Unified nonparametric equation-of-state inference from the neutron-star crust to perturbative-QCD densities

Abstract

Perturbative quantum chromodynamics (pQCD), while valid only at densities exceeding those found in the cores of neutron stars, could provide constraints on the dense-matter equation of state (EOS). In this work, we examine the impact of pQCD information on the inference of the EOS using a nonparametric framework based on Gaussian processes (GPs). We examine the application of pQCD constraints through a "pQCD likelihood," and verify the findings of previous works; namely, a softening of the EOS at the central densities of the most massive neutron stars and a reduction in the maximum neutron-star mass. Although the pQCD likelihood can be easily integrated into existing EOS inference frameworks, this approach requires an arbitrary selection of the density at which the constraints are applied. The EOS behavior is also treated differently on either side of the chosen density. To mitigate these issues, we extend the EOS model to higher densities, thereby constructing a "unified" description of the EOS from the neutron-star crust to densities relevant for pQCD. In this approach the pQCD constraints effectively become part of the prior. Since the EOS is unconstrained by any calculation or data between the densities applicable to neutron stars and pQCD, we argue for maximum modeling flexibility in that regime. We compare the unified EOS with the traditional pQCD likelihood, and although we confirm the EOS softening, we do not see a reduction in the maximum neutron-star mass or any impact on macroscopic observables. Though residual model dependence cannot be ruled out, we find that pQCD suggests the speed of sound in the densest neutron-star cores has already started decreasing toward the asymptotic limit; we find that the speed of sound squared at the center of the most massive neutron star has an upper bound of $\sim 0.5$ at the $90\%$ level.

Figures (16)

• Figure 1: Pressure vs chemical potential from the pQCD expansion of Eq. \ref{['eq:pqcd_expansion']} (bottom panel), and the corresponding relative uncertainty on the pressure (top panel) when $X$ is varied between $1/2$ and 2. In the top panel we also indicate the relative uncertainty in the pressure from a $\chi$EFT calculation. As in Ref. Komoltsev:2021jzg, for the $\chi$EFT uncertainty, we take the difference in the "soft" and "stiff" EOSs at $1.1\,n_\mathrm{sat}$ from Ref. Hebeler:2013nza. At around a chemical potential of $2.6\,\mathrm{GeV}$ the relative uncertainty from the pQCD calculation is equal to that of the $\chi$EFT calculation.
• Figure 2: Given a pre-existing candidate EOS (black line) up to some termination point $p=p_\mathrm{T}$ (black dot), the "maximized" pQCD constraint asks if it is possible to connect that EOS to the pQCD prediction at a single value $p=p_\mathrm{H}$ (red dot). This is done by considering EOSs that minimize (blue) and maximize (red) the pressure change from the termination point and checking if $\Delta p_\mathrm{min} \leq p_\mathrm{H} - p_\mathrm{T} \leq \Delta p_\mathrm{max}$ (this is easily visualized in the $\mu$--$n$ plane, since the change in pressure corresponds to the area under the EOS). If this condition is true, then the EOS is given a likelihood of 1, else it is given a likelihood of 0.
• Figure 3: Visualization of the "maximized" (left) and "modeled" (right) pQCD likelihoods with $n_\mathrm{T} = 10\,n_\mathrm{sat}$. (Left panel) For a particular choice of the renormalization scale $X$ and $\mu_\mathrm{H}$ (dots), the maximized pQCD likelihood, Eq. \ref{['eq:maximized_likelihood']}, defines an allowed region in the $\epsilon$--$p$ plane for the EOS at $n_\mathrm{T}$. We show this region for $X = 1/2$ (red) and $X = 2$ (blue), both with $\mu_\mathrm{H} = 2.6\,\mathrm{GeV}$. Marginalizing over X with a log-uniform distribution between $1/2$ and $2$, we obtain the black shaded region with darker regions having a higher likelihood. (Right panel) The modeled likelihood instead uses EOS extensions that start at the pQCD high-density prediction (whilst taking into account uncertainty on $X$) and are propagated to lower densities. A sample of these EOSs are shown in pink, taken from the "conditioned" GP of Ref. Komoltsev:2023zor (which also imposes $c^2_{s} \rightarrow 1/3$ at densities above $25\,n_\mathrm{sat}$). The point where these EOS extensions reach $n_\mathrm{T}$ is identified, and a KDE is built on the corresponding pressure and energy-density values, shown with the shaded black region.
• Figure 4: Posteriors for the EOS pressure and energy density (90% credible intervals) from applying the maximized (left) and modeled (right) pQCD constraints at $n_\mathrm{T}=10\,n_\mathrm{sat}$. Posteriors are truncated at that value, beyond which the EOS model changes. The energy density and pressure of individual EOSs at $10\,n_\mathrm{sat}$ are highlighted with black markers. The prior is shown in light gray; it contains no astrophysical input. These results demonstrate which EOSs are being ruled out by pQCD alone. (Left panel) As in Fig. \ref{['fig:max_vs_marg']}, we consider the maximized constraint at $X=1/2$ (red), 2 (blue), and marginalized over $X$ (black dashed). We also show the $X=1/2$ and $X=2$ allowed regions for the maximized likelihood from Fig. \ref{['fig:max_vs_marg']} (thin red and blue lines). To satisfy the maximized pQCD likelihood, the EOS at $10\,n_\mathrm{sat}$ must be within the allowed region. (Right panel) As in Fig. \ref{['fig:max_vs_marg']}, we include the modeled pQCD likelihood represented as a KDE. Only EOSs which overlap with this KDE at $10\,n_\mathrm{sat}$ have non-zero weight.
• Figure 5: Similar to Fig. \ref{['fig:pqcd_posterior']}, but for other choices of the termination density $n_\mathrm{T}$. We show the prior (gray shaded region) and posteriors for different values of $n_\mathrm{T}$ (pink, blue, and black) for the maximized (left) and modeled (right) constraint. For the maximized constraint we have marginalized over $X$. This comparison quantifies the impact of $n_\mathrm{T}$ on the posterior, i.e., the value of the density at which we switch from the low-density GP EOS model to the high-density EOS that is tailored to the pQCD limit.