Stability of spherically symmetric spacetimes in metric $f(R)$ gravity

We consider stability properties of spherically symmetric spacetimes of stars in metric $f(R)$ gravity. We stress that these not only depend on the particular model, but also on the specific physical configuration. Typically configurations giving the desired ${\gamma }_{\mathrm{PPN}}\approx 1$ are strongly constrained, while those corresponding to ${\gamma }_{\mathrm{PPN}}\approx 1/2$ are less affected. Furthermore, even when the former are found strictly stable in time, the domain of acceptable static spherical solutions typically shrinks to a point in the phase space. Unless a physical reason to prefer such a particular configuration can be found, this poses a naturalness problem for the currently known metric $f(R)$ models for accelerating expansion of the Universe.

Figure 1

The gradient term proportional to $\delta R^{\prime }$ in Eq. (16), normalized to $m_R^2/r_{\odot }$, for various $f(R)$ models ($R=\kappa \rho$): $-{\mu }^4/R$ (lower, solid blue), $-{\mu }^4/R+\alpha R^2/{\mu }^2$ (dashed green), Hu and Sawicki (dot-dashed red) [12], and $\alpha R\log (R/{\mu }^2)$ (upper, solid black). The actual density profile used in all figures corresponds to the known density profile of the Sun with a central density of $150\mathrm{g}/{\mathrm{cm}}^3$ and with a roughly exponential dependence on $r$. We have also superimposed a constant dark matter distribution on the profile of the Sun with ${\rho }_{\mathrm{DM}}=0.3\mathrm{GeV}/{\mathrm{cm}}^3$.

Figure 2

The parameter $F$ given as a function of the radius for various $f(R)$ models ($R=\kappa \rho$): $-{\mu }^4/R$ (lower, solid blue), $-{\mu }^4/R+\alpha R^2/{\mu }^2$ (dashed green), Hu and Sawicki (dot-dashed red) [12], and $\alpha R\log (R/{\mu }^2)$ (upper, solid black).

Figure 3

The mass squared $m_R^2$ in units $1/r_{\odot }^2$ for various $f(R)$ models ($R=\kappa \rho$): $-{\mu }^4/R$ (upper, solid blue), $-{\mu }^4/R+\alpha R^2/{\mu }^2$ (dashed green), Hu and Sawicki (dot-dashed red) [12], and $\alpha R\log (R/{\mu }^2)$ (lower, solid black). The horizontal solid gray line corresponds to the limit $1/r_{\odot }^2$. For the $-{\mu }^4/R$ and $\alpha R\log (R/{\mu }^2)$ models we have also plotted $1/3F_{,R}$ with dotted blue and dotted black lines, respectively. The upper panel displays the corresponding sign of the mass squared where we have excluded the $-{\mu }^4/R$ model for which $m_R^2$ is strictly negative.