Differentiating between Modified Gravity and Dark Energy

TL;DR

The paper tackles distinguishing modified gravity (MG) from dark energy (DE) as the source of cosmic acceleration by imposing a Birkhoff-like constraint and deriving a Schwarzschild-like metric from a given dust-dominated expansion a(t). It reframes the cosmology via a generalized Friedmann equation $H^2 = H_0^2 \;g(x)$ with $x = \frac{8\pi G\rho}{3H_0^2}$, showing how MG can imprint on both the growth of structure and the evolution of gravitational potentials, thereby affecting the ISW effect and potentially inducing measurable solar-system deviations through a characteristic scale $r_*$. The work demonstrates that, for the same expansion history, MG typically slows the growth of perturbations relative to DE and alters the ISW cross-correlation with galaxies, while predicting a sub-horizon modification to gravity that could be probed by high-precision future observations. Overall, the authors argue that joint constraints from the expansion history, structure growth, ISW measurements, and local gravity tests can robustly differentiate MG from DE, with implications extending beyond the specific Birkhoff-constrained construction.

Abstract

The nature of the fuel that drives today's cosmic acceleration is an open and tantalizing mystery. We entertain the suggestion that the acceleration is not the manifestation of yet another new ingredient in the cosmic gas tank, but rather a signal of our first real lack of understanding of gravitational physics. By requiring that the underlying gravity theory respects Birkhoff's law, we derive the modified gravitational force-law necessary to generate any given cosmology, without reference to the fundamental theory, revealing modifications of gravity at scales typically much smaller than today's horizon. We discuss how through these modifications, the growth of density perturbations, the late-time integrated Sachs--Wolfe effect, and even solar-system measurements may be sensitive to whether today's cosmic acceleration is generated by dark energy or modified gravitational dynamics, and are subject to imminent observational discrimination. We argue how these conclusions can be more generic, and probably not dependent on the validity of Birkhoff's law.

Figures (2)

• Figure 1: Growth of perturbations as a function of redshift in dark energy (DE) models and modified gravity (MG) models with identical expansion histories, given by Eq. (\ref{['gGT']}). The dashed line shows the ratio of density perturbation growth factors $D_+^{\rm MG}/D_+^{\rm DE}$, the solid line shows the ratio of the velocity perturbation growth factors $f^{\rm MG}/f^{\rm DE}$.
• Figure 2: The angular cross-correlation function between galaxies and CMB temperature anisotropies. The solid line shows the dark energy case, dashed line corresponds to modified gravity with the same expansion history.