Disentangling the primordial nature of stochastic gravitational wave backgrounds with CMB spectral distortions

The recent detection of a stochastic gravitational wave background (SGWB) at nanohertz frequencies by pulsar timing arrays (PTAs) has sparked a flurry of interest. Beyond the standard interpretation that the progenitor is a network of supermassive black hole binaries, many exotic models have also been proposed, some of which can potentially offer a better fit to the data. We explore how the various connections between gravitational waves and CMB spectral distortions can be leveraged to help determine whether a SGWB was generated primordially or astrophysically. To this end, we present updated $k$-space window functions which can be used for distortion parameter estimation on enhancements to the primordial scalar power spectrum. These same enhancements can also source gravitational waves (GWs) directly at second order in perturbation theory, so-called scalar-induced GWs (SIGWs), and indirectly through the formation of primordial black holes (PBHs). We perform a mapping of scalar power spectrum constraints into limits on the GW parameter space of SIGWs for $\delta$-function features. We highlight that broader features in the scalar spectrum can explain the PTA results while simultaneously producing a spectral distortion (SD) within reach of future experiments. We additionally update PBH constraints from $\mu$- and $y$-type spectral distortions. Refined treatments of the distortion window functions widen existing SD constraints, and we find that a future CMB spectrometer could play a pivotal role in unraveling the origin of GWs imprinted at or below CMB anisotropy scales.