Towards a fully consistent parameterization of modified gravity

TL;DR

This work advocates a fully consistent, broadly applicable parameterization for cosmological modifications to gravity, arguing that two‑function schemes based on a simple slip and a modified Poisson equation are not generally sufficient, especially beyond the quasistatic regime. The authors develop a Parameterized Post-Friedmann (PPF) formalism that decomposes the modification tensor U_{\mu u} into metric, additional degrees of freedom, and their interactions, and then express the perturbative parts δU_{\mu u} as a gauge‑invariant expansion in derivatives of the metric potentials. In the purely metric, no‑extra‑field case, the formalism reduces to two free functions (often reinterpreted as ĝ and ζ) that modify the Poisson equation and the Φ̂–Ψ̂ relation, with explicit constraints from the Bianchi identities; they show this two‑function approach can fail to capture higher‑derivative theories and horizon‑scale behavior. When extra degrees of freedom are present, a Generalized Dark Matter (GDM) closure with three parameters is proposed to map the additional fields to fluid‑like perturbations, enabling a model‑independent closure of the perturbation equations. The paper maps representative theories (scalar‑tensor, f(R), Einstein‑Aether, DGP) into the framework, highlighting that quasistatic tests are well described by the two‑function PPF limit, while large‑scale and nonlocal effects require the extended GDM or additional structure for accurate interpretation. Overall, the PPF approach provides a practical, physically grounded bridge between theory and data, with explicit consistency requirements and clear pathways to include a broad class of modified gravity models.

Abstract

There is a distinct possibility that current and future cosmological data can be used to constrain Einstein's theory of gravity on the very largest scales. To be able to do this in a model-independent way, it makes sense to work with a general parameterization of modified gravity. Such an approach would be analogous to the Parameterized Post-Newtonian (PPN) approach which is used on the scale of the Solar System. A few such parameterizations have been proposed and preliminary constraints have been obtained. We show that the majority of such parameterizations are only exactly applicable in the quasistatic regime. On larger scales they fail to encapsulate the full behaviour of typical models currently under consideration. We suggest that it may be possible to capture the additions to the `Parameterized Post-Friedmann' (PPF) formalism by treating them akin to fluid perturbations.