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

Abstract

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 realisation called Renormalization Group extended General Relativity (RGGR). Special attention is taken 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 parameterν , and this parameter is such that forν = 0 one fully recovers GR in the Solar System. Previously this parameter was constrained to be |ν | 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 Parametrised Post-Newtonian (PPN) formalism, and we use both the PPN formalism and the Laplace-Runge-Lenz technique to put new bounds onν , either considering or not the external potential effect. With the external potential effect the new bound reads |ν | 10 −16 . We discuss the possible consequences of this bound to the dark matter abundance in galaxies.

Will-Nordtvedt PPN formalism applied to renormalization group extensions of general relativity

“…This class is such that the RG scale is a function of the Newtonian potential, hence in particular it includes the RGGR proposal [46][47][48]. We also consider the External Potential Effect (EPE), which is an intrinsic effect of these proposals and which depends on the environment of the system [55].…”

“…[55], using a more heuristic approach within the less rigorous and simpler Eddington-Robertson-Schiff PPN version, it was found the bound |ν | 10…”

“…41). That is, the noncovariant approximation works as an order of magnitude approximation, at the 1PN order, to the covariant version [55]. The situation would be different in case α 3 would depend on ν 1 or ν 2 .…”

“…This choice is relevant for the passage from the covariant action (1) to the noncovariant RGGR formulation (further details are in Appendix A and in Refs. [47,55]). …”

“…The Λ term was not considered in the previous Solar System analysis of RGGR [52,54,55]. In Appendix B we comment on the possible effects of Λ and show that the Will-Nordtvedt PPN formalism in its standard form cannot handle the Λ term, in particular because Λ cannot be both an analytical function and be compatible with asymptotic flatness.…”