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Motivated by a recent work by Terashima (Phys. Rev. D60 084001), we revisitthe fluctuation-dissipation (FD) relation between the dissipative coefficientof a detector and the vacuum noise of fields in curved spacetime. In anexplicit manner we show that the dissipative coefficient obtained fromclassical equations of motion of the detector and the scalar (or Dirac) fieldsatisfies the FD relation associated with the vacuum noise of the field, whichdemonstrates that the Terashima's prescription works properly in the$N$-dimensional de Sitter spacetime. This practice is useful not only toreconfirm the validity of the use of the retarded Green function to evaluatethe dissipative coefficient from the classical equations of motion but also tounderstand why the derivation works properly, which is discussed in connectionwith previous investigations on the basis on the Kubo-Martin-Schwinger (KMS)condition. Possible application to black hole spacetime is also brieflydiscussed.Comment: 22 pages including 2 figures, to be published in Int. J. Mod. Phys.

EXPLICIT DERIVATION OF THE FLUCTUATION-DISSIPATION RELATION OF THE VACUUM NOISE IN THE N-DIMENSIONAL DE SITTER SPACE–TIME