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

Abstract

We investigate the scale dependence of the primordial power spectrum by combining \Planck, ACT DR6, SPT-3G, and eBOSS Lyman-$\alpha$ 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 ${\alpha }_s$ and its running ${\beta }_s$. 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 $({\alpha }_s,{\beta }_s)$, 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.

Comments: Matches the version accepted for publication in PRD. Typo fixed in the title