A generic ${\omega }_b$ tension in early-time solutions to the Hubble tension

I show that early-time (pre-recombination) solutions to the Hubble tension are generically expected to increase the preferred baryon density ${\omega }_b$. This puts these models in tension with Big Bang Nucleosynthesis (BBN), as measurements of primordial deuterium constrain ${\omega }_b$ at percent level. I show that existing analyses are in tension with the BBN determination of ${\omega }_b$, and that including a likelihood component for primordial deuterium deters two representative models from recovering a high $H_0$, and leads to worse fits to CMB, BAO, supernova, and BBN data than $\mathrm{\Lambda }$CDM.