Correlators in phase-ordering from Schr\"odinger-invariance

Systems undergoing phase-ordering kinetics after a quench into the ordered phase with $0<T<T_c$ from a fully disordered initial state and with a non-conserved order-parameter have the dynamical exponent ${z}=2$. The long-time behaviour of their single-time and two-time correlators, determined by the noisy initial conditions, is derived from Schr\"odinger-invariance and we show that the generic ageing scaling forms of the correlators follow from the Schr\"odinger covariance of the four-point response functions. The autocorrelation exponent $\lambda$ is related to the passage exponent $\zeta_p$ which describes the time-scale for the cross-over into the ageing regime. Both Porod's law and the bounds $d/2 \leq \lambda \leq d$ are reproduced in a simple way. The dynamical scaling in fully finite systems and of global correlators is found and the low-temperature generalisation $\lambda= d-2\Theta$ of the Janssen-Schaub-Schmittmann scaling relation is derived.