Causal hydrodynamic fluctuations in non-static and inhomogeneous backgrounds

To integrate hydrodynamic fluctuations, namely thermal fluctuations of hydrodynamics, into dynamical models of high-energy nuclear collisions based on relativistic hydrodynamics, the property of the hydrodynamic fluctuations given by the fluctuation-dissipation relation should be carefully investigated. The fluctuation-dissipation relation for causal dissipative hydrodynamics with the finite relaxation time is naturally given in the integral form of the constitutive equation by the linear-response theory. While, the differential form of the constitutive equation is commonly used in analytic investigations and dynamical calculations for practical reasons. We give the fluctuation-dissipation relation for the general linear-response differential form and discuss the restrictions to the structure of the differential form, which comes from the causality and the positive semi-definiteness of the noise autocorrelation, and also the relation of those restrictions to the cutoff scale of the hydrodynamic fluctuations. We also give the fluctuation-dissipation relation for the integral form in non-static and inhomogeneous background by introducing new tensors, the pathline projectors. We find new modification terms to the fluctuation-dissipation relation for the differential form in non-static and inhomogeneous background which are particularly important in dynamical models to describe rapidly expanding systems.