Graviton Production in Elliptical and Hyperbolic Universes

The problem of cosmological graviton creation for homogeneous and isotropic universes with elliptical ($\text{vae}=+1$) and hyperbolical ($\text{vae}=-1$) geometries is addressed. The gravitational wave equation is established for a self-gravitating fluid satisfying the barotropic equation of state $p=(\gamma -1)\rho$,whichisthesourceoftheEinstei{n}^{\mathrm{\prime }}sequationsplusacosmological$\Lambda$-term. The time dependent part of this equation is exactly solved in terms of hypergeometric functions for any value of $\gamma$ and spatial curvature $\text{vae}$. An expression representing an adiabatic vacuum state is then obtained in terms of associated Legendre functions whenever $\gamma\neq \frac{2}{3}; \frac{(2n+1)}{(2n-1)}$, where n is an integer. This includes most cases of physical interest such as $\gamma =0,4\mathrm{/}3,1$. The mechanism of graviton creation is reviewed and the Bogoliubov coefficients related to transitions between arbitrary cosmic eras are also explicitly evaluated.