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Vibrational instability of Population III very massive main-sequence stars due to the $\varepsilon$-mechanism

Very massive stars are thought to be formed in the early Universe because of a lack of cooling process by heavy elements, and might have been responsible for the later evolution of the Universe. We had an interest in vibrational stability of their evolution and carried out the linear nonadiabatic analysis of radial and nonradial oscillations for population III very massive main-sequence stars with $500-3000M_{\sun}$. We found that only the radial fundamental mode becomes unstable due to the $\varepsilon$-mechanism for these stars. The instability appears just after the CNO cycle is activated and the nuclear energy generation rate becomes large enough to stop the pre--main-sequence contraction, and continues during the early stage of the core hydrogen burning. Besides, we roughly estimated amount of mass loss due to the instability to know its significance.