Asteroseismology and Buoyancy Glitch Inversion with Fourier Spectra of Gravity Mode Period Spacings

We investigate the small, quasi-periodic modulations seen in the gravity-mode period spacings of pulsating stars. These ``wiggles’’ are produced by buoyancy glitches – sharp features in the buoyancy frequency ($N$) caused by composition transitions and the convective-radiative interface. Our method takes the Fourier transform of the period-spacing series, $FT(\mathrm{\Delta }{P}_k)$ as a function of radial order $k$. We show that $FT(\mathrm{\Delta }{P}_k)$ traces the radial derivative of the normalized glitch profile $\delta N\mathrm{/}N$ with respect to the normalized buoyancy radius; peaks in $FT(\mathrm{\Delta }{P}_k)$ therefore pinpoint jump/drop locations in $N$ and measure their sharpness. We also note that the Fourier transform of relative period perturbations (deviations from asymptotic values), $FT(\delta P\mathrm{/}P)$, directly recovers the absolute value of the glitch profile $\mathrm{\mid }\delta N\mathrm{/}N\mathrm{\mid }$, enabling a straightforward inversion for the internal structure. The dominant $FT(\mathrm{\Delta }{P}_k)$ frequency correlates tightly with central hydrogen abundance ($X_c$) and thus with stellar age for slowly pulsating B-stars, with only weak mass dependence. Applying the technique to MESA stellar models and to observed slowly pulsating B-stars and $\gamma$ Dor pulsators, we find typical glitch amplitudes $\delta N\mathrm{/}N\lesssim 0.01$ and derivative magnitudes $\lesssim 0.1$, concentrated at chemical gradients and the convective boundary. This approach enables fast, ensemble asteroseismology of g-mode pulsators, constrains internal mixing and ages, and can be extended to other classes of pulsators, with potential links to tidal interactions in binaries.