Applications of toric Calabi-Yau singularities to Cosmological model building in String Theory

String Theory is nowadays the best candidate to describe gravity at the quantum level together with the other interactions and matter fields. Therefore, it should be able to describe physics at the most fundamental level. In order to describe many phenomena consistently it is often necessary to provide some mechanism creating hierarchies between the different scales in the physical sys- tem. These kind of hierarchies admit a natural explanation in terms of warping on the internal space, which leads to suppression factors on the 4 dimensional field theory. In String Theory these warp suppressions can be obtained by the use of the so-called warped throats. In this thesis I will use toric Calabi-Yau singularities in order to build up warped throats where different physical phenomena happen. Most phenomena I focus on are related to dark energy, the source of accelerate expansion of the Universe on its very early stages and nowadays. I also present some new techniques in order to guess which throats admit orientifold actions, and use them to describe UV completions of stringy instantons in terms of gauge instantons in such geometries.