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

Background: The remarkable progress in recent multimessenger observations of both isolated neutron stars (NSs) and their mergers has provided some of the much needed data to improve our understanding about the equation of state (EOS) of dense neutron-rich matter. Various EOSs with or without some kinds of phase transitions from hadronic to quark matter (QM) have been widely used in many forward modelings of NS properties. Direct comparisons of these predictions with observational data sometimes also using χ² minimizations have provided very useful constraints on the model EOSs. However, it is normally difficult to perform uncertain quantifications and analyze correlations of the EOS model parameters involved in forward modelings especially when the available data are still very limited.

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 QM mass fraction and its radius in a 1.4 M_⊙ NS and predict their values in more massive NSs.

Method: Metamodelings 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 n p e μ 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 with and without using the Seidov stability condition for first-order phase transitions.

Results: In the default calculation with the Seidov stability condition, we find that (i) The most probable values of the hadron-quark transition density ρ_t/ρ₀ and the relative energy density jump there Δ ɛ /ɛ_t are ρ_t/ρ₀=1 .6_-0.4^+1.2 and Δ ɛ /ɛ_t=0 .4_-0.15^+0.20 at 68% confidence level, respectively. The corresponding probability distribution of QM fraction in a 1.4 M_⊙ NS peaks around 0.9 in a 10 km sphere. Strongly correlated to the PDFs of ρ_t and Δ ɛ /ɛ_t , the PDF of the QM speed of sound squared c_QM²/c² peaks at 0 .95_-0.35^+0.05 , and the total probability of being less than 1/3 is very small. (ii) The correlations between PDFs of hadronic and QM EOS parameters are very weak. While the most probable values of parameters describing the EOS of symmetric nuclear matter remain almost unchanged, the high-density symmetry energy parameters of neutron-rich matter are significantly different with or without considering the hadron-quark phase transition. Removing the Seidov condition, while there are appreciable and interesting changes in the PDFs of quark matter EOS parameters, the qualitative conclusions remain the same.

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.