First searches for gravitational waves from r-modes of the Crab pulsar
Binod Rajbhandari, Benjamin J. Owen, Santiago Caride, Ra Inta
FFirst searches for gravitational waves from r-modes of the Crab pulsar
Binod Rajbhandari, Benjamin J. Owen, Santiago Caride, ∗ and Ra Inta † Department of Physics and Astronomy, Texas Tech University, Lubbock, Texas, 79409-1051, USA
We present the first searches for gravitational waves from r -modes of the Crab pulsar, coherentlyand separately integrating data from three stretches of the first two observing runs of AdvancedLIGO using the F -statistic. The second run was divided in two by a glitch of the pulsar roughlyhalfway through. The frequencies and derivatives searched were based on radio measurements ofthe pulsar’s spin-down parameters as described in Caride et al., Phys. Rev. D , 064013 (2019).We did not find any evidence of gravitational waves. Our best 90% confidence upper limits ongravitational wave intrinsic strain were 1 . × − for the first run, 1 . × − for the first stretchof the second run, and 1 . × − for the second stretch of the second run. These are the first upperlimits on gravitational waves from r -modes of a known pulsar to beat its spin-down limit, and theydo so by more than an order of magnitude in amplitude or two orders of magnitude in luminosity. I. INTRODUCTION
Rapidly rotating neutron stars might be detectableemitters of long lived quasi-monochromatic radiationknown as continuous gravitational waves (GWs) [1]. Theemission mechanism for continuous waves could be anonaxisymmetric mass quadrupole (“mountain”) or acurrent-quadrupolar r -mode. Hence the detection of con-tinuous GWs might help reveal the underlying propertiesof neutron star interiors. The GW frequencies of manypulsars lie in the most sensitive band of the Laser In-terferometer Gravitational-wave Observatory (LIGO), sothese rapidly rotating neutron stars are attractive targetsfor continuous GW searches [2].The r -modes, whose frequencies are mainly determinedby the Coriolis force, are unstable to GW emission [3, 4]even allowing for various damping mechanisms [5]. Hencethey might amplify and sustain themselves, and might bethe most interesting possibility for continuous GWs. R -modes might play an important role in the spin-downs ofthe fastest young neutron stars [6] and in the regulation ofspin periods of some older accreting neutron stars [7, 8].Comparison of the GW frequency to the spin frequencydetermined from timing radio or x-ray pulses could mea-sure the compactness of a pulsar [9], and the existence of r -modes at certain frequencies could constrain the prop-erties of the neutron superfluid [10].Some GW searches, starting with Ref. [11], have setupper limits on r -mode GW emission. However most ofthese have been broad band searches for neutron stars notpreviously known as pulsars. The searches themselves didnot take any extra steps to account for r -mode ratherthan mass-quadrupole emission; rather the results couldbe interpreted in terms of r -modes [12].Caride et al. [13] showed that dedicated searches for r -modes from known pulsars are feasible. The key is ∗ Current address: Target Field, 1 Twins Way, Minneapolis,MN,55403 † Current address: Accelebrate, 925B Peachtree Street, NE, PMB378, Atlanta, GA 30309-3918, USA to search the right range of frequencies and frequencyderivatives, which are significantly different from themore often considered case of mass-quadrupole emission.Not surprisingly, as with other GW searches for pul-sars, the Crab is the first prospect to beat the spin-down limit using LIGO data. The spin-down limit as-sumes that all the rotational energy is lost in the formof GWs, and represents a milestone a search must beatto have a chance of detection. Recently Fesik andPapa [14] first published an r -mode pulsar search sim-ilar to that proposed by Caride et al. [13], for anotherpulsar which is interesting for different reasons; but theydid not beat its spin-down limit. The Crab is a rela-tively nearby pulsar with one of the fastest known spin-down rates, and thus it has one of the highest spin-down limits. Its rotational frequency changes at the rate − . × − Hz/s [15, 16]. The Crab’s pulse timing isconstantly measured by electromagnetic (EM) observa-tions, so the spin frequency evolution of the Crab is wellknown during the LIGO first observing run (O1) andsecond observing run (O2).For various reasons we know that the Crab is not emit-ting GW at or near the spin-down limit. Alford andSchwenzer [17] have argued that, if the r -mode instabilityoperates similarly in all neutron stars, the Crab is prob-ably spinning to slowly to be unstable in the presenceof common damping mechanisms. Observations of theCrab nebula indicate that most of the rotational energyis lost powering the nebula via synchrotron and inverseCompton radiation from the pulsar wind, and a few per-cent is lost in the narrow light beam [18]. Recent GWsearches [19, 20] have concluded that less than 0.01%of the Crab’s rotational energy loss is through mass-quadrupole gravitational radiation. The braking index n of the Crab (the logarithmic derivative of its spin-downwith respect to frequency) is 2.519. This is closer to the n = 3 expected for magnetic dipole radiation [21] thanto the n = 7 expected for r -mode emission [6]. Such alow braking index is another indicator that GW emissionis a small fraction of the spin-down limit. How small hasnot been quantified in a model-independent way or for r -modes. But Palomba [22] found that, for a class of rea-sonable mass-quadrupole models, the measured braking a r X i v : . [ g r- q c ] J a n Search Start time (UTC) End time (UTC) T span (days) T dat (days) SFTsO1 09/12/15 06:03:57 01/19/16 15:34:47 129.4 133.1 6389O2 (early) 11/30/16 18:01:57 03/27/17 16:28:25 117.1 128.9 6186O2 (late) 03/28/17 23:47:38 08/25/17 21:59:34 150.9 167.4 8035TABLE I. Start and end times of the three searches. The time difference between start and end is T span , while T dat is the livetime of the data. The latter can be greater than the former because there are two interferometers. index of the Crab means it is emitting GW at least afactor of a few below the spin-down limit. We take thisto suggest that any r -mode GW signal from the Crabmust be at least a factor of a few (in strain) below thespin-down limit. Therefore a GW search must beat thespin-down limit by a factor of a few in amplitude (anorder of magnitude in luminosity) to be interesting.Nevertheless, Caride et al. [13] showed that a searchof the Crab with interesting sensitivity is feasible, andwe confirm this. We performed searches for the Crabin O1 and O2 data (the publicly available LIGO datasets at the time of writing) using the matched filtering-based technique known as the F -statistic. While we didnot find any evidence of a GW signal, we were able to setupper limits beating the spin-down limit by an interestingamount over a wide parameter space. II. R-MODE SEARCH METHOD
Our search is based on a matched filtering techniqueknown as the F -statistic. Developed by Jaranowski et al. [23] for a single interferometer and by Cutler and Schutz[24] for multiple interferometers, it is a statistical pro-cedure for the detection of the continuous gravitationalwaves. The F -statistic accounts for the amplitude mod-ulation due to the daily rotation of Earth in a computa-tionally efficient manner. In the presence of a signal, 2 F is a non-central chi squared distribution with four degreesof freedom and the non-centrality parameter is approx-imately the power signal-to-noise ratio. The F -statisticis one half the log of the likelihood function maximizedover the unknown strain, phase constant, inclination an-gle and polarization angle. The main issue when usingthe F -statistic for this type of search, as described byCaride et al. [13], is to find the ranges of frequencies andfrequency derivatives to search.Pulsars are slowly spinning down due to GW emissionand other losses of rotational energy. The evolution of therotational frequency ν of a spinning down neutron star inthe frame of the solar system barycenter is approximatedby ν ( t ) = ν ( t ) + ˙ ν ( t ) ( t − t ) + 12 ¨ ν ( t ) ( t − t ) , (1)where t is a reference time (often the start time of theobservation) and dots indicate time derivatives. Gener-ally ... ν is not needed for less than a year of integration time [13]. The frequency evolution is precisely knownfrom electromagnetic observations. It might depart fromthe above approximation due to timing noise or glitches.Glitches are sudden increases in spin frequency followedby exponential recovery to the pre-glitch frequency [25].Since the Crab glitched during O2, we divided that searchinto pre- and post-glitch stretches. More on the Crab pul-sar timing for our searches will be discussed in Section III.Timing noise is residual phase wandering of pulses rela-tive to the normal spin down model. Timing noise willdeviate GW phases from Taylor series for time scales ofa year or longer [26]. Assuming the GW timing noise issimilar to the one observed in EM pulses, the mismatch ofthe Crab ephemeris during LIGO S5 run from the modelwith no timing noise is less than 1% [27]. So, for all oursearches (about four months of data each), timing noiseshould not significantly mismatch the templates from sig-nal.For a Newtonian star with spin frequency ν , the r -mode GW frequency is approximately f = ν [28]. Thefrequency ratio deviates from 4 / r -modes result-ing in avoided crossings [29]. These are small frequencybands where the simple relation between f and ν is dras-tically altered as modes change identities. Away fromavoided crossings, the r -mode frequency as a function ofspin frequency is approximately f = Aν − B (cid:18) νν K (cid:19) ν, (2)where ν K is the Kepler frequency. Here, as in Ref. [13],we neglect effects other than slow rotation and generalrelativity. The effect of rotation is to decrease the modefrequency in an inertial frame, so A and B are positive.The ranges of A and B are chosen as in Ref. [13]: Weconsider a range of compactness of neutron stars (0 . ≤ M/R ≤ .
31) [9] which comes from the uncertainty in theequation of state. The range of compactness gives a range1 . ≤ A ≤ .
57. The range of B is derived from therelation of f /ν to the ratio of the rotational energy ( T ) tothe gravitational potential energy ( W ) [30]. For differentpolytropic indices and compactnesses, the range is 1.23–1.95 times the rotational parameter ( T /W ). Caride et al. [13] converted
T /W into ( ν/ν k ) using M/R = 0 . B = 0.195. The minimumvalue of B is not well known, so it is taken to be zero.Our search covers the parameters ( f , ˙ f , ¨ f ). The pa-rameter ranges for our searches are given by Caride et al. [13] based on the considerations above: ν (cid:104) A min − B max ( ν/ν K ) (cid:105) ≤ f ≤ ν A max , (3)˙ ν (cid:104) f /ν − B max ( ν/ν K ) (cid:105) ≤ ˙ f ≤ ˙ νf /ν, (4)0 ≤ ¨ f ≤ ¨ νf /ν, (5)where, f is spin frequency, ˙ f and ¨ f are the first andsecond spin derivative, A min = 1 . , A max = 1 . , and B max = 0 . . We use the parameter space metric g ij to control thecomputational cost of our search. The proper distancebetween two templates is given by the mismatch ( µ ),which is the loss in signal-to-noise ratio when signal fallsexactly between two template waveforms [31]. Templatesused to filter the data are chosen with a spacing deter-mined by this mismatch. The template number is givenby dividing the proper volume of the parameter space bythe proper volume per template [13]: √ gν | ˙ ν | ¨ νB max ( ν/ν K ) (cid:2) A − A (cid:3)(cid:16) (cid:112) µ/ (cid:17) (6)where g is the (constant) metric determinant. The pa-rameter space metric is given by [31, 32] g ij = 4 π T i + j +2span ( i + 1)( j + 1)( i + 2)!( j + 2)!( i + j + 3) , (7)where i , j stand for parameters ( f, ˙ f , ¨ f ) = (0 , ,
2) and T span is the time spanned by the observation.We look at the highest values of the F -statistic (2 F ∗ )that survived the automated vetoes (described below).The probability that a given value of 2 F ∗ will be observedwhen no signal is present is given by [11]: P ( N ; 2 F ∗ ) = N P ( χ ; 2 F ∗ )[ P ( χ ; 2 F ∗ )] N − (8)where N is the number of templates (assuming statisti-cally independence) and P ( χ ; 2 F ∗ ) is the central χ cu-mulative distribution function with four degrees of free-dom. The above false alarm probability and correspond-ing threshold F ∗ assume gaussian noise, which is notalways the case. A high 2 F is not enough to claim thedetection, as instrumental lines might act as a periodicsignal. We follow each search with the interferometerconsistency veto as in (e.g.) Ref. [33], which discardscandidates when the joint 2 F value from the two inter-ferometers is less than for either single interferometer. III. THE CRAB PULSAR
The Crab pulsar is the remnant of a supernova explo-sion seen by Chinese astronomers in the year 1054 AD.
Search ν (Hz) ˙ ν (Hz/s) ¨ ν (Hz/s )O1 29 . − . × − . × − O2 (early) 29 . − . × − . × − O2 (late) 29 . − . × − . × − TABLE II. Timing of the Crab pulsar at the beginning of ourthree different searches. The timing is measured by JodrellBank Observatory [16] and interpolated to the start time ofeach LIGO run. The displayed ¨ ν is the maximum monthlyvalue observed during each run. This is our first target due to its high spin down limit,which is well above the LIGO O1 sensitivity curve [13].The rotational energy loss of the pulsar is given by˙ E = 4 π I zz ν ˙ ν ≈ . × W, where ( I zz = 10 kg m )is the principal moment of inertia [34]. The spin-downpower of the Crab is high enough that, even if r -modegravitational wave emission is only responsible for a smallfraction of it, the GWs could be detectable [13].The sky location of the Crab pulsar in J2000 coordi-nates is [16] α = 05 h m . , s δ = +22 ◦ (cid:48) . . (cid:48)(cid:48) (9)The spin frequency and its derivative [16] were ob-tained from the Jodrell Bank observatory monthlyephemeris and interpolated to the start dates of theLIGO run from the nearest dates of 09/16/2015 (O1),11/23/2016 (early O2) and 04/16/2017 (late O2). Therewere no glitches of the Crab during O1, so the whole runwas coherently integrated. The Crab glitched at 2017-03-27 22:04:48.000 UTC [25] during the O2 run, approx-imately halfway through. So we divided the O2 run intotwo roughly equal stretches. The glitch was very smallby Crab standards ( ∆ νν = 2 . × − ) [25], so we startedthe search of late O2 a few hours afterwards although thepost-glitch relaxation time could be days. Our code doesnot yet have the ability to combine pre- and post-glitchstretches, so we kept the searches separate and presentthem as such.Fast spinning down pulsars are contaminated by timingnoise [35], especially when ¨ ν appears negative. Duringour observational time, the monthly ¨ ν measurement wasfluctuating. This might be due to external torque fromthe magnetosphere and might not affect the high densityinterior which sources GWs. So we chose the maximum ¨ ν of our search to be the maximum value observed duringthe observational time and the minimum ¨ ν to be zero.The extra number of templates required by including ¨ ν isfactor of a few. Thus the searches were not too expensivedue to our probably over-wide range of ¨ ν. FIG. 1. Upper limits (90% confidence) on intrinsic strain vs.frequency, in 0.1 Hz bands, for our three searches are plottedbelow the legend. The points arranged in straight lines abovethe legend indicate “bad” bands (see text).
IV. SEARCH IMPLEMENTATION
We used data from the Gravitational Wave Open Sci-ence Center (GWOSC) [36], starting with time-domainstrain data sampled at 4 kHz. We downloaded all suchO1 data for the official duration of the run (from GPStimes 1126051217 to 1137254417) from both interferom-eters (H1 and L1), gating on only “CBC CAT1” vetoes.These indicate disastrous conditions for the instruments,such as loss of laser power. We ignored the other vetoesused in searches for binary black holes and neutron starsbecause they are aimed at short-duration disturbanceswhich are not significant for continuous wave searches.We then used the code lalapps MakeSFTDAG from LAL-Suite [37] to generate 1800 s long short Fourier trans-forms (SFTs), version 2 format, high pass filtered with aknee frequency of 7 Hz and windowed with the defaultTukey window. This produced 6,389 SFTs for O1 (3,474from H1 and 2,915 from L1). A similar procedure for O2produced 14,231 SFTs (7,242 from H1 and 6,989 fromL1). Our r -mode searches used a modified version of acode usually used to search for supernova remnants, mostrecently in Ref. [33], and was performed on the TexasTech University High Performance Computing Center’s“Quanah” cluster.The frequency bands of the searches were 41.2–46.6 Hz(O1) and 41.1–46.6 Hz (O2). The minimum and maxi-mum frequencies were rounded down and up respectivelyfrom the range in Eq. (3) because we used upper limitbands of a uniform 0.1 Hz. We used a bank of templateswith minimal match µ = 0 . . The ideal template num-bers for O1, early O2 and late O2 are 2 . × , 9 . × and 4 . × respectively. The actual search produced1 . × , 1 . × and 3 . × templates. Thenumber of templates of each search is around an orderof magnitude greater than the ideal number. This is be- cause the code uses extra templates to cover the bound-aries outside the parameter space [11]. The O1, early O2and late O2 searches used computational times of 1584,944 and 3892 core hours on Quanah respectively.Unlike in previous work such as Ref. [33], we did notuse the “Fscan veto,” based on spectrograms of the data,to eliminate candidates caused by nonstationarity andspectral lines. Applying that veto resulted in the elimi-nation of almost 1/4 of the frequency band of each search.This was due to the wide “wings” of the veto account-ing for varying Doppler shifts, spin-down values, and thewidth of the Dirichlet kernel (used in computing 2 F ).However, the same factors mean that template waveformshave less overlap with stationary instrumental lines thanin shorter searches done previously, and so eliminatingthis veto did not produce an unmanageable number ofcandidate signals (see below).After using the interferometer consistency veto, weconsidered as candidates batch jobs which produced 2 F values exceeding a threshold corresponding to a 5% falsealarm probability (95% detection confidence) in gaussiannoise. We found 29 search jobs with candidates: 16 inO1, 7 in early O2, and 6 in late O2. For each such job weinspected 2 F histograms and plots of 2 F vs. frequencyas in Ref. [38] and later works based on it. All candidatejobs showed distorted histograms and wide band distur-bances, indicating instrumental rather than astrophysicalorigin. Many events just barely passed the interferome-ter consistency veto, which is a lenient veto. Althoughwe did not use the known spectral lines (and combs) de-scribed in Ref. [39] as a priori vetoes, we found thatmany of our candidates were coincident with those linesor with new lines in the combs extending beyond thoselisted in Ref. [39].Therefore we do not claim any detection of gravita-tional waves. V. UPPER LIMITS
In the absence of a detection, we set upper limits ongravitational wave emission. The method is similar toRef. Lindblom and Owen [33] among others. Upper limitsare the weakest signal that can be detected from oursearch with a certain probability, in our case chosen tobe 90%. This means we set the false dismissal rate to10%, and the loudest 2 F observed (even if vetoed) willset the false alarm rate [38]. Upper limit frequency bandsare chosen to be small enough for the interferometer noiseto stay reasonably constant. The upper limit band for allthe searches is 0.1 Hz.First, we use a computationally inexpensive MonteCarlo integration to find the intrinsic strain amplitude( h ) that exceeds the loudest observed 2 F
90% of thetime. This includes marginalization over inclination andpolarization angles, which reduce the actual strain am-plitude in the data compared to h . Consistent with the F -statistic (and with most other directed searches), we FIG. 2. Upper limits (90% confidence) on r -mode amplitudevs. frequency, in 0.1 Hz bands, for our three searches. “Bad”bands (see text) have been dropped. used priors corresponding to an isotropic probability dis-tribution of the pulsar’s rotation axis. In the case of theCrab the axis is known fairly well [40] and this informa-tion could be used to improve the sensitivity of a GWsearch [41]. However the LALSuite code does not havethis capability.After the Monte Carlo, we use computationally expen-sive (20–30% of the cost of search) [38] software injectionsearches to validate the upper limit. In each upper limitband we inject 1000 signals with various h . For each h we use variable f, ˙ f , ¨ f and inclination and polarizationangles, and consider an injection “detected” if it pro-duces F > F ∗ . Comparing this to the Monte Carlo isanother check for contaminated frequency bands.Our upper limits on intrinsic strain in 0.1 Hz bands forour three searches are shown in Fig. 1. In some bandsinjections indicated that the true false dismissal rate washigher than 10%, typically due to many and/or strongspectral lines. In Fig. 1, for the sake of visibility, thesebad bands are given constant values well above the upperlimits and thus appear along horizontal lines above thefigure legend. The data files are included in the supple-mental material to this article [42]. The best 90% confi-dence level upper limits on intrinsic strain amplitude are1 . × − (O1), 1 . × − (early O2) and 1 . × − (late O2). O2 upper limits beat the spin-down limit bymore than an order of magnitude. Late O2 was the mostsensitive because it had the most data and the noise spec-trum was better (lower) than the other searches. Upperlimits can also be characterized in terms of a statisticalfactor Θ [32] of the form h = Θ (cid:112) S h /T dat , (10)where T dat is the data live time and S h is the powerspectral density of strain noise. We achieved an averageΘ (cid:39) , typical for coherent directed searches. Another figure of merit is the “sensitivity depth” [43] D = (cid:112) S h /h = (cid:112) T dat / Θ . (11)Our searches achieve D (cid:39)
100 Hz − / , comparable to nar-row band searches for known pulsars (among the bestdirected searches).We can also set upper limits on r -mode amplitude.The coversion of h to r -mode amplitude ( α ) is shownby Owen [12], who took the fiducial value of moment ofinertia ( I zz = 10 kg m ) and typical M = 1 . M (cid:12) . Weconvert the upper limit on intrinsic strain h to r -modeamplitude α using α = 0 . (cid:18) h − (cid:19) (cid:18) r (cid:19) (cid:18)
100 Hz f (cid:19) . (12)Upper limits on α are plotted (without the bad bands) inFig. 2. The best r -mode amplitude upper limits for ourdifferent searches are 0 .
085 (O1), 0 .
075 (early O2) and0 .
067 (late O2).
VI. CONCLUSION
Our searches of LIGO O1 and O2 data for the Crabpulsar did not detect r -mode GWs. However, we setthe first upper limits on r -mode GW emission from thispulsar. These upper limits beat the spin-down limit by anorder of magnitude in strain or two orders of magnitudein luminosity, and are the first to do so for r -modes from aknown pulsar. The corresponding upper limits on r -modeamplitude are not competitive with most predictions of r -mode saturation in young neutron stars such as Ref. [44].In the near future, with more and better data, we willbe able to extend this type of search to longer data setswith lower noise, thereby increasing sensitivity. We willalso be able to beat the spin-down limits of more pulsars.With improvements in our code, we will be able to takeadvantage of the spin axis orientation for those pulsars(such as the Crab) for which it is known and integratedata sets with glitches in them. ACKNOWLEDGMENTS
We are grateful to various members of the LSC con-tinuous waves working group, especially Ian Jones andKarl Wette, for helpful discussions over the years. Thiswork was supported by NSF grant PHY-1912625. Thisresearch has made use of data, software and/or web toolsobtained from the Gravitational Wave Open Science Cen-ter ( ), a service ofLIGO Laboratory, the LIGO Scientific Collaboration andthe Virgo Collaboration. LIGO is funded by the U.S. Na-tional Science Foundation. Virgo is funded by the FrenchCentre National de Recherche Scientifique (CNRS), theItalian Istituto Nazionale della Fisica Nucleare (INFN)and the Dutch Nikhef, with contributions by Polish andHungarian institutes. The authors acknowledge the HighPerformance Computing Center (HPCC) at Texas TechUniversity for providing computational resources thathave contributed to the research results reported within this paper ( ). Anearlier draft of this material was included in the firstauthor’s Ph.D. thesis (Texas Tech University, 2020, un-published). [1] K. Glampedakis and L. Gualtieri, Astrophys. Space Sci.Libr. , 673 (2018), arXiv:1709.07049 [astro-ph.HE].[2] K. Riles, Mod. Phys. Lett.
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