We investigate constraints on the Hubble constant ( H _ { 0 } ) using Baryon Acoustic Oscillations ( BAO ) and baryon density measurements from Big Bang Nucleosynthesis ( BBN ) .
We start by investigating the tension between galaxy BAO measurements and those using the Lyman- \alpha forest , within a Bayesian framework .
Using the latest results from eBOSS DR14 we find that the probability of this tension being statistical is \simeq 6.3 \% assuming flat \Lambda CDM .
We measure H _ { 0 } = 67.6 \pm 1.1 km s ^ { -1 } Mpc ^ { -1 } , with a weak dependence on the BBN prior used , in agreement with results from Planck Cosmic Microwave Background ( CMB ) results and in strong tension with distance ladder results .
Finally , we forecast the future of BAO + BBN measurements of H _ { 0 } , using the Dark Energy Spectroscopic Instrument ( DESI ) .
We find that the choice of BBN prior will have a significant impact when considering future BAO measurements from DESI .