Self-Bound Anisotropic Stars in $\mathcal{R}+α\mathcal{R}^2$ Gravity with a Hayward Exterior: Junction Conditions and NICER Constraints

Abstract

Analytic compact-star models in modified gravity usually borrow the junction conditions of general relativity and fix their constants with radii derived for a particular stellar model. Here we avoid both. We study $\mathcal{F}(\mathcal{R})=\mathcal{R}+α\mathcal{R}^{2}$ gravity with a Durgapal-Bannerji interior of embedding class one (Karmarkar condition) and a Hayward exterior. Without Birkhoff's theorem, the exterior is an assumption, and we measure its effect with the Hayward metric, which deforms Schwarzschild by a length $\ell$. The junction conditions of quadratic $\mathcal{F}(\mathcal{R})$ gravity fix all constants in closed form in terms of $M$, $R$ and $\ell$. The condition $p_r(R)=0$ is replaced by continuity of the trace of the extrinsic curvature. The Ricci scalar jumps at the surface, so there is a thin shell with negative surface energy density for $α>0$, a gravitational double layer, and a small radial stress. For this stress to be below 1% of the central pressure we need $α\lesssim0.2$ km$^{2}$ and $\ell\lesssim0.3M$. The interior behaves as a self-bound star. A single surface energy density, $ρ_s=(1.37\pm0.16)ρ_0$, reproduces PSR J0740+6620, J0437-4715, J0030+0451 and J0614-3329, and for $α=0.1$ km$^{2}$ the model predicts $R_{1.4}=11.5$ km and a maximum mass of $2.9 M_\odot$. Because $\ell$ enters surface observables only at order $(\ell/M)^{2}(M/R)^{3}$, the results carry over to a Schwarzschild exterior. Calibrating instead with strange-star radii of eclipsing X-ray pulsars gives $R_{1.4}\approx9$ km, which current data rule out.