Modeling the complete gravitational wave spectrum of neutron star mergers

Modeling the complete gravitational wave spectrum of neutron star mergers

Sebastiano Bernuzzi    Tim Dietrich    Alessandro Nagar TAPIR, California Institute of Technology, 1200 E California Blvd,Pasadena, California 91125, USA DiFeST, University of Parma, and INFN Parma, I-43124 Parma, Italy Theoretical Physics Institute, University of Jena, 07743 Jena, Germany Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France
July 12, 2019

In the context of neutron star mergers, we study the gravitational wave spectrum of the merger remnant using numerical relativity simulations. Postmerger spectra are characterized by a main peak frequency related to the particular structure and dynamics of the remnant hot hypermassive neutron star. We show that is correlated with the tidal coupling constant that characterizes the binary tidal interactions during the late-inspiral–merger. The relation depends very weakly on the binary total mass, mass-ratio, equation of state, and thermal effects. This observation opens up the possibility of developing a model of the gravitational spectrum of every merger unifying the late-inspiral and postmerger descriptions.

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


Direct gravitational wave (GW) observations of binary neutron stars (BNS) late-inspiral, merger and postmerger by ground-based GW inteferometric experiments can lead to the strongest constraints on the equation of state (EOS) of matter at supranuclear densities Damour et al. (2012); Read et al. (2013); Del Pozzo et al. (2013); Lackey and Wade (2015); Agathos et al. (2015); Bauswein and Janka (2012); Clark et al. (2014). There are two ways to set such constraints111 GW observations of BNS mergers can also constrain the source redshift Messenger and Read (2012); Messenger et al. (2014).: (I) measure the binary phase during the last minutes of coalescence using matched filtered searches Damour et al. (2012); Del Pozzo et al. (2013); Lackey and Wade (2015); Agathos et al. (2015); (II) measure the postmerger GW spectrum frequencies using burst searches Bauswein and Janka (2012); Clark et al. (2014).

Method (I) relies on the availability of waveform models that include tidal effects and are accurate up to merger Favata (2014); Lackey and Wade (2015); Agathos et al. (2015). Here, “up to merger” indicates the end of chirping signal in a precise sense that will be described below. Tidal interactions are significant during the late stages of coalescence at GW frequencies  Hz (for typical binary masses), and affect the phase evolution of the binary. The zero-temperature EOS is constrained by the measure of the quadrupolar tidal coupling constant (or equivalent/correlated parameters, e.g. Lackey and Wade (2015)) that accounts for the magnitude of the tidal interactions Damour and Nagar (2010); Damour et al. (2012).

Combining results from numerical relativity and the effective-one-body (EOB) approach to the general relativistic two-body problem Buonanno and Damour (1999, 2000); Damour et al. (2000); Damour (2001), one can show that the merger dynamics of every irrotational binary is characterized by the value of  Bernuzzi et al. (2014a). At sufficiently small separations, the relevant dependency of the dimensionless GW frequency on the EOS, binary mass, and mass-ratio is completely encoded in the tidal coupling constant222The spin dependence is approximately linear for small spins aligned with the orbital angular momentum.. A tidal effective-one-body model compatible with numerical relativity data up to merger was introduced in Bernuzzi et al. (2015), but no prescription is available to extend the model to the postmerger.

Method (II) relies on the high-frequency GW spectrum, and can, in principle, deliver a measure independent on (I) Clark et al. (2014). Binary configurations with total mass are expected to produce a merger remnant composed of a hot massive/hypermassive neutron star. The merger remnant has a characteristic GW spectrum composed of a few broad peaks around  kHz. The key observation here is that the main peak frequencies of the postmerger spectrum strongly correlate with properties (radius at a fiducial mass, compactness, etc.) of a zero-temperature spherical equilibrium star in an EOS-independent way Bauswein and Janka (2012); Bauswein et al. (2012). Thus, a measure of the peak frequency constraints the correlated star parameter. Recently, there has been intense research on this topic, and various EOS-independent relations were proposed Bauswein and Janka (2012); Bauswein et al. (2012); Hotokezaka et al. (2013); Bauswein et al. (2013, 2014); Takami et al. (2014); Bauswein and Stergioulas (2015); Takami et al. (2015). Most of the relations are constructed for equal-mass configurations and do not describe generic configurations for different total masses and mass-ratios, e.g. Hotokezaka et al. (2013); Bauswein et al. (2015). Additionally, the postmerger GW spectrum might be influenced in a complicated way by thermal effects, magnetohydrodynamical instabilities and dissipative processes.

In this paper we observe that the coupling constant can also be used to determine the main features of the postmerger GW spectrum in an EOS-independent way and for generic binary configurations, notably also in the unequal-mass case. The observation opens up the possibility of modeling the complete GW spectrum of neutron star mergers unifying the late-inspiral and postmerger descriptions. Geometrical units are employed throughout this article, unless otherwise stated. We use for the spectrum frequencies and for the instantaneous, time-dependent frequency.

Numerical Relativity GW Spectra.—

Figure 1: Simulations of BNS and GWs. Top: real part and amplitude of the GW mode and the associated dimensionless frequency versus the mass-normalized retarded time for a fiducial configuration, H4-135135. The signal is shifted to the moment of merger, , defined by the amplitude’s peak (end of chirping). Also shown is (twice) the dynamical frequency . Bottom: Snapshots of on the orbital plane, during the late inspiral (left), at simulation time corresponding to (middle), during the postmerger (right).
Figure 2: GWs spectra from BNS. The plot shows only a representative subset of the configurations of Table 1. Triangles mark frequencies corresponding to , circles mark frequencies.
Name EOS [] [kHz] []
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 1: BNS configurations and data. Columns: name, EOS, binary total mass , mass ratio , frequency in kHz, dimensionless frequency, tidal coupling constant . Configurations marked with  are stable MNS.

The numerical relativity data used in this work were previously computed in Bernuzzi et al. (2014a); Dietrich et al. (2015). In our simulations we solve Einstein equations using the Z4c formulation Bernuzzi and Hilditch (2010) and general relativistic hydrodynamics Font (2007). Our numerical methods are detailed in Dietrich et al. (2015); Hilditch et al. (2013); Bernuzzi et al. (2012a, b); Thierfelder et al. (2011); Brügmann et al. (2008). The binary configurations considered here are listed in Table 1. In the following we summarize the main features of the GW radiation obtained by BNS simulations.

We consider equal and unequal masses configurations, different total masses, and a large variation of zero-temperature EOSs parametrized by piecewise polytropic fits Read et al. (2009). Thermal effects are simulated with an additive thermal contribution in the pressure in a -law form, , where , is the rest-mass density and the specific internal energy of the fluid, see Shibata et al. (2005); Thierfelder et al. (2011); Bauswein et al. (2010). The initial configurations are prepared in quasicircular orbits assuming the fluid is irrotational.

Initial data are evolved for several orbits, during merger and in the postmerger phase for  milliseconds. A detailed discussion of the merger properties determined by different EOSs, mass, and mass-ratio is presented in Bernuzzi et al. (2014a); Dietrich et al. (2015). The binary configurations in our sample do not promptly collapse to a black hole after merger, but form either a stable massive neutron star (MNS) or an unstable hypermassive neutron star (HMNS), which collapses on a dynamical timescale  ms Baumgarte et al. (2000). Both HMNS and MNS remnants at formation are hot, differentially rotating, nonaxisymmetric, highly dynamical two-cores structures, e.g. Shibata et al. (2005); Stergioulas et al. (2011).

The typical GW signal computed in our simulations is shown in Fig. 1 for a fiducial configuration. We plot the real part and amplitude of the dominant multipole of the spin-weighted spherical harmonics decomposition of the GW, , versus the retarded time, . The figure’s main panel also shows the instantaneous and dimensionless GW frequency where . The bottom panels show snapshots of on the orbital plane, corresponding to three representative simulations times.

The waveform at early times is characterized by the well-known chirping signal; frequency and amplitude monotonically increase in time. The GW frequency reaches typical values , i.e.  kHz for a binary. The chirping signal ends at the amplitude peak, , which is marked in the figure by the middle vertical line. We formally define this time as the moment of merger, , and refer to the signal at as the postmerger signal. The GW postmerger signal is essentially generated by the structure of the remnant, see bottom right panel of Fig. 1. The frequency increases monotonically to as the HMNS becomes more compact and eventually approaches the collapse. Assuming the remnant can be instantaneously approximated by a perturbed differentially rotating star Stergioulas et al. (2011), the -mode of pulsation is strongly excited at formation and it is the most efficient emission channel for GWs.

The GW spectra are shown in Fig. 2 for a representative subset of configurations. Triangles mark frequencies corresponding to . Circles mark the main postmerger peak frequencies  kHz. The small frequency cut-off is artificial and related to the small binary separation of the initial data; physical spectra monotonically extend to lower frequencies. From the figure one also observes that: (i) there exists other peaks, expected by nonlinear mode coupling or other hydrodynamical interactions Shibata et al. (2005); Stergioulas et al. (2011); Bauswein and Stergioulas (2015); (ii) peaks are broad, reflecting the nontrivial time-evolution of the frequencies (see Fig. 1 and also the spectrogram in Bernuzzi et al. (2014b)); (iii) secondary peaks are present in most of the configurations, their physical interpretation has been discussed in Stergioulas et al. (2011); Takami et al. (2014); Bauswein and Stergioulas (2015); Dietrich et al. (2015). We postpone the analysis of these features to future work. In the following we focus only on the peak, which is the most robust and understood feature of the GW postmerger spectrum.

Figure 3: dimensionless frequency as a function of the tidal coupling constant . 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), and (bottom right). The black solid line is our fit (see Eq. (2) and Table 2); the grey area marks the 95% confidence interval.
+4.0302 +0.7538 +3.1956
0 +0.45500
Table 2: Fit coefficients of different quantities at and of with the template in (2).

Characterization of the postmerger GW spectra.—

Here, we show that correlates with the tidal coupling constant that parametrizes the binary tidal interactions and waveforms during the late-inspiral–merger. The relation depends very weakly on the binary total mass, mass-ratio, and EOS. We use a large data sample of 99 points including the data of Hotokezaka et al. (2013); Takami et al. (2015).

Let us first briefly summarize the definition of and its role in the merger dynamics.

Within the EOB framework, tidal interactions are described by an additive correction to the radial, Schwarzschild-like metric potential of the EOB Hamiltonian Damour and Nagar (2010). The potential represents the binary interaction energy. In order to understand its physical meaning, it is sufficient to consider the Newtonian limit of the EOB Hamiltonian, , where is the binary reduced mass, the momenta, and , with the relative distance between the stars (constants and are re-introduced for clarity). The tidal correction is parametrized by a multipolar set of relativistic tidal coupling constants , where label the stars in the binary Damour and Nagar (2010); Damour et al. (2012). The leading-order contribution to is proportional to the quadrupolar () coupling constants, where is the mass of star , the compactness, , and the dimensionless Love number Hinderer (2008); Damour and Nagar (2009); Binnington and Poisson (2009); Hinderer et al. (2010). The total coupling constant is defined as , and can be written as


assuming . The leading-order term of the tidal potential is simply .

A consequence of the latter expression for is that the merger dynamics is essentially determined by the value of  Bernuzzi et al. (2014a). All the dynamical quantities develop a nontrivial dependence on as the binary interaction becomes tidally dominated. The characterization of the merger dynamics via is “universal” in the sense that it does not require any other parameter such as EOS, , and . (There is, however, a dependency on the stars spins.) For example, at the reference point , the corresponding binary reduced binding energy , the reduced angular momentum , and the GW frequency can be fitted to simple rational polynomials Bernuzzi et al. (2014a)


with fit coefficients given in Table 2.

In view of these results, it appears natural to investigate the depedency of the postmerger spectrum on .

Our main result is summarized in Fig. 3, which shows the postmerger main peak dimensionless frequency as a function of for a very large sample of binaries. Together with our data we include those tabulated in Hotokezaka et al. (2013); Takami et al. (2015). The complete dataset spans the ranges , , and a large variation of EOSs. The peak location is typically determined within an accuracy of  kHz, see also Bauswein et al. (2012). Each of the four panels of Fig. 3 shows the same data; the color code in each panel indicates different values of (top left), EOS (top right), (bottom left), and (bottom right). The data correlate rather well with . As indicated by the colors and different panels, the scattering of the data does not correlate with variations of , EOS, , . The black solid line is our best fit to Eq. (2), where we set and fit also for , see Table 2. The fit 95% confidence interval is shown as a gray shaded area in Fig. 3.

We argue that the observed postmerger correlation with is a direct consequence of the merger universality. Although an analytical/approximate description of the postmerger dynamics is not available, the gauge-invariant curves contain, in analogy to the merger case, significant information about the system dynamics Bernuzzi et al. (2015). Specifically, we interpret as being generated by some Hamiltonian flow that continuously connects merger and postmerger. In terms of this Hamiltonian evolution, the values provide initial conditions for the dynamics of the MNS/HMNS; it is then plausible to assume that the postmerger correlation follows from these initial conditions by continuity. In order to assess this conjecture, we define the frequency given by the equation , notice that , and show that is the relevant dynamical frequency for both inspiral-merger and postmerger. Recalling that the standard quadrupole formula predicts that a generic source with geometry and rotating at frequency emits GWs at a frequency , we plot the latter in Fig. 1 and indeed observe that it corresponds to the main emission channel during the whole evolution. In practice, the gauge-invariant can be interpreted as the orbital frequency during the inspiral, and the angular frequency of the MNS/HMNS during postmerger. Furthermore, since merger remnants from larger binaries are less bound and have larger angular momentum support at formation, (so ) must be a monotonically decreasing function of , which is what one can observe in Fig. 3.

The frequency evolution is also expected to depend on angular momentum dissipation due to magnetic fields instabilities, e.g. Ciolfi et al. (2011); Zink et al. (2012); Kiuchi et al. (2014), cooling and shear viscosity Baumgarte et al. (2000). However, the available literature indicates these physical effects are negligible in first approximation, and we argue that they might result in frequency shifts . The stars rotation can instead play a relevant role via spin-orbit coupling effects: stars with dimensionless spin parameters can give frequency shifts  Bernuzzi et al. (2014b).


The result of this work, coupled with the modeling of the merger process given in Bernuzzi et al. (2014a, 2015), indicates the possibility to model the late-inspiral-merger-postmerger GW spectrum in a consistent way using as main parameter. In particular, an accurate late-inspiral-merger GW spectrum is given by a suitable frequency-domain representation, , of the waveform of Bernuzzi et al. (2015). The leading-order tidal contribution of such a spectrum reads with ; see Damour et al. (2012) for at 2.5 post-Newtonian order. A simple template for the postmerger spectrum for binaries with is then given by a single-peak-model and our fit for . The precise construction of such complete spectrum will be subject of future work. As mentioned in our discussion, it will be particular important to include spin effects, e.g. Bernuzzi et al. (2014b); Agathos et al. (2015).

The performance of the proposed model in a GW data-analysis context will be carefully evaluated in a separated study. In this respect, we suggest that an optimal strategy to constrain the EOS could be combining the late-inspiral measurement of type (I) with measurement of type (II). The inclusion of the postmerger model might lead to an improved estimate of , for the same number of observed events Del Pozzo et al. (2013); Lackey and Wade (2015); Agathos et al. (2015).


We thank Andreas Bauswein, Bernd Brügmann, Tibault Damour, Sarah Gossan, Tjonnie Li, David Radice, Maximiliano Ujevic, Loic Villain for comments and discussions. This work was supported in part by DFG grant SFB/Transregio 7 “Gravitational Wave Astronomy” and the Graduierten-Akademie Jena. S.B. acknowledges partial support from the National Science Foundation under grant numbers NSF AST-1333520, PHY-1404569, and AST-1205732. T. D. thanks IHES for hospitality during the development of part of this work. Simulations where performed at LRZ (Münich) and at JSC (Jülich).


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