Spectral modeling

The spectrum of each bin is obtained using the automatic online tool provided by Swift for spectral analysis (see “Data availability”). Each spectrum is analyzed using XSPEC⁴³, version 12.10.1, and the Python interface PyXspec. We discard all photons with energy E < 0.5 keV and E > 10 keV. The spectra are modeled with an absorbed power law, and for the absorption, we adopted the Tuebingen–Boulder model⁴⁴. If the GRB redshift is known, we use two distinct absorbers, one Galactic⁴⁵ and one relative to the host galaxy (the XSPEC syntax is tbabs*ztbabs*po). The column density N_H of the second absorber is estimated through the spectral analysis, as explained below. On the other hand, if the GRB redshift is unknown, we model the absorption as a single component located at redshift z = 0 (the XSPEC syntax is tbabs*po) and also in this case the value of N_H is derived from spectral analysis.

For the estimation of the host N_H, we consider only the late part of the XRT light curve following the SD phase. At late time with respect to the trigger, we do not expect strong spectral evolution, as verified in several works in the literature^46,47. Therefore, for each GRB, the spectrum of each bin after the SD is fitted adopting the same N_H, which is left free during the fit. Normalization and photon index are also left free, but they have different values for each spectrum. We call NHlate the value of N_H obtained with this procedure. In principle, the burst can affect the ionization state of the surrounding medium, but we assume that such effects are negligible and N_H does not change dramatically across the duration of the burst⁴⁸. Hence, we analyzed separately all the spectra of the SD using a unique value of NH=NHlate, which is fixed during the fit. Normalization and photon index, instead, are left free.

An alternative method for the derivation of N_H is the fitting of all the spectra simultaneously imposing a unique value of N_H that is left free. On the other hand, since N_H and photon index are correlated, an intrinsic spectral evolution can induce an incorrect estimation of N_H. For the same reason we do not fit the spectra adopting a free N_H, since we would obtain an evolution of photon index strongly affected by the degeneracy with N_H.

In this regard, we tested how our results about spectral evolution depend on the choice of N_H. On average, we found that the fits of the SD spectra remain good (stat/dof ≲1) for a variation of N_H of about 50%. As a consequence, the photon index derived by the fit would change at most of 30%. Therefore the error bars reported in all the plots α − F are possibly underestimated, but even considering a systematic error that corresponds to ~30% of the value itself would not undermine the solidity of the results.