arXiv:1810.05448v3 [astro-ph.HE] 21 Feb 2019 r eedn ntettlms fabnr eto trsyste star matter. neutron-star neutron for binary state a of of equation mass the total and the on dependent , are millisecond possibi strongly-magnetized and two holes a include black et objects (Abbott even central counterparts Post-merger (GW) comes electromagnetic 2017a,b). wave re- its evidence and (for gravitational Direct GW170817 (GRBs) recently-discovered 2014). bursts Berger the gamma-ray 2007; from short Nakar of see origin views m the star neutron are binary that ers shows evidence indirect of Plenty Introduction 1. ogvu fego Gn ta.21) n osbeX-ray possible a broadban 2019). and a 2018), 2018), at al. appearing et al. in- flare Metzger et (Geng to 2018; 201 al. afterglow suggested et 2017). al. longevous Li et is (Yu 2018; al. al. et remnant et Piro multiwavelength Ai star 2016; observed (perhaps neutron an al. fraction et terpret a (Gao good GW170817, afterglows a For GRB in short occurs shows of engine analysis plateaus a statistical an temporal with A with 2013). such also afterglows 2010, but al. GRB et 2017) short (Rowlinson relativis merger al. of et general star sample Piro with neutron large 2013; consistent binary Perna & of only (Giacomazzo simulations not sur- hydrodynamic is 2006). tic the engine al. over et consider- of (Dai place by pulsars kind millisecond GRBs taking post-merger short events of faces in reconnection flares magnetic X-ray ing an late-time 2011), explain al. 1998 et to Lu Dall’Osso & a 2004; (Dai M´esz´aros Dai into waves & 2001; blast injection Zhang relativistic energy post-burst rotational for count taking plateaus temporal by predicted afterglows the GRB as proposed was ) srnm & Astronomy eray2,2019 22, February ⋆ rvttoa ae rmps-egrradially-oscillati post-merger from waves Gravitational h eta ilscn usregn icuigmillisec (including engine millisecond central The -al [email protected] E-mail: epoiea re-fmgiueetmt fteeetrate event the of estimate order-of-magnitude an (G provide waves We gravitational resulting The bin oscillations. after radial pulsars th oscillating In radially simulations. existing, numerical possibly recent in are indicated objects as compact rotating Such atures, long-lived. rapidly or magnetized, short- are strongly remnants to lead mergers star e words. Key processes. astrophysical c other discussion the through Our formed detectors. next-generation or LIGO/Virgo bevtoso hr-uaingmarybrt n their and bursts gamma-ray short-duration of Observations 2019 22, February version: online Preprint 1 2 colo srnm n pc cec,NnigUniversity, Nanjing Science, Space and Astronomy of School e aoaoyo oenAtooyadAtohsc (Nanji Astrophysics and Astronomy Modern of Laboratory Key rvttoa ae usr:gnrl—sas eto st — neutron stars: — general pulsars: — waves gravitational ∼ 5dy fe hsmre vn Pr tal. et (Piro event merger this after 155days aucitn.aa34552-18 no. manuscript r eto trmres n n htti ehns a effi can mechanism this that find and mergers, star neutron ary s ol aeannngiil otiuint h high-fr the to contribution non-negligible a have could Ws) & sppr ehv netgtdrtto-nue gravitati rotation-induced investigated have we paper, is which .G Dai G. Z. lities: 50%) pulsars usr icuigmlieodmgeas,n atrwhet matter no magnetars), millisecond (including pulsars eylkl ohv infiatrda siltosadhigh and oscillations radial significant have to likely very This fegosso htago rcin(perhaps fraction good a that show afterglows nas eapidt ebr,rdal siltn,millis oscillating, radially newborn, to applied be also an ABSTRACT erg- that ond n ugs htsc Weet ol edtcal ihtea the with detectable be would events GW such that suggest and a,b; of m c- 8; l. d ajn 103 China; 210093, Nanjing d s - t gUiest) iityo dcto,China Education, of Ministry University), ng aia ia fet nGsfo nprln eto trb star neutron inspiraling from dy- GWs explored on (2018) effects Ho tidal & namical Andersson and (2013) al. et Weinberg inleeg ogaiainlbnigeeg sblw0 rota- below stellar is the energy of binding ratio a gravitational the instabilities to because neu- energy secular place rotating tional and take rapidly bar-mode to these unlikely dynamical For stars, constant). tron Boltzmann the is 07 ap ta.21) W rmps-egrradially- unexplored post-merger remain from stars GWs neutron 2018), Kiuc al. oscillating & et Shibata Zappa 2017; 20 al. Stergioulas 2017; et & Chatziioannou Bauswein 2015; al. 2015; et al. Takami et Bernuzzi sim neu (e.g., numerical through tions binary presented been from have mergers GWs star tron Although LIGO/Virgo advanced detectors. the that next-generation with show or detectable be and would events stars GW neutron radially-oscillating rotating wi pulsars . millisecond small newborn cann extremely of instabilities evolution from rotational these the increasing lea Therefore, – at values. – saturation seconds time to of long hundreds very few a a spend necessarily amplitudes sation l oe Psaot ta.21)aeatiue to attributed are 2013) quadrupola al. (including et multipole (Passamonti higher Lai modes 2013; and & Wasserman 2016) un Ho & al. Bondarescu 1998; et as 2009; Dai al. al. et such et Owen Bondarescu instabilities 1998; 2000; (Andersson spin-related r-modes (Piro remaining stable state of The equations 2017). neutron-matter realistic many for 0 etgtdb orsRvse l (2018). al. et Torres-Rivas detector by GW vestigated future with signals post-merger GW170817-like h ennshv ailplainamplitude radial-pulsation a 20 have al. et remnants Bauswein (e.g., The remnants star neutron post-merger . = 1 1 n ihtemperature high a and 1 1s. , 2 m nti ae,w netgt W rmps-egrrapidly- post-merger from GWs investigate we paper, this In fe hspprwssbitd h rset fstudying of prospects the submitted, was paper this After eetnmrclsmltosidct ailoscillation radial indicate simulations numerical Recent ⋆ r:oscillations ars: > o-ailplain epciey oee,terpul- their However, respectively. pulsations non-radial 2 T & & gmillisecond ng 10 0)o iayneutron binary of 50%) − nlrdainfrom radiation onal qec spectrum. equency 20MeV inl apthe damp ciently neirtemper- interior e h pulsar the her cn pulsars econd 1 Interestingly, . l

c / = dvanced α k S 2019 ESO m ∼ (where eein- were s taffect ot = 0 hage th . and 2 03 tal. et such 18). ula- ina- )f- r) in s . 15; 14 − re hi st k - - 2 Dai: Gravitational waves from newborn millisecond pulsars ries before the mergers,and Dall’Osso et al. (2015) studied GWs tems (Cheng & Dai 1998). The GW luminosity via this mecha- from post-merger neutron stars by considering mass quadrupole nism is given by moments that are induced by magnetic field amplification in the 2 4 interiors during the mergers. In order to assess the importance 1 G 45 9 Ω 2 4 2 6 E˙ GW = γ M R α ω (5) of GWs from post-merger radially-oscillating neutron stars, we 375 c5 4 − 5 ω     propose a physical model and find that they could have a non- 3 negligible contribution to the post-merger spectrum at a fre- 54 2 M 2 4 1 = 0.81 10 ηγκγ R6α 1P−3 ergs− , (6) quency & 3kHz. We also suggest that this model can be used × 2.5M − − to discuss GWs from newborn, radially oscillating, millisecond  ⊙  pulsars formed through other astrophysical processes. where Ω = 2π/P is the stellar angular rotation frequency with This paper is organized as follows. We first analyze three P being the rotational period (Chau 1967), and κγ (225γ ≡ − damping mechanisms of the radial oscillations, described in 36)/414. Therefore, the corresponding damping timescale is Section 2, and then we discuss the detectability of the result- 1 ing GWs in Section 3. We apply this analysis to GW170817 in E 2 M − 2 4 tGW = 6.3κγ− R6− P 3 ms. (7) Section 4, and discuss some implications of our model and give ≡ E˙ GW 2.5M − an order-of-magnitude estimate of the event rate in Section 5.  ⊙  Finally, we summarize our conclusions in Section 6. The second damping mechanism is pulsational magnetic ra- diation (PMR) due to a temporally-changing magnetic dipole moment m = BR3/2 ∝ R(t), where B is the stellar surface field 2. Damping mechanisms strength at| the| magnetic pole (Hoyle et al. 1964; Cameron 1965; Heintzmann & Nitsch 1972; Duncan 1989). The resulting elec- We consider a model in which a radially oscillating, strongly tromagnetic emission power is written as2 magnetized, rapidly rotating pulsar occurs just after the merger of two neutron stars. At this moment, any non-radial hydrody- 2 m¨ 2 B2R6α2ω4 E˙ = | | = namical effect is insignificant so that the pulsar remnant has PMR 3c3 6c3 a radial oscillation alone, as shown in recent numerical sim- 2 ulations (Bauswein et al. 2018). We also neglect the effect of 47 M 2 2 1 = 2.8 10 B14α 1 ergs− , (8) any non-radial instability (e.g., unstable r-mode and f-mode) be- × 2.5M −  ⊙  cause its initial pulsation amplitude is extremely small so that the stellar rotation remains unchanged for t . 1s (Andersson and the corresponding damping timescale is 1998; Owenetal. 1998; Ho&Lai 2000; Bondarescuetal. E 2009; Bondarescu & Wasserman 2013; Passamonti et al. 2013; t 1 8 104η B 2R 1 s (9) PMR ˙ = . γ 14− 6− . Dai et al. 2016). The radial oscillation energy of this ≡ EPMR × is calculated by (Chau 1967; Sawyer 1989) This timescale is much larger than tGW for any reasonable sur- face field strength (i.e., B . 1016 G). 3 2 2 2 51 M 2 2 2 E = MR α ω = 0.75 10 R6α 1ω4 erg, (1) The third damping mechanism is the bulk viscosity of 20 × 2.5M −  ⊙  neutron-star matter (Sawyer 1980, 1989). At the initial stage post merger, the newborn pulsar has a high temperature of where M and R are the stellar mass and radius respectively, T & 10 20MeV/k, at which neutrinos are trapped in the in- α ∆R/R is the radial pulsation amplitude, ω is the angular terior and− form an ideal Fermi-Dirac gas with chemical po- pulsation≡ frequency, and the convention Q = Q/10x is adopted x tential µ kT (Sawyer & Soni 1979). As a result, the radial in cgs units. By linearizing the differential hydrodynamic equa- ν pulsations≫ are damped through the following non-equilibrium tions describing the stellar non-equilibrium mechanical behavior reactions: p e n ν . According to these reactions, and considering only small-amplitude adiabatic oscillations, the + − + e Reisenegger & Goldreich↔ (1992) estimated the viscous damp- angular pulsation frequency turns out to be (Cox 1980) ing timescale for freely escaping neutrinos (see their Eq. [37]), which is certainly much larger than t . Cheng & Dai (1998) 4π 1/2 GW ω = (3γ 4)Gρ (2) derived the bulk viscosity coefficient (ζ) for trapped neutrinos 3 − 3   and the corresponding damping timescale 1/2 4 1/2 M 3/2 1 2 2 = 2.6 10 ηγ R−6 s− , (3) ρR 1/3 2/3 4/3 2/3 2 kT × 2.5M tV,N = 22Ye Yν− Yn− ρ15− R6 s, (10)  ⊙  ≡ 30ζ 10MeV   where γ is the adiabatic index of neutron-star matter, ηγ (3γ 3 ≡ − where Ye, Yν, and Yn are respectively the fractions of electrons, 4)/2, and ρ = M/(4πR /3) is the stellar average mass density. neutrinos, and neutrons. In the following, we calculate these Inserting Eq. (3) into Eq. (1), we obtain 1/3 2/3 4/3 fractions and Λ(l0,nb) Ye Yν− Yn− to obtain tV,N. For a 2 neutral mixture of free≡ non-relativistic neutron, non-relativistic 51 M 1 2 E = 5.1 10 ηγ R6− α 1 erg. (4) proton, relativistic electron and relativistic neutrino gases in the × 2.5M −  ⊙  interior of the post-merger neutron star, if the lepton fraction is This radial oscillation energy is lost through three mech- 2 We note that the factor multiplying c3 in the denominator of the anisms, which we discuss in detail below. The first damp- right term of the second equality sign should be six rather than twelve ing mechanism is rotation-induced gravitational radiation (Chau in Duncan (1989). 1967), which has been used to investigate GWs from phase tran- 3 The chiral angle θ in Eq. (5) of Cheng & Dai (1998) is taken to be sitions of accreting neutron stars in low-mass X-ray binary sys- zero in this paper, which implies no kaon condensation. Dai: Gravitational waves from newborn millisecond pulsars 3

3. Detectability of GWs

The strain of GWs can be estimated through using the

l =0.07 0 mass quadrupole approximation to the Einstein field equations

l =0.1 0 (Shapiro & Teukolsky 1983; Thorne 1987). This approximation l =0.15

3

0 10 shows that the GW strain is given by 2G Q¨ h 4 | |, (12) 2 ≃ c d 10

where d is the distance to the source (Shapiro & Teukolsky 1983), and Q is the time-dependent mass quadrupole moment (Chau 1967) 1

10 8π 45 9 Ω 2 Q = γ ραR5 cos(ωt). (13) 15 4 − 5 ω 1 2 3 4 5   

n /n b 0 Inserting Eq. (13) into Eq. (12), we have

1 M d − 1 3 2 3 4 3 22 2 2 / / / h 0.88 10− κγα 1P−3 R6 . (14) Fig. 1: Λ(l0,nb) Ye Yν− Yn− as a function of nb/n0 in Eq. − 2.5M 100Mpc ≡ ≃ × − (10) for different values of l0. The black, blue, and red lines are  ⊙   corresponding to l0 = 0.07, 0.1, and 0.15, respectively. The characteristic GW strain can be approximated by (Corsi & M´esz´aros 2009)

dt hc = fh h ftGW, (15) taken to be l0, then we have Yν = l0 Ye as well as the pro- s d f ≃ − ton fraction Yp = Ye = 1 Yn. The chemical equilibrium condi- p 1/3 2−/3 2/3 1/3 where f = ω/2π is the frequency of GWs. Inserting Eqs. (3), tion leads to ξYe + Yp = Yn + ξ(2Yν) , where the fac- (7), and (14) into Eq. (15), we find tor of two multiplying Yν in the last term is due to the exis- tence of only one coupled polarization state for ν and ξ 3/4 1 e 1/4 1/4 M d − 1/3 ≡ 22 2 1/3 ~ 1/3 hc 4.5 10− ηγ α 1R6 (16). (3π )− nb− (2mnc)/ = 3.1(nb/n0)− with nb being the ≃ × − 2.5M 100Mpc baryon number density and n0 being the nuclear baryon number  ⊙    density (Sawyer 1980). Therefore, we can calculate the fractions This equation shows that hc is not only independent of the of four kinds of particles and then Λ(l0,nb) for given l0 and nb. neutron-star rotational period but also weakly dependent of the Figure 1 presents Λ(l0,nb) as a function of nb for l0 = 0.07, 0.1, stellar radius. In addition, the inferred remnant mass M is gener- and 0.15. We can see from this figure that Λ(l0,nb) > 10, and ally close to 2.5M for most of the observed binary neutron star ⊙ thus from Eq. (10) that the viscous damping timescale tV,N is systems in the Galaxy (also see Table 1 of Baiotti & Rezzolla also much larger than tGW. 2017). Thus, the radial pulsation amplitude α could be found An alternative possibility for the central object post merger if hc is detected and if d is inferred at the inspiraling stage. is that it is a strange . This possibility was put for- On the other hand, if no GW signal from the radial oscilla- ward by Dai & Lu (1998b), recently implied from a statistic tions is detected, we can give an upper limit on α by compar- analysis of the observed plateau durations in the light curves ing hc with the sensitivity (i.e., noise level) of the GW detector, 1/2 of short GRB afterglows by Li et al. (2016), and very recently hrms = [ fSn( f )] , where Sn( f ) is the power spectral density suggested in numerical simulations by Most et al. (2018) and (PSD) of the detector noise. The PSD has been presented as a Bauswein et al. (2018). In this case, the bulk viscosity arises function of GW frequency for the advanced LIGO detector and from non-equilibrium non-leptonic reactions among quarks: u+ the Einstein Telescope (ET) respectively (e.g., Arun et al. 2005; d u + s, through which the viscous damping timescale of ra- Mishra et al. 2010; Sun et al. 2015; Gao et al. 2017). dial↔ oscillations in the strange quark star is estimated by

4 2 4. Application to GW170817 m c2 − kT t = 12ρ2 R2 s s, (11) V,Q 15 6 100MeV 10MeV The observations of GW170817 indicate that this event should     have arisen from an inspiral of two neutron stars, the total mass where ms is the strange quark mass (Dai & Lu 1996; Madsen of which is Mtot 2.74M (Abbott et al. 2017a). If the two ≃ ⊙ 1992, 2000). This timescale is still far beyond tGW for typical stars are assumed to have the same mass, then the gravitational 2 values of the relevant parameters (e.g., ρ15 1, R6 1, msc mass of each star becomes 1.37M . Because the baryonic ∼ ∼ ∼ ≃ ⊙ 100MeV, and kT & 10MeV). mass (Mb) and gravitational mass (Mg) of a neutron star sat- 2 Therefore, we can conclude that no matter whether the post- isfy a correlation, Mb = Mg + 0.075Mg (Timmes et al. 1996), merger radially oscillating object is a neutron star or a strange the total baryonic mass of the binary system turns out to be quark star, the damping mechanisms due to pulsational mag- Mb,tot 3.02M . In addition, the baryonic mass of the ejecta ≃ ⊙ netic radiation and bulk viscosity are both negligible and thus during the merger is found to be Mej 0.065M by fitting the the gravitational radiation must be the dominant damping mech- multiwavelength kilonova data (Villar≃ et al. 2017),⊙ so the gravi- anism. In the next section, we discuss the detectability of such tational mass of the neutron star left behind after the merger is GWs. derived as Mg,tot 2.49M . This is why the neutronstar remnant ≃ ⊙ 4 Dai: Gravitational waves from newborn millisecond pulsars mass M in the two sections above is scaled as 2.5M . The actual a millisecond pulsar remnant, then we can constrain the radial ⊙ gravitational mass of the neutron star remnant is dependent of pulsation amplitude α from Eq. (16), provided that hc is detected the mass ratio (q) of the pre-merger two neutron stars. However, and that d is inferred from the GWs at the inspiraling stage. In since q 1 for most of the observed binary neutron star systems addition, the frequency and duration of post-merger GWs would in the Galaxy∼ (Baiotti & Rezzolla 2017), the mass M of each be found from their waveforms. Furthermore, according to Eqs. neutron star remnant from these systems is inferred to be around (3) and (7), one would obtain the mass-radius relation and ro- 2.5M . Furthermore, since hc in Eq. (16) is weakly dependentof tational period of the pulsar remnant, which, together with the the stellar⊙ radius, the remnant structure would scarcely influence inferred pulsation amplitude α, would provide useful informa- our theoretical GW strain if α is fixed. tion for constraining the post-merger central object. The advanced LIGO/Virgo detectors searched for GWs We give here an order-of-magnitude estimate of the de- from the neutron star remnant after GW170817 but no tectable GW event rate. From Abbott et al. (2017a), the best signal was found (Abbottetal. 2017c, 2018). Recently, rate of GW170817-like events is approximated by Rtot 3 1 ∼ van Putten & Della Valle (2019) claimed the detection of a post- 1540Gpc− yr− . To be conservative, we assume that a half of merger signal candidate with a duration 1s but their estimated such events can produce neutron star remnants (Gao et al. 2016; GW energy is lower than the sensitivity estimates∼ of Abbott et al. Piro et al. 2017), so the rate of GW events from radial oscil- 3 1 (2017c). The upper limits on the GW strain were derived for lations is Rosc 800Gpc− yr− . Furthermore, since the sen- ∼ LIGO 22 ET two different observed periods of a GW signal by Abbott et al. sitivities of aLIGO and ET are hrms 3 10− and hrms 23 ∼ × ∼ (2017c). For example, the best upper limit on the root-sum- 2 10− for signals of 1 4kHz respectively (also see Figure 3 square of the GW strain emitted from 1 4kHz, for a signal of× Gao et al. 2017), from− Eq. (16), we obtain the detection hori- 50% 22 1/2 − . 1s, is h = 2.1 10 Hz at the 50% detection ef- zons of these detectors, dLIGO . 150Mpc and dET . 2.2Gpc for rss × − − ficiency (Abbott et al. 2017c). From Eq. (16), we find that the α 0.1, and thus we find the detectable event rates, RLIGO = ∼ 3 1 3 strain of GWs from the neutron star remnant after GW170817 is (4π/3)dLIGORosc 20yr− and RET = (4π/3)dETRosc 7 21 4 1 ∼ ∼ × hc 1.1 10− α 1 for M 2.5M , R6 1 and d 40Mpc 10 yr− . For a small radial pulsation amplitude α 0.03, how- ≃ × − ≃ ⊙ ≃ ≃ (hereafter γ = 2 is assumed so that ηγ = κγ = 1). The require- ever, the detection horizons of the detectors decrease∼ by a factor 50% 1 ment that hc . √ fhrss leads to a radial pulsation amplitude 3.3, so the detectable event rates become RLIGO 0.6yr− and ∼ 3 1 ∼ α . 0.6√ f3. Unfortunately, this limit seems too loose to pro- RET 2 10 yr− . Therefore, it seems that next-generation de- vide useful information. Post-merger GW emission from a sim- tectors∼ such× as ET would be able to detect a large numberof GW ilar event would be possibly detectable with next-generation de- events from radially oscillating pulsar remnants every year. tectors such as ET or when the advanced LIGO/Virgo detec- Besides binary neutron star mergers, millisecond pulsars in- tors reach their design sensitivity (Abbott et al. 2017c). Once cluding millisecond magnetars are produced through the other detected, a sample of such GW events would provide a unique astrophysical processes, for example, core collapse of massive probe for post-merger central objects. stars, -induced 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. 5. Discussion Of course, the complete waveforms of GWs from binary neutron star mergers are different from those of GWs from the other pro- Gravitational waves from post-merger radial oscillations may cesses, since the former include the GWs at the inspiraling stage have a non-negligible contribution to the high-frequency spec- and the latter have only GWs from the radial oscillations. This trum. On the one hand, such GWs are emitted at a fre- property could be used to distinguish between the binary neutron 3/2 1/2 quency f 4.1R6− (M/2.5M ) kHz from Eq. (3), which star merger and other processes. is very close≃ to the cutoff frequency⊙ shown in Figure 1 of We finally discuss the effects of γ. This parameter is actu- Chatziioannou et al. (2017). On the other hand, the characteris- ally dependent of the equation of state for neutron-star matter. 22 1/4 3/4 For realistic equations of state, γ is in the range of two to three tic strain hc 4.5 10− α 1R6 (M/2.5M ) from Eq. (16) − ⊙ ≃ × 24 (Haensel et al. 2002), so that ηγ 1 2.5 and κγ 1 1.54. is much stronger than the numerical peak strain hp 10− ∼ − ∼ − at a frequency & 3kHz in Chatziioannouet al. (2017)∼ if the Since hc is weakly dependent of ηγ and independent of κγ, from distance to the source is 100Mpc. In fact, GWs from post- Eq. (16), the characteristic strain of GWs is hardly affected by merger spin-related non-radial instabilities must be radiated at γ directly. However, from Eqs. (3) and (7), the frequency (dura- a frequency that is about twice as large as the Kepler rota- tion) of GWs increases (decreases) with increasing γ. Evenso, as tion limit of the neutron star remnants, that is, f 2 P long as the stellar period is in the order of 1ms and the surface / Kepler 16 ∼ 3/2 1/2 ≃ ≃ field strength is not beyond 10 G, the GW damping timescale 3.5R− (M/2.5M ) kHz, where PKepler is the stellar Kepler 6 ⊙ is much smaller than the viscous damping timescale and also rotational period (Haensel et al. 2009). Thus, the frequency of the PMR damping timescale (even though these timescales are GWs from radial oscillations is nearly equal to that from non- all dependent of the mass-radius relation). This conclusion is radial instabilities. Moreover, the pulsation amplitudes of non- always true for any realistic equation of state of dense matter radial instabilities must grow to some saturation values in a long above the nuclear density, no matter whether the post-merger period of at least a few hundreds of seconds (Andersson 1998; compact object is a neutron star or a strange quark star, and thus Ho & Lai 2000; Dai et al. 2016; Passamonti et al. 2013), so that our analysis in Section 2 is valid. GWs from them are much weaker than those from the radial os- cillations for t . 1s, as shown by comparing hc and hp. Observationally, a GW detector would first discover GWs 6. Summary 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 We have proposed a model to explore GWs from post-merger the detector subsequently discovers GWs from the post-merger radially oscillating, rapidly rotating pulsars after binary neutron central object and if such GWs are due to radial oscillations of star mergers, and found that rotation-induced gravitational radi- Dai: Gravitational waves from newborn millisecond pulsars 5 ation is the dominant damping mechanism of the radial oscilla- Cheng, K. S., & Dai, Z. G. 1998, ApJ, 492, 281 tions. Some other conclusions are summarized below. Corsi, A., & M´esz´aros, P. 2009, ApJ, 702, 1171 First, the resulting GWs have a frequency f Cox, J. P. 1980, Theory of Stellar Pulsation (Princeton University Press), Chapter 8 3/2 1/2 ≃ 4.1R6− (M/2.5M ) kHz, which is very close to the Dai, Z. G. 2004, ApJ, 606, 1000 cutoff frequency shown⊙ by numerical simulations, while the Dai, Z. G., & Lu, T. 1996, Z. Phys. A, 355, 415 characteristic strain h is not only independent of the neutron- Dai, Z. 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