A simple model of dust extinction in gamma-ray burst host galaxies

Gamma-ray burst (GRB) afterglows are powerful probes for studying the different properties of their host galaxies (e.g., the interstellar dust) at all redshifts. By fitting their spectral energy distribution (SED) over a large range of wavelengths, we can gain direct insights into the properties of the interstellar dust by studying the extinction curves. Unlike the dust extinction templates, such as the average Milky Way (MW) or the Small and Large Magellanic Cloud (SMC and LMC), the extinction curves of galaxies outside the Local Group exhibit deviation from these laws. Altogether, X-ray and gamma-ray satellites as well as ground-based telescopes, such as Neil Gehrels Swift Observatory (Swift) and Gamma-Ray Optical and Near-Infrared Detector (GROND), provide measurements of the afterglows from the X-ray to the NIR, which can be used to extract information on dust extinction curves along their lines of sight. The study presented in this paper undertakes such a photometric study, comprising a preparatory work for the SVOM mission and its ground-based follow-up telescope COLIBRI. We propose a simple parameterization of the dust extinction curve of GRB host galaxies. Our model is based on a power law form with the addition of a Loretzian-like Drude profile with two parameters: the extinction slope, $\gamma$, and the 2175 $\AA$ bump amplitude, $E_\rm{b}$. Using the seven GROND filter bands, we tested our dust extinction model and explored the parameter space in extinction and redshift by fitting SEDs of simplified simulations of GRB afterglow spectra based on different extinction curve templates. From a final sample of 10 real Swift/GROND extinguished GRBs, we determined the quantities of the dust extinction in their host and measured their extinction curves. The measured average extinction curve is equivalent to a quasi-featureless in-between SMC-LMC template.