Constraints on the running of the running of the scalar tilt from CMB anisotropies and spectral distortions

We use the recent observations of cosmic microwave background (CMB) temperature and polarizationanisotropies provided by the Planck satellite experiment to place constraints on the runningαs¼dns=d logkand the running of the runningβs¼dαs=d logkof the spectral indexnsof primordial scalarfluctuations. We findαs¼0.0110.010andβs¼0.0270.013at 68% C.L., suggesting the presence ofa running of the running at the level of two standard deviations. We find no significant correlation betweenβsand foregrounds parameters, with the exception of the point sources amplitude at 143 GHz,APS143, whichshifts by a half-sigma when the running of the running is considered. We further study the cosmologicalimplications of such a preference forαs;βs∼0.01by including in the analysis the lensing amplitudeAL,the curvature parameterΩk, and the sum of neutrino massesPmν. We find that when the running of therunning is considered Planck data are more compatible with the standard expectations ofAL¼1andΩk¼0but still hint at possible deviations. The indication forβs>0survives at two standard deviationswhen external data sets such as baryon acoustic oscillation surverys and CFHTLenS are included in theanalysis and persists at∼1.7standard deviations when CMB lensing is considered. We discuss thepossibility of constrainingβswith current and future measurements of CMB spectral distortions, showingthat an experiment like PIXIE could provide strong constraints onαsandβs.