tokens: calculations and the applicability of their formula for C-rich AGB stars with Z ini 10 4 should be checked by using the appropriate values. Also, the dependence of the mass-loss rate on the velocity amplitude of pulsation should be investigated, since the dependence is considered to be more sensitive for stars with larger C and R . The mass-loss rate derived from the DDW model is significantly smaller than the mass-loss rate assumed in stellar evolution calculations. If we use the derived mass-loss formula after the necessary condition for the efficient DDW is satisfied on the TP-AGB, the dredged-up carbon accumulates in the surface regions; accordingly, the effective temperature decreases and the mass-loss rate could increase. Although in the present calculations the derived mass-loss rate is inconsistent with the assumed mass-loss rate, it should be recalled here again that the hydrodynamical model can derive the properties of the DDW by specifying a set of input parameters, being independent of the stellar evolution model. Thus, the derived formulae, together with a necessary condition for the efficient DDW presented in Section 5.1 , could enable us to evaluate the mass-loss rate and dust yield during the C-rich AGB phase of stars with Z ini 10 4 in a manner consistent with stellar evolution, including whether the efficient DDW can operate on the C-rich AGB. 5.3 Implication for evolution of C-rich AGB stars and the dust-driven wind in the early Universe Investigations on the formation of stars in low-metallicity environments have revealed that the critical metallicity Z cri for the transition from Population III to Population II stars is as low as 10 9 10 7.5 , depending on the depletion factor of metal into dust (e.g. Omukai etal. 2005 , Schneider etal. 2006 , Chiaki etal. 2015 ). Although there is no information available for the initial mass function at the present time, intermediate-mass AGB stars with Z ini 10 4 can contribute to the enrichment of dust in the early Universe if the condition for the efficient DDW derived in the previous subsection is satisfied during the TP-AGB phase. However, the possibility of developing dust formation and a resulting DDW on the AGB is strongly influenced by the mass-loss history during the evolution, on which the time evolution of effective temperature as well as the elemental composition and opacity in the surface layer depend strongly through the number of TDU episodes and/or the occurrence of HBB. At present, we have no knowledge of the mass-loss mechanism and mass-loss rate before the onset of the DDW for the extremely low-metallicity stars considered in this article. In recent investigations focused on low-metallicity (1.6[Fe/H] 0.5) AGB stars observed in distant galaxies, Rosenfield etal. ( 2014 , 2016 ) showed that a mass-loss rate higher than SC05 is required during the AGB phases previous to the onset of the DDW, to reproduce the observed TP-AGB luminosity function as well as the number
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