Modified gravity in Arnowitt-Deser-Misner formalism

TL;DR

The paper introduces two ADM-based modified gravity frameworks motivated by Horava-Lifshitz theory: an f(R)-type with ultraviolet and infrared corrections to the Einstein term, and a K-essence-type with a nonlinear extrinsic-curvature term. In cosmology, the f(R) model yields a simple Friedmann equation with $H^4$ (UV) and $H^{-2}$ (IR) terms, which can both avoid the Big-Bang and Big-Rip, and can account for current acceleration without dark energy; the IR term behaves like a Cardassian modification with a late-time de Sitter endpoint. The K-essence model provides a dynamical dark-energy sector through an exponential F(X) that can cross the phantom divide and mimic ΛCDM at late times. In static vacuum spacetimes, both theories produce Schwarzschild or Schwarzschild–de Sitter solutions, ensuring compatibility with solar-system tests. Overall, these ADM-based modifications yield tractable cosmological dynamics and viable local solutions, warranting further exploration of their phenomenology and theoretical consistency.

Abstract

Motivated by Horava-Lifshitz gravity theory, we propose and investigate two kinds of modified gravity theories, the f(R) kind and the K-essence kind, in the Arnowitt-Deser-Misner (ADM) formalism. The f(R) kind includes one ultraviolet (UV) term and one infrared (IR) term together with the Einstein-Hilbert action. We find that these two terms naturally present the ultraviolet and infrared modifications to the Friedmann equation. The UV and IR modifications can avoid the past Big-Bang singularity and the future Big-Rip singularity, respectively. Furthermore, the IR modification can naturally account for the current acceleration of the Universe. The Lagrangian of K-essence kind modified gravity is made up of the three dimensional Ricci scalar and an arbitrary function of the extrinsic curvature term. We find the cosmic acceleration can also be naturally interpreted without invoking any kind of dark energy. The static, spherically symmetry and vacuum solutions of both theories are Schwarzschild or Schwarzschild-de Sitter solution. Thus these modified gravity theories are viable for solar system tests.

Figures (8)

• Figure 1: The ratio of densities for dark matter (circled line) and dark energy (solid line). The cosmic coincidence problem is relaxed. Here we put $\Omega_{m0}=0.25$.
• Figure 2: The evolution of the equation of state for dark energy. This is a phantom dark energy. Here we put $\Omega_{m0}=0.25$.
• Figure 3: The Hubble-redshift relations for $\Lambda \textrm{CDM}$ model (pointed line) and the IR model (solid line). Both models are consistent with the observational data. Here we put $\Omega_{m0}=0.25$.
• Figure 4: The evolution of decelerating parameters for $\Lambda \textrm{CDM}$ model (pointed line) and the IR model (solid line). Both models predict the transition redshift of the Universe from deceleration to acceleration at $z_T\simeq0.8$. Here we put $\Omega_{m0}=0.25$.
• Figure 5: Evolution of the equation of state for dark energy for $\xi=0.36,\ 0.66,\ 1.26$ up down. When $\xi<0.66$, the dark energy model behaves as quintom matter which can crosse phantom divide smoothly. When $\xi\geq 0.66$, the dark energy behaves as phantom matter which always have the equation of state $w<-1$. When $\xi=0$, it reduces to the cosmological constant. Here we put $\Omega_{m0}=0.25,\ \Omega_{r0}=8.1\cdot10^{-5}$.