Regular black hole and cosmological spacetimes in Non-Polynomial Gravity theories

General Relativity is known to suffer from singularities at short distances, which indicates the breakdown of its predictability, for instance at the center of black holes, and in the very early universe. This is one of the main reason to look for a Quantum Theory of Gravity, that would describe spacetime geometry as a quantum field, and possibly cure these classical singularities. However, no consensus on the topic has yet been reached, as many different approaches have been proposed, but none has yet received an experimental confirmation. This is in part due to the extraordinary small scale at which quantum gravitational effects are expected to become dominant, and to the technical difficulty to make unambiguous predictions. For this reason, many works have focused on the so-called effective approaches in which the possible high energy corrections to General Relativity are classified, and their theoretical and ob- servational predictions derived, with the idea that among these modifications, some could come as the semi-classical limits of quantum gravity theories. A way to discriminate between the different proposals is precisely the absence of singular geometries in their solutions. In the first two Chapters of this thesis, we will present such an effective approach, in which the action of General Relativity is modified at high energy by non-polynomial curvature invariants, which are constructed in such a way that the dynamical spherically symmetric sector of these theories (which contain both cosmological and non-rotating black hole spacetimes) yield second order field equations. These properties of the non-polynomial invariants follow from a peculiar algebraic identity satisfied by the Cotton tensor in this class of geometries. As we will see in the last two Chapters, having second order dynamical spherically symmetric field equations is necessary in order to recover some quantum corrected geometries that have been found from more fundamental approaches like Loop Quantum Cosmology and Asymptotic Safety, within its Einstein-Hilbert truncation. The existence of such gravitational models provides an interpretation of two-dimensional Horn- deski theory as describing the dynamical spherically symmetric sector of specific higher dimensional non-polynomial gravity theories. Therefore, it allows to have some concrete d-dimensional formu- lations of the two-dimensional Einstein-Dilaton and Lovelock Designer effective approaches that have been studied extensively, in particular to find and study the properties of non-singular black holes. This enables us to propose two four-dimensional effective-like actions, which are constructed in such a way that their dynamical spherically symmetric sectors decompose in the same way as those of General Relativity and Gauss-Bonnet gravity. In the remaining Chapters, we essentially investigate the solutions and properties of these theories. It is shown that the first one leads to regular (A)dS-core black hole solutions, with the correct quantum correction to their Newton potentials and logarithmic correction to their entropies. The charged generalization is considered, and a way to avoid the mass inflation instability of their inner horizons is found, provided that a bound between the mass and the charge is satisfied. In Chap. 4, we establish a reconstruction procedure able to find theories admitting as solutions the Modesto semi-polymeric black hole, as well as the D’Ambrosio-Rovelli and Visser-Hochberg geometries. All these black holes are regular and derived or inspired by quantum gravity results. They have many properties in common, as for example the fact that they automatically regularize the Coulomb singularity of a static electric field. Finally, the last Chapter is devoted to the theory whose dynamical spherically symmetric sector is a generalization of the one of Gauss-Bonnet gravity. It is shown that the Loop quantum cosmology bounce universe and some Asymptotic Safety black holes can be reconstructed from two members of these theories. In particular, the associated black hole solutions of the first are regular, and the associated cosmological solution of the second is as well, and describe a universe which is eternal in the past, and behaves as de Sitter spacetime in the limit of infinite past. Some generalizations of these results are provided, and the Mimetic gravity formulations of the cosmological solutions are found.