Effective Lagrangian and energy-momentum tensor in de Sitter space

The effective Lagrangian and vacuum energy-momentum tensor 〈T^〉 due to a scalar field in a de Sitter-space background are calculated using the dimensional-regularization method. For generality the scalar field equation is chosen in the form (^2++m^2) =0. If =16 and m=0, the renormalized 〈T^〉 equals g^ (960{^2a^4) }^-1, where a is the radius of de Sitter space. More formally, a general zeta-function method is developed. It yields the renormalized effective Lagrangian as the derivative of the zeta function on the curved space. This method is shown to be virtually identical to a method of dimensional regularization applicable to any Riemann space.