Following [@Gong:2011qe; @Gordon:2000hv; @DiMarco:2002eb; @Lalak:2007vi], we give the expression of adiabatic and entropy field perturbations ($Q^\sigma$ and $Q^s$) as: \[QsigmaQs\] Q\^=+e\^[2b]{} , Q\^s=-e\^[b]{}+e\^[b]{} , where $\dot\sigma=\sqrt{\dot\phi^2+e^{2b}\dot \chi^2}$ is the velocity of the background part of the adiabatic field $\sigma$. From the definition of $\cal R$ one can have \[curvature\] [R]{}=Q\^  in flat gauge. Similarly one can define the isocurvature perturbation as ${\cal S}=(H/\dot\sigma)Q^s$.

From action (\[einstein\]) as well as Eq. (\[QsigmaQs\]), one can finally obtain the equation of motion of $Q^\sigma$ and $Q^s$ after a long derivation [@Lalak:2007vi]: \[eomQsigma\] Q\^+3HQ\^+(+\_\^2)Q\^=-( Q\^s)\^+ Q\^s , where \_\^2=--(3H+) , and \[eomQs\] Q\^s+3HQ\^s+(+\_s\^2+)Q\^s= , where \[mus\] \_s\^2=V\_[,ss]{}+R\_F- , and the right hand side which is proportional to $k^2$ is negligible on large scales. Here $R_F$ is the scalar curvature in the [*field space*]{}, and in our model, we have $R_F=-2b_\phi^2$. Another useful relationship is between $V_{,s}$ and the slow-roll parameters, namely [@Lalak:2007vi]: \[Vs\] =H\_[s]{}-b\_\^3 .

By manipulating these equations, we can finally get a solution of $Q^\sigma$ and $Q^s$ as: Q\^Q\^s\~H\^[(1)]{}\_(-k) , where we have taken the slow-roll limit and assume that the correlation between $Q^\sigma$ and $Q^s$ is small. $H^{(1)}_{3/2}$ is the first kind Hankel function of order $3/2$. In the $k\rightarrow\infty$ limit, 