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.