Vacuum fluctuations of a quantized scalar field in a Robertson-Walker universe

The vacuum expectation value of the energy-momentum tensor of a quantized scalar field is calculated in a Robertson-Walker universe. The resulting divergences are regularized by averaging over an appropriate mass spectrum, which fulfills the same regularization conditions as needed for regularization of the vacuum energy-momentum tensor in Minkowski space. Up to higher orders in time derivatives and inverse radius of universe, there result three contributions to the vacuum energy-momentum density: a cosmological term, a term proportional to the Einstein tensor, and a term derivable by variation of a gravitational Lagrangian containing quadratic expressions of the curvature quantities $R$, $R_{\mu \nu }$, $R_{\mu \nu \rho \sigma }$. These contributions are estimated with the following result: There is no direct evidence that the collapse of a Robertson-Walker universe is averted at some realistic dimensions of the universe, or that gravitation is described in a natural way by the elastic properties of the vacuum of some particles. On the other hand, there is given some evidence that quantum corrections of the type shown must be taken into account in general relativity and are important at least for highly collapsed states of the early universe.