Seismic diagnosis for rapidly rotating upper-main-sequence g-mode pulsators: the combined effects of the centrifugal acceleration and differential rotation

Space-based asteroseismology has revolutionised our understanding of stellar structure, evolution, mixing, and rotation. In particular, intermediate-mass, main-sequence g-mode pulsators like gamma-Dor and slowly pulsating B-type (SPB) stars allow us to probe rotation and mixing at their convective core/radiative envelope interface with a high precision. This constitutes a gold mine for our global understanding of stellar rotation and related mixing. To fully exploit the information that is provided by detected g-mode pulsations, it is crucial to improve our understanding of how stellar rotation influences g-modes in rapidly rotating stars for which the action of the Coriolis and the centrifugal accelerations have to be taken into account. In this framework, the Traditional Approximation of Rotation (hereafter TAR) provides a flexible treatment of the adiabatic propagation of gravity modes modified by rotation (i.e. gravito-inertial modes including Rossby modes which propagate under the combined action of the buoyancy force and the Coriolis acceleration), which is extensively used for intensive seismic forward modelling. However, it has been built on the restrictive assumptions of spherical uniformly rotating stars. In this work, we generalise the TAR to take into account simultaneously the centrifugal deformation and differential rotation. We determine the validity domain of this generalised TAR using the state-of-the-art 2D stellar structure and evolution code ESTER. We then demonstrate how these new physics affect the pulsation-period spacings between consecutive g-mode pulsations, which are a common diagnostic that allow us to probe rotation and the chemicals mixing, for instance by convective overshoot or penetration, at the core boundary. We show that the effects induced by the centrifugal acceleration and the differential rotation are detectable using high-precision asteroseismic data. Finally, we discuss how this work can be generalised in a near future to include the effects of stellar magnetic fields and how it will lead to more realistic and accurate asteroseismic modelling of OBA-type stars