Growing evidence suggests that the first generation of stars may have been quite massive ( \sim 100 - 300 { \mathrm { M } _ { \odot } } ) .
Could these stars have left a distinct nucleosynthetic signature ?
We explore the nucleosynthesis of helium cores in the mass range { M _ { \mathrm { He } } } = 64 to 133 \mathrm { M } _ { \odot } , corresponding to main-sequence star masses of approximately 140 to 260 \mathrm { M } _ { \odot } .
Above { M _ { \mathrm { He } } } = 133 { \mathrm { M } _ { \odot } } , without rotation and using current reaction rates , a black hole is formed and no nucleosynthesis is ejected .
For lighter helium core masses , \sim 40 to 63 \mathrm { M } _ { \odot } , violent pulsations occur , induced by the pair instability and accompanied by supernova-like mass ejection , but the star eventually produces a large iron core in hydrostatic equilibrium .
It is likely that this core , too , collapses to a black hole , thus cleanly separating the heavy element nucleosynthesis of pair instability supernovae from those of other masses , both above and below .
Indeed , black hole formation is a likely outcome for all Population III stars with main sequence masses between about 25 \mathrm { M } _ { \odot } and 140 \mathrm { M } _ { \odot } ( M _ { He } = 9 to 63 \mathrm { M } _ { \odot } ) as well as those above 260 \mathrm { M } _ { \odot } .
Nucleosynthesis in pair-instability supernovae varies greatly with the mass of the helium core which determines the maximum temperature reached during the bounce .
At the upper range of exploding core masses , a maximum of 57 \mathrm { M } _ { \odot } of ^ { 56 } Ni is produced making these the most energetic and the brightest thermonuclear explosions in the universe .
Integrating over a distribution of masses , we find that pair instability supernovae produce a roughly solar distribution of nuclei having even nuclear charge ( Si , S , Ar , etc .
) , but are remarkably deficient in producing elements with odd nuclear charge - Na , Al , P , V , Mn , etc .
This is because there is no stage of stable post-helium burning to set the neutron excess .
Also , essentially no elements heavier than zinc are produced owing to a lack of s - and r -processes .
The Fe/Si ratio is quite sensitive to whether the upper bound on the IMF is over 260 \mathrm { M } _ { \odot } or somewhere between 140 and 260 \mathrm { M } _ { \odot } .
When the yields of pair-instability supernovae are combined with reasonable estimates of the nucleosynthesis of Population III stars from 12 to 40 \mathrm { M } _ { \odot } , this distinctive pattern of deficient production of odd-Z elements persists .
Some possible strategies for testing our predictions are discussed .