In the context of star formation through fragmentation of an extremely metal-deficient protogalactic cloud , the gravitational collapse of filamentary gas clouds is explored with one-dimensional numerical hydrodynamics coupled with non-equilibrium chemistry of H _ { 2 } and HD .
It is found that the cloud evolution is governed mainly by the initial central density ( n _ { c, 0 } ) and H _ { 2 } abundance ( x _ { H _ { 2 } , 0 } ) .
In particular , the evolution of low-density filaments ( n _ { c, 0 } \lesssim 10 ^ { 5 } cm ^ { -3 } ) bifurcates at a threshold H _ { 2 } abundance of x _ { H _ { 2 } ,cr } \simeq 3 \times 10 ^ { -3 } , beyond which HD cooling overwhelms H _ { 2 } cooling .
The contraction of a filament with n _ { c, 0 } \lesssim 10 ^ { 5 } cm ^ { -3 } and x _ { H _ { 2 } , 0 } \gtrsim x _ { H _ { 2 } ,cr } is strongly decelerated when the central density ( n _ { c } ) reaches a critical density of HD at which LTE level populations are achieved , and therefore the filament is expected to fragment at \sim 10 ^ { 7 } cm ^ { -3 } .
The fragment mass is assessed to be \approx 10 M _ { \odot } .
In contrast , the contraction of a filament with n _ { c, 0 } \lesssim 10 ^ { 5 } cm ^ { -3 } and x _ { H _ { 2 } , 0 } \lesssim x _ { H _ { 2 } ,cr } is regulated by H _ { 2 } cooling .
In this case , the filament tends to fragment at lower density as \sim 10 ^ { 4 } cm ^ { -3 } owing to the low critical density of H _ { 2 } , and the fragment mass is as high as \approx 10 ^ { 2 } M _ { \odot } .
For a high-density filament with n _ { c, 0 } \gtrsim 10 ^ { 5 } cm ^ { -3 } , the temperature stays at a relatively high value because both H _ { 2 } and HD cooling saturate , and the cloud evolution is governed by H _ { 2 } cooling .
The contraction of a high-density filament is accelerated by effective three-body H _ { 2 } formation when the density reaches 10 ^ { 8 - 9 } cm ^ { -3 } .
The fragmentation is not expected to take place until the cloud becomes opaque in H _ { 2 } lines at n _ { c, 0 } \sim 10 ^ { 12 - 13 } cm ^ { -3 } , so that the fragment mass is reduced to 1 - 2 M _ { \odot } .
As a result , the stellar initial mass function ( IMF ) could be bimodal and deficient in sub-solar mass stars , where the high mass peak is around 10 M _ { \odot } or 10 ^ { 2 } M _ { \odot } , dependently on n _ { c, 0 } and x _ { H _ { 2 } , 0 } .
If the protogalactic clouds are ionized by UV radiation or strong shocks , the H _ { 2 } abundance could exceed x _ { H _ { 2 } ,cr } \simeq 3 \times 10 ^ { -3 } by reactions of H + e \rightarrow H ^ { - } + { \it h \nu } and H + H ^ { - } \rightarrow H _ { 2 } + e .
The high mass peak would then be O ( 10 ) M _ { \odot } .