Time-averaging Polarimetric and Spectral Properties of Gamma-Ray Bursts

A comprehensive database of GRB polarimetric observations has been created in a recent work (Li et al., 2022) by extensively searching for GRBs in the literature with reported polarization measurements. A total of 73 bursts with polarization detections were included in the database, covering a broad wavelength range from radio to optical, X-ray, and $\gamma$-ray emission (see Table 1 in Li et al. 2022). The prompt emission data of these bursts were observed by various satellites (Fermi, Swift, BeppoSAX, and BATSE). Among these satellites, Fermi covers the broadest energy range in the observation, making it crucial for evaluating current spectral models. The GBM (8 KeV-40 MeV, Meegan et al. 2009) and the Large Area Telescope (LAT, 20 MeV- 300 GeV, Atwood et al. 2009), on board the NASA Fermi Gamma-ray Space Telescope, together provide unprecedented spectral coverage for 7 orders of magnitude in energy (from $\sim$8 keV to $\sim$300 GeV). Our statistical analysis in the current work includes the prompt $\gamma$-ray emission spectral analysis and explores the connection between the spectrum and polarization based on the Fermi-detected bursts. We specifically focus on GRBs with polarization measurements taken during the prompt emission in the $\gamma$-ray band to compare polarization and spectral properties simultaneously. Our sample has been refined to include 27 bursts (refer to Table 1 for details).

In order to diagnose the jet properties for a given burst, a refined time-integrated spectral analysis is required. Following the standard practice (Yu et al., 2019; Li, 2019; Burgess et al., 2019a; Dereli-Bégué et al., 2020; Li et al., 2021) provided by the Fermi Science Term, the spectral analysis is performed using a pure Python package called the the Multi-Mission Maximum Likelihood Framework (3ML,Vianello et al. 2015), and a Bayesian approach and Markov Chain Monte Carlo (MCMC) iterations are used to explore the best parameter space. The main steps of our spectral analysis include selecting detectors, sources, and background intervals; choosing the same observed epoch during the prompt emission phase for polarization and spectral data; calculating the significance ($S$, Vianello 2018) for each burst; fitting all the spectral data by using various GRB spectral models and their hybrid version, like power law, BB, CPL, Band function, power law+BB, CPL+BB, and Band+BB; using a fully Bayesian approach to obtain the best model parameters; and comparing models by using the AIC (Akaike 1974, defined as ${\rm AIC}=2k-2{\rm ln}\mathcal{L}$, where $k$ is the number of estimated parameters in the model and $\mathcal{L}$ is the maximized value of the likelihood function for the model) and BIC (Schwarz 1978, defined as ${\rm BIC}=k{\rm ln}N-2{\rm ln}\mathcal{L}$, where $\mathcal{L}$ is the maximum likelihood, $k$ is the number of parameters of the model, and $N$ is the number of data points). We use both AIC and BIC because BIC is recommended for nested models (e.g., Band versus Band+BB), while AIC is favored for models that are not nested (e.g., Band versus CPL). For a given set of models, the preferred model is the one that provides the lowest AIC and BIC scores. We consequently accept the Band+BB model as the preferred one if the difference between the BIC of Band+BB and the BIC of Band is less than -10, i.e., $\Delta{\rm BIC=BIC_{Band+BB}-BIC_{Band}<-10}$ (Li, 2023). We also double-check their corner–corner plots of the posteriors to determine well-constrained $\beta$ (Band-like) and unconstrained $\beta$ (CPL-like) to select the preferred model between Band and CPL models (Li, 2022).

Twenty-seven bursts were claimed to have a high-significance polarization measurement and GBM data taken at their prompt emission (Table 1, and Figure 1, Figure 2, and Figure 3), providing an “ideal” sample to study the possible connection between jet properties and GRB polarization straightforwardly.

Apart from one burst (GRB 160623A) found to have low statistical significance, which cannot ensure that the spectral fits are well determined and the type of outflow cannot be determined, the remaining 26 bursts are used in our analysis. For 10 bursts, the Band+BB model has an AIC/BIC-statistic improvement of at least 10 with respect to the Band alone and other models, which suggests Band+BB as the preferred model that would fit the data and a thermal component existing in the spectrum (Table 2). Our refined time-integrated spectral analysis suggests that the Band+BB model can best characterize the spectral shape of 10 out of 26 (38%) bursts; therefore, the HD-type outflow can be identified (see Table 2). This implies that one cannot rule out that the photosphere emission may also be the possible mechanism powering the high levels of polarization. For the remaining 16 out of 26 (62%) bursts, either a Band-like or CPL-like component has been observed and therefore can be attributed to the presence of a PFD-type outflow, accounting for the largest percentage of the sample (see Table 3). Notably, no bursts are identified as the “KED-type” outflow.

Our time-integrated spectral analysis indicates that ten bursts (38%) in our target sample belong to the HD-type outflow. This finding is quite interesting since only a subset of GRBs (a fairly low percentage) have an observed thermal component in their spectral analysis, as suggested by several statistical studies (e.g., Li, 2023). Recently, (Li, 2023) has made a great effort to collect a complete GRB sample in which all bursts were detected by Fermi/GBM with known redshift and created a spectral parameter catalog based on their model-wise properties. He discovered that $\sim 5\%$ (7/153) of the analyzed bursts were found to require a subdominant thermal component in their time-integrated spectral analysis, including GRB 110721A ($\pi=84^{+16}_{-28}\%$). Our results imply that high-degree polarization measurements may still be dominated by the nonthermal component of the HD jets, originating well above the photosphere.

Using the same observed epoch during the prompt emission phase, our target sample allows for a thorough comparison of the polarization properties between the HD- and PFD-type bursts. Figure 4 shows the distribution of polarization degree $\pi$, comparing the PFD-type bursts (gray color) with the HD-type bursts (red color). It is evident that the HD- and PFD-type bursts exhibit inconsistent peaks, with the HD-type bursts typically exhibiting higher values compared to the PFD-type bursts. Notably, the PFD-type bursts display a bimodal distribution, with a lower peak at $\pi$=11.9% and a higher peak at $\pi$=50.0%. Observing such a high degree of polarization originating from the HD-type jets, particularly when it surpasses the polarization degree measurements from the PFD-type jets, is quite unexpected. According to current theoretical models, the expectation is the opposite: the PFD-type jets should exhibit higher-degree polarization measurements than the HD-type jets. Here we offer several remarks regarding these results. (1) The high-degree polarization measurements in the HD-type jets should still be dominated by their nonthermal components. (2) The HD-type jets contain both a KED thermal component and a PFD non-thermal component, and the thermal flux ratio between these thermal and nonthermal components varies from burst to burst, leading to a more intricate jet composition and magnetic field structure in the HD-type jets compared to pure PFD-type jets. If the finding that the HD-type bursts exhibit a higher degree of polarization than the PFD-type bursts truly holds, it is both intriguing and surprising. One possible explanation is that in the outflow of the HD-type bursts, in radiating regions dominated by nonthermal components, a large-scale ordered magnetic field (or a stronger ordered magnetic field) is more easily formed due to some mechanism, resulting in higher observed polarization measurements. (3) To answer this question better, a well-sampled collection of accumulating thermal-dominated bursts (similar to 090902B) may carry a key clue. By directly comparing the statistical results of polarization properties between the thermally dominated bursts and nonthermally dominated bursts, a firm conclusion can be drawn.

Figure 5 displays scatter plots between $\pi$ and various GRB observed quantities, for instance, $\pi$ correlated with (i) the peak energy ($E_{\rm p}$) of the $\nu F_{\nu}$ prompt emission spectrum for all the bursts (Fig.5a), (ii) the BB temperature $kT$ (Fig.5b) for the HD-type bursts, (iii) the corresponding energy fluence $S_{\gamma}$ for all bursts (Fig.5c), (iv) the magnetization parameter $\sigma_{0}$ for the HD-type bursts (Fig.5d), and (v) the redshift $z$ (Fig.5e). Despite the presence of a thermal component, the thermal flux ratio ($F_{\rm BB}$/$F_{\rm tot}$) for these bursts is found to be less than 50% (see Table 2), indicating that the thermal components are subdominant. Interestingly, a recent study (Chattopadhyay et al., 2019) of 11 bright bursts detected by CZTI during its first year of operation revealed that four bursts (GRB 160106A, GRB 160509A, GRB 160802A, and GRB 160910A) required an additional thermal BB component to accurately fit their spectra, deviating from the Band model. For the remaining 16 bursts, either the Band or CPL model provided a good fit to the spectral data. Following the spectral analysis, we obtain the peak energy ($E_{\rm p}$) and BB temperature ($kT$), allowing us to calculate the magnetization parameter $\sigma_{0}$ using hybrid-spectrum observed properties and the method described in Gao & Zhang (2015) and Li (2020). A robust correlation was not identified, as shown in Figure 5. The most intriguing result that captures our attention is the distinct regions occupied by HD-type bursts and PFD-type bursts in the $\pi-E_{\rm p}$ and $\pi-S_{\gamma}$ planes. The former may be due to different $E_{\rm p}$ values expected by different polarization models (Toma et al., 2009b). The latter, on the other hand, may be attributed to the more complicated spectral shape of the HD-type bursts compared to the PFD-type bursts. In practice, more complicated models, which have more free parameters, require more source photons to establish a reliable fit (Li, 2022).

The state of polarization of a radiation field can be expressed in terms of the Stokes parameters $I$ (total intensity), $Q$ and $U$ (linear polarizations), and $V$ (circular polarization). Stokes parameters $Q$ and $U$ are differences in the flux for two orthogonal directions on the sky that are coordinate-dependent quantities (Rybicki & Lightman, 2008; Westfold, 1959), we define the local degree of linear polarization $\pi=\sqrt{Q^{2}+U^{2}}/I$ where

and $\theta_{p}$ is the local polarization position angle (PA). The global Stokes parameters are evaluated by the integration over flux $dF_{\nu}$ of each fluid element that contributes to the radiation at any given observer time as

leading to the global linear polarization $\Pi=\sqrt{Q_{g}^{2}+U_{g}^{2}}/I_{g}$ which is finally measured by an observer from the image of the GRB jet on the sky plane (Granot, 2003a).

During the prompt emission, we have an ultrarelativistic jet with bulk Lorentz factor $\Gamma\gg 1$, which leads to a strong beaming effect of the emitted radiation. In the ultrarelativistic regime, the Doppler factor can be approximated as

where the observed emission is mainly received from a region that is limited to a cone with angular size $\tilde{\theta}\lesssim 1/\Gamma$ around line of sight (LOS). In principle, different GRBs can be viewed from various observing angles $\theta_{\rm obs}$ with respect to the jet’s central axis. Those observers whose LOS intersects the surface of the jet can detect the GRB’s prompt phase. At the early-time prompt emission, when the LOS intersects the jet surface, if $\theta_{obs}/\theta_{j}\lesssim 1-(\Gamma\theta_{j})^{-1}$ and $\Gamma\theta_{j}\gtrsim\mathcal{O}(10)$ with $\theta_{j}$ as the half-opening angle of the ejecta, the jet’s edge remains invisible to the observer Gill et al. (2018).

The emission region in the prompt emission can be approximated as an expanding thin spherical shell of width $\Delta\ll R/\Gamma^{2}$ (in the lab frame) in which particles cool relatively fast compared to the dynamical time scale of the system. As the GRB jet has slowed down significantly during the afterglow, when the opening angle $\theta_{j}\simeq\Gamma^{-1}$ then the jet break happens, and the edge effects become important. The flux density measured by a distant observer from each fluid element in an infinite thin-shell approximation for the prompt emission is given by (Granot, 2005)

where $d_{L}(z)$ is the luminosity distance of the source and $d\tilde{\Omega}=d\tilde{\phi}d(\cos\tilde{\theta})$ is the solid angle with $\tilde{\theta}$ and $\tilde{\phi}$ as the polar angle and the azimuthal angle measured from the LOS, respectively.

The anisotropic synchrotron luminosity $L^{\prime}_{\nu^{\prime}}$ depends on the pitch angle $\chi$ and the frequency $\nu^{\prime}$(Lyutikov et al., 2003; Rybicki & Lightman, 2008), as

where $\chi$ is the angle between the electron’s velocity vector (perpendicular to the local direction of the magnetic field $\hat{B}^{\prime}$) and the observer’s LOS $\hat{n}^{\prime}$ in the comoving frame of the GRB jet ($\cos\chi=\hat{n}^{\prime}\cdot\hat{B}^{\prime}$). As long as the electron energy distribution is independent of the pitch angle, it was shown that $\epsilon=1+\alpha$, where $\alpha=-d\rm{log}(F_{\nu})/d\rm{log}(\nu)$ is the spectral index Granot (2003). Then, we have

This means that most of the power is emitted at the peak frequency $\nu_{p}^{\prime}$ Granot (2003); Gill et al. (2018). The above spectrum is associated with a specific LOS (fixed polar angle) for a thin shell radiating at a particular radius. In the case of considering the temporal evolution of prompt GRB polarization, a Band-like spectrum may apply to model the radiation over a broad range of radii Gill & Granot (2021).

The polarization measurements can help to probe the magnetic field configuration inside the GRB shock wave. Moreover, the degree of polarization depends on the GRB jet’s angular structure and the observer’s viewing angle from the jet symmetry axis (Lazzati et al., 2004). Specifically, the observation of the early-time polarization from the prompt emission phase may indicate the magnetic field configuration close to the GRB progenitor. The direction of the local polarization vector in the synchrotron emission is orthogonal to the LOS of the observer $\hat{n}$ and the local direction of the magnetic field $\hat{B}$ in the jet,

The degree of linear polarization generated in the synchrotron emission from an isotropic electron distribution with power-law energy spectrum ($n_{e}\propto\gamma^{-p}$), and for a given direction of the magnetic field is given by (Rybicki & Lightman, 2008; Westfold, 1959):

where $p_{\rm eff}=2\alpha+1$ is the effective power-law index of the electron distribution (Sari et al., 1998).

and, therefore,

This value of polarization may be associated with a very small region (pointlike emitter) in which the magnetic field has a specific orientation. Only in the case of an ordered magnetic field with the coherence length comparable to or larger than the visible surface of the emitting region can the highest value of the polarization can be generated. The photon index in the synchrotron radiation is limited to $-1/3\leqslant\alpha\lesssim 3/2$, which, regarding Eq. (8) leads to the maximum degree of polarization, $50\%\lesssim\Pi\lesssim 75\%$.

The local degree of linear polarization for a tangled or random field configuration for a thin ultrarelativistic shell modeling of the prompt emission by assuming $\alpha=1$ is obtained by averaging over all local magnetic field directions as Sari (1999); Gruzinov (1999)

where $b\equiv 2\langle B^{2}_{\|}\rangle/\langle B_{\perp}^{2}\rangle$ denotes the anisotropy of the magnetic field distribution as the ratio of the parallel $B_{\|}$ to the perpendicular $B_{\perp}$ components with respect to the shock direction and $\theta_{B}$ is the angle between the LOS from the observer and the direction of the shock. In the case of a globally ordered magnetic field configuration aligned with the jet direction ($B\rightarrow B_{\|}$, $b\rightarrow\infty$), Eq. (17) returns back to Eq. (8) and gives the maximum value of the linear polarization.

The magnetic field structure in KED and PFD flows has a different origin and can be classified into three categories (Lyutikov et al., 2003; Gill et al., 2018, 2021):

• •

  A locally ordered magnetic field ($B_{\text{ord}}$) with angular coherent length $\theta_{j}>\theta_{B}\gtrsim 1/\Gamma$.

• •

  A toroidal magnetic field ($B_{\text{trod}}$) which has an ordered axisymmetric configuration in the transverse direction with respect to the jet.

• •

  A tangent magnetic field that could, in principle, be parallel ($B_{\|}$) or perpendicular ($B_{\perp}$) to the local fluid velocity.

In the PFD, the magnetic field is dynamically dominated and usually has a large coherence length such as $B_{\text{trod}}$ which can be produced by a rotating central engine or in a high magnetized flow; other locally and globally ordered field configurations are also possible in this case. On the other hand, in KED we may have a tangled magnetic field structure with $B_{\perp}$ and/or $B_{\|}$ components; however, generating such an anisotropic field configuration in shock waves seems to be challenging (Gill & Granot, 2020). A globally ordered magnetic field may naturally be advected from near the central source, while random magnetic fields are generated in the shock dissipation region (Kumar & Zhang, 2015; Geng et al., 2018; Gill et al., 2018; Fan et al., 2008). The magnetic field structures that are generated at relativistic collisionless shocks, due to the two-stream instabilities are expected to be tangled within the shock plane (Medvedev & Loeb, 1999).

In general, the measured polarization is obtained by integrating the local Stokes parameters over the flux of the GRB jet as

Assuming it to have an axisymmetric flow and taking into account symmetry consideration, we see that $\bar{U}=0$, and consequently, the instantaneous total degree of the linear polarization is $\bar{\Pi}={|\bar{Q}|/I}$. We perform an integration over the equal time surface (EATS) for a single pulse as

in order to obtain pulse-integrated polarization of the prompt emission, which leads to

The above formula is valid for the prompt emission from an ultrarelativistic thin shell for an on-axis observer ($\theta_{\rm obs}=0$) where $y_{\rm max}=(\Gamma\theta_{\rm max})^{2}$ and $\theta_{\rm max}$ is defined as the maximum angle from the LOS Granot (2003a). The factor $\Lambda(y,\phi)$ is an average over the magnetic field orientations in the plane of the ejecta as

The polarization angle $\theta_{p}$ and $\Lambda(y,\phi)$ take different forms regarding the configuration of the magnetic field in the plane of the GRB jet; it is instructive to compare Equations (6) and Eq. (21). In the case of an ordered magnetic field $B_{\text{ord}}$ we have :

The time-integrated linear polarization in the presence of an ordered magnetic field in the plane normal to the jet velocity is plotted as a function of the spectral index in the right panel of Fig. (6). As it is seen, the polarization degree increases toward higher values of $\alpha$ and lower values of $y_{max}$, which can cover the observed polarization of GRB 110721A, GRB 110301A, GRB 160802A, GRB 170101B, GRB 180120A and GRB 180427A. For a configuration with the globally ordered magnetic field, high values of the linear polarization even larger than $50\%$ are obtainable.

In the left panel of Fig. (6), we see polarization values inferred from the theoretical model presented by Eq. (8) together with observed polarization data of our selected sample; here $\alpha$ is obtained from the spectral analysis given in §3. We indicate the polarization values of 16 GRBs, including 6 HD-type bursts and 10 PFD-type bursts, where 11 GRBs with low polarization significance are not displayed (see Table 1). Interestingly, it is found that HD-type bursts are better distributed along with the line predicted by Eq. (8) and additionally show higher values of polarization compared to PFD-type bursts.

The degree of polarization for a magnetic field with a locally tangled or random configuration is obtained by averaging over all directions of the local magnetic field within the plane of the shock (Granot & Königl, 2003a; Sari, 1999; Gruzinov, 1999; Nava et al., 2016). The presence of a random magnetic field leads to negligible values of net linear polarization measured by an on-axis observer. In the case of a random field behind the shock wave, only if the observer is off-axis and the circular symmetry is broken is nonzero net polarization measurable. The total linear polarization arising from the whole jet that is subjected to a random field with a direction perpendicular to the jet velocity is given by Granot (2003a)

where $\Theta(1-\zeta)$ is the Heaviside step function with $\zeta\equiv\theta_{\rm obs}/\theta_{j}$ as a parameter to define the observer’s point of view, and

In the above expressions, $y_{1,2}=(1\mp\zeta)^{2}y_{j}$ and $y_{j}=(\Gamma\theta_{j})^{2}$. The variation of the linear polarization in the presence of a random field configuration measured by an off-axis observer is displayed in Fig. (7). In the left panel, the spectral indices are selected to be consistent with the average values reported in Table (2) for our target sample and for $y_{j}=10$. In the right panel, $y_{j}$ is changed while $\alpha=1$, it is found that the appeared peak has a width in the order of $1/\sqrt{y_{j}}$. From Fig. (7), we see that the polarization degree is limited to small values for $\zeta<1$ while it is sharply increased for $\zeta\approx 1$ and finally reaches an asymptotic limit at $\zeta>\mathcal{O}(1)$. It is seen that the synchrotron radiation with $B_{\perp}$ can potentially generate a wide range of polarization values from low levels to moderate values that cover the observed values associated with our sample. In principle, various viewing angles $\theta_{\rm obs}$ and different angular structures of the jet affect the measured fluence of GRBs. Note that the fluence significantly decreases from outside the jet’s sharp edge, so high levels of polarization in off-axis jets may only be obtainable in very close bursts. The detectibility of GRB polarization needs high-fluence sources, and usually, the fluence rapidly drops below the detector threshold for a large off-axis observer. For example, the observed fluences of GRB 140206A are relatively higher than other sources (see Table 1) and due to its higher redshift, $\zeta$ cannot get large values.

As another possibility, the polarized emission may also originate from independent magnetic patches with various field orientations Li et al. (2022) where magnetic patches are locally coherent but distributed randomly in observed emission regions. In this case, the measured polarization from different patches is estimated as $\Pi=\Pi_{max}/\sqrt{N}$, where $N$ is the number of magnetic patches or, equivalently, multiple pulses where the coherence length of the magnetic field is as large as the emission region in a single pulse and the observed polarization is an average over multiple pulses (Gruzinov & Waxman, 1999; Granot & Königl, 2003b). The magnetic field that is generated within internal shock for KED jets usually has a coherence length much smaller than the angular size of the emission region, which causes negligible net polarization.

The time-resolved spectral analysis in §3.1 showed thermal-to-nonthermal (KED-to-PFD) transitions in our sample, where a subdominant component of the thermal emission during HD-type bursts is observed. The thermal component may also originate from photosphere emission due to repeated Compton scattering of photons with thermal electrons in plasma before escaping, and observing hard values of the spectral indices during the bursts can serve as hints that LOS is not highly off-axis, since high-latitude emission leads to a softer spectrum Lundman et al. (2013). The local degree of polarization due to the Compton scattering process is given by Rybicki & Lightman (1979):

where $\theta_{\rm sc}$ is the scattering angle between the incoming and scattered photons. For a plasma including charged particles with a velocity distribution, $\Pi_{\rm sc}$ is obtained by a weighted integrating over various directions of the incoming photons and velocities of the electrons. Compton scattering generates polarized emission especially close to the photosphere, where the anisotropy of the comoving scatter photons is large.

The observed high values of the polarization for HD bursts while they show the peak-KED pattern cannot be explained simply by the subphotospheric dissipation model based on Comptonization. The multiple scatterings at large optical depth regions lead to washing out the directionality of the polarization vectors Parsotan et al. (2020). While the photospheric emission of a relativistically expanding fireball has been traditionally considered as a mechanism to produce small values of polarization ($\Pi\lesssim 20\%$), however, introducing structures in the jet profile can increase the polarization up to $40\%$ Lundman et al. (2014a). A structured jet photosphere model is also considered in (Chang et al., 2014b, a, 2013) as a source of polarized photons via Compton scattering. If synchrotron emission is the origin of soft photons in the photospheric model, the observed polarization could even potentially be enhanced by approximately $50\%$ (Lundman et al., 2018b; Ito et al., 2014). To explain the strong polarized signals, models invoking dissipation of ordered magnetic fields are favored (Lyutikov et al., 2003; Zhang & Yan, 2011; McKinney & Uzdensky, 2012).

The jitter radiation emitted by ultrarelativistic electrons accelerating in a small-scale random magnetic field (Medvedev, 2000), can also generate a hard energy spectrum with a photon index as high as $\alpha=+0.5$. Due to the random distribution of the magnetic field, jitter radiation is highly symmetric in the electron radiative plane, leading to the vanishing polarization degree for an on-axis observer (Mao & Wang, 2013, 2017; Mao et al., 2018). The maximum level of polarization is obtainable when the emitting plane is viewed from the edge-on; it can even reach up to $90\%$ (Prosekin et al., 2016). However, for smaller off-axis viewing angles, which can yield measurable fluences, jitter radiation causes almost negligible polarization degree. Meanwhile, regardless of the viewing angle, the jitter radiation cannot produce the observed high degree of polarization close to the spectral peak energy of the jet.

To summarize, polarization features can be explained either by the synchrotron radiation in the ordered/random magnetic field (Granot, 2003b; Granot & Königl, 2003b; Nakar et al., 2003), the jet structure (Lazzati & Begelman, 2009), or the observer’s viewing angle with respect to the jet (Lazzati et al., 2004), even in the case of thermal radiation from the jet photosphere (Lundman et al., 2014b). For a hybrid spectrum that includes thermal and nonthermal components, we expect to see relatively high values of the polarization in the prompt emission, which can be produced by the synchrotron emission mechanism in the ordered magnetic field of the jet and for random field configurations only for off-axis observers (Gill et al., 2021). However, the spectral properties of our target sample demonstrated that off-axis observations, especially for the large viewing angle, are not the case, and the observed values of the polarization most probably are a hint of the ordered magnetic field originating from the central engine. Since polarization washout effects are gradually increased from PFD jets towards HD and KED jets due to thermal photons, we would expect that the inequality $\pi_{\rm KED}\lesssim\pi_{\rm HD}\lesssim\pi_{\rm PFD}$ is satisfied if other conditions are fixed for a given jet. However, as it was noted in §3, the polarization pattern of GRBs in our sample displays a bimodal distribution for PFD-type bursts with two peaks at $\pi=11.9\%$ and $\pi=50.0\%$, and also a higher peak at $\pi=65.9\%$ for HD-type bursts. Although this observation is not consistent with our expectations, we would like to stress that this behavior can still be explained by the geometrical effects and the jet structure for different configurations of the magnetic fields. Due to the different degrees of polarization predicted by different emission models in various energy bands, it is essential to have a high-sensitivity gamma-ray polarimeter with a wide bandpass to detect energy-dependent polarization signals and constrain different models (Zhang, 2014). However, because of several free parameters in polarization models, upcoming more precise observations and theoretical investigations are needed to discriminate between competing models in order to explain the observed joint polarization and spectral properties.

It should be noted that the time-integrated assumption could potentially smear out important details about the spectral evolution in GRBs, making it challenging to identify the nature of the outflow. Therefore, the time-resolved analysis, as discussed in (Burgess et al., 2019b), is crucial for capturing the dynamic changes in the spectrum and accurately identifying the emission mechanisms and outflow properties. By considering the temporal evolution of the spectral parameters, one can better distinguish between different physical processes and improve the classification of the outflow type in GRBs; this will be the subject of our future paper.