  We propose two effective parameters that fully characterise galactic-scale
structure formation at high redshifts ($z\gtrsim5$) for a variety of dark
matter (DM) models that have a primordial cutoff in the matter power spectrum.
Our description is within the recently proposed ETHOS framework and includes
standard thermal Warm DM (WDM) and models with dark acoustic oscillations
(DAOs). To define and explore this parameter space, we use high-redshift
zoom-in simulations that cover a wide range of non-linear scales from those
where DM should behave as CDM ($k\sim10\,h\,{\rm Mpc}^{-1}$), down to those
characterised by the onset of galaxy formation ($k\sim500\,h\,{\rm Mpc}^{-1}$).
We show that the two physically motivated parameters $h_{\rm peak}$ and $k_{\rm
peak}$, the amplitude and scale of the first DAO peak, respectively, are
sufficient to parametrize the linear matter power spectrum and classify the DM
models as belonging to effective non-linear structure formation regions. These
are defined by their relative departure from Cold DM ($k_{\rm
peak}\rightarrow\infty$) and WDM ($h_{\rm peak}=0$) according to the non-linear
matter power spectrum and halo mass function. We identify a region where the
DAOs still leave a distinct signature from WDM down to $z=5$, while a large
part of the DAO parameter space is shown to be degenerate with WDM. Our
framework can then be used to seamlessly connect a broad class of particle DM
models to their structure formation properties at high redshift without the
need of additional $N$-body simulations.
