Effective Lagrangian and energy-momentum tensor in de Sitter space

The effective Lagrangian and vacuum energy-momentum tensor $〈T^{\mu \nu }〉$ 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 $({\square }^2+\xi R+m^2)\varphi =0\mathrm{}$. If $\xi =\frac{1}{6}$ and $m=0\mathrm{}$, the renormalized $〈T^{\mu \nu }〉$ equals $g^{\mu \nu }{\left(960\mathrm{}{\pi }^2{a}^4\right)}^{-1\mathrm{}}$, 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.