  For the models of inflation driven by the potential energy of an inflaton
field $\phi$, the covariant Galileon Lagrangian $(\partial\phi)^2\Box \phi$
generally works to slow down the evolution of the field. On the other hand, if
the Galileon self-interaction is dominant relative to the standard kinetic
term, we show that there is no oscillatory regime of inflaton after the end of
inflation. This is typically accompanied by the appearance of the negative
propagation speed squared $c_s^2$ of a scalar mode, which leads to the
instability of small-scale perturbations. For chaotic inflation and natural
inflation we clarify the parameter space in which inflaton oscillates
coherently during reheating. Using the WMAP constraints of the scalar spectral
index and the tensor-to-scalar ratio as well, we find that the self coupling
$\lambda$ of the potential $V(\phi)=\lambda \phi^4/4$ is constrained to be very
much smaller than 1 and that the symmetry breaking scale $f$ of natural
inflation cannot be less than the reduced Planck mass $M_{\rm pl}$. We also
show that, in the presence of other covariant Galileon Lagrangians, there are
some cases in which inflaton oscillates coherently even for the self coupling
$\lambda$ of the order of 0.1, but still the instability associated with
negative $c_s^2$ is generally present.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology;High Energy Physics - Theory