Graviton production in elliptical and hyperbolic universes

The problem of cosmological graviton creation for homogeneous and isotropic universes with elliptical (epsilon=+1) and hyperbolical (epsilon=-1) geometries is addressed. The gravitational wave equation is established for a self-gravitating fluid satisfying the barotropic equation of state p=(gamma-1)rho, which is the source of Einstein's equations plus a cosmological Lambda term. The time-dependent part of this equation is exactly solved in terms of hypergeometric functions for any value of gamma and spatial curvature epsilon. An expression representing ari adiabatic vacuum state is then obtained in terms of associated Legendre functions whenever gamma not equal 2/3 [(2n+1)/(2n-1)], where n is an integer. This includes most cases of physical interest such as gamma=0, 4/3,1. The mechanism of graviton creation is reviewed and the Bogoliubov coefficients related to transitions between arbitrary cosmic eras are also explicitly evaluated.