Running of the spectral index: Reconciling the CMB with the Lyman-$α$ Forest

Abstract

We investigate the scale dependence of the primordial power spectrum by combining \Planck, ACT DR6, SPT-3G, and eBOSS Lyman-$α$ forest data, extending sensitivity to smaller comoving scales than those probed by the CMB alone. Within a parametrisation based on a Taylor expansion around the pivot scale, we constrain the running of the spectral index $α_s$ and its running $β_s$. By using eBOSS likelihoods exhibiting a suppression of small-scale power either in amplitude or spectral index, we show that the latter can be accommodated by correlated variations of $(α_s,β_s)$, leading to a preference for non-zero running. We show that inflationary potentials with localised features -- such as Gaussian dips, bumps, or axion-monodromy modulations -- can reproduce the inferred scale dependence while remaining compatible with current CMB constraints. We release the public {\tt PIPE} code to enable systematic tests of inflationary potentials against current CMB datasets.