Evolution of the curvature perturbation during and after multi-field   inflation
  We study the evolution of the curvature perturbation on the super-horizon scales starting from the inflationary epoch until there remains only a single dynamical degree of freedom, presureless matter, in the universe. We consider the cosmic inflation driven by a multiple number of the inflaton fields, which decay into both radiation and pressureless matter components. We present a complete set of the exact background and perturbation equations which describe the evolution of the universe throughout its history. By applying these equations to the simple but reasonable model of multi-field chaotic inflation, we explicitly show that the total curvature perturbation is continuously varying because of the non-adiabatic components of the curvature perturbation generated by the multiple inflaton fields throughout the whole evolution of the universe. We also provide an useful analytic estimation of the total as well as matter and radiation curvature perturbations, assuming that matter is completely decoupled from radiation from the beginning. The resulting isocurvature perturbation between matter and radiation is at most sub-percent level when the masses of the inflaton fields are distributed between $10^{-6}m_{pl}$ and $10^{-5}m_{pl}$. We find that this result is robust unless we use non-trivial decay rates, and that thus, in general, it is hard to obtain large matter-radiation isocurvature perturbation. Also, by using the $\delta{N}$ formalism, we point out that the inflationary calculation, especially when involving multiple inflaton fields, is likely to lose the potentially important post-inflationary evolution which can modify the resulting curvature perturbation. 
Ki-Young Choi, Jinn-Ouk Gong and Donghui Jeong