The BAO+BBN take on the Hubble tension
  Many attempts to solve the Hubble tension with extended cosmological models
combine an enhanced relic radiation density, acting at the level of background
cosmology, with new physical ingredients affecting the evolution of
cosmological perturbations. Several authors have pointed out the ability of
combined Baryon Acoustic Oscillation (BAO) and Big Bang Nucleosynthesis (BBN)
data to probe the background cosmological history independently of both CMB
maps and supernovae data. Using state-of-the-art assumptions on BBN, we confirm
that combined BAO, deuterium, and helium data are in tension with the SH0ES
measurements under the $\Lambda$CDM assumption at the 3.2$\sigma$ level, while
being in close agreement with the CMB value. We subsequently show that floating
the radiation density parameter $N_\mathrm{eff}$ only reduces the tension down
to the 2.6$\sigma$ level. This conclusion, totally independent of any CMB data,
shows that a high $N_\mathrm{eff}$ accounting for extra relics (either
free-streaming or self-interacting) does not provide an obvious solution to the
crisis, not even at the level of background cosmology. To circumvent this
strong bound, (i) the extra radiation has to be generated after BBN to avoid
helium bounds, and (ii) additional ingredients have to be invoked at the level
of perturbations to reconcile this extra radiation with CMB and LSS data.
