Evidence against a strong first-order phase transition in neutron star cores: Impact of new data

With the aim of exploring the evidence for or against phase transitions in cold and dense baryonic matter, the inference of the sound speed and equation of state for dense matter in neutron stars is extended in view of recent new observational data. The impact of the heavy ($2.35M_{\odot }$) black-widow pulsar PSR J0952-0607 and the unusually light supernova remnant HESS J1731-347 is inspected. In addition, a detailed reanalysis is performed of the low-density constraint based on chiral effective field theory and of the perturbative QCD constraint at asymptotically high densities, in order to clarify the influence of these constraints on the inference procedure. The trace anomaly measure, $\mathrm{\Delta }=1/3-P/ϵ$, is also computed and discussed. A systematic Bayes factor assessment quantifies the evidence (or nonevidence) of low averaged sound speeds $(c_s^2\le 0.1)$, a prerequisite for a phase transition, within the range of densities realized in the core of neutron stars. One of the consequences of including PSR J0952-0607 in the database is a further stiffening of the equation of state, resulting for a 2.1 solar-mass neutron star in a reduced central density of less than 5 times the equilibrium density of normal nuclear matter at the 68% level. The evidence against small sound speeds in neutron star cores is further strengthened. Within the inferred 68% posterior credible bands, only a weak first-order phase transition with a coexistence density interval $\mathrm{\Delta }n/n\lesssim 0.2$ would be compatible with the observed data.

Figure 1

Marginal prior probability distributions at the 95% and 68% levels of the squared speed of sound $c_s^2$ as a function of energy density $ϵ$. At each $ϵ$, there exist 95% and 68% prior credible intervals for $c_s^2(ϵ)$. These intervals are connected to obtain the prior credible bands. The solid line represents the median of the prior distribution. The dashed black line indicates the value $c_s^2=1/3$ of the conformal bound.

Figure 2

Marginal posterior probability distributions of the maximum neutron star mass $M_{\max }$. The nomenclature “Previous” refers to the Shapiro time delay, NICER, ChEFT, pQCD, and GW data listed in Table 1. The shift induced by adding the (nonrotating) mass information from the BW pulsar PSR J0952-0607 is also shown. The prior distribution is nearly uniform over a wide mass range.

Figure 3

Marginal posterior probability distributions at the 95% and 68% levels: squared speed of sound $c_s^2$ and pressure $P$ as a function of energy density $ϵ$, inferred from the data set “$\mathrm{Previous}+\mathrm{BW}$” (see Table 1). Also shown are the marginal posterior probability distributions for the mass-radius relation and the tidal deformability, $\mathrm{\Lambda }$, as a function of neutron star mass $M$ in units of the solar mass $M_{\odot }$. At each $ϵ$ or $M$, there exist 95% and 68% posterior credible intervals for $c_s^2(ϵ)$, $P(ϵ)$ or $R(M)$, $\mathrm{\Lambda }(M)$. These intervals are connected to obtain the posterior credible bands. Similarly, the medians of the posterior probability distributions at each $ϵ$ or $M$ are connected (solid lines). For squared speed of sound or pressure the dashed black lines indicate the value of the conformal limit or represent the APR EoS [44]. In the upper two figures bars mark the 68% credible intervals of the central energy densities of neutron stars with masses $M=1.4M_{\odot }$ and $2.3M_{\odot }$, respectively. The mass-radius relation is compared to the marginalized intervals at the 68% level from the NICER data analyses by Riley et al. (black) [7, 8] of PSR $\mathrm{J}0030+0451$ and PSR $\mathrm{J}0740+6620$. In addition the 68% mass-radius credible intervals of the thermonuclear burster 4U 1702-429 [79] are displayed as well as the 68% credible interval of $R(1.4M_{\odot })$ extracted from quiescent low-mass x-ray binaries [78] (blue), both of which are not included in the Bayesian analysis. $\mathrm{\Lambda }(M)$ is compared to the masses and tidal deformabilities inferred in Ref. [81] for the two neutron stars in the merger event GW170817 at the 90% level (black) as well as $\mathrm{\Lambda }(1.4M_{\odot })$ at the 90% level extracted from GW170817 [82] (blue).

Figure 4

Density profiles of neutron stars with masses of $M=1.4M_{\odot }$ and $M=2.1M_{\odot }$. The employed equation of state corresponds to the median of the credible band in Fig. 3, i.e., using the previously available and new data from the BW pulsar. The bars indicate the 68% credible intervals of the central densities and radii of neutron stars with mass $M=1.4M_{\odot }$ (blue) and $2.1M_{\odot }$ (orange), as listed in Table 2.

Figure 5

Similar to Fig. 3: posterior credible bands are displayed for the squared speed of sound, $c_s^2$, as a function of energy density $ϵ$, and the mass-radius relation $R(M)$, but now using only the “Previous” data in Table 1 without inclusion of the new PSR J0952-0607 (BW) information.

Figure 6

Maximum possible phase coexistence interval $(\mathrm{\Delta }n/n{)}_{\max }$ of constant pressure (where $n$ is the density at which the interval starts) extracted from the 68% and 95% posterior credible bands of $P(n)$. $(\mathrm{\Delta }n/n{)}_{\max }$ is displayed as a function of baryon density $n$ in units of the nuclear saturation density, $n_0=0.16{\mathrm{fm}}^{-3}$.

Figure 7

Bayes factors ${\mathcal{B}}_{c_{s,\min }^2\le 0.1}^{c_{s,\min }^2>0.1}$ comparing EoS samples with the following competing scenarios: (a) minimum squared speed of sound (following a maximum), with $c_{s,\min }^2$ larger than 0.1, excluding a strong first-order phase transition with a Maxwell construction, versus (b) EoS samples with $c_{s,\min }^2\le 0.1$. The Bayes factors are calculated for a given maximum neutron star mass $M$, i.e., the minimum speed of sound up to the corresponding mass is used. For illustration, the evidence classification from Refs. [57, 58] is indicated by dashed grey lines and annotated on the right-hand side.

Figure 8

Posterior 95% and 68% credible bands and medians for the trace anomaly measure $\mathrm{\Delta }=1/3-P/ϵ$ as a function of energy density $ϵ$.

Figure 9

Posterior credible bands for the squared speed of sound, $c_s^2$, as a function of energy density $ϵ$. The inference includes previous data as well as the new information from the BW pulsar. The low-density behavior of the posterior credible bands is compared to the N3LO ChEFT results from Refs. [34, 35] (grey) up to $300\mathrm{MeV}{\mathrm{fm}}^{-3}$.

Figure 10

Similar to Fig. 3: posterior credible bands for the squared speed of sound, $c_s^2$, as a function of energy density $ϵ$, based on the previous data as well as the new information from the BW pulsar. Here, however, the integral pQCD likelihood is implemented, as in Refs. [17, 37, 38], at $n_{\mathrm{NS}}=10n_0$ instead of ${ϵ}_{\mathrm{NS}}={ϵ}_{c,\max }$.

Figure 11

Similar to Fig. 3: posterior credible bands for the squared speed of sound, $c_s^2$, as a function of energy density $ϵ$, and mass-radius relation $R(M)$, but now including the information from the supernova remnant HESS J1731-34 in addition to the BW pulsar PSR J0952-0607. The resulting mass-radius relation is also compared to the marginalized intervals at the 68% level from the analysis of the low-mass supernova remnant HESS J1731-34 [40] (green).

Figure 12

PSR J0952-0607 mass measurement with 68% uncertainty [39] compared to the static case (without the $\nu \simeq 709\mathrm{Hz}$ rotation) computed using the prescription in Ref. [64].

Figure 13

Posterior 95% and 68% credible bands and medians for the baryon chemical potential $\mu$ as a function of baryon density $n$ in units of the nuclear saturation density, $n_0=0.16{\mathrm{fm}}^{-3}$.