Asteroseismic diagnostics for red giants with Kepler: measuring ε and small frequency separations in 16 000 stars

ABSTRACT Asteroseismic studies of red giants have primarily relied on two global parameters: the large frequency separation ($\Delta \nu$) and the frequency of maximum power ($\nu _{\rm max}$). Meanwhile, the p-mode phase shift ($\varepsilon$) and small frequency separations ($\delta \nu _{01}$, $\delta \nu _{02}$), which offer additional constraints on stellar interiors, remain underexplored due to measurement challenges. Here, we develop an automated pipeline based on collapsed échelle diagrams and apply it to $\sim 16\,000$ Kepler red giants, jointly measuring $\Delta \nu$, $\varepsilon$, $\delta \nu _{01}$, and $\delta \nu _{02}$ and assembling the largest homogeneous catalogue of these quantities to date, together with updated $\Delta \nu$ values and formal internal uncertainties. Using this catalogue, we quantify evolutionary trends across the red-giant branch and core-helium-burning phase. We find that $\delta \nu _{02}/\Delta \nu$ stays nearly constant for RGB stars and, for core-helium-burning stars, organizes into two sequences that are systematically offset but partially overlap, broadly separating stars in the red-clump and secondary-clump regimes. We also trace the mass- and metallicity-dependent helium-flash transition. Meanwhile, $\varepsilon$ follows a single $\Delta \nu$–$\varepsilon$ relation common to both evolutionary phases. Comparisons with stellar-evolution models reveal systematic offsets in $\varepsilon$ and $\delta \nu _{01}$, which we interpret as signatures of near-surface and outer-envelope modelling deficiencies. These comparisons further suggest that dipole-mode small separations are sensitive to mode-dependent surface terms in evolved stars. Overall, our results demonstrate that $\varepsilon$ and the small separations provide important diagnostics of core structure, convective-boundary mixing, and helium ignition that are complementary to those provided by $\Delta \nu$ and $\nu _{\rm max}$ alone. The resulting catalogue offers a reference for testing and calibrating future stellar-evolution models.