar matter, η_{γ} ≡ (3γ − 4)/2, and ρ = M/(4πR^{3}/3) is the stellar average mass density. Inserting Eq. (3) into Eq. (1), we obtain
This radial oscillation energy is lost through three mechanisms, which we discuss in detail below. The first damping mechanism is rotationinduced gravitational radiation (Chau 1967), which has been used to investigate GWs from phase transitions of accreting neutron stars in lowmass Xray binary systems (Cheng & Dai 1998). The GW luminosity via this mechanism is given by
where Ω = 2π/P is the stellar angular rotation frequency with P being the rotational period (Chau 1967), and κ_{γ} ≡ (225γ − 36)/414. Therefore, the corresponding damping timescale is
The second damping mechanism is pulsational magnetic radiation (PMR) due to a temporallychanging magnetic dipole moment m = BR^{3}/2 ∝ R(t), where B is the stellar surface field strength at the magnetic pole (Hoyle et al. 1964; Cameron 1965; Heintzmann & Nitsch 1972; Duncan 1989). The resulting electromagnetic emission power is written as^{2}
and the corresponding damping timescale is
This timescale is much larger than t_{GW} for any reasonable surface field strength (i.e., B ≲ 10^{16} G).
The third damping mechanism is the bulk viscosity of neutronstar matter (Sawyer 1980, 1989). At the initial stage post merger, the newborn pulsar has a high temperature of T ≳ 10 − 20 MeV/k, at which neutrinos are trapped in the interior and form an ideal Fermi–Dirac gas with chemical potential μ_{ν} ≫ kT (Sawyer & Soni 1979). As a result, the radial pulsations are damped through the following nonequilibrium reactions: p + e^{−} ↔ n + ν_{e}. According to these reactions, Reisenegger & Goldreich (1992) estimated the viscous damping timescale for freely escaping neutrinos (see their Eq. (37)), which is certainly much larger than t_{GW}. Cheng & Dai (1998) derived the bulk viscosity coefficient (ζ) for trapped neutrinos and the corresponding damping timescale^{3}
where Y_{e}, Y_{ν}, and Y_{n} are respectively the fractions of electrons, neutrinos, and neutrons. In the following, we calculate these fractions and to obtain t_{V, N}. For a neutral mixture of free nonrelativistic neutron, nonrelativistic proton, relativistic electron and relativistic neutrino gases in the interior of the postmerger neutron star, if the lepton fraction is taken to be l_{0}, then we have Y_{ν} = l_{0} − Y_{e} as well as the proton fraction Y_{p} = Y_{e} = 1 − Y_{n}. The chemical equilibrium condition leads to , where the factor of two multiplying Y_{ν} in the last term is due to the existence of only one coupled polarization state for ν_{e} and with n_{b} being the baryon number density and n_{0} being the nuclear baryon number density (Sawyer 1980). Therefore, we can calculate the fractions of four kinds of particles and then Λ(l_{0}, n_{b}) for given l_{0} and n_{b}. Figure 1 presents Λ(l_{0}, n_{b}) as a function of n_{b} for l_{0} = 0.07, 0.1, and 0.15. We can see from this figure that Λ(l_{0}, n_{b}) > 10, and thus from Eq. (10) that the viscous damping timescale t_{V, N} is also much larger than t_{GW}.
Fig. 1. as a function of n_{b}/n_{0} in Eq. (10) for different values of l_{0}. The black, blue, and red lines are corresponding to l_{0} = 0.07, 0.1, and 0.15, respectively. 

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An alternative possibility for the central object post merger is that it is a strange quark star. This possibility was put forward by Dai & Lu (1998b), recently implied from a statistic analysis of the observed plateau durations in the light curves of short GRB afterglows by Li et al. (2016), and very recently suggested in numerical simulations by Most et al. (2018) and Bauswein et al. (2018). In this case, the bulk viscosity arises from nonequilibrium nonleptonic reactions among quarks: u + d ↔ u + s, through which the viscous damping timescale of radial oscillations in the strange quark star is estimated by
where m_{s} is the strange quark mass (Dai & Lu 1996; Madsen 1992, 2000). This timescale is still far beyond t_{GW} for typical values of the relevant parameters (e.g., ρ_{15} ∼ 1, R_{6} ∼ 1, m_{s}c^{2} ∼ 100 MeV, and kT ≳ 10 MeV).
Therefore, we can conclude that no matter whether the postmerger radially oscillating object is a neutron star or a strange quark star, the damping mechanisms due to pulsational magnetic radiation and bulk viscosity are both negligible and thus the gravitational radiation must be the dominant damping mechanism. In the next section, we discuss the detectability of such GWs.
3. Detectability of GWs
The strain of GWs can be estimated through using the mass quadrupole approximation to the Einstein field equations (Shapiro & Teukolsky 1983; Thorne 1987). This approximation shows that the GW strain is given by
where d is the distance to the source (Shapiro & Teukolsky 1983), and Q is the timedependent mass quadrupole moment (Chau 1967)
Inserting Eq. (13) into Eq. (12), we have
The characteristic GW strain can be approximated by (Corsi & Mészáros 2009)
where f = ω/2π is the frequency of GWs. Inserting Eqs. (3), (7), and (14) into Eq. (15), we find
This equation shows that h_{c} is not only independent of the neutronstar rotational period but also weakly dependent of the stellar radius. In addition, the inferred remnant mass M is generally close to 2.5 M_{⊙} for most of the observed binary neutron star systems in the Galaxy (also see Table 1 of Baiotti & Rezzolla 2017). Thus, the radial pulsation amplitude α could be found if h_{c} is detected and if d is inferred at the inspiraling stage. On the other hand, if no GW signal from the radial oscillations is detected, we can give an upper limit on α by comparing h_{c} with the sensitivity (i.e., noise level) of the GW detector, h_{rms} = [fS_{n}(f)]^{1/2}, where S_{n}(f) is the power spectral density (PSD) of the detector noise. The PSD has been presented as a function of GW frequency for the advanced LIGO detector and the Einstein Telescope (ET) respectively (e.g., Arun et al. 2005; Mishra et al. 2010; Sun et al. 2015; Gao et al. 2017).
4. Application to GW170817
The observations of GW170817 indicate that this event should have arisen from an inspiral of two neutron stars, the total mass of which is M_{tot} ≃ 2.74 M_{⊙} (Abbott et al. 2017a). If the two stars are assumed to have the same mass, then the gravitational mass of each star becomes ≃1.37 M_{⊙}. Because the baryonic mass (M_{b}) and gravitational mass (M_{g}) of a neutron star satisfy a correlation, (Timmes et al. 1996), the total baryonic mass of the binary system turns out to be M_{b, tot} ≃ 3.02 M_{⊙}. In addition, the baryonic mass of the ejecta during the merger is found to be M_{ej} ≃ 0.065 M_{⊙} by fitting the multiwavelength kilonova data (Villar et al. 2017), so the gravitational mass of the neutron star left behind after the merger is derived as M_{g, tot} ≃ 2.49 M_{⊙}. This is why the neutron star remnant mass M in the two sections above is scaled as 2.5 M_{⊙}. The actual gravitational mass of the neutron star remnant is dependent of the mass ratio (q) of the premerger two neutron stars. However, since q ∼ 1 for most of the observed binary neutron star systems in the Galaxy (Baiotti & Rezzolla 2017), the mass M of each neutron star remnant from these systems is inferred to be around 2.5 M_{⊙}. Furthermore, since h_{c} in Eq. (16) is weakly dependent of the stellar radius, the remnant structure would scarcely influence our theoretical GW strain if α is fixed.
The advanced LIGO/Virgo detectors searched for GWs from the neutron star remnant after GW170817 but no signal was found (Abbott et al. 2017c, 2018). Recently, van Putten & Della Valle (2019) claimed the detection of a postmerger signal candidate with a duration ∼1 s but their estimated GW energy is lower than the sensitivity estimates of Abbott et al. (2017c). The upper limits on the GW strain were derived for two different observed periods of a GW signal by Abbott et al. (2017c). For example, the best upper limit on the rootsumsquare of the GW strain emitted from 1 to 4 kHz, for a signal ≲1 s, is at the 50% detection efficiency (Abbott et al. 2017c). From Eq. (16), we find that the strain of GWs from the neutron star remnant after GW170817 is h_{c} ≃ 1.1 × 10^{−21}α_{−1} for M ≃ 2.5 M_{⊙}, R_{6} ≃ 1 and d ≃ 40 Mpc (hereafter γ = 2 is assumed so that η_{γ} = κ_{γ} = 1). The requirement that leads to a radial pulsation amplitude . Unfortunately, this limit seems too loose to provide useful information. Postmerger GW emission from a similar event would be possibly detectable with nextgeneration detectors such as ET or when the advanced LIGO/Virgo detectors reach their design sensitivity (Abbott et al. 2017c). Once detected, a sample of such GW events would provide a unique probe for postmerger central objects.
5. Discussion
Gravitational waves from postmerger radial oscillations may have a nonnegligible contribution to the highfrequency spectrum. On the one hand, such GWs are emitted at a frequency kHz from Eq. (3), which is very close to the cutoff frequency shown in Fig. 1 of Chatziioannou et al. (2017). On the other hand, the characteristic strain from Eq. (16) is much stronger than the numerical peak strain h_{p} ∼ 10^{−24} at a frequency ≳3 kHz in Chatziioannou et al. (2017) if the distance to the source is 100 Mpc. In fact, GWs from postmerger spinrelated nonradial instabilities must be radiated at a frequency that is about twice as large as the Kepler rotation limit of the neutron star remnants, that is, kHz, where P_{Kepler} is the stellar Kepler rotational period (Haensel et al. 2009). Thus, the frequency of GWs from radial oscillations is nearly equal to that from nonradial instabilities. Moreover, the pulsation amplitudes of nonradial instabilities must grow to some saturation values in a long period of at least a few hundreds of seconds (Andersson 1998; Ho & Lai 2000; Dai et al. 2016; Passamonti et al. 2013), so that GWs from them are much weaker than those from the radial oscillations for t ≲ 1 s, as shown by comparing h_{c} and h_{p}.
Observationally, a GW detector would first discover GWs from an inspiral of two neutron stars and derive the masses of the two stars and the distance to the source, as in GW170817. If the detector subsequently discovers GWs from the postmerger central object and if such GWs are due to radial oscillations of a millisecond pulsar remnant, then we can constrain the radial pulsation amplitude α from Eq. (16), provided that h_{c} is detected and that d is inferred from the GWs at the inspiraling stage. In addition, the frequency and duration of postmerger GWs would be found from their waveforms. Furthermore, according to Eqs. (3) and (7), one would obtain the massradius relation and rotational period of the pulsar remnant, which, together with the inferred pulsation amplitude α, would provide useful information for constraining the postmerger central object.
We give here an orderofmagnitude estimate of the detectable GW event rate. From Abbott et al. (2017a), the best rate of GW170817like events is approximated by R_{tot} ∼ 1540 Gpc^{−3} yr^{−1}. To be conservative, we assume that a half of such events can produce neutron star remnants (Gao et al. 2016; Piro et al. 2017), so the rate of GW events from radial oscillations is R_{osc} ∼ 800 Gpc^{−3} yr^{−1}. Furthermore, since the sensitivities of aLIGO and ET are and for signals of 1 − 4 kHz respectively (also see Fig. 3 of Gao et al. 2017), from Eq. (16), we obtain the detection horizons of these detectors, d_{LIGO} ≲ 150 Mpc and d_{ET} ≲ 2.2 Gpc for α ∼ 0.1, and thus we find the detectable event rates, and . For a small radial pulsation amplitude α ∼ 0.03, however, the detection horizons of the detectors decrease by a factor ∼3.3, so the detectable event rates become R_{LIGO} ∼ 0.6 yr^{−1} and R_{ET} ∼ 2×10^{3} yr^{−1}. Therefore, it seems that nextgeneration detectors such as ET would be able to detect a large number of GW events from radially oscillating pulsar remnants every year.
Besides binary neutron star mergers, millisecond pulsars including millisecond magnetars are produced through the other astrophysical processes, for example, core collapse of massive stars, accretioninduced collapse of white dwarfs, and mergers of binary white dwarfs. These stars may have radial oscillations. Thus, our analysis in this paper can also be applied to such stars. Of course, the complete waveforms of GWs from binary neutron star mergers are different from those of GWs from the other processes, since the former include the GWs at the inspiraling stage and the latter have only GWs from the radial oscillations. This property could be used to distinguish between the binary neutron star merger and other processes.
We finally discuss the effects of γ. This parameter is actually dependent of the equation of state for neutronstar matter. For realistic equations of state, γ is in the range of two to three (Haensel et al. 2002), so that η_{γ} ∼ 1 − 2.5 and κ_{γ} ∼ 1 − 1.54. Since h_{c} is weakly dependent of η_{γ} and independent of κ_{γ}, from Eq. (16), the characteristic strain of GWs is hardly affected by γ directly. However, from Eqs. (3) and (7), the frequency (duration) of GWs increases (decreases) with increasing γ. Even so, as long as the stellar period is in the order of ∼1 ms and the surface field strength is not beyond 10^{16} G, the GW damping timescale is much smaller than the viscous damping timescale and also the PMR damping timescale (even though these timescales are all dependent of the massradius relation). This conclusion is always true for any realistic equation of state of dense matter above the nuclear density, no matter whether the postmerger compact object is a neutron star or a strange quark star, and thus our analysis in Sect. 2 is valid.
6. Summary
We have proposed a model to explore GWs from postmerger radially oscillating, rapidly rotating pulsars after binary neutron star mergers, and found that rotationinduced gravitational radiation is the dominant damping mechanism of the radial oscillations. Some other conclusions are summarized below.
First, the resulting GWs have a frequency kHz, which is very close to the cutoff frequency shown by numerical simulations, while the characteristic strain h_{c} is not only independent of the neutronstar rotational period but also weakly dependent of the stellar radius. Furthermore, for the same source, our derived strain is much stronger than given by numerical simulations at a frequency ≳3 kHz for α ∼ 0.03−0.1. Therefore, the GWs from radially oscillating neutron star remnants would have a dominant contribution to the highfrequency spectrum.
Second, the estimated event rate is so high that the GW events would be detectable with the advanced LIGO/Virgo or nextgeneration detectors such as ET. If detected, such GW events, together with their frequencies, durations and strains, would be used to constrain the properties of neutron star remnants, for example, the massradius relation (from Eq. (3)), rotational periods (from Eq. (7)) and radial pulsation amplitudes (from Eq. (16)). If not detected, on the other hand, an upper limit on α would be given.
Third, for GW170817, no postmerger signal was found by the aLIGO/Virgo detectors. This requires that , which seems too loose to provide any useful constraint on the neutron star remnant.
Finally, the other astrophysical formation processes of newborn radially oscillating, strongly magnetized millisecond pulsars include core collapse of massive stars, accretioninduced collapse of white dwarfs, and mergers of binary white dwarfs. Our model can also be applied to these newborn pulsars.
After this paper was submitted, the prospects of studying GW170817like postmerger signals with future GW detectors were investigated by TorresRivas et al. (2019).
We note that the factor multiplying c^{3} in the denominator of the right term of the second equality sign should be six rather than twelve in Duncan (1989).
The chiral angle θ in Eq. (5) of Cheng & Dai (1998) is taken to be zero in this paper, which implies no kaon condensation.
Acknowledgments
The author would like to thank the referee for helpful comments and suggestions, and He Gao, Maurice H. P. M. van Putten, YunWei Yu, and Bing Zhang for their discussions. This work is supported by the National Key Research and Development Program of China (grant No. 2017YFA0402600), the National Basic Research Program of China (“973 Program”, grant No. 2014CB845800), and the National Natural Science Foundation of China (grant No. 11573014 and 11833003).
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All Figures
Fig. 1. as a function of n_{b}/n_{0} in Eq. (10) for different values of l_{0}. The black, blue, and red lines are corresponding to l_{0} = 0.07, 0.1, and 0.15, respectively. 

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