Is The Trace Anomaly at its Minimum Value at Neutron Star Centers?

TL;DR

This work analyzes the near-center behavior of the EOS-parameter $φ = P/ε$ and the trace anomaly $Δ = 1/3 - φ$ in neutron-star cores using the EOS-independent IPAD-TOV framework. It demonstrates that $φ$ decreases monotonically outward from the center, with central behavior governed by the dimensionless central parameter $X = φ_c$ and robust against the details of the NS EOS. A local minimum in $φ$ is shown to be incompatible with a central peak in the speed-of-sound $s^2$, as encoded by the trace-anomaly relation; thus valley-then-peak structures in $φ$ are disfavored under current observations. The paper further discusses how high-density pQCD constraints can induce a central bump in massive NSs and introduces the complementary mean stiffness $Φ$ to connect the center-to-surface stiffness profile to global NS structure, providing EOS-independent constraints that tie nuclear physics, astrophysics, and QCD together.

Abstract

While the equation of state (EOS) $P(\varepsilon)$ of neutron star (NS) matter has been extensively studied, the EOS-parameter $φ= P/\varepsilon$ or equivalently the dimensionless trace anomaly $Δ= 1/3 - φ$, which quantifies the balance between pressure $P$ and energy density $\varepsilon$, remains far less explored, especially in NS cores. Its bounds and density profile carry crucial information about the nature of superdense matter. Physically, the EOS-parameter $φ$ represents the mean stiffness of matter accumulated from the stellar surface up to a given density. Based on the intrinsic structure of the Tolman--Oppenheimer--Volkoff equations, we show that $φ$ decreases monotonically outward from the NS center, independent of any specific input NS EOS model. Furthermore, observational evidence of a peak in the speed-of-sound squared (SSS) density-profile near the center effectively rules out a valley and a subsequent peak in the radial profile of $φ$ at similar densities, reinforcing its monotonic decrease. These model-independent relations impose strong constraints on the near-center behavior of the EOS-parameter $φ$, particularly demonstrating that the mean stiffness (or equivalently $Δ$) reaches a local maximum (minimum) at the center.

Figures (8)

• Figure 1: (Color Online). Three possible patterns of the EOS-parameter $\phi$ near NS centers. Panel (b) is excluded by the expansion of $\phi$ in $\widehat{r}$ around the center, while the pattern in panel (c) is incompatible with the presence of a peak in $s^2$ near $\widehat{r}=0$ since a valley in $\phi$ will generate a corresponding valley in $s^2$ at a similar energy density, see the analysis of Section \ref{['SEC_4']}. At the center, $\phi$ approaches $\mathrm{X}$ which is $\lesssim0.374$ to remain casual in NSs at the TOV configuration.
• Figure 2: (Color Online). The $\widehat{r}$-dependence of $\phi$ to order $\widehat{r}^4$ as $\phi\approx \mathrm{X}+\langle \varphi_2\rangle\widehat{r}^2+\langle \varphi_4\rangle \widehat{r}^4$. Inset shows the $\mathrm{X}$-dependence of $\langle\varphi_2\rangle$, $\langle\varphi_4\rangle$, $\widehat{r}_{\rm{vl}}^{\phi,2}/10$ and $a_4^{\rm{vl},\phi}/10$ (see Eq. (\ref{['def-a40']})). Although $\langle\varphi_4\rangle$ is positive, the quartic term $\langle\varphi_4\rangle\widehat{r}^4$ can hardly change the sign of $\phi-\mathrm{X}$ (near the center) due to the overall smallness of $\widehat{r}$. See the text for details of these quantities.
• Figure 3: (Color Online). The same as FIG. \ref{['fig_phi_aver']} but for $s^2-\phi$ to order $\widehat{r}^4$ as $s^2-\phi\approx s_{\rm{c}}^2-\mathrm{X}+\langle h_2\rangle\widehat{r}^2+\langle h_4\rangle \widehat{r}^4$ under the assumption that $\langle a_4\rangle\approx\langle a_6\rangle\approx0$. The inset shows the $\mathrm{X}$-dependence of $\langle h_2\rangle$ and $\langle h_4\rangle$, from which one can find although $\langle h_2\rangle$ is negative the higher order term $\langle h_4\rangle$ is positive.
• Figure 4: (Color Online). The requirement that there exists a local minimum in the EOS-parameter $\phi$ near NS centers (bisque hatched region) is basically incompatible with the requirement that a peaked $s^2$ exists near NS centers (lavender hatched region); indicated by the fact that there is no overlapped region for $a_4$ parameter from them (white band). The grey dotted line marks the position of $\mathrm{X}\approx0.374$. Two insets sketch the valley structure of the EOS-parameter $\phi$ and the peaked SSS $s^2$ near the center, respectively.
• Figure 5: (Color Online). A local minimum (valley) in $\phi$ within NS densities can be used to infer the existence of a valley in $s^2$ at a similar energy density (upper line), because $\widehat{\varepsilon}_{\phi}^{\rm{vl}}\lesssim1$ implies $\widehat{\varepsilon}_{\star}^{\rm{vl}}\lesssim2/3<1$; the inverse, however, does not necessarily hold, since $\widehat{\varepsilon}_{\phi}^{\rm{pk}} \approx3\widehat{\varepsilon}_{\star}^{\rm{pk}}/2$ may exceed unity even if $\widehat{\varepsilon}_{\star}^{\rm{pk}}\lesssim1$.