Generic features of the fluctuation dissipation relation in coarsening systems

The integrated response function in phase-ordering systems with scalar, vector, conserved and non conserved order parameter is studied at various space dimensionalities. Assuming scaling of the aging contribution $\chi_{ag} (t,t_w)= t_w ^{-a_\chi} \hat \chi (t/t_w)$ we obtain, by numerical simulations and analytical arguments, the phenomenological formula describing the dimensionality dependence of $a_\chi$ in all cases considered. The primary result is that $a_\chi$ vanishes continuously as $d$ approaches the lower critical dimensionality $d_L$. This implies that i) the existence of a non trivial fluctuation dissipation relation and ii) the failure of the connection between statics and dynamics are generic features of phase ordering at $d_L$.