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The recent NANOGrav finding of a common-spectrum process has invitedinterpretations as possible evidence of a primordial stochasticgravitational-wave background (SGWB) stronger than predicted by standardinflation+LCDM. Such an SGWB would contribute an extra radiation component tothe background Universe which may affect its expansion history. As such, it mayhelp alleviate the current Hubble tension, a novel connection betweengravitational waves and cosmology. We demonstrate this by considering acosmological model, the "standard inflation + stiff amplification" scenario,with two components added to the LCDM model: a stiff component (w=1) and theprimordial SGWB. Previously, we showed that even for standard inflation, theSGWB may be detectable at the high frequencies probed by laser interferometers,if it is amplified by a possible early stiff era after reheating. Models thatboost the SGWB enough to cause significant backreaction, however, must stillpreserve the well-measured radiation-matter equality, as precision cosmologydemands. For that, we calculate the fully-coupled evolution of the SGWB andexpansion history, sampling parameter space (tensor-to-scalar ratio, reheatingtemperature and temperature at stiff-to-radiation equality). We then perform ajoint analysis of the NANOGrav results and latest upper bounds from Planck, bigbang nucleosynthesis and Advanced LIGO-Virgo, to constrain the model. Theresulting blue-tilted, stiff-amplified SGWB is still too small to explain theNANOGrav results. However, if someday, Advanced LIGO-Virgo detects the SGWB,our model can explain it within standard inflation (without requiring aninitial blue tilt). Meanwhile, this model may bring current high-z measurementsof the Hubble constant within 3.4 sigma of the low-z measurements by SH0ES(from 4.4 sigma) and within 2.6 sigma of those by H0LiCOW (from 3.1 sigma),reducing the tension.

Precision cosmology and the stiff-amplified gravitational-wave background from inflation: NANOGrav, Advanced LIGO-Virgo and the Hubble tension