Phantom and non-phantom dark energy: The cosmological relevance of non-locally corrected gravity

TL;DR

The paper investigates cosmological dynamics of non-locally corrected gravity with a nonlocal term $f(\Box^{-1}R)$, focusing on a tractable local formulation via auxiliary fields and FRW dynamics. For the simple exponential form $f(\phi)\sim e^{\alpha\phi}$, it uncovers a one-parameter family of late-time dark energy attractors with $w_{\rm eff}=\frac{\alpha-1}{3\alpha-1}$ and stable behavior when $\frac{1}{3}<\alpha<\frac{2}{3}$, including phantom regimes for $\frac{1}{3}<\alpha<\frac{1}{2}$ and a de Sitter point at $\alpha=\tfrac{1}{2}$. The model can reproduce a dust-like era before settling to the attractor and achieves late-time phantom acceleration without negative kinetic energy fields. The work also discusses the entropy of de Sitter space in this non-local context, showing that entropy finiteness imposes nontrivial conditions on the matter content (e.g., phantom matter) and the parameter $\alpha$, highlighting subtleties in defining gravitational entropy for non-local theories.

Abstract

In this paper we have investigated the cosmological dynamics of non-locally corrected gravity involving a function of the inverse d'Alembertian of the Ricci scalar, $f(\Box^{-1} R))$. Casting the dynamical equations into local form, we derive the fixed points of the dynamics and demonstrate the existence and stability of a one parameter family of dark energy solutions for a simple choice, $f(\Box^{-1} R)\sim \exp(α\Box^{-1} R)$. The effective EoS parameter is given by, $w_{\rm eff}=({α-1})/({3α-1})$ and the stability of the solutions is guaranteed provided that $1/3<α<2/3$. For $1/3<α<1/2$ and $1/2<α<2/3$, the underlying system exhibits phantom and non-phantom behavior respectively; the de Sitter solution corresponds to $α=1/2$. For a wide range of initial conditions, the system mimics dust like behavior before reaching the stable fixed point. The late time phantom phase is achieved without involving negative kinetic energy fields. A brief discussion on the entropy of de Sitter space in non-local model is included.

Figures (2)

• Figure 1: The evolution of $w_{\rm eff}$ from the different initial conditions, dark black line corresponds to $\alpha =0.5$, $x=-1.99$, $y=5322.02$, gray to $\alpha =0.487$, $x=-1.99$, $y=1012.06$ and dashed to $\alpha =0.507$, $x =-1.99$,$y=1013.06$. At the fixed point $w_{\rm eff}$= $-1$, $-1.11$ and $-0.95$ respectively
• Figure 2: The phase portrait of the system in absence of matter for $\alpha=1/2$. The phase space splits into two disjoint regions: all the trajectories starting from $y>3$ converge to de Sitter where as they approach (0,0) for $y<3$ and $x>-4$.