  In this dissertation, we explore models based on the idea that there are two
metrics in spacetime: One describes the standard gravity, and the other
provides a geometry in which matter fields propagate. In order to do that, we
provide the essentials of Finsler geometry and the rules to induce a metric for
the propagation of matter. Such a description will cover some of the most
critical features related to the field necessary to do the induction, these
will arise in an attempt to build an action for this field. And finally, we
provide an example to study the homogeneous limit of background FLRW equations
for the cosmological model and the role of Lorentz symmetry breaking to provide
a graceful exit.
