$��$-Distortions or Running: A Guaranteed Discovery from CMB Spectrometry

We discuss the implications of a PIXIE-like experiment, which would measure $��$-type spectral distortions of the CMB at a level of $��_��=(1/n)\times 10^{-8}$, with $n\geq1$ representing an improved sensitivity (e.g. $n=10$ corresponds to PRISM). Using Planck data and considering the six-parameter $��$CDM model, we compute the posterior for $��_8\equiv��\times 10^{8}$ and find $��_8=1.57^{+0.11}_{-0.13}$ ($68\%\,\mathrm{CL}$). This becomes $��_{8} = 1.28^{+0.30}_{-0.52}$ ($68\%\,\mathrm{CL}$) when the running $��_\mathrm{s}$ of the spectral index is included. We point out that a sensitivity of about $3\times$ PIXIE implies a guaranteed discovery: $��$-distortion is detected or $��_\mathrm{s}\geq 0$ is excluded (both at $95\%\,\mathrm{CL}$ or higher). This threshold sensitivity sets a clear benchmark for CMB spectrometry. For a combined analysis of PIXIE and current Planck data, we discuss the improvement on measurements of the tilt $n_\mathrm{s}$ and the running $��_\mathrm{s}$ and the dependence on the choice of the pivot. A fiducial running of $��_\mathrm{s}=-0.01$ (close to the Planck best-fit) leads to a detection of negative running at $2��$ for $5\times$ PIXIE. A fiducial running of $��_\mathrm{s}=-0.02$, still compatible with Planck, requires $3\times$ PIXIE to rule out $��_\mathrm{s} = 0$ (at $95\%\,\mathrm{CL}$). We propose a convenient and compact visualization of the improving constraints on the tilt, running and tensor-to-scalar ratio.
10+2 pages, 8 figures, 2 tables. Matches published version