  Dark matter models in which the constituent particle is an ultra-light boson
have become part of the mainstream discussion in cosmology and astrophysics. At
the classical level, the models are represented by the dynamics of a (real or
complex) scalar field endowed with a potential that contains its
self-interactions, and for this reason are generically known as scalar field
dark matter models. Here, we revisit the properties of such a model with a cosh
potential and compare it with other known examples in the literature. Within
the cosmological context, the self-interaction in the potential induces a
radiation-like behavior at early times of the scalar field density, which is
followed by a proper matter-like behavior at the onset of rapid field
oscillations around the minimum of the potential. The solutions are found by
numerical means, and from them we obtain information about the cosmological
observables up to the level of linear density perturbations. We also study the
general properties of self-gravitating objects in the non-relativistic limit
and determine the role in them of the self-interaction obtained from the cosh
potential. An overall conclusion is that, for the range of values in its
parameters allowed by different constraints, a cosh potential behaves almost
indistinguishable from the simpler quadratic one, which also means that the two
models suffer the same tight constraints from cosmological and astrophysical
observations.
