First stars can only form in structures that are suitably dense, which can be parametrized by the minimum dark matter halo mass $M_{ \rm min}$. $M_{ \rm min}$ must plays an important role in star formation. The connection of long gamma-ray bursts (LGRBs) with the collapse of massive stars has provided a good opportunity for probing star formation in dark matter halos. We place some constraints on $M_{ \rm min}$ using the latest $Swift$ LGRB data. We conservatively consider that LGRB rate is proportional to the cosmic star formation rate (CSFR) and an additional evolution parametrized as $(1+z)^{ \alpha}$, where the CSFR model as a function of $M_{ \rm min}$. Using the $\chi^{ 2}$ statistic, the contour constraints on the $M_{ \rm min}$--$\alpha$ plane show that at the $1\sigma$ confidence level, we have $M_{ \rm min}<10^{ 10.5}$ $\rm M_{ \odot}$ from 118 LGRBs with redshift $z<4$ and luminosity $L_{ \rm iso}>1.8\times10^{ 51}$ erg $\rm s^{ -1}$. We also find that adding 12 high-\emph{ z} $(4<z<5)$ LGRBs (consisting of 104 LGRBs with $z<5$ and $L_{ \rm iso}>3.1\times10^{ 51}$ erg $\rm s^{ -1}$) could result in much tighter constraints on $M_{ \rm min}$, for which, $10^{ 7.7}\rm M_{ \odot}<M_{ \rm min}<10^{ 11.6}\rm M_{ \odot}$ ($1\sigma$). Through Monte Carlo simulations, we estimate that future five years of Sino-French spacebased multiband astronomical variable objects monitor (\emph{ SVOM}) observations would tighten these constraints to $10^{ 9.7}\rm M_{ \odot}<M_{ \rm min}<10^{ 11.3}\rm M_{ \odot}$. The strong constraints on $M_{ \rm min}$ indicate that LGRBs are a new promising tool for investigating star formation in dark matter halos.
Categories: astro-ph.HE astro-ph.GA