arXiv subjects

Bao-An Li

Publications and source records attributed to Bao-An Li.

At least 55 records · Page 3Linked to original sources

arXiv2023

Impact of Symmetry Energy on Sound Speed and Spinodal Decomposition in Dense Neutron-Rich Matter

Using a meta model for nuclear Equation of State (EOS) with its parameters constrained by astrophysical observations and terrestrial nuclear experiments, we examine effects of nuclear EOS especially its symmetry energy \esym term on the speed of sound squared $C^2_s(ρ)$ and the critical density $ρ_t$ where $C^2_s(ρ_t)$ vanishes (indicating the onset of spinodal decomposition) in both dense neutron-rich nucleonic matter relevant for relativistic heavy-ion collisions and the cold $n+p+e+μ$ matter in neutron stars at $β$-equilibrium. Unlike in nucleonic matter with fixed values of the isospin asymmetry $δ$, in neutron stars with a density dependent isospin profile $δ(ρ)$ determined consistently by the $β$ equilibrium and charge neutrality conditions, the $C^2_s(ρ)$ almost always show a peak and then vanishes at $ρ_t$. The latter strongly depends on the high-density behavior of \esym if the skewness parameter $J_0$ characterizing the stiffness of high-density symmetric nuclear matter (SNM) EOS is not too far above its currently known most probable value of about $-190$ MeV inferred from recent Bayesian analyses of neutron star observables. Moreover, in the case of having a super-soft \esym that is decreasing with increasing density above about twice the saturation density of nuclear matter, the $ρ_t$ is significantly lower than the density where the \esym vanishes (indicating the onset of isospin-separation instability and pure neutron matter formation) in neutron star cores.

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arXiv2023

Principal Component Analysis of Azimuthal Flow in Intermediate-Energy Heavy-Ion Reactions

Principal Component Analysis (PCA) via Singular Value Decomposition (SVD) of large datasets is an adaptive exploratory method to uncover natural patterns underlying the data. Several recent applications of the PCA-SVD to event-by-event single-particle azimuthal angle distribution matrices in ultra-relativistic heavy-ion collisions at RHIC-LHC energies indicate that the sine and cosine functions chosen {\it a priori} in the traditional Fourier analysis are naturally the most optimal basis for azimuthal flow studies according to the data itself. We perform PCA-SVD analyses of mid-central Au+Au collisions at $E_{\rm beam}/A$=1.23 GeV simulated using an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model to address the following two questions: (1) if the principal components of the covariance matrix of nucleon azimuthal angle distributions in heavy-ion reactions around 1 GeV/nucleon are naturally sine and/or cosine functions and (2) what if any advantages the PCA-SVD may have over the traditional flow analysis using the Fourier expansion for studying the EOS of dense nuclear matter. We find that (1) in none of our analyses the principal components come out naturally as sine and/or cosine functions, (2) while both the eigenvectors and eigenvalues of the covariance matrix are appreciably EOS dependent, the PCA-SVD has no apparent advantage over the traditional Fourier analysis for studying the EOS of dense nuclear matter using the azimuthal collective flow in heavy-ion collisions.

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arXiv2022

Long Range Plan: Dense matter theory for heavy-ion collisions and neutron stars

Since the release of the 2015 Long Range Plan in Nuclear Physics, major events have occurred that reshaped our understanding of quantum chromodynamics (QCD) and nuclear matter at large densities, in and out of equilibrium. The US nuclear community has an opportunity to capitalize on advances in astrophysical observations and nuclear experiments and engage in an interdisciplinary effort in the theory of dense baryonic matter that connects low- and high-energy nuclear physics, astrophysics, gravitational waves physics, and data science

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arXiv2022

Nuclear Equation of State and Single-nucleon Potential from Gogny-like Energy Density Functionals Encapsulating Effects of Nucleon-nucleon Short-range Correlations

Nucleon-nucleon short-range correlations (SRCs) induce a high momentum tail (HMT) in the single-nucleon momentum distribution function $n_{ǩ}^J(ρ,δ)$ in cold neutron-rich matter. While there are clear experimental evidences that the SRC/HMT effects are different for neutrons and protons and their strengths depend strongly on the isospin asymmetry of finite nuclei mostly based on electron-nucleus scattering experiments, much less is known experimentally about the SRC/HMT effects in the dense neutron-rich matter. To facilitate further explorations of SRC/HMT effects in dense neutron-rich matter especially with heavy-ion reactions involving high-energy radioactive beams as well as multimessenger observations of neutron stars and their mergers, by incorporating the SRC-induced HMT in $n_{ǩ}^J(ρ,δ)$ into a Gogny-like energy density functional we study SRC/HMT effects on the equation of state (EOS) especially its symmetry energy term and single-nucleon potential in the dense asymmetric nucleonic matter (ANM). Using a parametrization as a surrogate for the momentum-dependent kernel in the Gogny-like energy density functional (EDF) we derive analytical expressions for all components of the ANM EOS and their characteristics (e.g., magnitude, slope and curvature as well as nucleon effective mass) at saturation density $ρ_0$ as well as the momentum-dependent single-nucleon optical potential in neutron-rich matter using parameters characterizing nuclear interactions as well as the size, shape and isospin dependence of the HMT at $ρ_0$. Some consequences of the SRC/HMT effects on properties of neutron stars are also studied.

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arXiv2022

Probing high-density nuclear symmetry energy with $Ξ^{-}/Ξ^{0}$ ratio in heavy-ion collisions at $\sqrt{s_{NN}} \sim 3$ GeV

Recent beam energy scan (BES) experiments at RHIC by the STAR Collaboration (PLB {\bf 827},137003 (2022) and PRL {\bf 128}, 202303 (2022)) found that hadronic interactions dominate the collective flow and the proton cumulant ratios are driven by baryon number conservation in a region of high baryon density in $\sqrt{s_{NN}}$ = 3 GeV Au+Au reactions, indicating the dense medium formed in such collisions is likely hadronic matter. Within an updated ART (A Relativistic Transport) model with momentum dependent isoscalar and isovector single-nucleon mean-field potentials corresponding to different symmetry energies at suprasaturation densities, the $n/p$, $π^{-}/π^{+}$, $K_{s}^{0}/K^{+}$, $Σ^{-}/Σ^{+}$ and $Ξ^{-}/Ξ^{0}$ ratios are studied for central Au+Au collisions at $\sqrt{s_{NN}}$ = 3 GeV where the maximum central density reaches about $(3.6\sim 4.0)ρ_0$. The doubly strange $Ξ^{-}/Ξ^{0}$ ratio is found to have the strongest sensitivity to the variation of high-density nuclear symmetry energy. Thus, the $Ξ^{-}/Ξ^{0}$ ratio in relativistic heavy-ion reactions at $\sqrt{s_{NN}} \sim 3$ GeV may help probe sensitively the poorly known symmetry energy of dense neutron-rich matter critically important for understanding various properties of neutron stars.

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arXiv2022

Equation of State of Neutron-Rich Matter in $d$-Dimensions

Nuclear systems under constraints, with high degrees of symmetries and/or collectivities may be considered as moving effectively in spaces with reduced spatial dimensions. We first derive analytical expressions for the nucleon specific energy $E_0(ρ)$, pressure $P_0(ρ)$, incompressibility coefficient $K_0(ρ)$ and skewness coefficient $J_0(ρ)$ of symmetric nucleonic matter (SNM), the quadratic symmetry energy $E_{\rm{sym}}(ρ)$, its slope parameter $L(ρ)$ and curvature coefficient $K_{\rm{sym}}(ρ)$ as well as the fourth-order symmetry energy $E_{\rm{sym,4}}(ρ)$ of neutron-rich matter in general $d$ spatial dimensions (abbreviated as "$d$D") in terms of the isoscalar and isovector parts of the isospin-dependent single-nucleon potential according to the generalized Hugenholtz-Van Hove (HVH) theorem. The equation of state (EOS) of nuclear matter in $d$D can be linked to that in the conventional 3-dimensional (3D) space by the $ε$-expansion which is a perturbative approach successfully used previously in treating second-order phase transitions and related critical phenomena and more recently in studying the EOS of cold atoms. The $ε$-expansion of nuclear EOS in $d$D based on a reference dimension $d_{\rm{f}}=d-ε$ is shown to be effective with $-1\lesssimε\lesssim1$ starting from $1\lesssim d_{\rm{f}}\lesssim3$ in comparison with the exact expressions derived using the HVH theorem. Moreover, the EOS of SNM (with/without considering its potential part) is found to be reduced (enhanced) in lower (higher) dimensions, indicating in particular that the many-nucleon system tends to be deeper bounded but saturate at higher densities in spaces with lower dimensions. The links between the EOSs in 3D and $d$D spaces from the $ε$-expansion provide new perspectives to the EOS of neutron-rich matter.

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arXiv2022

A theoretical overview of isospin and EOS effects in heavy-ion reactions at intermediate energies

The isospin dependence of in-medium nuclear effective interactions is a fundamental issue in nuclear physics and has broad ramifications in astrophysics. Its uncertainties, especially the difference of neutron-proton interactions in the isosinglet and isotriplet channels, affect significantly the density and momentum dependence of the isovector single-nucleon potential and nucleon-nucleon short-range correlation in neutron-rich matter. Consequently, the neutron-proton effective mass splitting and the density dependence of nuclear symmetry energy are still rather uncertain. Heavy-ion reactions especially those involving rare isotopes is a useful tool for probing the isospin dependence of nuclear effective interactions through (1) the neutron-skin in coordinate and proton-skin in momentum of the initial state of colliding nuclei, (2) the density and momentum dependence of especially the isovector nuclear mean-field as well as (3) the isospin dependence of in-medium nucleon-nucleon cross sections. Observations of neutron stars especially since GW1710817 have also helped us significantly in understanding the isospin dependence of nuclear effective interactions. {\it We summarize here a review talk on these issues given at the 2021 International Workshop on multi-facets of EOS and Clustering. For details we refer the readers to the original publications and references therein}.

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arXiv2022

Investigating Effects of Relativistic Kinematics, Dimensionality, Interactions, and Short-Range Correlations on the Ratio of Quartic over Quadratic Nuclear Symmetry Energies

While ample evidence for the so-called empirical parabolic law of the Equation of State (EOS) of isospin asymmetric nuclear matter (ANM) has been obtained in many studies within both non-relativistic and relativistic nuclear many-body theories using various interactions, it has been unclear if there is any fundamental physics reason for the small quartic symmetry energy compared to the quadratic one even as the ANM approaches pure neutron matter. Within both relativistic and non-relativistic Free Fermi Gas (FFG) models in coordinate spaces of arbitrary dimension $d$ with and without considering Short-Range Correlations (SRC) as well as non-linear Relativistic Mean Field (RMF) models, we study effects of relativistic kinematics, dimensionality, interactions and SRC on the ratio $Ψ(ρ)$ of quartic over quadratic symmetry energies in ANM EOSs. We found that the ratio $Ψ(ρ)$ in the FFG model depends strongly on the dimension $d$. While it is very small already in the normal 3D space, it could be even smaller in spaces with reduced dimensions for sub-systems of particles in heavy-ion reactions and/or whole neutron stars due to constraints, collectivities and/or symmetries. We also found that the ratio $Ψ(ρ)$ could theoretically become very large only at the ultra-relativistic limit far above the density reachable in neutron stars. On the other hand, nuclear interaction directly and/or indirectly through SRC-induced high-momentum nucleons affect significantly the density dependence of $Ψ(ρ)$ compared to the relativistic FFG model prediction. The SRC affects significantly not only the kinetic energy of symmetric nuclear matter but also the ratio $Ψ(ρ)$ while the relativistic corrections are found negligible. The results may help better understand the EOS of dense neutron-rich matter.

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arXiv2022

Transport Model Comparison Studies of Intermediate-Energy Heavy-Ion Collisions

Transport models are the main method to obtain physics information from low to relativistic-energy heavy-ion collisions. The Transport Model Evaluation Project (TMEP) has been pursued to test the robustness of transport model predictions in reaching consistent conclusions from the same type of physical model. Calculations under controlled conditions of physical input and set-up were performed with various participating codes. These included both calculations of nuclear matter in a box with periodic boundary conditions, and more realistic calculations of heavy-ion collisions. In this intermediate review, we summarize and discuss the present status of the project. We also provide condensed descriptions of the 26 participating codes, which contributed to some part of the project. These include the major codes in use today. We review the main results of the studies completed so far. They show, that in box calculations the differences between the codes can be well understood and a convergence of the results can be reached. These studies also highlight the systematic differences between the two families of transport codes, known as BUU and QMD type codes. However, when the codes were compared in full heavy-ion collisions using different physical models, as recently for pion production, they still yielded substantially different results. This calls for further comparisons of heavy-ion collisions with controlled models and of box comparisons of important ingredients, like momentum-dependent fields, which are currently underway. We often indicate improved strategies in performing transport simulations and thus provide guidance to code developers. Results of transport simulations of heavy-ion collisions from a given code will have more significance if the code can be validated against benchmark calculations such as the ones summarized in this review.

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arXiv2021

Bayesian uncertainty quantification for nuclear matter incompressibility

Within a Bayesian statistical framework using the standard Skyrme-Hartree-Fock model, the maximum {\it a posteriori} (MAP) values and uncertainties of nuclear matter incompressibility and isovector interaction parameters are inferred from the experimental data of giant resonances and neutron-skin thicknesses of typical heavy nuclei. With the uncertainties of the isovector interaction parameters constrained by the data of the isovector giant dipole resonance and the neutron-skin thickness, we have obtained $K_0 = 223_{-8}^{+7}$ MeV at 68% confidence level using the data of the isoscalar giant monopole resonance in $^{208}$Pb measured at the Research Center for Nuclear Physics (RCNP), Japan, and at the Texas A&M University (TAMU), USA. Although the corresponding $^{120}$Sn data gives a MAP value for $K_0$ about 5 MeV smaller than the $^{208}$Pb data, there are significant overlaps in their posterior probability distribution functions.

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arXiv2021

Differential analysis of incompressibility in neutron-rich nuclei

Both the incompressibility \Ka of a finite nucleus of mass A and that ($K_{\infty}$) of infinite nuclear matter are fundamentally important for many critical issues in nuclear physics and astrophysics. While some consensus has been reached about the $K_{\infty}$, accurate theoretical predictions and experimental extractions of $K_τ$ characterizing the isospin dependence of \Ka have been very difficult. We propose a differential approach to extract the \Kt and \Ki independently from the \Ka data of any two nuclei in a given isotope chain. Applying this new method to the \Ka data from isoscalar giant monopole resonances (ISGMR) in even-even Pb, Sn, Cd and Ca isotopes taken by U. Garg {\it et al.} at the Research Center for Nuclear Physics (RCNP), Osaka University, Japan, we find that the $^{106}$Cd-$^{116}$Cd and $^{112}$Sn-$^{124}$Sn pairs having the largest differences in isospin asymmetries in their respective isotope chains measured so far provide consistently the most accurate up-to-date \Kt value of $K_τ=-616\pm 59$ MeV and $K_τ=-623\pm 86$ MeV, respectively, largely independent of the remaining uncertainties of the surface and Coulomb terms in expanding the $K_{\rm A}$, while the $K_{\infty}$ values extracted from different isotopes chains are all well within the current uncertainty range of the community consensus for $K_{\infty}$. Moreover, the size and origin of the "Soft Sn Puzzle" is studied with respect to the "Stiff Pb Phenomenon". It is found that the latter is favored due to a much larger (by $\sim 380$ MeV) \Kt for Pb isotopes than for Sn isotopes, while the \Ki from analyzing the \Ka data of Sn isotopes is only about 5 MeV less than that from analyzing the Pb data.

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arXiv2021

Imprints of high-momentum nucleons in nuclei on hard photons from heavy-ion collisions around the Fermi Energy

The short-range correlation (SRC) induced by the tensor force in the isosinglet neutron-proton interaction channel leads to a high-momentum tail (HMT) in the single-nucleon momentum distributions n(k) in nuclei. Owing to the remaining uncertainties about the tensor force, the shape of the nucleon HMT may be significantly different from the dilute interacting Fermi gas model prediction $n(k) \sim1/k^4$ similar to the HMT in cold atoms near the unitary limit. Within an isospin- and momentum-dependent Boltzmann-Uehling-Uhlenbeck transport model incorporating approximately the nucleon HMT, we investigate hard photon emissions in $^{14}$N+$^{12}$C and $^{48}$Ca+$^{124}$Sn reactions at beam energies around the Fermi energy. Imprints of different shapes of the HMT on the energy spectrum, angular distribution and transverse momentum spectrum of hard photons are studied. While the angular distribution does not carry any information about the shape of the nucleon HMT, the energy spectra and especially the mid-rapidity transverse momentum spectra of hard photons are found to bare strong imprints of the shapes of nucleon HMTs in the two colliding nuclei.

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arXiv2021

Impact of NICER's Radius Measurement of PSR J0740+6620 on Nuclear Symmetry Energy at Suprasaturation Densities

By directly inverting several neutron star observables in the three-dimensional parameter space for the Equation of State of super-dense neutron-rich nuclear matter, we show that the lower radius limit for PSR J0740+6620 of mass $2.08\pm 0.07~M_{\odot}$ from Neutron Star Interior Composition Explorer (NICER)'s very recent observation sets a much tighter lower boundary than previously known for nuclear symmetry energy in the density range of $(1.0\sim 3.0)$ times the saturation density $ρ_0$ of nuclear matter. The super-soft symmetry energy leading to the formation of proton polarons in this density region of neutron stars is clearly disfavoured by the first radius measurement for the most massive neutron star observed reliably so far.

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arXiv2021

Comparison of Heavy-Ion Transport Simulations: Mean-field Dynamics in a Box

Within the transport model evaluation project (TMEP) of simulations for heavy-ion collisions, the mean-field response is examined here. Specifically, zero-sound propagation is considered for neutron-proton symmetric matter enclosed in a periodic box, at zero temperature and around normal density. The results of several transport codes belonging to two families (BUU-like and QMD-like) are compared among each other and to exact calculations. For BUU-like codes, employing the test particle method, the results depend on the combination of the number of test particles and the spread of the profile functions that weight integration over space. These parameters can be properly adapted to give a good reproduction of the analytical zero-sound features. QMD-like codes, using molecular dynamics methods, are characterized by large damping effects, attributable to the fluctuations inherent in their phase-space representation. Moreover, for a given nuclear effective interaction, they generally lead to slower density oscillations, as compared to BUU-like codes. The latter problem is mitigated in the more recent lattice formulation of some of the QMD codes. The significance of these results for the description of real heavy-ion collisions is discussed.

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arXiv2021

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

New observational data of neutron stars since GW170817 have helped improve our knowledge about nuclear symmetry energy especially at high densities. We have learned particularly: (1) The slope parameter $L$ of nuclear symmetry energy at saturation density $ρ_0$ of nuclear matter from 24 new analyses is about $L\approx 57.7\pm 19$ MeV at 68\% confidence level consistent with its fiducial value, (2) The curvature $K_{\rm{sym}}$ from 16 new analyses is about $K_{\rm{sym}}\approx -107\pm 88$ MeV, (3) The magnitude of nuclear symmetry energy at $2ρ_0$, i.e. $E_{\rm{sym}}(2ρ_0)\approx 51\pm 13$ MeV at 68\% confidence level, has been extracted from 9 new analyses of neutron star observables consistent with 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 do not provide tight constraints on nuclear symmetry energy at densities above about $2ρ_0$, the lower radius boundary $R_{2.01}=12.2$ km from NICER's very recent observation of PSR J0740+6620 of mass $2.08\pm 0.07$ $M_{\odot}$ and radius $R=12.2-16.3$ km at 68\% confidence level sets a tight lower limit for nuclear symmetry energy at densities above $2ρ_0$, (5) Bayesian inferences of nuclear symmetry energy using models encapsulating a first-order hadron-quark phase transition from observables of canonical neutron stars indicate that the phase transition shift appreciably both the $L$ and $K_{\rm{sym}}$ to higher values but with larger uncertaintie , (6) The high-density behavior of nuclear symmetry energy affects significantly the minimum frequency necessary to rotationally support GW190814's secondary component of mass (2.50-2.67) $M_{\odot}$ as the fastest and most massive pulsar discovered so far.

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arXiv2021

Auxiliary Function Approach for Determining Symmetry Energy at Supra-saturation Densities

Nuclear symmetry energy $E_{\rm{sym}}(ρ)$ at density $ρ$ is normally expanded or simply parameterized as a function of $χ=(ρ-ρ_0)/3ρ_0$ in the form of $E_{\rm{sym}}(ρ)\approx S+Lχ+2^{-1}K_{\rm{sym}}χ^2+6^{-1}J_{\rm{sym}}χ^3+\cdots$ using its magnitude $S$, slope $L $, curvature $K_{\rm{sym}}$ and skewness $J_{\rm{sym}}$ at the saturation density $ρ_0$ of nuclear matter. Much progress has been made in recent years in constraining especially the $S$ and $L$ parameters using various terrestrial experiments and astrophysical observations. However, such kind of expansions/parameterizations do not converge at supra-saturation densities where $χ$ is not small enough, hindering an accurate determination of high-density $E_{\rm{sym}}(ρ)$ even if its characteristic parameters at $ρ_0$ are all well determined by experiments/observations. By expanding the $E_{\rm{sym}}(ρ)$ in terms of a properly chosen auxiliary function $Π_{\rm{sym}}(χ,Θ_{\rm{sym}})$ with a parameter $Θ_{\rm{sym}}$ fixed accurately by an experimental $E_{\rm{sym}}(ρ_{\rm{r}})$ value at a reference density $ρ_{\rm{r}}$, we show that the shortcomings of the $χ$-expansion can be completely removed or significantly reduced in determining the high-density behavior of $E_{\rm{sym}}(ρ)$. In particular, using two significantly different auxiliary functions, we show that the new approach effectively incorporates higher $χ$-order contributions and converges to the same $E_{\rm{sym}}(ρ)$ much faster than the conventional $χ$-expansion at densities $\lesssim3ρ_0$. Several quantitative demonstrations using Monte Carlo simulations are given.

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arXiv2021

Intrinsic Correlations Among Characteristics of Neutron-rich Matter Imposed by the Unbound Nature of Pure Neutron Matter

The unbound nature of pure neutron matter (PNM) requires intrinsic correlations between the symmetric nuclear matter (SNM) EOS parameters (incompressibility $K_0$, skewness $J_0$ and kurtosis $I_0$) and those (slope $L$, curvature $K_{\rm{sym}}$ and skewness $J_{\rm{sym}}$) characterizing the symmetry energy independent of any nuclear many-body theory. We investigate these intrinsic correlations and their applications in better constraining the poorly known high-density behavior of nuclear symmetry energy. Several novel correlations connecting the characteristics of SNM EOS with those of nuclear symmetry energy are found. In particular, at the lowest-order of approximations, the bulk parts of the slope $L$, curvature $K_{\rm{sym}}$ and skewness $J_{\rm{sym}}$ of the symmetry energy are found to be $L\approx K_0/3, K_{\rm{sym}}\approx LJ_0/2K_0$ and $J_{\rm{sym}}\approx I_0L/3K_0$, respectively. High-order corrections to these simple relations can be written in terms of the small ratios of high-order EOS parameters. The resulting intrinsic correlations among some of the EOS parameters reproduce very nicely their relations predicted by various microscopic nuclear many-body theories and phenomenological models constrained by available data of terrestrial experiments and astrophysical observations in the literature. The unbound nature of PNM is fundamental and the required intrinsic correlations among the EOS parameters characterizing both the SNM EOS and symmetry energy are universal. These intrinsic correlations provide a novel and model-independent tool not only for consistency checks but also for investigating the poorly known high-density properties of neutron-rich matter by using those with smaller uncertainties.

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arXiv2021

Bayesian inference of dense matter EOS encapsulating a first-order hadron-quark phase transition from observables of canonical neutron stars

[Purpose:] We infer the posterior probability distribution functions (PDFs) and correlations of nine parameters characterizing the EOS of dense neutron-rich matter encapsulating a first-order hadron-quark phase transition from the radius data of canonical NSs reported by LIGO/VIRGO, NICER and Chandra Collaborations. We also infer the quark matter (QM) mass fraction and its radius in a 1.4 M$_{\odot}$ NS and predict their values in more massive NSs. [Method:] Meta-modelings are used to generate both hadronic and QM EOSs in the Markov-Chain Monte Carlo sampling process within the Bayesian statistical framework. An explicitly isospin-dependent parametric EOS for the $npeμ$ matter in NSs at $β$ equilibrium is connected through the Maxwell construction to the QM EOS described by the constant speed of sound (CSS) model of Alford, Han and Prakash. [Results:] (1) The most probable values of the hadron-quark transition density $ρ_t/ρ_0$ and the relative energy density jump there $\De\ep/\ep_t$ are $ρ_t/ρ_0=1.6^{+1.2}_{-0.4}$ and $\De\ep/\ep_t=0.4^{+0.20}_{-0.15}$ at 68\% confidence level, respectively. The corresponding probability distribution of QM fraction in a 1.4 M$_{\odot}$ NS peaks around 0.9 in a 10 km sphere. Strongly correlated to the PDFs of $ρ_t$ and $\De\ep/\ep_t$, the PDF of the QM speed of sound squared $\cQMsq/c^2$ peaks at $0.95^{+0.05}_{-0.35}$, and the total probability of being less than 1/3 is very small. (2) The correlations between PDFs of hadronic and QM EOS parameters are very weak. [Conclusions:] The available astrophysical data considered together with all known EOS constraints from theories and terrestrial nuclear experiments prefer the formation of a large volume of QM even in canonical NSs.

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