Recent stellar evolution models show consistently that very massive metal-free stars evolve into red supergiants shortly before they explode. We argue that the envelopes of these stars, which will form pair-instability supernovae, become pulsationally unstable and that this will lead to extreme mass-loss rates despite the tiny metal content of the envelopes. We investigate the pulsational properties of such models and derive pulsationally induced mass-loss rates, which take the damping effects of the mass loss on the pulsations selfconsistently into account. We find that the pulsations may induce mass-loss rates of similar to 10(-4)-10(-2) M-circle dot yr(-1) shortly before the explosions, which may create a dense circumstellar medium. Our results show that very massive stars with dense circumstellar media may stem from a wider initial mass range than pulsational-pair instability supernovae. The extreme mass loss will cease when so much of the hydrogen-rich envelope is lost that the star becomes more compact and stops pulsating. The helium core of these stars therefore remains unaffected, and their fate as pair-instability supernovae remains unaltered. The existence of dense circumstellar media around metal-free pair-instability supernovae can make them brighter and bluer, and they may be easier to detect at high redshifts than previously expected. We argue that the mass-loss enhancement in pair-instability supernova progenitors can naturally explain some observational properties of superluminous supernovae: the energetic explosions of stars within hydrogen-rich dense circumstellar media with little Ni-56 production and the lack of a hydrogen-rich envelope in pair-instability supernova candidates with large Ni-56 production.
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