Solar System tests and chameleon effect in $f\mathbf{(}R\mathbf{)}$ gravity

Using a novel and self-consistent approach that avoids the scalar-tensor identification in the Einstein frame, we reanalyze the viability of $f(R)$ gravity within the context of solar-system tests. In order to do so, we depart from a simple but fully relativistic system of differential equations that describe a compact object in a static and spherically symmetric spacetime, and then we make suitable linearizations that apply to nonrelativistic objects such as the Sun. We then show clearly under which conditions the emerging chameleonlike mechanism can lead to a post-Newtonian parameter $\gamma$ compatible with the observational bounds. To illustrate this method, we use several specific $f(R)$ models proposed to explain the current acceleration of the Universe, and we show which of them are able to satisfy those bounds.

Figure 1

Behavior of the quantities $d{V}_{\mathrm{eff}}/dR$, $dV/dR$, and $d{V}_{\mathrm{mat}}/dR$ in units of ${10}^{-76}{\mathrm{cm}}^{-4}$ in the interstellar medium (as defined in the main text) with respect to $R/R_{\min }^{\mathrm{IM}}$ for the Starobinsky model given by Eq. (8) [20] with $q=2$. A similar behavior of these quantities is exhibited by the other $f(R)$ models (see Secs. 2a and 5).

Figure 2

Ricci scalar (in units of $R_{\min }^{\mathrm{in}}$) as a function of $r$ (in units of ${\mathcal{R}}_{\odot }$) computed from the Starobinsky model with $q=2$. Inside the Sun $R(r)\approx R_{\min }^{\mathrm{in}}$ and beyond ${\mathcal{R}}_{\odot }$ the Ricci scalar $R(r)\ll R_{\min }^{\mathrm{in}}$. The total solution behaves basically as a single step function. The red dotted line and the green dashed line indicate the values $R_{\min }^{\mathrm{cor}}$ and $R_{\min }^{\mathrm{IM}}$, respectively (both values are very small compared with $R_{\min }^{\mathrm{in}}$).

Figure 3

Same as Fig. 2, but plotted very near the Sun’s surface within the corona. The Ricci scalar decreases around 19 orders of magnitude from $R_{\min }^{\mathrm{in}}$ to $R_{\min }^{\mathrm{cor}}$ within a very thin region $\mathrm{\Delta }{\mathcal{R}}_{\odot }\sim 5\times {10}^{-11}{\mathcal{R}}_{\odot }$ (between $r={\mathcal{R}}_{\odot }$ and $r={\mathcal{R}}_{\odot }+\mathrm{\Delta }{\mathcal{R}}_{\odot }$).

Figure 4

The effect on $R(r)$ when decreasing the corona’s density by half. The higher the density the steeper the gradient of $R(r)$ provoking the screening effect to be more effective.

Figure 5

Deviation parameter $|\gamma -1|$ as a function of $r$ (in units of ${\mathcal{R}}_{\odot }$) computed from the Starobinsky model used in Fig. 2. The constraint from the Cassini mission is shown as a horizontal dotted red line.

Figure 6

Similar to Fig. 2 but taking $q=0.4$ and $\lambda =4.0$ for the Starobinsky model. Qualitatively, the Ricci scalar behaves similarly in both cases ($q=2$ and $q=0.4$); however, for $q=0.4$ it is several orders of magnitude larger than the case $q=2$, and thus, this second model fails the solar-system tests.

Figure 7

Similar to Fig. 5 using $q=0.4$ and $\lambda =4.0$. This Starobinsky model fails the solar-system tests: the predicted value for $|\gamma -1|$ is around 4 orders of magnitude larger than the observational bounds.

Figure 8

Similar to Fig. 2, but for the MJWQ model. Inside the Sun $R(r)\approx R_{\min }^{\mathrm{in}}$, but outside, the Ricci scalar does not reach a minimum $R_{\min }^{\mathrm{IM}}$ and decreases slowly.

Figure 9

Similar to Fig. 5, but for the MJWQ model. This model does not satisfy the observational bounds on $\gamma$ by more than 4 orders of magnitude.

Figure 10

Behavior of the terms involved in the potential’s derivative and the different linear approximations considered (in units of ${10}^{91}{\mathrm{cm}}^{-4}$) as a function of $R/R_{\min }^{\mathrm{in}}$ within the Sun. Here the plots correspond to the Starobinsky model (8) with $q=2$ but a similar behavior can be seen for other $f(R)$ models.