What can CMB observations tell us about the neutrino distribution
  function?
  Cosmic Microwave Background (CMB) observations have been used extensively to
constrain key properties of neutrinos, such as their mass. However, these
inferences are typically dependent on assumptions about the cosmological model,
and in particular upon the distribution function of neutrinos in the early
Universe. In this paper, we aim to assess the full extent to which CMB
experiments are sensitive to the shape of the neutrino distribution. We
demonstrate that Planck and CMB-S4-like experiments have no prospects for
detecting particular features in the distribution function. Consequently, we
take a general approach and marginalise completely over the form of the
neutrino distribution to derive constraints on the relativistic and
non-relativistic neutrino energy densities, characterised by $N_\mathrm{eff} =
3.0 \pm 0.4$ and $\rho_{\nu,0}^{\rm NR} < 14 \, \mathrm{eV}\,\mathrm{cm}^{-3}$
at 95% CL, respectively. The fact that these are the only neutrino properties
that CMB data can constrain has important implications for neutrino mass limits
from cosmology. Specifically, in contrast to the $\Lambda$CDM case where CMB
and BAO data tightly constrain the sum of neutrinos masses to be $\sum m_\nu <
0.12 \, \mathrm{eV}$, we explicitly show that neutrino masses as large as $\sum
m_\nu \sim 3 \, \mathrm{eV}$ are perfectly consistent with this data.
Importantly, for this to be the case, the neutrino number density should be
suitably small such that the bound on $\rho_{\nu,0}^\mathrm{NR} = \sum m_\nu
n_{\nu,0}$ is still satisfied. We conclude by giving an outlook on the
opportunities that may arise from other complementary experimental probes, such
as galaxy surveys, neutrino mass experiments and facilities designed to
directly detect the cosmic neutrino background.
