  We study the non-equilibrium relaxation of an elastic line described by the
Edwards-Wilkinson equation. Although this model is the simplest representation
of interface dynamics, we highlight that many (not though all) important
aspects of the non-equilibrium relaxation of elastic manifolds are already
present in such quadratic and clean systems. We analyze in detail the aging
behaviour of several two-times averaged and fluctuating observables taking into
account finite-size effects and the crossover to the stationary and equilibrium
regimes. We start by investigating the structure factor and extracting from its
decay a growing correlation length. We present the full two-times and size
dependence of the interface roughness and we generalize the Family-Vicsek
scaling form to non-equilibrium situations. We compute the incoherent cattering
function and we compare it to the one measured in other glassy systems. We
analyse the response functions, the violation of the fluctuation-dissipation
theorem in the aging regime, and its crossover to the equilibrium relation in
the stationary regime. Finally, we study the out-of-equilibrium fluctuations of
the previously studied two-times functions and we characterize the scaling
properties of their probability distribution functions. Our results allow us to
obtain new insights into other glassy problems such as the aging behavior in
colloidal glasses and vortex glasses.
