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We present the LWL formula which represents the long wavelengh limit of thesolutions of evolution equations of cosmological perturbations in terms of theexactly homogeneous solutions in the most general case where multiple scalarfields and multiple perfect fluids coexist. We find the conserved quantitywhich has origin in the adiabatic decaying mode, and by regarding this quantityas the source term we determine the correction term which corrects thediscrepancy between the exactly homogeneous perturbations and the $k \to 0$limit of the evolutions of cosmological perturbations. This LWL formula isuseful for investigating the evolutions of cosmological perturbations in theearly stage of our universe such as reheating after inflation and the curvatondecay in the curvaton scenario. When we extract the long wavelength limits ofevolutions of cosmological perturbations from the exactly homogeneosperturbations by the LWL formula, it is more convenient to describe thecorresponding exactly homogeneous system with not the cosmological time but thescale factor as the evolution parameter. By applying the LWL formula to thereheating model and the curvaton model with multiple scalar fields and multipleradiation fluids, we obtain the S formula representing the final amplitude ofthe Bardeen parameter in terms of the initial adiabatic and isocurvatureperturbations  Keywords:cosmological perturbations,long wavelength limit,reheating,curvaton  PACS number(s):98.80.Cq

Long wavelength limit of evolution of cosmological perturbations in the universe where scalar fields and fluids coexist