  We derive equations of motion of hydrodynamic fluctuations performing
perturbative expansion of the energy-momentum conservation equations around the
boost invariant solution in one-dimensional expanding system. In the course of
derivation, we do not assume any specific forms of constitutive equations for
shear stress tensor $\pi^{\mu \nu}$ and bulk pressure $\Pi$. Therefore, the
framework enables us to employ any constitutive equations beyond the
Navier-Stokes theory which satisfy the causality. Employing Israel-Stewart
equations as examples of the constitutive equations, we demonstrate the
dynamics of causal hydrodynamic fluctuations in (1+1)-dimensional Milne
coordinates. We observe that structure of energy density fluctuations is almost
frozen in the early stage of the expansion. Two-point correlations of energy
density fluctuations turn out to be closely related with the properties of the
medium such as sound velocity, viscosity, and relaxation time. Furthermore, we
show that two-particle correlation functions of final hadrons after freezeout
inherit correlations of thermodynamic variables and flow rapidity. This opens a
new door for an analysis of transport properties of the medium produced in
relativistic heavy ion collisions.
