Effective Lagrangian and energy-momentum tensor in de Sitter space

The effective Lagrangian and vacuum energy-momentum tensor 〈T^μν〉 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+ξR+m²)φ=0. If ξ=1/6 and $m=0$, the renormalized 〈T^μν〉 equals g^μν(960π²a⁴)^-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.