Measuring $\mu$ distortions from the thermal Sunyaev-Zeldovich effect

The thermal Sunyaev-Zeldovich (tSZ) effect is a spectral distortion of the cosmic microwave background (CMB) resulting from inverse Compton scattering of CMB photons with electrons in the medium of galaxy clusters. The spectrum of the tSZ effect is typically calculated assuming the spectrum of the CMB is a blackbody. However, energy or photon number injection at any epoch after photon creation processes become inefficient will distort the blackbody, potentially leading to a chemical potential or $\mu$ distortion for early injection. These primordial spectral distortions will therefore introduce a change in the tSZ effect, effectively a distortion of a distortion. While this effect is small for an individual cluster's spectrum, upcoming and proposed CMB surveys expect to detect tens of thousands of clusters with the tSZ effect. In this paper, we forecast constraints on the $\mu$-distortion monopole from the distortion of the tSZ spectrum of clusters measured by CMB surveys. We find that planned experiments have the raw sensitivity to place constraints on $\mu$ that are comparable to or better than existing constraints but control over foregrounds and other systematics will be critical.