Parameterized Beyond-Einstein Growth

TL;DR

The paper addresses the challenge of distinguishing dark energy from modifications to gravity by jointly constraining the expansion history and the growth of structure. It introduces a model-independent framework, the gravitational growth index $\gamma$, via the relation $G(a)=Omega_m(a)^\gamma-1$, and couples it to a parameterized expansion history $w(a)$ within Minimal Modified Gravity (MMG). Through analytic derivations and toy models (early dark energy, time-varying gravity, DGP braneworld, and scalar-tensor theories), it demonstrates that $\gamma$ is largely constant and separable from $w(a)$, while beyond-Einstein gravity can produce observable deviations (up to ~20%) from the GR baseline. The framework provides a simple, scalable benchmark for upcoming cosmological data to test gravity beyond Einstein, guiding interpretation and model discrimination while acknowledging limitations to linear regime and potential scale-dependent effects.

Abstract

A single parameter, the gravitational growth index γ, succeeds in characterizing the growth of density perturbations in the linear regime separately from the effects of the cosmic expansion. The parameter is restricted to a very narrow range for models of dark energy obeying the laws of general relativity but takes on distinctly different values in models of beyond-Einstein gravity. In analogy to the parameterized post-Newtonian (PPN) formalism for testing gravity, we extend and motivate the gravitational growth index, or Minimal Modified Gravity, approach to parameterizing beyond-Einstein cosmology. Using a simple analytic formalism, we show how the growth index parameter applies to early dark energy, time-varying gravity, DGP braneworld gravity, and scalar-tensor gravity.

Figures (1)

• Figure 1: In the parameterized beyond-Einstein approach, the growth index $\gamma$ tests gravity separately from the expansion and clearly distinguishes beyond-Einstein gravity from physical dark energy. The curves show the values of $\gamma$ change little as the expansion history, here represented by the equation of state $w(z=1)$, varies. Solid curves show the ratio of the numerical solution for $\gamma$ in the braneworld case to that in general relativity (GR), and in the quintessence case the ratio of the fitting form $\gamma=0.55+0.04\, [1+w(z=1)]$ to the numerical GR solution. (Dotted curves assume $\gamma=0.55$, the GR value in a cosmological constant model.)