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 polarization anisotropies provided by the Planck satellite experiment to place constraints on the running alpha(s) = dn(s)/d log k and the running of the running beta(s) = d alpha(s)/d log k of the spectral index n(s) of primordial scalar fluctuations. We find alpha(s) = 0.011 +/- 0.010 and beta(s) = 0.027 +/- 0.013 at 68% C.L., suggesting the presence of a running of the running at the level of two standard deviations. We find no significant correlation between beta(s) and foregrounds parameters, with the exception of the point sources amplitude at 143 GHz, A(143)(PS), which shifts by a half-sigma when the running of the running is considered. We further study the cosmological implications of such a preference for alpha(s), beta(s) similar to 0.01 by including in the analysis the lensing amplitude A(L), the curvature parameter Omega(k), and the sum of neutrino masses Sigma m(v). We find that when the running of the running is considered Planck data are more compatible with the standard expectations of A(L) = 1 and Omega(k) = 0 but still hint at possible deviations. The indication for beta(s) > 0 survives at two standard deviations when external data sets such as baryon acoustic oscillation surverys and CFHTLenS are included in the analysis and persists at similar to 1.7 standard deviations when CMB lensing is considered. We discuss the possibility of constraining beta(s) with current and future measurements of CMB spectral distortions, showing that an experiment like PIXIE could provide strong constraints on alpha(s) and beta(s).