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 1D 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 ($n_{ \rm c,0}$) and H$_2$ abundance ($x_{ \rm H_2,0}$). In particular, the evolution of low-density filaments bifurcates at a threshold H$_2$ abundance of $x_{ \rm H_2,cr}\simeq 3 \times 10^{ -3}$, beyond which HD cooling overwhelms H$_2$ cooling. The contraction of a low density filament with $x_{ \rm H_2, 0}\gtrsim x_{ \rm H_2,cr}$ is strongly decelerated when $n_{ \rm c}$ reaches a critical density of HD, and the filament is expected to fragment at $\sim 10^{ 7}$ cm$^{ -3}$. The fragment mass is lowered to be $\approx 10M_\odot$. In contrast, the contraction of a low density filament with $x_{ \rm H_2, 0}\lesssim x_{ \rm H_2,cr}$ is regulated by H$_2$ cooling. In this case, the fragment mass is as high as $\approx 10^2M_\odot$. For a high-density filament, the cloud evolution is governed by H$_2$ cooling. The fragmentation is not expected to take place until the cloud becomes opaque in H$_2$ lines at $n_{ \rm c,0}\sim 10^{ 12-13}$ cm$^{ -3}$, so that the fragment mass is reduced to 1-2 M$_\odot$. As a result, the stellar IMF could be bimodal and deficient in sub-solar mass stars, where the high mass peak is around $10M_\odot$ or $10^2M_\odot$, dependently on $n_{ \rm c,0}$ and $x_{ \rm H_2,0}$.
Categories: astro-ph