NANOGrav spectral index $\gamma =3$ from melting domain walls

We discuss cosmic domain walls described by a tension red-shifting with the expansion of the Universe. These melting domain walls emit gravitational waves with the low-frequency spectral shape ${\mathrm{\Omega }}_{gw}\propto f^2$ corresponding to the spectral index $\gamma =3$ favoured by the recent NANOGrav 15 yrs data. We discuss a concrete high-energy physics scenario leading to such a melting domain wall network in the early Universe. This scenario involves a feebly coupled scalar field, which can serve as a promising dark matter candidate. We identify parameters of the model matching the gravitational wave characteristics observed in the NANOGrav data. The dark matter mass is pushed to the ultra-light range below ${10}^{-11}-{10}^{-12}\text{eV}$ which is accessible through planned observations thanks to the effects of superradiance of rotating black holes.