Scalar runnings and a test of slow roll from CMB distortions

A future measurement of cosmic microwave \mu-distortions by an experiment with the specifications of PIXIE will provide an equivalent 3-sigma detection of the running of running of the spectral index of scalar perturbations, \beta = d\alpha/dln(k), if \mu > 7.75 E-8, covering much of the PIXIE sensitivity range. This corresponds to a resolution limit of \beta > 0.015 which is relatively large given any presumption of slow roll, a result of the current tight constraints on \alpha < 0 on CMB scales. We show that a detection of \beta at this level is in conflict with slow roll conditions if the primordial signal can be distinguished from any post-inflationary contamination.