2 Analysis

The time dilation of GRBs is essentially a measurement of the redshift vs. brightness relations. To use the 1 + z vs. P relation to constrain the distance scale, the following three effects have to be taken into account: the intrinsic luminosity function of GRBs, correction of the peakflux due to redshift of photons in the detector waveband (similar to the K-correction in the optical measurements of galaxy magnitude), and the uncertainty introduced by the specific cosmological models assumed.

We adopt the standard Friedmann-Lemaître model assuming a power law luminosity function of GRBs $\phi(L) = C \cdot L^{-\beta}$($L_{\rm min} < L < L_{\rm max}$). If this luminosity function is normalized, there are two remaining free parameters, the average luminosity <L> and the width of the luminosity function $K = L_{\rm max}/L_{\rm min}$. At each observed peak flux level P, the GRBs themselves can be located at a wide range of redshifts because of the above luminosity distribution. Therefore we calculate the corresponding average redshift <z> of GRBs as sampled from the above luminosity function. When averaging, these redshifts have to be appropriately weighted by the density distribution according to the $\log N-\log P$ curve, and possibly some form of cutoff of the GRB distribution at large redshifts where galaxies are still not formed.

We fit the time dilation data to the above model. The data utilizes the peak-to-peak time scales derived from the time intervals between statistically significant peak structures in the GRB time profiles (Norris et al. [1995]; Deng & Schaefer [1998]). It is found that time data of GRBs can be reasonably fit by models with a standard candle luminosity or a broad luminosity function. In the case of models with a luminosity function, the best fit <L> is plotted against the change of luminosity function width K as shown in Fig. 2 for a particular model with $\Omega_{0} = 0.3$. It is found that the best fit average luminosity rises as the luminosity function width is increased. The cosmological parameters are also varied from the values preferred by the inflationary scenario to the recent values measured by the high redshift supernova surveys.

The combination of the uncertainty in the luminosity function and the cosmological models translates into large variation of the best average luminosity <L>. The best fit ranges from $L_{0} = 1.1 \pm 0.3 \ 10^{57}\ {\rm photons} \ {\rm s}^{-1}$ for a cosmological model with $\Omega_{0} = 1.0, \Omega_{\Lambda} = 0.0$ and a standard candle luminosity L₀, spectral index $\alpha = 2.0$ to as high as $<L\gt\, = 8.7 \pm 3.6 \ 10^{57}\ {\rm photons} \ {\rm s}^{-1}$ for a cosmological model with $\Omega_{0} = 0.3, \Omega_{\Lambda} = 0.7$, and a power law luminosity function $\phi(L) = C \cdot L^{-\beta}$, $\alpha = 2.5$.

  

Figure 2: Best fit average luminosity <L> vs. the width K of the luminosity function in a particular cosmological model with $\Omega_{0} = 0.3$ and $\Omega_{\Lambda} = 0.7$, luminosity function index $\beta = 1.8$, and the spectral index of GRBs $\alpha = 2.0$. This graph shows that the best fit of <L> changes by a factor of $\sim 3$ with a broad luminosity function width K = 2000