Evolution of the curvature perturbation during and after multi-field inflation

Abstract

We study the evolution of the curvature perturbation on the super-horizon scalesstarting from the inflationary epoch until there remains only a single dynamicaldegree of freedom, presureless matter, in the universe. We consider the cosmicinflation driven by a multiple number of the inflaton fields, which decay into bothradiation and pressureless matter components. We present a complete set of the exactbackground and perturbation equations which describe the evolution of the universethroughout its history. By applying these equations to the simple but reasonablemodel of multi-field chaotic inflation, we explicitly show that the total curvatureperturbation is continuously varying because of the non-adiabatic components of thecurvature perturbation generated by the multiple inflaton fields throughout thewhole evolution of the universe. We also provide an useful analytic estimation ofthe total as well as matter and radiation curvature perturbations, assuming thatmatter is completely decoupled from radiation from the beginning. The resultingisocurvature perturbation between matter and radiation is at most sub-percent levelwhen the masses of the inflaton fields are distributed between 10−6mPl and 10−5mPl. We find that this result is robust unless we use non-trivial decayrates, and that thus, in general, it is hard to obtain large matter-radiationisocurvature perturbation. Also, by using the δN formalism, we point outthat the inflationary calculation, especially when involving multiple inflatonfields, is likely to lose the potentially important post-inflationary evolutionwhich can modify the resulting curvature perturbation.