5.5 Discussion and concluding remarks
6.1.1 NSNS coalescence and gravitational waves
The discovery of the first binary system consisting of two neutron stars represented and important landmark in the study of compact objects, close binary systems and general relativity [26]. Often called the Hulse-Taylor binary after its discoverers, the system has provided some of the most compelling evidence to date for the existence of gravitational waves and accuracy of Einstein’s general relativity; the orbital separation of the binary has decayed over time at the precise rate predicted by general relativity [43, 44]. To date, at least ten [28] binary neutron star systems1 have been discovered. Many of these systems have small enough orbital separations that neutron stars are expected to coalesce on timescales less than 1Gyr due to orbital decay from the emission of gravitational waves. The closest NSNS binary is PSR J0737-3039 which has an orbital period of less than 3 hours and has an expected time until merger of 85 million years [8].
1Astronomers generally refer to any binary star system with one neutron star component as a neutron-star binary;
neutron-star neutron-star (NSNS) systems are referred to as ‘binary neutron stars’ or ‘double neutron star’ systems.
Gravitational waves from the coalescence of compact binaries represent the most promising can- didates for the first direct observation of gravitational waves from the Laser Gravitational-Wave Observatory or LIGO [1, 3]. Close NSNS binary systems are the only compact object binaries which have been identified observationally; while the existence of black hole-neutron star (BHNS) and binary black hole (BBH) system is almost certain given our current picture of the evolution of binary star systems, they have yet to be identified in the field. Consequently, the rate of NSNS coalescences detectable by Advanced LIGO is the least uncertain (though BBH coalescences may have higher detection rates due to their being stronger sources at frequencies to which LIGO is the most sensitive) [2]. In addition, providing a strong field test of general relativity, a LIGO detection of a clean waveform from the inspiral and merger of two neutron stars has the potential to constrain the neutron star equation of state (EOS). As neutron stars have densities on the same order as that of an atomic nucleus2, a more precise constraint on the neutron star EOS would yield a fascinating probe of fundamental physics. An excellent review of our current constraints and understanding of the neutron star EOS may be found in a recent review by Jim Lattimer [28].
A first order, quantitative, measurement of the neutron star equation of state may be determined from the gravitational wave signal of a NSNS coalescence in two ways. First, in the final phase of the inspiral, each neutron star will obtain a tidal deformation due to its companion which, to leading order, will result in an induced quadrupole moment. The ratio of the applied tidal field to the resulting quadrupole deformation of a NS is called the tidal deformability and is closely related to a parameter known as the tidal Love number; given a fixed NS mass, this parameter provides a quantitative measurement of the NS EOS’s ‘stiffness’. Hinderer and Flanagan have shown that the evolution of the orbital phase for the coalescing NSNS system is altered by an amount which is directly dependent on the tidal deformability of the NSs [15, 23, 24]. Thus the direct measurement of the progression of the orbital phase from the gravitational waveform of a NSNS coalescence can directly measure the tidal Love number of the component NSs and therefore the ‘stiffness’ of the NS EOS. Secondly, a NSNS merger which does not immediately collapse to a black hole will leave behind a differentially rotating, highly excitedhypermassive neutron star (HMNS) remnant (cf. Ch. 8). As the HMNS remnant oscillates and settles down, it emits high frequency (2-4KHz) gravitational waves which produce a pronounced peak in the GW frequency spectrum. A comprehensive study by Bauswein et al. was able to find a relation between the value of this frequency peak and the radius of a 1.6Mthat is remarkably uniform over 38 different candidate neutron star EOSs [4]. Since the radius of a NS for a fixed mass is directly dependent on the ‘stiffness’ of the NS EOS, this yields a second way in which a quantitative measurement of the NS EOS may be accomplished via the gravitational wave signal from the merger of two neutron stars.
Although the latter (HMNS oscillation modes) result is more easily calculated by simulations,
2This density is called nuclear saturation density and has a value of 2.7×1014g cm−3.
Tidal Deformations
Postmerger HMNS Oscillation Modes
Figure 6.1: Plot courtesy of Sarah Gossan. Shows the amplitude spectral density of the design sensitivity noise floor,p
S(f), for the advanced LIGO, KAGRA, and advanced VIRGO gravitational wave detectors as black lines. Plotted as colored lines are the scaled frequency spectra, |˜h(f)|√f, of the gravitational waves from the inspiral and merger of neutron stars from the simulations by Sekiguchi et. al. [38], with the amplitude scaled for a distance to the source of 50 Mpc. The inspiral portion of the spectra are calculated via hybridization with the simulation data. As the binary inspirals and its frequency increases, it gets closer and closer to merger; the frequency at several times before merger are noted on the plot.
the former (tidal Love numbers) is more likely to be distinguished by a LIGO observation. This becomes clear when examining in Fig. 6.1 the gravitational wave frequency spectrum from a NSNS coalescence compared to the advanced LIGO noise curve. It is clear that the integral of the difference between the signal (colored lines) and the noise (black lines) is far greater for the inspiral portion where tidal deformations may be measured via the phase evolution. However, the differences in the phase evolution of the binary is more subtle than is able to be observed on this frequency plot. A NSNS binary simulation must run for many orbits and achieve high precision in the binary’s orbital phase in order to measure the tidal Love numbers. This can take up to 100ms of physical evolution time, compared to ∼20ms for measuring the postmerger HMNS oscillation frequency. Bernuzzi et.
al. has done an impressive study which carefully examines the precision of the inspiral and merger of an NSNS binary over 10 orbits [5]. In order to measure higher order effects beyond the tidal Love numbers from an NSNS inspiral, similar studies will have to be done which consist of more orbits and even greater phase accuracy. The initiation of such a study using the SpEC code is what we describe in this chapter.