  In this work, we study inflation in a particular scalar-vector-tensor theory
of gravitation without the $U(1)$ gauge symmetry. The model is constructed from
the more general action introduced in Heisenberg et al. (Phys Rev D 98:024038,
2018) using certain specific choices for the Lagrangians and the coupling
functions. Also, for this model we build the explicit form for the action, and
from it, we derive the general equations: the energy-momentum tensor and the
equations of motion, and using the flat FLRW background, we have analyzed if
it's possible to obtain an inflationary regime with it. Additionally, using
particular choices for the potential, the coupling functions, suitable
dimensionless coupling constants and initial conditions, it was possible verify
numerically that this model of inflation is viable. In this sense, we could
verify that the introduction of the coupling function $f(\phi)$ in our model of
inflation, allows us to reach a suitable amount of $e$-foldings $N$ for
sufficient inflation. This is a remarkable result, since without the coupling
function contribution, the amount of $e$-foldings is smaller at the end of
inflation, as has been demonstrated in Heisenberg et al. (2018). Also, the
no-ghosts and stability conditions that the model during inflation must
satisfy, i.e., absence of ghosts and Laplacian instabilities of linear
cosmological perturbations were obtained, furthermore these conditions were
verified numerically too.
