Effective Lagrangian and energy-momentum tensor in de Sitter space

Abstract

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