Running the running

We use the recent observations of Cosmic Microwave Background temperature and polarization anisotropies provided by the Planck satellite experiment to place constraints on the running $α_\mathrm{s} = \mathrm{d}n_{\mathrm{s}} / \mathrm{d}\log k$ and the running of the running $β_{\mathrm{s}} = \mathrm{d}α_{\mathrm{s}} / \mathrm{d}\log k$ of the spectral index $n_{\mathrm{s}}$ of primordial scalar fluctuations. We find $α_\mathrm{s}=0.011\pm0.010$ and $β_\mathrm{s}=0.027\pm0.013$ at $68\%\,\mathrm{CL}$, suggesting the presence of a running of the running at the level of two standard deviations. We find no significant correlation between $β_{\mathrm{s}}$ and foregrounds parameters, with the exception of the point sources amplitude at $143\,\mathrm{GHz}$, $A^{PS}_{143}$, which shifts by half sigma when the running of the running is considered. We further study the cosmological implications of such preference for $α_\mathrm{s},β_\mathrm{s}\sim0.01$ by including in the analysis the lensing amplitude $A_L$, the curvature parameter $Ω_k$, and the sum of neutrino masses $\sum m_ν$. 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 $Ω_k = 0$ but still hint at possible deviations. The indication for $β_\mathrm{s} > 0$ survives at two standard deviations when external datasets such as BAO and CFHTLenS are included in the analysis, and persists at $\sim 1.7$ standard deviations when CMB lensing is considered. We discuss the possibility of constraining $β_\mathrm{s}$ with current and future measurements of CMB spectral distortions, showing that an experiment like PIXIE could provide strong constraints on $α_\mathrm{s}$ and $β_\mathrm{s}$.

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