Overcoming the Circulation Problem for γ−ray Bursts in Cosmological Global Fitting Analysis

Due to the lack of low redshift long Gamma-Ray Bursts (GRBs), the circulation problem has been a severe obstacle for using GRBs as cosmological candles. In this paper, we present a new method to deal with such a problem in Markov Chain Monte Carlo (MCMC) global fitting analysis. Our methodology is similar to that of self-calibrations in using clusters of galaxies as cosmological probes. Assuming that a certain type of correlations between different observables exists in a subsample of GRBs, for the parameters involved in the correlation relation, we treat them as free parameters and determine them simultaneously with cosmological parameters through MCMC analysis on GRB data together with other observational data, such as SNe Ia, cosmic microwave background radiation (CMB) and large-scale structure (LSS). Then the circulation problem is naturally eliminated in this procedure. To demonstrate the feasibility of our method, we take the Ghirlanda relation as an example while keeping in mind the debate about its physical validity. Together with SNe Ia, WMAP and SDSS data, we include 27 GRBs with the reported Ghirlanda relation in our study, and perform MCMC global fitting. We consider the ΛCDM model and dynamical dark energy models with equation of state (EoS) wDE = w0 +w1(1− a) and the oscillating EoS wDE = w0 +w1 sin(w2 ln(a)), respectively. We also include the curvature of the universe in our analysis. In each case, in addition to the constraints on the relevant cosmological parameters, we obtain the best fit values as well as the distributions of the correlation parameters A and C. We find that the observational data sets other than GRBs can affect A and C considerably through their degeneracies with the cosmological parameters. With CMB+LSS+SNe+GRB data included in the analysis, the results on A and C for different cosmological models are in well agreement within 1σ range. The best fit value of A in all models being analyzed is A ∼ 1.53 with σ ∼ 0.08. For C, we have the best value in the range of 0.94 − 0.98 with σ ∼ 0.1. It is also noted that the distributions of A and C are generally broader than the priors used in many studies in literature. Our method can be readily applied to other GRB relations, which might be better physically motivated.