Cosmological searches for the neutrino mass scale and mass ordering
  In this thesis, I describe a number of recent important developments in
neutrino cosmology on three fronts. Firstly, focusing on Large-Scale Structure
(LSS) data, I will show that current cosmological probes contain a wealth of
information on the sum of the neutrino masses. I report on the analysis leading
to the currently best upper limit on the sum of the neutrino masses of
$0.12\,{\rm eV}$. I show how cosmological data exhibits a weak preference for
the normal neutrino mass ordering because of parameter space volume effects,
and propose a simple method to quantify this preference. Secondly, I will
discuss how galaxy bias represents a severe limitation towards fully
capitalizing on the neutrino information hidden in LSS data. I propose a method
for calibrating the scale-dependent galaxy bias using CMB lensing-galaxy
cross-correlations. Moreover, in the presence of massive neutrinos, the usual
definition of bias becomes inadequate, as it leads to a scale-dependence on
large scales which has never been accounted for. I show that failure to define
the bias appropriately will be a problem for future LSS surveys, and propose a
simple recipe to account for the effect of massive neutrinos on galaxy bias.
Finally, I discuss implications of correlations between neutrino parameters and
other cosmological parameters. In non-phantom dynamical dark energy models, the
upper limit on the sum of the neutrino masses becomes tighter than the
$\Lambda$CDM limit. Therefore, such models exhibit an even stronger preference
for the normal ordering, and their viability could be jeopardized should
near-future laboratory experiments determine that the mass ordering is
inverted. I then discuss correlations between neutrino and inflationary
parameters. I find that our determination of inflationary parameters is stable
against assumptions about the neutrino sector. (abridged)
