Screened Scalar Hair and the Weak-Lensing Separation of Black Holes from Neutron Stars in Quadratic $f(R)$ Gravity

Abstract

Quadratic $f(R)$ gravity carries a massive scalar degree of freedom, the scalaron, whose finite range $\lambda$ screens its influence on the geometry outside a compact object. We show that this screening severs the exterior of a black hole from that of a neutron star of the same mass. Correction to the Schwarzschild metric is Yukawa-suppressed, falling as $e^{-r/\lambda }$ in the coupling, and is thus invisible to any expansion in powers of that coupling; the exterior is moreover intrinsically isotropic, so that inverting the temporal potential alone, as one does for single-potential solutions, fails to solve the field equations. Applying the Gibbons & Werner method of the Gauss-Bonnet theorem to this geometry, we find the leading deflection angle to be exactly the general-relativistic, ($b$: impact parameter), scalaron contributions cancelling identically in the combination that bends light. First correction carries the unfamiliar signature ${\lambda }^{-1/2}{b}^{-3/2}{e}^{-b/\lambda }$, screened beyond the scalaron range and confirmed against direct quadrature to around 20% at $b=4\lambda$, improving to 7% at $b=12\lambda$. What survives is the ingredients that deliver it, namely, non-analyticity in the coupling, the intrinsically isotropic gauge, the closed-form screened correction and the pressure-weighted scalar charge, are obtained within a single, self-consistent derivation here. A static black hole carries no scalar hair, and lenses precisely as general relativity requires; a neutron star acquires a scalar charge weighted by the pressure supporting it against collapse, and does not. Weak lensing therefore closes as a discriminant of the theory, whilst the horizon, as against a material surface, remains one in principle. The observational advantage lies in the strong-field imaging of the photon sphere, and in the stellar interior, where the scalar charge is fixed by the equation of state.

arXiv comment9 pages, 4 figures