Graviton production in elliptical and hyperbolic universes.

Abstract

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