Modeling the complete gravitational wave spectrum of neutron star mergers

Sebastiano Bernuzzi1,2, Tim Dietrich3, and Alessandro Nagar41TAPIR, California Institute of Technology, 1200 E California Blvd,Pasadena, California 91125, USA

2DiFeST, University of Parma, and INFN Parma, I-43124 Parma, Italy3Theoretical Physics Institute, University of Jena, 07743 Jena, Germany and

4Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France(Dated: August 19, 2015)

In the context of neutron star mergers, we study the gravitational wave spectrum of the mergerremnant using numerical relativity simulations. Postmerger spectra are characterized by a mainpeak frequency f2 related to the particular structure and dynamics of the remnant hot hypermassiveneutron star. We show that f2 is correlated with the tidal coupling constant κT

2 that characterizes thebinary tidal interactions during the late-inspiral–merger. The relation f2(κT

2 ) depends very weaklyon the binary total mass, mass-ratio, equation of state, and thermal effects. This observation opensup the possibility of developing a model of the gravitational spectrum of every merger unifying thelate-inspiral and postmerger descriptions.

PACS numbers: 04.25.D-, 04.30.Db, 95.30.Sf, 95.30.Lz, 97.60.Jd 98.62.Mw

Introduction.– Direct gravitational wave (GW) ob-servations of binary neutron stars (BNS) late-inspiral,merger and postmerger by ground-based GW intefer-ometric experiments can lead to the strongest con-straints on the equation of state (EOS) of matter atsupranuclear densities [1–7]. There are two ways to setsuch constraints1: (I) measure the binary phase duringthe last minutes of coalescence using matched filteredsearches [1, 3–5]; (II) measure the postmerger GW spec-trum frequencies using burst searches [6, 7].

Method (I) relies on the availability of waveform mod-els that include tidal effects and are accurate up tomerger [4, 5, 10]. Here, “up to merger” indicates the endof chirping signal in a precise sense that will be describedbelow. Tidal interactions are significant during the latestages of coalescence at GW frequencies fGW & 400 Hz(for typical binary masses), and affect the phase evolu-tion of the binary. The zero-temperature EOS is con-strained by the measure of the quadrupolar tidal cou-pling constant κT2 (or equivalent/correlated parameters,e.g. [4]) that accounts for the magnitude of the tidal in-teractions [1, 11].

Combining results from numerical relativity and theeffective-one-body (EOB) approach to the general rela-tivistic two-body problem [12–15], one can show that themerger dynamics of every irrotational binary is charac-terized by the value of κT2 [16]. At sufficiently small sep-arations, the relevant dependency of the dimensionlessGW frequency on the EOS, binary mass, and mass-ratiois completely encoded in the tidal coupling constant2. Atidal effective-one-body model compatible with numeri-cal relativity data up to merger was introduced in [17],but no prescription is available to extend the model tothe postmerger.

1 GW observations of BNS mergers can also constrain the sourceredshift [8, 9].

2 The spin dependence is approximately linear for small spinsaligned with the orbital angular momentum.

Method (II) relies on the high-frequency GW spec-trum, and can, in principle, deliver a measure indepen-dent on (I) [7]. Binary configurations with total massM ≤ Mthr ∼ 2.9M are expected to produce a mergerremnant composed of a hot massive/hypermassive neu-tron star. The merger remnant has a characteristic GWspectrum composed of a few broad peaks around fGW ∼1.8 − 4 kHz. The key observation here is that the mainpeak frequencies of the postmerger spectrum stronglycorrelate with properties (radius at a fiducial mass, com-pactness, etc.) of a zero-temperature spherical equilib-rium star in an EOS-independent way [6, 18]. Thus,a measure of the peak frequency constraints the corre-lated star parameter. Recently, there has been intenseresearch on this topic, and various EOS-independent re-lations were proposed [6, 18–24]. Most of the relationsare constructed for equal-mass configurations and do notdescribe generic configurations for different total massesand mass-ratios, e.g. [19, 25]. Additionally, the post-merger GW spectrum might be influenced in a compli-cated way by thermal effects, magnetohydrodynamicalinstabilities and dissipative processes.

In this paper we observe that the coupling constant κT2can also be used to determine the main features of thepostmerger GW spectrum in an EOS-independent wayand for generic binary configurations, notably also in theunequal-mass case. The observation opens up the possi-bility of modeling the complete GW spectrum of neutronstar mergers unifying the late-inspiral and postmergerdescriptions. Geometrical units c = G = 1 are employedthroughout this article, unless otherwise stated. We usef for the spectrum frequencies and ω for the instanta-neous, time-dependent frequency.

Numerical Relativity GW Spectra.— The numericalrelativity data used in this work were previously com-puted in [16, 26]. In our simulations we solve Einsteinequations using the Z4c formulation [27] and general rel-ativistic hydrodynamics [28]. Our numerical methods aredetailed in [26, 29–33]. The binary configurations consid-

arX

iv:1

504.

0176

4v3

[gr

-qc]

17

Aug

201

5

2

1.0

0.5

0.0

0.5

1.0

2MΩ

Mω22

|Rh22/(Mν)|R[Rh22/(Mν)]

2000 1000 0 1000 2000(t−tmrg)/M

-30 0 30x

-30

0

30

y

-20 0 20x

-20

0

20

y

-20 0 20x

-20

0

20

y

9

7

5

3log10(ρ)

FIG. 1. Simulations of BNS and GWs. Top: real part and amplitude of the GW mode Rh22/(νM) and the associateddimensionless frequency Mω22 versus the mass-normalized retarded time t/M for a fiducial configuration, H4-135135. Thesignal is shifted to the moment of merger, tmrg, defined by the amplitude’s peak (end of chirping). Also shown is (twice) thedynamical frequency MΩ = ∂Eb/∂j ∼ Mω22/2. Bottom: Snapshots of log10 ρ on the orbital plane, during the late inspiral(left), at simulation time corresponding to tmrg (middle), during the postmerger (right).

ered here are listed in Table I. In the following we sum-marize the main features of the GW radiation obtainedby BNS simulations.

We consider equal and unequal masses configurations,different total masses, and a large variation of zero-temperature EOSs parametrized by piecewise polytropicfits [34]. Thermal effects are simulated with an additivethermal contribution in the pressure in a Γ-law form,Pth = (Γth − 1)ρε, where Γth = 1.75, ρ is the rest-massdensity and ε the specific internal energy of the fluid,see [32, 35, 36]. The initial configurations are preparedin quasicircular orbits assuming the fluid is irrotational.

Initial data are evolved for several orbits, duringmerger and in the postmerger phase for & 30 millisec-onds. A detailed discussion of the merger properties de-termined by different EOSs, mass, and mass-ratio is pre-sented in [16, 26]. The binary configurations in our sam-ple do not promptly collapse to a black hole after merger,but form either a stable massive neutron star (MNS) oran unstable hypermassive neutron star (HMNS), whichcollapses on a dynamical timescale τGW . 〈R〉4/〈M〉3 ≈

200 ms [37]. Both HMNS and MNS remnants at for-mation are hot, differentially rotating, nonaxisymmetric,highly dynamical two-cores structures, e.g. [35, 38].

The typical GW signal computed in our simulationsis shown in Fig. 1 for a fiducial configuration. Weplot the real part and amplitude of the dominant ` =m = 2 multipole of the s = −2 spin-weighted spheri-cal harmonics decomposition of the GW, R(h+− ih×) =∑`mRh`m −2Y`m(θ, φ), versus the retarded time, t. The

figure’s main panel also shows the ` = m = 2 instanta-neous and dimensionless GW frequency Mω22 = Mdφ/dtwhere φ = −arg(Rh22). The bottom panels show snap-shots of log10 ρ on the orbital plane, corresponding tothree representative simulations times.

The waveform at early times is characterized by thewell-known chirping signal; frequency and amplitudemonotonically increase in time. The GW frequencyreaches typical values ωGW = 2πfGW ≈ ω22 . 0.1 −0.2/M , i.e. fGW ∼ 0.8 − 1.6 kHz for a M = 2.7M bi-nary. The chirping signal ends at the amplitude peak,max |Rh22|, which is marked in the figure by the middle

3

1 2 3 4 5f [kHz]

100

101

√5/

(16π

)R|h

22(f

)|/M

MS1b-150100ALF2-140110H4-135135SLy-140120SLy-135135

FIG. 2. GWs spectra from BNS. The plot shows only a rep-resentative subset of the configurations of Table I. Trianglesmark frequencies fmrg corresponding to tmrg, circles mark f2frequencies.

vertical line. We formally define this time as the momentof merger, tmrg, and refer to the signal at t > tmrg as thepostmerger signal. The GW postmerger signal is essen-tially generated by the m = 2 structure of the remnant,see bottom right panel of Fig. 1. The frequency increasesmonotonically to Mω22 ∼ 0.2−0.5 as the HMNS becomesmore compact and eventually approaches the collapse.Assuming the remnant can be instantaneously approxi-mated by a perturbed differentially rotating star [38], thef -mode of pulsation is strongly excited at formation andit is the most efficient emission channel for GWs.

The GW spectra are shown in Fig. 2 for a representa-tive subset of configurations. Triangles mark frequenciesfmrg corresponding to tmrg. Circles mark the main post-merger peak frequencies f2 ∼ 1.8−4 kHz. The small fre-quency cut-off is artificial and related to the small binaryseparation of the initial data; physical spectra monoton-ically extend to lower frequencies. From the figure onealso observes that: (i) there exists other peaks, expectedby nonlinear mode coupling or other hydrodynamical in-teractions [23, 35, 38]; (ii) peaks are broad, reflecting thenontrivial time-evolution of the frequencies (see Fig. 1and also the spectrogram in [39]); (iii) secondary peaksare present in most of the configurations, their physicalinterpretation has been discussed in [22, 23, 26, 38]. Wepostpone the analysis of these features to future work.In the following we focus only on the f2 peak, which isthe most robust and understood feature of the GW post-merger spectrum.

Characterization of the postmerger GW spectra.—Here, we show that f2 correlates with the tidal coupling

TABLE I. BNS configurations and data. Columns: name,EOS, binary total mass M , mass ratio q, f2 frequency inkHz, dimensionless Mf2 frequency, tidal coupling constantκT2 . Configurations marked with ∗ are stable MNS.

Name EOS M [M] q f2 [kHz] Mf2 [×102] κT2

SLy-135135 SLy 2.70 1.00 3.48 4.628 74

SLy-145125 SLy 2.70 1.16 3.42 4.548 75

ENG-135135 ENG 2.70 1.00 2.86 3.803 91

SLy-140120 SLy 2.60 1.17 3.05 3.906 96

MPA1-135135 MPA1 2.70 1.00 2.57 3.418 115

SLy-140110 SLy 2.50 1.27 2.79 3.426 126

ALF2-135135 ALF2 2.70 1.00 2.73 3.630 138

ALF2-145125 ALF2 2.70 1.16 2.66 3.537 140

H4-135135 H4 2.70 1.00 2.50 3.325 211

H4-145125 H4 2.70 1.16 2.36 3.138 212

ALF2-140110 ALF2 2.50 1.27 2.38 2.931 216

MS1b-135135∗ MS1b 2.70 1.00 2.00 2.660 290

MS1-135135∗ MS1 2.70 1.00 1.95 2.593 327

MS1-145125∗ MS1 2.70 1.16 2.06 2.740 331

2H-135135∗ 2H 2.70 1.00 1.87 2.561 439

MS1b-140110∗ MS1b 2.50 1.27 2.08 2.487 441

MS1b-150100∗ MS1b 2.50 1.50 1.87 2.303 461

TABLE II. Fit coefficients of different quantities at tmrg andof Mf2 with the template in (2).

Q(κT2 ) Q0 n1 [×102] n2 [×105] d1 [×102]

Emrgb −0.1201 +2.9905 −1.3665 +6.7484

jmrg +2.8077 +4.0302 +0.7538 +3.1956

Mωmrg22 +0.3596 +2.4384 −1.7167 +6.8865

Mf2 +0.053850 +0.087434 0 +0.45500

constant κT2 that parametrizes the binary tidal inter-actions and waveforms during the late-inspiral–merger.The relation f2(κT2 ) depends very weakly on the binarytotal mass, mass-ratio, and EOS. We use a large datasample of 99 points including the data of [19, 24].

Let us first briefly summarize the definition of κT2 andits role in the merger dynamics.

Within the EOB framework, tidal interactions aredescribed by an additive correction AT (r) to the ra-dial, Schwarzschild-like metric potential A(r) of the EOBHamiltonian [11]. The potential A(r) represents the bi-nary interaction energy. In order to understand its phys-ical meaning, it is sufficient to consider the Newtonianlimit of the EOB Hamiltonian, HEOB ≈ Mc2 + µ

2p2 +

µ2 (A(r) − 1) + O(c−2), where µ = MAMB/(MA + MB)is the binary reduced mass, p the momenta, and A(r) =1 − 2

r = 1 − 2 GMc2rAB

, with rAB the relative distance be-

tween the stars (constants c and G are re-introduced forclarity). The tidal correction AT (r) is parametrized bya multipolar set of relativistic tidal coupling constants

4

2.5

3.0

3.5

4.0

4.5

5.0Mf 2

[×102

]Binary Mass M

2.4502.5002.5502.6002.650

2.7002.7502.8002.8502.900

EOSMS1bSLyENG2HH4APR4

MS1ALF2MPA1GNH3Γ2

100 200 300 400κT2

2.5

3.0

3.5

4.0

4.5

5.0

Mf 2

[×10

2]

Mass-ratio q1.0001.0771.0801.1541.160

1.1671.2311.2501.2731.500

100 200 300 400κT2

Γth1.6001.750

1.8002.000

FIG. 3. Mf2 dimensionless frequency as a function of the tidal coupling constant κT2 . Each panel shows the same dataset; the

color code in each panel indicates the different values of binary mass (top left), EOS (top right), mass-ratio (bottom left), andΓth (bottom right). The black solid line is our fit (see Eq. (2) and Table II); the grey area marks the 95% confidence interval.

κA,B(`) , where A,B label the stars in the binary [1, 11].

The leading-order contribution to AT (r) is proportionalto the quadrupolar (` = 2) coupling constants, κA2 =

2kA2 (XA/CA)5MB/MA where MA is the mass of star A,

CA the compactness, XA = MA/M , and kA2 the ` = 2dimensionless Love number [40–43]. The total ` = 2 cou-pling constant is defined as κT2 = κA2 + κB2 , and can bewritten as

κT2 = 2

(q4

(1 + q)5kA2C5A

+q

(1 + q)5kB2C5B

), (1)

assuming q = MA/MB ≥ 1. The leading-order term ofthe tidal potential is simply AT (r) = −κT2 r−6.

A consequence of the latter expression for AT (r) isthat the merger dynamics is essentially determined bythe value of κT2 [16]. All the dynamical quantities developa nontrivial dependence on κT2 as the binary interactionbecomes tidally dominated. The characterization of themerger dynamics via κT2 is “universal” in the sense thatit does not require any other parameter such as EOS, M ,and q. (There is, however, a dependency on the starsspins.) For example, at the reference point tmrg, the cor-responding binary reduced binding energy Emrg

b , the re-duced angular momentum jmrg, and the GW frequency

Mωmrg22 can be fitted to simple rational polynomials [16]

Q(κT2 ) = Q01 + n1κ

T2 + n2(κT2 )2

1 + d1κT2, (2)

with fit coefficients (ni, di) given in Table II.In view of these results, it appears natural to investi-

gate the depedency of the postmerger spectrum on κT2 .Our main result is summarized in Fig. 3, which shows

the postmerger main peak dimensionless frequency Mf2as a function of κT2 for a very large sample of bina-ries. Together with our data we include those tabu-lated in [19, 24]. The complete dataset spans the rangesM ∈ [2.45M, 2.9M], q ∈ [1.0, 1.5], and a large varia-tion of EOSs. The peak location is typically determinedwithin an accuracy of δf ∼ ±0.2 kHz, see also [18]. Eachof the four panels of Fig. 3 shows the same data; the colorcode in each panel indicates different values of M (topleft), EOS (top right), q (bottom left), and Γth (bottomright). The data correlate rather well with κT2 . As indi-cated by the colors and different panels, the scattering ofthe data does not correlate with variations of M , EOS, q,Γth. The black solid line is our best fit to Eq. (2), wherewe set n2 = 0 and fit also for Q0, see Table II. The fit95% confidence interval is shown as a gray shaded areain Fig. 3.

5

We argue that the observed postmerger correlationwith κT2 is a direct consequence of the merger univer-sality. Although an analytical/approximate descriptionof the postmerger dynamics is not available, the gauge-invariant Eb(j) curves contain, in analogy to the mergercase, significant information about the system dynam-ics [17]. Specifically, we interpret Eb(j) as being gener-ated by some Hamiltonian flow that continuously con-nects merger and postmerger. In terms of this Hamilto-nian evolution, the values (Emrg

b (κT2 ), jmrg(κT2 )) provideinitial conditions for the dynamics of the MNS/HMNS;it is then plausible to assume that the postmerger cor-relation follows from these initial conditions by conti-nuity. In order to assess this conjecture, we define thefrequency given by the equation MΩ = ∂Eb/∂j, noticethat Ωmrg = Ωmrg(κT2 ), and show that Ω is the relevantdynamical frequency for both inspiral-merger and post-merger. Recalling that the standard quadrupole formulapredicts that a generic source with m = 2 geometry androtating at frequency Ω emits GWs at a frequency 2Ω,we plot the latter in Fig. 1 and indeed observe that itcorresponds to the main emission channel ω22 during thewhole evolution. In practice, the gauge-invariant Ω canbe interpreted as the orbital frequency during the inspi-ral, and the angular frequency of the MNS/HMNS duringpostmerger. Furthermore, since merger remnants fromlarger κT2 binaries are less bound and have larger angularmomentum support at formation, Ω(κT2 ) (so f2) must bea monotonically decreasing function of κT2 , which is whatone can observe in Fig. 3.

The frequency evolution is also expected to depend onangular momentum dissipation due to magnetic fields in-stabilities, e.g. [44–46], cooling and shear viscosity [37].However, the available literature indicates these physicaleffects are negligible in first approximation, and we arguethat they might result in frequency shifts ∆f2 . δf2. Thestars rotation can instead play a relevant role via spin-

orbit coupling effects: stars with dimensionless spin pa-rameters & 0.05−0.1 can give frequency shifts & δf2 [39].Outlook.— The result of this work, coupled with the

modeling of the merger process given in [16, 17], indi-cates the possibility to model the late-inspiral-merger-postmerger GW spectrum in a consistent way using κT2 asmain parameter. In particular, an accurate late-inspiral-merger GW spectrum is given by a suitable frequency-domain representation, h(f) = A(f) exp [−iΨ(f)], of thewaveform of [17]. The leading-order tidal contributionof such a spectrum reads ΨT (f) = −39/4κT2 x

5/2 withx(f) ∝ f2/3; see [1] for ΨT (f) at 2.5 post-Newtonianorder. A simple template for the postmerger spectrumfor binaries with M ≤ Mthr is then given by a single-peak-model and our fit for f2. The precise constructionof such complete spectrum will be subject of future work.As mentioned in our discussion, it will be particular im-portant to include spin effects, e.g. [5, 39].

The performance of the proposed model in a GW data-analysis context will be carefully evaluated in a separatedstudy. In this respect, we suggest that an optimal strat-egy to constrain the EOS could be combining the late-inspiral measurement of type (I) with measurement oftype (II). The inclusion of the postmerger model mightlead to an improved estimate of κT2 , for the same numberof observed events [3–5].Acknowledgments.— We thank Andreas Bauswein,

Bernd Brugmann, Tibault Damour, Sarah Gossan, Tjon-nie Li, David Radice, Maximiliano Ujevic, Loic Vil-lain for comments and discussions. This work was sup-ported in part by DFG grant SFB/Transregio 7 “Gravita-tional Wave Astronomy” and the Graduierten-AkademieJena. S.B. acknowledges partial support from the Na-tional Science Foundation under grant numbers NSFAST-1333520, PHY-1404569, and AST-1205732. T. D.thanks IHES for hospitality during the development ofpart of this work. Simulations where performed at LRZ(Munich) and at JSC (Julich).

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