Model-independent Multivariable Gamma-Ray Burst Luminosity Indicator and Its Possible Cosmological Implications

Without imposing any theoretical models and assumptions, we present a multivariable regression analysis to several observable quantities for a sample of 15 gamma-ray bursts (GRBs). The observables used in the analysis include the isotropic gamma-ray energy (E(gamma),iso), the peak energy of the (nu)F(nu) spectrum in the rest frame (E...), and the rest-frame break time of the optical afterglow light curves (t...). A strong dependence of E(gamma),iso on E... and t... is derived, which reads E(gamma),iso/1052 ergs = (0.85 +- 0.21)(E.../100 keV)^1.94+-0.17(t.../1 day)^-1.24+-0.23 in a flat universe with (omega)M = 0.28 and H0 = 71.3 km s-1 Mpc-1. We also extend the analysis to the isotropic afterglow energies in the X-ray and optical bands, respectively, and find that they are essentially not correlated with E... and t.... Regarding the E(gamma),iso ...relationship as a luminosity indicator, we explore the possible constraints on the cosmological parameters using the GRB sample. Since there is no low-redshift GRB sample to calibrate this relationship, we weight the probability of using the relationship in each cosmology to serve as a standard candle by (chi)^2 statistics and then use this cosmology-weighted standard candle to evaluate cosmological parameters. Our results indicate that 0.05 < (omega)M < 0.50 at the 1 (sigma)level, with the most probable value of (omega)M being 0.28. The best value of (omega)(lambda)is 0.64, but it is less constrained. Only a loose limit of (omega)(lambda)< 1.2 is obtained at the 1 (sigma)level. In the case of a flat universe, the 1 (sigma)constraints are 0.13 < (omega)M < 0.49 and 0.50 < (omega)(lambda)< 0.85, respectively. The deceleration factor (q) and its cosmological evolution (dq/dz) are also investigated with an evolutionary form of q = q0 + z dq/dz. The best-fit values are (q0, dq/dz) = (1.00, 1.12), with -2.23 < q0 < 0.26 and -0.07 < dq/dz < 3.48 at the 1 (sigma)level. The inferred transition redshift between the deceleration and acceleration phases is 0.78... (1 (sigma)). Through Monte Carlo simulations, we find that the GRB sample satisfying our relationship observationally tends to be a soft and bright one and that the constraints on the cosmological parameters can be much improved either by enlarging the sample size or by increasing the observational precision. Although the sample may not expand significantly in the Swift era, a significant increase of the sample is expected in the long-term future. Our simulations indicate that with a sample of 50 GRBs satisfying our multivariable standard candle, one can achieve a constraint to the cosmological parameters comparable to that derived from 157 Type Ia supernovae. Furthermore, the detections of a few high-redshift GRBs satisfying the correlation could greatly tighten the constraints. Identifying high-z GRBs and measuring their E... and t... are therefore essential for the GRB cosmology in the Swift era.