Frequency deviations in universal relations of isolated neutron stars and postmerger remnants
FFrequency deviations in universal relations of isolated neutron stars and postmergerremnants
Georgios Lioutas,
1, 2, ∗ Andreas Bauswein,
1, 3 and Nikolaos Stergioulas GSI Helmholtzzentrum f¨ur Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany Department of Physics and Astronomy, Ruprecht-Karls-Universit¨at Heidelberg,Im Neuenheimer Feld 226, 69120 Heidelberg, Germany Helmholtz Research Academy Hesse for FAIR (HFHF),GSI Helmholtz Center for Heavy Ion Research, Campus Darmstadt, Germany Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (Dated: February 25, 2021)We relate the fundamental quadrupolar fluid mode of isolated non-rotating NSs and the dominantoscillation frequency of neutron star merger remnants. Both frequencies individually are known tocorrelate with certain stellar parameters like radii or the tidal deformability, which we furtherinvestigate by constructing fit formulae and quantifying the scatter of the data points from thoserelations. Furthermore, we compare how individual data points deviate from the corresponding fitto all data points. Considering this point-to-point scatter we uncover a striking similarity betweenthe frequency deviations of perturbative data for isolated NSs and of oscillation frequencies ofrapidly rotating, hot, massive merger remnants. The correspondence of frequency deviations inthese very different stellar systems points to an underlying mechanism and EoS information beingencoded in the frequency deviation. We trace the frequency scatter back to deviations of the tidalLove number from an average tidal Love number for a given stellar compactness. Our resultsthus indicate a possibility to break the degeneracy between NS radii, tidal deformability and tidalLove number. We also relate frequency deviations to the derivative of the tidal deformability withrespect to mass. Our findings generally highlight a possibility to improve GW asteroseismologyrelations where the systematic behavior of frequency deviations is employed to reduce the scatter insuch relationships and consequently increase the measurement accuracy. In addition, we relate the f − mode frequency of static stars and the dominant GW frequency of merger remnants. We find ananalytic mapping to connect the masses of both stellar systems, which yields particularly accuratemass-independent relations between both frequencies and between the postmerger frequency andthe tidal deformability. I. INTRODUCTION
Fluid oscillation frequencies are one of the most fun-damental properties of a stellar system. In the case ofisolated neutron stars (NSs), the fundamental f − modeis one of the main characteristics of the system and isparticularly important because it leads to strong emis-sion of gravitational waves (GWs) (see [1]). Similarly,binary neutron star (BNS) mergers which do not lead toa prompt collapse to a black hole, produce remnants inwhich fluid modes are excited. At a frequency f peak inthe range of a few kHz, the dominant fluid oscillation ofthe merger remnant is an efficient emitter of GWs, seee.g. [2–12] as well as the reviews [13–19] and referencestherein. Hence, it is an important target of current GWsearches. The sensitivity of the LIGO and Virgo GW de-tectors was not sufficient to detect the post-merger phaseof the BNS merger GW170817 [20, 21], or the likely BNSmerger GW190425 [22]. However, a post-merger detec-tion is expected to be achieved in the near future, eitherwith upgraded or next-generation detectors [12, 23–39].The frequency of fluid oscillations depends on the stel-lar structure. However, the equation of state (EoS) of NS ∗ [email protected] matter is only incompletely known e.g. [40–45]. In orderto decipher the high-density EoS, many different relationshave been proposed between the frequency of fluid oscilla-tions and stellar parameters which are uniquely linked tothe EoS. These relations are the basis of GW asteroseis-mology. There exists a variety of relations with differentindependent variables exhibiting a different degree of ac-curacy for both the f − mode in isolated NSs [46–49], aswell as for the dominant fluid oscillation in BNS mergerse.g. [7, 8, 10, 11, 50, 51]. In practice, these relations areobtained by fitting frequencies as function of some chosenstellar parameter for a sizable number of EoS models.In this paper we present a systematic comparison be-tween previously proposed relations based on a consistentdata set. We investigate the accuracy of these relationsfor isolated stars and merger remnants by quantifying thescatter in these relations. As a figure of merit, we usethe mean and maximum deviation of data points fromthe corresponding relations in Hz. By determining ab-solute values of frequency deviations for all relations, wecan quantitatively compare different relationships basedon the set of EoSs considered here. Work along this di-rection for the case of isolated NSs was carried out inthe past, but for a significantly smaller set of EoSs anda subset of the relations considered here [52].Furthermore, we focus on the exact distribution ofpoints with respect to these relations. This aspect is a r X i v : . [ a s t r o - ph . H E ] F e b largely unexplored and has not yet been addressed be-fore. Specifically, we investigate the point-to-point scat-ter comparing individual models, i.e. where a specificmodel is located with respect to a fit to the complete setof models. As the main result of this study, we point outthat the individual models follow a systematic behavior.In particular, we uncover a striking similarity betweenthe frequency deviations w.r.t. a fit to the full sample ofmodels in isolated stars and merger remnants describedby the same EoSs. The agreement in how individualpoints scatter is surprising, since the frequencies referto two very different systems and are obtained indepen-dently using different approaches and numerical codes.As a side note, this result further supports that the dom-inant fluid oscillation in BNS merger remnants is pro-duced by the f − mode [6, 13, 53, 54]. We further investi-gate the underlying mechanism for the frequency devia-tions in both systems. We find that it is directly relatedto the tidal Love number k [55–57] which indicates fu-ture applications for improved EoS constraints based onan understanding of the frequency deviations. Also, fre-quency deviations can be related to the derivative of thetidal deformability with respect to mass. Finally, witha deeper understanding of the frequency deviations weexplore direct relations between the f − mode frequencyof static stars and the dominant postmerger frequency ofBNS merger remnants.The paper is organized as follows: In Sec. II we de-scribe our data sets for both isolated NSs and mergerremnants, as well as the set of EoSs we employ in thisstudy. In Sec. III we systematically investigate the ac-curacy of proposed relations between stellar pulsationfrequencies and stellar parameters. Initially we focuson relations for the f − mode frequency in isolated NSsand then investigate also relations involving the domi-nant fluid oscillation frequency for BNS mergers. In Sec.IV we point out the similarity in how individual mod-els distribute with respect to the respective relation forisolated NSs and for BNS mergers. We further investi-gate the source of these deviations and highlight futureapplications. In Sec. V we introduce direct relations be-tween the f − mode and postmerger frequencies. Finally,in the last section we provide a summary and conclu-sions. Throughout the whole work we set c = G = 1,unless otherwise specified. II. PERTURBATIVE SETUP AND MERGERDATA
In this study we consider two different sets of data forstellar pulsations. We discuss frequencies of static iso-lated stars, which we determine based on perturbativecalculations. Moreover, we describe the oscillation fre-quencies of NS merger remnants. These data are basedon relativistic hydrodynamics calculations, where we ex-tract the frequency from the GW spectrum. In this sec-tion we provide more details on the data.
A. Linear perturbations
Neutron star pulsations can lead to GW emission.Since GWs carry away energy, they act as a dampingmechanism. In a perturbative approach the pulsationsare treated as damped linear oscillations, which are an-alyzed in terms of quasi-normal modes (QNMs). Thisansatz assumes a e iωt time dependency, where ω is thecomplex eigenfrequency of the QNM. The complex na-ture of the eigenfrequency accounts for the damping. Itreads ω = 2 πf pert + iτ damp , (1)where f pert is the pulsation frequency and τ damp thedamping time of the oscillation. Extensive reviews on theformulation of linear oscillations can be found in [1, 58].We focus on the fundamental ( f − )mode. We obtainthe frequencies using the code presented in [59]. We com-pute perturbative frequencies for stellar models in therange from 1 . M (cid:12) to 1 . M (cid:12) with a spacing of 0 . M (cid:12) for different EoSs. We do not include the most compactstellar models for a given EoS, because our main purposeis to compare with binary neutron star mergers, whichfor the binary mass range under consideration do notreach such high densities/compactness. The most mas-sive merger system we consider has a total mass of 3 M (cid:12) (see Section II B). The central rest-mass densities ρ c inthese systems are comparable to the rest-mass densitiesof static stars up to about 1 . M (cid:12) . B. BNS mergers data sets
We simulate binary neutron star mergers with a 3Dsmoothed particle hydrodynamics (SPH) code employ-ing the results from [8, 60–62]. The code adopts theconformal flatness condition [63, 64] to solve Einstein’sfield equations. We choose a resolution of about 300 , th = 1 .
75 (see [65] for a detailed discussion).We extract the dominant postmerger GW frequencies(hereafter f peak ) for a total of 57 equal-mass binary sys-tems. Among them 16 are 1 . . M (cid:12) systems, 19 are1 .
35 + 1 . M (cid:12) systems, 16 refer to 1 . . M (cid:12) systemsand finally 6 correspond to 1 . . M (cid:12) systems. Thereare only a few models with M tot = 3 M (cid:12) because formost EoSs these binary systems lead to a prompt col-lapse of the merger remnant [66]. A detailed overviewof which EoSs are simulated for the different binary sys-tems can be found in Table I. Section II C provides moreinformation on the different EoS models.Finally, we refer to [6, 53] for evidence that the rem-nant’s oscillation at f peak is indeed produced by the TABLE I. EoSs simulated for each binary system. The firstcolumn displays the masses of the binary system, while thesecond column lists all EoSs simulated for these particularmasses.System masses [ M (cid:12) ] Simulated EoSs1 . . .
35 + 1 .
35 ALF2, APR, BHBLP, BSK20,BSK21, DD2, DD2F, DD2Y,eosUU, LS220, LS375, GS2,NL3, SFHO, SFHOY, SFHX,SLY4, TM1, TMA1 . . . . f − mode, which motivates a comparison between the fre-quencies in static stars and in NS mergers. C. Equations of state
We consider a set of 20 EoSs (ALF2 [67, 68], APR[69], BHBLP [70], BSK20 [71], BSK21 [71], DD2 [72,73], DD2F [73–75], DD2Y [76, 77], eosUU [78], LS220[79], LS375 [79], GS1 [80], GS2 [80], NL3 [72, 81], SFHO[82], SFHOY [76, 77], SFHX [82], SLY4 [83], TM1 [84,85], TMA [85, 86]) for which we calculate perturbativefrequencies. Postmerger GW frequencies are computedfor a slightly smaller subset of EoSs (see Section II B).All of the EoSs in this study yield a maximum grav-itational mass larger than 1 . M (cid:12) , which is in agree-ment with current observational constraints at the twosigma level [87–91]. Most of the EoS models are com-patible with a tidal deformability of 1 . M (cid:12) stars beingsmaller than 800. Thus, they are in agreement with theless strict tidal deformability constraint from the analysisof the inspiral of GW170817 [20, 92]. Six EoSs (LS375,GS1, GS2, NL3, TM1, TMA) yield Λ . > M versus radius R relation for each EoS. EoSs which are excluded basedon current GW measurements are depicted with dashedcurves. Evidently, the EoS sample covers a broad rangein the M − R diagram. 10 11 12 13 14 15 R [km]0 . . . . . . M [ M (cid:12) ] SLY4APRGS2LS375GS1 SFHOYBHBLPLS220TMAeosUU NL3SFHXDD2YALF2DD2F BSK21BSK20TM1DD2SFHX
FIG. 1. Gravitational mass M versus radius R for all EoSsconsidered in this work. Dashed curves refer to EoSs incom-patible with current constraints on the tidal deformability. III. FREQUENCY RELATIONS OF THE f − MODE
In the case of isolated, static stars many relations havebeen proposed relating f pert to stellar parameters suchas the mass M , radius R , moment of inertia I and tidaldeformability Λ = k ( c RGM ) (see e.g. [46–49]). Theyexhibit a different level of accuracy. Similarly, in thecase of binary systems, such relations involving f peak havebeen proposed for systems of different masses and binarymass ratios ([7, 8, 10, 11, 50, 51]). These relations aresometimes presented for a broad range of masses, whilein some cases they focus on a fixed mass.Relations involving the f − mode frequency of either astatic star or a merger remnant can be used to extractinformation about the stellar parameters from GW ob-servations. Generally tighter relations allow for a betterdetermination of the involved parameters. Hence rela-tions with a smaller scatter are favored. In this sectionwe focus on quantifying the scatter in different relationsto enable an objective comparison between them.For static stars, we examine different relations pro-posed in the literature and newly introduced in this work.We discuss how the points scatter in each case. Further-more, we consider new relations involving only propertiesreferring to the innermost part of the star, containing90% of its mass. Our results are then applied to binarymergers as well.As figure of merit to assess the tightness of a given re-lation, we consider the maximum and mean deviationbetween the data points and the respective fit to thedata. The maximum deviation may be biased by themost extreme model and does not represent how most ofthe points scatter. However, it may provide a conserva-tive measure of the accuracy of a relation and thus anupper estimate of the error if the relation is employedin GW measurements. Conversely, the mean deviationcaptures the point distribution, but it may not be fullyrepresentative of the error because the EoS sample is nota statistical ensemble. Hence, the mean deviation may beless suited to describe the error when using such relationsto determine stellar parameters. We present both devi-ations, since a combination of the two provides a morecomplete picture. Generally, we find a consistent behav-ior of both measures.In terms of notation, given a set of data ( X i , Y i ) with N points, we denote deviations by δ X Y . We define devi-ations between the data points and the corresponding fitas δ X Y i = Y i − Y fit ( X i ) , (2)for which we express the maximum and average deviationas max ( δ X Y ) = max i ( | δ X Y i | ) , (3) δ X Y = (cid:80) Ni =1 | δ X Y i | N , (4)where | · | is the absolute value.
A. Isolated stars
We start by discussing relations between f pert and thestellar mass M and radius R . All relations that we de-scribe in the following are provided in Table II based onour data. Table II includes the deviations of the relations.A very well known relation was proposed by Ander-sson and Kokkotas in [46] between f pert and the meandensity of the star. Later, Tsui and Leung used a dif-ferent scaling, which accurately describes both f pert and τ damp , as well as other families of modes [47]. By this,the mass-scaled frequency M f pert is found to yield a tightcorrelation with the compactness
M/R .In Figs. 2a-2b we plot our f − mode data for both re-lations. The solid curves display second-order fits to ourdata in both diagrams. The purpose of these fits is toaccurately quantify how the data scatter around the re-spective function. Table II provides the fit parametersand the mean and maximum deviation of the data pointsfrom the fit. Based on the deviations it is evident thatthe second relation is more accurate, but still both fitsexhibit some scatter .Another relation was discussed by Lau et al. [48] in-volving the moment of inertia. They remarked that Note that for a fair comparison, throughout the whole paper, wecompare absolute frequencies, also for relations with mass-scaledfrequencies. .
04 0 .
05 0 .
06 0 . p M/R . . . . . f p e r t [ k H z ] (a) .
10 0 .
15 0 .
20 0 . M/R M × f p e rt [ M (cid:12) × k H z ] (b) FIG. 2. Relations between f pert and the mass and radius ofa non-rotating neutron star. Panel (a) displays the relationbetween the f − mode frequency and the mean density of thestar as proposed in [46]. In panel (b) we plot the mass-scaledfrequency versus the compactness as suggested in [47]. Inboth panels the solid curve shows a second-order fit based onour data. the previously suggested relations cannot describe quarkstars, because neutron stars and quark stars result in dif-ferent density profiles. They argued that the moment ofinertia I is sensitive to the matter distribution within thestar. Thus defining an effective compactness through I leads to a relation which successfully describes both typesof stars. We include a second-order fit, which was alsoemployed in [48], as well as a fourth-order fit. The rela-tion is very tight (see mean and maximum deviations inTable II). Its accuracy does not improve with the orderof the fit.Chan et al. [49] suggested that there should exist atight correlation between f pert and Λ (see also [93]) based TABLE II. Relations between f − mode frequencies of static stars and different stellar parameters. The tightness of the relationsis quantified by the average and maximum deviation between fit and underlying data. First column provides a reference tothe work where the relation of the respective form has been proposed. Frequencies are in kHz, masses in M (cid:12) , radii in units of GM (cid:12) /c and moments of inertia in units of G M (cid:12) /c . The tidal deformability Λ is dimensionless.Reference Position Fit Mean dev. Max dev.[Hz] [Hz][46] Fig. (2a) f pert = − .
133 + 47 . (cid:113) MR − . MR
31 102This work Fig. (4a) f pert = − . . (cid:113) M ( R ) − . M ( R )
12 34[47] Fig. (2b) Mf pert = − .
427 + 14 . MR + 14 . (cid:0) MR (cid:1)
19 49This work Fig. (4b) Mf pert = − .
626 + 13 . MR + 11 . (cid:16) MR (cid:17)
10 34[48] Text Mf pert = − .
117 + 3 . (cid:113) M I + 18 . M I . Mf pert = − .
117 + 4 . (cid:113) M I + 16 . M I + 6 . (cid:16) M I (cid:17) / − . (cid:16) M I (cid:17) . Mf pert = − .
656 + 12 . − / − . − / Mf pert = − .
24 + 7 . − / + 11 . − / − . − / + 15 . − / .
12 0 . Mf pert = 6 . − .
929 ln Λ + 0 .
033 (ln Λ) Mf pert = 5 . − .
281 ln Λ − .
121 (ln Λ) + 1 . × − (ln Λ) − . × − (ln Λ) .
14 0 . on the I-Love-Q relations [94], since f pert tightly corre-lates with the moment of inertia I [48]. Consequently,they consider a relation of the form M f pert (ln Λ).Moreover, Λ − / is directly related to the compactness.This motivates a relation of the form M f pert (Λ − / ),which we plot in Fig. 3 and model by a second-order fit.According to the mean and maximum deviation of 3 Hzand 17 Hz, respectively, this relation is very tight (seeTable II). For comparison, the mean and maximum de-viation for a second-order M f pert (ln Λ) relation are 4 Hzand 19 Hz respectively. Considering fourth-order fits, asproposed in [49], relations with Λ − / and those with ln Λare identically accurate (see also bottom panel of Fig. 3for relations w.r.t. Λ − / ).Interestingly, second-order fits involving Λ − / or ln Λare less accurate than the relation involving the momentof inertia I . However, increasing the order of the fitsleads to tighter relations for Λ, while the accuracy of themoment-of-inertia relation remains practically the sameregardless of the order. As a result, fourth-order relationswith Λ − / or ln Λ are more accurate than the fourth-order relation with I .Comparing entries in Table II, it is clear that relationswith Λ are more accurate than those involving M and R . One main difference between the tidal deformabilityand the compactness is that the first is less sensitive tothe low-density parts of the star, in particular the crust[95, 96]. One may thus pose the question of whetherthe scatter in relations involving M and R can be solelyattributed to the crust, since the radius R is somewhat 234 M × f p e rt [ M (cid:12) × k H z ] nd order4 th order0 . . . . − / − − − | M × f p e rt − ( M × f p e rt ) f i t | ( M × f p e rt ) f i t FIG. 3. Relation between mass-scaled f − mode frequencyand Λ − / . Solid curve displays a second-order fit based onour data. Bottom panel shows the fractional error for thesecond-order fit plotted in the upper panel, as well as thefourth-order fit discussed in the text. The caption refers tothe bottom panel. sensitive to the low-density EoS.In order to investigate this point we define a new effec-tive radius R for static stars. It refers to the radius ofa sphere containing 90% of the gravitational mass of theconfiguration. By disregarding the outer shell containing10% of the mass, we obtain a radius which is insensitiveto the crust and the low-density EoS. Using this defini-tion, we excise the outermost 1 . − .
41 km of the stellarconfiguration. An additional feature of this newly definedquantity is that, for all EoSs and models considered, thepressure at this radius corresponds to 3 −
5% of the cen-tral pressure p c . Thus, one could equivalently define afixed pressure surface of e.g. p ∗ = 0 . × p c .Based on the newly defined radius we introduce themean density and compactness of the correspondingsphere. In Fig. 4 we plot the relations shown in Fig.2, but employing the new quantities which omit the low-density material. Both relations become tighter. This isclearly shown in Table II, where we explicitly give rela-tions and characterize their quality by the correspondingmean and maximum deviation. In particular, for therelation involving the mean density the improvement issignificant. Both relations involving the redefined meandensity and compactness are practically identically ac-curate compared to each other. We conclude that f pert is an excellent measure of the mean density of the star,when referring to the interior part comprising 90% of itsmass.However, we note that the relation involving Λ − / is still more accurate than the relations involving R .Although low-density material has been removed, stillsome scatter is visible. We thus conclude that the scatterin these relations does not entirely result from the low-density description. This implies that the distributionof data points with respect to the fit may also containadditional information about high-density properties ofthe EoS, which affect the f − mode frequency. B. Merger remnants
In the case of BNS mergers, tight relations have beenfound for systems with fixed total binary mass relatingthe dominant postmerger frequency to radii of static starsof a fixed mass [7, 8, 97]. Employing radii of static stars isa choice which is empirically found to yield tight relationsbearing in mind that one cannot define the mass andradius of merger remnants in an unambiguous way. Mass-scaled relations are not as accurate [11, 13]. We considerrelations between f peak and R or Λ − / to investigatewhether these relations also become tighter as those forstatic stars. This pressure corresponds to 25 −
47% of the central rest-massdensity ρ c or, in terms of the nuclear saturation density ρ sat ,translates to (0 . − . × ρ sat . .
04 0 .
05 0 .
06 0 .
07 0 . r . × M ( R ) . . . . . f p e r t [ k H z ] (a) .
15 0 .
20 0 . . × M/R M × f p e rt [ M (cid:12) × k H z ] (b) FIG. 4. Same as in Fig. 2, but the stellar parameters refer to asphere containing only 90% of the mass of the correspondingconfiguration. Compared to Fig. 2 both relations get tighter.
Table III lists empirical relations of the form f peak ( R x ), f peak ( R x ) and f peak (Λ x ) for all binary systems consid-ered in this study. Here x stands for the mass of a staticstar. In order to choose an appropriate stellar mass foreach binary system, we consider the maximum rest-massdensities in the first few milliseconds of the postmergerphase (for a more extensive discussion see Appendix A).Then, we determine the mass of static stellar configu-rations, which have roughly comparable central densi-ties. For instance, we relate systems with a total massof 2 . M (cid:12) to static stars of 1 . M (cid:12) . Similarly, 1 .
6, 1 . . M (cid:12) static stars are chosen for binary systemswith a total mass of 2 .
7, 2 . M (cid:12) , respectively. Asan example, Fig. 5 displays the empirical relations for1 . . M (cid:12) systems.Quantifying deviations in terms of frequencies allowsus to compare the quality of all relations to each otherkeeping in mind that the quantitative results to some ex-tent depend on the chosen fiducial mass x . We find thatusing either R or Λ − / leads to tighter empirical re-lations compared to R (see Fig. 5 for an example with1 . . M (cid:12) mergers). R relations are marginallyless accurate than those with Λ − / . This is in line withthe findings for the relations of static stars considering alarge mass range . Furthermore, we point out that rela-tions between f peak and the tidal deformability of staticstars are more accurate for fiducial masses higher thanthe mass of the inspiraling stars for all binary systemsconsidered (see Table III). We in particular refer to themore thorough analysis of this aspect in Appendix A (seealso [50] for relations for a range of binary masses).The analysis shows that using a frequency measure-ment the determination of R is up to twice as accu-rate as that of R as the maximum deviation should beincluded as an error estimate. For all binary systems themean deviation in f peak ( R x ) is about 70m (based onthe inverted relations R x ( f peak )). Therefore, R of afixed mass static star can be determined with high accu-racy from an observation of an equal-mass binary system.We emphasize that R is as informative about the EoSas R . As R , the redefined radius R is uniquely linkedto the EoS and, moreover, is only sensitive to the high-density regime of the EoS. IV. CONNECTION BETWEEN f pert AND f peak FREQUENCIES
In this section we address how individual data pointsare distributed with respect to the corresponding rela-tion, i.e. with respect to the fit to all points.
A. Point scatter in f peak relations and f pert relations In Fig. 5, we plot empirical relations between f peak and 3 different stellar parameters, namely R , R andΛ − / for 1 . . M (cid:12) systems. Considering the exactlocation of individual points in the plots, the points de-viate from the respective fit in a very similar way in allpanels. EoSs which lie above the fit in one plot, typicallylie above the fit in the other relations as well. The sameholds for EoSs lying below the fit. The data apparentlyfollows the same systematic behavior in all three rela-tions. From the fact that f peak ( R ) shows the same We note that for 1 . . M (cid:12) systems the improvement is notas pronounced. This results from the fact that less systems areconsidered in this case, since many EoS models result in a promptcollapse and thus the data set is significantly smaller comparedto the other binary masses. For simplicity, in the following we will use the term “EoS” foractually referring to the resulting frequency/data point obtainedfrom a calculation for this EoS.
11 12 13 14 R . [km]2 . . . . . f p e a k [ k H z ] (a) . . . . . . R . [km]2 . . . . . f p e a k [ k H z ] (b) .
00 3 .
25 3 .
50 3 .
75 4 . / . . . . . . . f p e a k [ k H z ] (c) FIG. 5. Postmerger frequencies f peak as function of variousstellar parameters of static stars with different EoSs. R . (top panel) refers to the radius of a 1 . M (cid:12) non-rotating NS, R . (middle panel) is the radius of a star with artificiallyexcised low-density region (see Section III A) and Λ / . (bot-tom panel) is the fifth-root of the tidal deformability. Thefrequencies refer to 1 . . M (cid:12) binary systems. TABLE III. Fits (third column) to the data of postmerger frequencies for different total binary masses (first column) employingvarious independent variables given in the second column. Fourth and fifth column provide the average and maximum deviationbetween fit and data in absolute frequencies. Frequencies are in kHz, radii R and R in km and Λ / is dimensionless.Deviations for all relations are in Hz, so they can be directly compared to each other.Binary masses Independent Fit Mean dev. Max dev.[ M (cid:12) ] variable [Hz] [Hz]1 . . R f peak = 10 . − . R . + 0 . R .
41 1091 . . R f peak = 12 . − . R . + 0 . (cid:16) R . (cid:17)
31 581 . . / f peak = 9 . − . / . + 0 . / .
18 441 . . / f peak = 9 . − . / . + 0 . / .
39 721 .
35 + 1 . R f peak = 12 . − . R . + 0 . R .
48 841 .
35 + 1 . R f peak = 12 . − . R . + 0 . (cid:16) R . (cid:17)
31 601 .
35 + 1 .
35 Λ / f peak = 9 . − . / . + 0 . / .
26 611 .
35 + 1 .
35 Λ / f peak = 8 . − . / . + 0 . / .
46 881 . . R f peak = 12 . − . R . + 0 . R .
53 1511 . . R f peak = 15 . − . R . + 0 . (cid:16) R . (cid:17)
38 1301 . . / f peak = 11 . − . / . + 0 . / .
36 1241 . . / f peak = 9 . − . / . + 0 . / .
65 1591 . . R f peak = − .
89 + 6 . R . − . R .
29 761 . . R f peak = − .
534 + 2 . R . − . (cid:16) R . (cid:17)
25 641 . . / f peak = 3 .
74 + 0 . / . − . / .
30 731 . . / f peak = − .
88 + 8 . / . − . / .
52 109 trend (middle panel) we conclude that this general ob-servation of similar frequency deviations is insensitive tothe low-density regime of the star. Furthermore, we finda similar pattern in plots for other binary masses.We now compare in more detail the point scatter inthe data from merger simulations to that of perturbationcalculations of static stars. In Fig. 6 we show six plots.The upper left panel displays postmerger frequencies for1 .
35 + 1 . M (cid:12) binary systems versus the radii R . of1 . M (cid:12) static stars. The middle left panel is a plot of f pert versus R . . Hence, we show both frequencies asfunction of the same independent variable.We compare how individual points scatter around thefits by examining the location of each data point with re-spect to the corresponding fits. We depict EoSs which areon the same side of the fit in both plots as black crosses.Remarkably, most data points follow this behavior, al-tough the frequencies describe very different systems. Weuse yellow symbols for EoSs which lie on opposite sides ofthe respective fits in these two plots (upper and middleleft panel) and which thus do not follow the systematicbehavior. This is a rather strict classification, especiallyfor points which lie relatively close to the fit. For in-stance, changing the set of EoS models to construct the fit or choosing another functional ansatz for the fit, wouldlead to another fit function and thus possibly change thecharacter of the deviation. This is obvious for pointswhich are very close to the fit. Hence, one should notclassify such points as actual outliers, even if they do notformally fulfill the corresponding quantitative criterion.We thus refine the criterion to identify actual outliers.We introduce blue shaded bands around the fits. Theyextend 15 Hz towards both sides of the fits, resulting in atotal width of 30 Hz. EoSs which lie within these bandsin both plots are also displayed as black points and arenot considered outliers. If however models lie outsidethese blue shaded bands in at least one of the two plots,we mark them with either black or yellow symbols asdescribed above.Adding such a band is well justified. As shown in Ta-ble III, the mean deviation is 48 Hz for the f peak ( R . )relation. Consequently, most points lie more than 15 Hzaway from the fit and they occur outside the blue shadedregion at least in the f peak − R plot. Thus, only a smallnumber of EoSs, lying very close to the fit, is captured bythis criterion. For 1 .
35 + 1 . M (cid:12) mergers, only 4 pointsout of 19 lie within the band in both plots with R . .Based on the described classification, 17 out of 19 EoSs
11 12 13 14 15 R . [km]2 . . . . f p e a k [ k H z ] (a) . . . . / . . . . . f p e a k [ k H z ] (b)
11 12 13 14 15 R . [km]1 . . . . . f p e rt , . [ k H z ] (c) . . . . / . . . . . . f p e rt , . [ k H z ] (d)
11 12 13 14 15 R . [km]2 . . . . . . Λ / . (e) . . . . / . . . . . . . Λ / . (f) FIG. 6. Panels (a) and (c) show postmerger frequencies f peak for 1 .
35 + 1 . M (cid:12) binary systems and perturbative frequenciesfor 1 . M (cid:12) stars, f pert , . , respectively versus the radius of static 1 . M (cid:12) stars, R . . Panels (b) and (d) display f peak for1 .
35 + 1 . M (cid:12) binary systems and f pert , . as function of Λ / . for static models with 1 . M (cid:12) . Panels (e) and (f) provideΛ / . versus R . and Λ / . , respectively. In all plots the solid curve shows a second-order fit to the data points. We plot a bandwith a total width of 30 Hz around frequency fits. See the main text for an explanation of the symbols’ colors. f pert − R and f peak − R relations shown inFig. 6 (see also Fig. 10). We find a similar behavior forsystems of other binary masses, which we summarize inTable IV. We also refer to the later discussion of Fig.10 showing that in fact the deviations of all data pointsfollow the same trend.Finally, we show a similar comparison of postmergerfrequencies and perturbation frequencies of static starsin the upper and middle right panels of Fig. 6, but withΛ / . as independent variable. Note that we employthe perturbative frequency of a more massive star with1 . M (cid:12) (as in the middle left panel of Fig. 6). Frequen-cies deviate in the same way in both relations (only oneoutlier). We also notice a very similar distribution ofdata points in the upper left and upper right panel (cf.also Fig. 5) and also the middle panels, i.e. all four plots.We summarize the different comparisons in Table IVemploying the same criterion as described above to quan-tify the behavior of the scatter in these relations. We con-sider additional pairs of relations in Table IV and deter-mine the number of outliers for each of them. Through-out all pairs of relations and binary masses the numberof outliers is very small. This corroborates our observa-tion that data points referring to two different systemsscatter in a similar way (see also Fig. 10).The agreement is even more pronounced in cases wherethe independent variable ( R or Λ / ) refers to static starswith the same mass as the inspiraling stars. In these plotsthe data points on average deviate more from the respec-tive fit. Hence, the location of data points with respectto the fit is less sensitive to small changes of the fit andthe similarities in the frequency deviations become moreevident for overall larger deviations. This further sub-stantiates the observation that the location of individualdata points with respect to the fits, which represent somekind of average behavior, follows a systematic pattern de-termined by the EoS.In a broader sense, we find that in fact all points be-have consistently in plots like Fig. 6. Considering forinstance clusters of points, we recognize very similar pat-terns of the distribution of points in the correspondingplots. This general consistency between the behavior inboth sets of frequency data is indeed remarkable, con-sidering statistical fluctuations and uncertainties, whichstem from the complexity of merger simulations.The fact that we consider a large number of EoSs fordifferent binary masses makes the observation even moreremarkable. The fits are based on a significant number ofEoS models, which essentially cover the full viable rangein the M − R diagram and arguably somewhat beyond.Including a few additional EoSs will not significantly alterthe fit and thus will not strongly affect the distributionof the current data points with respect to it.We emphasize once more that the agreement of fre-quency deviations with respect to the fits in Fig. 6 andTable IV is very remarkable and by no means expected. f pert refers to the frequencies from pertubative calcula- tions of static, non-rotating stars with a mass of 1.6 M (cid:12) ,whereas f peak frequencies describe the dominant oscilla-tion mode of rapidly rotating, hot merger remnants ofsignificantly higher mass, which actually still undergo adynamical evolution while f peak is extracted. We wouldlike to make two further remarks.1. Notably, the merger frequencies are obtained from athree-dimensional relativistic hydrodynamical sim-ulation code, which is computationally much morecomplex than solving the equations of linearizedperturbations around a background equilibriummodel (see Sections II A and II B). Clearly, the lat-ter code, in comparison, yields more robust and ac-curate results. Therefore, it is generally encourag-ing that the hydrodynamical simulations with thecurrent resolution are apparently able to uncoverthe frequencies to a degree that the frequency devi-ations resolve some underlying physics. This doesnot necessarily mean that the accuracy of about10 Hz, i.e. the level of frequency deviations, re-flects the full systematic uncertainties involved inthe numerical model nor that the frequencies arefully converged with respect to the numerical res-olution. This said, we comment that data pointswhich do not follow the described behavior (yel-low symbols), may well be attributed to numericalartifacts since the quoted frequency accuracy is cer-tainly on the edge of what a code of this type canachieve. However, we argue below that also theoutliers behave in some way consistently.2. The striking similarity of frequency deviations verylikely points to an underlying mechanism responsi-ble for the frequency shift in a certain direction.This implies that the frequency deviation on itsown encodes additional information about the EoS,which is the only link between the two systems.In the following section we further investigate thispoint and identify which EoS properties, or equiv-alently NS parameters, are causing the frequenciesto deviate in a certain way. We stress that, at leastin principle, the deviations may be measurable. Ifthe fits can be constructed based on simulationswith sufficient precision, measurements of the fre-quency and the respective independent quantity in-form about the frequency deviation from the fit.The radius or tidal deformability could be obtainedeither from independent measurements or from thevery same merger event providing f peak . Clearly,these ideas require a high measurement accuracy.It may also be possible that the frequency devia-tions correlate with other features of the GW signalof a NS merger. We note that secondary frequen-cies apparently deviate in the same way as the mainpeak (see Fig. 6 in [53]).1 TABLE IV. Second and third columns list relations for which we compare frequency deviations w.r.t. the fit (see main text).Fourth column provides the number of data points lying on opposite sides of the relations and outside the 30 Hz band in bothplots for the binary system given in the first column.Binary masses Relation 1 Relation 2 Number of outliers[ M (cid:12) ]1 . . f pert, . ( R . ) f peak ( R . ) 1 / . . f pert, . (Λ / . ) f peak (Λ / . ) 0 / . . f pert, . ( R . ) f peak ( R . ) 1 / . . f peak (Λ / . ) f peak ( R . ) 1 / . . f peak ( R . ) f peak ( R . ) 1 / .
35 + 1 . f pert, . ( R . ) f peak ( R . ) 2 / .
35 + 1 . f pert, . (Λ / . ) f peak (Λ / . ) 1 / .
35 + 1 . f pert, . ( R . ) f peak ( R . ) 2 / .
35 + 1 . f peak (Λ / . ) f peak ( R . ) 1 / .
35 + 1 . f peak ( R . ) f peak ( R . ) 1 / . . f pert, . ( R . ) f peak ( R . ) 3 / . . f pert, . (Λ / . ) f peak (Λ / . ) 2 / . . f pert, . ( R . ) f peak ( R . ) 2 / . . f peak (Λ / . ) f peak ( R . ) 1 / . . f peak ( R . ) f peak ( R . ) 2 / . . f pert, . ( R . ) f peak ( R . ) 1 / . . f pert, . (Λ / . ) f peak (Λ / . ) 0 / . . f pert, . ( R . ) f peak ( R . ) 0 / . . f peak (Λ / . ) f peak ( R . ) 0 / . . f peak ( R . ) f peak ( R . ) 2 / B. Physical explanation for frequency deviationsand encoded EoS information
The fact that points scatter in a similar way in fre-quency versus radius/Λ plots, for two very different sys-tems, suggests that there is a physical reason behind it. Itis clear that the EoS determines where individual pointsoccur in the diagram. In order to investigate this as-pect, we focus on perturbative frequencies. We alreadydiscussed that f pert − R x and f peak − R x diagrams showvery similar patterns. The perturbative data refer to asimpler system, which is why we expect that relationsfor f pert are more accurate and reliable. Hence, they arebetter suited to identify what causes points to occur ata certain location.For static stars with different masses there is a verytight relation between the mass-scaled f pert and Λ − / (see Section III A, Fig. 3 and Table II). The correspond-ing relation for a fixed mass shown in Fig. 7 is very tightwith a maximum deviation of only 2 . . M (cid:12) stars (see Table V). We find a similarly high accuracyfor relations with other fixed masses. Hence, one canconsider f pert and Λ − / being practically equivalent.Comparing Figs. 6c and 7 we notice a drastically dif-ferent distribution of points. In the plot involving theradius the points significantly scatter around the respec-tive relation. Employing the tidal deformability instead 2 .
50 2 .
75 3 .
00 3 .
25 3 .
50 3 . / . . . . . . f p e rt , . [ k H z ] FIG. 7. f pert versus Λ / . for non-rotating 1 . M (cid:12) stars.There is minimal scatter with a maximum deviation of 2 . the data points hardly exhibit any scatter . We remark that the frequency deviations are not related to thelow-density regime of the EoS, which affects radii stronger than TABLE V. Relations between f − mode frequencies or tidal deformabilities Λ / of static stars and different stellar parametersfor a fixed mass. Third and fourth column provide the average and maximum deviation between fit and underlying data.Frequencies are in kHz, radii in km and the tidal deformability Λ is dimensionless. Deviations for relations involving f pert, . are in Hz, while deviations for the relation between the tidal deformabilities are dimensionless.Position Fit Mean dev. Max dev.[Hz] [Hz]Fig. (6c) f pert, . = 7 . − . R . + 0 . R .
15 36Fig. (6d) f pert, . = 5 . − . / . + 0 . / .
17 45Fig. (7) f pert, . = 4 . − . / . + 0 . / . . / . = 0 .
205 + 0 . / . + 0 . / . .
073 0 . − . − .
02 0 .
00 0 . δ R f pert , . [kHz] − . − . . . . . δ R Λ / . FIG. 8. Deviations δ R Λ / between data points and a second-order fit in a Λ / . − R . diagram versus frequency deviations δ R f pert , . in a f pert ( R . ) relation (see panels (e) and (c) ofFig. 6 respectively). Solid line displays a first-order fit to thedata. The comparison between these two figures indicateswhich EoS properties cause the frequency deviations.Figure 7 shows that there is an essentially exact rela-tion between f pert and Λ / (for fixed mass) meaningthat f pert can be equivalently replaced by Λ / in therelations in Fig. 6. This implies that deviations in the f pert , . − R . plot (panel (c) in Fig. 6) are tightly anti-correlated with deviations in a Λ / . − R . diagram (panel(e) in Fig. 6; compare also panel (a) and (e)).In Fig. 8 we verify that this is indeed the case for1 . M (cid:12) stars. We define deviations in terms of frequen-cies, denoted by δ R f pert , between data points and the the tidal deformability. In Section III A, we introduce a newlydefined radius, R , such that it is insensitive to the EoS atlower densities, and we find significantly tighter relations between f pert and this new measure. Still, the relations feature a sizablepoint-to-point scatter, from which we conclude that it does notentirely result from the low-density EoS. In this context the term“low-density” thus refers to the material in the outer shell of thestar containing 10% of its total mass second-order fit in panel (c) of Fig. 6. Similarly, devia-tions in terms of Λ / . , denoted as δ R Λ / in Fig. 8, are de-fined between data points and a second-order Λ / . ( R . )fit. The deviations are strongly anti-correlated and followa linear trend. We find a similar behavior of the devia-tions for any other fixed mass within the mass range ofstatic stars considered here.The fact that deviations in f pert ( R . ) and Λ / . ( R . )are tightly correlated implies that we can trace back andexplain frequency deviations in f pert , and ultimately in f peak , by the difference between R . and Λ / . (Fig. 6eexhibits the same pattern of deviations as Fig. 6c andsimilarly for panels (f) and (d)).Therefore, the frequency deviations, i.e. the scatter infrequency plots in Fig. 6, are directly linked to the tidallove number k , which describes the difference betweenΛ and R through Λ = k ( c RGM ) . k is known to roughly correlate with the inverse compactness ( R/M ) (for in-stance the relation Λ (cid:39) α ( c RGM ) with α = 0 . ± . k av2 = α ( c RGM )). More specif-ically, the scatter in Λ / . ( R . ), and thus the frequencydeviations, are determined by how much k deviates froman average k av2 estimated based on the compactness. Wethus directly link the frequency scatter to the detailedbehavior of k . This in turn implies that observationalconstraints on the frequency deviation, possibly only itssign, informs about properties of k e.g. by how muchit deviates from an average k given by the compactnessand can be employed to break the degeneracy betweenΛ, k and R .In the upper panel of Fig. 9 we plot k versus R/M for1 . M (cid:12) static models. We include a second-order fit tothe data. Data points roughly follow the fit, but they par-tially exhibit visible deviations from it. The gray shadedband shows the maximum deviation in each panel. As ar-gued, the deviations are related to frequency deviations δ R f pert .Following the above reasoning about the equivalencebetween frequency deviations δ R f pert and differences be-tween Λ . and R . , we introduce a correction to k ,which is proportional to δ R f pert . We obtain the propor-tionality constant b by a single fit to the deviations in the3 . . . . . k . . . . . k − b δ R f p e rt , . . . . R/M . . . . . k − b δ R f | fi t ( δ R f p e a k ) FIG. 9. Upper panel shows k as a function of R/M for1 . M (cid:12) static stars. Middle panel displays “corrected” tidalLove number k − b δ R f pert , . as function of R/M . Bottompanel presents “corrected” k using δ R f peak values via a first-order fit between δ R f pert , . and δ R f peak (see Fig. 10). Inthe middle and bottom panels b = − . − . The grayshaded band represents the maximum deviation in each panel.Solid curve is a second-order fit to k ( R/M ) and identical inall panels. upper panel in Fig. 9. The resulting relation is shown inthe middle panel, which includes the same second-orderfit from the top panel and exhibits a very tight correla-tion of the corrected k − b δ R f pert with R/M . For thisfigure we find b = − . − and observe a simi-lar behavior for other masses in the range 1 . − . M (cid:12) .Obviously, we can also include the correction in R/M by changing the independent variable, which becomes
R/M − b (cid:48) δ R f pert6 and directly determines k .In Fig. 10 we plot δ R f peak , the deviation of data pointsfrom the fit in terms of postmerger frequencies in panel Note that b (cid:48) (cid:54) = b . In order to obtain it one needs to quantify hor-izontal deviations of data points from the fit in the upper panelof Fig. 9. Fitting δ R f pert , . to these deviations determines b (cid:48) . −
50 0 50 δ R f peak [Hz] − δ R f p e r t , . [ H z ] FIG. 10. Frequency deviations as occurring in panels (a) and(c) of Fig. 6 respectively. The blue shaded box has a sidelength of 30 Hz and matches the bands introduced in Fig. 6.The solid blue line is a first-order fit to the data given by δ R f pert , . = 0 . δ R f peak . (a) of Fig. 6, versus δ R f pert , . . The blue shaded area isa box with a width of 30 Hz, which matches the band weintroduced in Fig. 6. The solid blue line is a first-orderfit to the data, which we refer to as δ R f | fit . Evidently,the points approximately follow the line, which can beused to obtain an estimate for δ R f pert based on δ R f peak .This estimate can then be employed to obtain a betterestimate for k , as shown in the bottom panel of Fig.9. Data points shifted by b δ R f | fit ( δ R f peak ) deviate lessfrom the k ( R/M ) fit with the average and maximumdeviation reduced by 33% and 36% respectively. In par-ticular, the improvement is significant for most points,especially those with
R/M < .
5. A single point with
R/M (cid:39) .
83 is the only one which still arguably deviatesfrom the fit. Furthermore, we find that using deviationsdefined on f peak versus radius plots for various differentchoices of the mass to which the radius refers, also leadsto improved relations for k . In particular, δ R . f peak produces even better results than the bottom panel ofFig. 9, which is rather interesting as R . can poten-tially be extracted from the analysis of the inspiral.Although estimating δ R f pert through the linear fit isnot accurate and measuring δ R f peak may be challeng-ing, knowing whether δ R f pert is positive or negative isalready useful: The sign of δ R f pert informs whether thecorresponding point lies above or below the respective k ( R/M ) fit. This suffices to reduce the error in de-termining k through the respective fit by half. In thiscontext we recall that frequencies of postmerger oscil-lations can be recovered with ∼
10 Hz accuracy withsufficient signal-to-noise ratio (SNR) with future ground-based detector configurations, which has been shown bysimulated injections [23, 24]. Hence, the prospects to in-fer frequency deviations rely mostly on the challenge toconstruct by calculations reliable theoretical relations be-4tween frequency and TOV properties, to which measuredfrequencies can be compared.In summary, these relations show that δ R f pert or δ R f peak can be used for a more accurate estimate of k (beyond a relation with the compactness C) and thus toestablish the exact relationship between tidal deformabil-ity and radius, which is for instance important for EoSconstraints from the GW inspiral. As we already men-tioned quantifying the exact frequency deviation for themerger data is challenging and may also explain the fewoutliers in Fig. 6 and Table IV. In this respect we alsorefer to Fig. 10, where one can clearly see that in fact all data points do follow the same trend including the twooutliers. This exemplifies that our criterion for definingoutliers above is arguably too conservative and could inprinciple be replaced by a better classification scheme.At any rate, the consistent behavior of all data pointsin Fig. 10 corroborates our observation that frequencydeviations are correlated. C. Frequency deviations and the tidaldeformability of high-mass neutron stars
Finally, we connect frequency deviations with the be-havior of the tidal deformability Λ( M ) as function ofmass. In Fig. 6d we plot the perturbative frequency f pert , . versus the tidal deformability Λ / . . However,as already discussed, f pert , . scales very tightly with thetidal deformability of the stellar system with the samemass Λ / . (see Fig. 7). Hence, Fig. 6d practically dis-plays the relation between Λ / referring to two differentmasses, namely Λ / . and Λ / . . The difference betweenthese two values of Λ / approximates the derivative ofΛ / w.r.t. the mass.The upper panel of Fig. 11 shows the derivative d Λ / /dM at M = 1 . M (cid:12) versus Λ / . . The datapoints in the upper panel follow a coarse trend describedby a fit (blue curve), but they exhibit some sizable scatterbecause the derivative may still be different for the sameΛ / . . However, it is clear that the tidal deformabilityof some higher mass NS does carry information aboutthe behavior of the slope of Λ( M ). We thus anticipatethat the frequency deviations in Fig. 6d (or any other fre-quency deviation correlated to it like δ Λ / . f peak ) can beemployed to remove the significant scatter in the upperpanel.Following a very similar procedure as in Fig. 9 theadditional information encoded in the frequency devia-tions can be included. Employing either δ Λ / . f pert , . or δ Λ / . f peak leads to tighter relations for the derivative d Λ / /dM (see middle and bottom panel of Fig. 11 re-spectively). In particular, in the case of δ Λ / . f pert , . the accuracy of the relation improves significantly. Themaximum deviation is reduced by 80%.The importance of Fig. 11 is that the observation of a − . − . − . − . d Λ / d M (cid:12)(cid:12)(cid:12) . − . − . − . − . d Λ / d M (cid:12)(cid:12)(cid:12) . − b δ Λ / . f p e rt , . . . / . − . − . − . − . d Λ / d M (cid:12)(cid:12)(cid:12) . − b δ Λ / . f p e a k FIG. 11. Upper panel shows the derivative d Λ / /dM at afixed mass equal to 1 . M (cid:12) as a function of Λ / . . Middlepanel presents a ”corrected” derivative using the deviations δ Λ / . f pert , . . Bottom panel displays a corrected derivativethrough deviations δ Λ / . f peak (see Fig. 6b). The values of thefit parameters are b = − .
293 kHz − and b = − .
029 kHz − respectively. The gray shaded area represents the maximumdeviation in each panel. Solid curve is a second-order fit tothe data points in the upper panel and identical in all panels. single BNS is in principle sufficient to determine both thetidal deformability and its derivative w.r.t. mass. Thismeans the properties of Λ( M ) at higher masses are acces-sible without explicitly measuring the tidal deformabilityat higher masses if information on the frequency devia-tions is available (possibly from the same event).We remark that it is not strictly necessary to pick themass 1 . M (cid:12) for the deviations δ Λ / . f pert , . . In prin-5ciple measuring f pert of any mass can be used to obtaina corrected value for the derivative. In practice, valuescloser to 1 . M (cid:12) (or generally the mass of the inspiral-ing stars) may even lead to a more significant improve-ment. We also note that similar figures can be obtainedfor other binary masses. Furthermore, we comment thatthe reasoning in this subsection may also be reversed. Itmay be conceivable to use information on the derivative d Λ / /dM , e.g. from measuring Λ in two BNS eventswith different mass, to provide a more accurate predic-tion of the postmerger frequency. V. DIRECT RELATIONS BETWEENFREQUENCIES OF STATIC STARS ANDMERGER REMNANTS
Both perturbative frequencies f pert and postmergerfrequencies f peak scale tightly with stellar parameters ofstatic stars such as the radius R and the tidal deforma-bility Λ (see Figs. 5, 6 and 7). Furthermore, as discussedin Sections IV A and IV B, data points deviate from suchrelations in a very similar way for f pert and f peak . Thisimplies that there should also exist a direct correlationbetween the f − mode frequency f pert and the dominantpostmerger oscillation frequency f peak . One may expectsuch relations to become particularly tight, because thefrequency deviations in f pert and f peak , which we foundto be correlated, may to some extent cancel/compensateeach other.Figure 12 presents a mass-independent relation be-tween f peak ( M tot ) scaled by the chirp mass M chirp7 and f pert ( M TOV ) scaled by the mass of the correspondingstatic star M TOV for all 57 equal-mass systems consideredin this work. We relate each binary configuration to astatic NS by choosing the mass of the static star such thatthe densities in both systems are comparable. In partic-ular, we find that the choice M TOV = √ × M tot / f pert ( M TOV ) (and Λ / ( M TOV ) in Fig. 13) forany mass M TOV by a cubic spline fit to our perturbativedata.We find a highly accurate relation between M chirp f peak ( M tot ) and M TOV f pert ( M TOV ). The av-erage and maximum deviation of the data from the fitare 35 Hz and 162 Hz respectively. This is very smallconsidering that this is a mass-independent relation,while the relations in Table III are in comparisononly slightly more accurate. The high accuracy of thisrelation further highlights the strong connection between For binary systems with individual star masses M and M , thechirp mass is defined as M chirp = ( M M ) / ( M + M ) / . Note that thechirp mass is fully equivalent to the total mass for equal-massbinaries. We employ M chirp since it may prove more useful in afuture work which includes asymmetric binaries. .
25 2 .
50 2 .
75 3 .
00 3 . M TOV × f pert [M (cid:12) × kHz]2 . . . . . M c h i r p × f p e a k [ M (cid:12) × k H z ]
5% error
FIG. 12. Dominant postmerger oscillation frequency f peak ( M tot ) scaled by the chirp mass M chirp ( M tot ) as a func-tion of the perturbative frequency f pert ( M TOV ) scaled by themass of the corresponding stellar configuration M TOV . Thequantities M TOV and f pert refer to a stellar model with mass √ × M tot / M tot . Black sym-bols refer to 1 . . M (cid:12) , blue to 1 .
35 + 1 . M (cid:12) , red to1 . . M (cid:12) and green to 1 . . M (cid:12) systems. The solidblack curve is a second-order fit to the data. The gray shadedarea illustrates the 5% error band. All points lie within theband. f peak and f pert over the whole range of densities realizedin postmerger remnants.Figure 12 further indicates that there is a tight mass-independent relation between M chirp f peak and the tidaldeformability of static stars. As discussed in Section III A(in particular Fig. 3), the mass-scaled f pert correlates ex-tremely tightly with Λ − / . Hence, we expect that asimilarly tight relation M chirp f peak (Λ / ) exists.In Fig. 13 we replace M TOV f pert ( M TOV ) byΛ / ( M TOV ) of the corresponding static model. Asexpected, we find a tight correlation between the data.The average and maximum deviation of the second-order fit to the data is 35 Hz and 162 Hz respectively,which is perfectly in line with the deviations of the M chirp f peak ( M TOV f pert ) relation. These deviations inthe mass-independent relation correspond to mass-scaleddeviations of 41 M (cid:12) × Hz and 197 M (cid:12) × Hz respectively,which is significantly more accurate than relation (4)in [51]. The fit parameters for both relations aresummarized in Table VI.In order to consider a broader parameter range, we ex-tend our data set by also including unequal-mass binariesand constructing relations of the same type. We directlyimport the unequal mass data from Table II in [50] (ex-cept for one EoS which is not considered in this study).We include a total of 40 unequal-mass binary systems,with mass ratios as low as 0.67. Each binary configura-tion is related to a static star through6
TABLE VI. Mass-independent relations between f peak or mass-scaled M chirp f peak and static star properties. First column liststhe figures presenting the corresponding relation, while third and fourth columns provide the average and maximum deviationof each relation in Hz respectively. The frequencies are in kHz, masses in M (cid:12) and the tidal deformability is dimensionless.Fig. Systems Fit Mean dev. Max dev.[Hz] [Hz]12 Equal-mass M chirp f peak = − .
207 + 1 . M TOV f pert + 0 .
109 ( M TOV f pert )
35 16213 Equal-mass M chirp f peak = 12 . − . / + 0 . /
35 16214 All M chirp f peak = − .
229 + 1 . M TOV f pert + 0 .
103 ( M TOV f pert )
36 172- All M chirp f peak = 12 . − . / + 0 . /
35 172 . . . . / . . . . . M c h i r p × f p e a k [ M (cid:12) × k H z ]
5% error
FIG. 13. Dominant postmerger oscillation frequency f peak ( M tot ) scaled by the chirp mass M chirp ( M tot ) versus thetidal deformability Λ / ( M TOV ) of static stars. The massesof static stars and symbol colors are as in Fig. 12. The solidblack curve is a second-order fit to the data. All points liewithin the gray 5% error band. M TOV = (cid:18) q / − (1 + q ) q (cid:19) / × M tot , (5)which for equal mass systems becomes M TOV = √ × M tot / M chirp f peak ( M tot ) and Λ / ( M TOV ). We includethe expressions for both relations and their respective av-erage and maximum deviations in Table VI.
VI. SUMMARY AND DISCUSSION
In this study we consider the frequency of the funda-mental quadrupolar fluid mode in isolated NSs and thedominant oscillation of postmerger remnants. We com-pute the oscillation frequencies f pert of isolated NSs with 2 .
25 2 .
50 2 .
75 3 .
00 3 . M TOV × f pert [M (cid:12) × kHz]2 . . . . . M c h i r p × f p e a k [ M (cid:12) × k H z ] .
2% error
FIG. 14. Same as in Fig. 12, but black symbols representequal-mass systems and red symbols denote unequal-masssymbols. M TOV follows from Eq. (5). The solid black curveis a second-order fit to the data. All data points lie withinthe gray 5 .
2% error band. a perturbative method. In contrast, we obtain the fre-quency f peak of the dominant postmerger oscillation froma full dynamical simulation. We consider a large sampleof different high-density EoSs for both stellar systemsand vary the masses in a considerable range.Considering these frequency data separately for bothtypes of objects we construct fits, which relate the fre-quency to stellar parameters of non-rotating NSs thatwe choose to characterize the EoS. We employ differ-ent stellar parameters like radii and the tidal deforma-bility as independent variables and assess the accuracyof these relations by quantifying the maximum and av-erage deviations of the individual data points from theleast-square fit to all data points. Some of those relationshave been proposed previously in the literature and byemploying the same set of data we can consistently com-pare between these fits and evaluate their accuracy. Byconstructing second-order fits we find that the relationinvolving the moment of inertia I is the most accurate,while relations with the tidal deformability as indepen-dent variable are only slightly less tight. Extending theserelations to higher order, in particular the relations be-tween the mass-scaled perturbative frequency and the7tidal deformability become even tighter and essentiallyexact for all practical purposes. For fixed masses, second-order relations of the form f pert (Λ / ) are practically ex-act throughout the whole mass range and thus one canuse f pert and Λ / interchangeably.Furthermore, we introduce a newly defined stellar ra-dius R , where we disregard the outer mass shells con-taining 10% of the total mass. By doing this we obtaina measure for the stellar compactness, which is largelyinsensitive to the low-density regime of the EoS (belowapproximately (1 . − . × g / cm ). Employing R , we observe that relations for isolated stars as wellas for postmerger remnants become generally tighter withregard to the mean and maximum deviations. For per-turbative results of isolated stars the deviations are morecomparable to those with Λ. These results indicate thatoscillation frequencies in both systems are predominantlydetermined by the high-density regime of the EoS. Likethe commonly defined radius R at the stellar surface, R is uniquely linked to the EoS, but unbiased by thelow-density part, which presumably has a smaller influ-ence on the oscillation frequencies. Thus, a determina-tion of R is likely more informative about the high-density EoS than R , as the latter may be “biased” by thelow-density EoS. Employing R relations may thus bepreferable in GW asteroseismology since it results in amore accurate determination if the scatter in the fit for-mulae is taken into account as source of error and sinceit represents a more direct measure of the EoS propertiesin the relevant density regime.However, we also notice that there are finite frequencydeviations in the f peak relations for any of the indepen-dent variable we tested, i.e R , R or the tidal deforma-bility. With regard to this scatter, the main finding ofthis study is that frequency deviations follow the verysame behavior in isolated NSs and in postmerger rem-nants if frequencies are considered with respect to thesame independent variable: If f pert for a given EoS modelis slightly increased with respect to the fit to all datapoints of the perturbative calculations of isolated stars,the postmerger oscillation frequency for this EoS also oc-curs at slightly higher frequency compared to fit to allmerger simulations. Similarly, data points for other EoSmodels exhibit slightly reduced frequencies in both stellarsystems.The consistent behavior of frequency deviations in re-lations describing isolated NSs on one hand and rela-tions for merger remnants on the other hand is very re-markable: We compare the frequency of a cold, isolated,non-rotating NS to oscillations of a hot, rapidly rotating,non-stationary, massive merger remnant. We observe thecorrespondence of frequency deviations in various rela-tions for different independent variables characterizingthe EoS, and for different (binary) masses. We identify,if at all, only a very small number of outliers with re-spect to this behavior, which is why it is unlikely thatwe describe a mere coincidence. Instead, the agreementof the frequency scatter points to some underlying physi- cal mechanism which is mediated by the EoS as the onlycommon ingredient in both types of calculations. Also,the relatively large number of tested EoS models sup-ports the argument of additional EoS information beingencoded in the frequency beyond the gross scaling of uni-versal relations.In this regard, we stress that we compare frequenciesfrom perturbative calculations for isolated NSs, whichone should consider as rather robust and converged re-sults, and frequencies which are extracted from complex,three-dimensional hydrodynamical simulations of the fullmerger process using a different numerical code. Also,the merger remnant has not yet reached a stationaryconfiguration when the dominant frequency peak of theGW emission is shaped. We note that the magnitudeof frequency deviations is typically of the order of some10 Hz. It is thus remarkable that the hydrodynamicalsimulations apparently resolve some systematic behaviorof the frequency deviations, which are of this magnitude.Since this level of precision is certainly challenging for ahydrodynamical code of this type, we may even speculatethat the few outliers we observed can be attributed to in-accuracies of the merger simulations and that frequencydeviations follow the indicated trends even more closely.We further investigate the source of frequency devia-tions in GW asteroseismology relations like for instance f ( R ). To this end we exploit the correspondence betweenthe frequency increase or decrease in isolated NSs andmerger remnants, and thus focus on explaining the slightfrequency shifts for static stars. Moreover, we employ thefact that for static NSs there is a practically exact rela-tion between the f-mode frequency and the tidal deforma-bility. This implies that frequency deviations in f ( R ) arefully equivalent to deviations in Λ( R ). Hence, we can at-tribute frequency shifts to the scatter in the relationshipbetween the tidal deformability and the stellar radius,which by definition is given by the tidal Love number k .The frequency deviations thus encode by how much thetidal Love number deviates from an approximate scalingof k with the stellar radius.This indicates new directions to exploit this result infuture measurements and theoretical studies particularlyin the context of merger remnants, where oscillation fre-quencies might be more likely to be measured, although f − mode frequencies of isolated stars may play a role dur-ing the inspiral phase [99–102] and in other astrophysicalsystems. At least in principle frequency deviations froman expected universal relation, reflecting the average be-havior of a large class of EoS models, can be measured.As an example, measuring the magnitude or at least thesign of a frequency deviation from a universal relation canbe employed to break the degeneracy between radius andtidal deformability and can thus lead to a more precisedetermination of the tidal Love number and ultimatelyproperties of the EoS. We show an explicit case where k is determined more precisely if additional information forthe frequency deviation is available. Also, understandingthe link between frequency shifts and stellar properties8can be used to construct tighter universal relations be-tween GW frequencies like f peak and stellar parametersby removing the frequency shifts. Hence, more infor-mation can be extracted from a measurement if moreaccurate asteroseismology relations are available.Along the same lines we explicitly show that a mea-surement of the postmerger frequency and a measure-ment of the tidal deformability in the same event canbe combined to yield information on the slope of Λ( M ).Here, we again consider the deviation between the mea-sured postmerger frequency and the one expected from auniversal relation for the given tidal deformability. Thisreflects the additional information about properties athigher densities being encoded in the postmerger rem-nant. This is in line with the observation that the domi-nant postmerger frequency shows a particularly tight cor-relation with the tidal deformability of a NS with a highermass compared to that of the inspiralling stars.In this respect we also refer to the extensive analy-sis of f peak (Λ) and f peak ( R ) relations in Appendix A. Inparticular, we point out that f peak relations for a fixedbinary mass M tot are tighter if one relates f peak to thetidal deformability of a more massive fiducial star with M > M tot /
2, i.e. a mass larger than that of the inspiral-ing star, similar to what has been observed for frequency-radius relations [8]. This is summarized by the compari-son in Tab. III.Since frequency deviations in static stars and mergerremnants are correlated, one can employ this correspon-dence to partially remove the scatter in plots which di-rectly relate the perturbative frequency of static stars andpostmerger GW frequencies. In fact, we find very accu-rate mass-independent relations. We emphasize that forsuch type of relationships there is the freedom to choosea fiducial mass of the static model corresponding to agiven binary mass. We identify a simple, analytic map-ping M TOV = √ × M tot / f − modes of isolatedNSs and in relations for the dominant oscillation fre-quency of merger remnants provides additional evidencethat the dominant oscillation in postmerger objects islinked to the fundamental quadrupolar fluid mode in linewith previous arguments [6, 13, 53, 54].Future work should confirm that other hydrodynamicalcodes find a similar behavior of the frequency deviationsin f peak . As mentioned one should keep in mind that re-solving f peak with this accuracy is certainly challenging and that the frequency deviations are small in compari-son to the typical FWHM of a few 100 Hz of postmergerGW peaks. Other simulations not finding similar fre-quency patterns would not automatically imply that sys-tematic frequency deviations are not real but instead thatthese numerical models are possibly more affected by nu-merical uncertainties. In future studies one may check forconsistency between the frequency deviations of mergersimulations and the frequency deviations of static starsfrom either perturbative calculations or simply from theexpected frequencies employing the very tight relationsbetween f-mode frequency and tidal deformability. Bythis one may benchmark the quality of simulation datain larger surveys. Moreover, we speculate that in fu-ture more accurate merger models may yield frequencydeviations that more closely follow the quantitative de-pendencies, which we observed in this study, similar tothose for perturbative frequencies. This aspect may alsobe addressed by perturbative calculations of differentiallyrotating NSs in equilibrium resembling merger remnants[103, 104].By purpose we did not include EoS models with astrong phase transition in this study, which should beconsidered in future work. The significant and suddensoftening of the EoS by a strong phase transition will leadto a strongly increased postmerger frequency, i.e. an ex-treme frequency deviation of some 100 Hz [105–107]. Theeffect on the different relations presented here will how-ever very sensitively depend on the onset density of thephase transition and at which mass the stellar structureis affected. Thus choice of the dependent and indepen-dent variables is critical (in an extreme case one quantitywould be affected by a phase transition, while anothervariable only being sensitive to lower densities does notcarry any information about the EoS softening). Consid-ering phase transitions would introduce several new effec-tive degrees of freedom like the onset density, the densityjump across the transition and the stiffening of the EoSbeyond the phase transition. Such a variety can hardly becovered by a few models to allow a comprehensive study.We thus omit such models since they would severely af-fect the different fits representing an average behaviorand thus the quantification of frequency deviations ofthe purely hadronic models. Physically, this approachis very well justified because the extreme frequency devi-ations by a strong phase transition would unambiguouslyindicate the presence of exotic forms of matter as arguedin [105] and thus caution that the considerations of thepresent study may not be applicable. Similarly, evidencefor a phase transition may be provided by other indepen-dent measurements or observations.More work should also be spend on concrete meth-ods to implement the findings of our study. This in-cludes extracting frequency deviations from GW signalsand developing improved relations for GW asteroseismol-ogy where the scatter is reduced by taking into accountthe particular dependencies of the frequency deviationson EoS properties. Other aspects involve mergers of un-9equal mass, which we did not cover, and the behaviorof subdominant GW peaks, which we only briefly men-tioned to follow a similar trend. ACKNOWLEDGMENTS
We are grateful to Reed Essick, Brynmor Haskell andJocelyn Read for useful comments during a first presen-tation of these results. G.L. and A.B. acknowledge sup-port by the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovationprogramme under grant agreement No. 759253. A.B.acknowledges support by Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) - Project-ID279384907 - SFB 1245 and DFG - Project-ID 138713538- SFB 881 (“The Milky Way System”, subproject A10).N.S. acknowledges support by the ARIS facility of GR-NET in Athens (SIMGRAV, SIMDIFF and BNSMERGEallocations) and the “Aristoteles Cluster” at AUTh, aswell as by the COST actions CA16214 “PHAROS”,CA16104 “GWVerse”, CA17137 “G2Net”and CA18108“QG-MM”. NS gratefully acknowledges the Italian Is-tituto Nazionale di Fisica Nucleare (INFN), the FrenchCentre National de la Recherche Scientifique (CNRS) andthe Netherlands Organization for Scientific Research, forthe construction and operation of the Virgo detector andthe creation and support of the EGO consortium.
Appendix A: Accuracy of relations between f peak and static stellar properties Throughout this work we discuss relations between f peak for different binary systems and stellar propertiesof static stars with a fixed fiducial mass (e.g. Table IIIand Figures 5, 6, 12 and 13). In principle, the fiducialmass of the static models can be chosen freely. However,different choices for the fiducial mass lead to relationsof different accuracy (see also [8]), and the choice of themass of the static model should be justified.We consider relations between f peak and three indepen-dent variables: the radius R , the radius R referringto 90% of the mass and the fifth-root of the tidal de-formability Λ / . In order to quantify the accuracy ofthe respective relations, we examine three different fig-ures of merit. Specifically, the average deviations, maxi-mum deviations and the sum of squared residuals of theleast-squares fit.In Fig. 15 we present the three accuracy metrics forthe relation f peak (Λ / M ) for 1.35+1.35 M (cid:12) binaries asfunction of the fiducial mass M . All three figures of meritare minimized within the mass range 1 . − . M (cid:12) .Thus, we identify this mass range as the optimal for thisparticular binary system and type of relation.We summarize the analysis for other binary massesand other relations in Table VII. We list the mass rangesof the fiducial stellar model for which relations between 1 . . . . . . . . . M [M (cid:12) ]0 . . . | f p e a k − f fi t p e a k ( Λ / M ) | [ k H z ] FIG. 15. Different figures of merit to quantify the accuracy of f peak (Λ / M ) relations for 1 .
35 + 1 . M (cid:12) systems as functionof chosen fiducial masses M . The gray curve illustrates themaximum deviation, blue curve depicts the average deviationand green curve displays the normalized sum of squared resid-uals of the least-squares fit. Dashed lines indicate minima ofthe curves of the respective color. f peak and the different independent variables becometightest. Evidently, for a fixed binary mass, relationsw.r.t. different independent variables become tighter forslightly different fiducial masses. In particular, relationsinvolving the radius tend to become more accurate forhigher fiducial masses than relations w.r.t. Λ / . Obvi-ously, the “optimal” fiducial mass, in the sense of mini-mizing the deviations in frequency relations, is higher formore massive binaries . In all cases the optimal fiducialmass is higher than the mass of the inspiralling stars.This reflects the fact that merger remnants are in com-parison more massive and that densities in the mergerremnant are higher because of compression.Based on Table VII, for each binary system there existsa fiducial mass range of about 0 . M (cid:12) for which f peak re-lations become particularly tight. In order to understandthis observation, we consider the central rest-mass den-sities ρ c of the fiducial static models and the maximumrest-mass densities in the merger remnants during thefirst few milliseconds after merging. There is no uniqueway to define a characteristic density of the remnant be-cause it is strongly oscillating and dynamically evolving.We pick the maximum value of the maximum density ρ maxmax which occurs over the first few oscillation cycles af-ter merging (see [51, 105]) . In Figure 16 we plot thesedensities for binary systems of a total mass of 2 . M (cid:12) This is not the case for 1 . . M (cid:12) systems, because the data setis significantly smaller since many EoS models promptly collapseto a black hole. An alternative definition is to extract an average density overthe initial few milliseconds of the postmerger evolution. Thesituation is rather similar to Fig. 16 in that case as well. TABLE VII. Mass ranges of fiducial masses which minimizefrequency deviations in relations between postmerger frequen-cies and stellar parameters of static stars for different binarysystems. First column lists the masses of the binary systems.Second column provides the independent variable, i.e. thestellar parameter of a fiducial NS, which is employed in therespective fit. Third column gives the mass range over whichthe three considered figures of merit are minimized (see text).Fourth column lists the maximum values that the averageand maximum deviations assume in the corresponding massrange.Binary masses Independent Optimal mass Mean/Max[ M (cid:12) ] variable range [ M (cid:12) ] dev. [Hz]1 . . R . − . < (33 , . . R . − . < (20 , . . / . − . < (18 , .
35 + 1 . R . − . < (38 , .
35 + 1 . R . − . < (22 , .
35 + 1 .
35 Λ / . − . < (27 , . . R . − . < (42 , . . R . − . < (30 , . . / . − . < (35 , . . R . − . < (33 , . . R . − . < (26 , . . / . − . < (30 , and a static star with a mass of 1 . M (cid:12) . The choice ofstatic star mass is motivated by the mass ranges in Ta-ble VII. Overall, we notice an agreement between the twodensities, which explains why such a choice for the fidu-cial mass appears to be optimal. For softer EoSs, i.e. athigher densities, the densities in the remnant are in rela-tion to those in the static stars slightly higher. This is inagreement with the results in Table VII showing that ahigher fiducial mass represents the optimal description ofmatter in 1 .
35 + 1 . M (cid:12) systems. Moreover, it indicates that the compression during the merger process is morepronounced for softer EoSs.In summary, we conclude that relations between f peak and different independent variables referring to stellarproperties of static stars become most accurate for dif-ferent values of the mass of the static star. Using differ-ent figures of merit we can identify the mass range whichleads to the tightest relations. Typically this mass rangerefers to static stars with central densities comparable totypical densities realized in the merger remnants duringthe first few milliseconds after merging. Finally, we re-mark that the exact distribution of frequency deviationswill depend on the chosen set of candidate EoSs and mayalso be affected by the numerical model. Thus, the opti-mal values of the fiducial mass might be slightly differentin other surveys. In any case the extrema in Fig. 15 arerelatively broad. Hence, the exact choice of the fiducial0 . . . ρ c (1 . (cid:12) ) [10 g cm − ]0 . . . . . ρ m a x m a x [ g c m − ] × ρ sat × ρ sat × ρ sat × ρ sat FIG. 16. Maximum rest-mass density ρ maxmax in the rem-nant during the first few milliseconds after merging for1 .
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