Quasi-universal relations for the gravitational-wave luminosity and energy emission of binary neutron star mergers: EOS universality of the prompt-collapse threshold, high-spin validity, and remnant characterization

The 2018 numerical-relativity study by Zappa, Bernuzzi, Radice, Perego, and Dietrich (Phys. Rev. Lett. 120, 111101) showed that key emission and remnant properties of binary neutron star mergers — the peak gravitational-wave luminosity $L_{\rm peak}$, the total radiated energy, and the remnant angular momentum — are largely controlled by the tidal coupling constant $\kappa_2^T$ and obey quasi-universal (equation-of-state-insensitive) fits with residuals of order 20–30%. It left open whether these relations can be completed into a fully EOS-universal description: no single prompt-collapse threshold in $\kappa_2^T$ common to all equations of state was found; the fits were validated only for spin magnitudes up to about 0.15; robustness under fuller physics was only partially probed — the subset of simulations with microphysical neutrino cooling or effective magnetic viscosity lies on the same relations as the purely hydrodynamic ones, but full magnetohydrodynamics and complete microphysical neutrino transport were untested; and the super-Keplerian Type III remnants' long-term evolution and outflows were unresolved.

Background

A binary neutron star (BNS) merger is the late inspiral and collision of two neutron stars, simulated in numerical relativity (NR) by solving the Einstein field equations coupled to relativistic fluid dynamics. The matter description enters through the equation of state (EOS), the (still uncertain) relation between density, pressure, and composition of matter above nuclear density; finite-temperature (microphysical) EOSs additionally track thermal and composition effects. Tidal interactions during the inspiral are parametrized to leading order by the tidal coupling constant $\kappa_2^T$, a mass-weighted combination of the stars' quadrupolar tidal deformabilities (Love numbers), which measures how easily each star is deformed by the companion's gravity; smaller $\kappa_2^T$ corresponds to more compact, less deformable stars. The gravitational-wave (GW) emission of a merger is characterized by, among others, the peak luminosity $L_{\rm peak}=\max_t(dE_{\rm GW}/dt)$ (conveniently quoted in units of the Planck luminosity $c^5/G\approx 3.63\times10^{59}$ erg/s) and the total radiated energy $e_{\rm GW}^{\rm tot}=E_{\rm GW}^{\rm tot}/(M\nu)$, where $M$ is the total mass and $\nu$ the symmetric mass ratio; the remnant angular momentum is tracked as $j_{\rm rem}=J_{\rm rem}/(M^2\nu)$. Merger outcomes are conventionally classified into Type I (prompt collapse to a black hole within about a millisecond), Type II (delayed collapse after a phase as a hypermassive neutron star, HMNS — a transient, differentially rotating remnant supported temporarily by rotation and heat), and Type III (a longer-lived supramassive neutron star, SMNS, or massive stable neutron star that does not collapse within the simulated time). A relation between observable or binary parameters is called quasi-universal when it holds across many different EOSs with small scatter, making it usable for EOS inference despite not knowing the EOS. Building on earlier work showing that quantities at merger are essentially fixed by $\kappa_2^T$ alone, Zappa et al. (2018) analyzed about 100 BNS simulations (8 EOSs, mass ratios $q=M_A/M_B\in[1,2.06]$, roughly 30 configurations with aligned or anti-aligned spins up to $|S|/m^2\approx0.15$) and found: (i) a fit $L_{\rm peak}(\nu,\kappa_2^T)$ with coefficient of determination $R^2=0.943$ but maximal residuals of order 30% (one outlier near 60%); (ii) a tight quadratic relation $e_{\rm GW}^{\rm tot}\approx c_2 j_{\rm rem}^2-c_1 j_{\rm rem}+c_0$ with residuals below 20% for irrotational binaries, rising to about 30% when spinning binaries are included, which in principle lets one infer the final black-hole spin from the radiated energy with third-generation (3G) ground-based GW detectors such as the Einstein Telescope and Cosmic Explorer — the class of planned detectors with sufficient sensitivity at kilohertz frequencies to measure postmerger signals; (iii) that the prompt-collapse threshold translates into limiting values of $\kappa_2^T$ that vary by roughly a factor of two across EOSs, so that no single universal threshold value exists; and (iv) that Type III remnants carry more angular momentum than the Keplerian (mass-shedding) limit of any uniformly rotating star of the same rest mass — they are super-Keplerian — which suggests massive subsequent outflows. The quoted fits were calibrated without full magnetohydrodynamics: 37 of the simulations included a microphysical treatment of neutrino cooling and four an effective treatment of the turbulent angular momentum transport arising from small-scale magnetohydrodynamical instabilities, and these runs lie on the same relations as the purely hydrodynamic ones, but full magnetohydrodynamics and complete microphysical neutrino transport were not covered.

Problem Statement

The problem is to complete the equation-of-state-insensitive (quasi-universal) quantitative description of the gravitational-wave energy emission of binary neutron star mergers and its remnants, left unfinished by Zappa et al. (2018): (i) determine whether the prompt-collapse threshold admits a characterization universal across equations of state — whether in the tidal coupling constant $\kappa_2^T$ or an equivalent combination of binary and tidal parameters — that remains valid at high mass ratios, where asymmetric binaries exhibit accretion-induced prompt collapse, and for aligned/anti-aligned stellar spins, which are known to shift threshold behavior; (ii) construct fits for the peak gravitational-wave luminosity $L_{\rm peak}$ and for the total radiated energy $e_{\rm GW}^{\rm tot}$ (including its relation $e_{\rm GW}^{\rm tot}\approx c_2 j_{\rm rem}^2-c_1 j_{\rm rem}+c_0$ to the remnant angular momentum $j_{\rm rem}$) whose residuals are substantially below the ~30% level of the original calibration and which are demonstrably valid for dimensionless spin magnitudes beyond the $|S|/m^2\approx0.15$ range in which they were calibrated; (iii) establish whether these empirical relations survive the inclusion of physics absent from the original simulations — full magnetohydrodynamics, and microphysical neutrino transport beyond the neutrino cooling already included in part of the original simulation sample — and quantify any shifts or added scatter; and (iv) determine the long-term evolution and ejecta budget of the super-Keplerian Type III remnants, whose angular momentum exceeds the mass-shedding limit of uniformly rotating configurations, including the mechanisms and timescales of the outflows they are expected to launch.

References

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