tokens: of gravitational wave after supernova explosion. In this study, since we consider only one progenitor model and one EOS, it is difficult to say how this relation is independent of the models, i.e. equation ( 7 ) cannot be used to analyse data from future gravitational wave observations. Even so, this relation seems to be independent of the models at least in the early phase, as shown in Fig. A2 in Appendix A . Anyway, additional models should be considered in the future. Figure 6. In the left-hand panel, frequencies of the f - and g 1 -modes are shown as a function of the square root of the PNS average density, where the thick solid and the thick dotted lines are the fitting formula given by equations ( 7 ) and ( 8 ). In the right-hand panel, frequencies of the f - and g 1 -modes are shown as a function of the surface gravity of the PNS, where the thick-solid line denote the fitting formula given by equation ( 9 ). The relation similar to equation ( 7 ) has already been proposed, as a function of x in Sotani Sumiyoshi ( 2019 ): and as a function of in the unit of M ⊙ km −2 in Torres-Forné et al. ( 2019b ): although in Torres-Forné et al. ( 2019b ) the ramp up signal is identified as g 2 -mode in their classification. Equation ( 8 ) is derived for the f -mode frequency after the avoided crossing with the g 1 -mode with the PNS models provided by the 1D numerical simulations, which are eventually collapsed into black hole. In the left-hand panel of Fig. 6 , we also plot the thick dotted line given by equation ( 8 ). From this figure, we find that the f -mode frequency is well identified by equation ( 8 ) up to f ≃ 1.4 kHz, which corresponds to T pb ≲ 0.6 s from Fig. 5 (or Fig. 2 ), but it deviates for T pb ≳ 0.6 sec. This may come from that the PNS models for deriving equation ( 8 ) become more compact in the later phase due to the massive progenitor model. On the other hand, in order to see the correspondence between the eigenmodes calculated in this study and equation ( 9 ), we show the f - and g 1 -mode frequencies as a function of in the right-hand panel of Fig. 6 . This fitting formula seems to correspond to the f -mode rather than the g 1 -mode in the very early phase and also in the late phase, at least for comparing to our results. Finally, we make a comment with respect to the previous results about the gravitational wave signals, especially obtained via the general relativistic 3D simulation (Kuroda et al. 2016 ), which is done with a 15 M ⊙ progenitor model (Woosley Weaver 1995 ) and SFHx EOS (Steiner et al. 2013 ). Unlike most of the other simulation results, several modes of gravitational wave signals have been found in this simulation. The characteristic gravitational wave frequencies extracted by the time-frequency analysis (Kawahara et al. 2018 ) are shown in the left-hand panel of Fig. 7 , where the signal A corresponds to the ramp up signals. Nevertheless, considering the results with the PNS asteroseismology shown in Fig. 2
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