Scalar field in standard cosmology: Time equation

The separated time equation relative to the generalized Klein-Gordon equati on in the Robertson-Walker space-time is integrated in the background of th e standard cosmology. The solutions are given in terms of series that are o btained by the usual integration method of differential equations with regu lar singularity. The normalization of the solutions implied by the request of second quantization of the scalar field is performed. The result exhaust s the request of providing an explicit complete set of normal modes solutio ns of the scaler field equation in the standard cosmology.