Is $Λ$CDM on the run? Reconciling the CMB with the Lyman-$α$ Forest

TL;DR

This work confronts the tension between CMB and Ly-$\alpha$ forest data by constraining the scale dependence of the primordial power spectrum beyond a simple power law. By combining Planck, ACT DR6, SPT-3G, and eBOSS Ly-$\alpha$ data, the authors constrain both the running $\alpha_s$ and the running of the running $\beta_s$, finding that eBOSS dramatically tightens these constraints and favors nonzero higher-order runnings, challenging vanilla slow-roll inflation. They test inflationary potentials with localized features (bumps, dips, and axion-monodromy) using a new public tool PIPE, showing that such features can reproduce the observed small-scale suppression while remaining compatible with CMB constraints. The results imply a persistent CMB–Ly-$\alpha$ tension that may point to high-energy structures in the inflaton potential, and PIPE enables rapid testing of arbitrary potentials against current data. If confirmed by future surveys, these findings could provide indirect evidence for nontrivial inflationary dynamics beyond standard slow-roll models.

Abstract

We present new constraints on 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. While ACT results previously indicated a mild preference for positive running of the spectral index, our joint analysis constrains both the running $α_s$ and its running $β_s$. Including eBOSS markedly tightens these constraints, yielding a $>2σ$ indication of nonzero $α_s$ and/or $β_s$, challenging predictions from vanilla slow-roll inflation potentials. Comparing reconstructed spectral parameters with theoretical models, we find that inflationary potentials with localised dips, bumps, or oscillations better reproduce the observed scale dependence. We release the public PIPE code to test arbitrary inflationary potentials against these datasets.

Figures (9)

• Figure 1: Joint and marginalized posterior probability distributions for $n_s$ and its running $\alpha_s = \mathrm{d}n_s/\mathrm{d}\ln k$, derived from the three likelihood combinations analysed in this work. Shaded contours represent the 68% and 95% credible regions. The dashed grey curve corresponds to the ACT+P constraint from Ref. ACT:2025tim, obtained when fitting only $\alpha_s$.
• Figure 2: Same as FIG. \ref{['fig:corner_norunrun']} but including $\beta_s = \mathrm{d}^2 n_s/\mathrm{d}\ln k^2$. Median values and corresponding 68% credible intervals for these parameters are reported in the Supplemental Material.
• Figure 3: Posteriors for $(\Delta_{\rm lin}^2,n_{\rm lin})$ when includin the running (left panel) and running-of-the-running (right panel) of the spectral index.
• Figure 4: Relative modulation of the potential $(V-V_\alpha)/V_\alpha$, as compared to the baseline model $V_\alpha\equiv V_0|\phi/M_p|^\alpha$, for each set of best-fit parameters $(V_0, \alpha, \ldots)$ listed in TABLE \ref{['tab:bestfits_main_new']}. Vertical, coloured, dashed lines denote the field value corresponding to where perturbations exit the horizon at the eBOSS scale.
• Figure S1: Linear matter power spectrum, rescaled both in amplitude and scale compared to the $\Lambda$CDM best fit from Planck Planck:2018vyg at the eBOSS scale, and compared to the eBOSS range of frequencies and credible region (grey-shaded area) as defined in Ref. Chabanier:2019eai.