*7.4. Improvements in Theoretical Modeling of Prompt GRB Polarization*

Pulse-integrated polarization from semi-analytic models of axisymmetric flows with different prompt GRB radiation mechanisms and B-field configurations have been presented in many works [20,24,108–111,134]. The same setup was used to make predictions for the time-dependent polarization for synchrotron emission in some works [84,322,323]. On the other hand, only a few works have attacked the problem using MC simulations [56–58] or radial integration of the transfer equations for the Stokes parameters [55]. Many of these have focused only on photospheric emission.

As the next decade may see the launch of more sensitive instruments to measure GRB polarization with high fidelity, it calls for time- and energy-dependent polarization predictions (Π(*E*, *t*), *θp*(*E*, *t*)) for more realistic outflow models, which would also predict the time-dependent flux density, *FE*(*t*).

One of the weaknesses of current theoretical models is the assumption of an axisymmetric flow, which is usually made for simplicity and convenience. This restricts the change in PA to only ∆*θ<sup>p</sup>* = 90◦ , whereas some observations do show, although not so convincingly yet, hints of gradual PA swings. To obtain a change in the PA other than ∆*θ<sup>p</sup>* = 90◦ or to get a gradually changing PA, the condition for axisymmetry must be broken, e.g., the magnetic field configuration/orientation and/or the emissivity can change as a function of (*θ*, *φ*).

One possibility is that the different pulses that contribute to the emission arise in "mini-jets" within the outflow e.g., [32,85,324–327]. In this case, the different directions of the mini-jets or bright patches w.r.t. the LOS (e.g., [109,230]) would cause the PA to also be different between the pulses even for a field that is locally symmetric w.r.t the local radial direction (e.g., *<sup>B</sup>*<sup>⊥</sup> or *<sup>B</sup>*<sup>k</sup> ) as well as for fields that are axisymmetric w.r.t to the center of each mini-jet (e.g., a local *B*tor for each mini-jet). Finally, broadly similar results would follow from an ordered field within each mini-jet (*B*ord), which are incoherent between different mini-jets. Time-resolved measurement in such a case would naturally yield a time-varying PA.

Alternatively, as shown by Granot and Königl [109] for GRB afterglow polarization, a combination of an ordered field component (e.g., *B*ord) and a random field, like *B*⊥, can give rise to a time-varying PA between different pulses (with a different ratio of the two field components) that cab, e.g., arise from internal shocks. The ordered field component here would be that advected from the central enginem and the random field component can be argued to be shock-generated. Notice that the ordered field component should not be axisymmetric in order for the position angle to smoothly vary.

Realistic theoretical predictions can be obtained by coupling radiation transfer modeling with MHD numerical simulations of relativistic jets after they break out of the confining medium. A step towards this direction was taken by Parsotan et al. [58] who used the MHD code FLASH to first obtain the jet's angular structure by injecting variable jets into stellar den-

sity profiles of Wolf–Rayet stars at core-collapse. They then used an MC code to carry out the radiation transfer of the Stokes parameters and obtain the time-resolved polarization for the photospheric emission (see Figure 10). In another recent work, Ito et al. [328] carried out global neutrino-hydrodynamic simulations of a relativistic jet launched in a binary NS merger scenario. The photospheric emission and polarization from the short GRB was then calculated using a relativistic MC code. While these works focused only on photospheric emission, polarization modeling for other radiation mechanisms performed in the same vein is lacking and can prove to be very fruitful.

MC radiation transfer and MHD numerical simulations of relativistic jets can be computationally expensive. They are nevertheless a useful tool that can be used to calibrate semi-analytic models by delineating the relevant parameter space expected in GRB jets. Ultimately, when high quality observations are made in this decade, fast and computationally inexpensive theoretical models will be required to carry out time-resolved spectro-polarimetric fits in a reasonable amount of time. This further stresses the need for a library of models, akin to Xspec [319] that is used routinely for spectral fitting or boxfit [329] for GRB afterglow lightcurve modeling, which can be conveniently used by observers. Combining the library of models with the 3ML framework for spectro-polarimetric data analysis will become a very powerful tool for GRB science.

In order to test the different model predictions, e.g., from different radiation mechanisms, on an equal footing, a single underlying theoretical framework should be devised for the jet structure and dynamics, which allows the same freedom in the different model parameters. Such an approach can help to isolate the dominant prompt GRB radiation mechanism when compared with observations.

To conclude, the next decade appears very promising for answering many fundamental questions in GRB physics. With the launch of several dedicated instruments capable of performing high-fidelity *γ*-ray and X-ray spectro-polarimetry, a larger sample of statistically significant prompt GRB polarization measurements will be obtained. Improvements in polarization data analysis using a single underlying framework that allows simultaneous fitting of both spectrum and polarization from different instruments will yield unbiased and high-quality results. More realistic theoretical models of both time- and energy-dependent polarization based on advanced numerical simulations will allow to better understand the true nature of GRB jets.

**Author Contributions:** Writing—original draft preparation—review and editing, R.G., M.K. and J.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded in part by the ISF-NSFC joint research program under grant no. 3296/19 (R.G. and J.G.) and the Swiss National Science Foundation (M.K.).

**Institutional Review Board Statement:** Not Applicable.

**Informed Consent Statement:** Not Applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Acknowledgments:** We thank Stefano Covino and Mark McConnell for a thorough read of an earlier version of the manuscript and for their comments and feedback.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Kai Wang 1,\* and Zi-Gao Dai 2,3**


**Abstract:** The prompt emission of most gamma-ray bursts (GRBs) typically exhibits a non-thermal Band component. The synchrotron radiation in the popular internal shock model is generally put forward to explain such a non-thermal component. However, the low-energy photon index *α* ∼ −1.5 predicted by the synchrotron radiation is inconsistent with the observed value *α* ∼ −1. Here, we investigate the evolution of a magnetic field during propagation of internal shocks within an ultrarelativistic outflow, and revisit the fast cooling of shock-accelerated electrons via synchrotron radiation for this evolutional magnetic field. We find that the magnetic field is first nearly constant and then decays as *B* 0 ∝ *t* −1 , which leads to a reasonable range of the low-energy photon index, −3/2 < *α* < −2/3. In addition, if a rising electron injection rate during a GRB is introduced, we find that *α* reaches −2/3 more easily. We thus fit the prompt emission spectra of GRB 080916c and GRB 080825c.

**Keywords:** gamma rays bursts; radiation mechanisms; non-thermal
