  At large scales of space and time, the nonequilibrium dynamics of local
observables in extensive many-body systems is well described by hydrodynamics.
At the Euler scale, one assumes that each mesoscopic region independently
reaches a state of maximal entropy under the constraints given by the available
conservation laws. Away from phase transitions, maximal entropy states show
exponential correlation decay, and independence of fluid cells might be assumed
to subsist during the course of time evolution. We show that this picture is
incorrect: under ballistic scaling, regions separated by macroscopic distances
develop long-range correlations as time passes. These correlations take a
universal form that only depends on the Euler hydrodynamics of the model. They
are rooted in the large-scale motion of interacting fluid modes, and are the
dominant long-range correlations developing in time from long-wavelength,
entropy-maximised states. They require the presence of interaction and at least
two different fluid modes, and are of a fundamentally different nature from
well-known long-range correlations occurring from diffusive spreading or from
quasi-particle excitations produced in far-from-equilibrium quenches. We
provide a universal theoretical framework to exactly evaluate them, an
adaptation of the macroscopic fluctuation theory to the Euler scale. We verify
our exact predictions in the hard-rod gas, by comparing with numerical
simulations and finding excellent agreement.
