Curvature of the energy per particle in neutron stars

Neutron stars (NSs) serve as laboratories for probing strongly interacting matter at the most extreme densities and nearly vanishing temperature. Their inner cores are expected to be dense enough to host deconfined quark matter. Utilizing state-of-the-art theoretical and multimessenger constraints, we statistically determine the bulk properties of dense NS matter. We show that the speed of sound can be expressed in terms of the slope and curvature of the energy per particle. We demonstrate that the restoration of conformal symmetry requires changing the sign of the curvature of the bulk energy per particle as a function of energy density. Furthermore, we find that such a sign change is closely related to the peak in the speed of sound. We argue that the curvature of the energy per particle may serve as an approximate order parameter that signifies the onset of strongly coupled conformal matter in the NS core.

Figure 1

The curvature of the energy per particle $\beta$ as a function of energy density. Shown are results at $1\sigma$ (68%) and $2\sigma$ (95%) confidence intervals. Additionally, marked are $1\sigma$ and $2\sigma$ estimates of energy densities for centers of $1.4M_{\odot }$ NSs (green), the peak of the speed of sound (purple), vanishing curvature of the energy per particle (gray), and centers of maximally massive NSs (blue). The orange band at high densities shows the pQCD constraint. The horizontal, dashed black line marks $\beta =0$.

Figure 2

Curvature of the energy per particle $\beta$ and its decomposition [see Eq. (6) for details] in the parametrized model from Eq. (10). Solid lines indicate the limiting values in the conformally symmetric phase (see text). Gray, dashed lines mark $\beta =0$.

Figure 3

Comparison of physical quantities obtained in the parametrized model in Eq. (10). Solid lines indicate the limiting values in the conformally symmetric phase (see text). Gray, dashed lines mark $\beta =0$. Circles indicate criteria for identification of conformal matter, $\gamma =1.75$ [15], $d_c=0.2$ [45], $\beta =0$ (this Letter).

Figure 4

The speed of sound $c_s^2$, polytropic index $\gamma$, trace anomaly $\mathrm{\Delta }$, measure of conformality $d_c$ as functions of energy density. Shown are results at $1\sigma$ (68%) and $2\sigma$ (95%) confidence intervals. The gray region shows $1\sigma$ and $2\sigma$ estimates of energy density for vanishing curvature of the energy per particle, i.e., $\beta =0$.