*4.4. Polarization from Multiple Overlapping Pulses*

Since GRBs are generally photon-starved, the only hope of obtaining a statistically significant polarization measurement often relies on integrating over broad segments of the prompt GRB lightcurve. Due to the highly variable nature of the prompt GRB emission, a given emission episode consists of multiple overlapping pulses. The properties of the emission region, e.g., bulk-Γ, B-field configuration, can change between different pulses and improper accounting of these changes in calculating the time-integrated polarization can lead to erroneous results.

In the simplest scenario, multiple pulses are produced by distinct patches or minijets within the observed region of size *R*/Γ of the outflow surface. These patches can be permeated by an ordered B-field the orientation of which is also mutually distinct among the different patches. A broadly similar B-field structure can also be obtained in both internal and external shocks due to macroscopic turbulence excited by, e.g., the Richtmyer–Meshkov instability, which arises in the interaction of shocks and upstream density inhomogeneities [68–70,229]. In the case of mini-jets, the bulk-Γ of the different jets can also be different by a factor of order unity, which will affect the size of the individual beaming cones. Since the Stokes parameters are additive for incoherent emission the timeintegrated net polarization of *N<sup>p</sup>* incoherent patches (in the visible region of angular size 1/Γ around the line of sight) is obtained from [63] (where the motivation was afterglow emission from a shock-generated field rather than incoherent patches or mini-jets).

$$
\Pi = \frac{\mathcal{Q}}{I} = \frac{\sum\_{i=1}^{N\_p} \mathcal{Q}\_i}{\sum\_{i=1}^{N\_p} I\_i} \sim \frac{\Pi\_{\text{max}}}{\sqrt{N\_p}} \,. \tag{17}
$$

The net polarization is significantly reduced for increasingly large numbers of patches due to the fact that the PA are randomly oriented, and when added together some cancellation occurs. This essentially represents a random walk for the polarized intensity *Q* while the total intensity adds up coherently. When multiple time-integrated segments of an emission episode are compared, the net polarization and PA will vary between them (the latter is possible as this is a non-axisymmetric global configuration). Alternatively, instead of ordered B-field patches, one can have a shock-produced B-field (e.g., *B*⊥) with a patchy shell or mini-jets that give different weights to different parts of the image and thereby produce a net polarization (see, e.g., [109,230]).

Another scenario that is worth considering is when multiple overlapping pulses are produced by episodic energization of the emission region, e.g., in the collision of multiple shells in the internal shock scenario where the ejection time of subsequent shells is different, such that the ejection time of the *i*th shell in the engine frame is *t*ej,i,z = *t*ej,i/(1 + *z*). The onset time of each pulse is then given by *t*onset,i,z = *t*ej,i,z + *t*0,*<sup>z</sup>* . The scenario of multiple pulses from a smooth top-hat jet is demonstrated in Figure 13 using simplifying assumptions, where all pulses have the same *R*<sup>0</sup> and Γ(*R*0) (so that the radial delay time *t*0,*<sup>z</sup>* for emission arising from different pulses is the same) and radial extent ∆*R*. In this case, the onset times of pulses is simply dictated by the different ejection times of the shells. The left panel shows the pulse profile, and the right panel shows the polarization calculated for the *B*tor field. Time-resolved polarization obtained from multiple temporal segments, where the emission episode is divided into one, two, or three equal duration segments, is shown to demonstrate the different levels of polarization obtained when using the multi-pulse or the single-pulse model. Therefore, when the emission consists of multiple overlapping pulses, it is important to compare the measurement with model predictions that account for multiple pulses.

**Figure 13.** (**Left**) Pulse profile of multiple overlapping pulses in an emission episode, shown here for a KED smooth top-hat jet. A single pulse is also shown for comparison. (**Right**) Temporal evolution of the polarization for a toroidal magnetic field (*B*tor) shown for both the single pulse and multiple pulses. Temporal segments over which polarization is obtained are calculated by dividing the pulse into one (red), two (blue), or three (green) part(s). See the caption of Figure 11 for explanation of different symbols. Figure adapted from [84].

#### *4.5. Most Likely Polarization Measurement*

As demonstrated in earlier sections, the prompt GRB polarization depends on (i) the underlying radiation mechanism, (ii) B-field structure (for synchrotron emission), (iii) bulk LF Γ (top-hat jet) or Γ*<sup>c</sup>* (structured jet), (iv) *θ<sup>j</sup>* (top-hat jet) or *θ<sup>c</sup>* (structured jet), (v) viewing angle *θ*obs, and (vi) angular structure, e.g., power-law indices *a* and *b* for a power-law structured jet (see Section 4.2). Due to variations in these parameters the polarization can vary between different pulses within the same GRB as well as between different GRBs. For an ultrarelativistic flow, three basic quantities naturally arise that affect the polarization, namely, (a) the normalized jet/core half-opening angle: *ξ* 1/2 *<sup>j</sup>* = Γ*θ<sup>j</sup>* (top-hat jet) or *ξ* 1/2 *<sup>c</sup>* = Γ*cθ<sup>c</sup>* (structure jet), (b) the normalized viewing angle: *q* = *θ*obs/*θ<sup>j</sup>* (tophat jet) or *q* = *θ*obs/*θ<sup>c</sup>* (structured jet), and (c) the normalized viewing-angle dependent fluence: ˜ *f*iso(*q*, *ξj*) = *Eγ*,iso(*q*, *ξj*)/*Eγ*,iso(0, *ξj*) (top-hat jet) or ˜ *f*iso(*q*, *ξc*) (structure jet), which is the ratio of the off-axis to on-axis isotropic-equivalent radiated energy or equivalently the fluence.

For different pulses emitted by the same GRB, it is natural to expect a considerable change in (iii), while the other parameters are likely to remain more or less fixed. In, e.g., a top-hat jet, this will change the parameter *ξ<sup>j</sup>* , and, for a given distribution of *ξ<sup>j</sup>* between several pulses, the total polarization, after integrating over multiple pulses, will be different from that obtained for a single pulse. When adding up the Stokes parameters of different pulses, an appropriate relative weight using, e.g., *Eγ*,iso (or more precisely the relative expected number of photons that will be detected), should be applied.

When comparing emission from different GRBs all of the above-mentioned quantities can in principle vary (or at least there is no strong evidence against this in the observed sample of GRBs). In this case, the fluence ratio is important in determining (i) whether for a given *θ*obs > *θ<sup>j</sup>* (top-hat jet) or *θ*obs > *θ<sup>c</sup>* (structured jet) the pulse will be bright enough to be observed by a given detector and (ii) for a given GRB out to which viewing angle it will be fluent enough for performing polarization measurements. For a top-hat jet, the fluence is strongly suppressed due to Doppler de-beaming when Γ(*θ*obs − *θj*) & 1, whereas, for a structured jet, the suppression in fluence is not as severe and emission from *q* . few to several can be detected if it is not suppressed due to compactness, as discussed earlier.

A distribution of polarization for a given radiation mechanism, while accounting for variations in the aforementioned quantities between different pulses from the same source and different GRBs, and its comparison with actual measurements can be used to answer some of the key questions of GRB physics. Such a distribution obtained from a Monte Carlo simulation (see [24] for more details) is shown in Figure 14 for a power-law jet and for different radiation mechanisms as well as different B-field configurations. As expected, the *B*tor field being ordered yields the highest polarization with 45% . Π . 60%. Therefore, if GRB jets feature a large-scale toroidal field, then most GRBs that are emitting synchrotron radiation will show Π ∼ 50%. For the other two B-field configurations, *B*<sup>⊥</sup> and *B*k , the expected polarization is small with Π . 10%, and one is most likely to find GRBs with negligible polarization. The same conclusion can be drawn for the Compton drag and photospheric radiation mechanisms. The polarization in the photospheric emission model can be Π . 15% when the flow features a much steeper bulk-Γ angular profile with √ *ξ<sup>c</sup>* = Γ*cθ<sup>c</sup>* ∼ few (see Figure 14 of [24] for more details). When comparing with observations, some of which have at least 3*σ* detection significance, no firm conclusions can be drawn at this point. Measurements made by IKAROS-GAP and AstroSat-CZTI find highly polarized GRBs with Π & 50%, although with large 1*σ* error bars. On the other hand, the POLAR data appear to indicate that GRBs are more likely to have significantly smaller polarization with most of their sample consistent with unpolarized sources. The apparent discord between the results of these works not only highlights the challenges involved in obtaining a statistically significant polarization measurement but also calls for the need to build more sensitive detectors.

**Figure 14.** (**Right**): Distribution of polarization from synchrotron emission for different B-field configurations, Compton drag (CD), as well as photospheric (Phot) emission in a power-law-structured jet obtained from a Monte Carlo simulation (with 10<sup>4</sup> samples). Measured polarizations with 1*σ* error bars from different instruments are shown for comparison. The measurement of Π = 66+<sup>26</sup> <sup>−</sup>27% (<sup>∼</sup> 5.3*σ*) from [231] obtained using AstroSat-CZTI is shown with a black dot with cyan error bars. Figure adapted from [24] where more details can be found. (**Left**): Zoomed-in version of the figure showing the several overlapping distributions for clarity (with a bin size smaller by a factor of 0.4).

#### *4.6. Energy Dependence of Polarization*

Polarization is energy dependent. This can be easily seen in emission mechanisms where the local polarization depends on the spectral index, e.g., in optically thin synchrotron radiation (see Equation (10)). The energy-dependent spectro-polarimetric evolution in this case is shown in the left panel of Figure 15; temporal evolution of polarization at a given energy and the pulse profile for the same case was shown earlier in Figure 12. The polarization is sensitive to the local spectral index, which, for a Band-like spectrum, changes near the spectral peak and asymptotes far away from it.

Energy-dependent polarization is possible also in emission mechanisms where the local polarization is independent of energy, such as Compton drag in the Thomson regime (where the energy-independent Equation (12) holds). A featureless power-law spectrum will have no energy dependence, but the energy-independent polarization would still depend on the spectral power-law index, Π = Π(*α*). This occurs since different *α*-values give different weights to different parts of the image between which the Doppler factor varies such that the same observed frequency corresponds to different comoving frequencies. For a non-featureless spectrum, the same effect can cause energy dependence in the polarization, e.g., for a Band spectrum, the relative weights of different parts of the image (and therefore also the polarization) will depend on the initial location of the observed frequency relative to the peak frequency along the LOS (i.e., on *x*<sup>0</sup> = *ν*/*ν*0).

Alternatively, if multiple spectral components from different radiation mechanisms having different levels of polarization contribute to the observed spectrum, the polarization of the total spectrum will change with energy. This is expected in some photospheric emission models [232] that posit that the spectral peak is dominated by the quasi-thermal photospheric component while the low and/or high energy wings may come from synchrotron emission (see, e.g., Figure 2 and discussion in Section 2.3.3). The right panel of Figure 15 presents such a case, where the polarization grows with decreasing energy owing to the dominance of flux by the synchrotron component. Near the spectral peak, the polarization vanishes. In this way, energy-resolved polarization measurements can be invaluable in understanding the GRB radiation mechanism.

**Figure 15.** (**Left**): Temporal evolution of the Band-like spectrum (*solid lines*; left *y*-axis) and the corresponding polarization (*dashed lines*; right *y*-axis) from synchrotron emission with a *B*tor field for a KED top-hat jet (THJ) with *ξ<sup>j</sup>* = (Γ*θ<sup>j</sup>* ) <sup>2</sup> = 10<sup>2</sup> and *q* = *θ*obs/*θ<sup>j</sup>* = 0.8, and *m* = 0. The different colours correspond to different normalized apparent times ˜*t* = *t*/*t*<sup>0</sup> where *t*<sup>0</sup> = 2(1 + *m*)Γ 2 0 *ct*/*R*<sup>0</sup> is the arrival time of the initial photons emitted from radius *R*<sup>0</sup> along the LOS. The peak frequency of the *νF<sup>ν</sup>* spectrum at this time is given by *ν*0. (**Right**): Multi-component GRB spectrum and its energy-resolved polarization. While the photospheric component dominates both spectral peak and at higher energies, the low-energy spectrum is produced by synchrotron emission. As a result, the polarization grows towards lower energies as the fraction of synchrotron photon grows. The light and dark shaded regions correspond to the energy ranges of *Fermi* GBM (NaI + BGO detectors, (8–30 MeV), and GAP (70–300 keV), respectively. Figure from [232].
