Improved constraints on primordial gravitational waves in light of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> tension and BICEP/Keck data

The Hubble tension that the standard $\Lambda$CDM model is suffering from can be resolved with pre-recombination early dark energy. We present first constraint on tensor-to-scalar ratio $r$ in corresponding Hubble-tension-free cosmologies using most recent BICEP/Keck cosmic microwave background (CMB) B-mode polarization data. find, combining Planck18 CMB and baryon acoustic oscillation data, models larger constant $H_0$ will have tighter upper bound $r$, resolution $H_0\sim73$ km/s/Mpc of tightens to $r<0.028\ (95\%\text{C.L.})$, $25\%$ than $r<0.036$. clarify origin this tightening bound.