The first stars are known to form in primordial gas , either in minihalos with about 10 ^ { 6 } M _ { \odot } or so-called atomic cooling halos of about 10 ^ { 8 } M _ { \odot } .
Simulations have shown that gravitational collapse and disk formation in primordial gas yield dense stellar clusters .
In this paper , we focus particularly on the formation of protostellar binary systems , and aim to quantify their properties during the early stage of their evolution .
For this purpose , we combine the smoothed particle hydrodynamics code GRADSPH with the astrochemistry package KROME .
The GRADSPH-KROME framework is employed to investigate the collapse of primordial clouds in the high-density regime , exploring the fragmentation process and the formation of binary systems .
We observe a strong dependence of fragmentation on the strength of the turbulent Mach number \mathcal { M } and the rotational support parameter \beta { } .
Rotating clouds show significant fragmentation , and have produced several Pop .
III proto-binary systems .
We report maximum and minimum mass accretion rates of 2.31 \times 10 ^ { -1 } M _ { \odot } yr ^ { -1 } and 2.18 \times 10 ^ { -4 } M _ { \odot } yr ^ { -1 } .
The mass spectrum of the individual Pop III proto-binary components ranges from 0.88 M _ { \odot } to 31.96 M _ { \odot } and has a sensitive dependence on the Mach number \mathcal { M } as well as on the rotational parameter \beta { } .
We also report a range from \sim 0.01 to \sim 1 for the mass ratio of our proto-binary systems .