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In this paper, we study a possible early universe source for the recentobservation of a stochastic gravitational wave background at the NANOGravpulsar timing array. The source is a tachyonic instability in a dark gaugefield induced by an axion-like particle (ALP), a known source for gravitationalwaves. We find that relative to the previous analysis with the NANOGrav12.5-year data set, the current 15-year data set favors parameter space with arelatively larger axion mass and decay constant. This favored parameter spaceis heavily constrained by $\Delta N_{\rm eff}$ and overproduction of ALP darkmatter. While there are potential mechanisms for avoiding the second problem,evading the $\Delta N_{\rm eff}$ constraint remains highly challenging. Inparticular, we find that the gravitational wave magnitude is significantlysuppressed with respect to the gauge boson dark radiation, which implies thatsuccessfully explaining the NANOGrav observation requires a large additionaldark radiation, violating the cosmological constraints.

Challenges in Interpreting the NANOGrav 15-Year Data Set as Early Universe Gravitational Waves Produced by ALP Induced Instability