  Observations of high-redshift galaxies have provided us with a rich tool to
study the physics at play during the epoch of reionisation. The luminosity
function (LF) of these objects is an indirect tracer of the complex processes
that govern galaxy formation, including those of the first dark-matter
structures. In this work, we present an extensive analysis of the UV galaxy LF
at high redshifts to extract cosmological and astrophysical parameters. We
provide a number of phenomenological approaches in modelling the UV LF and take
into account various sources of uncertainties and systematics in our analysis,
including cosmic variance, dust extinction, scattering in the halo-galaxy
connection, and the Alcock-Paczy\'{n}ski effect. Using UV LF measurements from
the Hubble Space Telescope together with external data on the matter density,
we derive the large-scale matter clustering amplitude to be
$\sigma_8=0.76^{+0.12}_{-0.14}$, after marginalising over the unknown
astrophysical parameters. We find that with current data this result is only
weakly sensitive to our choice of astrophysical modelling, as well as the
calibration of the underlying halo mass function. As a cross check, we run our
analysis pipeline with mock data from the IllustrisTNG hydrodynamical
simulations and find consistent results with their input cosmology. In
addition, we perform a simple forecast for future space telescopes, where an
improvement of roughly 30% upon our current result is expected. Finally, we
obtain constraints on astrophysical parameters and the halo-galaxy connection
for the models considered here. All methods discussed in this work are
implemented in the form of a versatile likelihood code, GALLUMI, which we make
public.
