  We show that higher-order actions for cosmological perturbations in the
multi-field DBI-inflation model are obtained by a Lorentz boost from the rest
frame of the brane to the frame where the brane is moving. We confirm that this
simple method provides the same third- and fourth- order actions at leading
order in slow-roll and in the small sound speed limit as those obtained by the
usual ADM formalism. As an application, we compute the leading order connected
four-point function of the primordial curvature perturbation coming from the
intrinsic fourth-order contact interaction in the multi-field DBI-inflation
model. At the third order, the interaction Hamiltonian arises purely by the
boost from the second-order action in the rest frame of the brane. The boost
acts on the adiabatic and entropy modes in the same way thus there exists a
symmetry between the adiabatic and entropy modes. But at fourth order this
symmetry is broken due to the intrinsic fourth-order action in the rest frame
and the difference between the Lagrangian and the interaction Hamiltonian.
Therefore, contrary to the three-point function, the momentum dependence of the
purely adiabatic component and the components including the entropic
contributions are different in the four-point function. This suggests that the
trispectrum can distinguish the multi-field DBI-inflation model from the single
field DBI-inflation model.
