Buchdahl limit of compact stars in presence of Weyl anomaly

TL;DR

The paper investigates how the Weyl anomaly trace affects the interior structure and maximum compactness of relativistic stars. The authors develop a static, spherically symmetric anisotropic star model with Weyl anomaly, enforce conformal flatness and trace cancellation to obtain an exact interior solution, and derive explicit metric functions, a vacuum-corrected mass function, and a Tolman-Oppenheimer-Volkoff equation for hydrostatic equilibrium. They match the interior solution to an exterior Schwarzschild-like vacuum, parameterizing the model by the anomaly parameter $β$ and the stellar compactness $C=\frac{2GM}{c^2 \mathcal{R}}$, and derive a modified Buchdahl bound depending on these quantities. Using the pulsar PSR J0740+6620 data from NICER and XMM-Newton, they constrain $β$ to negative values and obtain two branches $β\approx -22.3$ and $β\approx -198.5$ that yield MR relations in better agreement with the data than GR, though the model struggles to fit a low-mass pulsar without invoking non-minimal coupling. Overall, the work demonstrates that Weyl anomaly effects can quantitatively alter compact-star structure and provides observational tests of semi-classical gravity in neutron-star mass-radius observations.

Abstract

We setup an anisotropic compact star model in presence of Weyl ``trace" anomaly. We derive an exact interior solution which determines the contribution of the vacuum trace anomaly. We introduce a dimentionless parameter, $β$, to characterize this contribution. Applying appropriate matching conditions with the exterior solution, we determine the model parameters in terms of the Weyl anomaly parameter $β$ and the compactness parameter, $C=\frac{2GM}{c^2 \mathcal{R}}$ where $M$ and $\mathcal{R}$ are the mass and the radius of the star. We investigate the parameter space $\{β, C\}$ and the corresponding modifications of Buchdahl limit on the maximum compactness. We use astrophysical observations of mass and radius of the pulsar PSR J0740+6620 to constrain the Weyl anomaly parameter $β$. Also, we investigate the Mass-Radius diagram with other observational constraints from NICER and LIGO/Virgo collaboration.

Figures (3)

• Figure 1: \ref{['fig:fig1a']} The parameter space $\{\beta, C\}$ associated to Buchdahl limit, $p_r(0)\to \pm \infty$, of Weyl anomaly. For the GR case ($\beta=0$), the maximum compactness $C<\frac{8}{9}$ as indicated by the dotted horizontal line. For $\beta\neq 0$ the maximum compactness is given by the upper bound of the unshaded regions. Notably the maximum compactness reaches the BH limit (dash-dotted horizontal line) $C\to 1$ as $\beta\to -1$. At large $|\beta|$, the classical GR Buchdahl limit (dotted horizontal line) is recovered, but $\beta>0$ region is excluded. \ref{['fig:fig1b']} Buchdahl limit of the stellar compactness, at which $p_r$ diverges, is determined by the inequality $\zeta>0$, namely \ref{['eq:Buchdahl_ineq']}. At $\beta=0$ the maximum compactness of the GR theory $C\to 8/9$ is recovered. At $\beta\to -1$, the maximum compactness reaches the BH limit $C\to 1$. At large $|\beta|$ in the negative region, Buchdahl limit reduces to the GR limit $C\to 8/9^+$.
• Figure 2: The astrophysical constraints on the radius and the mass of the pulsar PSR J0740+6620 from updated NICER and XMM-Newton data $\mathcal{R}=12.49^{+1.28}_{-0.88}$ km and $M=2.073^{+0.069}_{-0.069}M_\odot$Salmi:2024aum. The Weyl anomaly with $\beta=-198.459$ (associated to \ref{['eq:const1']}) and $\beta=-22.299$ (associated to \ref{['eq:const2']}) fits the observation very well better than GR ($\beta=0$).
• Figure 3: MR diagram. We use the boundary condition that the surface density $\rho(r=\mathcal{R})=\rho_\text{S}=1.4 \rho_\text{sat}$, where the exclusion limits due to different physical conditions are given by shaded regions. \ref{['fig:beta1']} For $\beta=-198.459$, the curve does not cross Buchdahl limit where the maximum mass $M=7.24~M_\odot$. \ref{['fig:beta2']} For $\beta=-22.299$, the curve does not cross Buchdahl limit where the maximum mass $M=5.86~M_\odot$. \ref{['fig:NICER']} The MR curves for $\beta=-198.459$ and $\beta=-22.299$ with observational constraints from NICER and LIGO/Virgo.