Gravitational Waves from Neutron Stars: a Review

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Publications of the Astronomical Society of Australia (PASA) c Astronomical Society of Australia 2015; published by Cambridge University Press. doi: 10.1017/pas.2015.xxx.

Gravitational Waves from Neutron Stars: A Review

Paul D. Lasky1∗ 1Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC 3800, Australia

Abstract Neutron stars are excellent emitters of gravitational waves. Squeezing matter beyond nuclear densities invites exotic physical processes, many of which violently transfer large amounts of mass at relativistic velocities, disrupting spacetime and generating copious quantities of gravitational radiation. I review mechanisms for generating gravitational waves with neutron stars. This includes gravitational waves from radio and millisecond pulsars, magnetars, accreting systems and newly born neutron stars, with mechanisms including magnetic and thermoelastic deformations, various stellar oscillation modes and core superﬂuid turbulence. I also focus on what physics can be learnt from a gravitational wave detection, and where additional research is required to fully understand the dominant physical processes at play.

Keywords: Gravitational Waves – Neutron Stars

1 Introduction ing neutron stars, and primordial perturbations during inflation. Continuous gravitational waves are almost The dawn of gravitational wave astronomy is one of monochromatic signals generated typically by rotating, the most anticipated scientific advances of the coming non-axisymmetric neutron stars. decade. The second generation, ground-based gravita- The above laundry list of gravitational wave sources tional wave interferometers, Advanced LIGO (aLIGO prominently features neutron stars in their many guises. Aasi et al. 2015b) and Virgo (Acernese et al. 2015), are While supranuclear densities, relativistic velocities and due to start observing later in 2015 and in 2016, respec- enormous magnetic fields are exactly what makes neu- tively. The first aLIGO observing run is expected to be tron stars amenable to emitting gravitational waves of a few times more sensitive than initial LIGO, with a sufficient amplitude to be detectable on Earth, it is also full order of magnitude increase in strain sensitivity by these qualities that makes it difficult to provide accu- ∼ 2019 (for a review, see Aasi et al. 2013). rate predictions of the gravitational wave amplitudes, The inspiral and merger of compact binary systems and hence detectability, of their signals. A positive grav- (neutron stars and/or black holes) are commonly ex- itational wave detection from a neutron star would en- pected to be the first detections with aLIGO. The most gender great excitement, but it is the potential to un- robust predictions for event rates come from obser- derstand the interior structure of neutron stars that will vations of the binary neutron star population within make this field truly revolutionary. our galaxy, with an expected binary neutron-star de- In this review, I provide a detailed overview of many tection rate for the aLIGO/Virgo network at full sen- proposed gravitational wave generation mechanisms in sitivity between 0.4 and 400 per year (Abadie et al. neutron stars, including state-of-the-art estimates of the

arXiv:1508.06643v1 [astro-ph.HE] 26 Aug 2015 2010b). But there are many other exciting astrophysi- gravitational wave detectability. These include gravita- cal and cosmological sources of gravitational waves (for tional waves generated from magnetic deformations in a brief review, see Riles 2013). Loosely, these can be newly-born and older isolated radio pulsars (section2), divided into four categories: compact binary coales- accreting systems (section3), impulsive and continuous- cences, bursts, stochastic backgrounds and continuous wave emission from pulsar glitches (section4), magne- waves. Burst sources include gravitational waves gener- tar flares (section5), and from superfluid turbulence in ated in nearby supernova explosions, magnetar flares, the stellar cores (section6). As well as discussing grav- and cosmic string cusps. Stochastic backgrounds arise itational wave detectability, I also concentrate on what from the incoherent sum of sources throughout the Uni- physics can be learnt from a future, positive gravita- verse, including from compact binary systems, rotat- tional wave detection.

∗email: [email protected] 1 2 Lasky

It is worth stressing that this is not a review of all 1 are the projected strain sensitivities for aLIGO and gravitational wave sources in the audio band, but is ET (solid and dashed black curves respectively), and instead designed to review the theory of gravitational the strain sensitivity for the S5 run of initial LIGO (for wave sources from neutron stars. The article is but one details, see Aasi et al. 2014a). Age-based upper limits in a series highlighting Australia’s contribution to grav- on the gravitational wave strain can also be calculated itational wave research (Howell 2015; Kerr 2015; Slag- for neutron stars with unknown spin frequencies (Wette molen 2015). et al. 2008).

2 Magnetic Deformations 2.2 Magnetic field-induced ellipticities Spinning neutron stars possessing asymmetric deforma- As a general rule of thumb, a neutron star’s ellipticity tions emit gravitational waves. Such deformations can scales with the square of the volume-averaged magnetic be generated through elastic strains in the crust (Bild- field inside the star, hBi (e.g., Haskell et al. 2008), im- sten 1998; Ushomirsky et al. 2000), or strong magnetic plying the characteristic gravitational wave strain also 2 fields in the core (Bonazzola & Gourgoulhon 1996). scales as h0 ∼ hBi . The dipole, poloidal component of A triaxial body rotating about one of it’s principal the magnetic field at the surface of the star is inferred moments of inertia will emit radiation at frequency from the star’s spin period and it’s derivative, but very fgw = 2ν, where ν is the star’s spin frequency. More little is known about the interior field strength and/or precisely, a freely precessing, axisymmetric body, with configuration. principal moment of inertia, Izz, and equatorial ellip- A considerable body of work has been devoted to un- ticity, , emits a characteristic gravitational wave strain derstanding possible magnetic field configurations2, in (Zimmerman & Szedenits 1979) part to answer the question of gravitational wave detectability, but also to understand the inverse problem 2 2 4π G Izzfgw regarding what can be learnt from future gravitational h0 = c4 d wave observations. Purely poloidal and purely toroidal P −2 d −1 fields are known to be unstable on dynamical timescales = 4.2 × 10−26 , 10−6 10 ms 1 kpc (e.g., Wright 1973; Braithwaite & Spruit 2006; Braith- (1) waite 2007). Mixed fields, where a toroidal component threads the closed-field-line region of the poloidal field, where d is the distance to the source. For comparison, commonly termed ‘twisted-torus’ fields, are generally the smallest, upper limit on stellar ellipticity for young believed to be dynamically stable (e.g., Braithwaite & neutron stars in supernova remnants comes from LIGO Nordlund 2006; Ciolfi et al. 2009; Akgünet al. 2013), observations of Vela Jr. (G266.2-1.2), with ≤ 2.3 × −7 although it has been suggested this is dependent on the 10 (Aasi et al. 2014b). The overall ellipticity record- equation of state (Lander & Jones 2012; Mitchell et al. holder is for the millisecond pulsars J2124–3358 and −8 −8 2015). Moreover, a recent study of the effect twisted- J2129–5721 with ≤ 6.7 × 10 and ≤ 6.8 × 10 , re- torus fields have on crust-core rotational coupling dur- spectively (Aasi et al. 2014a). Typical ellipticity con- 4 6 ing neutron star spin down suggest these fields may straints for isolated radio pulsars are . 10 – 10 (Aasi be unstable on a spin down timescale (Glampedakis & et al. 2014a). Lasky 2015). While generating solutions to the equations of New- 2.1 Spin down limit tonian and general relativistic magnetohydrostatics is a noble task, it is no substitute for understanding the An absolute upper limit on the gravitational wave strain initial-value problem that brings one towards realis- from individual pulsars, known as the spin down limit, tic magnetic field configurations. In Newtonian simu- can be calculated assuming the observed loss of rota- lations with soft equations of state (typically more ap- tional energy is all going into gravitational radiation. plicable to main-sequence stars than to neutron stars), The left hand panel of Figure1 shows the current upper Braithwaite and co. showed magnetic fields evolve either limits on the gravitational wave strain from a search for to axisymmetric, twisted-tori (Braithwaite & Nordlund known pulsars (red stars; Aasi et al. 2014a), as well as 2006; Braithwaite 2009), or non-axisymmetric config- the spin down limits for the known pulsars in the ATNF urations (Braithwaite 2008), depending on the initial 1 catalogue (Manchester et al. 2005) . The observed grav- conditions. General relativistic magnetohydrodynamic itational wave upper limits beat the spin down limit for both the Crab and Vela pulsars, while a further five pul- 2In Australia, work on possible magnetic field configurations in sars are within a factor of five. Also plotted in Figure the cores of stars dates back to the 1960’s, where Monaghan (1965, 1966) calculated magnetic field equilibria in polytropes, which even included calculations of magnetic-field induced stel- 1http://www.atnf.csiro.au/people/pulsar/psrcat/ lar ellipticities.

PASA (2015) doi:10.1017/pas.2015.xxx Gravitational waves from neutron stars 3

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n -32 e

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-27 n n 10 i -34 i

a 10 a r r t t s s 10-28 10-36 100 101 102 103 100 101 102 103 gravitational wave frequency [Hz] gravitational wave frequency [Hz]

Figure 1. Left panel: Current upper limits on the gravitational wave strain from known pulsars (red stars; data from Aasi et al. 2014a) and the spin down limits for known pulsars in the ATNF catalogue (blue dots). Right panel: Gravitational wave strain predictions for known pulsars. The blue crosses and green squares are for normal neutron star matter with purely poloidal magnetic fields and Λ = 0.01, respectively [see equation (2) and surrounding text]. The red circles assume the neutron stars are colour-flavour-locked phase [CFL; equation (3)] with hBi = 10Bp. In both figures, the solid and dashed black curves show the projected strain sensitivity for aLIGO and ET respectively, and the grey curve is the strain sensitivity for the initial S5 run assuming a two-year coherent integration (e.g., see Dupuis & Woan 2005) simulations ubiquitously show the development of non- ternal toroidal component given by Λ = 0.01. The red axisymmetries (Kiuchi et al. 2011; Lasky et al. 2011, circles are described below, but with normal neutron 2012; Ciolfi et al. 2011; Ciolfi & Rezzolla 2012), however star matter the prospects for gravitational wave detec- these simulations begin with a restricted set of initial tion from magnetic field-induced non-axisymmetries are conditions. The long-term evolution to stable equilibria grim. One therefore has to hope that Nature has been in these systems is still an open question. kind, and has provided neutron stars with strong inter- The aforementioned uncertainty in possible mag- nal, toroidal components of the magnetic field. netic field configurations translates to uncertainty in gravitational wave predictions. The relative strengths of poloidal and toroidal components strongly affects 2.3 Generating large ellipticities the stellar ellipticity, and hence gravitational wave detectability (Haskell et al. 2008; Colaiuda et al. 2008; Certainly, one expects newly born neutron stars to have Ciolfi et al. 2009; Mastrano et al. 2011). Relatively stan- large internal toroidal fields and correspondingly large dard models suggest (e.g., Mastrano et al. 2011) ellipticities; strong differential rotation combined with turbulent convection in the nascent neutron star drives B 2 0.389 −7 p an α–Ω dynamo, which winds up a toroidal field as ≈ 4.5 × 10 14 1 − , (2) 10 G Λ strong as ∼ 1016 G (e.g., Duncan & Thompson 1992). where Bp is the poloidal component of the surface mag- Although the symmetry axis of the wound-up field is netic field, Λ is the ratio of poloidal-to-total magnetic aligned to the spin axis of the star, the ellipsoidal star field energy (i.e., 0 ≤ Λ ≤ 1, with Λ = 0, 1 correspond- will evolve on a viscous dissipation timescale to become ing to purely toroidal and purely poloidal fields rep- an orthogonal rotator through free-body precession, and sectively), and I have normalised to fiducial values of hence optimal emitter of gravitational waves (Cutler stellar mass and radius. Unfortunately, equation (2) is 2002; Stella et al. 2005; Dall’Osso et al. 2009). How pessimistic for gravitational wave detection with aLIGO such a strong field evolves over secular timescales is an 14 as the only neutron stars with Bp ∼ 10 G are the mag- open question. netars, but with spin periods of P ∼ 1 – 10 s they emit A recent spate of papers has shown that strong gravitational waves at frequencies too low. The right toroidal fields can also be achieved in systems in dy- hand panel of Figure1 shows predictions for the gravi- namical equilibrium by prescribing different forms for tational wave strain for known pulsars from the ATNF the azimuthal currents in the star (Ciolfi & Rezzolla pulsar catalogue (Manchester et al. 2005). The blue 2013), and also by invoking stratified, two fluid stellar crosses assume a purely poloidal field i.e., equation (2) models (i.e., where the neutrons form a superfluid con- with Λ = 1, while the green squares assume a strong in- densate and the protons are either a normal fluid or

PASA (2015) doi:10.1017/pas.2015.xxx 4 Lasky a superconductor; Glampedakis et al. 2012a; Lander stars in supernova remnants can be adapted to set lim- 2013). Interestingly, exotic states of matter in the core its on the amplitude of such oscillations (Owen 2010). such as crystalline colour-superconductors allow for sig- This was first done with a 12-day coherent search of nificantly higher ellipticities (Owen 2005; Haskell et al. S5 LIGO data targeting the neutron star in the super- 2007). Recently, Glampedakis et al.(2012b) showed nova remnant Cassiopeia A (Wette et al. 2008; Abadie that, if the ground state of neutron star matter is a et al. 2010a). The search has since been extended to nine colour-superconductor, then the colour-magnetic vor- young supernova remnants, with the most sensitive r- tex tension force leads to significantly larger mountains mode fractional amplitude being less than 4 × 10−5 for than for normal proton superconductors. They derived Vela Jr. (Aasi et al. 2014b). Such a limit is encroach- fiducial ellipticities for purely poloidal fields of3 ing ‘interesting values’ of the amplitude when compared to simulations that calculate the nonlinear saturation hBi 2SC −6 amplitude of various r-modes in young neutron stars ≈ 4.0 × 10 14 , (3) 10 G (Bondarescu et al. 2009; Aasi et al. 2014b). hBi CFL ≈ 1.2 × 10−5 , (4) 1014 G where 2SC and CFL respectively denote the elliptici- 3 Accreting systems ties if matter is in a two-flavour phase (i.e., only the u 3.1 Torque balance and d quarks form superconducting pairs) and a colour- flavour-locked phase (where all three quark species are An observational conundrum drives research into gravi- paired), and hBi is the volume averaged magnetic field tational wave emission from accreting systems; namely, in the core of the star. The red circles in Figure1 the absence of accreting pulsars with spin frequencies show gravitational wave predictions for a colour-flavour ν & 700 Hz (Chakrabarty et al. 2003; Patruno 2010). locked superconductor, equation (3), with the average The argument is simple: measured accretion rates allow internal field 10 times the observationally inferred sur- one to calculate the angular momentum being trans- face field, i.e., hBi = 10Bp. These results are fascinat- ferred to the neutron star, which should be spun up ing; a positive detection of gravitational waves from to frequencies at, or near, the breakup frequency (ν & magnetic deformations in neutron stars is a fundamen- 1 kHz for most equations of state Cook et al. 1994). The tal probe of the fundamental state of nuclear matter. common explanation is that the neutron stars are los- It is worth mentioning that the discussion in this sec- ing angular momentum through the emission of grav- tion pertains to the optimal case of a rigidly rotating, itational radiation (Papaloizou & Pringle 1978; Wag- axisymmetric body rotating about one of it’s principal oner 1984; Bildsten 1998), although alternatives exist, moments of inertia. Such a body emits monochromatic most notably invoking interactions between the accre- gravitational waves at a frequency of 2ν. If the neu- tion disk and the companion star’s magnetic field (e.g., tron star core contains a pinned superfluid, it will emit White & Zhang 1997; Patruno et al. 2012). also at the spin frequency, ν (Jones 2010). In general, Balancing the accretion torques with gravitational a non-aligned rotator will emit at ν, 2ν, and a num- wave emission allows one to estimate an upper limit on ber of frequencies straddling these values (Zimmerman the gravitational wave strain independent of the emis- 1980; Jones & Andersson 2002; van den Broeck 2005; sion mechanism (Wagoner 1984): Lasky & Melatos 2013). For radio pulsars, such modu- R3/4 F 1/2 300 Hz1/2 lations would also be present in other observables such hEQ = 5.5 × 10−27 10 X , (5) 0 1/4 as pulse time-of-arrivals and radio polarisation (Jones F? ν M1.4 & Andersson 2001; Jones & Andersson 2002). where R10 = R/10 km, M1.4 = M/1.4 M , FX is the X- −8 −2 −1 −1/2 ray flux and F? = 10 erg cm s . The ν scaling 2.4 Oscillations in young neutron stars arises only because of the spin-equilibrium assumption; The strongest emitters of gravitational waves from mag- slower rotators need larger ellipticities for torque bal- netic deformations are likely young, rapidly rotating ance. If one builds a mountain without being in spin neutron stars. Such stars may also emit through other equilibrium, then the gravitational wave strain is sim- channels, the most likely being the unstable r-mode, ply given by equation (1). whose restoring force is the Coriolis force; see section Equation (5) implies the loudest gravitational wave 3.2 for a detailed description of the physics of r-modes. emitters are the brightest X-ray sources such as the low- Targeted gravitational wave searches of young neutron mass X-ray binaries, the brightest of which being Scor- pius X-1 (Sco X-1). Watts et al.(2008) utilised observations of Sco X-1 and other known accreting systems 3It is worth noting that, in any kind of superconductor, the ellipticity scales linearly with the volume-averaged magnetic field to determine their gravitational wave detectability with (Cutler 2002), cf. hBi2 for normal matter. current and future interferometers showing that, even

PASA (2015) doi:10.1017/pas.2015.xxx Gravitational waves from neutron stars 5 at torque balance, most systems will be very difficult to waves, but carries with it positive angular momentum detect with aLIGO. from the star. This positive angular momentum is sub- The LIGO Scientific Collaboration has given periodic tracted from the negative angular momentum of the gravitational waves from Sco X-1 high priority, with mode, which becomes more negative, growing the am- multiple data analysis pipelines already fully developed plitude of oscillation. As the mode grows, it emits more (see Messenger et al. 2015, and references therein). The positive angular momentum, and continues to grow even unknown spin period of the neutron star in Sco X-1 and faster. Hence, a sufficiently rapidly rotating star is un- the variable accretion rate that drives spin-wandering stable to gravitational wave emission. of the neutron star complicate these searches, but the At all stellar rotation rates, r-modes are retrograde best gravitational wave upper limit utilises the Side- in the comoving frame, but prograde in the inertial band search (Sammut et al. 2014), which gives a 95% frame, implying they are generically unstable to the −25 upper limit of h0 . 8 × 10 Hz at 150 Hz (Aasi et al. CFS instability (Andersson & Kokkotas 1998; Fried- 2015a). This is still above the torque balance limit for man & Morsink 1998). A struggle therefore ensues be- EQ −26 1/2 tween the gravitational waves which drive up the os- Sco X-1, h0 ≈ 3.5 × 10 (300 Hz/ν) , but this is expected to be beaten with aLIGO observations (Whe- cillation amplitude and the viscous damping that acts lan et al. 2015). to suppress the mode amplitude. There exists a nar- Torque balance is an empirically derived limit that is row range in spin period and temperature for which independent of the mechanism generating the gravita- gravitational wave emission dominates over viscous dis- 9 tional radiation. From a theoretical perspective, there sipation: at low temperatures, T . (few) × 10 K, vis- are a number of ways in which sufficient energy can cous dissipation is dominated by shear viscosity, while 9 be lost to gravitational waves, including unstable oscil- at high temperatures, T & (few) × 10 K, bulk viscos- lation modes (Andersson 1998), and non-axisymmetric ity is the main culprit. Details of the r-mode instabil- deformations supported by the magnetic field (Cutler ity window therefore depend sensitively on relatively 2002; Melatos & Payne 2005) or elastic crust (Bildsten unknown neutron star physics, including microphysics 1998; Ushomirsky et al. 2000). and complicated crust-core interactions (for a review, The torque balance limit for accreting systems with see Andersson & Kokkotas 2001). This somewhat sim- known spin periods is shown in the left hand panel of ple picture describing the r-mode instability window is Figure2 (blue dots), and for Sco X-1 (red curve) which inconsistent with observed spins and temperatures of has an unknown spin period. The LIGO, aLIGO and low-mass X-ray binaries, implying the complete physi- ET sensitivity curves in this figure assume two years cal picture behind this mechanism is ill-understood (Ho integration; see Watts et al.(2008) for more detailed, et al. 2011; Haskell et al. 2012). realistic estimates of signal-to-noise ratios of these sys- Strohmayer & Mahmoodifar(2014) recently analysed tems. Note that current upper limits of gravitational RXTE observations of the accreting millisecond pul- wave emission from Sco X-1 utilise a semi-coherent, 10- sar XTE J1751-305, finding evidence for a coherent day search, which therefore yields less stringent upper oscillation mode during outburst that they attributed limits than the LIGO curve in Figure2(Aasi et al. to either an r- or g-mode (for which buoyancy is the 2015a). restoring force). The r-mode interpretation is the most interesting in the context of gravitational wave emission; had the outburst occurred during the S5 LIGO run it would have been marginally observable (Ander- 3.2 Unstable oscillation modes sson et al. 2014). However, doubt has been cast over Unstable oscillations modes, in particular r-modes, have the r-mode interpretation; Andersson et al.(2014) also received significant attention as potential sources of de- showed that the mode amplitude required to interpret tectable gravitational waves. These inertial modes – the observations necessarily leads to large spin-down of toroidal modes of oscillation for which the restoring the neutron star, which is inconsistent with the neutron force is the Coriolis force – are generically unstable to star’s observed spin evolution. a phenomena known as the Chandrasekhar-Friedman- Schutz (CFS) instability (Chandrasekhar 1970; Fried- 3.3 Mountains man & Schutz 1978). Consider an oscillation mode in a rotating star. The rotation induces both forward and On the other hand, permanent non-axisymmetries sup- backward propagating modes, but the backward prop- ported magnetically or elastically may generate a sig- agating modes are being dragged forward by the star’s nificant gravitational wave signature. Accreted mat- rotation. Although this retrograde mode continues to ter accumulates on the neutron star, is buried, com- move backwards in the star’s rotating frame, if the star pressed, and undergoes a range of nuclear reactions is rotating sufficiently fast it will be prograde in the (e.g., Haensel & Zdunik 1990). In the non-magnetic inertial frame. The mode loses energy to gravitational case, asymmetries in the accretion lead to compositional

PASA (2015) doi:10.1017/pas.2015.xxx 6 Lasky

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/ / -24

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a a 10 r r t t s 10-28 s 10-30 101 102 103 102 103 gravitational wave frequency [Hz] gravitational wave frequency [Hz]

Figure 2. Left panel: Gravitational wave torque balance limit for known accreting millisecond pulsars and systems with burst oscillations assuming gravitational wave emission at twice the neutron star spin period (blue dots). Also shown is the torque balance limit for Sco X-1 (red curve) which has unknown spin period. Data is collated from Watts et al.(2008) and Haskell et al.(2015). Right panel: Gravitational wave predictions for magnetic mountains on known accreting X-ray pulsars, where the range is for magnetic field 10 12 strengths at the onset of accretion of between B? = 10 and 10 G (for details of the calculation, see Haskell et al. 2015). In both panes, the solid and dashed black curves show the projected strain sensitivity for aLIGO and ET respectively, and the grey curve is the strain sensitivity for the initial S5 run, assuming two years of coherent integration time. For comparison, current observational upper −25 limits on Sco X1 from LIGO are . 8 × 10 at 150 Hz, which utilise a 10-day, semi-coherent analysis (Aasi et al. 2015a). and heating asymmetries, which induce stellar defor- nant scattering features in the X-ray spectrum may be mations. Approximate expressions for the quadrupolar able to discern magnetic or elastic mountains (Priymak deformation can be derived in terms of the quadrupo- et al. 2014; Haskell et al. 2015). lar component of the temperature variation and reac- tion threshold energies (Ushomirsky et al. 2000). Such deformations are limited by the maximum stress the 4 Pulsar Glitches crust can sustain before breaking (Ushomirsky et al. Pulsar glitches are sudden jumps in the neutron star 2000; Haskell et al. 2006; Johnson-McDaniel & Owen spin frequency with wide-ranging fractional amplitudes, −11 −4 2013), which typically gives larger gravitational wave 10 . ∆ν/ν . 10 (Melatos et al. 2008; Espinoza estimates than torque balance (Haskell et al. 2015). et al. 2011; Espinoza et al. 2014). The exact mecha- Accretion also affects the structure of the neutron nism driving a glitch is not fully understood, although star’s magnetic field which, in turn, changes the ac- it is clear that a sudden transfer of angular momentum cretion dynamics. The magnetosphere funnels accreted occurs between the rapidly rotating superfluid interior matter onto the poles, at which point it spreads to- and the outer crust (for a recent review, see Haskell & wards the equator, dragging, compressing, and burying Melatos 2015). Such large angular momentum transfer the local stellar magnetic field (Hameury et al. 1983; lends itself to the generation of gravitational radiation Melatos & Phinney 2001). Such fields can lead to stel- through a variety of avenues, as both broadband burst lar deformations considerably larger than those due to emission from the glitch and as a continuous wave signal the background magnetic field inferred from the exter- during the glitch recovery phase. nal, dipole (Payne & Melatos 2004; Melatos & Payne 2005; Vigelius & Melatos 2009), which may even generate gravitational waves detectable by aLIGO or the 4.1 Gravitational wave bursts Einstein Telescope (Priymak et al. 2011), particularly if Most theories of pulsar glitches rely on the general 12 the buried field is B & 10 G(Haskell et al. 2015). This mechanism introduced by Anderson & Itoh(1975). The is shown in the right-hand panel of Figure2, where the quantum mechanical nature of superfluids implies the gravitational wave signal from accreting systems with neutron star’s rotation is attributed to an array of known spin periods is shown assuming initial magnetic ∼ 1018 quantized superfluid vortices. These thread the 10 12 fields of B = 10 and 10 G (for details of this cal- entire star, but are pinned to lattice sites and/or crustal culation, see Haskell et al. 2015). Finally, it is worth defects, and hence are restricted from moving outwards remarking that, given a positive gravitational wave de- as the crust spins down through the usual electromag- tection, an in-principle measurement of cyclotron reso- netic torques. The superfluid core therefore retains a

PASA (2015) doi:10.1017/pas.2015.xxx Gravitational waves from neutron stars 7 higher angular velocity than the crust of the star, and 45 a differential lag builds up between these two components. A glitch occurs when ∼ 107 – 1015 vortices catas- trophically unpin and move outwards, thereby rapidly 35 transferring angular momentum to the crust. That so many vortices are required to unpin simulta- 25 neously implies an avalanche trigger process must be at work. Such an avalanche is likely non-axisymmetric, and hence capable of emitting gravitational waves through a 15 variety of channels. Warszawski & Melatos(2012) sim- ulated the motion of vortices, showing that a burst number of glitches of gravitational waves emitted through the current 5 quadrupole has a characteristic strain 10-32 10-30 10-28 10-26 10-24 10-22 10-20 3 −24 ∆Ω/Ω Ω gravitational wave amplitude h0 ≈ 10 10−7 102 rad s−1 ∆r −1 d −1 Figure 3. Predicted gravitational wave amplitudes for known × , (6) −2 10−2 m 1 kpc glitches using equation (6) with ∆r = 10 m (black, unfilled histogram) and ∆r = 10−4 (red, filled histogram). The vertical, where ∆r is the average distance travelled by a vor- dashed line is the gravitational wave strain upper limit derived tex during a glitch. The non-detection of gravitational for the August 2006 glitch of the Vela pulsar during the LIGO S5 run (Abadie et al. 2011b) waves from a glitch in the Vela pulsar during the fifth LIGO Science run in 2006 put an upper limit of h0 . 10−20 (Abadie et al. 2011b), implying an upper limit of −2 1996; Melatos 2012), thereby reducing the gravitational ∆r . 10 m and a lower limit on the glitch duration −4 wave strain estimates. Finally, it has recently been sug- & 10 ms. gested that vortex avalanches that are believed to trig- Figure3 shows a histogram of estimates for the grav- ger glitches leave behind long-lived, large-scale inhomo- itational wave strain using equation (6), where the geneities on the vortex distribution that are potentially glitches are taken from the ATNF glitch catalogue observable by aLIGO (Melatos et al. 2015). (Manchester et al. 2005)4 The vertical dashed line shows the empirical gravitational wave strain upper limit from the August 2006 glitch of the Vela pulsar (Abadie et al. 5 Magnetar Flares 2011b). Perhaps the most exotic of all neutron stars are the magnetars; isolated neutron stars with external dipole 14 4.2 Glitch recovery magnetic fields Bp & 10 G. The decay of the immense magnetic field powers irregular bursts in hard X-rays The motion of vortices is expected to excite hydro- and soft γ-rays, with typical peak luminosities ∼ 1038— dynamic oscillation modes such as f- and p-modes, 1043 erg s−1. Three giant flares have been observed in for which the pressure is the restoring force, and in- our Galaxy5, the largest of which was the extreme out- ertial r-modes (e.g., Andersson 1998; Andersson & burst from SGR 1806–20 on 27 December 2004 which Kokkotas 1998; Glampedakis & Andersson 2009). Al- emitted a total, isotropic energy of 2 × 1046 erg (Palmer though r-modes are expected to be the dominant emis- et al. 2005). sion mechanism, recent estimates of their detectability Giant flares are commonly attributed to catastrophic with ground-based interferometers is pessimistic (Sidery rearrangements of the magnetic field, either internal or et al. 2010). More optimistic estimates have been pro- external to the neutron star (e.g. Duncan & Thompson vided for gravitational waves from meridional Ekman 1992; Thompson & Duncan 1995). Strong coupling be- flows that are expected to couple the core and crust tween the magnetic field and the solid crust imply the rotation post-glitch (van Eysden & Melatos 2008; Ben- latter will stress, and potentially rupture (although, see nett et al. 2010), although it is worth noting the model Levin & Lyutikov 2012), exciting oscillation modes in is sensitive to many unknown, dimensionless parameters the star’s crust and core. Observations of quasi-periodic such as the Ekman number, normalised compressibility and Brunt-Väisäläfrequency. Moreover, strong strati- 5Interestingly, LIGO non-detections of gravitational waves from fication may suppress Ekman flows to thin boundary nearby short-duration, hard-spectrum gamma-ray bursts GRB layers in the outer regions of the star (Abney & Epstein 051103 (Abbott et al. 2008a) and GRB 070201 (Abadie et al. 2012) rule out compact binary coalescences as their progeni- tors, implying these are most likely extragalactic magnetar giant 4http://www.atnf.csiro.au/people/pulsar/psrcat/glitchTbl.html flares.

PASA (2015) doi:10.1017/pas.2015.xxx 8 Lasky oscillations (QPOs) in the tails of two giant flares (Is- 5 rael et al. 2005; Strohmayer & Watts 2005) has engen- dered excitement about the potential new field of neutron star asteroseismology. In particular, the interpre- 4 tation of QPOs as stellar magneto-elastic oscillations provide enticing potential to infer neutron star struc- 3 tural parameters (see Levin 2006; Glampedakis et al. 2006; Levin 2007; Sotani et al. 2007, 2008; Colaiuda 2 et al. 2009; van Hoven & Levin 2011a,b; Levin & van Hoven 2011; Gabler et al. 2013, 2014; Huppenkothen 1 et al. 2014c, and references therein). number of magnetars 0 5.1 KiloHertz gravitational waves 1026 1030 1034 1038 1042 1046 1050 First attempts to understand gravitational-wave emis- possible gravitational wave energy [erg] sion from magnetar flares were optimistic. Ioka(2001) assumed an instantaneous change in the star’s moment Figure 4. Predictions of the possible gravitational wave energy of inertia from a rearrangement of the global, internal emitted in f-mode oscillations if giant flares were to go off in each magnetic field can excite the neutron star’s fundamen- observed galactic magnetar given the calculations of (Ciolfi & Rezzolla 2012, shaded red histogram) and (Zink et al. 2012; Lasky tal f mode (typically at frequencies, f ∼ 1–2 kHz), a et al. 2012, empty black histogram). Plotted in blue are the opti- study that was backed up by Corsi & Owen(2011). mistic predictions of Ioka(2001) and Corsi & Owen(2011), and These studies estimated that the energy in gravitational the dashed black line is the upper limit on the f-mode gravita- 48 49 tional wave energy from the giant flare from SGR 1806-20 (Abadie waves could be Egw ∼ 10 –10 erg, comparable to the energy emitted in electromagnetic waves, and certainly et al. 2011a). detectable in the Advanced Detector Era. In contrast, Levin & van Hoven(2011) showed that, while the order-of-magnitude estimates of Ioka(2001) itational wave energy from the 2004 giant flare in SGR and Corsi & Owen(2011) are correct, in practice there is 1806-20 (Abadie et al. 2011a, see below). poor energy conversion into the f-mode, and prospects for gravitational wave detection in the Advanced Era 5.2 Low-frequency gravitational waves are pessimistic. The analytic calculation of Levin & van Hoven(2011) looked at the energy conversion from a Although the f-mode couples well to gravitational ra- flare triggered in the star’s magnetosphere. Zink et al. diation, one of the reasons it is not an optimal source (2012) and Ciolfi & Rezzolla(2012) performed com- for ground-based detectors is that it is damped through plementary, general relativistic magnetohydrodynamic the emission of gravitational radiation in . 0.1 s (De- simulations using their respective codes (Lasky et al. tweiler 1975; McDermott et al. 1988). Longer lasting, 2011; Ciolfi et al. 2011) for catastrophic reconfigurations lower frequency modes excited from magnetar flares of the internal magnetic field, also finding that the f- have been suggested as potential sources of detectable mode is not sufficiently excited to generate a detectable gravitational waves (Kashiyama & Ioka 2011; Zink et al. gravitational wave signature. Although these works had 2012; Lasky et al. 2012). To generate sufficient mass different predictions for the gravitational wave scaling motions, and therefore a detectable gravitational wave as a function of magnetic field strength, they both pre- signal, these modes need to be global core oscillations, dicted approximately 10 orders of magnitude less en- and hence are unlikely the cause of the observed QPOs. ergy being emitted in gravitational waves than Corsi & A key uncertainty for these low frequency modes is their Owen(2011). damping time, which underpins their gravitational wave Figure4 shows a histogram of the possible gravita- emission. If these modes form an Alfvéncontinuum of tional wave energies emitted in the f-mode from known frequencies, where the QPOs arise as the edges or turn- galactic magnetars6 using the calculations of Ciolfi & ing points of the continuum (Levin 2007), then reso- Rezzolla(2012) (shaded red histogram) and Zink et al. nant absorption should quickly redistribute energy, re- (2012); Lasky et al.(2012) (empty black histogram). In sulting in short lifetimes (. 1 s) for the QPOs (Levin blue are the gravitational wave predictions from Ioka 2007; Levin & van Hoven 2011; Gabler et al. 2011; van (2001) and Corsi & Owen(2011), while the dashed black Hoven & Levin 2012; Huppenkothen et al. 2014c), and line gives the observed upper limit on the f-mode grav- pessimistic estimates for gravitational wave emission. On the other hand, if global Alfvénwaves are excited 6The known magnetars are taken from the McGill Magnetar cat- in the core (Zink et al. 2012) then they may live for alogue (Olausen & Kaspi 2014). 100’s of seconds, and be potentially detectable by third-

PASA (2015) doi:10.1017/pas.2015.xxx Gravitational waves from neutron stars 9 generation interferometers, or even aLIGO in the case axisymmetric, and therefore emits stochastic gravita- of extremely strong (∼ 1016 G) magnetic fields in the tional waves. The signal was first calculated in Per- core (Glampedakis & Jones 2014). alta et al.(2006a) by numerically solving the spheri- 6 cal Couette-flow problem with Re . 10 . A subsequent analytic calculation (Melatos & Peralta 2010) made a 5.3 Current observational limits more robust prediction of the gravitational wave am- At the time of the giant flare from SGR 1806-20, the plitude and spectrum, showing in particular that the 4 km Hanford detectors was the only interferometer op- peak of the gravitational wave signal occurs near the 46 erating. An upper limit of Egw < 7.7 × 10 erg was inverse of the turbulence decoherence timescale, τc. For achieved around the 92.5 Hz QPO, with a significantly reasonable neutron star parameters, this decoherence weaker constraint placed in the kHz range where one timescale is expects the f-mode. The most sensitive gravitational −1 ∆Ω wave search of the giant flare was from the AURIGA τc ≈ 26 −1 ms, (7) bar detector (Baggio et al. 2005). They searched for 10 rad s exponentially decaying signals (with decay time 0.1 s) where ∆Ω is the difference in angular frequency between in a small frequency band around 900 Hz. Subsequent the crust and the core of the neutron star, which can LIGO/Virgo searches of regular flares from magnetars potentially be related to the star’s angular frequency, have yielded comparable gravitational wave limits to Ω, through the Rossby number, Ro = ∆Ω/Ω. For many the original 2004 burst (Abbott et al. 2008b; Abadie pulsars, this places the peak of the gravitational wave et al. 2011a). signal in LIGO’s most sensitive band, although it is While galactic giant flares only occur approximately worth noting there is considerable uncertainty in ∆Ω. once per decade, the regularity of magnetar flare storms Finally, the root-mean-square of the gravitational wave implies they may provide an attractive alternative. If strain is such flares also excite normal stellar modes, it is easy to 3 −1 ∆Ω d imagine that stacking gravitational wave data from mul- h ≈ 8 × 10−28 . (8) rms −1 1 kpc tiple flares could allow for sufficient integration time. 10 rad s Indeed, such a gravitational wave search has been de- Figure5 shows the peak gravitational wave predic- signed (Kalmus et al. 2009) and implemented to search tions for the superfluid turbulence model for observed for gravitational waves in LIGO/Virgo data (Abbott galactic pulsars and millisecond pulsars with Ro = 10−1 et al. 2009). These attempts have further been buoyed (black points) and 10−2 (red points). It is worth stress- by the successful detection of QPOs in ordinary mag- ing that these values of the crust-core shear, ∆Ω, are netar flares (Huppenkothen et al. 2014a,b). most likely exaggerated; in the same paper Melatos & −2 Peralta(2010) showed that Ro . 10 for typical millisecond pulsars. Calculations of vortex (Seveso et al. 6 Superfluid Turbulence 2012) and flux-tube (Link 2003) pinning suggest ∆Ω −2 Soon after the realisation that neutron star matter should vary, but will generally be . 10 . One there- should form a superfluid condensate (Baym et al. 1969, fore concludes that turbulence excited in newly born and references therein), Greenstein(1970) suggested protoneutron stars, or extremely close rapid rotators, this superfluid should be in a highly turbulent state7. are the only possible source for Advanced Era gravita- Such turbulence can be either hydrodynamic (e.g., Per- tional wave interferometers. The blue square and star in alta et al. 2005, 2006a,b, 2008) or quantum mechanical Figure5 show the peak gravitational wave emission for (e.g., Tsubota 2009; Andersson et al. 2003; Mastrano hypothetical neutron stars situated at d = 10 pc with & Melatos 2005; Andersson et al. 2007; Link 2012a,b). spin periods p = 3 ms and 10 ms, and Ro= 10−2 and Regardless, the turbulence is likely driven by differen- 10−1, respectively, the former of which would be de- tial rotation between the superfluid core (with Reynolds tectable by aLIGO. number Re ∼ 1011) and the solid crust that is being slowly spun down by electromagnetic torques. 6.2 Stochastic background A cosmological population of neutron stars with tur- 6.1 Individual sources bulent cores produces a stochastic gravitational wave Although the turbulence is axisymmetric when av- background that peaks in aLIGOs most sensitive fre- eraged over long times, it is instantaneously non- quency band (Lasky et al. 2013). Although there are large uncertainties in the expected amplitude of the 7The beautifully succinct abstract is seldom seen today: ‘The background due to a lack of understanding of ∆Ω, neutron superfluid in most neutron stars should be in a highly turbulent state. If so, this turbulence drastically alters its rota- the shape of the spectrum is relatively unique, with tional properties’ (Greenstein 1970). it being well-approximated by a piecewise power-law

PASA (2015) doi:10.1017/pas.2015.xxx 10 Lasky

signiﬁcant amplitudes from isolated neutron stars. This ] 10-22

2 review highlights many of those mechanisms where we /

1 have some understanding of the key, physical processes. − 10-24 z It also highlights the vast uncertainty of many of these H

[ predictions, showing that they rely on knowledge of neu- 10-26 y

t tron star physics beyond current capabilities. But this is i

v -28 what makes the ﬁeld so fascinating; a positive detection i

t 10 i of gravitational waves from any of the mechanisms dis- s

n -30 e 10 cussed in this article will allow an unprecedented view s into the heart of neutron stars, where some of the most n

i -32

a 10 exotic physics in the Universe takes place. r t s 10-34 0 1 2 3 10 10 10 10 8 ACKNOWLEDGEMENTS gravitational wave frequency [Hz] PDL is supported by the Australian Research Council Dis- covery Project DP140102578. I am extremely grateful to Figure 5. Peak gravitational wave amplitudes from superfluid Bryn Haskell, Daniel Price, Kostas Glampedakis, and Karl turbulence from galactic pulsars with Ro = ∆Ω/Ω = 10−1 (black Wette for useful comments on the manuscript. points) and 10−2 (red points). The blue star and square are hypothetical, nearby (d = 10 pc) rapid rotators with spin periods p = 3 ms and 10 ms, and Ro = 10−2 and 10−1, respectively. REFERENCES Aasi, J., Abadie, J., Abbott, B. P., Abbott, R., Ab- α bott, T., Abernathy, M. R., Accadia, T., Acernese, Ωgw(fgw) = Ωαfgw, with α = 7 and α = −1 for fgw < F., Adams, C., Adams, T., & et al. 2014a, Astrophys. fc and fgw > fc, respectively. Here, fc is the population- J., 785, 119 weighted average of 1/τc, and Ωgw(fgw) is the energy density in the gravitational wave background as a frac- Aasi, J., Abadie, J., Abbott, B. P., Abbott, R., Abbott, tion of the closure energy density of the Universe. T. D., Abernathy, M., Accadia, T., Acernese, F., & It is worth noting that turbulent convection in main et al. 2013, ArXiv e-prints sequence stars also produces a gravitational wave sig- Aasi, J., Abbott, B. P., Abbott, R., Abbott, T., Aber- nal (Bennett & Melatos 2014). Interestingly, the loudest nathy, M. R., Acernese, F., Ackley, K., Adams, C., gravitational wave signal detectable on Earth may come Adams, T., Adams, T., & et al. 2014b, submitted for −4 from the Sun as, for frequencies fgw . 3 × 10 Hz, the publication to ApJ Earth lies in the Sun’s near zone, in which the wave Aasi, J., Abbott, B. P., Abbott, R., Abbott, T., Aber- strain scales significantly more steeply with distance, nathy, M. R., Acernese, F., Ackley, K., Adams, C., −5 −1 hrms ∝ d (cf. ∝ d in the far zone; Cutler & Lind- Adams, T., Addesso, P., & et al. 2015a, Phys. Rev. blom 1996; Polnarev et al. 2009). This gravitational D, 91, 062008 wave signal is most relevant in the micro- to nHz regime, Aasi, J., Abbott, B. P., Abbott, R., Abbott, T., Aber- and therefore most applicable to pulsar timing experi- nathy, M. R., Ackley, K., Adams, C., Adams, T., ments. However, the gravitational-wave scaling at low- Addesso, P., & et al. 2015b, Classical and Quantum frequencies is uncertain as Kolmogorov scaling for tur- Gravity, 32, 074001 −8 bulence breaks down below fgw . 10 Hz. Abadie, J., Abbott, B. P., Abbott, R., Abernathy, M., Accadia, T., Acernese, F., Adams, C., Adhikari, R., Affeldt, C., Allen, B., & et al. 2011a, Astrophys. J., 7 Conclusions 734, L35 Ground-based gravitational wave interferometers are al- Abadie, J., Abbott, B. P., Abbott, R., Abernathy, M., ready contributing to our understanding of interesting Adams, C., Adhikari, R., Ajith, P., Allen, B., Allen, astrophysical phenomena. But non-detections will only G., Amador Ceron, E., & et al. 2010a, Astrophys. J., progress the field so far; the first direct detection of 722, 1504 gravitational radiation will herald a new scientific field Abadie, J., Abbott, B. P., Abbott, R., Adhikari, R., of study, and will allow new insight into the most exotic Ajith, P., Allen, B., Allen, G., Amador Ceron, E., regions of our Universe. Amin, R. S., Anderson, S. B., & et al. 2011b, Phys. While the first detection with aLIGO is expected to Rev. D, 83, 042001 be from the inspiral and merger of a compact binary Abadie, J., Abbott, B. P., Abbott, R., & et al. 2010b, system, there are many unknown, and ill-understood, Class. Q. Grav., 27, 173001 mechanisms that can generate gravitational waves with Abadie, J., Abbott, B. P., Abbott, T. D., Abbott, R.,

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