Linearized $f(R)$ gravity: Gravitational radiation and Solar System tests

We investigate the linearized form of metric $f(R)$-gravity, assuming that $f(R)$ is analytic about $R=0$ so it may be expanded as $f(R)=R+a_2{R}^2/2+\dots$. Gravitational radiation is modified, admitting an extra mode of oscillation, that of the Ricci scalar. We derive an effective energy-momentum tensor for the radiation. We also present weak-field metrics for simple sources. These are distinct from the equivalent Kerr (or Schwarzschild) forms. We apply the metrics to tests that could constrain $f(R)$. We show that light deflection experiments cannot distinguish $f(R)$-gravity from general relativity as both have an effective post-Newtonian parameter $\gamma =1$. We find that planetary precession rates are enhanced relative to general relativity; from the orbit of Mercury we derive the bound $|a_2|\lesssim 1.2\times {10}^{18}{\mathrm{m}}^2$. Gravitational-wave astronomy may be more useful: considering the phase of a gravitational waveform we estimate deviations from general relativity could be measurable for an extreme-mass-ratio inspiral about a ${10}^6{M}_{\odot }$ black hole if $|a_2|\gtrsim {10}^{17}{\mathrm{m}}^2$, assuming that the weak-field metric of the black hole coincides with that of a point mass. However Eöt-Wash experiments provide the strictest bound $|a_2|\lesssim 2\times {10}^{-9}{\mathrm{m}}^2$. Although the astronomical bounds are weaker, they are still of interest in the case that the effective form of $f(R)$ is modified in different regions, perhaps through the chameleon mechanism. Assuming the laboratory bound is universal, we conclude that the propagating Ricci scalar mode cannot be excited by astrophysical sources.

Figure 1

Region of parameter space in which $f(R)$ theories can be distinguished from GR when the central BH has spin $a=0$ (left panel) or $a=0.5$ (right panel). Each curve corresponds to a particular specification of the detectability criterion given in (129) in the text, as identified in the key. Dashed lines correspond to measurements of the vertical epicyclic frequency, while solid lines represent measurements of the radial epicyclic frequency. The region below the curve could be distinguishable in a LISA observation with that detectability value.