Density rate as a function of redshift for several known transient classes compared to our sample. Left panel: density rate as a function of redshift for FXRT 8/9/13 (cyan star) and CDF-S XT1/FXRT 14 (red square), and upper limits (black triangle) for FXRTs without measured redshifts. We also show the density rate of CDF-S XT2 (purple square; Xue et al. 2019), CC-SNe (dotted orange line; Madau & Dickinson 2014), the redshift-dependent intrinsic event rate densities of LGRBs (blue-filled region; taken from Sun et al. 2015 and Wanderman & Piran 2010, normalized to the local universe value and corrected for jet-aperture as ρ_0, LGRBs  ∼  250–500 yr⁻¹ Gpc⁻³), LL-LGRBs (green-filled region; taken from Zhang 2018, normalized to the local universe value and corrected for jet-aperture as ρ_0, LGRBs  ∼  100–200 yr⁻¹ Gpc⁻³), SGRBs considering a merger delay Gaussian model (red-filled region; taken from Sun et al. 2015 and Wanderman & Piran 2015, normalized to the local universe value and corrected for jet-aperture as ρ_0, SGRBs  ∼  13–75 yr⁻¹ Gpc⁻³), and TDEs (gray-filled region, taken from Sun et al. 2015, normalized to the local universe as ρ_0, TDEs  ∼  10⁴–10⁵ yr⁻¹ Gpc⁻³ at luminosities ∼10⁴²–10⁴⁴ erg s⁻¹). Right panel: local density rate for FXRT 8/9/13 considering they are related to GRBs (cyan star), TDEs (blue star), or FXRT 14/CDF-S XT1 (red square) (see Sect. 6.3). As a comparison, we also plot the local event rate of CDF-S XT2-like events (purple square; Xue et al. 2019), GW 170817 (green square; Abbott et al. 2017b), and GRB 170817A (magenta square; Zhang et al. 2018), as well other kinds of transients, such as the new SBO candidate in XMM-Newton data (yellow square and circle; Xu et al. 2008; Novara et al. 2020), the TDE rate of SMBH–MS TDEs (black triangle; Sun et al. 2015) and the IMBH–WD TDE rate (orange triangle and circle; Malyali et al. 2019; Tanikawa et al. 2021), the CC-SN rate (magenta triangle), the merger rate of BNS systems (light green circle; Abbott et al. 2021c), and the merger rate of neutron star and BH systems (gray circles; Abbott et al. 2021b). The local event rate of LGRBs, LL-LGRBs, and SGRBs are plotted (blue, green, and red horizontal shaded regions, corrected for the jet aperture factor; Zhang 2018).