Some Cosmological Solutions of a New Nonlocal Gravity Model

Abstract

In this paper, we investigate a nonlocal modification of general relativity (GR) with action $S = \frac{1}{16\pi G} \int [ R- 2\Lambda + (R-4\Lambda) \, \mathcal{F}(\Box) \, (R-4\Lambda) ] \, \sqrt{-g}\; d^4x ,$ where $\mathcal{F} (\Box) = \sum_{n=1}^{+\infty} f_n \Box^n$ is an analytic function of the d'Alembertian $\Box$. We found a few exact cosmological solutions of the corresponding equations of motion. There are two solutions which are valid only if $\Lambda \neq 0, \, k = 0,$ and they have not analogs in Einsten's gravity with cosmological constant $\Lambda$. One of these two solutions is $ a (t) = A \, \sqrt{t} \, e^{\frac{\Lambda}{4} t^2} ,$ that mimics properties similar to an interference between the radiation and the dark energy. Another solution is a nonsingular bounce one -- $ a (t) = A \, e^{\Lambda t^2}$. For these two solutions, some cosmological aspects are discussed. We also found explicit form of the nonlocal operator $\mathcal{F}(\Box)$, which satisfies obtained necessary conditions.