Constructing Love-Q-Relations with Gravitational Wave Detections
CConstructing Love-Q-Relations with Gravitational Wave Detections
Anuradha Samajdar , and Tim Dietrich , Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands Department of Physics, Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands and Institut für Physik und Astronomie, Universität Potsdam,Haus 28, Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany (Dated: February 20, 2020)Quasi-universal relations connecting the tidal deformability and the quadrupole moment of in-dividual neutron stars are predicted by theoretical computations, but have not been measuredexperimentally. However, such relations are employed during the interpretation of gravitationalwaves and, therefore, have a direct impact on the interpretation of real data. In this work, we studyhow quasi-universal relations can be tested and measured from gravitational wave signals connectedto binary neutron star coalescences. We study a population of binary neutron star systems andfind that Advanced LIGO and Advanced Virgo at design sensitivity could find possible deviationsof predicted relations if the observed neutron stars are highly spinning. In the future, a networkof third generation (3G) detectors will be able to even allow a measurement of quasi-universal re-lations. Thus, the outlined approach provides a new test of general relativity and nuclear physicspredictions.
I. INTRODUCTION
The observation of GW170817 proved that gravita-tional waves (GWs) serve as a new observational windowto probe matter at supranuclear densities and to decodethe unknown equation of state (EOS) governing the neu-tron star’s interior [1–3]. Already from this single detec-tion, it was possible to place constraints on the supranu-clear EOS, e.g., [1, 2, 4–11] and to disfavor some of thetheoretical predictions. The recent detection of anotherbinary neutron star (BNS) merger, GW190425 [12] how-ever, does not shed additional light on EOS informationbecause of its high mass [13, 14]. Nevertheless, rate esti-mates for BNS coalescences ( − − yr − [12])show that we can expect many more BNS signals to bedetected in the near future.During a BNS coalescence, each neutron star under-goes tidal deformation due to the influence of the otherstar’s gravitational field. This tidal deformability is im-printed in the emitted GW signal and carries informationabout the internal structure of the star. The main quan-tity characterizing these tidal deformations is the tidalpolarizability Λ = 2 k / (3 C ) with k being the tidal Lovenumber describing the static quadrupolar deformation ofone neutron star in the gravitoelectric field of the com-panion and C being the compactnesses of the star atisolation.In addition, a spinning neutron star undergoesdeformation, encoded in an additional spin-induced quadrupole moment. For rotating neutron stars, thequadrupole moments vary as Q (cid:39) − Qχ m with χ and m being the dimensionless spin and the mass of theobject; see [15] for a first discussion and Ref. [16] foran upgrade and update of [15]. Here, Q is a parameterconnected to the internal structure of the neutron stardepending on the supranuclear EOS. For a given EOS,this relation may be written as Q (cid:39) − Q ( m ) χ . Thecorresponding imprint in the GW phasing from Q was computed in [17]. Refs. [18, 19] laid out the importanceof the quadrupole moment on the measurability ofparameters in GW signals for highly spinning NSsand [20] investigated possible effects on GW signalsintroduced by the spin-induced quadrupole moments bycombining information from multiple signals. Finally,[21] used the measurement of spin-induced quadrupolemoments as a probe to distinguish between a binaryblack hole signal within general relativity and a signalarising from a binary of exotic compact objects. Theanalysis was further extended to a Bayesian approachin [22], the only work which samples directly on thespin-induced quadrupole moment parameters.Most analyses performed on GW signals GW170817and GW190425 inferred the quadrupole moment of eachneutron star from their tidal deformabilities, by leav-ing the latter as free parameters and using the EOS-insensitive relations [2, 3, 12] to determine the spin-induced quadrupole moment. Quasi-universal relationsconnecting the tidal deformability and the spin-inducedquadrupole moment of neutron stars have been first in-troduced by Yagi and Yunes [23] and have been improvedby incorporating information from GW170817 [24].While these EOS-insensitive relations are to second orderin the slow-rotation approximation essentially indepen-dent of the NS spin, additional deviations may occur forfast rotating neutron stars [23]. However, we point outthat these relations have been employed for the analy-sis of GW170817 and GW190425 even beyond the neu-tron star’s breakup spin. Therefore, we want to ask thequestion whether it is possible to verify and potentiallymeasure the relation between the quadrupole momentand the tidal deformability from real GW data. For thispurpose, we use the Yagi-Yunes relation [23, 25] that con-nects the quadrupole moment to the tidal deformability: ln Q = a i + b i ln Λ + c i ln Λ + d i ln Λ + e i ln Λ , (1) a r X i v : . [ g r- q c ] F e b with the fitting parameters a i = 0 . , b i = 0 . , c i = 0 . , d i = − . × − , and e i = 1 . × − .We use the quadrupole moments of the individual stars asfree parameters and sample on them during the analysisinstead of relying on the existing quasi-universal relationsto infer them from their corresponding tidal deformabil-ity parameters. While this increases the dimensionalityof the problem and leads to larger uncertainties in theobserved parameters, it also allows to test and find re-lations between the quadrupole moment and the tidaldeformability. II. METHODS
We perform a Bayesian analysis for parameter estima-tion using the
LALInference module [26] available in the
LALSuite [27] package. We employ the nested samplingalgorithm to estimate posterior probability distributionfunctions [28, 29] which further encode information aboutthe parameters. The parameter set of a BNS source con-sists of { m , m , χ , χ , θ, φ, ι, ψ, D L , t c , ϕ c , Λ , Λ } . m i is the mass of the i th object, χ i = (cid:126)S i m i · ˆ L is the dimen-sionless spin parameter aligned with the direction of theorbital angular momentum ˆ L , θ and φ are the angularcoordinates denoting the sky location, ι and ψ are theangles describing the binary’s orientation with respectto the line of sight, D L is the luminosity distance tothe source, t c and ϕ c are the time and phase at the in-stance of coalescence, and Λ i are the dimensionless tidaldeformability parameters. In addition, our parameterset also includes the spin induced quadrupole moments dQ = Q − and dQ = Q − .For our simulations, we employ the aligned spin wave-form model IMRPhenomD_NRTidalv2 [30]. Unlike in [30],our model contains amplitude tidal corrections andhigher-order spin-squared and spin-cubed terms at 3.5PNalong with their corresponding spin-induced quadrupolemoments, in addition to the spin-induced quadrupole mo-ment terms at 2PN and 3PN. We simulate sourcesin random noise realizations. The component masses liebetween 1.0 M (cid:12) and 2.0 M (cid:12) . Their tidal deformabilitiesare computed assuming the ALF2 EOS [31], which is ahybrid EOS with the variational-method APR EOS fornuclear matter [32] transitioning to color-flavor-lockedquark matter. ALF2 has been picked since it is inagreement with recent multi-messenger constraints onthe EOS [10]. The sources are distributed uniformlyin co-moving volume between 15 Mpc and 150
Mpc withrandomly chosen inclination angles and random sky lo-cations. The dimensionless spin components are dis-tributed uniformly between − . and . , while thesevalues are significantly larger than observed in BNS sys-tems, neutron stars not bound in BNS systems can ro-tate very rapidly, e.g., PSR J1807 − Hz [33, 34]. Furthermore, the recent ob-servation of GW190425 [12] whose estimated individualmasses are inconsistent with the population of observed p r o b . d e n s i t y d i s t . Λ Λ ˜Λ 10 20 30 Q Q Q FIG. 1. Posterior probability distributions of Λ , Λ , ˜Λ , Q , Q from our set of injections. This particularsetup has a signal-to-noise-ratio of 33.45. The neutron starmasses are m = 1 . , m = 1 . , the dimensionlessspins are χ = 0 . , χ = − . . Employing the ALF2EOS, the tidal deformabilities are Λ = 431 , Λ = 1501 .The injected values are shown as vertical dashed lines. Inparticular due to the large spin of the primary object, thissetup is one of the few cases for which the individual tidaldeformabilities and quadrupole moments can be determinedwith the advanced LIGO and advanced Virgo network. galactic BNSs showed that an extrapolation from our lim-ited number of known galactic BNS systems is unreliableso that we include also higher spins in our investiga-tion. We consider two injection sets for our simulatedsources; (i) one where the injected quadrupole-monopolemoments computed from the quasi-universal relation inEq. (1), i.e., Q injection = Q Yagi − Yunes , and (ii) one wherethe injected quadrupole-monopole moments do not fol-low the quasi-universal relation; the injected momentshere are half the values computed from Eq. (1) as an ar-bitrary choice of a modified quasi-universal relation, i.e., Q injection = 1 / × Q Yagi − Yunes . Modified relations mayoccur in alternate theories of gravity like the dynamicalChern Simons theory [35]; cf. e.g. [25]. In both kindsof injections, the quadrupole moments dQ i are sampleduniformly between [0 , and the tidal deformabilities Λ i are sampled uniformly between [0 , . As for the otherparameters, we sample the chirp mass uniformly between0.7 M (cid:12) and 2 M (cid:12) , the mass ratio m /m is sampled uni-formly between 1/8 and 1, and the spin components aresampled uniformly between [ − . , . . III. RESULTS
Testing existing quasi-universal relations:
Based on the methods discussed before, we extractfrom our simulated BNS population the individual tidaldeformabilities of the two stars ( Λ and Λ ) and thespin-induced quadrupole moments Q and Q . As anexample, we show the recovery of one injection in Fig. 1.For the shown example, the injected values of Q , are i )123 l n ( Q i ) i )123 l n ( Q i ) FIG. 2. Recovered Λ , and Q , values from our simulatedpopulation of 120 BNS systems for a 2G detector network.The shown errorbars mark the σ -credible interval, whilethe individual markers refer to the -percentile. Fainthercrosses refer to data with larger uncertainties.Top panel: The injection set based on the quasi-universal re-lation Eq. (1). The dashed line refers to the quasi-universalrelation predicted by Yagi and Yunes, Eq. (1).Bottom panel: The dashed purple line refers to the Yagi-Yunes quasi-universal relation , the blue dashed line the mod-ified relation where Q is reduced by with respect toEq. (1), i.e., to the values used for the injection set. determined from Eq. (1), i.e., we assume the correctnessof the theoretically derived quasi-universal relationsfor the injection. We find that the vast majority ofdetections will not allow us to determine reliably theindividual parameters Λ , , Q , . This is understandablesince the individual parameters enter in the GW phasedescription in special combinations, e.g., tidal effects aredominated by the tidal deformability parameter ˜Λ = 1613 (cid:88) i =1 , Λ i m i M (cid:16) − m i M (cid:17) , (2)see e.g. [36] and Fig. 1 for an illustration. Unfortunately,for the interpretation of quasi-universal relations for sin-gle neutron stars, we have to measure accurately the pa-rameters of the individual stars.In Fig.2 (top panel) we show all 240 recovered values,for both components of Q i and Λ i , together with their σ -credible interval, where we point out that in partic-ular the lower bound on the tidal deformabilities andquadrupole moments are partially driven by the choiceof our prior, i.e., that Λ i ≥ and Q ≥ . Simulations whose individual parameters return the prior are shownas faded. Only a few simulations have large enoughsignal-to-noise ratios as well as high individual spins sothat about 15 out of a total of 240 individual parame-ters can be measured reliably. Among these sources, thelowest component spin is ∼ . . In almost all of thesecases, these parameters belong to the more massive starin the binary system since its tidal deformability andspin-induced quadrupole moment dominate.For all systems for which Q i can be measured, thepredicted quasi-universal relation connecting Q − Λ lieswithin the σ -credible interval, which shows that, inprinciple, an assessment of the robustness of Eq. (1) ispossible. Probing new Λ - Q relations: To answer thequestion if we would be able to detect a violation ofEq. (1), we have analysed the same set of injections,i.e., identical parameters except for a reduction ofthe quadrupole moments Q , by . We show therecovered parameters in Fig. 2 (bottom panel). Asbefore, most of the simulations do not allow a reliableextraction of the quadrupole moments and the individualtidal deformabilities, however, for highly spinning andclose systems, we find a set of data which are not inagreement with the Eq. (1) (purple line), but with themodified relation for which Q new = Q/ . Obviously, theparticular choice of Q new is arbitrary, however, it showsthat large enough deviations from existing theoreticalpredictions might already be measurable with the secondgeneration (2G) GW detectors [37]. Construction of Λ - Q relations with 3G de-tectors: Finally, we simulate these sources in noisegenerated with envisaged sensitivities of future thirdgeneration (3G) detectors. For the 3G detectors, we usethe noise curve of the Einstein Telescope (ET) detectorwith its ET-D configuration [38], a cryogenic detectorto be built underground within the next decade inEurope [39], referred to as ‘ET’. Ref. [40] introducedthe idea of an interferometer available within similartimelines in the USA, also known as ‘Cosmic Explorer’(CE). Unlike ET, CE is planned to be a ground-baseddetector with an arm length of 40 km. For our configu-ration, we choose a detector network including the ETdetector, with its xylophone configuration (located atthe Virgo site) and two CE-type detectors (located atthe two LIGO sites) [41]. The 3G detectors will have theability to reach lower cutoff frequencies of f low ∼ Hz,which means that sources like those considered before,i.e., for 2G detector network, will spend many morecycles in the 3G detectors’ band, therefore improvingboth the signal-to-noise ratio as well as the durationfor which the signal is visible in band. Due to limitedcomputational resources, we keep the lower frequencycutoff with the 3G detectors same as the simulationswith design sensitivity of advanced LIGO and advancedVirgo, i.e., f low = 28 Hz. While this means that we i )0 . . . . . l n ( Q i ) i )0 . . . . . l n ( Q i ) FIG. 3. Recovered Λ , and Q , values from our simulatedpopulation of 120 BNS systems for which log( Q i ) > . and ∆ log( Q ) < for a 3G detector network. The shown errorbars( ∆ log Λ ) mark the σ -credible interval. Fainther crosses re-fer to data with larger uncertainties. The dashed purple linerefers to Eq. (1), the blue dashed line to the quasi-universalfor which Q i got reduced by , and the green solid linerefers to the best fit of the data.Top panel: The injection set is based on Eq. (1). Bottompanel: The injection set is based on our modified quasi-universal relation. are not using the full potential of the future detectorsand that we artificially reduce the maximum SNR [42],this procedure leads to a conservative result, i.e., theresult will be better with future data-analysis techniques.Employing the 3G network described above, we presentthe extracted values of Λ i and Q i for our two injectionsets in Fig. 3, where we restrict to using the data forwhich (i) log( Q ) ≤ . , larger values are basically notexpected and an indicator that the prior is recovered and(ii) we remove all datapoints for which ∆ log( Q ) > ,where ∆ log( Q ) refers to the width of the σ credible in-terval in the log-log plot, Fig. 3. We find clearly that therecovered source parameters cluster around the respec-tive, injected quasi-universal relations.For a quantitative measure, we try to extract a phe-nomenological Q − Λ relations directly from our recovereddataset. We fit the datapoints shown in Fig. 3 accordingto ln Q = ˆ a i + ˆ b i ln Λ . (3) For the fitting, we use weights that are indirectly propor-tional to the size of the σ credible interval of Q i , i.e.,setups in which the induced quadrupole-moment is mea-sured more accurately are favored. Different to Eq. (1)we decided to remove higher order terms since the mea-surement uncertainties do not allow any reliable deter-mination of terms ∝ log(Λ) k with k > . We find ˆ a = − . , ˆ b = 0 . for the dataset shown inthe top panel, i.e., those simulations employing the Yagi-Yunes relation, and ˆ a = − . , ˆ b = 0 . for our mod-ified quasi-universal relation. IV. CONCLUSION
We have tested if future GW detections might allow usto extract phenomenological relations between the spin-induced quadrupole moment and the tidal deformabilityof individual neutron stars. For this purpose, we havestudied a simulated population of
BNS systems for a2G detector network and a 3G detector network.We find that at design sensitivity a reduction of in the quadrupole moment would be visible, we antic-ipate that smaller deviations might not be observable.However, this means that Advanced LIGO and AdvancedVirgo might be able to detect possible deviations from ex-isting, theoretically-predicted, quasi-universal relations.However, one would need a 3G detector network for amore reliable measurement. We find that with a net-work of 2 Cosmic Explorer-like detectors and 1 EinsteinTelescope, we would be able to extract quasi-universalrelations from the neutron star properties inferred fromthe analysis of the gravitational wave signals.In the hypothetical scenario in which the extractedquasi-universal relations are not in agreement with theo-retical predictions, this would either indicate a violationof general relativity or that our current description of theinterior of neutron stars is insufficient.
ACKNOWLEDGMENTS
We thank Sebastian Khan and the LIGO-Virgo collab-orations’ extreme matter group for helpful discussions.We thank N. V. Krishnendu and Nathan K. Johnson-McDaniel for helpful feedback on the draft and goingthrough it carefully. We also thank Nathan K. Johnson-McDaniel and An Chen for support setting up the 3Ginjections. AS and TD are supported by the researchprogramme of the Netherlands Organisation for Scien-tific Research (NWO). TD acknowledges support by theEuropean Union’s Horizon 2020 research and innovationprogram under grant agreement No 749145, BNSmerg-ers. The authors are grateful for computational resourcesprovided by the LIGO Laboratory and supported by theNational Science Foundation Grants PHY-0757058 andPHY-0823459. [1] B. P. Abbott et al. (Virgo, LIGO Scientific), Phys. Rev.Lett. , 161101 (2017), arXiv:1710.05832 [gr-qc].[2] B. P. Abbott et al. (Virgo, LIGO Scientific), (2018),arXiv:1805.11581 [gr-qc].[3] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X9 , 011001 (2019), arXiv:1805.11579 [gr-qc].[4] E. Annala, T. Gorda, A. Kurkela, and A. Vuorinen,Phys. Rev. Lett. , 172703 (2018), arXiv:1711.02644[astro-ph.HE].[5] C. D. Capano, I. Tews, S. M. Brown, B. Margalit, S. De,S. Kumar, D. A. Brown, B. Krishnan, and S. Reddy,(2019), arXiv:1908.10352 [astro-ph.HE].[6] A. Bauswein, O. Just, H.-T. Janka, and N. Stergioulas,Astrophys. J. , L34 (2017), arXiv:1710.06843 [astro-ph.HE].[7] S. De, D. Finstad, J. M. Lattimer, D. A. Brown,E. Berger, and C. M. Biwer, (2018), arXiv:1804.08583[astro-ph.HE].[8] B. Margalit and B. D. Metzger, Astrophys. J. , L19(2017), arXiv:1710.05938 [astro-ph.HE].[9] E. R. Most, L. R. Weih, L. Rezzolla, and J. Schaffner-Bielich, (2018), arXiv:1803.00549 [gr-qc].[10] M. W. Coughlin et al. , (2018), arXiv:1805.09371 [astro-ph.HE].[11] D. Radice and L. Dai, (2018), arXiv:1810.12917 [astro-ph.HE].[12] B. P. Abbott et al. (LIGO Scientific, Virgo), (2020),arXiv:2001.01761 [astro-ph.HE].[13] M. W. Coughlin et al. , Astrophys. J. , L19 (2019),arXiv:1907.12645 [astro-ph.HE].[14] M. W. Coughlin, T. Dietrich, S. Antier, M. Bulla, F. Fou-cart, K. Hotokezaka, G. Raaijmakers, T. Hinderer, andS. Nissanke, (2019), arXiv:1910.11246 [astro-ph.HE].[15] W. G. Laarakkers and E. Poisson, Astrophys. J. , 282(1999), arXiv:gr-qc/9709033 [gr-qc].[16] G. Pappas and T. A. Apostolatos, Phys. Rev. Lett. ,231104 (2012), arXiv:1201.6067 [gr-qc].[17] E. Poisson, Phys. Rev. D57 , 5287 (1998), arXiv:gr-qc/9709032 [gr-qc].[18] I. Harry and T. Hinderer, Class. Quant. Grav. , 145010(2018), arXiv:1801.09972 [gr-qc].[19] A. Samajdar and T. Dietrich, Phys. Rev. D100 , 024046(2019), arXiv:1905.03118 [gr-qc].[20] M. Agathos, J. Meidam, W. Del Pozzo, T. G. F.Li, M. Tompitak, J. Veitch, S. Vitale, andC. Van Den Broeck, Phys. Rev.
D92 , 023012 (2015),arXiv:1503.05405 [gr-qc].[21] N. V. Krishnendu, K. G. Arun, and C. K. Mishra, Phys.Rev. Lett. , 091101 (2017), arXiv:1701.06318 [gr-qc].[22] N. V. Krishnendu, M. Saleem, A. Samajdar, K. G. Arun,W. Del Pozzo, and C. K. Mishra, Phys. Rev.
D100 ,104019 (2019), arXiv:1908.02247 [gr-qc]. [23] K. Yagi and N. Yunes, Science , 365 (2013),arXiv:1302.4499 [gr-qc].[24] Z. Carson, K. Chatziioannou, C.-J. Haster, K. Yagi,and N. Yunes, Phys. Rev.
D99 , 083016 (2019),arXiv:1903.03909 [gr-qc].[25] K. Yagi and N. Yunes, Phys. Rev.
D88 , 023009 (2013),arXiv:1303.1528 [gr-qc].[26] J. Veitch et al. , Phys. Rev.
D91 , 042003 (2015),arXiv:1409.7215 [gr-qc].[27] LIGO Scientific Collaboration, “LIGO Algorithm Library- LALSuite,” free software (GPL) (2018).[28] J. Veitch and A. Vecchio, Phys. Rev.
D81 , 062003 (2010),arXiv:0911.3820 [astro-ph.CO].[29] J. Skilling, Bayesian Anal. , 833 (2006).[30] T. Dietrich, A. Samajdar, S. Khan, N. K. Johnson-McDaniel, R. Dudi, and W. Tichy, Phys. Rev. D100 ,044003 (2019), arXiv:1905.06011 [gr-qc].[31] M. Alford, M. Braby, M. W. Paris, and S. Reddy, Astro-phys. J. , 969 (2005), arXiv:nucl-th/0411016 [nucl-th].[32] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall,Phys. Rev.
C58 , 1804 (1998), arXiv:nucl-th/9804027[nucl-th].[33] D. R. Lorimer, Living Rev. Rel. , 8 (2008),arXiv:0811.0762 [astro-ph].[34] J. M. Lattimer, Ann. Rev. Nucl. Part. Sci. , 485 (2012),arXiv:1305.3510 [nucl-th].[35] S. Alexander and N. Yunes, Phys. Rept. , 1 (2009),arXiv:0907.2562 [hep-th].[36] E. E. Flanagan and T. Hinderer, Phys. Rev. D77 , 021502(2008), arXiv:0709.1915 [astro-ph].[37] We expect that this observation does not depend on theparticular EOS, but that only the deviation from Eq. (1)is important.[38] S. Hild et al. , Class. Quant. Grav. , 094013 (2011),arXiv:1012.0908 [gr-qc].[39] M. Punturo et al. , Classical and Quantum Gravity ,194002 (2010).[40] S. Dwyer, D. Sigg, S. W. Ballmer, L. Barsotti, N. Maval-vala, and M. Evans, Phys. Rev. D91 , 082001 (2015),arXiv:1410.0612 [astro-ph.IM].[41] We note that we include an older CE noise curve andthat we do not take a frequency-dependent response intoaccount, we expect that due to our relatively large initialfrequency, the effect of the frequency-dependence of theresponse function is small. We refer the reader to [43] foradditional details.[42] Although we use a f low = 28 Hz for the 3G network, weobtain SNR values of about up to , i.e., about to30