A Parameterized Post-Friedmann Framework for Modified Gravity

TL;DR

The paper introduces a parameterized post-Friedmann (PPF) framework to model scalar modifications to gravity that drive cosmic acceleration without dark energy, unifying super-horizon, quasi-static linear, and non-linear regimes under a metric and energy-momentum-conserving description. It provides a linear theory parameterization with functions g, f_zeta, f_G and a transition scale c_Gamma, and validates the approach against $f(R)$ and DGP models; it also outlines a halo-model based non-linear ansatz to describe GR restoration inside halos in the absence of simulations. The framework enables model-independent tests of gravity using structure growth, lensing, and ISW effects, while offering a tractable path to incorporate non-linearities as simulations become available. Overall, PPF serves as a practical, testable template linking background expansion, linear perturbations, and non-linear clustering in modified gravity scenarios.

Abstract

We develop a parameterized post-Friedmann (PPF) framework which describes three regimes of modified gravity models that accelerate the expansion without dark energy. On large scales, the evolution of scalar metric and density perturbations must be compatible with the expansion history defined by distance measures. On intermediate scales in the linear regime, they form a scalar-tensor theory with a modified Poisson equation. On small scales in dark matter halos such as our own galaxy, modifications must be suppressed in order to satisfy stringent local tests of general relativity. We describe these regimes with three free functions and two parameters: the relationship between the two metric fluctuations, the large and intermediate scale relationships to density fluctuations and the two scales of the transitions between the regimes. We also clarify the formal equivalence of modified gravity and generalized dark energy. The PPF description of linear fluctuation in f(R) modified action and the Dvali-Gabadadze-Porrati braneworld models show excellent agreement with explicit calculations. Lacking cosmological simulations of these models, our non-linear halo-model description remains an ansatz but one that enables well-motivated consistency tests of general relativity. The required suppression of modifications within dark matter halos suggests that the linear and weakly non-linear regimes are better suited for making complementary test of general relativity than the deeply non-linear regime.

Figures (9)

• Figure 1: Super-horizon (SH) metric evolution of $\Phi_{-}=(\Phi-\Psi)/2$ for various choices of $g=g_{0}a$. For $g<0$, $\Phi_{-}$ can actually grow near the onset of acceleration. The expansion history is fixed by $w_{\rm eff}=-1$ and $\Omega_{m}=0.24$ in all cases.
• Figure 2: Fractional difference between the quasi-static (QS) vs super-horizon (SH) metric at $a=1$ for a metric ratio of $g=g_0 a$. The two evolutions differ unless $w_{\rm eff}=-1$ and $g_{0}=0$. $\Omega_{m}=0.24$ for all cases.
• Figure 3: Evolution and scale dependence of the metric ratio $g$ in $f(R)$ models compared with the PPF fit. Here $B_{0}=0.4$, $w_{\rm eff}=-1$ and $\Omega_{m}=0.24$.
• Figure 4: Evolution and scale dependence of $\Phi_-$ in $f(R)$ models compared with the PPF fit. Here $B_{0}=0.4$, $w_{\rm eff}=-1$ and $\Omega_{m}=0.24$.
• Figure 5: Evolution and scale dependence of the metric ratio $g$ in the DGP model compared with a PPF fit. Here $\Omega_{m}=0.24$ and the PPF parameter $c_g=0.4$.