In this paper we will construct a linearized metric solution for an electrically charged system in a ghost-free infinite derivative theory of gravity which is valid in the entire region of spacetime. We will show that the gravitational potential for a point-charge with mass m is non-singular, the Kretschmann scalar is finite, and the metric approaches conformal-flatness in the ultraviolet regime where the non-local gravitational interaction becomes important. We will show that the metric potentials are bounded below one as long as two conditions involving the mass and the electric charge are satisfied. Furthermore, we will argue that the cosmic censorship conjecture is not required in this case. Unlike in the case of Reissner-Nordström in general relativity, where |Q| ≤ m/Mp has to be always satisfied, in ghost-free infinite derivative gravity |Q| > m/Mp is also allowed, such as for an electron.

We study (covariant) scalar-vector-tensor (SVT) perturbations of infinite derivative gravity (IDG), at the quadratic level of the action, around conformally flat, covariantly constant curvature backgrounds that are not maximally symmetric spacetimes (MSS). This extends a previous analysis of perturbations done around MSS, which were shown to be ghost-free. We motivate our choice of backgrounds that arise as solutions of IDG in the UV, avoiding big bang and black hole singularities. Contrary to MSS, in this paper we show that, generically, all SVT modes are coupled to each other at the quadratic level of the action. We consider simple examples of the full IDG action, and we illustrate this mixing and also a case where the action can be diagonalized and a ghost-free spectrum constructed. Our study is widely applicable for both nonsingular cosmology and black hole physics where backgrounds depart from MSS. In the appendixes, we provide SVT perturbations around conformally flat and arbitrary backgrounds which can serve as a compendium of useful results when studying various higher derivative gravity models.

We study a class of higher derivative, nonlocal gravity with a cosmological constant which admits homogeneous and isotropic nonsingular bouncing universes in the absence of matter. We show that, at the linearized level around the bounce, it is possible to constrain the nonlocal form factor such that there is only a scalar propagating degree of freedom, and no vector or tensor modes. The scalar mode can be made free from perturbative ghost instabilities and has oscillatory and bounded evolution across the bounce.

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We show that it is possible to realize a cosmological bouncing solution in an anisotropic but homogeneous Bianchi-I background in a class of non-local, infinite derivative theories of gravity. We show that the anisotropic shear grows slower than in general relativity during the contraction phase, peaks to a finite value at the bounce point, and then decreases as the universe asymptotes towards isotropy and homogeneity, and ultimately to de Sitter. Along with a cosmological constant, the matter sector required to drive such a bounce is found to consist of three components — radiation, stiff matter and k-matter (whose energy density decays like the inverse square of the average scale factor). Generically, k-matter exerts anisotropic pressures. We will test the bouncing solution in local and non-local gravity and show that in the latter case it is possible to simultaneously satisfy positivity of energy density and, at least in the late time de Sitter phase, avoid the introduction of propagating ghost/tachyonic modes.