Direct Inference of Nuclear Equation-of-State Parameters from Gravitational-Wave Observations

Direct Inference of Nuclear Equation-of-State Parameters from Gravitational-Wave Observations

Overview

This paper presents a method for directly inferring parameters of the nuclear equation of state (EOS) from gravitational-wave (GW) observations of binary neutron star mergers. The key innovations are:

• Development of machine learning (ML) emulators for the computationally expensive Tolman-Oppenheimer-Volkoff (TOV) equations that govern neutron star structure. These emulators enable rapid evaluation of the EOS parameters during the GW data analysis.
• Implementation of these emulators in a Bayesian inference framework to directly sample the EOS parameters from the GW strain data of the binary merger GW170817.
• Extraction of constraints on the nuclear matter parameters $K\mathrm{sat}$, $L\mathrm{sym}$, and $K_\mathrm{sym}$ as well as the high-density speed of sound from the GW170817 event.
• Propagation of the inferred EOS parameter posteriors to derive constraints on other neutron star observables like the maximum mass, radius, and tidal deformability.

Methodology

Equation of State Model

• The EOS is modeled using a metamodel approach below twice nuclear saturation density, with parameters $K\mathrm{sat}$, $L\mathrm{sym}$, and $K_\mathrm{sym}$ varied.
• At higher densities, a speed-of-sound model is used, with either 2 or 7 discrete squared sound speed parameters.
• Priors on the EOS parameters are chosen to match constraints from nuclear theory and experiments.

Emulator Construction

• Multilayer perceptron neural networks are used to build emulators for the TOV equations, mapping the EOS parameters to the tidal deformability $\Lambda$.
• The emulators achieve 0.1% accuracy in predicting $\Lambda$ for a 1.4 solar mass neutron star, with evaluation times of tens of milliseconds.
• Separate classifiers are built to identify EOS parameter sets that produce maximum neutron star masses below 2 solar masses, which are outside the training range.

GW Inference

• Bayesian inference is performed on the GW170817 data using the <cit.> framework.
• The sampling parameters include the GW waveform parameters as well as the EOS parameters.
• The GW waveform model uses the SEOBNRv4T_surrogate template, with the tidal deformabilities computed using the emulators.
• Nested sampling with 8000 live points is used to obtain the posterior distributions.

Results

Sampling Convergence and Emulator Efficiency

• Comparisons between the emulator and full TOV solver results show excellent agreement in the inferred GW and EOS parameters with 8000 live points.
• The emulators provide a speed-up of nearly two orders of magnitude over the full TOV solver.

GW Inference Posteriors

• For the 5-parameter EOS model, the inference constrains $K\mathrm{sat} \lesssim 290$ MeV, $L\mathrm{sym} \lesssim 106$ MeV, and $K_\mathrm{sym} \lesssim 26$ MeV at the 90% credible level.
• The data favors a high squared sound speed at $3n_0$, with values below $\sim 0.2$ ruled out.
• The higher density sound speeds are largely unconstrained, as the neutron stars in GW170817 did not probe those densities.
• Results are similar for the 10-parameter EOS model.

Indirect Inference Constraints

• Propagating the inferred EOS parameter posteriors, the maximum neutron star mass is constrained to $2.32 \pm 0.21 M_\odot$.
• The radius of a 1.4 solar mass neutron star is constrained to $11.8^{+1.1}_{-0.7}$ km.
• The tidal deformability of a 1.4 solar mass neutron star is constrained to $335^{+362}_{-113}$.

Discussion

• The use of emulators enables computationally feasible large-scale Bayesian inference studies of GW data, saving substantial computational time and energy.
• The inferred constraints on nuclear matter parameters like $L\mathrm{sym}$ and $K\mathrm{sym}$ are consistent with nuclear theory and experimental estimates.
• The framework can be extended to study the impact of exotic degrees of freedom in the EOS by using different parameterizations.
• Future work will incorporate constraints from multiple GW events to further improve the nuclear EOS constraints.

Conclusion

This work demonstrates a method to directly infer nuclear equation of state parameters from gravitational-wave observations of binary neutron star mergers. The use of machine learning emulators enables efficient Bayesian inference, leading to constraints on the nuclear matter properties as well as neutron star bulk observables like mass and radius.