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

ETHOS -- An effective parametrization and classification for structure formation: the non-linear regime at $z\gtrsim5$