Aging is - almost - like equilibrium

We study and compare equilibrium and aging dynamics on both sides of the ideal glass transition temperature $T_{MCT}$. In the context of a mean field model, we observe that all dynamical behaviors are determined by the energy distance $\epsilon$ to threshold - i.e. marginally stable - states. We furthermore show the striking result that after eliminating age and temperature at the benefit of $\epsilon$, the scaling behaviors above and below $T_{MCT}$ are identical, reconciling {\it en passant} the mean field results with experimental observations. In the vicinity of the transition, we show that there is an exact mapping between equilibrium dynamics and aging dynamics. This leads to very natural interpretations and quantitative predictions for several remarkable features of aging dynamics: waiting time-temperature superposition, interrupted aging, dynamical heterogeneity.