Cosmological Solutions of a Nonlocal Square Root Gravity

In this paper we consider modification of general relativity extending $R - 2 \Lambda$ by nonlocal term of the form $\sqrt{R-2\Lambda}\, \mathcal{F}(\Box)\, \sqrt{R-2\Lambda} ,$ where $\mathcal{F}(\Box)$ is an analytic function of the d'Alembert operator $\Box$. We have found some exact cosmological solutions of the corresponding equations of motion without matter and with $\Lambda \neq 0$. One of these solutions is $a (t) = A \, t^{\frac{2}{3}} \, e^{\frac{\Lambda}{14} t^2} ,$ which imitates properties similar to an interplay of the dark matter and the dark energy. For this solution we calculated some cosmological parameters which are in a good agreement with observations. This nonlocal gravity model has not the Minkowski space solution. We also found several conditions which function $\mathcal{F}(\Box)$ has to satisfy.