TOI-2458 b: A mini-Neptune consistent with in situ hot Jupiter formation

We report on the discovery and spectroscopic confirmation of TOI-2458 b, a transiting mini-Neptune around an F-type star leaving the main-sequence with a mass of $M_\star=1.05 \pm 0.03$ M$_{\odot}$, a radius of $R_\star=1.31 \pm 0.03$ R$_{\odot}$, an effective temperature of $T_{\rm eff}=6005\pm50$ K, and a metallicity of $-0.10\pm0.05$ dex. By combining TESS photometry with high-resolution spectra acquired with the HARPS spectrograph, we found that the transiting planet has an orbital period of $\sim$3.74 days, a mass of $M_p=13.31\pm0.99$ M$_{\oplus}$ and a radius of $R_p=2.83\pm0.20$ R$_{\oplus}$. The host star TOI-2458 shows a short activity cycle of $\sim$54 days revealed in the HARPS S-index and H$\alpha$ times series. We took the opportunity to investigate other F stars showing activity cycle periods comparable to that of TOI-2458 and found that they have shorter rotation periods than would be expected based on the gyrochronology predictions. In addition, we determined TOI-2458's stellar inclination angle to be $i_\star\,=\,10.6_{-10.6}^{+13.3}$ degrees. We discuss that both phenomena (fast stellar rotation and planet orbit inclination) could be explained by in situ formation of a hot Jupiter interior to TOI-2458 b. It is plausible that this hot Jupiter was recently engulfed by the star. Analysis of HARPS spectra has identified the presence of another planet with a period of $P\,=\,16.55\pm0.06$ days and a minimum mass of $M_p \sin i=10.22\pm1.90$ M$_{\oplus}$. Using dynamical stability analysis, we constrained the mass of this planet to the range $M_{c} \simeq (10, 25)$ M$_{\oplus}$.