We study cosmological perturbations generated from quantum fluctuations in multi-field inflationary scenarios in generalized Einstein theories, taking both adiabatic and isocurvature modes into account. In the slow-roll approximation, explicit closed-form long-wave solutions for field and metric perturbations are obtained by the analysis in the Einstein frame. Since the evolution of fluctuations depends on specific gravity theories, we make detailed investigations based on analytic and numerical approaches in four generalized Einstein theories: the Jordan-Brans-Dicke (JBD) theory, the Einstein gravity with a non-minimally coupled scalar field, the higher-dimensional Kaluza-Klein theory, and the R+R2 theory with a non-minimally coupled scalar field. We find that solutions obtained in the slow-roll approximation show good agreement with full numerical results except around the end of inflation. Due to the presence of isocurvature perturbations, the gravitational potential Φ and the curvature perturbation ζ do not remain constant on super-horizon scales. In particular, we find that negative non-minimal coupling can lead to strong enhancement of ζ in both the Einstein and higher derivative gravity, in which case it is difficult to unambiguously decompose scalar perturbations into adiabatic and isocurvature modes during the whole stage of inflation.
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