μdistortions or running: A guaranteed discovery from CMB spectrometry

We discuss the implications of a PIXIE-like experiment, which would measure μ-type spectral distortions of the cosmic microwave background (CMB) at a level of σ_μ=(1/n)×10^-8, with n≥1 representing an improved sensitivity (e.g. $n=10$ corresponds to PRISM). Using Planck data and considering the six-parameter ΛCDM model, we compute the posterior for μ₈≡μ×10⁸ and find μ₈=1.57_-0.13+0.11 (68% C.L.). This becomes μ₈=1.28_-0.52+0.30 (68% C.L.) when the running αₛ of the spectral index is included. We point out that a sensitivity of about 3×PIXIE implies a guaranteed discovery: μ distortion is detected or αₛ≥0 is excluded (both at 95% C.L. 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 nₛ and the running αₛ and the dependence on the choice of the pivot. A fiducial running of αₛ=-0.01 (close to the Planck best fit) leads to a detection of negative running at 2σ for 5×PIXIE. A fiducial running of αₛ=-0.02, still compatible with Planck, requires 3×PIXIE to rule out αₛ=0 (at 95% C.L.). We propose a convenient and compact visualization of the improving constraints on the tilt, running and tensor-to-scalar ratio.