Analysis of the duration$-$hardness ratio plane of gamma-ray bursts with skewed distributions

It was recently shown that the $T_{90}-H_{32}$ distributions of gamma-ray bursts from CGRO/BATSE and Fermi/GBM are well described by a mixture of only two skewed components, making the presumed third, intermediate class unnecesary. The Swift/BAT, Konus-Wind, RHESSI and Suzaku/WAM data sets are found to be consistent with a two-class description as well.


METHODOLOGY
The methodology is the same as in (Tarnopolski 2019). Two-and three-component mixtures of the following bivariate distributions are fitted: regular Gaussian (2G and 3G), skew-normal (2SN and 3SN), Student t (2T and 3T), and skew-Student (2ST and 3ST). The fits are compared using the small sample Akaike (Hurvich & Tsai 1989) and Bayesian Information Criteria (AIC c and BIC). AIC c is liberal, and has a tendency to overfit. BIC is much more stringent, and tends to underfit. Therefore, when the two IC point at different models, the truth lies somewhere in between. (See Tarnopolski 2019 for details.) The fitting is performed using the R package mixsmsn 2 (Prates et al. 2013).

RESULTS
The results are displayed in Figs. 1-4. For Swift and Konus no clear answer is obtained, however both IC point at skewed distributions (see bottom panels of Figs. 1 and 2). For Swift, the BIC yields 2ST and 2SN, while AIC c gives 3ST and 3SN. Henceforth, the lack of a third component in the data cannot be confidently ruled out; on the other hand, its presence is also not unambiguously supported. Konus gives remarkably similar results.
In case of RHESSI (see Fig. 3), both IC point unequivocally at 2-component mixtures, however BIC prefers symmetric distributions (2G and 2T), while AIC c hints at skewed ones (2ST and 2SN). Suzaku, the smallest data set examined, can be with no doubt well modeled with only 2 components, with 2G being the simplest model (see Fig. 4).

DISCUSSION
GRBs from BATSE and Fermi can be confidently divided into only two classical groups, short and long; the elusive soft-intermediate class is not necessary to satisfactorily describe the data (Tarnopolski 2015(Tarnopolski , 2016c(Tarnopolski ,a, 2019. In case of Swift and Konus, however, no firm conclusion can be formulated-the IC point at either two or three classes. The smallest data sets-RHESSI and Suzaku-can be adequately construed as consisting of two groups, although due to the smallness of these samples, the more subtle structure in the T 90 − H 32 plane can simply be not traced prominently enough. The asymmetry of the data, manifested via skewed distributions, might come from a non-symmetric distribution of the envelope masses of the progenitors of the long GRBs or other inherently asymmetrical distributions of physical parameters governing the progenitors or GRBs themselves; from the impact of the redshift distribution on the observables; or a combination of the listed possibilities (Tarnopolski 2015;Zitouni et al. 2015;Tarnopolski 2016cTarnopolski ,a,b, 2019.

CONCLUSIONS
No definite signs of the putative third GRB class are visible in the examined data. On the other hand, the Swift and Konus data yield inconclusive. It is desirable to have the exact shape of the observed distributions de- rived from a physical theory, or inferred on the grounds of statistics, which has not been convincingly realized thus far.
The author is grateful to Péter Veres for the Fermi data, Dmitry Svinkin for the Konus data, Norisuke Ohmori for the Suzaku data, Amy Lien for help with the Swift data, and JakubŘípa and NataliaŻywucka for discussions. Support by the Polish National Science Center through an OPUS grant No. 2017/25/B/ST9/01208 is acknowledged.