Towards a measurement of the spectral runnings

Single-field slow-roll inflation predicts a nearly scale-free power spectrum of perturbations, as observed at the scales accessible to current cosmological experiments. This spectrum is slightly red, showing a tilt $(1-n_s)\sim 0.04$. A direct consequence of this tilt are nonvanishing runnings $\alpha_s=\mathrm d n_s/\mathrm d\log k$, and $\beta_s=\mathrm d\alpha_s/\mathrm d\log k$, which in the minimal inflationary scenario should reach absolute values of $10^{-3}$ and $10^{-5}$, respectively. In this work we calculate how well future surveys can measure these two runnings. We consider a Stage-4 (S4) CMB experiment and show that it will be able to detect significant deviations from the inflationary prediction for $\alpha_s$, although not for $\beta_s$. Adding to the S4 CMB experiment the information from a WFIRST-like or a DESI-like survey improves the sensitivity to the runnings by $\sim$ 20\%, and 30\%, respectively. A spectroscopic survey with a billion objects, such as the SKA, will add enough information to the S4 measurements to allow a detection of $\alpha_s=10^{-3}$, required to probe the single-field slow-roll inflationary paradigm. We show that only a very-futuristic interferometer targeting the dark ages will be capable of measuring the minimal inflationary prediction for $\beta_s$. The results of other probes, such as a stochastic background of gravitational waves observable by LIGO, the Ly-$\alpha$ forest, and spectral distortions, are shown for comparison. Finally, we study the claims that large values of $\beta_s$, if extrapolated to the smallest scales, can produce primordial black holes of tens of solar masses, which we show to be easily testable by the S4 CMB experiment.