Checking the Empirical Relations with the Current Localized Fast Radio Bursts

Although fast radio bursts (FRBs) were discovered more than a decade ago, and they have been one of the active fields in astronomy and cosmology, their origins are still unknown. An interesting topic closely related to the origins of FRBs is their classifications. Different classes of FRBs require different physical mechanisms. If some empirical relations are found for different classes of FRBs, they might justify the classifications scenario and help us to reveal the physical mechanisms behind. On the other hand, FRBs are actually a promising probe for cosmology, since their redshifts could be $z\sim 3$ or even higher. Similar to the cosmology of type Ia supernovae (SNIa) or Gamma-ray bursts (GRBs), some empirical relations might also play an important role in the FRB cosmology. In the literature, some new classifications of FRBs different from repeaters and non-repeaters were proposed recently. In particular, it was suggested to classify FRBs into the ones associated with old or young stellar populations, and some empirical relations have also been found for them, respectively. One of these empirical relations (namely $L_\nu-E$ relation) without dispersion measure (DM) has been used to calibrate FRBs as standard candles for cosmology. This shows the potential of the new classification and the empirical relations for FRBs. Nowadays, more than 50 FRBs have been well localized, and hence their redshifts $z$ are observationally known. So, it is of interest to check the empirical relations with the actual data of current localized FRBs. We find that many empirical relations still hold, and in particular the one used to calibrate FRBs as standard candles for cosmology stands firm. This is beneficial to the FRB cosmology.