Long GRBs as a Tool to Investigate Star Formation in Dark Matter Halos
  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.
