Application of quantum chaos methods to the oscillations of rapidly rotating stars

Asteroseismology aims at inferring internal properties of stars from the analysis of their oscillation mode frequencies. This analysis can be greatly facilitated by an a priori information on the basic structure of the oscillation spectrum given by an asymptotic formula. Up to now, the only existing asymptotic formula for stellar oscillations was derived in the spherically symmetric case. For a rapidly rotating star, spherical symmetry is broken by the centrifugal force, and thus an adequate asymptotic theory was missing. Yet, rapid rotation is common among non-evolved intermediate mass and massive pulsating stars, and many of them are found in the data of asteroseismology space missions CoRoT and Kepler. The ray limit of pressure waves that causes stellar oscillations can be described by a Hamiltonian dynamical system. It was shown that this system undergoes a transition, as the rotation rate of the star increases, from an integrable to a mixed system where stable and chaotic regions coexist in phase space. In this thesis, it is shown how to obtain semi-analytical formulas for regular frequency spacings in the pressure mode spectrum of rapidly rotating stars by using ray theory and techniques from quantum chaos. These formulas relate regular frequency spacings to physical properties of the star, which provides a new theoretical tool for the asteroseismology of rapidly rotating stars.