Gravity darkening and tidally perturbed stellar pulsation in the misaligned exoplanet system WASP-33

WASP-33 is one of the few $\delta$ Sct stars with a known planetary companion. By analyzing the stellar oscillations, we search for possible star-planet interactions in the pattern of the pulsation. We made use of the Transit and Light Curve Modeller (TLCM) to solve the light curve from the Transiting Exoplanet Survey Satellite (TESS). We include gravity darkening into our analysis. The stellar oscillation pattern of WASP-33 clearly shows signs of several tidally perturbed modes. We find that there are peaks in the frequency spectrum that are at or near the $3$rd, $12$th and $25$th orbital harmonics ($f_{orb} \sim 0.82$ d$^{-1}$). Also, there is a prominent overabundance of pulsational frequencies rightwards of the orbital harmonics, being characteristic of a tidally perturbed stellar pulsation, which is an outcome of star-planet interactions in the misaligned system. There are peaks in both the $\delta$ Sct and $\gamma$ Dor ranges of the Fourier spectrum, implying that WASP-33 is a $\gamma$ Dor -- $\delta$ Sct hybrid pulsator. The transit light curves are best fitted by a gravity darkened stellar model, and the planet parameters are consistent with earlier determinations.