Solar System constraints on renormalization group extended general relativity: The PPN and Laplace-Runge-Lenz analyses with the external potential effect

General relativity extensions based on renormalization group effects are motivated by a known physical principle and constitute a class of extended gravity theories that have some unexplored unique aspects. In this work we develop in detail the Newtonian and post-Newtonian limits of a realization called renormalization group extended general relativity (RGGR). Special attention is given to the external potential effect, which constitutes a type of screening mechanism typical of RGGR. In the Solar System, RGGR depends on a single dimensionless parameter ${\overline{\nu }}_{\odot }$, and this parameter is such that for ${\overline{\nu }}_{\odot }=0$ one fully recovers GR in the Solar System. Previously this parameter was constrained to be $|{\overline{\nu }}_{\odot }|\lesssim {10}^{-21}$, without considering the external potential effect. Here we show that under a certain approximation RGGR can be cast in a form compatible with the parametrized post-Newtonian (PPN) formalism, and we use both the PPN formalism and the Laplace-Runge-Lenz technique to put new bounds on ${\overline{\nu }}_{\odot }$, either considering or not the external potential effect. With the external potential effect the new bound reads $|{\overline{\nu }}_{\odot }|\lesssim {10}^{-16}$. We discuss the possible consequences of this bound on the dark matter abundance in galaxies.

Figure 1

The Newtonian potential generated by the Sun ${\varphi }_{\odot }$ across the Solar System, and the value of the Newtonian potentials generated by the Galaxy at the Solar System (${\varphi }_{\mathrm{MW}}{|}_{\odot }$). The letters “P” and “A” after Mercury refer to its perihelion and aphelion.

Figure 2

The relation between $|\delta G|=|G-1|\approx 2\nu \ln \mu$ [from Eq. (2)] and $|\mathrm{\Phi }|$ [from Eq. (a6)] for four different values of $\nu$.

Figure 3

The relation between $\alpha$ and $\mathrm{\Phi }$ for different values of $\nu$. It shows that the noncovariant approximation, where $\alpha$ is a constant, can be a good approximation for many systems, since large changes of $\mathrm{\Phi }$ lead to much smaller changes in $\alpha$.

Figure 4

The relative errors (b3) introduced by the use of the approximation (8), in the context of Mercury’s orbit with the external potential of the Galaxy. The range of $\mathrm{\Phi }$ in this plot spans the variation of the Newtonian potential through the planet orbit added by ${10}^{-6}$. Dividing or multiplying the latter value by 5, does not change the value of ${\log }_{10}ϵ$ significantly. The plot indicates that the noncovariant scale setting works as a good approximation for the covariant one, in the context of Mercury’s orbit.