Search for Continuous Gravitational Waves from Scorpius X-1 in LIGO O2 Data
Yuanhao Zhang, Maria Alessandra Papa, Badri Krishnan, Anna L. Watts
DDraft version November 10, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Search for Continuous Gravitational Waves from Scorpius X-1 in LIGO O2 Data
Yuanhao Zhang,
1, 2
Maria Alessandra Papa,
1, 2, 3
Badri Krishnan,
1, 2 and Anna L. Watts Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut), D-30167 Hannover, Germany Leibniz Universit¨at Hannover, 30167 Hannover, Germany Department of Physics, University of Wisconsin, Milwaukee, WI 53201, USA Anton Pannekoek Institute for Astronomy, University of Amsterdam, Postbus 94249, NL-1090 GE Amsterdam, the Netherlands
ABSTRACTWe present the results of a search in LIGO O2 public data for continuous gravitational waves fromthe neutron star in the low-mass X-ray binary Scorpius X-1. We search for signals with ≈ constantfrequency in the range 40-180 Hz. Thanks to the efficiency of our search pipeline we can use a longcoherence time and achieve unprecedented sensitivity, significantly improving on existing results. Thisis the first search that has been able to probe gravitational wave amplitudes that could balance theaccretion torque at the neutron star radius. Our search excludes emission at this level between 67.5 Hzand 131.5 Hz, for an inclination angle 44 ◦ ± ◦ derived from radio observations (Fomalont et al. 2001),and assuming that the spin axis is perpendicular to the orbital plane. If the torque arm is ≈
26 km –a conservative estimate of the Alfv´en radius – our results are more constraining than the indirect limitacross the band. This allows us to exclude certain mass-radius combinations and to place upper limitson the strength of the star’s magnetic field. We also correct a mistake that appears in the literature inthe equation that gives the gravitational wave amplitude at the torque balance (Abbott et al. 2017b,2019a) and we re-interpret the associated latest LIGO/Virgo results in light of this.
Keywords: neutron stars — gravitational waves — continuous waves — Sco X-1 — accretion, accretiondisks INTRODUCTIONFast spinning neutron stars are promising sources ofcontinuous gravitational waves in the frequency range20 Hz - 2 kHz. The emission is typically generated bya non-axisymmetry in the star with respect to its ro-tation axis. The simplest example is the presence of anequatorial ellipticity that deforms the star into a triaxialellipsoid rotating around the principal moment of inertiaaxis (Jaranowski et al. 1998).
Corresponding author: Yuanhao [email protected] author: Maria Alessandra [email protected]@[email protected]
The strength of the gravitational wave signal is pro-portional to the ellipticity of the star. The maximum el-lipticity that a neutron star could support before break-ing has been estimated to lie in the 10 − − − rangefor neutron stars made of normal matter and a few or-ders of magnitude higher for exotic matter (Horowitz &Kadau 2009; Johnson-McDaniel & Owen 2013; Baiko &Chugunov 2018; Gittins et al. 2020). The minimum el-lipticity is harder to estimate: we expect some ellipticitydue to magnetic deformation, but the precise value de-pends strongly on the assumed magnetic field strengthand configuration (see for example Haskell et al. 2008;Mastrano et al. 2011; Suvorov et al. 2016). Woan et al.(2018) have argued for a minimum ellipticity ∼ − based on the spin-down of millisecond pulsars (due toeither magnetic field effects or some other source of el-lipticity such as crustal deformation).For accreting neutron stars, the accretion process pro-vides a potential additional source of asymmetry, par- a r X i v : . [ a s t r o - ph . H E ] N ov Zhang et al. ticularly if accreting material is channeled unevenlyonto the surface by the star’s magnetic field. Thiscan lead to thermal and compositional gradients in thecrust that generate a crustal ‘mountain’ (Bildsten 1998;Ushomirsky et al. 2000; Haskell et al. 2006; Singh et al.2020). Accretion-induced deformation of the star’s mag-netic field might also result in asymmetries (Melatos &Payne 2005; Vigelius & Melatos 2009). Accretion couldalso drive the excitation of some kind of internal oscil-lation that results in gravitational wave emission (An-dersson et al. 1999; Haskell 2015). Uncertainty aboutthe accretion process and the stellar response makes ithard to compute firm estimates for the expected size ofthe resulting ellipticities, but they could be large enoughfor the resulting gravitational wave emission to be de-tectable with the current generation of detectors (Lasky2015).What effect might such a gravitational wave torquehave on an accreting neutron star? It has long beennoted (Papaloizou & Pringle 1978; Wagoner 1984) thatneutron stars in low mass X-ray binaries, in spite of hav-ing accreted matter for millions of years, spin well belowthe maximum possible spin frequency (Cook et al. 1994;Haensel et al. 2009), with the fastest accreting neutronstar spinning at 620 Hz (Hartman et al. 2003; Patruno& Watts 2012; Watts 2012; Patruno et al. 2017). Sincegravitational wave torques scale with a high power of thefrequency, as the spin rate increases, they naturally pro-vide a mechanism that kicks-in more strongly than othermechanisms, preventing further spin-up. This has ledto the idea of torque balance, where gravitational waveand accretion torques reach equilibrium, preventing fur-ther spin-up and ensuring continuous gravitational waveemission (Bildsten 1998). Indeed Gittins & Andersson(2019) have shown that a synthetic population of neu-tron stars evolved without the gravitational wave torquecontribution does not produce the observed spin distri-bution.The accretion torque on a neutron star having mass M is N acc = ˙ M (cid:112) GM r m , (1)where G is the gravitational constant, r m is the torquearm and ˙ M the accretion rate. The correct value to usefor r m is not known a priori, but is typically assumedto be either the neutron star radius R or the radiusat which the star’s magnetic field starts to disrupt theaccretion flow.The maximum accretion luminosity is GM ˙ MR . If somefraction X of this is radiated away by an X-ray flux F X observed at a distance d , then X GM ˙ MR = 4 πd F X → ˙ M = 1 X πd F X RGM . (2) The gravitational wave intrinsic amplitude h at a dis-tance d , for a gravitational wave signal at twice the spinfrequency of the star (which is the case if the elliptic-ity is caused by a magnetic or crustal mountain) andbalancing the accretion torque, is h torq.bal. = (cid:115) G π c ˙ E GW d f with ˙ E GW = πf GW N acc , (3)where f GW is the gravitational wave frequency. Substi-tuting Eq. 1 and 2 in Eq. 3 one finds h torq.bal. = (cid:115) Xc F X Rf GW (cid:114) Gr m M == 3 . × − (cid:18) . M (cid:12) M (cid:19) (cid:16) r m
10 km (cid:17) ×× (cid:18) F X /X . × − erg cm − s − (cid:19) (cid:18) R
10 km (cid:19) (cid:18)
600 Hz f GW (cid:19) . (4)We note that Eq. 15 in Abbott et al. (2019a) and Eq. 10in Abbott et al. (2017b) are incorrect and yield the cor-rect numerical value only if r m = R . In those paperssuch mistake propagates to the Alfv´en radius torquebalance amplitude curve of Fig. 5 (yellow curve in Ab-bott et al. 2019a), which is over-estimated. This in turnmakes it look like the constrained inclination angle up-per limits from that search (for ι = ι orb ≈ ◦ ) probethe Alfv´en radius torque balance limit, when in fact theydo not.For ease of notation we define M = (cid:16) M . M (cid:12) (cid:17) R = (cid:0) R
10 km (cid:1) r m = (cid:0) r m
10 km (cid:1) B = (cid:0) B G (cid:1) f GW = (cid:16) f GW
600 Hz (cid:17) F X = (cid:16) F X . × − erg cm − s − (cid:17) d = (cid:16) d . (cid:17) (5)and re-write Eq. 4 as h torq.bal. = 3 . × − X − M − r m F X R f GW1 − . (6)Scorpius X-1 (Sco X-1) is the brightest persistent X-ray source after the Sun and hence, given the scalingof gravitational wave amplitude with X-ray flux, it isa particularly promising continuous wave source. The earch for continuous GWs from ScoX-1 . × − erg / cm / s used in Eq. 4 is thelong-term average X-ray luminosity of Sco X-1 measuredfrom Earth (see Watts et al. 2008, for details of how thisvalue was derived ). This value yields a torque balance h well within the reach of searches for continuous wavesfrom known pulsars (Abbott et al. 2019b, 2020; Niederet al. 2019, 2020).Many searches have targeted continuous gravitationalwave emission from Sco X-1 (Meadors et al. 2017; Ab-bott et al. 2017a,c,d, 2019a, only since 2017), but nonehave yet been sensitive enough to probe the torque bal-ance amplitudes of Eq. 4. This is because in contrastto the known pulsars targeted in Abbott et al. (2019b,2020); Nieder et al. (2019, 2020), the rotation frequencyand frequency derivative of the Sco X-1-neutron star,as well as some binary parameters, are unknown. Thismeans that a broad range of waveforms must be testedagainst the data, and this degrades the attainable sen-sitivity, through the increased trials factor.Another aspect that makes the Sco X-1 signal searchchallenging is its computational cost: as illustrated inWatts et al. (2008) our ignorance of the system param-eters results in a parameter space so broad that themost sensitive search method, a coherent matched fil-ter over the entire observation time, is computationallyprohibitive. This is a frequent predicament in searchesfor continuous gravitational waves and the standard so-lution is to adopt semi-coherent search methods, whereone trades sensitivity in favour of computational effi-ciency (Messenger et al. 2015; Dergachev & Papa 2019).In semi-coherent searches the observation time is par-titioned in segments spanning the same duration. If datafrom several instruments is used, these partitions are ≈ coincident in time. The most important quantity is theduration of such partitions, T coh . The larger T coh is, themore sensitive and the more computationally expensivethe search is going to be.We use for this search a cross-correlation method(Whelan et al. 2015, and references therein). Thanksto the much improved computational efficiency of ournew search (Meadors et al. 2018), we are able to use asignificantly longer T coh than ever used before and reachunprecedented levels of sensitivity. In particular for thefirst time a search is sensitive to signals at the torquebalance limit at both the stellar radius and for reason-able estimates of the magnetospheric radius. THE SEARCH The flux of Sco X-1 during the O2 observations was comparableto the earlier observations used to generate the flux estimate, seehttp://maxi.riken.jp/star data/J1619-156/J1619-156.html.
We use LIGO O2 open data from the Hanford andLivingston detectors (LIGO 2019; Vallisneri et al. 2015)between GPS time 1167984930 (January 2016) and GPStime 1187733514 (August 2016). Overall we have 5090hours of data, 2496 from Livingston and 2594 from Han-ford.We search for a nearly monochromatic signal fromthe neutron star in Sco X-1 – below we qualify this as-sumption further. At the detector the signal appearsfrequency-modulated due to the relative motion betweenthe star and the detector, and amplitude-modulated dueto the sensitivity-response of the detectors, which de-pends on the line-of-sight direction and hence for a fixedsource changes with time. If all the source parameterswere known, the gravitational waveform at the detectorwould also be known, and the search would be a per-fectly matched filter, like those carried out for knownpulsars. This is not the case.The low-mass X-ray binary (LMXB) Sco X-1 consistsof a 1 . +1 . − . M (cid:12) neutron star and a 0 . +0 . − . M (cid:12) compan-ion star (Wang et al. 2018, 95% confidence intervals).No accretion-powered pulsations or thermonuclear burstoscillations have been so far detected from the neutronstar, so its spin frequency is unknown. The orbital pa-rameters projected semi-major axis, a sin i , time of as-cending nodes, T asc , and orbital period, P orb , are con-strained within ranges larger than our search resolu-tion on those parameters, so these need to be explicitlysearched (Wang et al. 2018).The search parameters are given in Table 1. We searchfor gravitational wave signal frequencies between 40 Hzand 180 Hz. The computational cost per unit frequencyinterval is smaller at lower frequencies, so concentratingcomputational resources in the lower frequency rangemakes for the highest return in sensitivity. In fact thisis the frequency range in which we can match the torquebalance limit, even with an unrestricted prior on thestar’s inclination angle.We do not explicitly search over frequency derivatives,reflecting the assumption that the system is close toequilibrium. With our search set-up we have measuredan average loss in SNR at the 15% level for gravitationalwave first frequency derivative | ˙ f GW | (cid:39) × − Hz/s.This sets the scale for the maximum rate of change ofthe spin frequency that would not affect our ability todetect a signal, at | ˙ f spin | (cid:46) × − Hz/s. We recallthat for crustal mountains f GW = 2 f spin .The orbital parameter ranges are taken from Table 2of Wang et al. (2018). T asc is propagated to 1178556229GPSs which is ≈ the weighted middle of the LIGO dataobservation span. We note that this is 206 epochs afterthe T asc in (Abbott et al. 2019a). Following Eq. 5 of Zhang et al.
Table 1.
Waveform parameter rangesParameter Range Grid spacing f GW (Hz) [40 , ∼ × − a sin i (lt-s) [1 . , . ∼ .
17 [lt-s Hz] f GW T asc (GPS s) a ± × ∼ f GW a sin i P orb (s) 68023 . ± × . ∼
18 [lt-s] f GW a sin i a Time of ascension has been propagated to May 1116:43:31 UTC 2017, close to the weighted-middle of thegravitational wave data, in order to make the metricapproximately diagonal (Whelan et al. (2015)). The re-lation between T asc and the epoch of inferior conjunctionof the companion star T presented in Wang et al. (2018)is T asc = T − P orb / Galloway et al. (2014), we expand the uncertainty asso-ciated with T asc to 139 seconds and then consider the3 σ confidence interval.The grid spacings in every dimension dλ are chosen sothat the loss due to signal-template mismatch is at the m = 25% level. The spacings are estimated based on themetric g λλ as dλ = (cid:113) mg λλ . Expressions for the metriccan be found in Whelan et al. (2015). This approachresults in an overestimate of the actual mismatch (Allen2019), and in fact we measure an overall average SNRloss of 16%. The grid spacings are given in Table 1.Our search employs a fixed T coh (cid:39)
19 hrs, which isa factor of 4 . RESULTS3.1.
Upper limits on GW amplitude
As no significant candidate is found, we set upperlimits at the 95% confidence level, on the gravitationalwave intrinsic amplitude h at the detectors, in half-Hz bands. The upper limits are determined by addingfake signals with a fixed amplitude h to the data, andby measuring the detection efficiency, C ( h ). The detec-tion criterion is determined by the value of the detectionstatistic of the most significant result in the band. Theprocedure is repeated for various values of h and a sig-moid fit is used to determine the value corresponding to95% confidence: C ( h ) = .
95 (Fesik & Papa 2020).Two sets of upper limits are derived, reflecting twoassumptions: 1) an arbitrary value of the inclinationangle, with cos ι uniformly distributed − ≤ cos ι ≤ ι is equal to the orbitalinclination angle and hence drawn from a Gaussian dis-tribution with mean 44 ◦ and standard deviation 6 ◦ (Fo-malont et al. 2001; Wang et al. 2018). The latter sce-nario is equivalent to assuming that the spin axis of theneutron star is perpendicular to the orbital plane. The ι = 44 ◦ ± ◦ is a more favourable inclination than averagefor coupling to the gravitational wave detector (Jara-nowski et al. 1998, see for instance Eq.s 21 and 22 ) andthe resulting upper limits are a factor ≈ T coh in the lowfrequency range, which explains why at lower frequencyit is comparatively more sensitive than at higher fre-quency. The Abbott et al. (2019a) search is less sensitivethan a cross-correlation search but is more robust to de- earch for continuous GWs from ScoX-1 Figure 1.
95% confidence upper limits on the intrinsic gravitational wave amplitude in half-Hz bands. We assume orbitalinclination at 44 ◦ ± ◦ (lower black points) and arbitrary inclination (upper black points). The lower dashed and solid curvesare the torque-balance upper limits, based on estimates of the mass accretion rate and assuming the accretion torque to be atthe neutron star radius (lower solid red curve) or at the Alfv´en radius (upper dashed red curve). For comparison we show theupper limits from previous results (the three fainter upper curves) and draw the incorrect torque balance Alfv´en radius upperlimit that was reported (dash-dot line). Figure 2. h torq.bal. ( ξ, X ) = h torq.bal. (1 , ξ / X / , and thismultiplicative factor is shown here. viations of the signal waveform from the assumed model (Suvorova et al. 2016a,b). In particular the method ofAbbott et al. (2019a) is robust with respect to loss ofphase coherence in the signal.One of the ways in which the signal could lose phasecoherence with respect to the template waveforms ofthe search is through spin-wandering. This is a non-deterministic “jitter” in the spin of star, caused, for in-stance, by small changes in the mass accretion rate. Theresulting frequency variation depends on the accretiontorque, hence on the spin frequency of the star, its mo-ment of inertia, the ratio between the torque arm andthe co-rotation radius and the mass accretion rate.Based on RXTE/ASM observations of Sco X-1,Mukherjee et al. (2018) have explored different system-parameter combinations and the gravitational wave fre- Zhang et al. quency changes that may accumulate over different ob-servation periods, due to spin wandering. Their resultsindicate that, in our frequency range, the maximum fre-quency change during an observation time of 2 × s(our observation time) is less than 2 µ Hz (our frequencyresolution) for the vast majority of the simulated sys-tems. This means that the sensitivity of this searchshould not be impacted by spin-wandering effects.Our results improve on existing ones by more thana factor ≈ ≈ Interpretation in terms of torque balance model
Our results are also remarkable in absolute terms be-cause they probe gravitational wave amplitudes thatcould support emission at the torque balance level. It isthe first time that this milestone is reached.From Eq.s 4 or 6 we see that the torque balance gravi-tational wave amplitude depends on the torque arm andit is smallest at the star surface. If this minimum torquebalance amplitude is larger than our h upper limitsit means that our search should have detected a signal;The fact that it has not, means that we can exclude suchmass-radius combination:[ M − R ] excl ≥ X h . × − f GW F − . (7)The lower panel of Figure 4 shows the mass-radius re-gions excluded by the ι ≈ ◦ gravitational wave upperlimits for f GW = 117 . X = 1.If the torque arm is larger than the star radius, thetorque balance amplitude increases, and our gravita-tional wave upper limits constrain the magnetic fieldstrength of the mass-radius combinations not excludedby 7. We illustrate this point in the next paragraphs.We take the torque arm to be at the magnetosphericradius r m = max( ξr A , R ), with 0 . ≤ ξ ≤ r A theAlfv´en radius r A = 25 . X B R M d − F X − km , (8)where B is the normalised polar magnetic fieldstrength, defined in Eq. 5. We note that in the gravita-tional wave literature the Alfv´en radius has often beenplaced at 35 km, corresponding to ˙ M = 10 − M (cid:12) / yr,or X = 0 . M = 2 × − M (cid:12) / yr, for a fiducial 1.4 M (cid:12) and 10 kmradius neutron star. By combining Eq. 4 and Eq. 8 we find the torque-balance amplitude when r m ≥ R : h torq.bal. = 4 . × − ξ X × F X M − R B d − f GW − B ≥ . d ξ − / ( F /X ) / ( R M ) − / , (9)the last equation simply reflecting the condition r m ≥ R .We note that h torq.bal. ( ξ, X ) = h torq.bal. (1 , ξ / X / andthis factor is plotted in Fig. 2 to aid evaluate how thetorque balance amplitude changes under different as-sumptions for torque arm r m and the accretion lumi-nosity.When this torque balance amplitude is larger than our h upper limits it means that our search should havedetected a signal; The fact that it has not, means that wecan exclude the associated mass-radius-magnetic fieldstrength combinations:[ M R B ] excl ≥ X ξ h . × − f GW F − d . (10)This translates, for every mass-radius, into an upperlimit on the magnetic field strength.Figure 4 shows the magnetic field upper limits fromEq. 10 from the ι ≈ ◦ gravitational wave upper limitsfor f GW = 117 . X = 1 and ξ = 1, for differentequations of state. The upper limits for different gravi-tational wave frequencies can be easily derived from thegravitational wave upper limit values using Eq. 10. Forthe specific example shown in Figure 4, provided thatthe field is higher than ∼ × G, the torque balancelimit can be matched for all of the considered equationsof state, but magnetic fields above ∼ × G can beruled out.At Zhang et al. (2020, and Suppl. Mat.) we provideplots like the one of Figure 4 for gravitational wave fre-quencies in the searched range, at 2 Hz intervals.The gravitational wave upper limits marginalised overall possible inclination angles lead to less stringent con-straints on the physical parameters of the neutron star:the torque balance amplitude with torque arm at theneutron star surface is smaller than our upper limits forall equations of state, so no mass-radius combination canbe ruled out. Torque-balance amplitudes larger thanour upper limits can only be obtained for larger torquearms corresponding to magnetic field strengths (cid:39) G, which are higher than those expected from LMXBs.If the gravitational wave signal is due to a triaxialellipsoid rotating around a principal moment of inertiaaxis I , say along the ˆ z axis, the gravitational wave in-trinsic amplitude h is proportional to the ellipticity ε earch for continuous GWs from ScoX-1 Figure 3.
Upper limits on the ellipticity of the neutronstar, derived from the gravitational wave intrinsic amplitudeupper limits. of the star: h = π Gc Iεf d with ε = I xx − I yy I (11)We convert the h upper limits into ellipticity upperlimits with Eq. 11, with d = 2 . I = 10 kg m . We also derive the ellipticity requiredfor torque balance under the two previous assumptionson the lever arm. All these quantities are plotted inFig. 3, as a function of the gravitational wave signalfrequency.Above ∼
90 Hz the values of the ellipticity that we areexploring are a few × − and smaller. Deformationswhich are this large may be sustained by a neutron starcrust (Johnson-McDaniel & Owen 2013), although veryrecent work suggests that the maximum deformationsmay be smaller (Gittins et al. 2020). DISCUSSIONThis search has placed upper limits on stable GWemission that are tighter than the level predicted bytorque balance models for Sco X-1, for ι ∼ ◦ . Thisconclusion is robust to spin wandering at the level ex-pected for this source. If the accretion torque is appliedat the neutron star surface, the GW frequency rangefor which the torque balance limit is beaten is between67.5-131.5 Hz, for a 1.4 M (cid:12) and 10 km radius fiducialstar. If on the other hand the torque is applied at amagnetospheric radius at 25.6 km (see Eq.s 5 and 8),then the range for which the limit is beaten is the en-tire searched range, 40-180 Hz, for the fiducial star, asshown in Fig. 1. If we consider a wider range of masses and radii, con-sistent with our current best understanding of viableequation of state models, we are able to place constraintson mass-radius-magnetic field strength combinations: • independently of the magnetic field value, our re-sults exclude certain mass-radius combinations.Our tightest limits come for spin periods of ∼ f GW ∼
96 Hz, at twice the spin frequency)with a narrow range of allowed masses extendingonly between 1.9-2.2 M (cid:12) and magnetic fields largerthan ∼ × G being ruled out for all consideredequations of state. • if the magnetic field is larger than0 . d ξ − / ( F /X ) / ( R M ) − / ( r m > R ) wecan place upper limits on the magnetic fieldstrength. The upper limit on the magnetic field ishighest for the highest mass in the range. Stifferequations of state have a smaller range of masses(and magnetic field strengths) for which balancecan still be possible at the level of our upperlimits, than softer equations of state. We findthat the field must be smaller than (4 − × G, depending on frequency (but excluding toodisturbed frequency ranges, e.g. 60 Hz), for allequations of state models considered.It is the first time that constraints on the magneticfield, mass and radius are obtained through continu-ous wave observations. This is interesting because themagnetic field is in general very poorly constrained andbecause observations like these probe mass-radius andmagnetic fields through an entirely different mechanismthan gravitational wave binary inspiral signals (see e.g.Abbott et al. 2019d; Capano et al. 2020).If the spin of Sco X-1 is such that it is in therange where the limit is beaten (half the GW frequencyfor mountain models) and torque balance applies, thismeans that GW emission is not strong enough to bal-ance the assumed accretion torque. This implies thatthe accretion torque must be less strong than predictedby the models presented in this paper, which could hap-pen if, for example, strong radiation pressure modifiesthe structure of the inner disk (Andersson et al. 2005)or due to the effect of winds (Parfrey et al. 2016).The result also puts limits on the size of ther-mal/compositional crustal or magnetic mountains inSco X-1. Limits can also be placed on internal oscilla-tion amplitude for models where that is the mechanismthat provides the GW torque (for a different range ofspin frequencies since the relationship between spin andGW frequency is different for mode models).
Zhang et al.
Figure 4.
We have assumed f GW = 117 . ≈
17 ms spin period), ι = 44 ◦ ± ◦ , r m = r A ( ξ = 1), X = 1 and torquebalance. Top panel : the largest magnetic field consistent with our null result. The solid lines correspond to the equations ofstate from ¨Ozel & Freire (2016, http://xtreme.as.arizona.edu/NeutronStars/index.php/dense-matter-eos/ ). The dashedlines indicate stars of constant compactness
GM/Rc equal to 0.33 (upper), 0.29 (middle) and 0.25 (lower). We have consideredmasses in the range 1 to 3 M (cid:12) , radii between 8 and 13 km and we have dropped any equation of state with a maximum masslower than 1.9 M (cid:12) , consistent with observations (Antoniadis et al. 2013; Cromartie et al. 2019) and with estimates from densematter theory and experiment, (see for example Hebeler et al. 2013; Kurkela et al. 2014). The lowest value of B for each curvecorresponds to B ( r m = R ). Lower panel : mass-radius relations for the equations of state considered above (solid lines) andfor star configurations of constant compactness (dashed lines). The line that delimits the shaded region shows the mass-radiuscombinations that satisfy Eq. 7, i.e. that are consistent with our upper limits when r m = R . Below the shaded region the torquebalance gravitational wave amplitude with r m = R , is larger than our upper limits, so these configurations are excluded by ournull results. Above the shaded region the torque balance gravitational wave amplitude with r m = R , is smaller than our upperlimits, so these configurations cannot be excluded if r m = R . If, however, r m > R , i.e. a magnetic field above ≈ × G, thecorresponding torque balance becomes larger than our upper limits and this allows us to constrain the magnetic field (as shownin the top panel). earch for continuous GWs from ScoX-1 × − , for r m ∈ [10 − . × − . We are howeverquite far from having a complete ephemeris for Sco X-1. The next best thing would be to know the rota-tion frequency of the neutron star. The reason is thatthe torque balance amplitude decreases with frequency(so the sensitivity requirement increases, to match thetorque balance limit), and the sensitivity of the searchesdecreases with frequency due to the shot noise in thedetectors and to the increased template resolution per-Hz searched. These factors make it difficult to searchvery broad frequency bands. If it were possible to iden- tify the spin frequency, for example via the detection ofweak or intermittent pulsations (a major goal for futurelarge-area X-ray telescopes Watts et al. 2019; Ray et al.2019), we might be able to carry out a search like thisone, that could begin to probe the torque balance limitwhen the noise level at 1 kHz reaches its design value of ∼ . × − / √ Hz (Abbott et al. 2018) and with ∼ two years of data. ACKNOWLEDGMENTSThe computation of the work was run on the ATLAScomputing cluster at AEI Hannover AEI (2017) fundedby the Max Planck Society and the State of Niedersach-sen, Germany. A.L.W. acknowledges support from ERCConsolidator Grant No. 865768 AEONS (PI: Watts).This research has made use of data, software and/orweb tools obtained from the LIGO Open Science Cen-ter (https://losc.ligo.org), a service of LIGO Laboratory,the LIGO Scientific Collaboration and the Virgo Collab-oration. LIGO is funded by the U.S. National ScienceFoundation. Virgo is funded by the French Centre Na-tional de Recherche Scientifique (CNRS), the Italian Is-tituto Nazionale della Fisica Nucleare (INFN) and theDutch Nikhef, with contributions by Polish and Hun-garian institutes.APPENDIX A. OUTLIER TABLEREFERENCES Zhang et al.
Table 2.
Table of outlier clusters, as described in the text.Cluster ID freq [Hz] detection statistic description0 40.883985 180.55 known line in H1/L1 & too broad in freq.1 42.852488 19.05 known line in H1/L1 & too broad in freq.2 43.338994 12.99 fails single/multi-detector statistic comparison & too broad in freq.3 44.524761 35.08 known line in H1/L1 & too broad in freq.4 46.089393 415.72 fails single/multi-detector statistic comparison & too broad in freq.5 47.679448 15.54 known line in H1/L1 & too broad in freq.6 55.565183 11.96 known line in H1/L17 58.188448 17.06 fails single/multi-detector statistic comparison & too broad in freq.8 59.516247 42.19 known line in H1/L1 & too broad in freq.9 60.005969 185.90 known line in H1/L1 & too broad in freq.10 61.814927 11.50 known line in H1/L1 & too broad in freq.11 63.994606 68.80 known line in H1/L1 & too broad in freq.12 64.025584 13.24 same as cluster 11 & too broad in freq.13 64.279235 14.38 fails single/multi-detector statistic comparison & too broad in freq.14 68.463375 21.05 known line in H1/L1 & too broad in freq.15 68.492385 18.08 known line in H1/L1 & too broad in freq.16 69.564208 41.68 known line in H1/L1 & too broad in freq.17 69.649598 13.55 known line in H1/L1 & too broad in freq.18 69.764916 16.83 known line in H1/L1 & too broad in freq.19 83.301663 18.59 fails single/multi-detector statistic comparison & too broad in freq.20 85.989651 43.63 known line in H1/L1 & too broad in freq.21 99.332771 11.80 fails single/multi-detector statistic comparison22 99.382711 11.54 fails single/multi-detector statistic comparison23 99.984856 18.68 known line in H1/L1 & too broad in freq.24 105.249255 15.40 fails single/multi-detector statistic comparison & too broad in freq.25 105.467399 12.97 known line in H1/L1 & too broad in freq.26 105.603586 13.66 known line in H1/L1 & too broad in freq.27 119.886974 13.63 known line in H1/L1 & too broad in freq.28 119.934821 13.04 known line in H1/L1 & too broad in freq.29 120.002786 11.84 known line in H1/L1 & too broad in freq.30 176.308216 13.28 fails single/multi-detector statistic comparison & too broad in freq.31 179.994556 12.76 known line in H1/L1Antoniadis, J., et al. 2013, Science, 340, 6131,doi: 10.1126/science.1233232Baiko, D. A., & Chugunov, A. I. 2018, MNRAS, 480, 5511,doi: 10.1093/mnras/sty2259Bildsten, L. 1998, Astrophys. J. Lett., 501, L89,doi: 10.1086/311440Capano, C. D., Tews, I., Brown, S. M., et al. 2020, NatureAstron., 4, 625, doi: 10.1038/s41550-020-1014-6Cook, G. B., Shapiro, S. L., & Teukolsky, S. A. 1994,ApJL, 423, L117, doi: 10.1086/187250Covas, P., et al. 2018, Phys. Rev. D, 97, 082002,doi: 10.1103/PhysRevD.97.082002 Cromartie, H., et al. 2019, Nature Astron., 4, 72,doi: 10.1038/s41550-019-0880-2Dergachev, V., & Papa, M. A. 2019, Phys. Rev. Lett., 123,101101, doi: 10.1103/PhysRevLett.123.101101Fesik, L., & Papa, M. A. 2020, The Astrophysical Journal,895, 11, doi: 10.3847/1538-4357/ab8193Fomalont, E., Geldzahler, B., & Bradshaw, C. 2001,Astrophys. J., 558, 283, doi: 10.1086/322479Galloway, D. K., Premachandra, S., Steeghs, D., et al. 2014,Astrophys. J., 781, 14, doi: 10.1088/0004-637X/781/1/14Gittins, F., & Andersson, N. 2019, Mon. Not. Roy. Astron.Soc., 488, 99, doi: 10.1093/mnras/stz1719 earch for continuous GWs from ScoX-1 Gittins, F., Andersson, N., & Jones, D. 2020.https://arxiv.org/abs/2009.12794Haensel, P., Zdunik, J. L., Bejger, M., & Lattimer, J. M.2009, A&A, 502, 605, doi: 10.1051/0004-6361/200811605Hartman, J. M., Chakrabarty, D., Galloway, D. K., et al.2003, in AAS/High Energy Astrophysics Division Zhang et al.
Zhang, Y., Papa, M. A., & Krishnan, B. 2020,Supplemental materials to the paper