General relativity (GR), i.e. Einstein theory of gravity, is recognized as one of the best physical theories -- with nice theoretical properties and significant phenomenological confirmations. Nevertheless, GR is not a complete theory of gravity and there are many attempts to modify it. One of the actual approaches towards more complete theory is nonlocal modified gravity. Nonlocal gravity model, which we consider here without matter, is given by the 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 $\Lambda$ -- cosmological constant. $\mathcal{F}(\Box) = \sum_{n=1}^\infty f_n \Box^n$ is an analytic function of the corresponding d'Alambertian $\Box .$ Derivation of equations of motion is presented in [1].
We plan to present a brief review of general properties, and then discuss exact cosmological solutions. One of the exact cosmological solutions is $a(t) = A \, t^{\frac{2}{3}} \, e^{\frac{\Lambda}{14}\, t^2}$, which mimics the dark matter and the dark energy, see [2]. Computed cosmological parameters are in a good agreement with standard model of cosmology.
References
[1] I. Dimitrijevic, B. Dragovich, Z. Rakic and J. Stankovic, Variations of infinite derivative modified gravity, Springer Proc. Math. Stat. 263 (2018) 91-111.
[2] I. Dimitrijevic, B. Dragovich, A.S. Koshelev, Z. Rakic and J. Stankovic, Cosmological solutions of a nonlocal square root gravity, Phys. Lett. B 797 (2019) 134848.