  We study aging dynamics in two non-disordered spin models with multi-spin
interactions, following a sudden quench to low temperature. The models are
relevant to the physics of supercooled liquids. Their low temperature dynamics
resemble those of kinetically constrained models, and obey dynamical scaling,
controlled by zero-temperature critical points. Dynamics in both models are
thermally activated, resulting in multi-stage relaxation towards equilibrium.
We study several two-time correlation and response functions. We find that
equilibrium fluctuation-dissipation relations are generically not satisfied
during the aging regime, but deviations from them are well described by
fluctuation-dissipation ratios, as found numerically in supercooled liquids.
These ratios are purely dynamic objects, containing information about the
nature of relaxation in the models. They are non-universal, and can even be
negative as a result of activated dynamics. Thus, effective temperatures are
not well-defined in these models.
