The recent nonlocal generalization of Einstein's theory of gravitation reduces in the Newtonian regime to a nonlocal and nonlinear modification of Poisson's equation of Newtonian gravity. The nonlocally modified Poisson equation implies that nonlocality can simulate dark matter. Observational data regarding dark matter provide limited information about the functional form of the reciprocal kernel, from which the original nonlocal kernel of the theory must be determined. We study this inverse problem of nonlocal gravity in the linear domain, where the applicability of the Fourier transform method is critically examined and the conditions for the existence of the nonlocal kernel are discussed. This approach is illustrated via simple explicit examples for which the kernels are numerically evaluated. We then turn to a general discussion of the modified Poisson equation and present a formal solution of this equation via a successive approximation scheme. The treatment is specialized to the gravitational potential of a point mass, where in the linear regime we recover the Tohline-Kuhn approach to modified gravity.
New Cosmological Solutions in Nonlocal Modified Gravity
考虑了具有$Λ$项的非定域修正引力的宇宙学性质，其中非定域性是$R \mathcal{F}(\Box) R$型的。利用形式为$\Box R = r R +s,$的解析式，我们找到了几个a（t）非奇异的反弹宇宙学解，它们对应于空间曲率参数k的所有三个值。我们还从F（R）理论的角度讨论了这种改进的引力模型。
我们考虑非局部修正引力的某些方面，其中非局部性是$R \mathcal{F}(\Box) R$类型。特别地，使用形式为$\Box R = c R^γ,$的矩阵，我们找到了空间平坦FLRW度量的几个$R(t)$解。有奇异和非奇异反弹的解决方案。在宇宙时间晚期，标量曲率R（t）处于低状态，标度因子a（t）减速。当k =-1时，R（t）= 0满足所有方程。
