  We use state-of-art measurements of the galaxy luminosity function (LF) at
z=6, 7 and 8 to derive constraints on warm dark matter (WDM), late-forming dark
matter (LFDM) and ultra-light axion dark matter (ULADM) models alternative to
the cold dark matter (CDM) paradigm. To this purpose we have run a suite of
high-resolution N-body simulations to accurately characterise the low mass-end
of the halo mass function and derive DM model predictions of the high-z
luminosity function. In order to convert halo masses into UV-magnitudes we
introduce an empirical approach based on halo abundance matching which allows
us to model the LF in terms of the amplitude and scatter of the ensemble
average star formation rate halo mass relation of each DM model, $\langle {\rm
SFR}({\rm M_{ h}},z)\rangle$. We find that independent of the DM scenario the
average SFR at fixed halo mass increases from z=6 to 8, while the scatter
remains constant. At halo mass ${\rm M_{h}}\gtrsim 10^{12}\,{\rm M}_\odot$
h$^{-1}$ the average SFR as function of halo mass follows a double power law
trend that is common to all models, while differences occur at smaller masses.
In particular, we find that models with a suppressed low-mass halo abundance
exhibit higher SFR compared to the CDM results. Using deviance statistics we
obtain a lower limit on the WDM thermal relic particle mass, $m_{\rm
WDM}\gtrsim 1.5$ keV at $2\sigma$. In the case of LFDM models, the phase
transition redshift parameter is bounded to $z_t\gtrsim 8\cdot 10^5$ at
$2\sigma$. We find ULADM best-fit models with axion mass $m_a\gtrsim 1.6\cdot
10^{-22}$ eV to be well within $2\sigma$ of the deviance statistics. We remark
that measurements at $z=6$ slightly favour a flattening of the LF at faint
UV-magnitudes. This tends to prefer some of the non-CDM models in our
simulation suite, although not at a statistically significant level to
distinguish them from CDM.
