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We show, using the quench action approach [Caux&Essler Phys. Rev. Lett. 110 (2013)], that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient $s_0^{\Psi_0}(\lambda)$. This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of $s_0^{\Psi_0}(\lambda)$ in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density-density correlation in the interacting Lieb-Liniger gas after a quench from a Bose-Einstein condensate. This shows an approach to equilibrium with power law $t^{-3}$ which turns out to be independent of the post-quench interaction and of the considered observable.

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