Abstract
Neutronstar 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 phasetransitionlike behavior, treated as arbitrarily rapid crossovers in our framework.
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Introduction
For macroscopic systems in thermal equilibrium, the equation of state (EoS) is a central quantity that reflects not only the basic thermodynamic properties of the medium but also its active degrees of freedom and thus the underlying phase structure. For neutron stars (NSs)—extreme astrophysical objects containing the densest matter found in the presentday Universe—the EoS is closely related to their measurable macroscopic properties due to a link provided by General Relativity^{1,2}. This has made NSs a unique laboratory for ultradense strongly interacting matter, with astrophysical observations informing modelagnostic studies of the EoS and attempts to determine the phase of matter inside the cores of NSs of different masses^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24}.
Quantum Chromodynamics (QCD) predicts that at very high densities, strongly interacting matter no longer resides in a phase where individual nucleons can be identified^{25,26,27,28}. Instead, the active degrees of freedom become a set of elementary particles, quarks and gluons, that form a new phase dubbed quark matter (QM). In recent years, abinitio calculations in nuclear^{29,30,31,32,33} and particle^{34,35,36,37} theory have established that the respective EoSs of nuclear matter (NM) and QM are qualitatively distinct. While the hadronic EoS is controlled and characterized by the nucleon mass scale, its QM counterpart exhibits nearconformal properties, independent of any characteristic mass scales^{13,38}.
This fundamental difference leaves an imprint on various thermodynamic quantities associated with the equation of state, such as the normalized trace anomaly Δ ≡ (ϵ − 3p)/(3ϵ), its logarithmic rate of change with respect to the energy density \({\Delta }^{{\prime} }\equiv {{{{{{{\rm{d}}}}}}}}\Delta /{{{{{{{\rm{d}}}}}}}}\ln \epsilon\), the polytropic index \(\gamma \equiv {{{{{{{\rm{d}}}}}}}}\ln p/{{{{{{{\rm{d}}}}}}}}\ln \epsilon\), the speed of sound squared \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\equiv {{{{{{{\rm{d}}}}}}}}p/{{{{{{{\rm{d}}}}}}}}\epsilon\), and the pressure normalized by that of a system of free quarks, p/p_{free}. As illustrated in Table 1, these quantities indeed take markedly different values in (both low and highdensity) NM and in nearconformal QM at asymptotically high density.
At very high baryon densities, where weakcoupling methods produce reliable results, strongly interacting matter displays nearconformal properties, with conformality only mildly broken by subdominant loop effects and by the small values of the up, down, and strange quark masses. As summarized in Table 1 and discussed in detail in^{13,38,39}, the characteristics of this system include a small positive normalized trace anomaly Δ, a similarly small but negative \({\Delta }^{{\prime} }\), a sound speed just below its conformal value, \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\,\lesssim \,1/3\), and a polytropic index approaching its conformal limit from above, γ ≳ 1. Such values are common to many ultrarelativistic systems, but in stark contrast with those of the hadronic phase, also summarized in Table 1. In the latter case, be it at low densities where robust abinitio results are available^{3,29} or at higher densities where one must resort to phenomenological models (see, e.g.,^{13,40}), the properties of hadronic matter are dominated by the nucleon mass scale that strongly breaks conformal symmetry. At densities exceeding approx. 3n_{s} there is no longer agreement between different model predictions, but as we discuss in detail under Methods, the vast majority of viable models remain strongly nonconformal even there.
It would clearly be very interesting to contrast the above expectations with a modelagnostic inference of the NSmatter EoS, informed by abinitio theoretical results and astrophysical observations. With few recent exceptions^{24,41,42}, existing studies of this kind, however, suffer from at least one of two limitations: either they fail to take into account highdensity information from perturbativeQCD (pQCD) calculations, recently demonstrated to significantly constrain the NSmatter EoS down to realistic core densities^{24,43,44}, or they implement observational constraints in the form of hard cutoffs, limiting the measurements that can be employed.
In this work, we remedy the above shortcomings by generalizing our earlier analyses^{6,13,20} to a Bayesian framework. This enables us to take advantage of altogether 12 simultaneous NS massradius (MR) measurements (see the Methods section) and make quantitative statements about the likelihood of a transition from hadronic to QM within stable NSs. Our analysis is performed using two independent frameworks, the parametric speedofsound interpolation of^{13,20} and the nonparametric Gaussian process (GP) regression of^{24,44}, which utilizes the highdensity constraint from pQCD in a considerably more conservative fashion. Both methods are used to construct a prior for the EoS, connecting a lowdensity result provided by Chiral Effective Field Theory (CEFT)^{3,32} to a highdensity limit given by pQCD^{34,37}. This prior P(EoS), introduced in detail under Methods, is then conditioned using a likelihood function incorporating astrophysical measurements,
with \(P({{{{{{{\rm{data}}}}}}}} {{{{{{{\rm{EoS}}}}}}}})={\Pi }_{i=1}^{n}P({{{{{{{{\rm{data}}}}}}}}}_{i} {{{{{{{\rm{EoS}}}}}}}})\) corresponding to the uncorrelated individual likelihoods of various NS measurements, indexed here by i. These data, reviewed under Methods, include mass measurements of the heaviest pulsars^{45,46,47,48,49}, the LIGO and Virgo tidaldeformability constraints from the NSNS merger event GW170817^{50,51}, and a number of individual massradius measurements using Xray observations of pulsating, quiescent, and accreting neutron stars by the NICER and other collaborations^{17,52,53,54,55,56,57}. As described under Methods, we also vary the number of independent parameters in our parametric interpolation to verify that our results are not impacted by an overly restricted prior. Additionally, we perform one analysis with a polytropic construction to further assess the robustness of our approach. The main results from our work take the form of posterior distributions for different physical quantities evaluated at the centers of NSs of various masses, which we compare to theoretical expectations.
Results
Conformality criterion and theoretical expectations
The approach of an individual quantity towards its conformal value provides a necessary but not sufficient condition for the restoration of (approximate) conformal symmetry in a physical system. In order to establish the conformalization of strongly interacting matter inside NS cores, one should therefore simultaneously track several quantities and in addition ensure that the system stays conformal at higher densities. Here, a particularly useful pair turns out to be Δ and \({\Delta }^{{\prime} }\). They are related to γ and \({c}_{s}^{2}\) via
implying that when both ∣Δ∣ and \( {\Delta }^{{\prime} }\) are small, γ and \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\) also approach their conformal limits 1 and 1/3. In addition, a small value of \( {\Delta }^{{\prime} }\) naturally ensures that Δ will remain approximately constant at higher densities.
In order to draw a demarcation line between the non and nearconformal regimes, we combine Δ and \({\Delta }^{{\prime} }\) into a single quantity, adopting the criterion
for the identification of nearconformal matter at a given density. We justify the value 0.2 as follows. First, as noted in Table 1, it represents a natural choice between the values this parameter takes in NM and conformal systems. Second, at a discontinuous firstorder phase transition (FOPT), where \({c}_{s}^{2}=\gamma=0\) and \({\Delta }^{{\prime} }=1/3\Delta\) (see Eq. (2)), the quantity d_{c} can be shown to be bounded from below by \(1/(3\sqrt{2})\,\approx \,0.236\), so that our criterion prevents FOPTs from masquerading as conformalized matter. We choose to round this value down to 0.2 to provide numerical tolerance for the approximate version of FOPTs (i.e. arbitrarily rapid crossovers) considered in our analysis. Finally, as shown in Table 1, this corresponds approximately to the largest value obtained for this quantity in pQCD. We note that the specific value 0.2 is a choice, but that our qualitative conclusions are insensitive to its small variations.
While QM is nearconformal, not all matter that is nearconformal is QM. Therefore, establishing conformalization of matter in the cores of NSs does not, in principle, imply the presence of the deconfined phase. One quantity not fixed by conformal symmetry is the pressure normalized by the free (noninteracting) pressure p/p_{free}. Its value is related to the effective number of active degrees of freedom (particle species) N_{eff} in both weakly and strongly coupled systems. At weak coupling, Dalton’s law states that the pressure is the sum of partial pressures, which establishes the proportionality of N_{eff} ≡ N_{f}N_{c}p/p_{free}. This relation can be also derived in strongly coupled conformal field theories (CFTs)^{58,59} as well as in QCD at high temperatures^{60,61}. Because this quantity is sensitive to the number of degrees of freedom but insensitive to the interaction strength, we expect that QM—be it weakly or strongly coupled—will have to exhibit a slowly varying p/p_{free} of order one. To this end, we may use its behavior to distinguish QM from other possible nearconformal phases at NScore densities.
From Table 1, we see that in highdensity pQCD matter p/p_{free} is reduced from unity to approximately 0.6 through perturbative corrections. The normalized pressure has been extensively studied also in the context of hightemperature quarkgluon plasma (QGP), where it has played a key role in establishing that deconfined matter has been successfully produced in heavyion collisions (see, e.g.,^{60,61,62}). Nonperturbative lattice simulations have shown that above the pseudocritical temperature, its value saturates to a constant around 0.8. In other hightemperature quantum field theories in the stronglycoupled limit, p/p_{free} often takes fractional values smaller than one^{59,63,64,65,66,67}, and for instance in \({{{{{{{\mathcal{N}}}}}}}}=4\) Super YangMills theory at infinite ’t Hooft coupling, p/p_{free} = 3/4^{59}. In certain lowerdimensional CFTs, the value of this quantity is moreover related to the central charge of the theory that counts the active degrees of freedom^{58,68}.
Posterior distributions
The main results of our analysis are displayed in Figs. 1, 2, 3, 4, where we show the behavior of a number of physical quantities as functions of baryon number density, baryon chemical potential, or stellar mass. Starting from Figs. 1, 2, we show credible intervals (CIs) for d_{c} and the triplet \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2},\gamma\), and Δ as functions of the baryon number density n, obtained using both a foursegment speedofsound interpolation (\({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\)) and GP regression. Note that for the \({c}_{s}^{2}\) interpolation, we display results obtained with a foursegment interpolation which we have found to be the optimal choice, being sufficiently versatile to describe complex EoS behaviors (see Fig. 5 in “Methods” for results with a varying number of segments) and still computationally manageable.
Concretely, using the \({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\) ensemble with the very conservative criterion introduced in Eq. (3), we find the posterior probability for conformalized matter being present in maximally massive stars to be approximately 88%, while the corresponding figures for 2M_{⊙} and 1.4M_{⊙} NSs are only 11% and 0%, respectively. The same probability for maximally massive stars obtained using the nonparametric GP interpolation is 75%, which should be considered a robust lower limit for the quantity given the very conservative way this method handles the highdensity constraint. We note that these results are quantitatively stable between different computational choices (see Methods), and that they reflect the very conservative nature of the new criterion. Had we instead used the γ < 1.75 criterion introduced in^{13}, the posterior probability of conformalization in maximally massive stars would have been 99.8% for the \({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\) and 97.8% for the GP ensemble (see also^{41}). Finally, we note that the GP method allows a quantitative examination of the impact of the pQCD constraint on our results, a topic covered more extensively in^{24,69}. As seen in Fig. 6 under “Methods”, this impact is substantial, reflected in the likelihood of QM cores in TOV stars dropping from 75 to 50% should one not include pQCD input in the analysis.
In Fig. 3, we next show a detailed investigation of the posterior distributions of the quantities displayed in Fig. 2. The results are shown both as twodimensional CIs for the joint distributions between pairs of quantities and as onedimensional probability density functions, all displayed for both the \({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\) and GP ensembles. Shown in gray is finally the region corresponding to our conformal criterion d_{c} < 0.2, which is again seen to be considerably more restrictive than the criterion γ < 1.75 used in previous works^{13,41}. For γ and \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\), we find good quantitative agreement with previous results, with the inner cores of maximally massive stars exhibiting conformalized behavior with high credence. For all three quantities, our results as functions of M/M_{TOV} are moreover in good agreement with recent works^{22,38,39,42,69,70}, with the approach towards conformal behavior as M → M_{TOV} clearly visible. Finally, from the γ vs. \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\) panels, we observe that the majority of probability weight for M_{TOV} stars that lies outside the gray region resides at small values of γ and \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\). This is indicative of FOPTlike behavior, implying that should the cores of M_{TOV} stars not contain conformalized matter, then they are destabilized by a FOPT, consistent with observations reported in^{13}.
Having established the likely conformalization of matter within the cores of maximally massive NSs, we move to an analysis of the number of active degrees of freedom. In Fig. 4, we show the behavior of the pressure normalized by its noninteracting limit, p/p_{free}, as a function of baryon chemical potential μ. Inspecting this result, we see a clear flattening of the normalized pressure in the vicinity of the cores of maximally massive NSs, with a value p/p_{free} = 0.40 ± 0.03 with 68% credence (corresponding to N_{eff} = 3.6 ± 0.3) that is about 2/3 of the central pQCD value in Table 1. That the value of p/p_{free} within maximally massive NSs is both so close to that of weakly coupled QM and appears to evolve very slowly gives us confidence in labeling this phase deconfined QM, although it should be noted that there exist a number of hadronic models, where p/p_{free} obtains similar values close to the TOV density^{40}. As we discuss in Methods, in a large majority of the latter models, the quantity is already decreasing at this density, which is at odds with the behavior we witness. It is, however, interesting to ask whether the slightly reduced value of the quantity within maximally massive NSs may signal that matter in these stellar cores features stronglycoupled characteristics.
Finally, let us briefly note two interesting issues that go somewhat beyond the scope of our current work. First, in^{71} it was recently suggested that the posterior probability of QM cores sharply decreases somewhat below M_{TOV} ≈ 2.3M_{⊙}, so that if NSs more massive than this limit are discovered, QM cores can be ruled out. We have failed to reproduce this effect in our results, but instead find the probability of QM cores to have only a mild dependence of M_{TOV}. Second, some explicit FOPTs beginning at densities below 2n_{sat} have recently been shown to have a noticeable impact on hard limits for the EoS^{72}. While a generalization of our calculation to include explicit discontinuities in the EoS is clearly an interesting topic for further work, we again note that the criterion for nearconformality we have adopted in this work is very conservative. Moreover, the EoSs that^{72} identified to extend beyond previous hard limits were found to have very low posterior probabilities and lie right at the edge of the 90% credibility bound for the tidaldeformability constraint. We hence expect that including such EOSs with explicit FOPTs within an analysis such at our present one would result in only mild shifts in the posteriors. For this reason, we trust that the estimates we have obtained for the probability of finding conformalized matter inside massive NSs are reliable.
Discussion
In the context of heavyion collisions, the properties of the equation of state (EoS) have played a key role in establishing the production of deconfined hot quarkgluon plasma. While there is no discontinuous transition in this case, the two phases can still be clearly demarcated based on their qualitatively differing material properties. At low temperatures, the EoS exhibits the qualitative features of hadronic matter while at high T it is characterized by approximate conformal symmetry and a number of active degrees of freedom corresponding to deconfined quarks and gluons. Note, however, that approximate conformal symmetry does not imply weak coupling: at the temperatures reached in collider experiments, weakcoupling methods remain poorly convergent, and the system exhibits stronglycoupled behavior. Nevertheless, the transition to QGP is identifiable from the restoration of the conformal symmetry, closely related to that of chiral symmetry.
While the highdensity matter in NSs in general significantly differs from hightemperature QGP, the pattern of conformalsymmetry restoration can be similarly used as an indicator of phase change inside NSs. In this Letter, we have established that the cores of the most massive neutron stars are very likely characterized by approximate conformal symmetry. We see signs of conformalization across multiple microscopic properties characterizing the EoS, including the speed of sound \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\), polytropic index γ, normalized trace anomaly Δ, and its logarithmic derivative (with respect to energy density) \({\Delta }^{{\prime} }\). Moreover, we have introduced a new measure for the degree of conformality, \({d}_{{{{{{{{\rm{c}}}}}}}}}\equiv \sqrt{{\Delta }^{2}+{({\Delta }^{{\prime} })}^{2}}\), the behavior of which exhibits a striking transition between 1.4M_{⊙} and maximally massive NSs, and we have determined that the number of active degrees of freedom appears to be consistent with theoretical expectations for the properties of QM. Concretely, we find that within the cores of maximally massive NSs, the ratio p/p_{free} is about 2/3 of its value in weakly coupled quark matter, leading us to the conclusion that the most massive neutron stars very likely harbor cores of deconfined QM. An 88% probability (or 75% with the GP method), obtained with a very conservative definition of nearconformality, provides a quantitative estimate of the remaining uncertainties, of which the most important one is related to the possible presence of a destabilizing firstorder phase transition.
To conclude, we note that in the context of heavyion collisions, evidence for the successful production of deconfined matter consisted of multiple arguments, of which the behavior of the EoS was but one. Similarly, it would be desirable to find other observables and arguments that may either support or contradict the evidence provided by our analysis of the EoS; for concrete suggestions, see e.g.^{73,74,75,76,77,78}. Indeed, just as in the case of heavyion collisions, to compellingly establish the existence of QM in the cores of massive NSs, we need multiple lines of evidence all supporting the same picture.
Methods
Our different prior EoS ensembles are constructed by joining together three different pieces: a lowdensity CEFT result applied up to approx. 1.2n_{sat}, an intermediate density interpolator, and a highdensity part obtained from pQCD that is used either from μ = 2.6 GeV onward or robustly translated down to 10n_{sat} using the results of^{43}. Here, the CEFT part depends of altogether five parameters in a way specified below, while the pQCD pressure only depends on one free parameter, the renormalization scale \(\bar{\Lambda }\) of the \(\overline{{{{{{{{\rm{MS}}}}}}}}}\) scheme. Finally, in the intermediatedensity region we employ two independent approaches, one built with piecewisedefined polytropes or the speedofsound functions of^{13} and the other based on a nonparametric Gaussian process regression introduced in^{24}.
In the case of the parametric interpolation, we sample the parameter space with a MarkovChainMonteCarlo method using the EMCEE sampler^{79}. In practice, we use the affineinvariant stretchmove algorithm with 125 parallel “walkers” to generate MarkovChainMonteCarlo chains with N_{chain}/τ ~ 3 × 10^{3} independent samples. Here, τ ~ 2 × 10^{4} is the autocorrelation time, which can sometimes be very long due to the highdimensional parameter space. We have tested the numerical convergence of each calculation by generating subsamples of the chains (with a jackknife resampling) and then verifying that the resulting credibility regions do not change. In addition, for the particular case of the foursegment \({c}_{s}^{2}\) interpolation we have verified the convergence by generating 50% more samples and noting that all the results remain unchanged. Finally, we smooth the posterior distributions with kernel density estimation using Silverman’s rule to estimate the kernel bandwidth.
Below, we will in turn cover the construction of our low, high and intermediatedensity EoSs, as well as the implementation of the astrophysical constraints. After this, we perform a detailed quantitative comparison between the three frameworks used at intermediate densities (speedofsound and polytropic interpolation as well as Gaussian Processes), while in the last part of “Methods”, we review the results of an analysis of currently available model EoSs for highdensity NM.
Lowdensity EoS
At low densities, up to slightly above the nuclear saturation density n_{sat}, the NSmatter EoS can be reliably determined using the welltested machinery of modern nuclear theory, including the systematic effective theory framework of CEFT^{29,30,31,32,33}. Specifically, we take the crust EoS from^{80}, while starting from the crustcore transition around 0.5n_{sat}, we use the results of^{32} for pure neutron matter (PNM) based on the N3LO abinitio calculations using the Λ = 500 MeV potentials from^{31}, reported up to 0.195 fm^{−3}, or slightly above 1.2n_{sat}.. The upper limit for the pressure is chosen so that all (sampled) PNM EoSs have nonnegative speeds of sound squared throughout the lowdensity interval considered, and we similarly ensure the same condition after beta equilibration.
To transform the PNM results into a betastable form, we utilize the Skyrmemotivated ansatz introduced in^{3}, which expresses the sum of the interaction and kinetic energies in the form
Here, x ≡ n_{p}/n denotes the proton fraction, \(\bar{n}\equiv n/{n}_{{{{{{{{\rm{sat}}}}}}}}}\), and α, α_{L}, η, and η_{L} are parameters that can be determined from the saturation properties of symmetric nuclear matter (see^{3} for discussion and numerical values). The normalization constant T_{SNM} on the other hand stands for the Fermi energy of symmetric nuclear matter (SNM, x = 1/2) at n = n_{sat},
where m_{B} is the (mean) baryon mass, taken equal to that of the neutron m_{N} ≈ 939.565 MeV. Finally, we add to Eq. (4) an additional phenomenological term,
where we have introduced two further phenomenological parameters \({\bar{n}}_{0}\) and ζ_{L}. Their presence can be seen to lead to a considerable improvement in the model’s ability to describe the CEFT data for PNM.
Following the original study^{3}, we next require that the energy and pressure satisfy

1.
ϵ(n_{sat}, 1/2) = − 16 MeV,

2.
p(n_{sat}, 1/2) = 0,
where ϵ = ϵ_{1} + ϵ_{2} and p = n^{2}∂ϵ/∂n, and which aids us in the selection of the model parameters α and η. At the same time, we do not fix the parameter Γ using a single value of the incompressibility parameter for SNM at saturation
as done in^{3}, but rather consider Γ a free parameter following^{19}. Moreover, with the mass difference between the proton and neutron approximated to zero, we note that the proton fraction x ≡ n_{p}/n can be solved from the equation
where \({\mu }_{e}=\root 3 \of {3{\pi }^{2}xn}\) is the electron chemical potential in the ultrarelativistic limit^{3}.
In practice, we draw a large sample of the PNM EoSs of^{32} using a prior distribution for the five PNM model parameters: α_{L}, η_{L}, Γ, ζ_{L}, and \({\bar{n}}_{0}\). This distribution is then further approximated using a Gaussian mixture model to get a simpler and smoother multidimensional distribution, which is used to represent the CEFT result in building the NSmatter EoSs.
Highdensity EoS
Due to the asymptotic freedom of QCD, at very high densities the behavior of cold (T = 0) strongly interacting matter can be approached from the deconfined side using the machinery of modern perturbative thermal field theory. Here, we rely on stateoftheart perturbativeQCD (pQCD) calculations for threeflavor quark matter, presented in detail in^{34,37}.
In the limit of beta equilibrium and strictly vanishing temperature, the pQCD pressure depends on only two parameters: the baryon chemical potential μ and the \(\overline{{{{{{{{\rm{MS}}}}}}}}}\) renormalization scale \(\bar{\Lambda }\). The latter is often scaled by the central value of two times the quark chemical potential 2μ/3 to create a dimensionless parameter \(X\equiv 3\bar{\Lambda }/(2\mu )\), which is typically varied from 1/2 to 2. For the prior probability density on X, we use
with \({X}_{\min }=1/2\) and \({X}_{\max }=2\). The pQCD pressure is then used for baryon chemical potentials μ ≥ μ_{pQCD} = 2.6 GeV, corresponding to baryon densities n_{pQCD} ≳ 40n_{sat}.
Intermediate densities
As noted in the main text, at intermediate densities we use two independent and complementary methods for constructing the full EoSs, largely following earlier works^{13,20,24}. As none of the methods we use here is new, we leave a more extensive technical introduction to the original references.
In the case of parametric interpolation, we use both a piecewisepolytropic or piecewiselinear\({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\) EoSs with a varying number N of intermediate segments between the CEFT and pQCD regimes. The resulting EoSs contain 2N + 1 or 2N free parameters in the polytropic and \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\) cases, respectively.
Finally, for the nonparametric GP calculations, we exactly follow the prior construction of^{24}.
Astrophysical measurements and constraints
In addition to the theoretical constraints discussed above, our EoS models have been further conditioned using astrophysical observations. We list and discuss these measurements here.
We first discuss the pulsar mass measurements. We demand that an EoS be compatible with the latest mass measurements of massive pulsars. So far, two NS systems have been observed likely containing a twosolarmass NS, PSR J0348+0432 (M = 2.01 ± 0.04 M_{⊙}, 68% confidence interval)^{46} and PSR J0740+6620 (M = 2.08 ± 0.07 M_{⊙}, 68% credible interval)^{49}. We model these mass measurements as normally distributed, i.e. \({M}_{0432}/{M}_{\odot } \sim {{{{{{{\mathcal{N}}}}}}}}(2.01,0.0{4}^{2})\) and \({M}_{6620}/{M}_{\odot } \sim {{{{{{{\mathcal{N}}}}}}}}(2.08,0.0{7}^{2})\), respectively.
We then require that the TOV mass derived from a target EoS has to be greater than the observed masses of these two stars; otherwise, the EoS in question is discarded.
We now discuss the tidal deformability measurements. We use GW data from the first binary NS merger event GW170817 collected by the LIGO/Virgo collaboration^{50}. In particular, we use the marginalized 2d posterior distribution for the tidal deformabilities of the two merger components, given in Fig. 1 of^{51}. The individual tidal deformabilities for our EoSs are on the other hand calculated using formulae given in ref. ^{81} assuming that both compact objects are NSs. Note that the LIGO/Virgo collaboration has also detected another possible NSNS merger event GW190425^{82}. It is, however, unfortunately too faint to be of use to constrain the NSmatter EoS and is therefore omitted. Moreover, the collaboration has also detected three possible BHNS merger events: GW190814^{83}, GW200105, and GW200115^{84}. It is, however, very likely that GW190814 is just another BHBH event.
In our analysis, we use two prior variables, the chirp mass of the binary
and the mass ratio q = m_{2}/m_{1}, where m_{1} > m_{2}, instead of the individual masses m_{1} and m_{2} of the binary components. The LIGO/Virgo collaboration has determined the binary chirp mass to be \({{{{{{{\mathcal{M}}}}}}}}=1.186\pm 0.001\,{M}_{\odot }\) (90% symmetric credibility limits)^{85}, which we take to be normally distributed, i.e. \({{{{{{{\mathcal{M}}}}}}}}/{M}_{\odot } \sim {{{{{{{\mathcal{N}}}}}}}}(1.186,3.7\times 1{0}^{7})\). The distribution of the mass ratio q is on the other hand chosen to be uniform between 0.4 and 1, i.e. \(q \sim {{{{{{{\mathcal{U}}}}}}}}(0.4,1)\). Finally, we discard all parameter candidates that do not satisfy

1.
m_{1}, m_{2} ≤ M_{TOV} and

2.
Λ(m_{1}), Λ(m_{2}) ≤ 1600.
We now turn to the massradius measurements from Xray observations. The generated EoSs are also constrained by 12 astrophysical NS massradius measurements that are introduced in Table 2 together with the chosen mass priors. The corresponding radius of an individual star can be determined using the TOV equations when the EoS is generated and a candidate mass is drawn from a uniform prior distribution, \({m}_{i} \sim {{{{{{{\mathcal{U}}}}}}}}({m}_{\min },{m}_{\max })\), where \({m}_{\min }\) and \({m}_{\max }\) are the minimum and maximum masses, respectively. The limits of each mass prior are selected such that they encapsulate the complete range of the massradius measurement, with a maximal range of m_{i}/M_{⊙} ∈ [0.5, 2.8]. When possible, tighter limits are selected because they lead to more efficient sampling of the distribution We also ignore the additional factor to the prior, \(\sim {({m}_{{{{{{{{\rm{TOV}}}}}}}}}{m}_{\min })}^{1}\), correcting for the mass selection bias, as the factor is to a good approximation constant for our choice of \({m}_{\min }\approx 0.5{M}_{\odot }\)^{10,15}.
Proceeding to the individual measurements considered, we first use the 2d massradius probability distributions of the pulsars PSR J0030 + 0451 and PSR J0740 + 6620 as reported by the NICER mission. The PSR J0030 + 0451 measurement is based on publicly available data from^{56} employing their ST+PST model, while the PSR J0740 + 6620 measurement is similarly based on publicly available data from^{17}. From the latter reference, we employ their combined NICER+XMM constraints together with the inflated crossinstrument calibration error. The differences between this choice and other possible models would result in small radius differences of ΔR ≲ 0.1 km.
In addition to the above, we also incorporate 2d massradius measurements from three different Xray bursters. The most accurate constraint corresponds to the neutron star in the binary system 4U J1702 − 429 and is derived using direct atmosphere model fits to timeevolving energy spectra^{54}. Here, we use their model D measurement corresponding to a fit with a free mass, radius, and composition. The other two neutron star massradius measurements, corresponding to the binary systems 4U 1724 − 307 and SAX J1810.8 − 260^{86}, are derived using a cooling tail method analysis^{87}. We do not consider other possible bursting sources because of the large uncertainty related to the unconstrained accretion disk state^{88}. We also note that no rotational effects are modeled in^{89}; these can introduce an additional uncertainty of order 0.5 km into the radius of a rapidly rotating source.
Lastly, we use massradius measurements of seven neutron stars in quiescent lowmass Xray binary systems^{52,53}. We only use systems with reliable distance measurements, which enables breaking the degeneracy between the observed flux and emitted luminosity (i.e., the size of the emitting region and the source distance). We use models that assume the surface to be uniformly emitting, indicating the absence of hot/cold spots. The atmosphere composition is selected manually (from H vs. He) using a composition that gives a radius of R ≈ 12 km; varying the composition causes a large (~2 × ) change in R, so the selection of a realistic composition is relatively unambiguous for all of the used sources.
Finally, we discuss our implementation of multimessenger constraints arising from GW170817. The timing and spectral properties of the short gammaray burst (sGRB) and kilonova from the GW170817 event are also used to set additional constraints for the EoS. These constraints are based on the astrophysical jet generation and launching models that have demonstrated that the jet launching from a black hole merger remnant is strongly favored for sGRB events. Additionally, the prolonged existence of a supramassive/hypermassive NS (SMNS /HMNS, respectively) remnant is disfavored because of the observed moderate kinetic energy of the kilonova/sGRB: an SMNS/HMNS inside a thick disk would release substantial rotational energy into the surrounding kilonova, strongly modifying its appearance. No such modifications, such as a prolonged prompt emission phase or strong blueshifted winds, were detected from GW170817. This implies that the merger remnant either (i) collapsed immediately into a black hole or (ii) formed an SMNS or HMNS remnant that then shortly after collapsed into a black hole (see, e.g.^{20} and references therein).
The short gammaray burst from GW170817 was detected with a lag of ≲ 2 s after the NS merger, measured as the difference between the peak of the gravitational wave signal and the arrival of the first γrays. We therefore require that the remnant must (at least) form a supramassive NS and therefore has a total (baryon) mass exceeding the (baryon) TOV mass, m_{b,1} + m_{b,2} > m_{b,TOV}. We assume here zero ejecta for the merger process, following previous works (see^{20} for references and discussion). Additionally, we note that the short lag of ≲ 2 s favors the HMNS scenario, given that SMNSs are expected to be more longlived with typical lifetimes of ≳ 10s. Enforcing the HMNS scenario would lead to an even more stringent constraint, m_{1} + m_{2} > m_{SMNS}, but following^{20}, here we only consider the more relaxed constraints following from SMNS formation.
Comparison of the constructed EoSs
Our fiducial EoS consists of a piecewiselinear \({c}_{{{{{{{{\rm{s}}}}}}}}}^{2}\) interpolant with N = 4 intermediate segments as a function of the baryon chemical potential μ, for which we use the shorthand notation \({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\). We have benchmarked the robustness of the corresponding results to other interpolation methods, including a varying number of interpolation segments, with results of this comparison displayed in Fig. 5. Comparing the parameter regions that the models can sample for \({d}_{{{{{{{{\rm{c}}}}}}}}},{c}_{{{{{{{{\rm{s}}}}}}}}}^{2},\gamma,p\), and the NS massradius relation, we find that four segments provides the minimum viable interpolation accuracy. The biggest shortcoming of the computationally mosteconomic threesegment model (i.e., \({c}_{{{{{{{{\rm{s}}}}}}}},3}^{2}\)) is the lack of EoSs that mimic phase transitions at densities of n ≈ 5 − 10n_{sat}, whereas only negligible deviations are found between the foursegment and fivesegment models. We note in passing that the polytropic interpolation (p_{N}) gives very similar results when the number of segments N is at least four.
A similar comparison of the effects of various astrophysical observations on our results is shown in Fig. 6 and Fig. 7, where we separately consider the GP and \({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\) methods, respectively. In particular, we compare the resulting Bayesian posterior distributions for calculations with no astrophysical measurements (denoted as “no obs.”); only massmeasurements from radio pulsars ("r”); pulsar mass measurements, GW deformabilities, and the SMNS hypothesis for GW170817 ("r + GW”); and finally pulsar mass measurements, GW deformabilities, the SMNS hypothesis for GW170817, and Xray measurements ("r+GW+Xray”). Our fiducial calculations are based on the measurement set “r+GW+Xray”, which includes all measurements. We note that especially d_{c} does not strongly depend on the inclusion of Xray measurements, and that there are some interesting quantitative differences between the responses of the \({c}_{{{{{{{{\rm{s}}}}}}}},4}^{2}\) and GP ensembles to the astrophysical measurements. This may be partially related to the fact that in Fig. 6, the highdensity constraint from pQCD is only included in the fifth additional column, owing to the fact that in this approach, it is possible to turn this effect on and off at will. Once all observations and the pQCD input have been included, our final results for different physical quantities obtained using the two methods largely agree.
Properties of nuclear matter models at high density
In this final part of the Methods section, we provide brief justification for our claim that viable models of dense nuclear matter behave differently from our results, reflected in part in the values appearing in the “Dense NM" column of Table 1. To study this question in a systematic fashion, we have analyzed a large set of publicly available nuclearmatter models, determining the characteristic ranges of various thermodynamic quantities they predict. The result of this analysis is given in Fig. 8, which display results for all hadronic T = 0 EoSs appearing in the CompOSE database^{90}. (See also a more detailed analysis in the framework of relativistic meanfield models^{91} with qualitatively similar results for \({c}_{s}^{2},\gamma\) and Δ.) Note that here we do not discard EoSs that fail to satisfy given hard observational limits (unlike in the discussion of^{13}), but rather impose a color coding for the curves corresponding to their relative likelihoods in comparison with the most likely EoS in our GP ensemble. We also emphasize that the EoSs from the CompOSE database we analyze are not required to conform to our lowdensity CEFT prior; the latter appears only in the priors of our EoS ensembles and not in the likelihood function that we use here.
Our first observation from (Fig. 8a) is that the different NM models appear to be in qualitative agreement with each other for densities n ≲ 3n_{s}, where they all display strongly nonconformal behavior: a large or rapidly changing trace anomaly, a large value of the polytropic index γ, a rapidly rising value of the speed of sound \({c}_{s}^{2}\), and most indicatively, a large value of the conformal distance d_{c}. Above this density, the agreement is quickly lost (visible in particular in d_{c}), which is why the ranges we report for different quantities in Table 1 are chosen to correspond to n = 3n_{s}. Proceeding towards TOV densities, denoted by dots at the end of the curves in the four plots, some model predictions even end up at nearconformal values, but none of these models carries a nonnegligible likelihood in our analysis. Discarding them, the lower end of the range we give for d_{c} in Table 1 is observed to stay valid at all densities, so that sizable d_{c} values can be considered a characteristic prediction of NM models. Finally, in panel b of the same figure we display the behavior of the normalized pressure in the same NM models, noting that in most cases the values of the quantity are undergoing a slow decline at the TOV point. This appears to be at odds with the general trend of the interpolated EoS band.
Data availability
Thinned versions of the EoS ensembles generated for this study (interpolations and GP) have been deposited to the Zenodo data repository and are available separately for the interpolations^{92} and the GP method^{93}.
Code availability
All codes written for the generation and analysis of the EoS ensembles are available from the authors upon request.
References
Tolman, R. C. Static solutions of Einstein’s field equations for spheres of fluid. Phys. Rev. 55, 364 (1939).
Oppenheimer, J. R. & Volkoff, G. M. On massive neutron cores. Phys. Rev. 55, 374 (1939).
Hebeler, K., Lattimer, J. M., Pethick, C. J. & Schwenk, A. Equation of state and neutron star properties constrained by nuclear physics and observation. Astrophys. J. 773, 11 (2013).
Kurkela, A., Fraga, E. S., SchaffnerBielich, J. & Vuorinen, A. Constraining neutron star matter with Quantum Chromodynamics. Astrophys. J. 789, 127 (2014).
Most, E. R., Weih, L. R., Rezzolla, L. & SchaffnerBielich, J. New constraints on radii and tidal deformabilities of neutron stars from GW170817. Phys. Rev. Lett. 120, 261103 (2018).
Annala, E., Gorda, T., Kurkela, A. & Vuorinen, A. Gravitationalwave constraints on the neutronstarmatter Equation of State. Phys. Rev. Lett. 120, 172703 (2018).
Tews, I., Margueron, J. & Reddy, S. Critical examination of constraints on the equation of state of dense matter obtained from GW170817. Phys. Rev. C. 98, 045804 (2018).
Landry, P. & Essick, R. Nonparametric inference of the neutron star equation of state from gravitational wave observations. Phys. Rev. D. 99, 084049 (2019).
Capano, C. D. et al. Stringent constraints on neutronstar radii from multimessenger observations and nuclear theory. Nat. Astron. 4, 625 (2020).
Miller, M. C., Chirenti, C., & Lamb, F. K. Constraining the equation of state of highdensity cold matter using nuclear and astronomical measurements. arXiv:1904.08907 [astroph.HE] (2019a).
Essick, R., Landry, P. & Holz, D. E. Nonparametric inference of neutron star composition, equation of state, and maximum mass with GW170817. Phys. Rev. D. 101, 063007 (2020).
Raaijmakers, G. et al. Constraining the dense matter equation of state with joint analysis of NICER and LIGO/Virgo measurements. Astrophys. J. Lett. 893, L21 (2020).
Annala, E., Gorda, T., Kurkela, A., Nättilä, J. & Vuorinen, A. Evidence for quarkmatter cores in massive neutron stars. Nat. Phys. 16, 907 (2020).
Dietrich, T. et al. Multimessenger constraints on the neutronstar equation of state and the Hubble constant. Science 370, 1450 (2020).
Landry, P., Essick, R. & Chatziioannou, K. Nonparametric constraints on neutron star matter with existing and upcoming gravitational wave and pulsar observations. Phys. Rev. D. 101, 123007 (2020).
AlMamun, M. et al. Combining Electromagnetic and GravitationalWave Constraints on NeutronStar Masses and Radii. Phys. Rev. Lett. 126, 061101 (2021).
Miller, M. C. et al. The Radius of PSR J0740+6620 from NICER and XMMNewton Data. Astrophys. J. Lett. 918, L28 (2021).
Essick, R., Tews, I., Landry, P. & Schwenk, A. Astrophysical constraints on the symmetry energy and the neutron skin of Pb208 with minimal modeling assumptions. Phys. Rev. Lett. 127, 192701 (2021).
Raaijmakers, G. et al. Constraints on the dense matter equation of state and neutron star properties from NICER’s mass–radius estimate of PSR J0740+6620 and multimessenger observations. Astrophys. J. Lett. 918, L29 (2021).
Annala, E. et al. Multimessenger constraints for ultradense matter. Phys. Rev. X 12, 011058 (2022).
Huth, S. et al. Constraining neutronstar matter with microscopic and macroscopic collisions. Nature 606, 276 (2022).
Altiparmak, S., Ecker, C. & Rezzolla, L. On the sound speed in neutron stars. Astrophys. J. Lett. 939, L34 (2022).
Lim, Y. & Holt, J. W. Neutron star radii, deformabilities, and moments of inertia from experimental and ab initio theory constraints of the ^{208}Pb neutron skin thickness. Galaxies 10, 99 (2022).
Gorda, T., Komoltsev, O. & Kurkela, A. Abinitio QCD calculations impact the inference of the neutronstarmatter equation of state. Astrophys. J. 950, 107 (2023).
Ivanenko, D. D. & Kurdgelaidze, D. F. Hypothesis concerning quark stars. Astrophysics 1, 251 (1965).
Ivanenko, D. & Kurdgelaidze, D. F. Remarks on quark stars. Lett. Nuovo Cim. 2, 13 (1969).
Fritzsch, H., GellMann, M. & Leutwyler, H. Advantages of the color octet gluon picture. Phys. Lett. B 47, 365 (1973).
Shuryak, E. V. Quantum chromodynamics and the theory of superdense matter. Phys. Rept. 61, 71 (1980).
Tews, I., Krüger, T., Hebeler, K. & Schwenk, A. Neutron matter at nexttonexttonexttoleading order in chiral effective field theory. Phys. Rev. Lett. 110, 032504 (2013).
Lynn, J. E. et al. Chiral threenucleon interactions in light nuclei, neutronα scattering, and neutron matter. Phys. Rev. Lett. 116, 062501 (2016).
Drischler, C., Hebeler, K. & Schwenk, A. Chiral interactions up to nexttonexttonexttoleading order and nuclear saturation. Phys. Rev. Lett. 122, 042501 (2019).
Drischler, C., Furnstahl, R. J., Melendez, J. A. & Phillips, D. R. How well do we know the neutronmatter equation of state at the densities inside neutron stars? a bayesian approach with correlated uncertainties. Phys. Rev. Lett. 125, 202702 (2020).
Keller, J., Hebeler, K. & Schwenk, A. Nuclear equation of state for arbitrary proton fraction and temperature based on chiral effective field theory and a Gaussian process emulator. Phys. Rev. Lett. 130, 072701 (2023).
Kurkela, A., Romatschke, P. & Vuorinen, A. Cold quark matter. Phys. Rev. D. 81, 105021 (2010).
Gorda, T., Kurkela, A., Romatschke, P., Säppi, S. & Vuorinen, A. Nexttonexttonexttoleading order pressure of cold quark matter: Leading logarithm. Phys. Rev. Lett. 121, 202701 (2018).
Gorda, T., Kurkela, A., Paatelainen, R., Säppi, S. & Vuorinen, A. Cold quark matter at N3LO: Soft contributions. Phys. Rev. D. 104, 074015 (2021).
Gorda, T., Kurkela, A., Paatelainen, R., Säppi, S. & Vuorinen, A. Soft interactions in cold quark matter. Phys. Rev. Lett. 127, 162003 (2021).
Fujimoto, Y., Fukushima, K., McLerran, L. D. & Praszalowicz, M. Trace anomaly as signature of conformality in neutron stars. Phys. Rev. Lett. 129, 252702 (2022).
Marczenko, M., McLerran, L., Redlich, K. & Sasaki, C. Reaching percolation and conformal limits in neutron stars. Phys. Rev. C. 107, 025802 (2023).
Malik, T., Ferreira, M., Albino, M. B. & Providência, C. Spanning the full range of neutron star properties within a microscopic description. Phys. Rev. D. 107, 103018 (2023).
Han, M.Z., Huang, Y.J., Tang, S.P. & Fan, Y.Z. Plausible presence of new state in neutron stars with masses above 0.98MTOV. Sci. Bull. 68, 913 (2023).
Jiang, J.L., Ecker, C. & Rezzolla, L. Bayesian analysis of neutronstar properties with parameterized equations of state: The role of the likelihood functions. Astrophys. J. 949, 11 (2023).
Komoltsev, O. & Kurkela, A. How perturbative QCD constrains the equation of state at neutronstar densities. Phys. Rev. Lett. 128, 202701 (2022).
Gorda, T., Komoltsev, O., Kurkela, A. & Mazeliauskas, A. Bayesian uncertainty quantification of perturbative QCD input to the neutronstar equation of state. JHEP 06, 002.
Demorest, P., Pennucci, T., Ransom, S., Roberts, M. & Hessels, J. Shapiro delay measurement of a two solar mass neutron star. Nature 467, 1081 (2010).
Antoniadis, J. et al. A massive pulsar in a compact relativistic binary. Science 340, 1233232 (2013).
Fonseca, E. et al. The nanograv nineyear data set: Mass and geometric measurements of binary millisecond pulsars. Astrophys. J. 832, 167 (2016).
Cromartie, H. T. et al. Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar. Nat. Astron. 4, 72 (2019).
Fonseca, E. et al. Refined mass and geometric measurements of the highmass PSR J0740+6620. Astrophys. J. Lett. 915, L12 (2021).
Abbott, B. P. et al. GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017).
Abbott, B. P. et al. GW170817: Measurements of neutron star radii and equation of state. Phys. Rev. Lett. 121, 161101 (2018).
Shaw, A. W. et al. The radius of the quiescent neutron star in the globular cluster M13. Mon. Not. R. Astron. Soc. 476, 4713 (2018).
Steiner, A. W. et al. Constraining the mass and radius of neutron stars in globular clusters. Mon. Not. Roy. Astron. Soc. 476, 421 (2018).
Nättilä, J. et al. Neutron star mass and radius measurements from atmospheric model fits to Xray burst cooling tail spectra. Astron. Astrophys. 608, A31 (2017).
Miller, M. C. et al. PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter. Astrophys. J. Lett. 887, L24 (2019).
Riley, T. E. et al. A NICER view of PSR J0030+0451: Millisecond pulsar parameter estimation. Astrophys. J. Lett. 887, L21 (2019).
Riley, T. E. et al. A NICER view of the massive pulsar PSR J0740+6620 informed by radio timing and XMMNewton spectroscopy. Astrophys. J. Lett. 918, L27 (2021).
Cardy, J. L. Scaling and Renormalization in Statistical Physics. Cambridge Lecture Notes in Physics (Cambridge University Press, 1996).
Gubser, S. S., Klebanov, I. R. & Tseytlin, A. A. Coupling constant dependence in the thermodynamics of N=4 supersymmetric YangMills theory. Nucl. Phys. B 534, 202 (1998).
Borsanyi, S. et al. Full result for the QCD equation of state with 2+1 flavors. Phys. Lett. B 730, 99 (2014).
Bazavov, A. et al. Equation of state in (2+1)flavor QCD. Phys. Rev. D. 90, 094503 (2014).
Gardim, F. G., Giacalone, G., Luzum, M. & Ollitrault, J.Y. Thermodynamics of hot stronginteraction matter from ultrarelativistic nuclear collisions. Nat. Phys. 16, 615 (2020).
Sachdev, S. Polylogarithm identities in a conformal field theory in threedimensions. Phys. Lett. B 309, 285 (1993).
Drummond, I. T., Horgan, R. R., Landshoff, P. V. & Rebhan, A. Foam diagram summation at finite temperature. Nucl. Phys. B 524, 579 (1998).
Romatschke, P. Finitetemperature conformal field theory results for all couplings: O(N) model in 2+1 dimensions. Phys. Rev. Lett. 122, 231603 (2019). [Erratum: Phys.Rev.Lett. 123, 209901 (2019)].
DeWolfe, O. & Romatschke, P. Strong Coupling Universality at Large N for Pure CFT Thermodynamics in 2+1 dimensions. JHEP 10, 272, arXiv:1905.06355 [hepth].
Romatschke, P. Fractional degrees of freedom at infinite coupling in large Nf QED in 2+1 dimensions. Phys. Rev. Lett. 123, 241602 (2019).
Cardy, J. L. Operator content of twodimensional conformally invariant theories. Nucl. Phys. B 270, 186 (1986).
Ecker, C. & Rezzolla, L. Impact of largemass constraints on the properties of neutron stars. Mon. Not. Roy. Astron. Soc. 519, 2615 (2022).
Ecker, C. & Rezzolla, L. A general, scaleindependent description of the sound speed in neutron stars. Astrophys. J. Lett. 939, L35 (2022).
Takatsy, J., Kovacs, P., Wolf, G. & SchaffnerBielich, J. What neutron stars tell about the hadronquark phase transition: A Bayesian study. Phys. Rev. D. 108, 043002 (2023).
Gorda, T., Hebeler, K., Kurkela, A., Schwenk, A. & Vuorinen, A. Constraints on strong phase transitions in neutron stars. Astrophys. J. 955, 100 (2023).
Alford, M. G., Rajagopal, K. & Wilczek, F. QCD at finite baryon density: Nucleon droplets and color superconductivity. Phys. Lett. B 422, 247 (1998).
Alford, M. G., Schmitt, A., Rajagopal, K. & Schäfer, T. Color superconductivity in dense quark matter. Rev. Mod. Phys. 80, 1455 (2008).
Alford, M. G., Mahmoodifar, S. & Schwenzer, K. Large amplitude behavior of the bulk viscosity of dense matter. J. Phys. G 37, 125202 (2010).
Most, E. R. et al. Signatures of quarkhadron phase transitions in generalrelativistic neutronstar mergers. Phys. Rev. Lett. 122, 061101 (2019).
Fujimoto, Y., Fukushima, K., Hotokezaka, K. & Kyutoku, K. Gravitational wave signal for quark matter with realistic phase transition. Phys. Rev. Lett. 130, 091404 (2023).
CasalderreySolana J., Mateos D., SanchezGaritaonandia M. MegaHertz Gravitational Waves from Neutron Star Mergers. arXiv:2210.03171 [hepth] (2022).
ForemanMackey, D., Hogg, D. W., Lang, D. & Goodman, J. emcee: The MCMC hammer. Publ. Astronomical Soc. Pac. 125, 306 (2013).
Baym, G., Pethick, C. & Sutherland, P. The Ground state of matter at high densities: Equation of state and stellar models. Astrophys. J. 170, 299 (1971).
Landry, P. & Poisson, E. Relativistic theory of surficial Love numbers. Phys. Rev. D. 89, 124011 (2014).
Abbott, B. P. et al. GW190425: Observation of a compact binary coalescence with total mass ~ 3.4M_{⊙}. Astrophys. J. Lett. 892, L3 (2020).
Abbott, R., Abbott, T. D., Abraham, S. & Acernese, F., LIGO Scientific, Virgo. GW190814: Gravitational waves from the coalescence of a 23 solar mass black hole with a 2.6 solar mass compact object. Astrophys. J. Lett. 896, L44 (2020).
Abbott, R. et al. Observation of gravitational waves from two neutron star–black hole coalescences. Astrophys. J. Lett. 915, L5 (2021).
Abbott, B. P. et al. Properties of the binary neutron Star Merger GW170817. Phys. Rev. X 9, 011001 (2019).
Nättilä, J., Steiner, A. W., Kajava, J. J. E., Suleimanov, V. F. & Poutanen, J. Equation of state constraints for the cold dense matter inside neutron stars using the cooling tail method. Astron. Astrophys. 591, A25 (2016).
Suleimanov, V. F. et al. The direct cooling tail method for Xray burst analysis to constrain neutron star masses and radii. Mon. Not. R. Astron. Soc. 466, 906 (2017).
Kajava, J. J. E. et al. The influence of accretion geometry on the spectral evolution during thermonuclear (type I) Xray bursts. Mon. Not. R. Astron. Soc. 445, 4218 (2014).
Suleimanov, V. F., Poutanen, J. & Werner, K. Observational appearance of rapidly rotating neutron stars. Xray bursts, cooling tail method, and radius determination. Astron. Astrophys. 639, A33 (2020).
Antonopoulou, D. et al. CompOSE: a repository for neutron star equations of state and transport properties. Eur. Phys. J. A 58, 254 (2022).
Providência, C., Malik, T., Albino, M. B., & Ferreira, M. Neutron star equation of state: identifying hadronic matter characteristics. arXiv:2307.05086 [nuclth].
Hirvonen J., Nättilä J., & Annala, E. K. O. Ensembles for neutronstarmatter equationofstate interpolations. https://doi.org/10.5281/zenodo.10102436 (2023).
Komoltsev, O., Kurkela, A., & Gorda, T. GP ensemble of neutron star equations of state. https://doi.org/10.5281/zenodo.10101447 (2023).
Acknowledgements
We would like to thank Kenji Fukushima, Sophia Han, Carlos Hoyos, Niko Jokela, and Anna Watts for useful discussions. The work has been supported by the Finnish Cultural Foundation (EA), the Academy of Finland grants no. 1303622 and 354533 (EA, JH, and AV), the European Research Council grant no. 725369 (EA, JH, and AV), the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – projectID 279384907 – SFB 1245 (TG), the State of Hesse within the Research Cluster ELEMENTS, projectID 500/10.006 (TG), and a joint Columbia University/Flatiron Institute Research Fellowship (JN). The authors also acknowledge CSC  IT Center for Science, Finland, for computational resources (project 2003485).
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We are listed in alphabetical order. All authors (E.A., T.G., J.H., O.K., A.K., J.N., A.V.) conducted the research together and participated in the preparation and revision of the manuscript. The GP framework was originally developed by O.K. and A.K. and implemented here by O.K. and T.G.. The MCMC interpolation framework was originally developed by J.N. and implemented here by J.H., E.A., and J.N.
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Annala, E., Gorda, T., Hirvonen, J. et al. Strongly interacting matter exhibits deconfined behavior in massive neutron stars. Nat Commun 14, 8451 (2023). https://doi.org/10.1038/s4146702344051y
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DOI: https://doi.org/10.1038/s4146702344051y
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