1 Introduction

Recent advances in optical counterpart observations have greatly reduced the uncertainties in the distance to Gamma Ray Bursts(GRBs). Based on some theoretical models, the GRBs are produced at the end of the life time of massive stars. Since the life time of the massive stars is relatively small, it is argued (Totani [1997], [1998]) that the GRB density should be similar to that of the star formation rate (SFR). This model is plausible as it puts the GRBs at larger distance scales $z_{\rm dim} = 5 \sim 6$ (Wijers [1997]) than the redshift of $z \sim 1$for the dim bursts in the conventional model, thus solving the problem that there is a lack of normal galaxies in the error boxes if they are at redshift of $z \sim 1$ (Schaefer [1997]).

Such a scenario can be confronted directly with the experimental evidence at hand. It is unclear to what extent this model is supported by the recent determination of the redshifts of a few GRB counterparts, GRB 970508 at 0.85 < z < 2.1 (Metzger [1997]), GRB 971214 at $z \sim 3.4$ (Kulkarni [1998]) and GRB 980703 at $z \sim 0.966$ (Djorgovski [1998]). Two of these apparently faint events have host galaxies with $R \sim 25\;{\rm mag}$ (Castro-Tirado [1997]; Kulkarni [1998]), while the third is similar in brightness but have a bright host galaxy with $R \sim 22 \;{\rm mag}$, therefore GRBs might have a broad luminosity function.

An alternative approach will be using other indirect methods of distance measurement to determine the distance to GRBs and hence the intrinsic luminosity. Totani ([1998]) examined the GRB number vs. brightness relation and attempted to constrain the GRB density rate to probe the star formation history. However, when the standard candle luminosity assumption is relaxed (Krumholz [1998]) and an intrinsic luminosity function is introduced, it is found that the GRB number vs. peak flux relation can be accommodated by a very broad range of models with either comoving densities or distributions tracing star formation history.

In this paper, we use the observed effect of time dilation (Norris et al. [1995]; Deng & Schaefer [1998]) to constrain the distance scale of GRBs and compare the result to that predicted by the SFR models.

  

Figure 1: The time dilation in the peak-to-peak time scales of GRB data with the solid curve being the best fit to a cosmological model with $\Omega_{0} = 0.3$ and $\Omega_{\Lambda} = 0.7$, a power law luminosity function with $<L\gt\, = 5.4 \pm 2.0 \ 10^{57} \ {\rm photons} \ {\rm s}^{-1}$with $\beta = 1.8$, $K = L_{\rm max}/L_{\rm min} = 2000.0$ and the spectral index of GRBs $\alpha = 2.0$. The unit of peak flux P is photons s-1cm-2 from 50 to 300 keV integrated over the 256 ms data, while the unit of the peak-to-peak time scale $\tau_{i}$ is second