PHYSICAL REVIEW D 101, 075028 (2020)
Continuous gravitational waves and magnetic monopole signatures from single neutron stars
† ‡ P. V. S. Pavan Chandra,1,* Mrunal Korwar,2, and Arun M. Thalapillil 1, 1Indian Institute of Science Education and Research, Homi Bhabha Road, Pashan, Pune 411008, India 2Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
(Received 2 December 2019; accepted 1 April 2020; published 15 April 2020)
Future observations of continuous gravitational waves from single neutron stars, apart from their monumental astrophysical significance, could also shed light on fundamental physics and exotic particle states. One such avenue is based on the fact that magnetic fields cause deformations of a neutron star, which results in a magnetic-field-induced quadrupole ellipticity. If the magnetic and rotation axes are different, this quadrupole ellipticity may generate continuous gravitational waves which may last decades, and may be observable in current or future detectors. Light, milli-magnetic monopoles, if they exist, could be pair-produced nonperturbatively in the extreme magnetic fields of neutron stars, such as magnetars. This nonperturbative production furnishes a new, direct dissipative mechanism for the neutron star magnetic fields. Through their consequent effect on the magnetic-field-induced quadrupole ellipticity, they may then potentially leave imprints in the early stage continuous gravitational wave emissions. We speculate on this possibility in the present study, by considering some of the relevant physics and taking a very simplified toy model of a magnetar as the prototypical system. Preliminary indications are that new- born millisecond magnetars could be promising candidates to look for such imprints. Deviations from conventional evolution, and comparatively abrupt features in the early stage gravitational waveforms, distinct from other astrophysical contributions, could be distinguishable signatures for these exotic monopole states.
DOI: 10.1103/PhysRevD.101.075028
I. INTRODUCTION instabilities, mountains, oscillations or glitches in the angular velocity (see for instance [13–15] and references Recent observation of gravitational waves (GWs) by the therein). There has been rapid progress in this area, with LIGO-VIRGO collaboration [1,2] has ushered in a new era many recent searches [16–18], and future third-generation of multimessenger astronomy. Apart from its significant GW detectors, such as the Einstein Telescope, expected astrophysical [3–5] and cosmological [6,7] implications, to significantly improve the sensitivity and reach in the gravitational wave astronomy also has the potential to relevant frequency bands [19–22]. All-sky surveys, illuminate many important questions in fundamental phys- looking for continuous gravitational waves, also hold great ics [8–12]. A fast emerging area in this context is the promise, with their ability to detect hitherto unknown endeavor to detect continuous GWs from single neutron sources [23–26]. stars. As opposed to GW signals from binary coalescence, Magnetic fields are known to cause a star to become which are short lived, the continuous gravitational waves oblate or prolate, depending on the field configuration are due to intrinsic deformations or other phenomena of [27,28]. This generates a quadrupole moment and asso- the compact star itself, and may last decades or centuries. ciated quadrupole ellipticity. In cases where the rotation The cause for these continuous GWs may be due to and magnetic axes do not coincide, this opens up the various distinct phenomena—stellar seismic activity, mode possibility of generating continuous gravitational waves [29–31]. As opposed to gravitational waves from binary *[email protected] coalescences, these waveforms will last for much longer † — [email protected] durations days or years. This enables the application of a ‡ [email protected] plethora of signal processing techniques in their analyses and understanding. The LIGO-VIRGO collaboration is Published by the American Physical Society under the terms of already searching earnestly for such signals from pulsars the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to [18]. Future third-generation detectors are expected to the author(s) and the published article’s title, journal citation, increase the reach much further and into the niche fre- and DOI. Funded by SCOAP3. quency ranges of such signals [19].
2470-0010=2020=101(7)=075028(15) 075028-1 Published by the American Physical Society CHANDRA, KORWAR, and THALAPILLIL PHYS. REV. D 101, 075028 (2020)
Magnetic monopoles have so far not been observed in II. GRAVITATIONAL WAVES FROM SINGLE nature. They are however a very generic prediction of many NEUTRON STARS quantum field theories [32,33] and may be awaiting A. Continuous gravitational waves discovery. Current bounds on magnetic monopoles come from colliders [34–37], terrestrial and balloon observations Let us briefly review the standard theory behind the [38–40], considerations of galactic magnetic field attenu- generation of continuous GWs [66,67]. Isolated neutron ation [41–43], searches in bulk matter [44,45], and limits stars may emit GWs through various processes [15].A on monopole-catalyzed proton decay in compact stars neutron star may sustain a deformation in some cases, and [46–48]. Very interesting limits have also been placed if not axisymmetric with respect to its rotation axis, then on heavy magnetic monopoles by considering their non- emit GWs. Such sustained distortions, due to the elasticity – perturbative production in heavy ion collisions and in the of the neutron star crust [68 71], are generically termed extreme magnetic fields of neutron stars [49]. neutron star mountains. Neutron star mountains may be We are specifically interested in the case of milli-magnetic caused by thermal gradients [68,72] or magnetic fields – monopoles (MMM), with masses below Oð1 eVÞ.Theyare [29 31,73]. We will be interested in the latter, in the context monopoles with fractional effective magnetic charges, and of MMMs, and will elaborate on this further in Sec. II B. which appear in many Standard Model (SM) extensions, In the transverse traceless gauge and an asymptotically especially those involving kinetic mixing [50] with a gauge- Cartesian and mass centered coordinate system (S) [66,67], singlet dark sector. There are previous works that have the leading contribution to the gravitational wave amplitude considered milli-magnetic monopoles [51–54], in various is given by [74,75] contexts. Recently, it was also demonstrated that using 1 2 energetic arguments from a magnetar, one may place very TT ˆ G ̈ r h ¼ Λ ; ðnˆÞ Q t − : ð1Þ stringent, nontrivial bounds on the magnetic charge of such ij r ij kl c4 kl c light MMMs [54]. Similar bounds have also been placed on ˆ ˆ ˆ ˆ ˆ light milli-electrically charged particles [55],forwhichthe Here, for propagation direction n and PijðnÞ¼δij − ninj, relevant pair-production and astrophysical considerations are ˆ one defines the transverse projection operator as Λ ; ¼ very different from MMMs. ij kl Pˆ Pˆ − 1 Pˆ Pˆ . Q is the mass quadrupole moment of the If MMMs exist, they may be nonperturbatively pair ik jl 2 ij kl produced [56,57], via Schwinger pair production, in the object. extreme magnetic fields of a neutron star, such as a magnetar Pulsars and magnetars are rotating neutron stars. If they [58,59]. This causes a decay of the magnetic field hitherto are endowed with a quadrupole moment, there is the different from conventional mechanisms operational in a possibility of generating continuous GWs. The case of neutron star. The modified magnetic field evolution in turn interest to us is where the deformations are such that there is a privileged direction—as in cases of a magnetic-field- may affect the time evolution of the quadrupole ellipticity, ’ assuming the concerned neutron star crustal strains are induced deformation (see Sec. II B). Here, the star s below the breaking limit [60,61]. This opens up an avenue magnetic moment furnishes a privileged direction, as for probing these exotic states by their imprints on the illustrated in Fig. 1. We also neglect any precession. Such deformations are usually parametrized either by a gravitational waves emitted. A time evolution of the mag- ε ¼ð − Þ netic-field-induced quadrupole ellipticity, and its impact on surface ellipticity S Requator Rpolar =Rpolar [27] or by – gravitational wave emissions, has been considered previ- a quadrupole ellipticity, defined as [29 31] ously, in other contexts [62–65]. We would like to explore if Q MMMs could potentially leave markers in the gravitational ε ¼ − : ð2Þ waveforms, from single neutron stars, that are distinguish- Q I able from common astrophysical features. In Sec. II we briefly review the relevant, well-known Here, I is the mean moment of inertia about the rotation theoretical underpinnings behind the generation of con- axis, defined in terms of angular momentum J as I ¼ J=Ω. ε ε tinuous gravitational waves, from single neutron stars, and S and Q quantify slightly different physics, geometrical outline how magnetic fields may generically lead to mass and bulk distortions respectively, and coincide only for a quadrupole moments. In Sec. III we then briefly review star with a constant-density equation of state [30]. ε ’ how MMMs may be incorporated in SM extensions, Q, which quantifies the star s bulk deformation, is the ε involving kinetic mixing, and also the relevant theoretical most relevant quantity in our case. Contributions to Q, background on Schwinger pair production of MMMs. With purely due to stellar rotations, will not contribute to the foundations laid, in Sec. IV we then present our continuous GWs. For the case of magnetic deformations, analyses and main results. We summarize and conclude with the privileged direction for the deformations making in Sec. V. There, we also highlight some of the short- two of the mass quadrupole moment eigenvalues equal, we ε˜ comings of the study, along with a few future directions. may write the relevant quadrupole ellipticity Q as [29]
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6 Ω2 G ˜ NS h0 ¼ − Q33 : ð5Þ c4 r þ and × denote the two polarizations. r is the distance to ¼ − r θ the source and the retarded time is defined as tr t c. is the line-of-sight angle to the observer, measured from the rotation axis. Through Eq. (3), note that Eq. (4) indeed has ε˜ a dependence on Q. From above, we see that for a general Ω 2Ω wobble angle, GWs may be emitted at NS or NS frequencies. Equation (4) is valid under the assumption that the magnetic field and angular velocity do not change significantly during a single period of the neutron star’s rotation. This “slow-roll” assumption is generally true for most neutron stars and will specifically be valid for the cases we study. The GW amplitude (h0) may be directly related to the strain (ΔL=L) of the GW detector arms. The reach in h0, for Advanced LIGO1 and the proposed Einstein telescope,2 are around 10−24–10−26 and 10−26–10−27 respectively [13,15,18,20], in the 10–100 Hz frequency range of FIG. 1. A representation of a neutron star, with its rotation and interest. This is assuming a year of phase-coherent obser- magnetic field axes misaligned with respect to each other. The vations and signal integration times [13,15]. There have quadrupole deformation due to the magnetic field [27,28] is been many pioneering searches already for continuous exaggerated for clarity. The presence of a quadrupole ellipticity, GWs [16–18], and future third-generation GW detectors with respect to the rotation axis, leads to the generation of are expected to significantly improve the sensitivities in the continuous gravitational waves. niche frequency bands [19–22]. Equation (4) may now be used in detail, to understand 3 Q˜ how the magnetic-field-induced deformations affect con- ε˜ ¼ − 33 ð Þ Q : 3 tinuous GWs, and how specifically modifications induced 2 I3 by the production of MMMs will impact it. As we will 2Ω remark later, we will specifically concentrate on the NS Here, Q˜ is the mass quadrupole moment due to the frequency mode, without much loss of generality, for magnetic field, in a frame of reference (S˜) where it is making our estimates. This choice will help us express diagonal. I3 is the principal moment of inertia about the the GW amplitude h0 almost solely in terms of observable rotation axis. The S and S˜ coordinate system quantities are parameters, like the neutron star time period and spin- related by Q ¼ RQ˜ RT, where R is an appropriate rotation down rate. matrix. Consider now a neutron star, rotating with an angular B. Magnetic-field-induced quadrupole moments Ω speed NS, whose rotational and magnetic field axes are In this subsection, let us now briefly review the rudi- misaligned by a wobble angle α. Then, from Eq. (1),we mentary ideas behind magnetic-field-induced stellar defor- may derive the leading GW waveform to be [29] mations [27,28]. It has long been known that a magnetic field threading a star may induce a mass quadrupole moment 1 [27,28]. The underlying physics behind this phenomena may ¼ α α θ θ Ω be understood based on simple energetic arguments. hþ h0 sin 2 cos sin cos cos NStr Consider a special case for the potential deformation, in a 1 þ 2θ simple model for the neutron star—a perfect sphere, of − α cos 2Ω sin 2 cos NStr ; radius R, comprising an incompressible fluid [27]. Assume 1 that there is a uniform magnetic field in the interior and a ¼ α α θ Ω dipolar magnetic field in the exterior. The respective field h× h0 sin 2 cos sin sin NStr profiles are − α θ 2Ω ð Þ sin cos sin NStr : 4 1https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?.submit= Identifier&docid=T1800044&version=5 2https://workarea.et-gw.eu/et/WG4-Astrophysics/base-sensitivity/ In the above expressions, we have defined et_b_spectrum.png/view
075028-3 CHANDRA, KORWAR, and THALAPILLIL PHYS. REV. D 101, 075028 (2020)
2 2 4 B ¼ B0 cos θ Bθ ¼ −B0 sin θ ðr
Note that in contrast to δE , this is second order in ζ and is D ¼ 36π5ð12−π2Þ B in which case 5ðπ2−6Þ3 . thus always positive. The total change in energy is obtained Considering the values of the deformation parameter, in by summing the magnetic and gravitational energy con- these examples, it seems D ∼ ½10−1; 102 . These ranges for tributions. This gives D are also believed to be typical for more realistic equations of state and field configurations [29,73], and 3 ζ 2 GM2 9 δE ¼ þ ζB2R2: ð10Þ we will use them for making our estimates. 25 20 0 ε˜ R R There are a few observational upper bounds on Q, for neutron stars in their early stages. X-ray light curves from ζ ≪ ε˜ Note from above that, for R, the sign of the net short gamma ray bursts have been used to constrain Q of change in energy will be determined directly by the sign postmerger stable neutron stars, giving mean bounds in the of ζ. range [15,76] To obtain the most stable configuration, we need to δ minimize E, and if it comes out to be negative, would ε˜Obs: GRB ≲ 10−2 − 10−1: ð15Þ suggest an energetically more favorable configuration [27]. Q Minimization gives For pulsars in their later stages, there are constraints from continuous GW searches by the LIGO-VIRGO collabora- ζ¯ 15 2 4 9 2 ¼ − B0R ¼ − B0 ð Þ tion, giving fiducial ellipticity bounds in the range 2 : 11 R 8 GM 2 B ½10−2; 10−8 [16–18]. Theoretical models suggest bounds
075028-4 CONTINUOUS GRAVITATIONAL WAVES AND … PHYS. REV. D 101, 075028 (2020) on fiducial ellipticities of compact stars in the range Fμν → F˜ μν; F˜ μν → −Fμν 10−2–10−7 [31,68–71]; depending on the stellar mass, eJν → gKν;gKν → −eJν: ð19Þ hadron composition, epoch, equation of state and theoreti- cal approximations used. Interestingly, there is even pos- ε˜ 10−9 α sibly an indication for a lower bound on Q, of about , The addition of the K term introduces magnetic from analyses of millisecond pulsars [77]. We will always monopoles. work with values well below the mean bounds in Eq. (15). The theoretical underpinnings for milli-magnetic In summary, the elastic properties of the neutron star monopoles, in the context of kinetic mixings, were dis- crust [31,68–71], and presence of very strong magnetic cussed in [54], and put on a firmer theoretical foundation fields, may lead generically to the presence of sustained later in [79]. Among the theoretical subtleties, in incorpo- deformations, resulting in a nonzero quadrupole ellipticity. rating magnetic monopoles directly in a quantum field As remarked earlier, there may even be a time evolution of theory, is the fact that it is not possible to write a local, the magnetic-field-induced quadrupole ellipticity in these Lorentz invariant Lagrangian containing both electric and early phases. This is a plausible scenario assuming that the magnetic charges [80–83]. We briefly review the theoretical concerned crustal stresses and strains, due to the magnetic framework [79] for incorporating MMMs, through kinetic pressure, are below the breaking limit [60,61]. An evolving mixing, as a specific example of incorporating MMMs into quadrupole ellipticity has been previously studied, in other beyond Standard Model extensions. This will also help fix GW contexts [62–65], and we would like to explore if the notations. presence of MMMs may leave imprints on this quadrupole One theoretical strategy to incorporate magnetic monop- ellipticity evolution, and consequent GW generation. oles, by Zwanziger [82], contains two gauge potentials Aα ˜ and Aα, with a local Lagrangian, but without any manifest III. MILLI-MAGNETIC MONOPOLES AND Lorentz invariance [82,84]. In this formulation, one of the NONPERTURBATIVE PRODUCTION gauge potentials, Aα, couples locally to the electric current ˜ A. Milli-magnetic monopoles Jα, while the other, Aα, couples to the magnetic current Kα. and theoretical foundations The Lagrangian density takes the form [79,82,84] Magnetic monopoles are yet to be observed in nature. α μ 1 They nevertheless seem to be a very generic prediction of n n βν A A A˜ A˜ νγδ A˜ A L ¼ − η ðF Fμν þ F FμνÞ − ϵμ ðFανF many quantum field theories and model frameworks (see 2n2 αβ αβ 2 γδ for instance [78], and related references). 4π A A˜ μ ˜ μ In conventional Maxwellian electrodynamics, the homo- − FανFγδÞ − eJμA − KμA : ð20Þ ⃗ e geneous equation ∇ · B⃗¼ 0, or equivalently the Bianchi identity of the field tensor Fαβ, presupposes the nonexist- A A˜ ˜ ˜ ence of magnetic monopoles. In this framework, the Here, Fαβ ¼ ∂αAβ − ∂βAα and Fαβ ¼ ∂αAβ − ∂βAα are the manifestly covariant equations in vacuum take the form respective field tensors. nα is an arbitrary four vector, corresponding to the direction of the Dirac string in certain ∂ μν ¼ 0 ∂ ˜ μν ¼ 0 ð Þ μF ; μF : 16 gauge choices. The presence of nα, projects out two, on- shell photon polarizations, breaking manifest Lorentz ˜ μν 1 μνρσ Here, F ¼ 2 ϵ Fρσ is the dual field tensor, and the invariance [79,82,84]. It has been argued that physical μν μ ν ν μ Bianchi identity implies F ¼ ∂ A − ∂ A .Asiswell observables of the theory are independent of nα [83]. The known, the vacuum equations are symmetric under the above Lagrangian density correctly gives the modified duality transformation Maxwell’s equations in Eq. (18), with the definition
μν ˜ μν ˜ μν μν F → F ; F → −F : ð17Þ α n ˜ ¼ ð A − A − ε β γ A Þ ð Þ Fμν 2 nμFαν nνFαμ μνα n Fγβ : 21 Once we introduce an electric source, say Jα, this n symmetry is lost. To consider restoration of the symmetry, we may speculate the addition of an analogous magnetic Let us now understand how MMMs may specifically be source term Kα. The equations then take the form included, in this framework, in the context of kinetic mixing [50]. For this, consider the Lagrangian density μν ν ˜ μν ν ∂μF ¼ −eJ ; ∂μF ¼ −gK ; ð18Þ [79] incorporating kinetic mixing with a dark sector (whose low-energy states are all Standard Model gauge singlets; which are clearly symmetric under the transformations labeled by subscript “D”)
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α μ 1 n n βν A A A˜ A˜ νγδ A˜ A A A˜ L ⊃− η ðF Fμν þ F FμνÞ − ϵμ ðFανF − FανF Þ MMM 2n2 αβ αβ 2 γδ γδ 4π α μ 1 μ ˜ μ n n βν A A A˜ A˜ νγδ A˜ A A A˜ − eJμA − KμA − η ðF F þ F F Þ − ϵμ ðF F − F F Þ e 2n2 Dαβ Dμν Dαβ Dμν 2 Dαν Dγδ Dαν Dγδ 2 4π α μ mDA μ μ ˜ μ n n βν A A A˜ A˜ − A μA − e J μA − K μA þ χ η ðF Fμν − F FμνÞ: ð Þ 2 D D D D D D D 2 Dαβ Dαβ 22 eD n
˜ FA and FA are the field tensors corresponding to the dark g D D ˜ ξ ≡ χ D : ð25Þ gauge potentials AD and AD. JD and KD are the dark electric g and magnetic currents, with eD being the dark electric ≡ 4π charge. e and eD are in general independent parameters of Here, we have defined gD =eD. With respect to our ξ ≡ χ the model. Without loss of generality, we take the nα four- photon, MMMs therefore have magnetic charges g gD. vector to be the same in both the sectors; this can always be We will express all analyses and limits with respect to ξ achieved with appropriate gauge transformations. The two henceforth. sectors are connected by kinetic mixing, via the last term in μν Eq. (22). This term is equivalent to χ=2FμνF , from the D B. Nonperturbative pair production definition in Eq. (21). The mass term for A μ breaks the D of milli-magnetic monopoles SOð2Þ symmetry of the kinetic terms and is uniquely responsible for MMMs [79]. In quantum electrodynamics, when the field strengths are μ very large, one may have nonperturbative production of Considering AD to be massive, after field redefinitions, we get magnetic monopoles that have effective milli- electrically or magneticallly charged particles, through the magnetic charges [54,79], at low energies. Explicitly, Schwinger pair-production mechanism [56,57,85–87]. This consider the field redefinitions is a distinct phenomena compared to, for instance, pertur- bative electron-positron pair production (γ þ γ → eþ þ e−). → þ χ ˜ → ˜ Aμ Aμ ADμ; Aμ Aμ For field strengths comparable to the particle masses, the ˜ ˜ ˜ nonperturbative rates may be exponentially enhanced. A μ → A μ; A μ → A μ − χAμ: ð23Þ D D D D For zero temperature and homogeneous magnetic fields, Note that the above field transformations ensure that the as compared to the Compton wavelength and separation of ˜ the particles, the average MMM pair-production rate, per visible-sector gauge potentials (Aμ, Aμ) do not get mass ð1Þ unit volume, is given by [56,57] terms, and hence U EM remains unbroken. After these field redefinitions, making the kinetic terms canonical, the ξ2g2B2 πm2 relevant interaction terms become Γ0 ¼ exp − : ð26Þ 8π3 ξgB 4π μ μ μ ˜ μ L ⊃ eJμA þ eχJμA þ e J μA þ KμA int: D D D D e The zero temperature rate assuming a magnetic field of 4π 4πχ 1016 G is shown in Fig. 2. This is the first term in the ˜ μ ˜ μ þ K μA − K μA : ð24Þ e D D e D vacuum decay rate [56,57,88]. Recently, this computation D D was also extended to strong coupling and finite temper- After making the kinetic terms canonical, one now has an atures [89]. 4πχ ˜ μ effective interaction of the form =eDKDμA . This makes We are interested in light, milli-magnetically charged the dark-sector magnetic monopoles milli-magnetically monopoles of mass m ≪ Oð1 eVÞ, with effective magnetic charged under the visible photon, with an interaction charges ξg ≪ 1, as in Eq. (24). We assume that 4πχ χ g ≲ g ≡ 4π=e, and that any higher order instanton cor- strength of =eD. in general is an arbitrary, irrational D number. This is the origin of the fractional magnetic charge, rections to the MMM pair-production rates [56,57,88,89] and of MMMs. Naively, χ being an irrational number may may be neglected, to good approximation. Also note that seem to violate the Dirac charge quantization condition at for the MMM mass ranges we consider, the Compton λmax ≲ 1 low energies. The emergence of milli-magnetically charged wavelengths ( Compt: m) are such that local magnetic particles, through kinetic mixing, is nevertheless still con- field inhomogeneities in the neutron star may be neglected, sistent with a global Dirac quantization condition [51,79]. to leading order. Moving forward, let us henceforth define all MMM Based on theoretical models and measurements, currently charges with respect to the visible sector g ≡ 4π=e. observed neutron stars are believed to have mean surface Towards this end, define the MMM charge parameter ξ as temperatures of the order of 106 K. It is believed that in the
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ξgB T ðm; ξ;BÞ ≡ : ð27Þ C 2m
Below this critical temperature, the thermal enhancements turn off and the rate subsequently follows the zero temper- ature rate, given by Eq. (26). The critical temperature estimates for our regions of interest are illustrated in Fig. 3. The thermal rate, at a finite temperature T ≡ β−1, may be approximated as [99,101]
Γ ðm; ξ;B;TÞ T X∞ ð−1Þpþ1ξ2 2 2 π 2 ≃ g B − p m 8π3 2 exp ξ p¼1 p gB ∞ X Xnmax ðξgBÞ2 þ ΘðT − T Þ 2ð−1Þp C ð2πÞ3=2ð βÞ1=2ϑ2 p¼0 n¼1 nm " 2 −1 2 FIG. 2. The pair-production rates per unit volume nβξgB 4 m ½Γ 1 −3 −1 × 1 − exp − 2πðp þ 1Þ (log10 0= m s ), for milli-magnetic monopoles at zero 2m 2ξgB temperature, are shown. The magnetic field has been taken to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1016 be G. The zero temperature rates bracket the true rates that nT nm n2T2 may be operational in systems with a finite temperature. − 2 arcsin C þ 1 − C ; ð28Þ T 2T T2 early stages of their formation, the mean temperatures may have been even higher (∼1011 K). In the standard cooling following the notion of an electromagnetic dual to Schwinger scenario for neutron stars, it is presumed that a neutron pair production by an electric field [56,89,99,101]. Θð Þ ≡ b2 star when formed has internal temperatures approaching Here, x is the Heaviside step function, nmax m= 11 ðξ βÞc ¼ b c ϑ ¼ 2πð þ 1Þ − 2 ðnTCÞ 10 K or more, and subsequently cools down by various gB T=TC , and p arcsin T . processes—neutrino emissions (through the Urca and bxc denotes the integer less than or equal to x. This explicit modified Urca processes), neutrino pair bremsstrahlung, analytic expression derived in the worldline instanton thermal photon emissions and so on (see, for instance, framework, utilizing a saddle-point approximation, is valid [90,91] and references therein). The rate of cooling differs for the semiclassical parameter ξgB=m2 ≲ 2π [56,89,99, widely during the many stages, with timescales varying from 101,103]. Note that the enhancement is present only when seconds to thousands of years. The neutron star mean T>TC, as already mentioned, and changes abruptly below temperature is thought to evolve from around 1011 to it. In fact, Eq. (28) seems to suggest that the rate also changes 4 ¼ 10 K over a few million years [90,91]. abruptly at all integer multiples of TC, owing to nmax b c Thus, a more relevant quantification of the MMM T=TC . We will utilize the above expression, in regions production rate, at least in the initial phases of the neutron satisfying ξgB=m2 ≲ 2π, to estimate Schwinger pair- star’s life, should try to incorporate the effects of this production rates at finite temperatures. finite temperature. As mentioned earlier, there has been Note that at a characteristic worldline sphaleron temper- tremendous progress recently in computing Schwinger ature, much higher than TC, the pair production transitions pair-production rates at finite temperature, both for electri- from a quantum tunneling phenomena to a classical, thermal cally charged as well as for strongly coupled magnetic process, described by a worldline sphaleron [100]. The monopoles [49,89,92–102]. There is currently some dis- characteristic worldline sphaleron temperature [100], where agreement on the exact functional form of the worldline this transition occurs, is greater than ∼1011 K for the instanton (see for instance discussions in [49,89,98–102]). parameter space of interest to us. Since the neutron star is Nevertheless, there seem to be a few generic predictions— believed to cool to around 1011 K within just a few seconds an exponential enhancement in the pair-production rate of its formation, we are mostly outside the sphaleron regime. relative to zero temperature rates, and a critical temper- For the MMM and dark photon mass ranges we will ature below which the thermal enhancements switch off consider, the MMM Compton wavelength and string sep- [49,89,94,95,98,99,101]. aration between monopole and antimonopole [54,79,104] The critical temperature (TC) is a function of the are also such that the magnetic field spatial inhomogeneities magnetic field, monopole mass and magnetic charge may be neglected, to good approximation. The temporal [49,89,94,95,98,99,101] variation of the magnetic field is also very gradual, and its
075028-7 CHANDRA, KORWAR, and THALAPILLIL PHYS. REV. D 101, 075028 (2020) effects may similarly be neglected while computing rates, to IV. EFFECTS OF MILLI-MAGNETIC leading order. MONOPOLES ON GRAVITATIONAL WAVES The additional magnetic field dissipation, due to With the basic concepts in place from the previous Schwinger pair production of MMMs, may cause a sections, we may now undertake a study of what potential deviation in the time evolution of the gravitational wave affects MMMs may have on continuous gravitational amplitude, and frequency, relative to the conventional case. waves from single neutron stars. The fact that the nonperturbative pair-production rate The MMMs are generally confined objects with a string reverts to the zero temperature rate, below a characteristic connecting the monopole and antimonopole [54,79,104]. temperature T [49,89,94,95,98,99,101], also opens up an C They behave like magnetically charged objects only intriguing possibility. As the neutron star cools down beyond a particular distance Oð1=m Þ. This suggests a during its lifetime, if milli-magnetic monopoles exist, there DA characteristic lower value for the dark photon mass m . could potentially be an abrupt change in the monopole DA There is also an upper bound to m that must be production rate, in the vicinity of T , that relatively DA C considered. The external magnetic field will accelerate brusquely affects the gravitational wave amplitude and the MMMs out of the magnetar, as long as the string frequency subsequent to it. As emphasized before, T itself C tension between the pair produced MMMs [Oðm2 Þ]is is a function of the magnetic field, monopole mass and DA smaller than the external electromagnetic force. The gravi- magnetic charge ξ. Note that as the MMMs we are tational forces on the MMMs, due to the neutron star, are considering have very small masses and tiny magnetic many orders of magnitude smaller than the Lorentz forces, charges, we do not expect them to drastically affect the and hence do not furnish any further bounds. These ordinary thermal evolution or dynamic processes in the requirements altogether translate finally to [54] neutron star in a very significant way. These comparatively abrupt features in the waveform 1 pffiffiffiffiffiffiffiffi would be a universal signature, potentially visible across ≲ ≲ ξ ð Þ mDA gB: 29 different magnetar systems, in their early phase continuous RNS gravitational wave emissions. They should also be distinct from signals originating due to typical astrophysical phe- For the parameter space of interest, the upper bound gives ≲ 108 −1 nomena, and hence potentially distinguishable. As may be mDA km , which may be trivially incorporated. deduced from Fig. 3, for a field of 1016 G, the critical Neutron stars have typical radii ∼10 km and we set the temperature may be as high as 108 K, in the viable ðm; ξÞ lower limit for the dark photon mass by it. This will also parameter space of interest. make robust our assumption of magnetic field homogeneity, relative to the particle Compton wavelength and separation. We will work assuming the above two bounds for mDA. Lower dark photon masses and corresponding modifications may be readily incorporated phenomenologically, by assum- ing an exponential suppression [54] of the external field, as felt by the monopole and antimonopole. The subsequent history of the MMMs, after they are pair produced and expelled by the magnetic field, is not important, as they do not return energy back into the magnetic fields. As mentioned earlier, due to the tiny MMM mass and charge, any direct imposition on the thermal or dynamical evolution of the neutron star should also be very marginal, after production. This is in sharp contrast to heavy magnetic monopoles, if they exist, that may be captured and trapped by neutron stars, and which may impact the internal neutron star processes and dynam- ics more drastically. For instance, these heavy magnetic monopoles may efficiently catalyze nucleon decays in the neutron star [46–48]. It is also distinct from interesting scenarios where very heavy dark matter states could be ½ 1 captured by neutron stars, sometimes through multiple FIG. 3. Plot of log10 TC= K is shown, for a fixed magnetic field of 1016 G. Certain regions are irrelevant, due to the scatterings, heating them up kinetically or through sub- exponential suppression of Schwinger pair-production rates. sequent annihilations [105,106]. In such cases, measuring Mean energetic arguments from magnetars [54] also render the temperatures of very old neutron stars could lead to very regions with ξ ≳ 10−17 (gray band) unviable, for m ≲ 1 eV. interesting constraints [105,106].
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It was pointed out recently, in [54], that by considering an average magnetar field of 1015 G, monopole antimono- pole pair-production rates bracketed by the zero temper- ature rate, and an assumed magnetar active lifetime of 104 yrs, one may place strong bounds on viable MMMs. For magnetars with magnetic fields in the range 1015–1016 G, and for various dark photon masses, such energetic considerations give limit estimates of
ξ ≲ 10−17; ð30Þ for m ≲ Oð1 eVÞ. Following [54], we will explicitly compute the limit on ξ and impose it, at each MMM mass of interest, before utilizing that point to study the evolution of the gravitational wave amplitude. Let us now turn to the GW waveforms that could be expected. To be concrete, let us focus specifically on the 2Ω GW mode with frequency NS. Assuming the dominance of electromagnetic dipole radiation, from Eq. (4), the 2Ω amplitude corresponding to the NS frequency mode may be expressed as
2 _ 2 2Ω þ 8 R P 1 þ cos θ h NS; ¼ D NS ; 0 5 cr P 2 2 _ 2Ω 8 R P h NS;× ¼ D NS cos θ: ð31Þ 0 5 cr P
Note that when expressed in terms of the observables P_ and 2Ω P in this fashion, the amplitude at frequency NS is independent of the moment of inertia and the unknown wobble angle α. This is an advantage to considering this specific frequency mode, as we had alluded to earlier. There θ is a dependence on the line-of-sight angle , that just gives FIG. 4. Estimates for the magnetic-field-induced GW ampli- Oð1Þ an factor, and may be ignored for our order of tudes, from a few representative pulsar (top) and magnetar magnitude estimates. The dominance of electromagnetic (bottom) candidates. The relevant parameter values were taken dipole radiation may be explicitly checked for reasonable from the ATNF pulsar [107] and McGill magnetar [108] data- ε˜ −1 2 values of Q, and we shall comment further on this later. bases. D is varied in the range ½10 ; 10 . From Eq. (31), the order of magnitude estimate for the GW amplitude gives intuit, from Eq. (32), that one must search for candidate compact stars with aforementioned characteristics. 2 _ 2Ω R kpc s P h NS ≃ 10−31D NS : ð32Þ This may be further sharpened by estimating the typical 0 10 km r P 10−11 GW amplitudes one may expect from observed pulsars and magnetars, due to their assumed magnetic-field-induced As we had remarked earlier, in Sec. II A, the sensitivity in quadrupole ellipticities, for reasonable ranges of the defor- strain (h0) for advanced LIGO and the proposed Einstein mation parameter D. These estimates are shown in Fig. 4, telescope, are around 10−24–10−26 and 10−26–10−27 respec- for a few representative pulsar and magnetar candidates. 3 tively [13,15,18,20], in the 10–100 Hz frequency range of The parameter values were taken from the ATNF pulsar 4 relevance to these continuous GWs. This is assuming one- [107] and McGill magnetar [108] catalogs. Estimates in year signal integration times [13,15]. We note therefore Fig. 4 suggest that magnetars with large time periods from above that the amplitude is typically very small, (∼10 s) and conventional radio pulsars with relatively small except when the compact object is spinning rapidly, _ undergoing rapid braking with large P or has large 3https://www.atnf.csiro.au/research/pulsar/psrcat/ magnetic-field-induced deformations. One may therefore 4http://www.physics.mcgill.ca/∼pulsar/magnetar/main.html
075028-9 CHANDRA, KORWAR, and THALAPILLIL PHYS. REV. D 101, 075028 (2020) magnetic fields (∼1011 G), or equivalently small P_ ,may above values have been found to capture relevant effects not be the most promising candidates to look for persistent [119]. A toy model of the magnetic field evolution, as GWs; or for that matter MMM imprints in them. encapsulated by Eq. (33), has also been seen to semi- Based on these broad inspections, perhaps the most quantitaively reproduce [119] essential results from more promising candidates are a class of newly born magnetars, detailed magnetothermal simulations [119–121]. A similar in their early stages of evolution—the so-called millisecond evolution equation was also considered recently in [49],to magnetars [109–116]. Millisecond magnetars are new-born set interesting limits on strongly coupled, heavy magnetic neutron stars with very high magnetic fields and very small monopoles. time periods, and have already been speculated to be The last term in Eq. (33) is due to the Schwinger pair promising sources for continuous GWs [13,15,112]. production of MMMs, and is derived from energy con- They have also garnered much interest recently, in the servation arguments. Specifically, it is obtained by equating context of fast radio bursts [116,117]. The other reason for the loss of energy from the electromagnetic field, to the optimism, while considering these candidates, is that the energy needed for Schwinger pair production and to the internal magnetic fields and temperatures are presumed to work done in accelerating the monopole antimonopole be much higher, during the early stages of the magnetar’s pairs outward. Vm is the active volume over which MMMs formation; relative to their mean values taken over the are being nonperturbatively pair produced, and is taken to entire magnetar lifetime. This opens up the possibility that be the volume of the neutron star. l is the mean distance detectable signatures may still be present in the early over which MMMs are being accelerated by the magnetic stages. The mean temperature of the neutron star is also field, after production, and is equated to the diameter of the varying very rapidly in the early epochs, and as we shall neutron star. The Schwinger pair production of the MMMs discuss later, this increases the possibility of MMM causes a nonperturbative decay of the magnetic flux. This is induced abrupt features in the GW waveforms. We there- a potentially new source of flux decay in neutron stars, fore explore imprints on gravitational waves from milli- different from classical processes. Energy is being second magnetars, induced by MMMs; with magnetic expended from the magnetic field during pair production charges below the bound set by mean energetic limits, and during their expulsion. as in Eq. (30). Equation (33) must be solved in tandem with the neutron Let us therefore look at the effects of MMM non- star spin-down equation, perturbative pair production in a very simplified toy model, 4 for a newly born millisecond magnetar. Consider specifi- 5 R 64 Ω_ ≃− NS B2 ðtÞΩ3 ðtÞ − GM R2 ε˜2 ðtÞΩ5 ðtÞ: NS 12 NS NS 25 NS NS Q NS cally the magnetic field evolution in this toy model, MNS assuming an external dipolar and uniform internal magnetic ð34Þ field, which attempts to capture the salient features. The simplified evolution equation [49,59,118–121] may be In this spin-down equation, we have assumed that the written as π magnetic axis is orthogonal to the rotation axis, i.e., α ¼ 2 ð Þ ð Þ ð Þ 2 ð Þ [115]. Note from Eq. (4) that this choice would also cause dBNS t BNS t − τ BNS t B t ≃ t= dyn: − − NS continuous gravitational emissions solely at 2Ω frequen- τ e τ ð0Þτ NS dt dyn: ohm BNS hall cies. In the above expression, the neutron star has been 2ξglVm idealized to an almost spherical object, with moment of − Γ ðm; ξ;B ðtÞ;TðtÞÞ: ð33Þ 2 2 3 T NS inertia ∼ M R . The first term in Eq. (34) is due to RNS 5 NS NS electromagnetic dipole radiation, and the second term The various terms try to crudely encapsulate the char- incorporates the gravitational quadrupole radiation. The acteristic timescales of the various relevant processes that latter term incorporates braking due to GW emissions and is ε˜2 ð Þ are operational. proportional to Q t . The GW emission contribution is ε˜ The first term is a dynamo term [59] that is believed to be small compared to the dipole term, for all Q values of operational for the first few seconds of a neutron star’s interest to us, as may be explicitly verified. It hence birth, after which it winds down. It amplifies and regen- validates the assumption in Eq. (31). We neglect effects erates the magnetic field in the magnetar. The second and due to precession, in the time evolution. third terms are the Ohmic and Hall drift terms that When there is nonperturbative pair production of contribute conventionally to the decay of the magnetic MMMs, the full gravitational waveform is plausibly fields in a neutron star. Following standard literature, we affected, relative to the conventional case, in both ampli- take the dynamo, Ohmic and Hall drift time constants as tude and frequency. As seen from Eqs. (3), (4), (5) and (12), τ ¼ 10 τ ¼ 106 τ ¼ 104 dyn: s, ohm yrs and hall yrs [118,119] the amplitude of the waveform is modified directly due to respectively. The respective time constants are in reality the refinement of the quadrupole ellipticity. It is also Ω ð Þ nontrivial functions of temperature and density, but the affected indirectly through the adjustments in NS t ,
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ε˜ 10−4 induced via the modified magnetic field evolution of state. This gives an initial Q of about . This magnitude Eq. (33) and by the GW emission term in Eq. (34). The seems to be consistent with typical expectations, for latter effects also modify the frequency of the emitted millisecond magnetars [117]. The distance to the source 2Ω ð Þ −19 gravitational waveform NS t : is taken as 1 kpc. For a magnetic charge of ξ ¼ 10 , the MMM masses have been taken to be 15, 20, and 25 meV. ð Þ ∝ ε˜ ð ÞΩ ð Þ2 h0 t Q t NS t ; The magnetic charge adopted for these masses satisfies the Ω_ ð Þ ∝ 2 ð Þ ε˜2 ð Þ ð Þ limit from mean energetic arguments, as derived in [54]. NS t BNS t ; Q t : 35 The parameter space points also satisfy ξgB=m2 ≲ 2π, ε˜ ð Þ ∝ ð Þ2 making Eq. (28) valid, and hence directly usable in Remembering that Q t BNS t , ultimately all the altered characteristics are a consequence of the MMM Eq. (33). The dark photon mass has been taken as ¼ 103 −1 modified magnetic field evolution, condensed in the sim- mDA m , which is consistent with current limits plified Eq. (33). Thus, a revised modulation in the fre- (see for instance discussions in [123,124], and references quency and amplitude envelope of the GW waveform therein). should be a consequence of MMM production in general. Using Eq. (32), the results of these numerical evolutions On a related note, observe from Eq. (33) that during the are displayed in Fig. 5. As is clearly seen from these curves, first many seconds after the millisecond magnetar’s birth the amplitudes deviate drastically from the conventional ’ (say around time t0) one may in some instances have a case, in the first few decades of the millisecond magnetar s ε˜ D _ ð Þ ∼ 0 birth. If Q, or equivalently , is even smaller, the main steady state situation (BNS t0 ). This may be prompted by a near cancellation of the positive dynamo and negative difference will be that the GW amplitudes will fall below MMM contributions: their detectability much earlier in the epoch. As already mentioned, assuming a higher mean temperature would
B ðt0Þ 2ξglV cause more conspicuous deviations with respect to conven- NS −t0=τ m e dyn: ∼ Γ ðm; ξ;B ðt0Þ;Tðt0ÞÞ: ð36Þ τ R3 T NS tional evolution. For the MMM masses and charges dyn: NS adopted in Fig. 5, the neutron star temperature, for the This quasi-steady-state, if achieved, should also reflect in time period displayed, is always higher than the respective ð ξ ð ÞÞ the persistent GW emissions during these brief intervals; critical temperatures TC m; ;B t . Thus, for these param- before the dynamo shuts off after Oð10 sÞ. The timescales eter points, one does not expect, nor see, any relatively for the Ohmic and Hall-drift processes are much longer, and should not play a significant role at these very early times. The possibility of such a steady state was also effectively leveraged in [49], to place very interesting lower bounds on the mass of heavy magnetic monopoles. To explore further, we numerically solve Eqs. (33) and (34), with a starting point taken as 10 yrs after the millisecond magnetar formation [109–114,116]; in a binary neutron star merger or supernovae explosion. For the 0 ¼ 1016 Ω0 ¼ estimates, initial starting values of BNS G, NS 2π ð30 Þ 0 ¼ 4 5 106 = ms and TNS;pole . × K, as well as temper- ature evolution profiles, are taken following representative values in the literature [113,114,116,120]. The neutron star equatorial temperature is usually much lower than the polar FIG. 5. Evolution of the gravitational amplitude, a decade into temperature [120] and the internal temperatures are the birth of the millisecond magnetar. The MMM charge has been believed to be much higher. Discounting magnetic fields, fixed at 10−19, and the MMM masses have been taken at 15 meV the interior temperature is thought to be related to the (dashed), 20 meV (dot-dashed), and 25 meV (dotted). The surface temperature via an approximate scaling that evolution of the gravitational wave amplitude, when there are ∼ 2 no MMMs, is shown as a solid line. The initial conditions for the roughly goes as TNS;in TNS;surf: [122]. To reduce model polar temperature (4.5 × 106 K), time period (30 ms) and mean assumptions, to the extent possible, we will take the 1016 neutron star polar temperature prediction [120] as a crude magnetic field ( G) were taken from representative values in the literature [116,120]. The distance to the source is assumed to proxy for the mean neutron star temperature. Assumption be 1 kpc. D has been assumed to be 81, corresponding to an of a higher mean temperature would cause a further n ¼ 1 polytropic equation of state. The amplitude must poten- enhancement to the thermal Schwinger pair-production tially be observable in third generation gravitational wave rate, and would only cause more pronounced deviations detectors, like the Einstein telescope, which are expected to from conventional evolution. D is taken to be 81, corre- have a sensitivity of 10−26–10−27, in the 10–100 Hz frequency sponding to the case of an n ¼ 1 polytropic equation of range, assuming integration times of one year [13,15,20].
075028-11 CHANDRA, KORWAR, and THALAPILLIL PHYS. REV. D 101, 075028 (2020) abrupt features in the gravitational wave amplitudes. Note discontinuous character. During these epochs, they should also that the mean energetic arguments [54] for these also fall in the sensitivity ranges of future third generation MMM masses, and corresponding limits on ξ based on gravitational wave detectors. it, are still relevant. The thermal Schwinger pair-production If they exist, these MMM imprints on GWs must be an rates are very prolific in the early epochs, but almost almost universal feature across different newly born milli- completely switch off once the magnetic field value second magnetars. They must have a very unique pattern decreases below the critical field value ∼m2=ξg; this correlated with the temperature and magnetic field evolu- happens after just a few decades. Thus, taken as an average tion, and hence should be potentially distinguishable from over the entire lifetime of the magnetar, the mean energetic many other astrophysical phenomena. At the moment, it is arguments should still furnish meaningful and interesting difficult to quantitatively demonstrate this in a satisfactory limits, while still being consistent with the enhanced rates manner, through an explicit rate computation and evolu- and prominences in the early stages. tion, even in the simplified toy model. This is because, in In general, as emphasized in Sec. III B, one should the potentially interesting ðm; ξÞ regions where such abrupt expect to see comparatively abrupt features in the gravi- features may show up, we have ξgB=m2 ≫ 2π. Therefore, tational wave amplitude and frequency. They would have a in these regions, all the known analytic expressions for distinct pattern, correlated with temperature and magnetic thermal Schwinger pair production break down, and their field evolution. The presence or absence of such abrupt applicability is unclear [49,89,94–102]. patterns, in the GW waveform, would of course depend on the ðm; ξÞ values of the MMMs that may exist in nature. V. SUMMARY AND CONCLUSIONS More specifically, such abrupt patterns may appear if the The search for continuous gravitational waves from mean temperature of the neutron star T ðtÞ falls below the NS neutron stars is well under way [16–18]. Exotic particle MMM critical temperature T ðtÞ at some point in time C states beyond the Standard Model have the potential to (equivalently, it may manifest through some evolution of a leave their imprints on these waveforms. In this work, we temperature gradient, across neutron star layers). After this speculated on the effect of milli-magnetic monopoles on crossover there should be a relatively abrupt change in the persistent gravitational wave signals, sourced by single MMM pair-production rates, and hence a relatively abrupt neutron stars. change in the gravitational wave amplitude and frequency Magnetic fields are known to cause distortions from evolution. Assume one is starting at an initial time t0, with spherical symmetry, in compact astrophysical objects, ð Þ ð Þ ð Þ generating a quadrupole moment [27,28]. If the magnetic TNS t0 >TC t0 : 37 and rotation axes are misaligned, this may produce detect- For a crossover to occur, a necessary criterion that the able gravitational wave signals. Milli-magnetic monopoles monotonically decreasing mean temperature and mean may be copiously pair produced in the extreme magnetic magnetic field profiles should satisfy, during some point fields of neutron stars, such as magnetars; through the subsequent to t0,is Schwinger pair-production mechanism [56,57]. This causes an additional attenuation of the magnetic field, T_ ðtÞ ξg NS ≳ : ð38Þ relative to conventional field decay mechanisms opera- B_ ðtÞ 2m tional in a magnetar. Consequently, through a modification of the quadrupole moment time evolution, this may leave Here, the dot denotes a first time derivative. imprints in the continuous gravitational waves, during the For the gravitational waves to be detectable, such a early stages of a neutron star’s life. A time evolution of the crossing should also occur in the early stages of the neutron star quadrupole moment has been considered ’ millisecond magnetar s birth. Depending on the allowed previously in other contexts [62–65]. We found that the ε˜ values of Q, this may mean a time frame of seconds to most promising candidate compact objects are a class of decades, following birth. An MMM imprint detection is newly born magnetars, the so-called millisecond magnetars also more plausible during the early stages, since the [109–114,116]. In addition to deviations from conventional internal magnetic fields are at their highest (implying large evolution, an imprint may potentially be present, as com- pair-production rates), and the temperatures are also vary- paratively discontinuous features, in the gravitational wave- ing rapidly [implying Eq. (38) is more prone to be form amplitude and frequency, in the early phases of a satisfied]. As seen from Fig. 3, in the viable ξ range, for millisecond magnetar’s life. Since the temperatures are −5 MMM masses m ≲ 10 , the critical temperatures can vary rapidly evolving in the early stages, and the internal 5 8 from 10 –10 K. As the neutron star is expected to cool magnetic fields during these periods are also at their from 1011 to 106 K, over its initial phase of a few hundred highest, these early times hold much promise. These years, if MMMs exist with the above-mentioned masses signatures, if they exist as evidence for milli-magnetic and charges, they may leave imprints in the amplitude monopoles, should be universally seen across new-born and frequency evolution that have a comparatively millisecond magnetars, with a very distinct pattern, and
075028-12 CONTINUOUS GRAVITATIONAL WAVES AND … PHYS. REV. D 101, 075028 (2020) may therefore be potentially distinguishable from other further sharpen future studies. Another crucial question is astrophysical signatures. regarding how prevalent millisecond magnetars are A more detailed implementation of the neutron star [113,114,116], and what their detection prospects are, magnetothermal evolution [119–121], incorporating milli- across the lifetime of advanced LIGO and future third magnetic monopole nonperturbative production, should generation GW detectors. We hope to address some of these help further clarify and add to the ideas of the present in future works. study. Another crucial aspect is reaching a consensus on the functional form of the thermal Schwinger pair-production ACKNOWLEDGMENTS rates [49,89,98–102] and striving to extend them to regions beyond the weak-field regime [56,125,126]. This would We thank Martin Hendry, Anson Hook, Adam Martin, facilitate quantitative analyses in all regions of the viable Dipanjan Mitra, Sunil Mukhi and Prasad Subramanian for ðm; ξÞ parameter space, and directly probing the presence discussions. A. T. would like to thank the organizers of the of abrupt features in the GW waveforms. Incorporating Gordon Research Conference on Particle Physics 2019, effects due to field inhomogeneities [103] and finite where parts of this work were completed, and would also chemical potentials [93,127], to account for the baryon like to acknowledge support from an SERB Early Career environment and finite densities in a neutron star, would Research Award.
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