Continuous gravitational waves from neutron stars: current status and prospects
CContinuous gravitational waves from neutronstars: current status and prospects
Magdalena Sieniawska ∗ and Michał Bejger Nicolaus Copernicus Astronomical Center, Polish Academy ofSciences, Bartycka 18, 00–716 Warszawa, Poland
Abstract
Gravitational waves astronomy allows us to study objects and events invisi-ble in electromagnetic waves. It is crucial to validate the theories and models ofthe most mysterious and extreme matter in the Universe: the neutron stars. Inaddition to inspirals and mergers of neutrons stars, there are currently a few pro-posed mechanisms that can trigger radiation of long-lasting gravitational radiationfrom neutron stars, such as e.g., elastically and / or magnetically driven deforma-tions: mountains on the stellar surface supported by the elastic strain or magneticfield, free precession, or unstable oscillation modes (e.g., the r-modes). The as-trophysical motivation for continuous gravitational waves searches, current LIGOand Virgo strategies of data analysis and prospects are reviewed in this work. Gravitational-wave (GW) astronomy has been one of the fastest-growing fields in astro-physics since the first historical detection of a binary black-hole (BH) system GW150814(Abbott et al., 2016a). In addition to studying the nature of gravitation itself, it may beused to infer information about the astrophysical sources emitting the GWs. This re-view concentrates on a specific kind of prospective GWs: persistent (continuous) grav-itational waves (CGWs), emitted by neutron stars (NSs). The article is arranged asfollows. Section 1 gathers introductory material: Section 1.1 presents the basics of theGWs theory, Section 1.2 contains a brief overview of GWs detections, Section 1.3 de-scribes properties of NSs and features of hitherto detected NSs-related GWs—a binaryNS merger GW170817 (Abbott et al., 2017d), Section 1.4 gathers general informationabout CGWs, whereas Section 1.5 is devoted to the main data analysis methods used inCGWs searches. Following sections describe the main CGWs emission mechanisms:elastic deformations (Section 2), magnetic field (Section 3), oscillations (Section 4),free precession (Section 5). Finally, in Section 6 contains summary and discussion. ∗ Correspondence: [email protected] a r X i v : . [ a s t r o - ph . H E ] N ov .1 Basics of the Gravitational Radiation Theory According to the general theory of relativity (Einstein, 1916, 1918), GWs are pertur-bations in the curvature of space-time, travelling with the speed of light. To producewaves, just as in the case of electromagnetic (EM) waves, accelerated movement ofcharges (masses) is needed. The lowest allowed multipole is the quadrupole, as themonopole is forbidden by the mass conservation and the dipole by the momentumconservation. A non-negligible time-varying quadrupole moment may be provided bye.g., binary systems: BHs or NSs, rotating non-axisymmetric objects (i.e., deformedNSs) or non-spherical explosions (supernovæ). According to the quadrupole formulaat the lowest order (Einstein, 1916, 1918), GW amplitude strain tensor h i j at position r is h i j = Gc r ¨ Q TTi j (cid:18) t − rc (cid:19) , where Q TTi j ( x ) = (cid:90) ρ (cid:32) x i x j − δ i j r (cid:33) d x , (1)is the mass-quadrupole moment in the transverse-traceless (TT) gauge , evaluated atthe retarded time ( t − r / c ) and ρ is the matter density in a volume element d x , at theposition x i ; c and G is the speed of light and gravitational constant, respectively. Ex-tension of the above h i j expression includes second-order multipole moment, calledcurrent-quadrupole moment, given by Thorne (1980). It describes the dynamics of themass currents that can lead to GWs emission caused by e.g., the r-mode instability(discussed in Section 4).Equation (1) shows that an axisymmetric NS rotating along its axis will not emitGWs, because its mass-quadrupole moment will not vary in time. The GW luminosity L GW is L GW = − dEdt = G c (cid:104) ... Q i j ... Q i j (cid:105) ∼ G (cid:15) I ν c , (2)with (cid:104) ... (cid:105) brackets denoting time averaging, and a dimensionless parameter (cid:15) quantify-ing the level of asymmetry. The moment of inertia along the rotational axis I scaleswith NS mass M and radius R as I ∼ MR ; ν is NS rotational frequency. An estimateof the GW strain amplitude is thus h ∼ G (cid:15) I ν c d , (3)which is inversely proportional to the distance to source d . Propagation of the GWs invacuum is governed by a standard wave equation: (cid:32) ∂ ∂ t − ∇ (cid:33) h i j = (cid:3) h i j = , (4)for which the simplest solution is a plane wave solution: h i j ( t , x ) = A i j cos( ω GW t − k × x + α ( i )( j ) ) (5) In Einstein’s theory, for weak gravitational fields, space-time can be described as a metric: ˜ g ij ≈ η ij + h ij , where η ij is Minkowski metric and h ij corresponds to (small) GW perturbation. In the TT gauge theperturbation is purely spatial h i =
0, and traceless h ii =
0. From the Lorentz gauge condition one can implythat the spatial metric perturbation is transverse: ∂ i h ij = k is a wave 3-vector defining the propagation direction, related to the wavelength λ as λ | k | = π , x is a 3-vector of coordinates, A i j is constant amplitude and α ( i )( j ) is theconstant initial phase. ω GW = π f GW is rotational (angular) frequency, while f GW is afrequency of the gravitational wave.In the TT gauge, the above equation can be rewritten in the following form thatdepends on two independent polarisations of the GW: plus ‘ + ’ and cross ‘ × ’ (see ex-planation in Jaranowski & Królak 2009): h TTi j ( t , x ) = h TTi j ( t − z / c ) = h + ( t , x ) h × ( t , x ) 00 h × ( t , x ) − h + ( t , x ) 00 0 0 0 (6)Here h + and h × are the polarisations of a plane wave moving in the + z direction: h + ( t , x ) = A + cos (cid:20) ω GW (cid:18) t − zc (cid:19) + α + (cid:21) , h × ( t , x ) = A × cos (cid:20) ω GW (cid:18) t − zc (cid:19) + α × (cid:21) . (7)Full derivation can be found e.g., in Misner et al. (1973); Bonazzola & Gourgoul-hon (1996); Jaranowski & Królak (2009); Prix (2009).The response of the GW detector h ( t ) to the wave described with above formulæcan be expressed as (Schutz & Tinto, 1987; Thorne, 1987; Jaranowski & Królak, 1994,2009): h ( t ) = F × ( t ) h × ( t ) + F + ( t ) h + ( t ) (8)where F + and F × are the detector’s antenna-pattern (beam-pattern) functions that de-scribe its sensitivity to the wave polarisation h + and h × (Zimmermann & Szedenits,1979; Bonazzola & Gourgoulhon, 1996; Jaranowski et al., 1998): F + ( t ) = sin ζ (cid:16) a ( t ) cos(2 ψ ) + b ( t ) sin(2 ψ ) (cid:17) , (9) F × ( t ) = sin ζ (cid:16) b ( t ) cos(2 ψ ) − a ( t ) sin(2 ψ ) (cid:17) . (10)Note that h + and h × depend on the mechanism of gravitational radiation and will bediscussed later in this review, while F + and F × are periodic functions with the periodequal to one sidereal day (for ground-based detectors such as the LIGO or Virgo), dueto the rotation of the Earth. Additionally, F + and F × depend on the wave polarisationangle ψ , the angle between detector’s arms ζ (usually ζ = π/ a ( t ) and b ( t ), which depend on the location and orientation of thedetectors on Earth and the position of the GWs source on the sky (full representationof a ( t ) and b ( t ) can be found e.g., in Jaranowski & Królak 2009, Appendix C). Orderof magnitude of frequency variations due to the daily and annual motion of the Earthis thus 10 − Hz and 10 − Hz, respectively.
First indirect evidence of the existence of GWs was deduced from the observationsof stars: binary systems containing white dwarfs (Paczy´nski, 1967), and later neutron3tars, most notably the Hulse-Taylor pulsar (catalogue number: PSR B1913 +
16, Hulse& Taylor 1975). The loss of orbital energy manifests itself as the shrinkage of the orbitand in the result as a drop of the orbital period (Peters & Mathews, 1963):˙ P GR = − π G / c (cid:18) P π (cid:19) − / (1 − e ) − / (cid:32) + e + e (cid:33) M M ( M + M ) − / , (11)where P — orbital period, e —orbital eccentricity, M and M —masses of the compo-nents. In 30 years of observations (1975-2005) observed orbital decay was consistentup to 0 . ± .
21% level with the theoretical prediction for the emission of GWs (Weis-berg & Taylor, 2005). The observed orbital decay (Taylor & Weisberg, 1982, 1989;Weisberg et al., 2010) is in an excellent agreement with the expected loss of energydue to the GW radiation as described by the general relativity.In 2015, the real GW astronomy began with the first direct detection of the bi-nary BH coalescence, GW150914 (Abbott et al., 2016a). The signal was registered bytwo LIGO interferometers (Aasi et al., 2015b) and analysed jointly by the LIGO andVirgo Collaborations (LVC). Thus far, nine more binary BH mergers (Abbott et al.,2016b, 2017a,b,c, 2018b) were reported. Virgo Collaboration operates a third detec-tor of the global network (Acernese et al., 2014). A second breakthrough came withthe LVC observation of the binary NS merger (Abbott et al., 2017d). A network ofthree GW detectors cooperated with many EM observatories, performing first observa-tions of GWs and a broad spectrum of EM waves from the same source (Abbott et al.,2017e,f). These unique, multi-messenger observations allowed for the first ‘standardsiren’ measurement of a Hubble constant (Abbott et al., 2017g), measurement of theGW propagation speed (Abbott et al., 2017d), discovery and study of the closest kilo-nova event (Abbott et al., 2017h), estimation of the progenitor properties (Abbott et al.,2017d,i, 2019a) and study of the post-merger remnant (Abbott et al., 2017j, 2019h).Recently, LVC published the first catalogue with all the previous detections andsource parameters’ estimates .At the time of writing, second-generation of interferometers: the Advanced LIGO(aLIGO, Harry et al. 2010) and Advanced Virgo (AdV, Acernese et al. 2014) - are col-lecting data. ‘Second-generation’ refers to the strongly improved versions of the ini-tial, first-generation detectors. First-generation observatories were the following GWinterferometers: TAMA (300-m arms) near Tokyo, Japan (Takahashi, 2004), GEO600(600-m arms) near Hannover, Germany (Willke, 2002), Virgo (3-km arms) near Pisa,Italy (Freise et al., 2005), and LIGO (two instruments with 4-km arms each) in Han-ford and Livingston, US (Abramovici et al., 1992). Soon, next detectors will jointhe global GW network: the Japanese collaboration that built TAMA is now testingthe 2nd-generation underground and cryogenic detector KAGRA—KAmioka GRAv-itational wave telescope (Akatsu et al., 2017) and the third LIGO interferometer willbe placed in India and operated by the Indian Initiative in Gravitational-wave Obser-vations (IndIGO) in near future (Unnikrishnan, 2013). Next step will be to design,build and operate third-generation detectors, with the planned sensitivity an order ofmagnitude better than the second-generation detectors. A European consortium is inthe design stage of the Einstein Telescope (ET)—the underground trio of triangular . http://gwplotter.com ). The expected CGWs am-plitudes ranges of pulsars are marked with the pastel-blue region. For comparison,the GW150914 signal strain is represented by the pink outline.interferometers, each 10 km long (Sathyaprakash et al., 2012). Comparison of the cur-rent and future ground-based detectors characteristic strains in the context of CGWssensitivities (denoted as ’Pulsars’ on the plot) is shown in Figure 1.Underground interferometers will probe GW frequencies down to ∼ . × km long) of threesatellites that will be placed in a solar orbit at the same distance from the Sun as theEarth (Amaro-Seoane et al., 2017). The mean linear distance between the formationand the Earth will be 5 × km.Currently, ongoing upgrades of the existing detectors, new detectors in the network,and improvements of the data analysis methods will lead to an increase in sensitivityand better quality of the detector data and, as a result, to the detections of the weakersignals. GW sources may be divided in to four categories, depending on the dura-tion and strength of the signal, as shown in Table ?? : continuous gravitational waves(CGWs), subject to this review; stochastic background GW which is a mixture of alarge number of independent sources; inspirals and mergers of binary systems; burstsources e.g., supernovæ explosions or magnetar flares. Search strategies for each type5nown waveform Unknown waveformLong-lived Rotating neutron stars Stochastic background(continuous) h ∼ − h ∼ − Short-lived Compact binaries coalescences Supernovæ( T ∼ . h ∼ − h ∼ − Table 1: General taxonomy of GW sources and their expected GW strain order-of-magnitude amplitude h .depend on the duration of GWs emission, knowledge of the signal model, characteristicamplitudes of the GWs, and the computational resources.So far only compact objects’ inspirals were detected in the LIGO-Virgo observa-tional runs. It is expected that in the next LIGO and Virgo observing seasons, moresubtle signals, such as CGWs or a stochastic background, will be detected. Steady im-provement of the search methods and sensitivity of the detectors was demonstrated inthe past: searches for CGWs (Abbott et al., 2017k, 2018c, 2019b,c), stochastic back-ground (Abbott et al., 2017l, 2018d) and burst signals (Abbott et al., 2017m, 2019d).Even though no significant detections were claimed, astrophysically interesting upperlimits were determined. They will be discussed in the next sections. Pioneering hypothesis of existence of the dense stars, that look like giant atomic nu-clei, was given by Landau (1932), even before the discovery of a neutron . Exis-tence of the NSs, as remnants of the supernovæ explosions, was proposed by Baade &Zwicky (1934), just after the breakthrough discovery of a neutron by Chadwick (1932).This hypothesis waited more than 30 years to be confirmed, when Jocelyn Bell Burnellfirst discovered a ‘rapidly pulsating radio source’ (Hewish et al., 1968a,b), which wasinterpreted subsequently as a fast-spinning NS that emits a beam of electromagneticradiation similarly to a lighthouse. Since then, the nature of NSs is still a mystery.The extremely large densities and pressures present inside NSs cannot be reproducedand tested in terrestrial laboratories. We also do not have a credible theoretical descrip-tion of how matter behaves above the saturation density of nuclear matter n = . − (density at which energy per nucleon of infinite, symmetric nuclear matter has aminimum).By measurements of the NSs masses M and radii R one can, in principle, determineproperties of the matter inside NSs, such as the relation between its pressure P anddensity ρ (the currently unknown equation of state of dense matter, EOS; for a textbookintroduction to the subject, see Haensel et al. 2007). Bijective functions P ( ρ ) and M ( R )are schematically represented on Fig. 2.Putting constraints on EOS from observations can be done using the fact that M ( R )is a bijective function to P ( ρ ), typically by solving hydrostatic equilibrium for a spher-ically symmetric distribution of mass, named Tolman-Oppenheimer-Volko ff (TOV) For the chronology of the events see Yakovlev et al. (2013). M ( R ) relation( right ), from which currently unknown P ( ρ ) relation ( left ), so-called equation of state,can be established.equation (Tolman, 1939; Oppenheimer & Volko ff , 1939): dP ( r ) dr = − Gr (cid:32) ρ ( r ) + P ( r ) c (cid:33) (cid:32) M ( r ) + π r P ( r ) c (cid:33) (cid:32) − GM ( r ) c r (cid:33) − , (12)constrained by the equation for the total gravitational mass inside the radius r : dM ( r ) dr = πρ ( r ) r . (13)Descriptions of the various methods used in research on EOS determination can befound in e.g., the review by Özel & Freire (2016). From the EM observations we knowthat any realistic EOS should support NSs with masses around 2 solar masses (Demor-est et al., 2010; Antoniadis et al., 2013; Fonseca et al., 2016). Uncertainties in the NSsradii and crust properties reflecting our limited knowledge of the EOS are discussede.g., by Fortin et al. (2016).Although this review focuses on the long-duration CGWs, for the sake of complete-ness, in the remainder of this section, we will provide a brief description of the tidaldeformability e ff ect imprinted on the GWs emitted during the last orbits of a binaryNS coalescence. So far, one measurement of this kind—the GW170817 event—wassuccessfully performed and published (Abbott et al., 2017d, 2018f, 2019a), whereasa second event, dubbed S190425z, was recently reported by the LVC via the publicdatabase service (LIGO / Virgo GraceDB, 2019). For the recent theoretical studies con-cerning the interpretation of the GW170817 tidal deformability measurement and itsrelation to the dense-matter EOS, see Annala et al. (2018); Burgio et al. (2018); De etal. (2018); Fattoyev et al. (2018); Lim & Holt (2018); Malik et al. (2018); Most et al.(2018); Paschalidis et al. (2018); Raithel (2018); Han & Steiner (2019); Montana et al.(2019); Sieniawska et al. (2019a).In a binary system, a quadrupole moment of each NS is induced by the companionNS, due to the presence of the external tidal field ε i j (Misner et al., 1973; Hinderer,7008): Q i j = − λ td ε i j , (14)where the proportionality factor λ td is called the tidal deformability parameter, ex-pressed in the lowest order approximation as λ td = R k . (15)The parameter k is the quadrupole ( l =
2) tidal Love number (Love, 1911; Flana-gan & Hinderer, 2008): k = x (1 − x ) (cid:0) − y + x ( y − (cid:1)(cid:16) x (cid:0) − y + x (5 y − (cid:1) + x (cid:16) − y + x (3 y − + x (1 + y ) (cid:17) + − x ) (cid:0) − y + x ( y − (cid:1) ln(1 − x ) (cid:17) − , (16)with the compactness of object x = GM / Rc ( M denoting the gravitational mass, R —radius), and y —a solution of dydr = − y r − + π Gr / c ( P / c − ρ )( r − GM ( r ) / c ) y + (cid:32) G / c ( M ( r ) + π r P / c ) √ r ( r − GM ( r ) / c ) (cid:33) + r − GM ( r ) / c − π Gr / c r − GM ( r ) / c ρ + P / c + (cid:16) ρ + P / c (cid:17) c ρ dP / d ρ , (17)evaluated at the stellar surface r = R (Flanagan & Hinderer, 2008; Van Oeveren &Friedman, 2017). For convenience, a dimensionless value of the tidal deformability Λ is often defined as Λ = λ td (cid:16) GM / c (cid:17) − . (18)Tidal contribution to the GW signal are extracted from the last stages of the in-spiral phase. What is actually measured is, in the first approximation, the e ff ective tidal deformability ˜ Λ , a mass-weighted combination of individual dimensionless tidaldeformabilities Λ , Λ , defined as˜ Λ = M + M ) M Λ + ( M + M ) M Λ ( M + M ) , (19)with M , M denoting the component gravitational masses . In principle, by solvingthe above equations, one can relate the measurable value of ˜ Λ to the EOS parameters P and ρ . Note that the most precisely measured quantity from an inspiral phase is a so-called chirp mass: M = ( M M ) / ( M + M ) / . Using Newton laws of motion, Newton universal law of gravitation, and Einsteinquadrupole formula, one can see that M depends only on GW frequency f GW and its derivative ˙ f GW — .4 General Information about Continuous Gravitational Waves According to the ATNF (Australia Telescope National Facility) Pulsar Database (Manch-ester et al., 2005), more than 2700 pulsars are known. Assuming that CGW signalprimarily corresponds to twice the NS spin frequency (in case of elastic deformations,see Section 2) or close to 4 / ν and ˙ ν . We also know from evolutionary arguments that the Galaxycontains ∼ –10 NSs (Narayan, 1987; Camenzind, 2005), of which ∼ ν max ≈
700 Hz. Two possibilities were considered in Haskell et al. (2018a): (a) ν max corresponds to a maximal allowed spin, above which the centrifugal forces causes massshedding and destroy the star (also called the Keplerian frequency). As was shownin Haskell et al. (2018a), ν max cannot be less than ≈ ≈
700 Hz is not consistent with minimal physicalassumptions on hadronic physics; (b) presence of additional spin-down torques actingon the NSs, possibly CGWs emission.It means that in our Galaxy numerous promising CGWs sources are hidden andawaiting for detections of their gravitational signatures.Rotating non-axisymmetric NSs (Ostriker & Gunn, 1969; Melosh, 1969; Chau,1970; Press & Thorne, 1972; Zimmermann, 1978) are expected to emit CGWs dueto the existence of time-varying mass-quadrupole moment. Such signals have smallercharacteristic amplitudes than signals emitted from compact binary mergers, but theiroverall duration is much longer. In the case of almost-monochromatic CGWs, theirintegrated signal-to-noise ratio (
S NR ) increases with the observation time T as S NR ∝ h (cid:114) TS , (20)where h is the instantaneous GW strain amplitude and S is the amplitude spectraldensity of the detector’s data signals’ frequency. For comparison, GW150914 lastedfor T ∼ . h ∼ − , yielding S NR ∼
24. For CGWs, the expected amplitude is a few order ofmagnitudes smaller, h ∼ − , but the data collection lasts for T of the order of months quantities determined directly from the observational data: M = c G (cid:20)(cid:16) (cid:17) π − f − GW ˙ f GW (cid:21) / . Informationabout the individual masses is taken from the waveforms filtering, including post-Newtonian expansion.That is why M determination has the smallest errors, while M , M estimations are model-dependent andgenerate relatively big errors, e.g., for the GW170817 event individual masses (for the low-spin priors) wereestimated as: M ∈ (1 . , .
60) M (cid:12) and M ∈ (1 . , .
36) M (cid:12) , while chirp mass M = . ± . (cid:12) (Abbott et al., 2018f, 2019a). The ATNF Pulsar Database website: .
9r even years. This is one of the incentives for consideration of the CGWs as seriouscandidates for the future detections (Brady et al., 1998; Jaranowski & Królak, 2000).By simple manipulation with Equation (20), one can roughly estimate the obser-vational time that is needed to observe signals considered in this paper. Let us put T ∝ S ( S NR / h ) . While writing this paper, currently ongoing O3 observational runtakes place and LIGO detector reaches √ S ∼ − Hz − / (see Abbott et al. 2019i alsofor comparison of sensitivities of Virgo and KAGRA detectors, as well as predictionsfor the future observing seasons). Typically, a threshold for the CGW detection is set S NR th =
5. For the most promising scenario of CGW emission (triaxial ellipsoid, seeSection 2), h ∼ − (see Equation (27)). As a result, in order to detect such a signal,one needs T ∼
300 days of good quality, coherent data (note that O3 run is scheduledfor 1 year, what makes the future CGW searches very promising). Analogously forthe magnetic field distortions model (Equation (35)), where h ∼ − , almost 80 mlnyears of data is needed! Detectability of the CGWs signals depends on the observational time (Equation (20)),but also on the balance between computational cost of the accurate data analysis andcomputational resources. For some isolated NSs the relevant parameters, such as skyposition (e.g., right ascension α and declination δ ), rotational frequency ν ( f GW is,in case of an elastic deformation, a mixture of 1 ν and 2 ν , see Section 2, or f GW = / ν for r-modes, see Section 4) and spin-down ˙ ν are known from EM observations .For these objects, dedicated targeted searches are performed, in order to check if aCGW signal is associated with known parameters of the individual pulsar (Nieder etal., 2019). Such searches are computationally easy to perform. A slight modificationto the targeted searches are so-called narrow-band searches, which allow for a smallmismatch between the frequency parameters known from the EM observations and theexpected GW signal.Another type of search strategy, in which position of the source in the sky is as-sumed, but the frequency parameters are unknown, is called a directed search. It maybe applicable to e.g., a core-collapse supernovæ remnants. Example of a young andrelatively close supernova remnant, for which spin frequency (and its derivatives) areunknown, is the Cassiopeia A remnant (Fesen et al., 2006). As was shown in Wetteet al. (2008), CGW strain depends on the assumed age a and distance d . Additionalintrinsic parameters such as NS mass or its equation of state increase uncertainty forthe GW strain of the strongest possible signal h age , by 50%: h age ≈ . × − (cid:32) .
30 kpc d (cid:33) (cid:32)
300 yr a (cid:33) / . (21) Of course, the whole picture is more complex when binary system is considered since in that case alsothe binary orbital parameters that additionally modulate the CGW signal have to be taken into account.In this review we focus only on the isolated NSs. Leverage of searches for CGWs signals from isolatedobjects, in order to identify and follow-up signals from NSs in binary systems were investigated in Singh etal. (2019). ff erentCGWs searches strategies.Additionally, young and hot NSs may become unstable and undergo various dy-namical processes (e.g., cooling processes and oscillations—see Section 4). Full real-istic description of the CGWs emission, in presence of multiple physical phenomenamay require inclusion of higher frequency derivatives and in general is extremely hardto model. Nevertheless, searches were performed for the CGWs from known super-novæ remnants (Abadie et al., 2010; Zhu et al., 2016; Ming et al., 2016, 2018, 2019;Abbott et al., 2019h), as well as in the direction towards the Galactic Centre region,where massive stars (progenitors of supernovæ explosions) are found in stellar clus-ters (Aasi et al., 2013; Dergachev & Papa, 2019c).The most computationally intensive searches are all-sky (blind) searches, since onlyminimal assumptions are made to search for the signals from a priori unknown sources.Such a search requires well-optimised and robust tools, because the less is known aboutthe source, the smaller sensitivity of the search can be achieved and bigger computa-tional cost is required. All types of searches are summarised in Figure 3. Severalmethods (described below) were so far tested and used in blind searches on mock andreal data. Each all-sky search has di ff erent advantages in the sensitivity vs. robustnessagainst complexity of the assumed CGWs emission models, as was compared in Walshet al. (2016).Several search methods were developed for the CGWs signals originated in isolatedNSs (very extensive comparison of the sensitivity of various searches can be foundin Dreissigacker et al. 2018): • The F -statistic method introduced in Jaranowski et al. (1998). The F -statistic11s obtained by maximizing the likelihood function with respect to four unknownparameters of the simple CGW model of rotating NSs—CGW amplitude h , ini-tial phase of the wave Φ , inclination angle of NS rotation axis with respect tothe line of sight ι , and polarisation angle of the wave ψ (which are henceforthcalled the extrinsic parameters). This leaves a function of only four remainingparameters: f GW , ˙ f GW , δ and α (called the intrinsic parameters). Thus the di-mension of the parameter space that we need to search decreases from 8 to 4.To reduce computational cost and improve method e ffi ciency, the F -statisticcan be evaluated on the 4-dimensional optimal grid of the intrinsic parame-ters (Pisarski & Jaranowski, 2015). As was shown in Equation (20), strengthof the signal depends on the observational time: on the one hand by increas-ing T one can expect a detection of weaker signals, on the other hand how-ever, analysing long-duration data requires substantial computational resources,e.g., for Polgraw time-domain F-statistic pipeline , computational costfor an all-sky search scales as ∼ T log( T ). Promising strategies to solve thisproblem are hierarchical semi-coherent methods, in which data is broken intoshort segments. In the first stage, each segment is analysed with the F -statisticmethod. In second stage, the short time segments results are combined incoher-ently using a certain algorithm. Several methods were proposed for the secondstage: search for coincidences among candidates from short segments (Abbottet al., 2007, 2009a), stack-slide method (Brady et al., 1998; Brady & Creighton,2000; Cutler et al., 2005), PowerFlux method (Abbott et al., 2008, 2009b) withthe latest significant search sensitivity improvements for O1 data (Dergachev& Papa, 2019a,b), global correlation coordinate method (Pletsch, 2008; Pletsch& Allen, 2009), Weave method (Wette et al., 2018; Walsh et al., 2019). Inde-pendently of the details, the main goal of the F -statistic method is to find themaximum of F ( f GW , ˙ f GW , δ, α ) function, and hence the parameters associatedwith the signal. Several optimisation procedures (such as optimal grid-based ornon-derivative algorithms) were implemented in such analyses (Astone et al.,2010a; Shaltev & Prix, 2013; Sieniawska et al., 2019b). F -statistic can be evalu-ated on the time-domain data (Jaranowski et al., 1998; Astone et al., 2010a; Aasiet al., 2014; Pisarski & Jaranowski, 2015; Abbott et al., 2017k, 2019b) and thefrequency-domain data (Brady et al., 1998; Brady & Creighton, 2000; Abbott etal., 2004, 2007, 2009a, 2017p). The main di ff erence between these two conceptsis that in the time-domain the information is distributed across the entire data set,while the frequency-domain analysis focuses on the part of the data around thefrequency at which the peak appears. The data is initially calibrated in the time-domain and to be used by the frequency-domain methods, usually it is convertedwith the Fourier Transform methods. • The Hough transform (Hough, 1959, 1962) is a widely used method to detectpatterns in images. It can be applied to detect the CGWs signals in specificrepresentations of the data: on the sky (Krishnan et al., 2004), and in frequency-spin-down plane (Antonucci et al., 2008; Astone et al., 2014). Both types of theHough transform method, called
SkyHough and
FrequencyHough , are typically Project repository: https://github.com/mbejger/polgraw-allsky . F -statistic are matched-filteringtype methods. Due to limited computational power, they require division of datainto relatively short segments. Interesting application of the Hough transformto the unknown sources searches was introduced in Miller et al. (2018). This Generalised FrequencyHough algorithm is sensitive to the braking index n ,a quantity that determines the frequency behaviour of an expected signal as afunction of time. In general, the evolution (decrease) of rotational frequency isdescribed as ˙ ν = − K ν n , (22)where K is a positive constant. Time derivative of the above equation providesthe dependency of the braking index on measurable quantities (from EM obser-vations, e.g., Espinoza et al. 2011; Hamil et al. 2015; Lasky et al. 2017; Anders-son et al. 2018), frequency and its higher derivatives: n = ν | ¨ ν | ˙ ν . (23)Value of the braking index reveals a spin-down mechanism: n = n =
3, if the spin-down is dom-inated by dipole radiation (as in the case of dipolar EM field); n = n = n = Einstein@Home project (Abbott etal., 2009a), a volunteer-based distributed computing project devoted to searchingfor CGWs. • The 5-vector method (Astone et al., 2010b), in which detection of the signal isbased on matching a filter to the signal + and × polarization Fourier components.The antenna response function depends on Earth sidereal angular frequency Ω ⊕ and results in a splitting of the signal power among five angular frequencies ω GW , ω GW ± Ω ⊕ and ω GW ± Ω ⊕ , where ω GW = π f GW . This method is typically usedfor narrowband and targeted searches. • The Band Sampled Data (BSD) method, is dedicated for the directed searches,or those assuming limited sky regions, such as the Galactic Centre (Piccinni etal., 2019). The application of this method results in a gain in sensitivity at a fixedcomputational cost, as well as gain in robustness with respect to source parameteruncertainties and instrumental disturbances. From the cleaned, band-limited anddown-sampled time series, collection of the overlapped short Fourier Transformsis produced. Then, the inverse Fourier Transform allows removing overlap, edge Project webpage: http://einstein.phys.uwm.edu . ff ects. Demodulation of the signal from the Doppler and spin-down e ff ects can be done e.g., by using heterodyne technique (see below). Whilein the F -statistic method one could manipulate with the search sensitivity by in-creasing the observation time, BSD method works in Fourier-domain and analo-gously S NR can be improved by increasing length of frequency bands (for com-parison: bandwidth in F -statistic method is typically ∼ .
25 Hz and in BSD ∼ • The time-domain heterodyne method (Dupuis & Woan, 2005) is a targeted searchwhich uses the EM measurements of ν , ˙ ν and ¨ ν (model of the phase evolution,Equation (26), assumes k = h , ψ , Φ and ι . Due to the Earth’s rotation, amplitude of the signal recorded byan interferometric detector is time-varying since the source moves through theantenna pattern (see Equations (8)–(10)). These variations, in the heterodynemethod, are used to find characteristic frequency which is the instantaneous sig-nal frequency, register at the detector. Additionally, frequency of the signal seenin the detector is a ff ected due to the Earth motion. Second important step ofthe demodulation is to remove the Doppler shifts (correct signal time-of-arrival).A targeted search is performed with a simple Bayesian parameter estimation:first the data is heterodyned with an expected phase evolution and binned toshort (e.g., 1 min) samples. Then, marginalisation over the unknown noise levelis performed, assuming Gaussian and stationary noise over su ffi ciently long (e.g.,of the order of 30 minutes) periods. 95% upper limit is defined, inferred by theanalysis, in terms of a cumulative posterior, with uniform priors on orientationand strain amplitude. At the end the parameter estimation is done by numericalmarginalisation. E ff ective and commonly used algorithm for the last marginal-isation stage is called Markov Chain Monte Carlo (Abbott et al., 2010; Ashton& Prix, 2018), in which the parameter space is explored more e ffi ciently andwithout spending much time in the areas with very low probability densities.As was mentioned previously, this paper is about CGWs emission from isolatedNSs. However, it is worth to mention that NSs in binary systems are also consideredas a serious candidate for sources of CGWs. NSs in binary systems have an addi-tional modulation due to the NSs movement around the binary barycenter (at leastthree additional parameters). To deal with the high computational cost, due to thebigger parameter space, search strategies usually rely on semi-coherent methods andare dedicated for known candidates (targeted / directed searches). Main proposed algo-rithms are: TwoSpect method (Goetz & Riles, 2011; Meadors et al., 2017), CrossCorrmethod (Whelan et al., 2015; Meadors et al., 2018), Viterbi / J -statistic (Suvorova etal., 2016, 2017) or the Rome narrow-band method (Leaci et al., 2017). Several di-rected searches for CGWs from NSs in known binary systems, such as Scorpius X-1,were already performed in the past (Abbott et al., 2017o; Meadors et al., 2017).14 Elastic Deformations
The simplest model of the NS CGWs emitter is described by a rigidly rotating alignedtriaxial ellipsoid, radiating purely quadrupolar waves (Figure 4). As was mentioned inSection 1.1, signal expected to be observed on the Earth will have two polarisations h × ( t ) and h + ( t ), which depend on the emission mechanism (Equation (8)): h + ( t ) = h (cid:16) + cos ι (cid:17) cos( Φ ( t ) + Φ ) (24) h × ( t ) = h cos ι sin( Φ ( t ) + Φ ) , (25)where ι is an inclination angle of NS rotation axis with respect to the line of sight, Φ ( t ) + Φ is the phase of the wave ( Φ being the initial phase), expressed as a truncatedTaylor series, Φ ( t ) = s (cid:88) k = f ( k ) GW t k + ( k + + n × r d ( t ) c s (cid:88) k = f ( k ) GW t k ( k )! , (26)where f ( k ) GW is a k -th frequency time-derivative at the Solar System Barycentre (SSB)evaluated at t = f (1) GW = ˙ f , f (2) GW = ¨ f , ...), n is a constant unit vector in the directionof the NS in the SSB reference frame (it, therefore, depends on the sky position of thesource) and r d is a vector joining the SSB with the detector.The parameter h is a constant GW strain, which can be estimated from Einstein’squadrupole formula (Equation (1)). For the triaxial ellipsoid model it is given by thefollowing formula: h = π Gc I f GW (cid:15) d = . × − (cid:18) (cid:15) − (cid:19) (cid:18) P
10 ms (cid:19) − (cid:32) d (cid:33) − , (27)where the deformation (also called ellipticity) is quantified by (cid:15) = ( I − I ) / I . Quan-tities I , I , I are the moments of inertia along three principal axes of the ellipsoid,with I aligned with the rotation axis (see Figure 4). The symbol d denotes the dis-tance to the source, and f GW = ν is the GW frequency, equal to twice the rotationalfrequency of the star . The NS angular frequency is given by relation ω = πν = π/ P ,where P is the spin period.From the right-hand side of Equation (27) one can notice that the ellipticity (cid:15) de-pends on the interior properties of the NS. It carries information on how easy it is todeform NS or, equivalently, about the sti ff ness of the matter. So far, exact values of (cid:15) We consider here density perturbations, which a ff ect the spherical shape of the star δρ = Re { δρ lm ( r ) Y lm ( θ, φ ) } , where Y lm ( θ, φ ) denotes spherical harmonics. The multipole moment of the perturba-tion along radius coordinate r is Q lm = (cid:82) δρ lm ( r ) r l + dr . Here we focus only on the lowest-order perturbation Q , consistent with l = m =
2, for which f GW = ν . Note that in this section we consider the simplestmodel, in which rotational and I axes are aligned. In general they may be misaligned, producing additionalCGW radiation at 1 ν frequency, whose strength depends on the angle between rotational and I axes andis maximal when they are perpendicular (Bonazzola & Gourgoulhon, 1996). Such cases are consider inSection 3 and 5. Searches in the LVC data for the CGW radiation at both 1 ν and 2 ν were performed in thepast (Abbott et al., 2019g). I (cid:39) a ( x ) × MR , (28)where the compactness is redefined as x = ( M / M (cid:12) )(km / R ), and a ( x ) functions arecharacteristic for the hadronic NSs and strange stars (SSs, composed of deconfinedquarks, discussed below in this section): a S S ( x ) = + x ) / , a NS ( x ) = x / (0 . + x ) for x (cid:54) . , + x ) / x > . . (29)According to Owen (2005), maximum ellipticity (cid:15) NSmax depends on breaking strainof the crust σ max , mass M and radius R in the following manner: (cid:15) NSmax = . × − (cid:18) σ max − (cid:19) (cid:32) . (cid:12) M (cid:33) . (cid:18) R
10 km (cid:19) . (cid:34) + . (cid:32) M . (cid:12) (cid:33) (cid:32)
10 km R (cid:33)(cid:35) − . (30)For the typical NS parameters ( M = . (cid:12) , R =
10 km), the maximum ellipticity canbe approximated as (Ushomirsky et al., 2000): (cid:15)
NSmax ≈ × − (cid:18) σ max − (cid:19) . (31)However, σ max values are also ambiguous: if the NS crust is a perfect crystal withno defects, the σ max ≈ . σ max ≈ − − − (Ruderman, 1992; Kittel, 2005). Nevertheless,Equation (31) suggests that even for the extreme value of σ max ≈ .
1, one can expect (cid:15)
NSmax (cid:46) × − . 16o far, we have assumed that NSs are made of hadronic matter, whose EOS islargely unknown. Another hypothesis about the dense NS matter was proposed in theliterature almost fifty years ago and is still waiting to be proven: Ivanenko & Kur-dgelaidze (1965); Itoh (1970); Fritzsch et al. (1973); Baym & Chin (1976); Keister &Kisslinger (1976); Chapline & Nauenberg (1977); Fechner & Joss (1978) considered apossibility that in extreme conditions, e.g., under enormous pressure P, baryon matter isunstable. Protons and neutrons are made of two types of quarks: u —up and d —down(proton = uud ; neutron = udd ) and forces between quarks are mediated by gluons. Itis intuitive that the mixture of protons and neutrons can deconfine only into 2-flavourquark matter (baryons deconfine into quasi-free u and d quarks). Hypothetical objectsmade of such matter are now called quark stars (QS).However, when chemical potential grows, about half of d quarks can be trans-formed into s —strange quarks. Such conversion is due to the weak-interaction pro-cess, in contrast to the conversion of baryons into quarks, which is a strong-interactionprocess (see Chapter 8 in Haensel et al. 2007 for the review). Special attention willbe paid to the sub-class of QS, called strange stars (SSs), in which strange quarks— s coexist with quarks d and u (Bodmer, 1971; Witten, 1984; Haensel et al., 1986; Mad-sen, 1998). This unique 3-flavour mixture ( which is in weak-interaction equilibrium)would be more stable than 2-flavour quark matter, but also more stable than hadronicmatter such as, e.g., iron Fe (which is one of the most tightly bound nuclei). For thedetailed information see a review by Weber (2005). Some authors hypothesize that SSsmay be in a solid state (see e.g., Xu 2003), having also a solid crust made of ’normal’matter. Contribution of such crust to the total ellipticity will be order of a few percentcorrection to that of the internal layers.Similarly to Equation (30), one can find maximal ellipticity for the solid SSs (Owen,2005): (cid:15)
S Smax = × − (cid:18) σ max − (cid:19) (cid:32) . (cid:12) M (cid:33) (cid:18) R
10 km (cid:19) (cid:34) + . (cid:32) M . (cid:12) (cid:33) (cid:32)
10 km R (cid:33)(cid:35) − . (32)From the above, one sees that (cid:15) S Smax is expected to be a few orders of magnitudeslarger than (cid:15)
NSmax . Broad study on maximal ellipticity for multiple EOS for the New-tonian and relativistic stars was performed in Haskell et al. (2007); Mannarelli et al.(2007, 2008); Knippel & Sedrakian (2009); Johnson-McDaniel & Owen (2013), in-cluding superconducting quark matter (SQM) for which (cid:15)
S QMmax can be smaller than 10 − .Observational determination of (cid:15) max value would put strong constraints on the type andproperties of the dense matter inside compact objects.For more than 200 of known pulsars with known frequency and frequency deriva-tive from EM observations, one can make the following assumption to illustrate theorder of magnitude. Let us assume that the observed pulsars are spinning down solelydue to loss of the gravitational energy. Such a hypothetical object is denoted as the gravitar (Knispel & Allen, 2008). For a given detector sensitivity, an indirect spin-down limit can be thus established (Jaranowski et al., 1998): h sd = . × − (cid:32) d (cid:33) (cid:115)(cid:32) f GW (cid:33) (cid:32) − ˙ f GW − Hz / s (cid:33) (cid:32) I kg × m (cid:33) . (33)17ombination of Equations (27) and (33) results in a corresponding spin-down limitfor the fiducial equatorial ellipticity: (cid:15) sd = . (cid:32) I kg × m (cid:33) − (cid:32) Hz ν (cid:33) (cid:32) h sd − (cid:33) (cid:32) d (cid:33) . (34)So far, in the O1 and O2 LVC observing runs, no CGWs signals were detected,but instead interesting upper limits in a broad range of frequencies for known pulsarswere set (Abbott et al., 2017n, 2019c). For example, the currently best limits for Crabpulsar (J0534 + h sd = (1 . ± . × − resulting in (cid:15) sd = . × − .Additionally, in the O2 data analysis, for some of the pulsars (J1400-6325, J1813-1246,J1833-1034, J1952 + + ∼
1% or less energy in CGW (for more details see Table IVin Abbott et al. (2019c)).Prospective detection of CGWs, consistent with a deformed NSs, will be a break-through in several topics, mostly related to the properties of matter under huge pres-sures. It will put strong constraints on EOS, and likely may yield a distinction betweencompact stars built of the hadronic and strange matter. It will also allow for testing ofthe outer NS layer—the crust. An analysis of the crustal failure and its other propertiesmay reveal not only astrophysically important information, but also can be valuablein engineering, e.g., mechanics and material strength studies. Finally, it allows mea-suring deformation itself, describe the exact NS shape and mechanisms that triggereddeviations from a spherical shape.
In a widely accepted pulsar model, inferred from the EM observations, misalignmentbetween the global dipole magnetic field axis and the rotation axis is responsible forobserved pulsations. It is likely that for some range of conditions (described in thissection), an asymmetry in the magnetic field distribution in the interior of NS may leadto the emission of CGWs.The idea that magnetic stresses can deform a star and lead to the CGWs emis-sion was originally proposed by Chandrasekhar & Fermi (1953) and was consideredlater in Bonazzola & Gourgoulhon (1996); Jones & Andersson (2002); Cutler (2002);Haskell et al. (2008); Kalita & Mukhopadhyay (2019). The simplest model was con-sidered in Gal’tsov et al. (1984); Gal’tsov & Tsvetkov (1984), where a NS was approx-imated by a rigidly rotating Newtonian incompressible fluid body, with the uniforminternal magnetic field and the dipole external magnetic field. Inclination of the mag-netic dipole moment with respect to the rotation axis is given in this model by angle χ .Note that for misaligned rotational and magnetic axes, time-varying quadrupole mo-ment will also be present for I = I . In such conditions, the NS is a biaxial ellipsoid,illustrated in Figure 5. The ellipticity is then described by the ratio (cid:15) = ( I − I ) / I ,18nd the resulting CGW strain will take form (Bonazzola & Gourgoulhon, 1996): h = . × − β sin ( χ ) (cid:18) R
10 km (cid:19) (cid:32) d (cid:33) (cid:32) P (cid:33) (cid:32) ˙ P − (cid:33) , (35)where coe ffi cient β measures e ffi ciency of the magnetic structure in distorting the star(magnetic distortion factor). For a simplified model (incompressible fluid, uniforminternal magnetic field), parameter β equals 1 /
5. Information about the induced de-formation is encoded in β factor, as (cid:15) = β M / M , where M is the magnetic dipolemoment and M has the dimension of a magnetic dipole moment in order to make β dimensionless. It is clear that h grows when χ → χ cannotbe too small because it will break the formula down), and / or for large β . As was shownin Bonazzola & Gourgoulhon (1996), such emission occurs mostly on f GW = ν . Onecan also notice, by comparison of the prefactors of Equations (27) and (35) that theexpected CGW strain from magnetic e ff ects is almost four orders of magnitude smallerthan in the case of triaxial ellipsoid. For the above model, in case of the incompressibleNS with poloidal magnetic field ellipticity is expressed as (Haskell et al., 2008): (cid:15) ≈ A (cid:18) R
10 km (cid:19) (cid:32) M . (cid:12) (cid:33) − (cid:32) ¯ B G (cid:33) , (36)where ¯ B is the volume averaged magnetic field and factor A is a constant, characteristicto the model (e.g., A = − in case of a uniform density star, A = × − for thepolytrope n = A = − × − for the polytrope n = .It was shown that purely poloidal or purely toroidal magnetic fields are unsta-ble (Wright, 1973; Braithwaite & Spruit, 2006; Braithwaite, 2007), and that the realisticdescription of NS requires mixed field configurations (Braithwaite & Nordlund, 2006;Ciolfi et al., 2009; Lander & Jones, 2012), giving an estimate of ellipticity (Mastranoet al., 2011): (cid:15) mixed ≈ . × − (cid:32) B poloidal G (cid:33) (cid:32) − . Ξ (cid:33) , (37)where B poloidal is the poloidal component of the surface magnetic field, and Ξ is the ratioof poloidal-to-total magnetic field energy: Ξ =
Ξ = G (Shapiro& Teukolsky, 1983), so the e ff ect of magnetic deformation is significantly smaller thanthe one discussed in Section 2. NSs with magnetic fields typically of the order of 10 Gare called magnetars . Known magnetars rotate with periods of P ∼ ff erent EOS and bulk viscosity mod-els in Dall’Osso et al. (2018) show that, while the first-year spin-down of a newborn Note that in Equation (36) the function has a positive sign, which means that the poloidal magnetic fieldtends to distort a NS into an oblate shape. For a toroidal field the expression changes sign, making the NSshape prolate. χ measures the misalignment between the magnetic field axis and rotation axis.In principle for χ (cid:44)
0, CGWs emission will be produced even if I = I . Note that onthis schematic picture (cid:126) B represents the dominant toroidal field, reflecting the fact thatNS shape is prolate along the magnetic axis.NS is most likely dominated by EM processes, reasonable values of internal toroidalfield B toroidal and the external dipolar field B external can lead to detectable GWs emis-sion, for an object in our Galaxy. While the centrifugal force distorts a NS into anoblateshape, the internal toroidal magnetic field makes them prolate. B toroidal determines theNS shape if B toroidal (cid:38) . × (cid:18) ν
300 Hz (cid:19) G . (38)The above formula is important not only for the newborn NSs, but also for the NSsin binary systems, where accretion processes are active and a ff ecting magnetic field.That may happened e.g., in low-mass X-ray binaries (LMXB)—systems in which masstransfer between the companion and NS is via Roche-lobe overflow. Material in accre-tion disc formed around NS is heated so much that it shines brightly in X-rays. NSsin LMXB initially possess B external ∼ –10 G, B toroidal ∼ –10 G and theirspin and magnetic axes are nearly aligned ( θ ≈ B external below 10 G. However, in the interior, B toroidal remains unchanged. Dissipation processes dominate and at some point mag-netic axis rapidly ‘flips’ orthogonally to the rotational axis. For the full understandingof the LMXB evolution and GWs emission, one should consider comparison betweentimescales of four processes: (i) spin-down due to the GWs emission:1 τ GW = . × − (cid:18) (cid:15) B − (cid:19) (cid:18) ν (cid:19) s − ; (39)20ii) spin-down due to the EM process:1 τ EM = . × − (cid:18) B external G (cid:19) (cid:18) ν (cid:19) s − ; (40)(iii) timescale on which accretion can significantly change the NS angular momen-tum: 1 τ ACC = . × − (cid:32) ˙ M − M (cid:12) / yr (cid:33) (cid:32)
300 Hz ν (cid:33) s − , (41)where ˙ M is the accretion rate; (iiii) dissipation timescale on which the instability acts:1 τ DIS S = . × − (cid:32) Y (cid:33) (cid:18) ν
300 Hz (cid:19) (cid:18) (cid:15) B − (cid:19) s − , (42)where Y = τ DIS S (cid:15) B P is a parameter, which was found hard to estimate and (cid:15) B is thedeformation caused by the toroidal field and defined as: (cid:15) B = − (cid:32) GM R (cid:33) − (cid:90) π B toroidal dV (43)As it was shown in Dall’Osso et al. (2018), for an optimal range of (cid:15) B ∼ (1 − × − and birth spin period (cid:46) h is directly related to the X-rayluminosity (Wagoner, 1984; Bildsten, 1998; Ushomirsky et al., 2000): h ≈ . × − (cid:18) R
10 km (cid:19) / (cid:32) M . (cid:12) (cid:33) / (cid:32)
300 Hz ν (cid:33) / (cid:32) F x − erg cm − s − (cid:33) / , (44)where F x is the observed X-ray flux; note that here the information about the dis-tance is encoded in F x and both—GWs and X-rays—are falling inversely proportionalto d . Most of LMXB have spins in the relatively narrow range 270 Hz (cid:46) ν (cid:46) ∼ h Sco X-10 ≈ × − (cid:32)
540 Hz ν (cid:33) / . (45)Unfortunately, Sco X-1 spin frequency is not well constrained, it is neverthelessone of the prime targets for LVC and its GW signal was searched for in the past (Ab-bott et al., 2007; Aasi et al., 2015a; Abbott et al., 2019f). During the latest and the21ost sensitive search (Abbott et al., 2019f), no evidence for the GWs emission wasfound; 95% confidence upper limits were set at h = . × − , assuming themarginalisation over the source inclination angle.As was mentioned several times in this section, CGWs created by the magneticprocesses typically have amplitudes smaller than the sensitivity of Advanced LIGO andAdvanced Virgo. However, it may be within the reach of a future improved network ofdetectors. Successful detection of such signals would be an amazing tool for probingthe magnetic fields in NSs: their composition (poloidal, toroidal), strength and evo-lution. Indirectly one could test deformability and compressibility of the NSs. Thesestudies will be also complementary to EM observations of special NSs classes: mag-netars, young NSs and NSs in LMXB systems. Closer look through the GW analysiscould possibly allow for testing of accretion, precession and spin evolution processes. In asteroseismology, the Rossby modes (also called r-modes, Rossby 1939) are a sub-set of inertial waves caused by the Coriolis force acting as restoring force along thesurface. In the NSs the r-modes are triggered by the Chandrasekhar-Friedman-Schutzinstability (Chandrasekhar, 1970; Friedman & Schutz, 1975). This instability is drivenby GW back-reaction—it creates and maintains hydrodynamic waves in the fluid com-ponents, which propagate in the opposite direction to that of the NS rotation, producingGWs. So far, the EM observations of the oscillations in NSs are insu ffi cient in the con-text of EOS, because it is impossible to directly observe the modes or thermal radiationfrom the surface. Details of the methods and limitations of EM asteroseismologicalobservations can be found e.g., in Cunha et al. (2007). It is expected that the CGWsobservations will complement our understanding of the compact objects oscillations.The r-modes were first proposed as a source of GWs from newborn NSs (Owenet al., 1998) and from accreting NSs (Bildsten, 1998; Andersson et al., 1999). It wasshown that the r-modes can survive only in a specific temperature window, in whichthey remain unstable: at too low temperatures, dissipation due to shear viscosity dampsthe mode and when the matter is too hot, bulk viscosity will prevent the mode fromgrowing, as shown in the schematic plot in Figure 6. R-mode instability window is openfor NSs with temperatures between 10 –10 K (Owen et al., 1998; Bildsten, 1998;Bondarescu et al., 2007; Haskell, 2015). This information allows testing properties ofthe interiors of NSs and theoretical models that describe matter interactions in suchconditions. The simplest model of the CGW emission, triggered by the unstable r-modes in the newborn NSs, was introduced in Owen et al. (1998), where authors shownthat CGW amplitude in this case can be expressed by: h = . × − α (cid:32) ω √ π G ¯ ρ (cid:33) (cid:32)
20 Mpc d (cid:33) , (46)where α is dimensionless r-mode amplitude and ¯ ρ is the mean density. Whole energy of the mode is transferred to the NS spin-down and loss of the canonical angular momen-tum of the mode. ff ect the spinevolution of rapidly spinning objects (Haskell, 2015; Haskell & Patruno, 2017). Thismechanism was proposed not only for young and hot NSs, but also for the old, accret-ing NSs. In the LMXB, together with matter also the angular momentum is ‘accreted’,resulting in spinning up the object. This is thought to be the mechanism by which old,long-period NSs are recycled to millisecond periods (Alpar et al., 1982). In principle,long-lasting persistent accretion may spin NSs up to the Keplerian frequency (see pre-vious Section), but a spin frequency cut-o ff of about 700 Hz is observed (Chakrabartyet al., 2003). One of the proposed mechanisms to balance the spin evolution is theGWs emission due to the r-mode instability (Bildsten, 1998; Andersson et al., 1999).When the r-mode amplitude, defined in Equation (53), is very large, the system will belocated deeply inside the instability window, but also the evolution will be very rapidand the system will rapidly evolve out of the window (Levin , 1999; Spruit, 1999). Onthe contrary, when the r-mode amplitude is low, the evolution is slower, but system willremain close to the instability curve (Heyl, 2002). As a consequence, in both scenariosit is highly unlikely to observe a LMXB deeply inside the instability window. How-ever, according to Haskell (2015), our theoretical understanding is not fully consistentwith observations: a large number of observed LMXBs is located deeply in the r-modeinstability window (Haskell et al., 2012). Mechanisms, which can potentially explainthis inconsistency, are e.g., the presence of exotic particles in the core (Alford et al.,2012; Haskell et al., 2012), viscous damping at the crust / core boundary layer inter-face (Glampedakis & Andersson, 2006; Ho et al., 2011) or strong superfluid mutualfriction (Andersson et al., 2006; Haskell et al., 2009). An advantage of the LMXB sys-tems is that they are observed in the EM spectrum and some systems were tentativelyinterpreted as sources of r-modes recently, e.g., by Andersson et al. (2018). The studyof r-modes provides a huge opportunity to test various models of NSs interiors and thephysics behind their spin evolution.For the CGWs triggered by r-modes oscillations, the (Newtonian) relation for thelowest order (strongest) mode is f GW = / ν . This relation is independent on EOSand it is a good approximation for the slowly rotating NSs. However, according to nu-merical results, inclusion of relativistic corrections may increase f GW by a few tens ofpercent (Lockitch et al., 2003; Pons et al., 2005; Idrisy et al., 2015; Jasiulek & Chirenti,2017). Exact value depends mostly on the object compactness C (dimensionless mass-radius ratio: C ≈ . (cid:16) M . (cid:12) (cid:17) (cid:16)
10 km R (cid:17) , for NSs typically 0 . (cid:46) C (cid:46) . ff erential rotation (Stavridis et al., 2007; Idrisy Unlike in the case of the elastic deformations, what was shown in Section 2, r-modes a ff ect equation ofmotion δ v = α ( r / R ) l R ω Y Blm exp ( i ω r t ), where α is dimensionless amplitude and ω r = − ml ( l + ω is the angularfrequency in co-rotating frame. The only non-trivial solution is for l = m =
2, giving ω r = − ω , what can betransferred to the angular frequency ω i in the inertial frame: ω i = − ω + ω = ω . Dividing this expressionby 2 π gives f GW = ν . ff ness of the crust (Bildsten & Ushomirsky,2000; Levin & Ushomirsky, 2001) and EOS (Lockitch et al., 2003; Pons et al., 2005;Lattimer & Prakash, 2007; Idrisy et al., 2015). Additionally, as it was shown in Arras etal. (2003), consideration of the non-linear coupling forces among internal modes leadsto the result that r-mode signal from both newly born NSs and LMXB in the spin-downphase of Levin’s limit cycle (Levin , 1999) will be detectable by enhanced LIGO de-tectors out to ∼ ff ects in supra-thermalregime for hadronic and quark EOS were studied by Alford et al. (2010). Nonethe-less, for practical purposes (such as CGWs searches), a Newtonian approximation istypically used.Potentially, the detection of r-modes can confirm or rule out the existence of SSs,introduced and discussed in Section 2. The viscosity coe ffi cients of NSs and SSs di ff ersignificantly and as a consequence, the r-modes in a SS are unstable at lower temper-atures (Chatterjee & Bandyopadhyay, 2008; Alford et al., 2010; Haskell et al., 2012).The existence, evolution and properties of SSs is still an open question, widely dis-cussed in the literature (see e.g., Jaikumar et al. 2006; Blaschke et al. 2002). As shownin Mytidis et al. (2015), a hypothetical r-mode detection can put constraints on themoment of inertia of the compact object and bring closer to understanding the EOS.The growth time of the instability for r-modes can be relatively short (Andersson When viscosity is dominated by normal matter, then the NS enters into a limit cycle of spin-up byaccretion and spin-down by the r-mode. The timescales discussed here are related to the n = l =
2: the sim-
24 Kokkotas, 2001; Haskell, 2015): τ GW = − (cid:32) M . (cid:12) (cid:33) − (cid:18) R
10 km (cid:19) − (cid:18) P (cid:19) s , (47)while the characteristic damping timescales associated with the bulk viscosity, τ bv ,and shear viscosity, τ sv are τ bv = . × (cid:32) M . (cid:12) (cid:33) (cid:18) R
10 km (cid:19) − (cid:18) P − s (cid:19) (cid:18) T K (cid:19) − s , (48) τ sv = . × (cid:32) M . (cid:12) (cid:33) − (cid:18) R
10 km (cid:19) (cid:18) T K (cid:19) s , (49)where T is the NS core temperature. In this minimal model it is assumed that at hightemperatures, bulk viscosity due to modified URCA reactions provides the main damp-ing mechanism, while at low temperatures the main contribution is from shear viscos-ity, due to electron-electron scattering processes. Instability curve from Figure 6 canbe calculated with the following simple formula:1 τ GW + τ diss = , (50)where 1 /τ diss = /τ bv + /τ sv + additional processes . Several additional mechanismswere considered in the literature, for example the crust / core velocity di ff erence (Levin& Ushomirsky, 2001; Glampedakis & Andersson, 2006), exotic particles in the core(Andersson et al., 2010; Alford et al., 2012), or strong superfluid mutual friction (An-dersson et al., 2006; Haskell et al., 2014). These models can lead to very significantchanges in the shape, width and depth of the instability window.Another natural and astrophysically motivated targets are the binary systems inwhich the accretion is present. CGW strain from the r-modes excited in accretingsystems can be estimated as (Owen, 2010; Chugunov, 2019): h (cid:38) . × − (cid:18) R
10 km (cid:19) (cid:18) T K (cid:19) (cid:18) ν
600 Hz (cid:19) − / (cid:32) M . (cid:12) (cid:33) − / (cid:32) d (cid:33) − . (51)GWs emitted by a NS destabilised by r-modes carry the information about theinternal structure and physical phenomena inside the star and are, in principle, a verypowerful tool for testing extreme conditions of the NSs interiors, providing informationthat cannot be obtained by other means. Constraints on the EOS were considered plest illustrative model. General expressions of the timescales are (Lindblom et al., 1998): τ GW = − π G ω l + c l + ( l − l [(2 l + (cid:16) l + l + (cid:17) l + R (cid:82) ρ r l + dr ; τ sv = ( l − l + R (cid:82) η r l dr R (cid:82) ρ r l + dr − , where η = ρ / T − is the shear viscosity factor; τ bv ≈ R l − ( l + (cid:82) ζ (cid:12)(cid:12)(cid:12) δρρ (cid:12)(cid:12)(cid:12) d x R (cid:82) ρ r l + dr − , where ζ = . × − (cid:16) ω (cid:17) ρ T is thebulk viscosity factor. In principle all these timescales are sensitive to EOS due to the occurrence of density ρ in the equations,
25y e.g., Mytidis et al. (2015); the superfluid layer and crust-core interface propertiesby Haskell (2015); rotational frequency evolution in accreting binaries by Anderssonet al. (1999), spin-down of the young NSs by Alford & Schwenzer (2014).Similarly to Section 2, one can define the spin-down limit for r-modes, obtained byassuming that all of the observed change in spin frequency is due to the GWs emis-sion (Owen, 2010): h sd = d (cid:115) I ˙ P P (cid:39) . × − (cid:32) d (cid:33) (cid:32) | ˙ f GW | − Hz / s (cid:33) / (cid:32)
100 Hz f GW (cid:33) / , (52)and the corresponding r-mode amplitude parameter (Lindblom et al., 1998): α sd (cid:39) . (cid:32)
100 Hz f GW (cid:33) / (cid:32) | ˙ f GW | − Hz / s (cid:33) / . (53)From the point of view of detecting the r-modes, several methods were used so farto look for their signatures in the LVC data. For example, the F -statistic method de-scribed in Section 1.5 is so general that can be applied also for r-modes CGWs searches,e.g., during supernovæ remnant searches (Abbott et al., 2019h). No signal was found,but interesting fiducial r-mode amplitudes upper limits ( α sd ≈ × − ) were set.Searches for r-modes from pulsars with known sky position require at least threefree parameters f GW , ˙ f GW , ¨ f GW . Fortunately, for some pulsars these parameters are wellmeasured. As shown in Caride et al. (2019), the selection of appropriate ranges offrequencies and spin-down parameters is crucial. In such a case, for most pulsars fromATNF Pulsar Database, number of required GWs templates is ∼ –10 , comparablein terms of the computational cost to the CGWs searches performed in the past in theO1 and O2 observational runs for triaxial ellipsoid models (see Section 2).Additionally, a special type of oscillations may come from the newborn NSs. Fol-lowing the gravitational collapse, the proto–neutron star (PNS) radiates its bindingenergy (about 0.1 M (cid:12) ) via neutrino emission in a timescale of tens of seconds. A smallfraction of this energy can be released through violent oscillations leading to GWsemission. As shown in Ferrari et al. (2003), initial amplitude of such oscillations forthe Galactic PNSs should be larger than h (cid:38) − that it is detectable by the LVC,what is marginally consistent with numerical studies of the axisymmetric collapse ofthe core of a massive star (Dimmelmeier et al., 2002). However, such a signal is ex-pected to last for a very brief moment (tens of seconds) and its waveform will dependon multiple variables, such as the poorly known high-temperature dense-matter EOS,resulting with a computationally expensive and uncertain data analysis results. Addi-tionally, Galactic supernovæ events are rare (approximately one per century), makingthe PNSs GWs unlikely in the LVC data.It is worth metioning that r-modes are not the only oscillation type that may, in prin-ciple, lead to the CGWs emission. Initially, the Chandrasekhar-Friedman-Schutz insta-bility was considered a source of the fundamental fluid mode: the f-mode. However,to produce CGWs from the f-mode instability, very fast rotational frequency is needed,around 95% of the Keplerian frequency (Friedman, 1983; Ipser & Lindblom, 1991), aswell as hot, T (cid:38) × K in accordance with recent models for the observed cooling26f Cassiopeia A (Page et al., 2011; Shternin et al., 2011), non-superfluid matter (Ipser& Lindblom, 1991; Passamonti et al., 2013). Such a situation could be possible onlyif superfluid core is not formed yet, which would correspond to rapidly spinning, new-born NSs. Nevertheless, further works by Gaertig et al. (2011); Doneva et al. (2013)unveil that relativistic, massive ( M ≈ (cid:12) ) NSs with realistic EOS can support a widerinstability window than it was initially thought, especially for the l = m = Historically, freely precessing NSs were first considered as good sources of CGWsemission by Zimmermann (1978); Alpar & Pines (1985). Let us adopt a free precessingNS model consisting of a shell containing a liquid. For a perfectly elastic shell and theinviscid fluid, there will be no energy dissipation: such star will precess with a constantangular velocity and wobble angle.However, a realistic model requires realistic elasticity and fluidity description, as wellas the internal energy dissipation and external, astrophysical torque mechanisms bal-ancing energy dissipation. Internal dissipation means here that energy losses inside NSare converting the mechanical energy into heat. Strong interactions between vorticesand the normal component (mutual friction, mentioned already in Section 4), are ef-ficient dissipative channels that can dumped precession (Shaham, 1977; Sedrakian etal., 1999; Haskell & Sedrakian, 2018b). Reasoning for a mutual friction in the caseof type-I superconducting protons is di ffi cult because of lack of a model-independentpredictions for the domain structure and size of type-I superconductor (Haskell & Se-drakian, 2018b). However, as it was shown in Wasserman (2003), if the magneticstresses are large enough, precession is inevitable. Such enormous magnetic stressescan arise if the core is a type-II superconductor or from toroidal fields ∼ G, if thecore is a normal conductor (Wasserman, 2003). Observational evidence of free preces-sion can put strong constraints for the mutual friction and confirm weak coupling ofthe superfluid to the normal component.Additionally, if the magnetic and rotational axes are misaligned, NS will pre-cess (Mestel & Takhar, 1972). That process can be especially important in the earlylife of the millisecond magnetars (Lander & Jones, 2017, 2018). For the very youngNSs, before the crust solidifies, any elastic component in the moment of inertia tensoris allowed and e ff ective ellipticity (cid:15) B (introduced in Section 3, Equation (43)) comes In the case of precession, general expression for the ellipticity is defined as (cid:15) = ( I − I ) / I , wheremoment-of-inertia tensor is given by I ij = (cid:82) ρ ( r δ ij − x i x j ) dV and I = (8 π/ (cid:82) ρ r dr is the moment of χ from Fig-ure 5, between axis of rotation and I becomes the angle around which the rotationalaxis moves around. It is commonly called the wobble angle and denoted by θ in Fig-ure 7. Rotational frequency of the slow precession Ω p (assuming rigid rotation of theuniform-density object) can be expressed as (Mestel & Takhar, 1972): Ω p = ω(cid:15) B cos( θ ) , (54)where ω = πν . For non-rigid rotation, the centrifugal force can deform a star by itself;a quantity corresponding to size of a centrifugal distortion is (cid:15) α and both deformationcomponents are related to the global parameters of the object (Lander & Jones, 2017,2018): (cid:15) α ∼ ω R GM ∼ . (cid:18) R
10 km (cid:19) (cid:18) ν (cid:19) (cid:32) M (cid:12) (cid:33) − , (55) (cid:15) B ∼ B R GM ∼ . × − (cid:18) R
10 km (cid:19) (cid:18) B G (cid:19) (cid:32) M (cid:12) (cid:33) − . (56)Dynamical evolution of the magnetic field in the newborn NSs is crucial to un-derstand their GWs emission. Interplay between energy loss due to the inclinationangle evolution ˙ θ and the internal energy-dissipation rate due to viscous processes (seeSection 3) may lead to the precession damping and change of angle θ . As it can be de-duced from Equation (35), maximal CGWs emission can be expected when rotationaland magnetic axes are orthogonal to each other. Ref. Lander & Jones (2018) shows thatbelow 10 G, all NSs at some point of evolution have orthogonal rotational and mag-netic field axes, regardless of their birth spin frequency. Above 10 G only those NSsthat are born spinning fast enough can enter the orthogonalisation region. Addition-ally, energy and angular momentum are carried away as GWs from a freely precessingNSs (Bertotti & Anile, 1973). If the body is rigid enough, θ diminishes monotonically,but elastic behaviour may instead increase θ to π/ θ , pumping mechanisms, such ase.g., accretion, are increasing θ . As shown in Lamb et al. (1975) accretion torque canbe e ff ective in exciting large amplitude precession of a rigid body with the excitationtimescale ( τ ACC , introduced in Section 3) comparable to the spin-up timescale. The cri-terion for excitation is τ excitation < τ damping . The evolution of the wobble angle can betherefore expressed as ˙ θ = θτ excitation − θτ damping , (57)where in the simplest model one can assume τ excitation = τ ACC and τ damping = τ CP .In principle, inequality τ excitation < τ damping can be used to set upper limits for the GWsamplitudes (for the NSs with known or assumed rotational frequency, accretion rate andcrustal breaking strain). Unfortunately, such a signal will be a few orders of magnitudetoo weak to be detected by the LVC and the Einstein Telescope (Jones & Andersson,2002). inertia of the spherically symmetric density field ρ . θ and precession spin Ω p .GW strain for free precession is given by Zimmermann (1978); Jones & Andersson(2002); Van Den Broeck (2005): h = Gc ω ∆ I d θ d ∼ − (cid:18) θ . (cid:19) (cid:32) d (cid:33) (cid:18) ν
500 Hz (cid:19) , (58)where ∆ I d = I − I is the strain-induced distortion of the whole star, also called ef-fective oblateness. Wobble angle θ cannot be too big, because too big a deformationof the object (too large breaking strain σ max introduced in Section 2) may destroy NS,so θ max ∼ σ max . Generally, the CGWs emission originating in free precession pro-cess is present at frequencies f GW , = ν , f GW , = ν + ν p , where ν p is the precessionfrequency (Zimmermann, 1978); after including second order expansion ( ∼ θ ) also f GW , = ν + ν p ), according to Van Den Broeck (2005). This third frequency will beseen in GWs spectra with h almost two orders of magnitude smaller than first andsecond frequencies. According to Van Den Broeck (2005), f GW , should be detectableif NS spin-down age is much less than 10 yr and observability of f GW , and f GW , de-pends on the crustal breaking strain, which is very uncertain parameter, see Section 2.To extract information about the wobble angle θ and the deformation (cid:15) at least two ofthe above-mentioned, characteristic frequencies need to be detected.Free precession may be much longer lived ( ∼ years) than initially thought (Cut-ler & Jones, 2000), resulting with weak CGWs emission h ∼ − or smaller. Ac-cording to Jones & Andersson (2002), it is impossible to find astrophysical pumpingmechanisms capable of producing CGWs detectable by an Advanced LIGO.Some hints about the precession were delivered by EM observations of the radiopulsars. The most convincing evidence for free precession is provided by 13 yearsobservations of PSR B1828-11. Variations of arrival-time residuals from PSR B1828-11 were interpreted as a precession, with precession period ≈
250 days (Stairs et al.,2000; Akgün et al., 2006). Results obtained from observational data are consistent with29he model of precession of a triaxial rigid body, with a slight statistical preference fora prolate NS shape (Akgün et al., 2006). Also analysis of the timing data from anothersource, PSR B1642-03, also were interpreted as a precessing NS, with characteristicvariation from 3 to 7 years (Cordes , 1993; Shabanova et al., 2001).Free precession may cause deformations of the object, subsequently allowing test-ing the NS dense-matter properties, such as breaking strain, viscosity, rigidity and elas-ticity. CGW detection from free precession will allow also for better understanding ofprocesses present inside the NSs, such as heating, superfluidity, dissipation, mecha-nisms of the torque and its evolution.
In recent years we have witnessed the first direct detections of gravitational waves (Ab-bott et al., 2016b, 2017a,b,c,d, 2018b). It started a new observational field: the gravita-tional wave astronomy. Those detections corresponded to observation of coalescencesof the binary systems: violent and energetic phenomena involving black-holes, impos-sible to detect with traditional astronomical methods, as well as electromagneticallybright binary neutron stars system coalescence. As the interferometers improve theirsensitivity (Harry et al., 2010; Acernese et al., 2014; Moore et al., 2015), we expectother types of signals to be detected. One promising scenario is the detection of contin-uously emitted, periodic and almost-monochromatic GWs produced by rotating NSs.Several mechanisms can be responsible for such a GWs emission: elastically and / ormagnetically driven deformations (mountains on the NS surface supported by the elas-tic strain or magnetic field), thermal asymmetries due to accretion, free precession orinstabilities leading to modes of oscillation (r-modes).Discovery of a persistent source will be the capstone of GW astronomy will al-low for repeatable observations, which will result in acquiring more knowledge andbetter understanding of the NSs interiors, especially on their crust physics and elasticproperties at low temperatures (study of cold EOS as opposed to hot EOS in binary NScollisions). In general, these NSs may have di ff erent masses and matter conditions thanthe objects in binary NS inspirals and mergers, which leads to valuable GWs observa-tions in di ff erent dynamical regimes and, additionally, to various tests of the generalrelativity by exploiting the fact that detectors are moving with respect to the source(e.g., studies of independent wave polarizations). In principle, NSs observed in CGWscould be also a valuable tool in the detectors’ strain calibration, astrophysical distanceladder calibration (useful in cosmology) and measurements of fundamental aspects ofgravity (Pitkin et al., 2016; Isi et al., 2017).The most promising CGWs emission scenario (giving the highest GW amplitude)assumes deformations on the NS surface. Because of that, most of the LVC e ff ort isconcentrated on searches of signals consistent with the model of triaxial ellipsoid radi-ating CGWs at twice the spin frequency (Abbott et al., 2017k, 2018c, 2019b, 2017c).However, in the future, when LIGO and Virgo detectors will reach their design sensitiv-ity and new instruments (ground- and space-based, Akatsu et al. 2017; Amaro-Seoaneet al. 2017; Sathyaprakash et al. 2012; Unnikrishnan 2013) will join the network, test-ing of other models and CGWs sources will be possible.30dditionally, by combining CGWs and EM observations one can break the de-generacy in some expressions considered in this paper. For example, for the triaxialellipsoid model, by knowing distance and spin frequency from EM observations andCGW amplitude from GWs detections, immediately one can obtain size of the defor-mation. This quantity is a few order of magnitudes di ff erent for NSs and SSs and car-ries information about the crust properties. For the strongly magnetised objects, suchas magnetars, newborn NSs or NSs in LMXB systems joint EM and CGWs analysiswill allow for testing magnetic field composition, strength and evolution. Also unstabler-modes carry information about the NSs interiors: their superfluid layer, viscosity andtemperature. Finally, the precession—in the case of detection may expose informationabout magnetic field, elasticity and superfluidity of the NSs.Potential CGWs detection will allow for testing NSs matter (their EOS, crustalproperties, deformability), processes inside the object (heating, superfluidity, instabil-ities, magnetic field evolution), close environment of the stars (their magnetic field,accretion processes). Such unique analysis will be essential also in our understandingof NSs evolution, especially for the newborn objects. Very young NSs undergo severaldynamical mechanisms and usually are unstable - such changes are di ffi cult to observein traditional, EM telescopes. GW astronomy will open a new door for NSs studies andmay be a crucial piece in solving the mystery of NSs nature and EOS. Funding
The work was partially supported by the Polish National Science Centre grants no.2016 / / E / ST9 / / / M / ST9 / / / T / ST9 / Acknowledgements
The authors thank Brynmor Haskell, Karl Wette, David Keitel for useful insights andcomments that greatly improved the manuscript and Maria Alessandra Papa for sug-gesting useful references. The authors thank Damian Kwiatkowski for help with lan-guage editing.
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