On the absence of the usual weak-field limit, and the impossibility of embedding some known solutions for isolated masses in cosmologies with f(R) dark energy

TL;DR

This paper demonstrates that in f(R) gravity theories, constructing inhomogeneous cosmological models with realistic weak-field limits is generally impossible unless the large-scale behavior mimics Einstein's equations with a cosmological constant, challenging standard perturbative approaches.

Contribution

It shows the fundamental difficulties in embedding known solutions into f(R) cosmologies and highlights the incompatibility of weak-field limits with late-time acceleration in these theories.

Findings

Matching conditions force large-scale behavior to resemble Einstein's equations with a cosmological constant.

No known spherically symmetric vacuum solutions can be embedded in FLRW backgrounds in R^n theories.

Standard weak-field perturbations around Minkowski space are incompatible with late-time cosmic acceleration in analytic f(R) models.

Abstract

The problem of matching different regions of spacetime in order to construct inhomogeneous cosmological models is investigated in the context of Lagrangian theories of gravity constructed from general analytic functions f(R), and from non-analytic theories with f(R)=R^n. In all of the cases studied, we find that it is impossible to satisfy the required junction conditions without the large-scale behaviour reducing to that expected from Einstein's equations with a cosmological constant. For theories with analytic f(R) this suggests that the usual treatment of weak-field systems as perturbations about Minkowski space may not be compatible with late-time acceleration driven by anything other than a constant term of the form f(0), which acts like a cosmological constant. In the absence of Minkowski space as a suitable background for weak-field systems, one must then choose and justify some other solution to perform perturbative analyses around. For theories with f(R)=R^n we find that no known spherically symmetric vacuum solutions can be matched to an expanding FLRW background. This includes the absence of any Einstein-Straus-like embeddings of the Schwarzschild exterior solution in FLRW spacetimes.