How Perturbative QCD Constrains the Equation of State at Neutron-Star Densities

We demonstrate in a general and analytic way how high-density information about the equation of state (EOS) of strongly interacting matter obtained using perturbative quantum chromodynamics constrains the same EOS at densities reachable in physical neutron stars. Our approach is based on utilizing the full information of the thermodynamic potentials at the high-density limit together with thermodynamic stability and causality. This requires considering the pressure as a function of chemical potential $p(\mu )$ instead of the commonly used pressure as a function of energy density $p(\epsilon )$. The results can be used to propagate the perturbative quantum chromodynamics calculations reliable around $40n_s$ to lower densities in the most conservative way possible. We constrain the EOS starting from only a few times the nuclear saturation density $n\gtrsim 2.2n_s$, and at $n=5n_s$ we exclude at least 65% of otherwise allowed area in the $\epsilon \text{−}p$ plane. This provides information complementary to astrophysical observations that should be taken into account in any complete statistical inference study of the EOS. These purely theoretical results are independent of astrophysical neutron-star input, and hence, they can also be used to test theories of modified gravity and beyond the standard model physics in neutron stars.

Figure 1

Theoretical constraints to $\epsilon$ and $p$ arising from low-density (CET) and high-density input (PQCD). The green lines show the envelope of allowed values of $p(\epsilon )$; dashed green lines arise from trivially imposing causality of the EOS, while the thick solid lines correspond to newly identified integral constraints arising from imposing simultaneously the high-density limit for $p$, $n$, and $\mu$. The blue regions correspond to allowed values $\epsilon$ and $p$ at fixed density $n$; the red regions would be otherwise allowed but are excluded by the high-density input. At $n=5n_s$, 75% of otherwise allowed values of $\epsilon$ and $p$ are excluded by the high-density input.

Figure 2

Density as a function of the chemical potential determines the EOS. At any given point $\{\mu ,n\}$, causality imposes a minimal slope to any EOS passing that point ${\partial }_{\mu }n(\mu )\ge (n/\mu )$ denoted by the arrows. Thermodynamic consistency demands that $n(\mu )$ is a monotonic function, and that the area under the curve in the plot is given by $\mathrm{\Delta }p$. These conditions cannot be simultaneously fulfilled by any EOS that passes to the red area at any density and are therefore excluded. The lines $\mathrm{\Delta }{p}_{\min }$ and $\mathrm{\Delta }{p}_{\max }$ display the constructions defined in Eqs. (5) and (8).

Figure 3

3D rendering of the constraints in the $\mu \text{−}n\text{−}p$ phase space. Each triangle is a slice of the $p\text{−}n$ constraints for fixed $\mu$. Note that $n_{\max }$, $n_{\min }$, and $n_c$, which are defined in Eqs. (6), (9), and (11), respectively, are projection on the $\mu \text{−}n$ plane of the lines denoted with the same names in the figure above.

Figure 4

Constraints to $p(\epsilon )$ for different limiting values of the sound speed. The green envelope corresponds to $c_s^2=1$, while the blue envelope corresponds to the conformal limit $c_s^2=1/3$. At the specific value of the sound speed $(c_s^2{)}_{\lim }^{\min }$, the allowed region degenerates into a single line.

Figure 5

The expected impact of future calculations on the theoretical constraints to $\epsilon$ and $p$ at NS densities obtained by extrapolating the CET and PQCD calculations to intermediate densities. Numbers in square brackets show the end point of extrapolation of the CET and PQCD calculations.