The Traditional Approximation of Rotation (TAR) is a treatment of the hydrodynamic equations of rotating and stably stratified fluids in which the action of the Coriolis acceleration along the direction of the entropy and chemical stratifications is neglected because it is weak in comparison with the buoyancy force. The dependent variables in the equations for the dynamics of gravito-inertial waves (GIWs) then become separable into radial and horizontal parts as in the non-rotating case. The TAR is built on the assumptions that the star is spherical (i.e. its centrifugal deformation is neglected) and uniformly rotating. We study the feasibility of carrying out a generalisation of the TAR to account for the centrifugal acceleration in the case of strongly deformed uniformly and rapidly rotating stars (and planets), and to identify the validity domain of this approximation. We built analytically a complete formalism that allows the study of the dynamics of GIWs in spheroidal coordinates which take into account the flattening of rapidly rotating stars by assuming the hierarchies of frequencies adopted within the TAR in the spherical case and by deriving a generalised Laplace tidal equation for the horizontal eigenfunctions of the GIWs and their asymptotic wave periods, which can be used to probe the structure and dynamics of rotating deformed stars with asteroseismology. Using 2D ESTER stellar models, we determine the validity domain of the generalised TAR as a function of the rotation rate of the star normalised by its critical angular velocity and its pseudo-radius. This generalisation allows us to study the signature of the centrifugal effects on GIWs in rapidly rotating deformed stars. We found that the effects of the centrifugal acceleration in rapidly rotating early-type stars on GIWs are theoretically detectable in modern space photometry using observations from Kepler.
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