Neutron Star Equation Of State Research Articles

Measuring the neutron star equation of state with gravitational waves: The first forty binary neutron star merger observations

Gravitational waves from binary neutron star coalescences contain rich information about matter at supranuclear densities encoded by the neutron star equation of state. We can measure the equation of state by analyzing the tidal interactions between neutron stars, which is quantified by the tidal deformability. Multiple merger events are required to probe the equation of state over a range of neutron star masses. The more events included in the analysis, the stronger the constraints on the equation of state. In this paper, we build on previous work to explore the constraints that LIGO and Virgo are likely to place on the neutron star equation of state by combining the first 40 binary neutron star detections, a milestone we project to be reached during the first year of accumulated design-sensitivity data. We carry out Bayesian inference on a realistic mock dataset of binaries to obtain posterior distributions for neutron star tidal parameters. In order to combine posterior samples from multiple observations, we employ a random forest regressor, which allows us to efficiently interpolate the likelihood distribution. Assuming a merger rate of $1540\text{ }\text{ }{\mathrm{Gpc}}^{\ensuremath{-}3}\text{ }\text{ }{\mathrm{yr}}^{\ensuremath{-}1}$ and a LIGO-Virgo detector network operating for 1 year at the sensitivity of the third-observation run, plus an additional 8 months of design sensitivity, we find that the radius of a $1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ neutron star can be constrained to better than $\ensuremath{\sim}10%$ at 90% confidence. The pressure at twice the nuclear saturation density can be constrained to better than $\ensuremath{\sim}45%$ at 90% confidence. We show that these results are robust to our choice of parametrization for the neutron star equation of state.

Bayesian modeling of the nuclear equation of state for neutron star tidal deformabilities and GW170817

We present predictions for neutron star tidal deformabilities obtained from a Bayesian analysis of the nuclear equation of state, assuming a minimal model at high-density that neglects the possibility of phase transitions. The Bayesian posterior probability distribution is constructed from priors obtained from microscopic many-body theory based on realistic two- and three-body nuclear forces, while the likelihood functions incorporate empirical information about the equation of state from nuclear experiments. The neutron star crust equation of state is constructed from the liquid drop model, and the core-crust transition density is found by comparing the energy per baryon in inhomogeneous matter and uniform nuclear matter. From the cold $\beta$-equilibrated neutron star equation of state, we then compute neutron star tidal deformabilities as well as the mass-radius relationship. Finally, we investigate correlations between the neutron star tidal deformability and properties of finite nuclei.

New Neutron Star Equation of State with Quark–Hadron Crossover

Abstract We present a much improved equation of state for neutron star matter, QHC19, with a smooth crossover from the hadronic regime at lower densities to the quark regime at higher densities. We now use the Togashi et al. equation of state, a generalization of the Akmal–Pandharipande–Ravenhall equation of state of uniform nuclear matter, in the entire hadronic regime; the Togashi equation of state consistently describes nonuniform as well as uniform matter, and matter at beta equilibrium without the need for an interpolation between pure neutron and symmetric nuclear matter. We describe the quark matter regime at higher densities with the Nambu–Jona–Lasinio model, now identifying tight constraints on the phenomenological universal vector repulsion between quarks and the pairing interaction between quarks arising from the requirements of thermodynamic stability and causal propagation of sound. The resultant neutron star properties agree very well with the inferences of the LIGO/Virgo collaboration, from GW170817, of the pressure versus baryon density, neutron star radii, and tidal deformabilities. The maximum neutron star mass allowed by QHC19 is 2.35 M ⊙, consistent with all neutron star mass determinations.

Constraining the Neutron Star Equation of State Using Multiband Independent Measurements of Radii and Tidal Deformabilities.

Using a Bayesian approach, we combine measurements of neutron star macroscopic observables obtained by astrophysical and gravitational observations to derive joint constraints on the equation of state (EOS) of matter at supranuclear density. In our analysis, we use two sets of data: (i)the masses and tidal deformabilities measured in the binary neutron star event GW170817, detected by LIGO and Virgo, and (ii)the masses and stellar radii measured from observations of nuclear bursts in accreting low-mass x-ray binaries. Using a phenomenological parametrization of the equation of state, we compute the posterior probability distributions of the EOS parameters, using which we infer the posterior distribution for the radius and the mass of the two neutron stars of GW170817. The constraints we set on the radii are tighter than previous bounds.

Inspiralling eccentric binary neutron stars: Orbital motion and tidal resonance

We study the orbital evolution of eccentric binary neutron stars. The orbit is described as a Quasi-Keplarian orbit with perturbations due to tidal couplings. We find that the tidal interaction between stars contributes to orbital precession, in addition to the Post-Newtonian procession. The coupling between the angular and radial motion of the binary also excites a series of harmonics in the stars' oscillation. In the small eccentricity limit, this coupling mainly gives rise to an additional orbital resonance, with the orbital frequency being one third of the f-mode frequency. For a binary with initial eccentricity $\sim 0.2$ at $50$Hz orbital frequency, the presence of this tidal resonance introduces $\sim 0.5$ phase shift in the gravitational waveform till merger, subject to uncertainties in neutron star equation of state and the distribution of binary component masses. Such phase shift in the late-inspiral stage is likely detectable with third-generation gravitational-wave detectors.

Parallelized inference for gravitational-wave astronomy

Bayesian inference is the workhorse of gravitational-wave astronomy, for example, determining the mass and spins of merging black holes, revealing the neutron star equation of state, and unveiling the population properties of compact binaries. The science enabled by these inferences comes with a computational cost that can limit the questions we are able to answer. This cost is expected to grow. As detectors improve, the detection rate will go up, allowing less time to analyze each event. Improvement in low-frequency sensitivity will yield longer signals, increasing the number of computations per event. The growing number of entries in the transient catalog will drive up the cost of population studies. While Bayesian inference calculations are not entirely parallelizable, key components are embarrassingly parallel: calculating the gravitational waveform and evaluating the likelihood function. Graphical processor units (GPUs) are adept at such parallel calculations. We report on progress porting gravitational-wave inference calculations to GPUs. Using a single code - which takes advantage of GPU architecture if it is available - we compare computation times using modern GPUs (NVIDIA P100) and CPUs (Intel Gold 6140). We demonstrate speed-ups of $\sim 50 \times$ for compact binary coalescence gravitational waveform generation and likelihood evaluation and more than $100\times$ for population inference within the lifetime of current detectors. Further improvement is likely with continued development. Our python-based code is publicly available and can be used without familiarity with the parallel computing platform, CUDA.

Accelerated detection of the binary neutron star gravitational-wave background

Most gravitational-wave signals from binary neutron star coalescences are too weak to be individually resolved with current detectors. We demonstrate how to extract a population of sub-threshold binary neutron star signals using Bayesian parameter estimation. Assuming a merger rate of one signal every two hours, we find that this gravitational-wave background can be detected after approximately three months of observation with Advanced LIGO and Virgo at design sensitivity, versus several years using the standard cross-correlation algorithm. We show that the algorithm can distinguish different neutron star equations of state using roughly seven months of Advanced LIGO and Virgo design-sensitivity data. This is in contrast to the standard cross-correlation method, which cannot.

Differentially rotating strange star in general relativity

Rapidly and differentially rotating compact stars are believed to be formed in binary neutron star merger events, according to both numerical simulations and the multi-messenger observation of GW170817. The lifetime and evolution of such a differentially rotating star, is tightly related to the observations in the post-merger phase. Various studies on the maximum mass of differentially rotating neutron stars have been done in the past, most of which assume the so-called $j$-const law as the rotation profile inside the star and consider only neutron star equations of state. In this paper, we extend the studies to strange star models, as well as to a new rotation profile model. Significant differences are found between differentially rotating strange stars and neutron stars, with both differential rotation laws. A moderate differential rotation rate for neutron stars is found to be too large for strange stars, resulting in a rapid drop in the maximum mass as the differential rotation degree is increased further from $\hat{A}\sim2.0$, where $\hat{A}$ is a parameter characterizing the differential rotation rate for $j$-const law. As a result the maximum mass of a differentially rotating self-bound star drops below the uniformly rotating mass shedding limit for a reasonable degree of differential rotation. The continuous transition to the toroidal sequence is also found to happen at a much smaller differential rotation rate and angular momentum than for neutron stars. In spite of those differences, $\hat{A}$-insensitive relation between the maximum mass for a given angular momentum is still found to hold, even for the new differential rotation law. Astrophysical consequences of these differences and how to distinguish between strange star and neutron star models with future observations are also discussed.

Computing fast and reliable gravitational waveforms of binary neutron star merger remnants

Gravitational waves have been detected from the inspiral of a binary neutron-star, GW170817, which allowed constraints to be placed on the neutron star equation of state. The equation of state can be further constrained if gravitational waves from a post-merger remnant are detected. Post-merger waveforms are currently generated by numerical-relativity simulations, which are computationally expensive. Here we introduce a hierarchical model trained on numerical-relativity simulations, which can generate reliable post-merger spectra in a fraction of a second. Our spectra have mean fitting factors of 0.95, which compares to fitting factors of 0.76 and 0.85 between different numerical-relativity codes that simulate the same physical system. This method is the first step towards generating large template banks of spectra for use in post-merger detection and parameter estimation.

Constraints on hybrid neutron stars equation of state from neutron stars merging

Using recent gravitational and electromagnetic constraints on the neutron star matter equation of state (EOS) coming from the merge of two neutron stars, we study the physical conditions for which a quark deconfinement phase transition in cold neutron star matter is consistent with these new measurements. To this end, we consider several microscopic EOSs based on various ab initio approaches to describe the confined hadronic phase, and combine them with two phenomenological quark matter EOSs for the deconfined phase. The low and high density phases are then joined up through a mixed phase determined by a Gibbs construction. For each EOS we calculate the dimensionless binary deformability parameter \(\tilde{\Lambda}\) which can be directly related to the constraints derived from the gravitational waves detection. We find that in order to see any difference between the pure hadronic and the hadron-quark EOS for neutron stars with mass in the range (1.4-1.6) \(M_{\odot}\) through the calculation of \(\tilde{\Lambda}\), the EOSs of both hadronic and quark matter should be quite stiff, otherwise the variation on \( \tilde{\Lambda}\) can be valued just on neutron star masses above 1.8 \( M_{\odot}\) which currently are not constrained by present gravitational waves data. We find in addition that the softening of the hadronic EOS induced by the quark deconfinement phase transition can change the compatibility of a given hadronic EOS with the constraints obtained from neutron stars merging.

Black-Hole Remnants from Black-Hole-Neutron-Star Mergers.

Observations of gravitational waves and their electromagnetic counterparts may soon uncover the existence of coalescing compact binary systems formed by a stellar-mass black hole and a neutron star. These mergers result in a remnant black hole, possibly surrounded by an accretion disk. The mass and spin of the remnant black hole depend on the properties of the coalescing binary. We construct a map from the binary components to the remnant black hole using a sample of numerical-relativity simulations of different mass ratios q, (anti)aligned dimensionless spins of the black hole a_{BH}, and several neutron star equations of state. Given the binary total mass, the mass and spin of the remnant black hole can therefore be determined from the three parameters (q,a_{BH},Λ), where Λ is the tidal deformability of the neutron star. Our models also incorporate the binary black hole and test-mass limit cases and we discuss a simple extension for generic black-hole spins. We combine the remnant characterization with recent population synthesis simulations for various metallicities of the progenitor stars that generated the binary system. We predict that black-hole-neutron-star mergers produce a population of remnant black holes with masses distributed around 7 M_{⊙} and 9 M_{⊙}. For isotropic spin distributions, nonmassive accretion disks are favored: no bright electromagnetic counterparts are expected in such mergers.

Scalar charges and scaling relations in massless scalar–tensor theories

The timing of binary pulsars allows us to place some of the tightest constraints on modified theories of gravity. Perhaps some of the most interesting and well-motivated extensions to general relativity (GR) are scalar–tensor theories (STT), in which gravity is mediated by the metric tensor and a scalar field. These theories predict large deviations from GR in the presence of neutron stars through a phenomenon known as scalarization. Neutron stars in STT develop scalar charges, which directly enter the timing model for binary pulsars. In this paper, we calculate and tabulate these scalar charges in two popular, massless scalar tensor theories for a collection of neutron star equations of state that are compatible with constraints placed by the recent, gravitational wave observations of a binary neutron star coalescence. We then study these scalar charges and explore analytic scaling relations that allow us to predict their value in a large region of parameter space. Our results allow for the quick evaluation of the scalar charge in a large region of scalar–tensor theory parameter space, which has applications for gravitational wave tests of STT, as well as binary pulsar experiments.

Thermal X-ray emission identified from the millisecond pulsar PSR J1909–3744

Context. Pulsating thermal X-ray emission from millisecond pulsars can be used to obtain constraints on the neutron star equation of state, but to date only five such sources have been identified. Of these five millisecond pulsars, only two have well-constrained neutron star masses, which improve the determination of the radius via modelling of the X-ray waveform. Aims. We aim to find other millisecond pulsars that already have well-constrained mass and distance measurements that show pulsed thermal X-ray emission in order to obtain tight constraints on the neutron star equation of state. Methods. The millisecond pulsar PSR J1909–3744 has an accurately determined mass, M = 1.54 ± 0.03 M⊙ (1σ error) and distance, D = 1.07 ± 0.04 kpc. We analysed XMM-Newton data of this 2.95 ms pulsar to identify the nature of the X-ray emission. Results. We show that the X-ray emission from PSR J1909–3744 appears to be dominated by thermal emission from the polar cap. Only a single component model is required to fit the data. The black-body temperature of this emission is $ {kT}=0.26^{0.03}_{0.02} $ keV and we find a 0.2–10 keV un-absorbed flux of 1.1 × 10−14 erg cm−2 s−1 or an un-absorbed luminosity of 1.5 × 1030 erg s−1. Conclusion. Thanks to the previously determined mass and distance constraints of the neutron star PSR J1909–3744, and its predominantly thermal emission, deep observations of this object with future X-ray facilities should provide useful constraints on the neutron star equation of state.

NEoS: neutron star equation of state from hadron physics alone

We contribute a publicly available set of tables and code to provide equations of state (EoS) for matter at neutron star densities. Our EoS are constrained only by input from hadron physics and fundamental principles, without feedback from neutron star observations, and so without relying on general relativity (GR). They can therefore be used to test GR itself, as well as modified gravity theories, with neutron star observables, without logical circularity. We have adapted state of the art results from NN chiral potentials for the low-density limit, pQCD results for the asymptotically high-density EoS, and use monotony and causality as the only restrictions for intermediate densities, for the EoS sets to remain as model-independent as is feasible today.

Reconstruction of the neutron star equation of state from w -quasinormal modes spectra with a piecewise polytropic meshing and refinement method

In this paper we present a new approach to the inverse problem for relativistic stars using quasinormal modes and the piecewise polytropic parametrization of the equation of state. The algorithm is a piecewise polytropic meshing and refinement method that reconstructs the neutron star equation of state from experimental data of the mass and the $wI$-quasinormal modes. We present an algorithm able to numerically calculate axial quasinormal modes of neutron stars in an efficient way. We use an initial mesh of $27440$ equations of state in a $4$-volume of piecewise polytropic parameters that contains most of the candidate equations of state used today. The refinement process drives us to the reconstruction of the equation of state with a certain precision. Using the reconstructed equation of state, we calculate predictions for tidal deformability and slow rotation parameters (moment of inertia and quadrupole moment, for example). In order to check the method with an explicit example, we use as input data a few (five) configurations of a given equation of state. We reconstruct the equation of state in a quite good approximation, and then we compare the curves of physical parameters from the original equation of state and the reconstructed one. We obtain a relative difference for all of the parameters smaller than $2.5\%$ except for the tidal deformability, for which we obtain a relative difference smaller than $6.5\%$. We also study constraints from GW170817 event for the reconstructed equation of state.

Constraints on the neutron star equation of state from GW170817

The first detection of gravitational waves from a neutron star-neutron star merger, GW170817, has opened up a new avenue for constraining the ultradense-matter equation of state (EOS). The deviation of the observed waveform from a point-particle waveform is a sensitive probe of the EOS controlling the merging neutron stars' structure. In this topical review, I discuss the various constraints that have been made on the EOS in the year following the discovery of GW170817. In particular, I review the surprising relationship that has emerged between the effective tidal deformability of the binary system and the neutron star radius. I also report new results that make use of this relationship, finding that the radius inferred from GW170817 lies between 9.8 and 13.2 km at 90% confidence, with distinct likelihood peaks at 10.8 and 12.3 km. I compare these radii, as well as those inferred in the literature, to X-ray measurements of the neutron star radius. I also summarize the various maximum mass constraints, which point towards a maximum mass < 2.3 M_sun, depending on the fate of the remnant, and which can be used to additionally constrain the high-density EOS. I review the constraints on the EOS that have been performed directly, through Bayesian inference schemes. Finally, I comment on the importance of disentangling thermal effects in future EOS constraints from neutron star mergers.

Nonparametric inference of the neutron star equation of state from gravitational wave observations

We develop a non-parametric method for inferring the universal neutron star (NS) equation of state (EOS) from gravitational wave (GW) observations. Many different possible realizations of the EOS are generated with a Gaussian process conditioned on a set of nuclear-theoretic models. These synthetic EOSs are causal and thermodynamically stable by construction, span a broad region of the pressure-density plane, and can be selected to satisfy astrophysical constraints on the NS mass. Associating every synthetic EOS with a pair of component masses $M_{1,2}$ and calculating the corresponding tidal deformabilities $\Lambda_{1,2}$, we perform Monte Carlo integration over the GW likelihood for $M_{1,2}$ and $\Lambda_{1,2}$ to directly infer a posterior process for the NS EOS. We first demonstrate that the method can accurately recover an injected GW signal, and subsequently use it to analyze data from GW170817, finding a canonical deformability of $\Lambda_{1.4} = 160^{+448}_{-113}$ and $p(2\rho_{\mathrm{nuc}})=1.35^{+1.8}_{-1.2}\times 10^{34}~\mathrm{dyn}/\mathrm{cm}^2$ for the pressure at twice the nuclear saturation density at 90$\%$ confidence, in agreement with previous studies, when assuming a loose EOS prior. With a prior more tightly constrained to resemble the theoretical EOS models, we recover $\Lambda_{1.4} = 556^{+163}_{-172}$ and $p(2\rho_{\mathrm{nuc}})=4.73^{+1.4}_{-2.5}\times 10^{34}~\mathrm{dyn}/\mathrm{cm}^2$. We further infer the maximum NS mass supported by the EOS to be $M_\mathrm{max}=2.09^{+0.37}_{-0.16}$ ($2.04^{+0.22}_{-0.002}$) $M_\odot$ with the loose (tight) prior. The Bayes factor between the two priors is $B^{\mathcal{A}}_{\mathcal{I}} \simeq 1.12$, implying that neither is strongly preferred by the data and suggesting that constraints on the EOS from GW170817 alone may be relatively prior-dominated.

Multiwavelength studies of gravitational wave sources: Physics and phenomenology

The discovery of binary neutron star merger GW170817 from its gravitational wave (GW) signature, together with its accompanying electromagnetic (EM) emission spanning gamma‐ray to radio, marked the birth of GW + EM multimessenger astrophysics. The radioactively powered thermal kilonova, which dominated the ultraviolet to infrared in the hours to weeks after the merger, indicates that such mergers are the site of heavy‐element nucleosynthesis, likely extending to the third r‐process peak. The prompt gamma‐ray flash, and late‐time nonthermal (X‐ray to radio) emission, indicate that the merger also produced an ultrarelativistic jet, thus tying this event to the phenomena of short‐duration gamma‐ray bursts. In the future, observations of further mergers promise to establish their contribution to global nucleosynthesis, allow investigation of jet launching and structure, provide independent estimates of the cosmological parameters, constrain the neutron star equation of state, and address questions of fundamental physics.

Finite-temperature Extension for Cold Neutron Star Equations of State

Abstract Observations of isolated neutron stars place constraints on the equation of state (EOS) of cold, neutron-rich matter, while nuclear physics experiments probe the EOS of hot, symmetric matter. Many dynamical phenomena, such as core-collapse supernovae, the formation and cooling of proto-neutron stars, and neutron star mergers, lie between these two regimes and depend on the EOS at finite temperatures for matter with varying proton fractions. In this paper, we introduce a new framework to accurately calculate the thermal pressure of neutron–proton–electron matter at arbitrary density, temperature, and proton fraction. This framework can be expressed using a set of five physically motivated parameters that span a narrow range of values for realistic EOS and are able to capture the leading-order effects of degenerate matter on the thermal pressure. We base two of these parameters on a new approximation of the Dirac effective mass, with which we reproduce the thermal pressure to within ≲30% for a variety of realistic EOS at densities of interest. Three additional parameters, which are based on the behavior of the symmetry energy near the nuclear saturation density, allow us to extrapolate any cold EOS in β-equilibrium to arbitrary proton fractions. Our model thus allows a user to extend any cold nucleonic EOS, including piecewise polytropes, to arbitrary temperature and proton fraction for use in calculations and numerical simulations of astrophysical phenomena. We find that our formalism is able to reproduce realistic finite-temperature EOS with errors of ≲20% and offers a 1–3 orders-of-magnitude improvement over existing ideal-fluid models.

A magnetar-powered X-ray transient as the aftermath of a binary neutron-star merger.

Mergers of neutron stars are known to be associated with short γ-ray bursts1-4. If the neutron-star equation of state is sufficiently stiff (that is, the pressure increases sharply as the density increases), at least some such mergers will leave behind a supramassive or even a stable neutron star that spins rapidly with a strong magnetic field5-8 (that is, a magnetar). Such a magnetar signature may have been observed in the form of the X-ray plateau that follows up to half of observed short γ-ray bursts9,10. However, it has been expected that some X-ray transients powered by binary neutron-star mergers may not be associated with a short γ-ray burst11,12. A fast X-ray transient (CDF-S XT1) was recently found to be associated with a faint host galaxy, the redshift of which is unknown13. Its X-ray and host-galaxy properties allow several possible explanations including a short γ-ray burst seen off-axis, a low-luminosity γ-ray burst at high redshift, or a tidal disruption event involving an intermediate-mass black hole and a white dwarf13. Here we report a second X-ray transient, CDF-S XT2, that is associated with a galaxy at redshift z=0.738 (ref. 14). The measured light curve is fully consistent with the X-ray transient being powered by a millisecond magnetar. More intriguingly, CDF-S XT2 lies in the outskirts of its star-forming host galaxy with a moderate offset from the galaxy centre, as short γ-ray bursts often do15,16. The estimated event-rate density of similar X-ray transients, when corrected to the local value, is consistent with the event-rate density of binary neutron-star mergers that is robustly inferred from the detection of the gravitational-wave event GW170817.