General relativity (GR) is a physical theory considered as one of the most beautiful mathematically and the most successful phenomenologically. Although very successful, GR is not a complete theory of gravity and there are many attempts to modify it. One of very promising directions of research is nonlocal modified gravity. We consider here the nonlocal de Sitter model $\sqrt{dS}$ which is given by the following action: $S = \frac{1}{16 \pi G}\int \sqrt{-g} \big(R - 2\Lambda + \sqrt{R-2\Lambda}\ \mathcal{F}(\Box)\ \sqrt{R-2\Lambda}\big) d^4x ,$ where $R$ is scalar curvature and $\Lambda$ is cosmological constant. $\mathcal{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 $\Box = \nabla_\mu \nabla^\mu = \frac{1}{\sqrt{-g}} \partial_\mu (\sqrt{-g} g^{\mu\nu}\partial_\nu)$ and its inverse $\Box^{-1}$. Derivation of equations of motion (EoM) for gravitational field $g_{\mu\nu}$ is presented in detail in [2]. To solve the corresponding equations of motion, we first solve eigenvalue problem $\Box \sqrt{R-2\Lambda} = q \sqrt{R-2\Lambda},$ where $ q =\zeta \Lambda \quad (\zeta \in \mathbb{R})$ is an eigenvalue and $\sqrt{R-2\Lambda}$ is an eigenfunction of 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, see [1]. Some other solutions are examples of the nonsingular bounce ones in flat, open and closed universe. There are also singular and cyclic solutions.
References
[1] I. Dimitrijevic, B. Dragovich, Z. Rakic, J. Stankovic, {\it Nonlocal de Sitter Gravity and its exact cosmological solutions}, JHEP, 12, 054; doi: https://doi.org/10.1007/JHEP12(2022)054
[2] I. Dimitrijevic, B. Dragovich, Z. Rakic and J. Stankovic, {\it Variations of infinite derivative modified gravity,} Springer Proc. in Mathematics $\&$ Statistics {\bf 263} (2018) 91--111.