Nonlocal de Sitter gravity and its exact cosmological solutions

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=root R-2 lambda root R-2 lambda ->root R-2 lambda root F(?)R-2 lambda, where F(?)=1+ n-expressionry sumexpressiontion (+infinity)(n=1)(f(n)?(n)+f(-n)?(-n)) is an analytic function of the d'Alembert-Beltrami operator ? and its inverse ?(-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(?) 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 ?root R-2 lambda=q root R-2 lambda, where q = zeta lambda (zeta is an element of R) is an eigenvalue and root R-2 lambda 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.