Is the $H_0$ tension suggesting a 4th neutrino's generation?
  Flavour oscillations experiments are suggesting the existence of a sterile,
$4$th neutrino's generation with a mass of an eV order. This would mean an
additional relativistic degree of freedom in the cosmic inventory, in
contradiction with recent results from the Planck satellite, that have
confirmed the standard value $N_{eff} \approx 3$ for the effective number of
relativistic species. On the other hand, the Planck best-fit for the
Hubble-Lema\^itre parameter is in tension with the local value determined with
the Hubble Space Telescope, and adjusting $N_{eff}$ is a possible way to
overcome such a tension. In this paper we perform a joint analysis of three
complementary cosmological distance rulers, namely the CMB acoustic scale
measured by Planck, the BAO scale model-independently determined by Verde {\it
et al.}, and luminosity distances measured with JLA and Pantheon SNe Ia
surveys. Two Gaussian priors were imposed to the analysis, the local expansion
rate measured by Riess {\it et al.}, and the baryon density parameter fixed
from primordial nucleosynthesis by Cooke {\it et al.}. For the sake of
generality, two different models are used in the tests, the standard
$\Lambda$CDM model and a generalised Chaplygin gas. The best-fit gives $N_{eff}
\approx 4$ in both models, with a Chaplygin gas parameter slightly negative,
$\alpha \approx -0.04$. The standard value $N_{eff} \approx 3$ is ruled out
with $\approx 3\sigma$.
