Nonlocal de Sitter gravity and its exact cosmological solutions

TL;DR

This paper introduces a nonlocal de Sitter gravity model with exact cosmological solutions, revealing novel behaviors like mimicking dark matter and dark energy, and including nonsingular bounce, cyclic, and singular solutions.

Contribution

It formulates a new nonlocal gravity model with exact solutions, extending local de Sitter gravity to include nonlocal effects and diverse cosmological behaviors.

Findings

Exact solutions mimic dark matter and dark energy effects

Discovery of nonsingular bounce solutions in various universe geometries

Presence of cyclic and singular solutions due to nonlocality

Abstract

This paper is devoted to a simple nonlocal de Sitter gravity model and its exact vacuum cosmological solutions. In the Einstein-Hilbert action with $\Lambda$ term, we introduce nonlocality by the following way: $R - 2 \Lambda = \sqrt{R-2\Lambda}\ \sqrt{R-2\Lambda} \to \sqrt{R-2\Lambda}\ F(\Box)\ \sqrt{R-2\Lambda} ,$ where ${F} (\Box) = 1 + \sum_{n= 1}^{+\infty} \big( f_n \Box^n + f_{-n} \Box^{-n} \big) $ is an analytic function of the d'Alembert-Beltrami operator $\Box$ and its inverse $\Box^{-1}$. By this way, $R$ and $\Lambda$ enter with the same form into nonlocal version as they are in the local one, and nonlocal operator $F(\Box)$ is dimensionless. The corresponding equations of motion for gravitational field $g_{\mu\nu}$ are presented. The first step in finding some exact cosmological solutions is solving the equation $\Box \sqrt{R-2\Lambda} = q \sqrt{R-2\Lambda} , $ where $ q =\zeta \Lambda \quad (\zeta \in \mathbb{R})$ is an eigenvalue and $\sqrt{R-2\Lambda}$ is an eigenfunction of the operator $\Box .$ We presented and discussed several exact cosmological solutions for homogeneous and isotropic universe. One of these solutions mimics effects that are usually assigned to dark matter and dark energy. Some other solutions are examples of the nonsingular bounce ones in flat, closed and open universe. There are also singular and cyclic solutions. All these cosmological solutions are a result of nonlocality and do not exist in the local de Sitter case.