Causal hydrodynamic fluctuations in a one-dimensional expanding system

We derive stochastic equations of motion of hydrodynamic fluctuations, performing perturbative expansion of the energy-momentum conservation equations around the boost invariant solution in a one-dimensional expanding system. In the course of derivation, we do not assume any specific forms of constitutive equations for shear stress tensor ^ and bulk pressure. 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 on an event-by-event basis. We observe that the structure of energy density and flow rapidity fluctuations is almost frozen in the early stage of the expansion. Two-point correlations of energy density fluctuations turn out to be closely related to 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 freeze-out 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.