slow-roll approximation, the equation of motion (\[eom\]) can be solved to give: - , -e\^[-2b]{} , 3H\^2V(,) .

The perturbation of the model
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Calculation of the perturbations in Einstein frame
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In this subsection, we first calculate the perturbations of our model in its Einstein frame, namely in form of a two-field inflation model (\[einstein\]). The analysis of two-field inflation model has been well-developed and the detailed calculation can be found in e.g. [@Gong:2011qe; @Gordon:2000hv; @DiMarco:2002eb; @Lalak:2007vi], and here to be more concise we will only summarize their results which is needed for our later study. First of all, the perturbed metric can in general be formulated as: \[perturbmetric\] ds\^2&=&-(1+2)dt\^2+2a(t)\_idtdx\^i\
&&+a\^2(t)\[(1-2)\_[ij]{}+2\_i\_j E\]dx\^idx\^j , while the field can also be perturbed as \_0(t)+(x) , \_0(t)+(x) .

It is convenient to define variables of perturbation that are invariant under gauge transformation. The often-used gauge invariant variables are: &&-\[a\^2(-/a)\]\^ ,\
&&+a\^2H(-/a) ,\
R&&-q ,\
Q\^i&&\^i+ ,    (\^i=,) , where $\Phi=\Psi$ is the Newtonian potential, $\cal R$ is the comoving curvature perturbation, and $Q^i$ is the gauge-invariant perturbation the $i$-th field. Moreover, $\delta q$ defined according to the relation $\partial_i\delta q=\delta T^0_i$ where $T^0_i$ is the $(0,i)$ component of the energy-momentum tensor of 