Neutron Star Radius Research Articles

Waveform systematics in the gravitational-wave inference of tidal parameters and equation of state from binary neutron-star signals

Gravitational-wave signals from binary neutron-star coalescences carry information about the star's equation of state in their tidal signatures. A major issue in the inference of the tidal parameters (or, directly, of the equation of state) is the systematic error introduced by the waveform approximants. We use a bottom-up approach based on gauge-invariant phase analysis and the Fisher information matrix to investigate waveform systematics and identify biases in parameter estimation. A mock analysis of 15 different binaries indicates that systematics in current waveform models dominate over statistical errors at signal-to-noise ratio (SNR) $\ensuremath{\gtrsim}80$ for LIGO and Virgo at design sensitivity. This implies biases in the inference of the reduced tidal parameter that are larger than the statistical 90% credible intervals. For example, while the neutron-star radius could be constrained at the $\ensuremath{\sim}5%$ level at SNR 80, systematics can be at the $\ensuremath{\sim}10%$ level. We apply our approach to GW170817 ($\mathrm{SNR}\ensuremath{\sim}30$) and confirm that no significant systematic effects are present. Using frequencies below $\ensuremath{\lesssim}1\text{ }\text{ }\mathrm{kHz}$ for the analysis, we estimate a neutron-star radius of ${12.5}_{\ensuremath{-}1.8}^{+1.1}\text{ }\text{ }\mathrm{km}$. The latter is consistent with an electromagnetic-informed prior and the recent NICER measurement. Exploring $\mathrm{SNR}\ensuremath{\gtrsim}100$ in view of third-generation detectors, we find that all the current waveform models lead to differences of at least $1\ensuremath{\sigma}$ in the inference of the reduced tidal parameter. We conclude that current waveform models, including those from numerical relativity, are insufficient to infer the equation of state in the loudest (and potentially most informative) events that will be observed by advanced and third-generation detectors.

Anisotropic quark stars in Einstein-Gauss-Bonnet theory

Recent progress in the determination of both masses and radii of neutron stars has put strong constraints on the equation of state (EoS) above the nuclear saturation density. Within a confining quark matter model, we propose an anisotropic star consisting of a homogeneous and unpaired charge-neutral 3-flavor interacting quark matter with O(ms4) corrections in the context of Einstein-Gauss-Bonnet gravity theory. This generalized model depends only on three free parameters: the bag constant B, the interaction parameter a and the Gauss-Bonnet coupling constant α. Given the underlying EoS, we show the possibility of obtaining the maximal neutron star mass which satisfies the recent observational data for PSR J0751+1807. The numerical analysis of mass-radius relations supports the existence of other massive pulsars with a maximum mass consistent and common radii in the range of R≲(11∼14) Km [1]. Furthermore, we discuss the mass vs central mass density (M−ρc) relation for stability, compactness and binding energy in this gravity theory. Our results thus provide circumstantial evidence in favor of super-massive pulsars in EGB gravity.

Measuring the masses and radii of neutron stars in low-mass X-ray binaries: Effects of the atmospheric composition and touchdown radius

Context. X-ray bursts (XRBs) are energetic explosive events that have been observed in low-mass X-ray binaries (LMXBs). Some Type-I XRBs exhibit photospheric radius expansion (PRE) and these PRE XRBs are used to simultaneously estimate the mass and the radius of a neutron star in LMXB. Aims. The mass and radius estimation depends on several model parameters, most of which are still uncertain. Here, we focus on the effects of the chemical composition of the photosphere, which determines the opacity during the PRE phase, and the touchdown radius, which can be larger than the neutron star radius. We investigate how these two model parameters affect the mass and radius estimation in a systematic way and whether there is any statistical trend for these two parameters as well as whether there is a possible correlation between them. Methods. We used both a Monte Carlo (MC) sampling and a Bayesian analysis to examine the effects of the photospheric composition and the touchdown radius. We applied these two methods to six LMXBs exhibiting PRE XRBs. With both methods, we solved the Eddington flux equation and the apparent angular area equation, both of which include the correction terms. For the MC sampling, we developed an iterative method in order to solve these two equations more efficiently. Results. We confirm that the effects of the photospheric composition and the touchdown radius are similar in the statistical and analytical estimation of mass and radius, even when the correction terms are considered. Furthermore, in all of the six sources, we find that a H-poor photosphere and a large touchdown radius are favored statistically regardless of the statistical method. Our Bayesian analysis also hints that touchdown can occur farther from the neutron star surface when the photosphere is more H-poor. This correlation could be qualitatively understood with the Eddington flux equation. We propose a physical explanation for this correlation between the photospheric composition and the touchdown radius. Our results show that when accounting for the uncertainties of the photospheric composition and the touchdown radius, it is most likely that the radii of the neutron stars in these six LMXBs are less than 12.5 km. This value is similar to that of the bounds placed on the neutron star radius based on the tidal deformability measured from the gravitational wave signal.

Systematics of prompt black-hole formation in neutron star mergers

This study addresses the collapse behavior of neutron star (NS) mergers expressed through the binary threshold mass ${M}_{\text{thres}}$ for prompt black-hole (BH) formation, which we determine by relativistic hydrodynamical simulations for a set of 40 equation of state (EOS) models of NS matter. ${M}_{\text{thres}}$ can be well described by various fit formulas involving stellar parameters of nonrotating NSs, which are employed to characterize the EOS models. Using these relations we compute which constraints on NS radii and the tidal deformability are set by current and future merger detections that reveal information about the merger product. We systematically investigate the impact of the binary mass ratio $q={M}_{1}/{M}_{2}$ and assemble various fits, which make different assumptions about a-priori knowledge. This includes fit formulas for ${M}_{\text{thres}}$ for a fixed mass ratio or a range of $q$ if this parameter is not known very well. Also, we construct relations describing the threshold to prompt collapse for different classes of candidate EOSs, which for instance do or do not include models with a phase transition to quark matter. In particular, we find fit formulas for ${M}_{\text{thres}}$ including an explicit $q$ dependence, which are valid in a broad range of $0.7\ensuremath{\le}q\ensuremath{\le}1$ and which are nearly as tight as relations for fixed mass ratios. For most EOS models except for some extreme cases the threshold mass of asymmetric mergers is equal or smaller than the one of equal-mass binaries. Generally, the impact of the binary mass asymmetry on ${M}_{\text{thres}}$ becomes stronger with more extreme mass ratios, while ${M}_{\text{thres}}$ is approximately constant for small deviations from $q=1$, i.e., for $0.85\ensuremath{\le}q\ensuremath{\le}1$. The magnitude of the reduction of ${M}_{\text{thres}}$ with the binary mass asymmetry follows a systematic EOS dependence. We also describe in more detail that a phase transition to deconfined quark matter can leave a characteristic imprint on the collapse behavior of NS mergers. The occurrence of quark matter can reduce the stability of the remnant and thus the threshold mass relative to a purely hadronic reference model. Comparing specifically the threshold mass and the combined tidal deformability ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}}_{\text{thres}}$ of a system with ${M}_{\text{thres}}$ can yield peculiar combinations of those two quantities, where ${M}_{\text{thres}}$ is particularly small in relation to ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}}_{\text{thres}}$. Since no purely hadronic EOS can yield such a combination of ${M}_{\text{thres}}$ and ${\stackrel{\texttildelow{}}{\mathrm{\ensuremath{\Lambda}}}}_{\text{thres}}$, a combined measurement or a constraint on both quantities can indicate the onset of quark deconfinement. Finally, we point out new univariate relations between ${M}_{\text{thres}}$ and stellar properties of high-mass NSs, which can be employed for direct EOS constraints or consistency checks in combination with other measurements.

Hybrid neutron stars in the mass‐radius diagram

AbstractWe present a systematic investigation of the possible locations for the special point (SP), a unique feature of hybrid neutron stars in the mass‐radius diagram. The study is performed within the two‐phase approach where the high‐density (quark matter) phase is described by the constant‐sound‐speed (CSS) equation of state (EoS) and the nuclear matter phase around saturation density is varied from very soft (APR) to stiff (DD2 with excluded nucleon volume). Different construction schemes for the deconfinement transition are applied: Maxwell construction, mixed phase construction, and parabolic interpolation. We demonstrate for the first time that the SP is invariant not only against changing the nuclear matter EoS, but also against variation of the construction schemes for the phase transition. Since the SP serves as a proxy for the maximum mass and accessible radii of massive hybrid stars, we draw conclusions for the limiting masses and radii of hybrid neutron stars.

Probing Dense Nuclear Matter in the Laboratory: Experiments at FAIR and NICA

The poorly known properties of high-density strongly-interacting matter govern the structure of neutron stars and the dynamics of neutron star mergers. New insight has been and will be gained by astronomical observations, such as the measurement of mass and radius of neutron stars, and the detection of gravitational waves emitted from neutron star mergers. Alternatively, information on the Nuclear Matter Equation-of-State (EOS) and on a possible phase transition from hadronic to quark matter at high baryon densities can be obtained from laboratory experiments investigating heavy-ion collisions. Detector systems dedicated to such experiments are under construction at the “Facility for Antiproton and Ion Research” (FAIR) in Darmstadt, Germany, and at the “Nuclotron-based Ion Collider fAcility” (NICA) in Dubna, Russia. In heavy-ion collisions at these accelerator centers, one expects the creation of baryon densities of up to 10 times saturation density, where quark degrees-of-freedom should emerge. This article reviews the most promising observables in heavy-ion collisions, which are used to probe the high-density EOS and possible phase transition from hadronic to quark matter. Finally, the facilities and the experimental setups will be briefly described.

Nuclear pasta and symmetry energy in the relativistic point-coupling model

Nonuniform structure of low-density nuclear matter, known as nuclear pasta, is expected to appear not only in the inner crust of neutron stars but also in core-collapse supernova explosions and neutron-star mergers. We perform fully three-dimensional calculations of inhomogeneous nuclear matter and neutron-star matter in the low-density region using the Thomas-Fermi approximation. The nuclear interaction is described in the relativistic mean-field approach with the point-coupling interaction, where the meson exchange in each channel is replaced by the contact interaction between nucleons. We investigate the influence of nuclear symmetry energy and its density dependence on pasta structures by introducing a coupling term between the isoscalar-vector and isovector-vector interactions. It is found that the properties of pasta phases in the neutron-rich matter are strongly dependent on the symmetry energy and its slope. In addition to typical shapes like droplets, rods, slabs, tubes, and bubbles, some intermediate pasta structures are also observed in cold stellar matter with a relatively large proton fraction. We find that nonspherical shapes are unlikely to be formed in neutron-star crusts, since the proton fraction obtained in $\beta$ equilibrium is rather small. The inner crust properties may lead to a visible difference in the neutron-star radius.

From Finite Nuclei to Neutron Stars: The Essential Role of High-Order Density Dependence in Effective ForcesSupported by the National Key R&D Program of China (Grant No. 2018YFA0404403), the National Natural Science Foundation of China (Grant Nos. 11975032, 11835001, 11790325, and 11961141003).

A unified description of finite nuclei and equation of state of neutron stars presents both a major challenge and also opportunities for understanding nuclear interactions. Inspired by the Lee–Huang–Yang formula of hard-sphere gases, we develop effective nuclear interactions with an additional high-order density dependent term. While the original Skyrme force SLy4 is widely used in studies of neutron stars, there are not satisfactory global descriptions of finite nuclei. The refitted SLy4’ force can improve descriptions of finite nuclei but slightly reduces the radius of neutron star of 1.4M ⊙ with M ⊙ being the solar mass. We find that the extended SLy4 force with a higher-order density dependence can properly describe properties of both finite nuclei and GW170817 binary neutron stars, including the mass-radius relation and the tidal deformability. This demonstrates the essential role of high-order density dependence at ultrahigh densities. Our work provides a unified and predictive model for neutron stars, as well as new insights for the future development of effective interactions.

Limiting masses and radii of neutron stars and their implications

We combine equation of state of dense matter up to twice nuclear saturation density ($n_{\rm sat}=0.16\, \text{fm}^{-3}$) obtained using chiral effective field theory ($\chi$EFT), and recent observations of neutron stars to gain insights about the high-density matter encountered in their cores. A key element in our study is the recent Bayesian analysis of correlated EFT truncation errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the $\chi$EFT expansion. We refine the bounds on the maximum mass imposed by causality at high densities, and provide stringent limits on the maximum and minimum radii of $\sim1.4\,{\rm M}_{\odot}$ and $\sim2.0\,{\rm M}_{\odot}$ stars. Including $\chi$EFT predictions from $n_{\rm sat}$ to $2\,n_{\rm sat}$ reduces the permitted ranges of the radius of a $1.4\,{\rm M}_{\odot}$ star, $R_{1.4}$, by $\sim3.5\, \text{km}$. If observations indicate $R_{1.4}<11.2\, \text{km}$, our study implies that either the squared speed of sound $c^2_{s}>1/2$ for densities above $2\,n_{\rm sat}$, or that $\chi$EFT breaks down below $2\,n_{\rm sat}$. We also comment on the nature of the secondary compact object in GW190814 with mass $\simeq 2.6\,{\rm M}_{\odot}$, and discuss the implications of massive neutron stars $>2.1 \,{\rm M}_{\odot}\,(2.6\,{\rm M}_{\odot})$ in future radio and gravitational-wave searches. Some form of strongly interacting matter with $c^2_{s}>0.35\, (0.55)$ must be realized in the cores of such massive neutron stars. In the absence of phase transitions below $2\,n_{\rm sat}$, the small tidal deformability inferred from GW170817 lends support for the relatively small pressure predicted by $\chi$EFT for the baryon density $n_{\rm B}$ in the range $1-2\,n_{\rm sat}$. Together they imply that the rapid stiffening required to support a high maximum mass should occur only when $n_{\rm B} \gtrsim 1.5-1.8\,n_{\rm sat}$.

Quark-hadron crossover equations of state for neutron stars: Constraining the chiral invariant mass in a parity doublet model

We construct an equation of state (EOS) for neutron stars by interpolating hadronic EOS at low density and quark EOS at high density. A hadronic model based on the parity doublet structure is used for hadronic matter and a quark model of Nambu--Jona-Lasinio type is for quark matter. We assume crossover between hadronic matter and quark matter in the the color-flavor locked phase. The nucleon mass of the parity doublet model has a mass associated with the chiral symmetry breaking, and a chiral invariant mass $m_0$ which is insensitive to the chiral condensate. The value of $m_0$ affects the nuclear EOSs at low density, and has strong correlations with the radii of neutron stars. Using the constraint to the radius obtained by LIGO-Virgo and NICER, we find that $m_0$ is restricted as $600\,\mathrm{MeV}\lesssim m_0 \lesssim 900\,\mathrm{MeV}$.

Unified weak and strong coupling framework for nuclear matter and neutron stars

Ab initio methods using weakly interacting nucleons give a good description of condensed nuclear matter up to densities comparable to the nuclear saturation density. At higher densities strong interactions between overlapping nucleons become important; we propose that the interactions will continuously switch over to follow a holographic model in this region. In order to implement this, we construct hybrid equations of state (EOSs) where various models are used for low-density nuclear matter, and the holographic V-QCD model is used for nonperturbative high-density nuclear matter as well as for quark matter. We carefully examine all existing constraints from astrophysics of compact stars and discuss their implications for the hybrid EOSs. Thanks to the stiffness of the V-QCD EOS for nuclear matter, we obtain a large family of viable hybrid EOSs passing the constraints. We find that quark matter cores in neutron stars are unstable due to the strongly first-order deconfinement transition and predict bounds on the tidal deformability as well as on the radius of neutron stars. By relying on universal relations, we also constrain characteristic peak frequencies of gravitational waves produced in neutron star mergers.

Equation of state of asymmetric nuclear matter and the tidal deformability of neutron star

Neutron star (NS) is a unique astronomical compact object where the four fundamental interactions have been revealed from the observation and studied in different ways. While the macroscopic properties of NS like mass and radius can be determined within the General Relativity using a realistic equation of state (EOS) of NS matter, such an EOS is usually generated by a nuclear structure model like, e.g., the nuclear mean-field approach to asymmetric nuclear matter. Given the radius of NS extended to above 10 km and its mass up to twice the solar mass, NS is expected to be tidally deformed when it is embedded in a strong tidal field. Such a tidal effect was confirmed unambiguously in the gravitation wave signals detected recently by the LIGO and Virgo laser interferometers from GW170817, the first ever direct observation of a binary NS merger. A nonrelativistic mean-field study is carried out in the present work within the Hartree-Fock formalism to construct the EOS of NS matter, which is then used to determine the tidal deformability, gravitational mass, and radius of NS. The mean-field results are compared with the constraints imposed for these quantities by the global analysis of the observed GW170817 data, and a strong impact by the incompressibility of nuclear matter on the hydrostatic configuration of NS is shown.

Updated universal relations for tidal deformabilities of neutron stars from phenomenological equations of state

Equation of state (EOS) insensitive relations, so-called universal relations, between the neutron star (NS) compactness, its multipolar tidal deformability coefficients, and between the tidal parameters for binary systems are essential to break degeneracies in gravitational wave data analysis. Here, we validate and recalibrate these universal relations using a large set of almost 2 million phenomenological EOSs that are consistent with current observations. In doing so, we extend universal relations to a larger region of the EOS parameter space, most notably to softer EOSs and larger compactnesses. We show that waveform models that neglect higher-than-leading-order tidal deformations of the NSs accumulate as much as $3.5$ radians of dephasing from $20\,{\rm Hz}$ to merger. We also perform a full Bayesian parameter estimation of the GW170817 data, and we compare the NS radius constraints produced using universal relations from the literature and the updated fits we propose here. We find that the new fits yield a NS radius that is smaller by about 500 meters. This difference is less than the statistical uncertainty on the radius at the signal-to-noise-ratio of GW170817, but it is significantly larger than the precision anticipated for next-generation detectors.

Neutron stars within the Bogoliubov quark-meson coupling model

A quark-meson coupling model based on the quark model proposed by Bogoliubov for the description of the quark dynamics is developed and applied to the description of neutron stars. Starting from a su(3) symmetry approach, it is shown that this symmetry has to be broken in order to satisfy the constraints set by the hypernuclei and by neutron stars. The model is able to describe observations such as two solar mass stars or the radius of canonical neutron stars within the uncertainties presently accepted. If the optical potentials for $\Lambda$ and $\Xi$ hyperons in symmetric nuclear matter at saturation obtained from laboratory measurements of hypernuclei properties are imposed the model predicts no strangeness inside neutron stars.

Neutron stars phenomenology with scalar–tensor inflationary attractors

In this work we shall study the implications of a subclass of E -models cosmological attractors, namely of a -attractors, on hydrodynamically stable slowly rotating neutron stars. Specifically, we shall present the Jordan frame theory of the a -attractors, and by using a conformal transformation we shall derive the Einstein frame theory. We discuss the inflationary context of a -attractors in order to specify the allowed range of values for the free parameters of the model based on the latest cosmic-microwave-background-based Planck 2018 data. Accordingly, using the notation and physical units frequently used in theoretical astrophysics contexts, we shall derive the Tolman–Oppenheimer–Volkoff equations in the Einstein frame. Assuming a piecewise polytropic equation of state, the lowest density part of which shall be chosen to be the WFF1, or APR or the SLy EoS, we numerically solve the Tolman–Oppenheimer–Volkoff equations using a double shooting python-based “LSODA” numerical code. The resulting picture depends on the value of the parameter a characterizing the a -attractors. As we show, for large values of a , which do not produce a viable inflationary era, the M – R graphs are nearly identical to the general relativistic result, and these two are discriminated at large central densities values. Also, for large a -values, the WFF1 equation of state is excluded, due to the GW170817 constraints on the radius of an M ∼ 1 . 6 M ⊙ neutron star, which must be larger than R = 10 . 6 8 − 0 . 04 + 15 km and on the radius corresponding to the maximum mass which must be larger than R = 9 . 6 − 0 . 03 + 0 . 14 km. In addition, the small a cases produce larger masses and radii compared to the general relativistic case and are compatible with the GW170817 constraints on the radii of neutron stars. A notable feature is that as the parameter a decreases, the radii of the static hydrodynamically stable neutron stars increase. Our results indicate deep and not yet completely understood connections between non-minimal inflationary attractors and neutron stars phenomenology in scalar–tensor theory.

Constraining neutron star radii in black hole–neutron star mergers from their electromagnetic counterparts

ABSTRACT Mergers of black hole (BH) and neutron star (NS) binaries are of interest since the emission of gravitational waves (GWs) can be followed by an electromagnetic (EM) counterpart, which could power short gamma-ray bursts. Until now, LIGO/Virgo has only observed a candidate BH–NS event, GW190426_152155, which was not followed by any EM counterpart. We discuss how the presence (absence) of a remnant disc, which powers the EM counterpart, can be used along with spin measurements by LIGO/Virgo to derive a lower (upper) limit on the radius of the NS. For the case of GW190426_152155, large measurement errors on the spin and mass ratio prevent from placing an upper limit on the NS radius. Our proposed method works best when the aligned component of the BH spin (with respect to the orbital angular momentum) is the largest, and can be used to complement the information that can be extracted from the GW signal to derive valuable information on the NS equation of state.

Combining Electromagnetic and Gravitational-Wave Constraints on Neutron-Star Masses and Radii.

We perform a joint Bayesian inference of neutron-star mass and radius constraints based on GW170817, observations of quiescent low-mass x-ray binaries (QLMXBs), photospheric radius expansion x-ray bursting sources, and x-ray timing observations of J0030+0451. With this dataset, the form of the prior distribution still has an impact on the posterior mass-radius curves and equation of state (EOS), but this impact is smaller than recently obtained when considering QLMXBs alone. We analyze the consistency of the electromagnetic data by including an "intrinsic scattering" contribution to the uncertainties, and find only a slight broadening of the posteriors. This suggests that the gravitational-wave and electromagnetic observations of neutron-star structure are providing a consistent picture of the neutron-star mass-radius curve and the EOS.

New equations of state constrained by nuclear physics, observations, and QCD calculations of high-density nuclear matter

We present new equations of state for applications in core-collapse supernova and neutron star merger simulations. We start by introducing an effective mass parametrization that is fit to recent microscopic calculations up to twice saturation density. This is important to capture the predicted thermal effects, which have been shown to determine the proto--neutron star contraction in supernova simulations. The parameter range of the energy-density functional underlying the equation of state is constrained by chiral effective field theory results at nuclear densities as well as by functional renormalization group computations at high densities based on QCD. We further implement observational constraints from measurements of heavy neutron stars, the gravitational wave signal of GW170817, and from the recent NICER results. Finally, we study the resulting allowed ranges for the equation of state and for properties of neutron stars, including the predicted ranges for the neutron star radius and maximum mass.

Neutron star properties from astrophysical observations

Recently, simultaneous measurements of the masses and the radii of neutron stars from X-ray observations by the Neutron Star Interior Composition Explorer (NICER) have been reported. In addition, gravitational wave detections from the mergers of neutron star binaries allowed us to estimate the tidal deformability of neutron stars. Based on these recent observations and our works, we review their implications for the masses, radii and tidal deformabilities of neutron stars. We also discuss the basic principles of the polytropic equations of state, including sound velocities, and the implications of those equations to the properties of neutron stars.

Optimized Statistical Approach for Comparing Multi-messenger Neutron Star Data

Abstract The neutron star equation of state is now being constrained from a diverse set of multi-messenger data, including gravitational waves from binary neutron star mergers, X-ray observations of the neutron star radius, and many types of laboratory nuclear experiments. These measurements are often mapped to a common domain for comparison with one another or are used to constrain the predictions of theoretical equations of state. We explore here the statistical biases that can arise when such multi-messenger data are compared or combined across different domains. We find that placing Bayesian priors individually in each domain of measurement can lead to biased constraints. We present a new prescription for defining Bayesian priors consistently across different experiments, which will allow for robust cross-domain comparisons. Using the first two binary neutron star mergers as an example, we show that a uniform prior in the tidal deformability can produce inflated evidence for large radii, while a uniform prior in the radius points toward smaller radii. Finally, using this new prescription, we provide a status update on multi-messenger constraints on the neutron star radius.