Asteroseismology of evolved stars with EGGMiMoSA I. Theoretical mixed-mode patterns from the subgiant to the RGB phase

Context. In the context of an ever increasing amount of highly precise data, thanks to the numerous space-borne missions, came a revolution in stellar physics. This data allowed asteroseismology to thrive and improve our general knowledge of stars. Important results were obtained about giant stars owing to the presence of `mixed modes' in their oscillation spectra. These modes carry information about the whole stellar interior, enabling the comprehensive characterisation
of their structure.
Aims. The current study is part of a series of papers that provide a technique to coherently and robustly analyse the mixed-modes frequency spectra and characterise the stellar structure of stars on both the subgiant branch and red-giant branch (RGB). In this paper we aim at de ning seismic indicators, relevant of the stellar structure, as well as studying their evolution along a grid of models.
Methods. The proposed method, EGGMiMoSA, relies on the asymptotic description of mixed modes. It de nes appropriate initial guesses for the parameters of the asymptotic formulation and uses a Levenberg-Marquardt minimisation scheme in order to adjust the complex mixed-modes pattern in a fast and robust way.
Results. We are able to follow the evolution of the mixed-modes parameters along a grid of models from the subgiant phase to the RGB bump, therefore extending previous works. We show the impact of the stellar mass and composition on the evolution of these parameters. We observe that the evolution of the period spacing ∆π_1 , pressure offset $\epsilon_p$ , gravity
offset $\epsilon_g$ , and coupling factor q as a function of the large frequency separation ∆ν is little affected by the chemical
composition and that it follows two different regimes depending on the evolutionary stage. On the subgiant branch, the stellar models display a moderate core-envelope density contrast. Therefore, the evolution of ∆π_1 , $\epsilon_p$ , $\epsilon_g$ , and q signifficantly changes with the stellar mass. Furthermore, we demonstrate that, for a given metallicity and with proper
measurements of the period spacing ∆π_1 and large frequency separation ∆ν , we may unambiguously constrain the stellar mass, radius and age of a subgiant star. Conversely, as the star reaches the red-giant branch, the core-envelope density contrast becomes very large. Consequently, the evolution of $\epsilon_p$ , $\epsilon_g$ and q as a function of ∆ν becomes independent of the stellar mass. This is also true for ∆π_1 in stars with masses $\lessim 1.8~M_{\odot}$ because of core electron degeneracy. This degeneracy in ∆π_1 is lifted for higher masses, again allowing for a precise measurement of the stellar age. Overall, our computations qualitatively agree with previous observed and theoretical studies.
Conclusions. The method provides automated measurements of the adjusted parameters along a grid of models and opens the way to the precise seismic characterisation of both subgiants and red giants. In the following papers of the series, we will explore further refinements to the technique as well as its application to observed stars.