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.

中文翻译：

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一个新的非局域引力模型的一些宇宙学解

在本文中，我们研究了广义相对论 (GR) 的非局部修正，其动作 $S = \frac{1}{16\pi G} \int [ R- 2\Lambda + (R-4\Lambda) \, \ mathcal{F}(\Box) \, (R-4\Lambda) ] \, \sqrt{-g}\; d^4x ,$ 其中 $\mathcal{F} (\Box) = \sum_{n=1}^{+\infty} f_n \Box^n$ 是 d'Alembertian $\Box$ 的解析函数。我们找到了相应运动方程的一些精确宇宙学解。有两种解仅当 $\Lambda \neq 0, \, k = 0,$ 并且它们在具有宇宙常数 $\Lambda$ 的爱因斯坦引力中没有类似物时才有效。这两个解决方案之一是 $ a (t) = A \, \sqrt{t} \, e^{\frac{\Lambda}{4} t^2} ,$ 模拟类似于辐射之间干扰的属性和暗能量。另一种解决方案是非奇异反弹 - $ a (t) = A \, e^{\Lambda t^2}$。对于这两个解决方案，讨论了一些宇宙学方面。我们还发现了非局部运算符 $\mathcal{F}(\Box)$ 的显式形式，它满足获得的必要条件。