Functional biases in GRB's spectral parameter correlations

Gamma Ray Bursts (GRBs) show evidence of different spectral shapes, light curves, duration, host galaxies and they explode within a wide redshift range. However, the most of them seems to follow very tight correlations among some observed quantities relating to their energetic. If true, these correlations have significant implications on burst physics, giving constraints on theoretical models. Moreover, several suggestions have been made to use these correlations in order to calibrate GRBs as standard candles and to constrain the cosmological parameters. We investigate the cosmological relation between low energy $\alpha$ index in GRBs prompt spectra and the redshift $z$. We present a statistical analysis of the relation between the total isotropic energy $E_{iso}$ and the peak energy $E_p$ (also known as Amati relation) in GRBs spectra searching for possible functional biases. Possible implications on the $E_{iso}$ vs $E_p$ relation of the $\alpha$ vs $(1+z)$ correlation are evaluated. We used MonteCarlo simulations and the boostrap method to evaluate how large are the effects of functional biases on the $E_{iso}$ vs $E_p$. We show that high values of the linear correlation coefficent, up to about 0.8, in the $E_{iso}$ vs $E_p$ relation are obtained for random generated samples of GRBs, confirming the relevance of functional biases. Astrophysical consequences from $E_{iso}$ vs $E_p$ relation are then to be revised after a more accurate and possibly bias free analysis.