Model-independently calibrating the luminosity correlations of gamma-ray bursts using deep learning

Gamma-ray bursts (GRBs) detected at high redshift can be used to trace the Hubble diagram of the Universe. However, the distance calibration of GRBs is not as easily as that of type Ia supernovae (SNe Ia). For the calibrating method based on the empirical luminosity correlations, there is an underlying assumption that the correlations should be universal over the whole redshift range. In this paper, we investigate the possible redshift dependence of six luminosity correlations with a completely model-independent deep learning method. We construct a network combining the Recurrent Neural Networks (RNN) and the Bayesian Neural Networks (BNN), where RNN is used to reconstruct the distance-redshift relation by training the network with the Pantheon compilation, and BNN is used to calculate the uncertainty of the reconstruction. Using the reconstructed distance-redshift relation of Pantheon, we test the redshift dependence of six luminosity correlations by dividing the full GRB sample into two subsamples (low-$z$ and high-$z$ subsamples), and find that only the $E_p-E_{\gamma}$ relation has no evidence for redshift dependence. We use the $E_p-E_{\gamma}$ relation to calibrate GRBs, and the calibrated GRBs give tight constraint on the flat $\Lambda$CDM model, with the best-fitting parameter $\Omega_{\rm M}$=0.307$^{+0.065}_{-0.073}$.