Solar system tests and neutron stars in $f(R)$ gravity revisited

TL;DR

This work performs a fully nonlinear, nonperturbative analysis of metric f(R) gravity in static, spherically symmetric spacetimes to test solar-system chameleon screening for cosmologically motivated models and to assess neutron-star structure in a purely quadratic modification f(R)=R+aR^2. The framework yields second-order equations for the Ricci scalar and the metric, enabling a direct comparison of interior and exterior solutions without conformal transformations. The solar-system results indicate numerical obstacles due to extreme density contrasts, leaving the presence of screening inconclusive in the tested models, while the neutron-star analysis shows that R-squared gravity can produce NS masses up to ≈2.34 M⊙ with realistic equations of state, potentially matching the heaviest observed NS without exotic EOS. These findings highlight the complementary roles of nonperturbative methods across weak- and strong-field regimes and stress the need to consider solar-system, binary-pulsar, and gravitational-wave constraints in evaluating the viability of f(R) theories.

Abstract

By implementing a full non-linear treatment of $f(R)$ gravity in static and spherically symmetric spacetimes, we analyze two scenarios. The first one within the context of the solar-system tests where we try to recover the chameleon effects without any approximations in the equations (e.g. linearization) from $f(R)$ models that are compatible with cosmology. The second scenario deals with a quadratic $f(R)$ model that is tested in neutron stars. This scenario, which is associated with strong gravity, is completely independent from the first one, but exploits the fact that the equations and formalism are basically the same in both applications. The difference between the two goals lies mainly in the values of the constants involved in the specific $f(R)$ models and the equation of state (EOS) of the central object (Sun or neutron star), but the numerical techniques and the general form of the field equations remain valid in both situations. For the neutron star problem we employ for the first time and in the context of $f(R)$ gravity a multiple algebraic polytropic EOS that mimics accurately realistic EOS in several density ranges. By doing so we avoid the numerical interpolation needed when a realistic EOS is given in tabulated form. Furthermore, we compare our results with the latest data, which includes the most massive neutron star known to date of about $2.35 M_\odot$ from PSRJ0952-0607.

Figures (23)

• Figure 1: Left panel: Radial profiles for the Ricci scalar for different values of the central pressure $p_0$. The red dots indicate the location of the star's radius where the pressure vanishes. Colors from green to violet indicate increasing central pressures, from $p_0=10^{-3}\rho_0 c^2$ to $p_0=0.5\rho_0 c^2$. Right panel: Ricci scalar at $r=0$ (associated with the solutions depicted in the left panel) as a function of $p_0$. The plots belong to the MJWQ model with $\alpha_M=2$ and fixed density $\rho_0=25R_1/(\kappa c^2)$. In these units, the asymptotic de Sitter value is $R_1/R_*\approx 6.1461833318$ (horizontal black dashed lines) where $R_*:= H_0^2/c^2$.
• Figure 2: Solutions for the MJWQ $f(R)$ model. Top left panel: Metric potentials $n(r)$ (dotted lines) and $m(r)$ (solid lines) for a star with $\rho_0= 10^7 R_1/(2\kappa c^2)$ and central pressures from $p_0=10^{-3}\rho_0 c^2$ to $p_0=0.3\rho_0 c^2$ (green to violet colors). The red dots indicate the location of the star's surface where the pressure vanishes. The metric functions show the de Sitter behavior asymptotically $n\sim 1 - \Lambda_{eff} r^3/3$ and $m\sim n^{-1}$. Top right panel: Product of the metric potentials $m(r)\times n(r)$ for the same configurations. Contrary to GR, the product deviates from one outside the star. Bottom left panel: Radial profiles for the Ricci scalar. Bottom right panel: Ricci scalar at the center of the star (associated with the solutions at the bottom left panel) as function of the central pressure (color stars). The corresponding GR values $-\kappa T_0= -\kappa (3p_0-\rho_0 c^2)$ (Y) are shown for reference.
• Figure 3: Left panel: Ricci scalar for the Starobinsky model with central pressures from $p_0=10^{-3}\rho_0 c^2$ to $p_0=0.3\rho_0 c^2$ (colors from green to violet) and fixed density $\rho_0= R_1/(4\kappa c^2)$. The asymptotic de Sitter value is $R_1/R_*\approx 6.82795191$, where $R_*:= H_0^2/c^2$. Right panel: the central values $R_0$ associated with the solutions of the left panel (color stars) plotted with respect to the central pressures $p_0$. For reference the corresponding GR values $-\kappa T_0= -\kappa (3p_0-\rho_0 c^2)$ are shown (Y).
• Figure 4: Similar to Fig.\ref{['FIG::STBS_JAIME_MultiplePressure_0.5']} but taking $\rho_0= 25R_1/(\kappa c^2)$.
• Figure 5: Left panel: Ricci scalar associated with solutions for the Hu-Sawicki $f(R)$ model at a fixed density $\rho_0= R_1/(4\kappa c^2)$ and central pressures ranging from $p_0=10^{-3}\rho_0 c^2$ to $p_0=0.3\rho_0 c^2$ (colors from green to violet indicate increasing central pressures). The de Sitter minimum is at $R_1/R_*\approx 8.931080$ (dashed line). The red dots indicate the location of the star's surface. Right panel:. Ricci scalar at the center of the star (corresponding to solutions of left panel) as a function of the central pressure (color stars). Like in previous figures, the GR value is plotted for reference (Y).