  Structure formation at small cosmological scales provides an important
frontier for dark matter (DM) research. Scenarios with small DM particle
masses, large momenta or hidden interactions tend to suppress the gravitational
clustering at small scales. The details of this suppression depend on the DM
particle nature, allowing for a direct link between DM models and astrophysical
observations. However, most of the astrophysical constraints obtained so far
refer to a very specific shape of the power suppression, corresponding to
thermal warm dark matter (WDM), i.e., candidates with a Fermi-Dirac or
Bose-Einstein momentum distribution. In this work we introduce a new analytical
fitting formula for the power spectrum, which is simple yet flexible enough to
reproduce the clustering signal of large classes of non-thermal DM models,
which are not at all adequately described by the oversimplified notion of WDM.
We show that the formula is able to fully cover the parameter space of sterile
neutrinos (whether resonantly produced or from particle decay), mixed cold and
warm models, fuzzy dark matter, as well as other models suggested by effective
theory of structure formation (ETHOS). Based on this fitting formula, we
perform a large suite of N-body simulations and we extract important nonlinear
statistics, such as the matter power spectrum and the halo mass function.
Finally, we present first preliminary astrophysical constraints, based on
linear theory, from both the number of Milky Way satellites and the
Lyman-$\alpha$ forest. This paper is a first step towards a general and
comprehensive modeling of small-scale departures from the standard cold DM
model.
