For the models of inflation driven by the potential energy of an inflaton field φ, the covariant Galileon Lagrangian (φ) 2φ 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 λ of the potential V(φ) = λφ 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 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 λ of the order of 0.1, but still the instability associated with negative c s 2 is generally present.
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