Gravitational waves from the late inspiral, merger, and ringdown of a black hole-neutron star (BHNS) system can provide information to constrain the neutron star equation of state (EOS). To better understand neutron stars, it is critical to establish the EOS, which relates the density and pressure throughout a neutron star. Scientists aim to determine these physical characteristics of distant gravitational-wave sources by extracting information from observed gravitational waves. With construction of more sensitive gravitational-wave detectors currently ongoing, it is likely that gravitational waveforms from compact binaries will be observed this decade.
When a black hole and a neutron star (or two neutron stars or black holes) spiral toward each other and merge, the waveform from the observed gravitational wave varies based on physical parameters of the inspiraling objects. The parameter Λ, describing the neutron star's deformability, can be extracted from the observed waveforms. Λ is directly related to the radius, mass and l=2 Love number of the neutron star. Λ1/5 is considered equivalent to the neutron star's radius. Notably, Λ is the parameter that determines the departure from point-particle dynamics during the inward spiral of the objects. When tidal interactions between the inspiraling objects occur, the phase of the gravitational wave drifts an amount proportional to Λ.
Since the gravitational waveform depends on the EOS, simulations of neutron star and black hole mergers were performed by systematically varying the free parameters of a parametrized EOS. BHNS mergers, each with a different EOS, were compared to binary black hole (BBH) mergers simulated with the effective-one-body (EOB) formalism. For every simulated EOS, a waveform comparison was performed for BHNS and BBH mergers with the same mass ratio. The goal was to determine how the EOS affects the departure from an EOB waveform. This analysis showed that, for each EOS, the two waveforms differed based on how much the neutron star was disturbed or distorted during the merger. This factor was based on the neutron star's size. For neutron stars with a small radius, the black hole did not significantly distort the neutron star, and it crossed the event horizon intact. As a result, the merger and ringdown produced similar waveforms in the BHNS and EOB BBH cases. However, larger neutron stars were completely tidally disrupted just before merger, resulting in a supressed merger and ringdown waveform when compared to the EOB case.
The dependence of the waveform on the EOS was also seen by decomposing each waveform into amplitude and phase. At early times, the BHNS and EOB BBH waveforms were nearly identical. However, depending on the physical properties of the EOS, the BHNS waveform began to depart from its corresponding EOB BBH waveform a few milliseconds before the maximum amplitude was reached. The departure was monotonic in Λ. Hence, neutron stars with large values of Λ had waveforms with smaller maximum amplitudes. The strong tidal interactions lead to higher frequency orbit, which caused earlier mergers.
Next-generation telescopes will capture gravitational waveforms with sufficient accuracy to determine Λ, therefore constraining the equation of state p=p(ρ) to a hypersurface of constant Λ. Once Advanced LIGO is operational and detecting gravitational waves, it is expected that Λ1/5 can be extracted to 10-40% accuracy from observed waveforms for merging systems with mass ratios of 2 to 3 within 100 Mpc. With the Einstein Telescope, it will be possible to determine the EOS parameters with an order of magnitude better accuracy.