Covariant c-flation: a variational approach

We study how a minimal generalization of Einstein's equations, where the speed of light ($c$), gravitational constant ($G$) and the cosmological constant ($\Lambda$) are allowed to vary, might generate a dynamical mechanism to explain the special initial condition necessary to obtain the homogeneous and flat universe we observe today. Our construction preserves general covariance of the theory, which yields a general dynamical constraint in $c$, $G$ and $\Lambda$. We re-write the conditions necessary in order to solve the horizon and flatness problems in this framework. This is given by the shrinking of the comoving particle horizon of this theory which leads to $\omega < -1/3$, but not necessarily to accelerated expansion like in inflation, allowing also a decelerated expansion, contraction and a phase transition in $c$, in the case of null $\Lambda$. We are able to construct the action of this theory, that describes the dynamics of a scalar field that represents $c$ or $G$ (and $\Lambda$). This action is general and can be applied to describe different cosmological solutions. We present here how the dynamics of the field can be used to solve the problems of the early universe cosmology by means of different ways to c-inflate the horizon in the early universe, solving the old puzzles of the cosmological standard model. Without a cosmological constant, we show that we can describe the dynamics of the scalar field representing $c$ given a potential, and derive the slow-roll conditions that this potential should obey. In this setup we do not have to introduce an extra unknown scalar field, since the degree of freedom associated to the varying constants plays this role, naturally being the field that is going to be responsible for inflating the horizon in the early universe.