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

TL;DR

This work develops and solves the weak-field, static, spherically symmetric problem in a quadratic $f(R)$ gravity model ($f(R)=R+\alpha R^2$), deriving a fourth-order modified Poisson equation whose solution decomposes into a Newtonian term plus Yukawa-type corrections with a characteristic screening length $\alpha^{-1}$. Imposing regularity at the origin and asymptotic flatness yields a unique potential $h_{00}(r)$ fully determined by the mass density, and explicit results are presented for several classical and novel spherical density profiles. The modified potentials are then employed to compute circular velocity curves, which are compared to the observed rotation curve of NGC 3198; a chi-squared analysis indicates improvements over Newtonian gravity in the inner to intermediate regions ($r\lesssim 30$ kpc), while velocities decline at larger radii due to Yukawa suppression. The analytic framework and density-model results provide a controlled means to test deviations from General Relativity at galactic scales and to explore the phenomenology of $f(R)$ gravity in astrophysical environments.

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 $α$ is analyzed, and the Newtonian limit of General Relativity is consistently recovered as $α\to \infty$. 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.

Figures (11)

• Figure 1: Modified gravitational potential $h_{00}(r)$ for a spherical shell of radius $R=1$ in units of $L$, for different values of the parameter $\alpha$ (in $L^{-1}$). The dashed curve represents the Newtonian limit ($\alpha\to\infty$), while solid curves correspond to finite $\alpha$, showing the transition from Yukawa-corrected to classical gravity. We set $M=1$ (in $L$).
• Figure 2: Modified gravitational potential $h_{00}(r)$ generated by a Gaussian mass distribution with width $\epsilon = 0.1$ (in units of $L$), shown for several values of the modified gravity parameter $\alpha$ (in $L^{-1}$). Solid curves correspond to finite $\alpha$, illustrating the Yukawa-type deviations induced by the quadratic $f(R)$ correction, while the dashed curve denotes the Newtonian limit $\alpha \to \infty$. The mass is fixed to $M = 1$ in units of $L$.
• Figure 3: Radial behavior of the modified gravitational potential $h_{00}(r)$ generated by a homogeneous sphere of radius $R = 1$ (in units of $L$) for several values of the parameter $\alpha$ (in $L^{-1}$). Solid curves illustrate the impact of finite-$\alpha$ corrections to the Newtonian potential, while the dashed curve corresponds to the General Relativity limit $\alpha \to \infty$. The mass density is fixed to $\rho_0 = 1$ (in $L^{-2}$).
• Figure 4: Modified gravitational potential $h_{00}(r)$ for the Plummer model, computed numerically varying $\alpha$. The dashed curve represents the Newtonian potential ($\alpha\to\infty$), while solid curves show $f(R)$-corrected profiles. $M=1$ (in $L$), $b=1$ (in $L$).
• Figure 5: Modified potential $h_{00}(r)$ for the Hernquist density, for some values of $\alpha$. The dashed curve represents the Newtonian potential. $\rho_0=1$ (in $L$) and $r_s=1$ (in $L$).