The sharpness of the quark-hadron transition and the properties of hybrid stars

TL;DR

The paper addresses how the sharpness of the hadron–quark transition in neutron stars affects the equation of state and observable properties. It introduces a Gaussian crossover between a hadronic EOS and a quark EOS, parameterized by $\delta_0$ (overpressure) and $\sigma$ (width), while enforcing causality and $M_{\rm max}>2\,M_{\odot}$. Across a range of $\delta_0$ and $\sigma$, the study finds that smoother transitions soften the EOS below the transition and stiffen it above, with the maximum mass remaining largely unchanged but the radius and tidal deformability decreasing for higher-mass stars; the speed of sound shows diverse behavior, including dips, peaks, and oscillations. The results are robust across hadronic/quark models and align with expectations from microscopic quark–hadron pasta phase analyses, implying general implications for interpreting NICER and GW observations. The framework provides a general, model-independent lens for linking transition sharpness to neutron-star structure and could accommodate hyperons or other degrees of freedom in future extensions.

Abstract

We investigate the effects of the sharpness of the phase transition between hadronic matter and quark matter on various properties of neutron stars. We construct hybrid equations of state by combining a hadronic model with a quark model using a Gaussian function. This approach introduces a smooth transition characterized by two parameters: one representing the overpressure relative to the first-order phase transition point, and the other related to the range over which the hybrid region extends in baryon chemical potential. We find that the sharpness of the phase transition significantly influences the equation of state, which can deviate by several tens of $\text{MeV fm}^{-3}$ from the one with a sharp first-order transition. The speed of sound exhibits diverse behaviors, including drastic drops, pronounced peaks, and oscillatory patterns, depending on the sharpness parameters. In terms of stellar structure, while the maximum neutron star mass remains largely unaffected by the sharpness of the phase transition, the stellar radii can vary significantly. Smoother transitions lead to a leftward shift (up to 1 km) of the mass-radius curve segment corresponding to hybrid stars. The tidal deformability decreases with smoother transitions, especially for higher-mass stars. Our results are quite general and do not qualitatively depend on the specific hadronic and quark matter models employed. In fact, the hybrid equation of state and stellar properties derived from microscopic models of quark-hadron pasta phases display the same behavior as described above.

Figures (5)

• Figure 1: Physical interpretation of $\delta(\mu_B)$ and $\lambda(\mu_B)$. The curves were obtained with the Soft hadronic EOS and the vMIT EOS with $B=105 ~ \text{MeV fm}^{-3}$ and $G_V = 0.18 ~ \mathrm{fm^2}$. Both EOSs intersect at $\mu_c = 1401 ~\mathrm{MeV}$ in the $P$ versus $\mu_B$ plane. The dotted line represents the crossover/mixed EOS with parameters $\delta_0 = 5 ~ \text{MeV fm}^{-3}$ and $\sigma = 100 ~ \mathrm{MeV}$. The maximum of $\delta(\mu_B)$ occurs at $\mu_c$.
• Figure 2: The results were obtained by using the Soft hadronic EOS and the vMIT EOS with $B=105 ~ \text{MeV fm}^{-3}$ and $G_V = 0.18 ~ \mathrm{fm^2}$. In all cases we used $\mu_c = 1401 ~\mathrm{MeV}$, which is the baryon chemical potential at which the pressure of the pure phases intersect. For the parameters ($\delta_0 [\text{MeV fm}^{-3}]$, $\sigma [ \mathrm{MeV}]$) we used: (2, 50), (5, 100) and (10,150). We also show the sharp (first-order) phase transition.
• Figure 3: Same as in the previous figure but using the Stiff hadronic EOS and the vMIT EOS with $B=100 ~ \text{MeV fm}^{-3}$ and $G_V = 0.21 ~ \mathrm{fm^2}$. In this case we obtain $\mu_c = 1378 ~\mathrm{MeV}$.
• Figure 4: Mass-radius relation for the EOSs shown in Figs. \ref{['fig2']} and \ref{['fig3']}: (a) Model 1 and (b) Model 2. Current astrophysical constraints are also included for reference (details provided in the text).
• Figure 5: Dimensionless tidal deformability as a function of stellar mass for the EOSs shown in Figs. \ref{['fig2']} and \ref{['fig3']}: (a) Model 1 and (b) Model 2. The vertical bar represents observational data from the GW170817 event detected by LIGO/Virgo, as referenced in the text.