Modified Gravity and Cosmic Acceleration: Now and in the Early Universe

We review our previous works which explored the problems of dark energy and cos- mological constant problem in the context of modified gravity. In the first part, we present extensions of mimetic dark matter. The latter is a Weyl-invariant scalar-tensor theory able to describe dark matter on cosmological scales. In our works we have extended the mimetic construction to vector fields and Yang-Mills gauge fields. The resulting theories provided a novel Weyl-invariant and higher derivative formulations of unimodular gravity. We also introduced a mixture of this mimetic dark energy with mimetic dark matter and showed that it results in k-essence like scalar theory. In the second part we reviewed a minimal modification of Einstein equations, in which their trace part is made trivial. This results in the Newton constant becoming a global degree of freedom. Consequently, the Newton constant is subjected to quantum fluctuations and uncertainty relations. We find that the same applies to the effective Planck constant in a certain classically equivalent formulation of this gravity modification. Finally, we present an analysis of tensor perturbations in the recently proposed models of minimally varying cosmological constant. In these theories, extending Einstein-Cartan gravity, the cosmological constant is allowed to..