Explicit Derivation of the Fluctuation-Dissipation Relation of the Vacuum Noise in the N-dimensional de Sitter Spacetime

Motivated by a recent work by Terashima (Phys. Rev. D60 084001), we revisit the fluctuation-dissipation (FD) relation between the dissipative coefficient of a detector and the vacuum noise of fields in curved spacetime. In an explicit manner we show that the dissipative coefficient obtained from classical equations of motion of the detector and the scalar (or Dirac) field satisfies the FD relation associated with the vacuum noise of the field, which demonstrates that the Terashima's prescription works properly in the $N$-dimensional de Sitter spacetime. This practice is useful not only to reconfirm the validity of the use of the retarded Green function to evaluate the dissipative coefficient from the classical equations of motion but also to understand why the derivation works properly, which is discussed in connection with previous investigations on the basis on the Kubo-Martin-Schwinger (KMS) condition. Possible application to black hole spacetime is also briefly discussed.