  Physical aging deals with slow property changes over time caused by molecular
rearrangements. This is relevant for non-crystalline materials like polymers
and inorganic glasses, both in production and during subsequent use. The
Narayanaswamy theory from 1971 describes physical aging - an inherently
nonlinear phenomenon - in terms of a linear convolution integral over the
so-called material time $\xi$. The resulting "Tool-Narayanaswamy (TN)
formalism" is generally recognized to provide an excellent description of
physical aging for small, but still highly nonlinear temperature variations.
The simplest version of the TN formalism is single-parameter aging according to
which the clock rate $d\xi/dt$ is an exponential function of the property
monitored [T. Hecksher et al., J. Chem. Phys. 142, 241103 (2015)]. For
temperature jumps starting from thermal equilibrium, this leads to a
first-order differential equation for property monitored, involving a
system-specific function. The present paper shows analytically that the
solution to this equation to first order in the temperature variation has a
universal expression in terms of the zeroth-order solution, $R_0(t)$. Numerical
data for a binary Lennard-Jones glass former probing the potential energy
confirm that, in the weakly nonlinear limit, the theory predicts aging
correctly from $R_0(t)$ (which by the fluctuation-dissipation theorem is the
normalized equilibrium potential-energy time-autocorrelation function).
