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We study the halo mass function and inner halo structure at high redshifts($z\geq5$) for a suite of simulations within the structure formation ETHOSframework. Scenarios such as cold dark matter (CDM), thermal warm dark matter(WDM), and dark acoustic oscillations (DAO) of various strengths are containedin ETHOS with just two parameters $h_{\rm peak}$ and $k_{\rm peak}$, theamplitude and scale of the first DAO peak. The Extended Press-Schechter (EPS)formalism with a smooth-$k$ filter is able to predict the cut-off in the halomass function created by the suppression of small scale power in ETHOS models(controlled by $k_{\rm peak}$), as well as the slope at small masses that isdependent on $h_{\rm peak}$. Interestingly, we find that DAOs introduce alocalized feature in the mass distribution of haloes, resulting in a massfunction that is distinct in shape compared to either CDM or WDM. We find thatthe halo density profiles of ${\it all}$ ETHOS models are well described by theNFW profile, with a concentration that is lower than in the CDM case in a waythat is regulated by $k_{\rm peak}$. We show that the concentration-massrelation for DAO models can be well approximated by the mass assembly modelbased on the extended Press-Schechter theory, which has been proposed for CDMand WDM elsewhere. Our results can be used to perform inexpensive calculationsof the halo mass function and concentration-mass relation within the ETHOSparametrization without the need of $N-$body simulations.

The halo mass function and inner structure of ETHOS haloes at high redshift