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 ${\alpha }_{\mathrm{s}}=\mathrm{d}{n}_{\mathrm{s}}/\mathrm{d}\log k$ and the running of the running ${\beta }_{\mathrm{s}}=\mathrm{d}{\alpha }_{\mathrm{s}}/\mathrm{d}\log k$ of the spectral index $n_{\mathrm{s}}$ of primordial scalar fluctuations. We find ${\alpha }_{\mathrm{s}}=0.011\pm 0.010$ and ${\beta }_{\mathrm{s}}=0.027\pm 0.013$ 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 ${\beta }_{\mathrm{s}}$ and foregrounds parameters, with the exception of the point sources amplitude at 143 GHz, $A_{143}^{\mathrm{PS}}$, 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 ${\alpha }_{\mathrm{s}},{\beta }_{\mathrm{s}}\sim 0.01$ by including in the analysis the lensing amplitude $A_L$, the curvature parameter ${\mathrm{\Omega }}_k$, and the sum of neutrino masses $\sum m_{\nu }$. 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 ${\mathrm{\Omega }}_k=0$ but still hint at possible deviations. The indication for ${\beta }_{\mathrm{s}}>0$ 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 $\sim 1.7$ standard deviations when CMB lensing is considered. We discuss the possibility of constraining ${\beta }_{\mathrm{s}}$ with current and future measurements of CMB spectral distortions, showing that an experiment like PIXIE could provide strong constraints on ${\alpha }_{\mathrm{s}}$ and ${\beta }_{\mathrm{s}}$.

• Received 15 May 2016

DOI:https://doi.org/10.1103/PhysRevD.94.023523