  We consider the unitary time evolution of a one-dimensional cloud of
hard-core bosons loaded on a harmonic trap potential which is slowly released
in time with a general ramp $g(t)$. After the identification of a typical
length scale $\ell(t)$, related to the time ramp, we focus our attention on the
dynamics of the density profile within a first order time-dependent
perturbation scheme. In the special case of a linear ramp, we compare the first
order predictions to the exact solution obtained through Ermakov-Lewis
dynamical invariants. We also obtain an exact analytical solution for a cloud
released from a harmonic trap with an amplitude that varies as the inverse of
time. In such situation, the typical size of the cloud grows with a power law
governed by an exponent that depends continuously on the initial trap
frequency. At high enough initial trap amplitude, the exponent acquires an
imaginary part that leads to the emergence of a log-periodic modulation of the
cloud expansion.
