  A scalar field $\phi$ endowed with a trigonometric potential has been
proposed to play the role of Dark Matter. A deep study of the cosmological
evolution of linear perturbations, and its comparison to the Cold Dark Matter
(CDM) and Fuzzy Dark Matter (FDM) cases (scalar field with quadratic
potential), reveals an enhancement in the amplitude of the mass power spectrum
for large wave numbers due to the non-linearity of the axion-like potential.
For the first time, we study the scale-dependence on physical quantities such
as the growth factor $D_k$, the velocity growth factor $f_k$, and $f_k
\sigma_8$. We found that for $z<10$, all these quantities recover the CDM
evolution, whereas for high redshift there is a clear distinction between each
model (FDM case, and axion-like potential) depending on the wavenumber $k$ and
on the decay parameter of the axion-like potential as well. A semi-analytical
Halo Mass Function is also revisited, finding a suppression of the number of
low mass halos, as in the FDM case, but with a small increment in the amplitude
of the variance and halo mass function due to the non-linearity of the
axion-like potential. Finally, we present constraints on the axion mass and the
axion decay parameter by using data of the Planck Collaboration 2018 and
Lyman-$\alpha$ forest.
