How Well Do We Know the Neutron-Matter Equation of State at the Densities Inside Neutron Stars? A Bayesian Approach with Correlated Uncertainties

Nuclear Charge Radii of Silicon Isotopes

The nuclear charge radius of ^{32}Si was determined using collinear laser spectroscopy. The experimental result was confronted with ab initio nuclear lattice effective field theory, valence-space in-medium similarity renormalization group, and mean field calculations, highlighting important achievements and challenges of modern many-body methods. The charge radius of ^{32}Si completes the radii of the mirror pair ^{32}Ar-^{32}Si, whose difference was correlated to the slope L of the symmetry energy in the nuclear equation of state. Our result suggests L≤60  MeV, which agrees with complementary observables.

An updated nuclear-physics and multi-messenger astrophysics framework for binary neutron star mergers

The multi-messenger detection of the gravitational-wave signal GW170817, the corresponding kilonova AT2017gfo and the short gamma-ray burst GRB170817A, as well as the observed afterglow has delivered a scientific breakthrough. For an accurate interpretation of all these different messengers, one requires robust theoretical models that describe the emitted gravitational-wave, the electromagnetic emission, and dense matter reliably. In addition, one needs efficient and accurate computational tools to ensure a correct cross-correlation between the models and the observational data. For this purpose, we have developed the Nuclear-physics and Multi-Messenger Astrophysics framework NMMA. The code allows incorporation of nuclear-physics constraints at low densities as well as X-ray and radio observations of isolated neutron stars. In previous works, the NMMA code has allowed us to constrain the equation of state of supranuclear dense matter, to measure the Hubble constant, and to compare dense-matter physics probed in neutron-star mergers and in heavy-ion collisions, and to classify electromagnetic observations and perform model selection. Here, we show an extension of the NMMA code as a first attempt of analyzing the gravitational-wave signal, the kilonova, and the gamma-ray burst afterglow simultaneously. Incorporating all available information, we estimate the radius of a 1.4M⊙ neutron star to be [Formula: see text] km.

Strongly interacting matter exhibits deconfined behavior in massive neutron stars

Neutron-star cores contain matter at the highest densities in our Universe. This highly compressed matter may undergo a phase transition where nuclear matter melts into deconfined quark matter, liberating its constituent quarks and gluons. Quark matter exhibits an approximate conformal symmetry, predicting a specific form for its equation of state (EoS), but it is currently unknown whether the transition takes place inside at least some physical neutron stars. Here, we quantify this likelihood by combining information from astrophysical observations and theoretical calculations. Using Bayesian inference, we demonstrate that in the cores of maximally massive stars, the EoS is consistent with quark matter. We do this by establishing approximate conformal symmetry restoration with high credence at the highest densities probed and demonstrating that the number of active degrees of freedom is consistent with deconfined matter. The remaining likelihood is observed to correspond to EoSs exhibiting phase-transition-like behavior, treated as arbitrarily rapid crossovers in our framework.

Plausible presence of new state in neutron stars with masses above 0.98MTOV

We investigate the neutron star (NS) equation of state (EOS) by incorporating multi-messenger data of GW170817, PSR J0030 + 0451, PSR J0740 + 6620, and state-of-the-art theoretical progresses, including the information from chiral effective field theory (χEFT) and perturbative quantum chromodynamics (pQCD) calculation. Taking advantage of the various structures sampling by a single-layer feed-forward neural network model embedded in the Bayesian nonparametric inference, the structure of NS matter's sound speed c_s is explored in a model-agnostic way. It is found that a peak structure is common in the c_s² posterior, locating at 2.4-4.8ρ_sat (nuclear saturation density) and c_s² exceeds c²/3 at 90% credibility. The non-monotonic behavior suggests evidence of the state deviating from the hadronic matter inside the very massive NSs. Assuming the new/exotic state is featured as it is softer than typical hadronic models or even with hyperons, we find that a sizable (⩾10^-3M_⊙) exotic core, likely made of quark matter, is plausible for the NS with a gravitational mass above about 0.98M_TOV, where M_TOV represents the maximum gravitational mass of a non-rotating cold NS. The inferred M_TOV=2.18_-0.13^+0.27M_⊙ (90% credibility) is well consistent with the value of 2.17_-0.12^+0.15M_⊙ estimated independently with GW170817/GRB 170817A/AT2017gfo assuming a temporary supramassive NS remnant formed after the merger. PSR J0740 + 6620, the most massive NS detected so far, may host a sizable exotic core with a probability of ≈0.36.

Nuclear Equation of State for Arbitrary Proton Fraction and Temperature Based on Chiral Effective Field Theory and a Gaussian Process Emulator

We calculate the equation of state of asymmetric nuclear matter at finite temperature based on chiral effective field theory interactions to next-to-next-to-next-to-leading order. Our results assess the theoretical uncertainties from the many-body calculation and the chiral expansion. Using a Gaussian process emulator for the free energy, we derive the thermodynamic properties of matter through consistent derivatives and use the Gaussian process to access arbitrary proton fraction and temperature. This enables a first nonparametric calculation of the equation of state in beta equilibrium, and of the speed of sound and the symmetry energy at finite temperature. Moreover, our results show that the thermal part of the pressure decreases with increasing densities.

The Symmetry Energy: Current Status of Ab Initio Predictions vs. Empirical Constraints

Infinite nuclear matter is a suitable laboratory to learn about nuclear forces in many-body systems. In particular, modern theoretical predictions of neutron-rich matter are timely because of recent and planned experiments aimed at constraining the equation of state of isospin-asymmetric matter. For these reasons, we have taken a broad look at the equation of state of neutron-rich matter and the closely related symmetry energy, which is the focal point of this article. Its density dependence is of paramount importance for a number of nuclear and astrophysical systems, ranging from neutron skins to the structure of neutron stars. We review and discuss ab initio predictions in relation to recent empirical constraints. We emphasize and demonstrate that free-space nucleon–nucleon data pose stringent constraints on the density dependence of the neutron matter equation of state, which essentially determines the slope of the symmetry energy at saturation.

Ab initio predictions link the neutron skin of 208Pb to nuclear forces

Heavy atomic nuclei have an excess of neutrons over protons, which leads to the formation of a neutron skin whose thickness is sensitive to details of the nuclear force. This links atomic nuclei to properties of neutron stars, thereby relating objects that differ in size by orders of magnitude. The nucleus 208Pb is of particular interest because it exhibits a simple structure and is experimentally accessible. However, computing such a heavy nucleus has been out of reach for ab initio theory. By combining advances in quantum many-body methods, statistical tools and emulator technology, we make quantitative predictions for the properties of 208Pb starting from nuclear forces that are consistent with symmetries of low-energy quantum chromodynamics. We explore 109 different nuclear force parameterizations via history matching, confront them with data in select light nuclei and arrive at an importance-weighted ensemble of interactions. We accurately reproduce bulk properties of 208Pb and determine the neutron skin thickness, which is smaller and more precise than a recent extraction from parity-violating electron scattering but in agreement with other experimental probes. This work demonstrates how realistic two- and three-nucleon forces act in a heavy nucleus and allows us to make quantitative predictions across the nuclear landscape.

Constraining neutron-star matter with microscopic and macroscopic collisions

Interpreting high-energy, astrophysical phenomena, such as supernova explosions or neutron-star collisions, requires a robust understanding of matter at supranuclear densities. However, our knowledge about dense matter explored in the cores of neutron stars remains limited. Fortunately, dense matter is not probed only in astrophysical observations, but also in terrestrial heavy-ion collision experiments. Here we use Bayesian inference to combine data from astrophysical multi-messenger observations of neutron stars1-9 and from heavy-ion collisions of gold nuclei at relativistic energies10,11 with microscopic nuclear theory calculations12-17 to improve our understanding of dense matter. We find that the inclusion of heavy-ion collision data indicates an increase in the pressure in dense matter relative to previous analyses, shifting neutron-star radii towards larger values, consistent with recent observations by the Neutron Star Interior Composition Explorer mission5-8,18. Our findings show that constraints from heavy-ion collision experiments show a remarkable consistency with multi-messenger observations and provide complementary information on nuclear matter at intermediate densities. This work combines nuclear theory, nuclear experiment and astrophysical observations, and shows how joint analyses can shed light on the properties of neutron-rich supranuclear matter over the density range probed in neutron stars.

Spectral density reconstruction with Chebyshev polynomials

Accurate calculations of the spectral density in a strongly correlated quantum many-body system are of fundamental importance to study its dynamics in the linear response regime. Typical examples are the calculation of inclusive and semiexclusive scattering cross sections in atomic nuclei and transport properties of nuclear and neutron star matter. Integral transform techniques play an important role in accessing the spectral density in a variety of nuclear systems. However, their accuracy is in practice limited by the need to perform a numerical inversion which is often ill-conditioned. In the present work we extend a recently proposed quantum algorithm which circumvents this problem. We show how to perform controllable reconstructions of the spectral density over a finite energy resolution with rigorous error estimates. An appropriate expansion in Chebyshev polynomials allows for efficient simulations also on classical computers. We apply our idea to obtain the local density of states for graphene in a magnetic field as a proof of principle. This paves the way for future applications in nuclear and condensed matter physics.

Translating Neutron Star Observations to Nuclear Symmetry Energy via Deep Neural Networks

One of the most significant challenges involved in efforts to understand the equation of state of dense neutron-rich matter is the uncertain density dependence of the nuclear symmetry energy. In particular, the nuclear symmetry energy is still rather poorly constrained, especially at high densities. On the other hand, detailed knowledge of the equation of state is critical for our understanding of many important phenomena in the nuclear terrestrial laboratories and the cosmos. Because of its broad impact, pinning down the density dependence of the nuclear symmetry energy has been a long-standing goal of both nuclear physics and astrophysics. Recent observations of neutron stars, in both electromagnetic and gravitational-wave spectra, have already constrained significantly the nuclear symmetry energy at high densities. The next generation of telescopes and gravitational-wave observatories will provide an unprecedented wealth of detailed observations of neutron stars, which will improve further our knowledge of the density dependence of nuclear symmetry energy, and the underlying equation of state of dense neutron-rich matter. Training deep neural networks to learn a computationally efficient representation of the mapping between astrophysical observables of neutron stars, such as masses, radii, and tidal deformabilities, and the nuclear symmetry energy allows its density dependence to be determined reliably and accurately. In this work, we use a deep learning approach to determine the nuclear symmetry energy as a function of density directly from observational neutron star data. We show, for the first time, that artificial neural networks can precisely reconstruct the nuclear symmetry energy from a set of available neutron star observables, such as masses and radii as measured by, e.g., the NICER mission, or masses and tidal deformabilities as measured by the LIGO/VIRGO/KAGRA gravitational-wave detectors. These results demonstrate the potential of artificial neural networks to reconstruct the symmetry energy and the equation of state directly from neutron star observational data, and emphasize the importance of the deep learning approach in the era of multi-messenger astrophysics.

Information Content of the Parity-Violating Asymmetry in ^{208}Pb

The parity-violating asymmetry A_{PV} in ^{208}Pb, recently measured by the PREX-2 Collaboration, is studied using modern relativistic (covariant) and nonrelativistic energy density functionals. We first assess the theoretical uncertainty on A_{PV} which is intrinsic to the adopted approach. To this end, we use quantified functionals that are able to accommodate our previous knowledge on nuclear observables such as binding energies, charge radii, and the dipole polarizability α_{D} of ^{208}Pb. We then add the quantified value of A_{PV} together with α_{D} to our calibration dataset to optimize new functionals. Based on these results, we predict a neutron skin thickness in ^{208}Pb r_{skin}=0.19±0.02  fm and the symmetry-energy slope L=54±8  MeV. These values are consistent with other estimates based on astrophysical data and are significantly lower than those recently reported using a particular set of relativistic energy density functionals. We also make a prediction for the A_{PV} value in ^{48}Ca that will be soon available from the CREX measurement.

Astrophysical Constraints on the Symmetry Energy and the Neutron Skin of ^{208}Pb with Minimal Modeling Assumptions

The symmetry energy and its density dependence are crucial inputs for many nuclear physics and astrophysics applications, as they determine properties ranging from the neutron-skin thickness of nuclei to the crust thickness and the radius of neutron stars. Recently, PREX-II reported a value of 0.283±0.071  fm for the neutron-skin thickness of ^{208}Pb, implying a slope parameter L=106±37  MeV, larger than most ranges obtained from microscopic calculations and other nuclear experiments. We use a nonparametric equation of state representation based on Gaussian processes to constrain the symmetry energy S_{0}, L, and R_{skin}^{^{208}Pb} directly from observations of neutron stars with minimal modeling assumptions. The resulting astrophysical constraints from heavy pulsar masses, LIGO/Virgo, and NICER clearly favor smaller values of the neutron skin and L, as well as negative symmetry incompressibilities. Combining astrophysical data with PREX-II and chiral effective field theory constraints yields S_{0}=33.0_{-1.8}^{+2.0}  MeV, L=53_{-15}^{+14}  MeV, and R_{skin}^{^{208}Pb}=0.17_{-0.04}^{+0.04}  fm.

Nuclear Physics and Astrophysics Constraints on the High Density Matter Equation of State

(1) This review has been written in memory of Steven Moszkowski who unexpectedly passed away in December 2020. It has been inspired by our many years of discussions. Steven’s enthusiasm, drive and determination to understand atomic nuclei in simple terms of basic laws of physics was infectious. He sought the fundamental origin of nuclear forces in free space, and their saturation and modification in nuclear medium. His untimely departure left our job unfinished but his legacy lives on. (2) Focusing on the nuclear force acting in nuclear matter of astrophysical interest and its equation of state (EoS), we take several typical snapshots of evolution of the theory of nuclear forces. We start from original ideas in the 1930s moving through to its overwhelming diversity today. The development is supported by modern observational and terrestrial data and their inference in the multimessenger era, as well as by novel mathematical techniques and computer power. (3) We find that, despite the admirable effort both in theory and measurement, we are facing multiple models dependent on a large number of variable correlated parameters which cannot be constrained by data, which are not yet accurate, nor sensitive enough, to identify the theory closest to reality. The role of microphysics in the theories is severely limited or neglected, mostly deemed to be too difficult to tackle. (4) Taking the EoS of high-density matter as an example, we propose to develop models, based, as much as currently possible, on the microphysics of the nuclear force, with a minimal set of parameters, chosen under clear physical guidance. Still somewhat phenomenological, such models could pave the way to realistic predictions, not tracing the measurement, but leading it.

Progress in Constraining Nuclear Symmetry Energy Using Neutron Star Observables Since GW170817

The density dependence of nuclear symmetry energy is among the most uncertain parts of the Equation of State (EOS) of dense neutron-rich nuclear matter. It is currently poorly known especially at suprasaturation densities partially because of our poor knowledge about isovector nuclear interactions at short distances. Because of its broad impacts on many interesting issues, pinning down the density dependence of nuclear symmetry energy has been a longstanding and shared goal of both astrophysics and nuclear physics. New observational data of neutron stars including their masses, radii, and tidal deformations since GW170817 have helped improve our knowledge about nuclear symmetry energy, especially at high densities. Based on various model analyses of these new data by many people in the nuclear astrophysics community, while our brief review might be incomplete and biased unintentionally, we learned in particular the following: (1) The slope parameter L of nuclear symmetry energy at saturation density ρ0 of nuclear matter from 24 new analyses of neutron star observables was about L≈57.7±19 MeV at a 68% confidence level, consistent with its fiducial value from surveys of over 50 earlier analyses of both terrestrial and astrophysical data within error bars. (2) The curvature Ksym of nuclear symmetry energy at ρ0 from 16 new analyses of neutron star observables was about Ksym≈−107±88 MeV at a 68% confidence level, in very good agreement with the systematics of earlier analyses. (3) The magnitude of nuclear symmetry energy at 2ρ0, i.e., Esym(2ρ0)≈51±13 MeV at a 68% confidence level, was extracted from nine new analyses of neutron star observables, consistent with the results from earlier analyses of heavy-ion reactions and the latest predictions of the state-of-the-art nuclear many-body theories. (4) While the available data from canonical neutron stars did not provide tight constraints on nuclear symmetry energy at densities above about 2ρ0, the lower radius boundary R2.01=12.2 km from NICER’s very recent observation of PSR J0740+6620 of mass 2.08±0.07M⊙ and radius R=12.2–16.3 km at a 68% confidence level set a tight lower limit for nuclear symmetry energy at densities above 2ρ0. (5) Bayesian inferences of nuclear symmetry energy using models encapsulating a first-order hadron–quark phase transition from observables of canonical neutron stars indicated that the phase transition shifted appreciably both L and Ksym to higher values, but with larger uncertainties compared to analyses assuming no such phase transition. (6) The high-density behavior of nuclear symmetry energy significantly affected the minimum frequency necessary to rotationally support GW190814’s secondary component of mass (2.50–2.67) M⊙ as the fastest and most massive pulsar discovered so far. Overall, thanks to the hard work of many people in the astrophysics and nuclear physics community, new data of neutron star observations since the discovery of GW170817 have significantly enriched our knowledge about the symmetry energy of dense neutron-rich nuclear matter.

Implications of PREX-2 on the Equation of State of Neutron-Rich Matter

Laboratory experiments sensitive to the equation of state of neutron rich matter in the vicinity of nuclear saturation density provide the first rung in a "density ladder" that connects terrestrial experiments to astronomical observations. In this context, the neutron skin thickness of ^{208}Pb (R_{skin}^{208}) provides a stringent laboratory constraint on the density dependence of the symmetry energy. In turn, an improved value of R_{skin}^{208} has been reported recently by the PREX collaboration. Exploiting the strong correlation between R_{skin}^{208} and the slope of the symmetry energy L within a specific class of relativistic energy density functionals, we report a value of L=(106±37)  MeV-which systematically overestimates current limits based on both theoretical approaches and experimental measurements. The impact of such a stiff symmetry energy on some critical neutron-star observables is also examined.

Accurate Determination of the Neutron Skin Thickness of ^{208}Pb through Parity-Violation in Electron Scattering

We report a precision measurement of the parity-violating asymmetry A_{PV} in the elastic scattering of longitudinally polarized electrons from ^{208}Pb. We measure A_{PV}=550±16(stat)±8(syst) parts per billion, leading to an extraction of the neutral weak form factor F_{W}(Q^{2}=0.00616  GeV^{2})=0.368±0.013. Combined with our previous measurement, the extracted neutron skin thickness is R_{n}-R_{p}=0.283±0.071  fm. The result also yields the first significant direct measurement of the interior weak density of ^{208}Pb: ρ_{W}^{0}=-0.0796±0.0036(exp)±0.0013(theo)  fm^{-3} leading to the interior baryon density ρ_{b}^{0}=0.1480±0.0036(exp)±0.0013(theo)  fm^{-3}. The measurement accurately constrains the density dependence of the symmetry energy of nuclear matter near saturation density, with implications for the size and composition of neutron stars.

A Modern View of the Equation of State in Nuclear and Neutron Star Matter

Background: We analyze several constraints on the nuclear equation of state (EOS) currently available from neutron star (NS) observations and laboratory experiments and study the existence of possible correlations among properties of nuclear matter at saturation density with NS observables. Methods: We use a set of different models that include several phenomenological EOSs based on Skyrme and relativistic mean field models as well as microscopic calculations based on different many-body approaches, i.e., the (Dirac–)Brueckner–Hartree–Fock theories, Quantum Monte Carlo techniques, and the variational method. Results: We find that almost all the models considered are compatible with the laboratory constraints of the nuclear matter properties as well as with the largest NS mass observed up to now, 2.14−0.09+0.10M⊙ for the object PSR J0740+6620, and with the upper limit of the maximum mass of about 2.3–2.5M⊙ deduced from the analysis of the GW170817 NS merger event. Conclusion: Our study shows that whereas no correlation exists between the tidal deformability and the value of the nuclear symmetry energy at saturation for any value of the NS mass, very weak correlations seem to exist with the derivative of the nuclear symmetry energy and with the nuclear incompressibility.