1. Adiabatic and isocurvature conditions There are only two kinds of perturbations that can be generated. The first one is the adiabatic perturbation discussed previously; it is a perturbation along the late time classical trajectories of the scalar fields during inflation. When the primordial perturbations enter our horizon they perturb the matter density with a generic adiabatic condition , which is satisfied when the density contrast of the individual species is related to the total density contrast δ$_{k}$ [24, 25] where b stands for baryons, c for cold dark matter, γ for photons and ν for neutrinos. The other type is the isocurvature perturbation. During inflation this can be viewed as a perturbation orthogonal to the unique late time classical trajectory. Therefore, if there were N fluctuating scalar fields during inflation, there would be N − 1 degrees of freedom which would contribute to the isocurvature perturbation [216–219]. The isocurvature condition is known as δρ = 0: the sum total of all the energy contrasts must be zero. The most general density perturbations is then given by a linear combination of an adiabatic and an isocurvature density perturbations. In a comoving gauge, see Eq. (22), ζ = − Hδφ/ ˙ φ holds good even for multi-field inflation models, provided we identify each field component of φ along the slow-roll direction. There also exists a relationship between the comoving curvature perturbations and the number of e-foldings, N , given by [213, 220] 2. Adiabatic perturbations due to multi-field where N is measured by a comoving observer while passing from flat hypersurface (which defines δφ ) to the comoving hypersurface (which determines ζ , where it remains conserved on large scales even for multi-field case [183].). The repeated indices are summed over and the subscript a denotes a component of the inflaton. A more intuitive discussion has been given in [10, 158, 211]. 22 1 3 δ$_{kb}$ = 1 3 δ$_{kc}$ = 1 4 δ$_{kν}$ = 1 4 δ$_{kγ}$ = 1 4 δ$_{k}$ , (49) ζ = δN = ∂N ∂φ$_{a}$ δφ$_{a}$ , (50)