Spherically Symmetric Potentials in Quadratic $f(R)$ Gravity

Abstract

We study the gravitational potential generated by static, spherically symmetric matter distributions in a quadratic $f(R)$ gravity model. In the weak-field regime, the linearized field equations lead to a fourth-order modified Poisson equation whose solutions contain Newtonian and Yukawa-type contributions. Imposing regularity at the origin and asymptotic flatness uniquely fixes the integration constants, yielding potentials fully determined by the mass density. Analytical expressions are derived for several classical profiles, including Plummer, Hernquist, and Navarro-Frenk-White (NFW), as well as for new analytic density models introduced in this work. The dependence on the quadratic gravity parameter $\alpha$ is analyzed, and the Newtonian limit of General Relativity is consistently recovered as $\alpha \to \infty$. As an application, circular velocity curves are computed and compared with the observed rotation curve of NGC 3198. A chi-squared analysis shows that the linearized quadratic $f(R)$ model provides improved fits relative to the Newtonian case in the inner and intermediate galactic regions $r \lesssim 30$ kpc, while predicting a decline at larger radii due to Yukawa suppression.