Nonlocal de Sitter gravity and its exact cosmological solutions

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 Λ term, we introduce nonlocality by the following way: R-2Lambda =sqrt{R-2Lambda}kern1em sqrt{R-2Lambda}to sqrt{R-2Lambda}kern1em Fleft(square right)kern1em sqrt{R-2Lambda} , where Fleft(square right)=1+{sum}_{n=1}^{+infty}left({f}_n{square}^n+{f}_{-n}{square}^{-n}right) is an analytic function of the d’Alembert-Beltrami operator □ and its inverse □−1. By this way, R and Λ enter with the same form into nonlocal version as they are in the local one, and nonlocal operator F(□) is dimensionless. The corresponding equations of motion for gravitational field gμν are presented. The first step in finding some exact cosmological solutions is solving the equation square sqrt{R-2Lambda}=qsqrt{R-2Lambda} , where q = ζΛ (ζ ∈ R) is an eigenvalue and sqrt{R-2Lambda} is an eigenfunction of the operator □. 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.