The power of general relativity

We study the cosmological and weak-field properties of theories of gravity derived by extending general relativity by means of a Lagrangian proportional to $R^{1+\delta }$. This scale-free extension reduces to general relativity when $\delta \to 0$. In order to constrain generalizations of general relativity of this power class, we analyze the behavior of the perfect-fluid Friedmann universes and isolate the physically relevant models of zero curvature. A stable matter-dominated period of evolution requires $\delta >0$ or $\delta <-1/4$. The stable attractors of the evolution are found. By considering the synthesis of light elements (helium-4, deuterium and lithium-7) we obtain the bound $-0.017<\delta <0.0012$. We evaluate the effect on the power spectrum of clustering via the shift in the epoch of matter-radiation equality. The horizon size at matter-radiation equality will be shifted by $\sim 1\%$ for a value of $\delta \sim 0.0005$. We study the stable extensions of the Schwarzschild solution in these theories and calculate the timelike and null geodesics. No significant bounds arise from null geodesic effects but the perihelion precession observations lead to the strong bound $\delta =2.7\pm 4.5\times {10}^{-19}$ assuming that Mercury follows a timelike geodesic. The combination of these observational constraints leads to the overall bound $0\le \delta <7.2\times {10}^{-19}$ on theories of this type.

Figure 1

Critical density, ${\Omega }_0$, as a function of $\delta$. The solid line corresponds to pressureless dust and the dashed line to blackbody radiation.

Figure 2

Phase plane of cosmological solutions for $\omega =0$ and $\delta =0.1$.

Figure 3

Phase plane of cosmological solutions for $\omega =0$ and $\delta =-0.1$.

Figure 4

The $Z=0$ plane of the phase space defined by $X$, $Y$ and $Z$ for $\delta =0.1$ and $c_1=0.5$. The bold lines show the critical points in this plane and the diagonal lines show the unphysical trajectories confined to this plane. The dotted line is $Y=2X$ and separates the critical points where $\xi \to \infty$ from the points where $\xi \to -\infty$.

Figure 5

The surface at infinity of the phase space defined by $X$,$Y$ and $Z$ for $\delta =0.1$. The shaded areas show regions I and II. Region III is unshaded.