4.1.1. Synchrotron Emission from Different Magnetic Field Structures

Synchrotron emission is generally partially linearly polarized. The local polarization emerging from a given point on the outflow depends on the geometry of the local Bfield and distribution of the emitting electrons, both in energy, *γemec* 2 , and pitch angle, *χ* <sup>0</sup> = arccos(*B*ˆ<sup>0</sup> · *β*ˆ0 *e* ), where *β*ˆ<sup>0</sup> *e* is a unit vector pointing in the direction of the electron velocity. In the case of power-law electrons, with distribution *ne*(*γe*) ∝ *γ* −*p e* for *γ<sup>e</sup>* > *γ*min, and with isotropic velocity distribution so that all pitch-angles are sampled during the emission, the maximum local polarization for a locally ordered B-field depends on the spectrum [108,220]

$$
\Pi\_{\text{max}} = \frac{\alpha + 1}{\alpha + 5/3} = \frac{p\_{\text{eff}} + 1}{p\_{\text{eff}} + 7/3}.\tag{10}
$$

Here *α*(*ν*) = −*d* log *Fν*/*d* log *ν* is the *local* spectral index, and *p*eff = 2*α* + 1 is the effective power-law index of the electron distribution. Since the local value of *α* (and therefore also of *p*eff) smoothly varies with *ν*, the maximum polarization, Πmax, also varies smoothly with *ν* across the spectral breaks of the synchrotron spectrum. The *asymptotic* spectral index is different for different power-law segments (PLSs) of the well-studied [87,88] broken power-law synchrotron spectrum. We have *α* = 1/2, *p*eff = 2, and Πmax = 9/13 ≈ 0.692 for *ν<sup>c</sup>* < *ν* < *ν<sup>m</sup>* (fast cooling); *α* = (*p* − 1)/2, *p*eff = *p* and Πmax = (*p* + 1)/(*p* + 7/3) for *ν<sup>m</sup>* < *ν* < *ν<sup>c</sup>* (slow cooling); *α* = *p*/2, *p*eff = *p* + 1 and Πmax = (*p* + 2)/(*p* + 10/3) for *ν* > max(*νc*, *νm*) (either slow or fast cooling). For *ν* < min(*νm*, *νc*), there is no *p*eff since emission in this PLS arises from all cooling electrons that are emitting below their typical (optically thin) synchrotron frequency. In this case, *α* = −1/3 and Πmax = 1/2, the lowest local polarization obtained from synchrotron emission. On the other hand, shock-acceleration theory suggests that 2 . *p* . 3, which means that the maximum local polarization in synchrotron is limited to Πmax . 75%.

When the magnetic fields are tangled or switch direction on angular scales 1/Γ, e.g., in the *B*<sup>⊥</sup> case, the local polarization must be averaged over different B-field orientations. This has been calculated for an infinitely thin ultrarelativistic shell, while assuming *α* = 1, for a tangled B-field [109,222,223]

$$\frac{\Pi\_{\rm md}}{\Pi\_{\rm max}} = \frac{(b-1)\sin^2\theta'}{2 + (b-1)\sin^2\theta'} = \begin{cases} \frac{-\sin^2\theta'}{1 + \cos^2\theta'} & (b = 0, B \to B\_\perp) \\ 1 & (b = \infty, B \to B\_{\parallel}) \end{cases} \tag{11}$$

where ˜*θ* 0 is the polar angle measured from the LOS in the comoving frame (this holds for a radial flow and more generally ˜*θ* <sup>0</sup> → arccos(*n*ˆ 0 · *n*ˆ 0 sh)). The level of anisotropy of the B-field is quantified by the parameter *b* = 2h*B* 2 k i/h*B* 2 ⊥ i, which represents the ratio of the average energy densities in the two field orientations. The factor of two simply reflects the two independent directions of the *B*<sup>⊥</sup> component, such that *b* = <sup>1</sup> for a field that is isotropic in three dimensions.

The polarization map over the GRB image on the plane of the sky is shown in Figure 6 for different B-field structures (for Γ 1). Only the area contained within the beaming cone, shown by the red circle, contributes dominantly to the emission. Outside of it, the intensity is strongly suppressed by relativistic beaming, which scales as a power of the Doppler factor. This effect is reflected by the decrease with the angle ˜*θ* from the LOS (shown by the red "+" symbol) in the size of the black arrows, which correspond to the magnitude of the polarized intensity. When the jet possesses axial symmetry (and for synchrotron emission the same requirement holds also for the global magnetic field structure), then the image and polarization map are symmetric to reflection along the line connecting the jet symmetry axis to the LOS. Therefore, it is natural to choose a reference direction for measuring the local PA ¯*θ<sup>p</sup>* either along this line or transverse to it (in the figure, ¯*θ<sup>p</sup>* as well as

*θ<sup>p</sup>* are measured from the latter, i.e., the horizontal direction). For such a choice, *U* = 0 i.e., the local Stokes parameter *U*¯ ∝ sin(2 ¯*θp*) vanishes when integrated over the GRB jet image, and therefore the global polarized intensity is entirely given by Stokes *Q*, i.e., the integration of *Q*¯ ∝ cos(2 ¯*θp*) over the image, where the sign of *Q*¯ for each fluid element depends on the local PA ¯*θp*. The different B-field configurations produce completely different polarization maps, with distinct patterns of regions contributing predominantly either to polarization along the line connecting the jet symmetry axis to the LOS (orange–yellow, with local polarization Π¯ < 0) or transverse to it (blue–white, with local polarization Π¯ > 0), as shown by the color map. When averaged over the entire GRB image, these are the only two directions of polarization that can be obtained in an axisymmetric flow in which the magnetic field also possesses axial symmetry about the jet axis, such that it would represent a change of 90◦ in the PA when the direction of polarization switches from one to the other.

**Figure 6.** Polarization map for different B-field configurations shown on the surface of a top-hat jet (for Γ 1). The jet symmetry axis marked with a black "+" symbol and the observer's LOS is marked with a red "+" symbol. The region where the LOS is within the beaming cone of the local emission (i.e., from which the radiation is beamed towards us) is within the red circle, outside of which the the polarized intensity (as shown by the size of the black arrows) declines sharply. The red line segments show the direction and polarized intensity, now without the de-beaming suppression. Green lines show the orientation of the magnetic field lines (in the cases *B*ord and *B*tor where it is locally ordered). The color map shows *Q*¯ ∝ cos(2 ¯*θp*), with ¯*θ<sup>p</sup>* being the local polarization angle measured counter-clockwise from the horizontal axis, which corresponds to the level at which each point is polarized either along the line connecting the LOS with the jet symmetry axis (orange-yellow dominated) or transverse to it (blue-white dominated).

An example of a B-field configuration that does not possess such axial symmetry is *B*ord. When *B*ord is not oriented along the line connecting the jet symmetry axis to the LOS or perpendicular to it, then this breaks the symmetry of the image polarization map, thereby enabling other directions of net global polarization to occur, and the corresponding PA can vary continuously with a finite Π [109].

The level of net polarization after averaging over the GRB image depends on the level of symmetry of the polarization map around the LOS. In the case of *B*⊥, and likewise for *B*k , the polarization map is symmetric around the LOS and therefore averaging over the GRB image would yield zero net polarization (Π = 0) due to complete cancellation for a spherical flow (or well within a top-hat jet, Γ(*θ<sup>j</sup>* − *θ*obs) 1). This symmetry is naturally broken in *B*ord and *B*tor where the local B-field is ordered and provides a particular direction (transverse to the local B-field direction and to the propagation direction of the photon) along which the polarization vector aligns. Another way to break the symmetry is by having the LOS close to the edge of the jet, with *θ<sup>j</sup>* − Γ <sup>−</sup><sup>1</sup> . *θ*obs . *θ<sup>j</sup>* + Γ <sup>−</sup><sup>1</sup> <sup>⇔</sup> <sup>Γ</sup>|*θ*obs <sup>−</sup> *<sup>θ</sup><sup>j</sup>* | . 1 (1 − *ξ* −1/2 *<sup>j</sup>* . *q* . 1 + *ξ* −1/2 *<sup>j</sup>* ⇔ *ξ* 1/2 *j* |*q* − 1| . 1), so that some part of the beaming cone lies outside of the jet surface. The missing emission, which would otherwise contribute towards cancellation, leads to only partial cancellation and yields a net finite polarization, |Π| > 0. The sign of net polarization is decided by whichever region, either blue–white or orange–yellow, makes the dominant contribution to the polarized flux. In Figure 6, Π < 0 for both *B*ord and *B*tor, whereas Π ≈ 0 for *B*⊥.

Pulse-integrated polarization as a function of *q* is shown in Figure 7 for different B-field configurations and different *ξ<sup>j</sup>* , where the latter describes how wide or narrow the jet aperture is compared to the beaming cone. The polarization curves look very different for the three different field configurations, but there are some features that are worth pointing out. The polarization vanishes when the observer is looking down the jet axis, i.e., when *θ*obs = 0 (*q* = 0), in all cases due to complete cancellation (such a cancellation would not occur for *B*ord, which is not shown in Figure 7). For *q* > 0, polarization grows rapidly for *B*tor (for which it saturates at *ξ* −1/2 *<sup>j</sup>* . *q* . 1 − *ξ* −1/2 *<sup>j</sup>* ⇔ Γ <sup>−</sup><sup>1</sup> . *<sup>θ</sup>*obs . *<sup>θ</sup><sup>j</sup>* <sup>−</sup> <sup>Γ</sup> −1 ) but slowly for both *<sup>B</sup>*<sup>⊥</sup> and *<sup>B</sup>*<sup>k</sup> . It reaches a local maxima when the LOS is close to the jet edge, i.e., as before, when |*q* − 1| . *ξ* −1/2 *<sup>j</sup>* ⇔ Γ|*θ*obs − *θ<sup>j</sup>* | . 1, and declines sharply for *B*<sup>⊥</sup> and *B*tor when the LOS exceeds one beaming cone outside of the jet, i.e., *θ*obs & *θ<sup>j</sup>* + Γ −1 (*q* > 1 + *ξ* −1/2 *j* ). The *<sup>B</sup>*<sup>k</sup> case yields a different behavior where Π becomes maximal when the jet is viewed from outside its edge. In all cases, when *q* > 1 + *ξ* −1/2 *<sup>j</sup>* ⇔ Γ(*θ*obs − *θj*) > 1 the fluence drops off very sharply for a top-hat jet. So, even though a large Π is expected for *<sup>B</sup>*<sup>k</sup> , it will be challenging to detect. Finally, a change in the PA by 90◦ occurs when *θ*obs ≈ *θ<sup>j</sup>* (*<sup>q</sup>* ≈ 1) for *<sup>B</sup>*<sup>⊥</sup> and *<sup>B</sup>*<sup>k</sup> , at which point Π = 0.

It is clear from Figure 7 that only the *B*tor case, an ordered field scenario, yields high levels of polarization when the LOS passes within the aperture of the jet. Since all distant GRBs must be viewed with *q* < 1, otherwise they will be too dim to detect, a measurement of 50% . Π . 65% will strongly indicate the presence of an ordered field component. On the other hand, if the B-field configuration is more like *<sup>B</sup>*<sup>⊥</sup> or *<sup>B</sup>*<sup>k</sup> , then most GRBs will show negligible polarization.

**Figure 7.** Pulse-integrated polarization of synchrotron emission for different B-field configurations shown for different LOSs (*q*) and size of the beaming cone w.r.t to the jet aperture (*ξ<sup>j</sup>* = (Γ*θ<sup>j</sup>* ) 2 ). The spectral index was fixed to *α* = 3/4, where a larger *α* produced a larger Π. Figure adapted from [24] but originally produced in [108].

#### 4.1.2. Photospheric Emission from a Uniform Jet

A photospheric spectral component can arise and even dominate the spectral peak in scenarios where energy is dissipated below the photosphere. At the photosphere, radiation decouples from matter and is able to stream freely towards the observer. However, in a matter-dominated flow in which the baryon rest mass energy density, *ρ* 0 *c* 2 , is much larger than that of the radiation field, *U*0 *γ* , the radiation field becomes highly anisotropic at the photosphere [55]. At the last scattering surface, this produces significant *local* polarization at each point of the observed part of the flow. Nevertheless, upon averaging over the GRB image the net polarization is expected to be negligible in an axisymmetric uniform flow since there is no preferred direction for the polarization vector. To obtain finite net polarization, an inhomogeneous outflow with gradients in bulk-Γ (and to a lesser extent in comoving emissivity *L* 0 *ν* <sup>0</sup>) across the beaming cone are needed. This scenario is discussed in Section 4.2.

Alternatively, if the flow is radiation-dominated, i.e., *U*0 *<sup>γ</sup> ρ* 0 *c* 2 , as shown by Beloborodov [55], the comoving angular distribution of the radiation field is preserved in the ultrarelativistic limit as the flow goes from being optically thick to thin. This occurs due to

the fact that radiation always tries to push the plasma to an equilibrium Lorentz frame in which the radiative force on the plasma vanishes. As a result, the radiation field accelerates the plasma to a bulk LF Γ(*R*) ∝ *R*, which is a special Lorentz frame in which the (comoving) direction of freely streaming photons w.r.t the local radial direction remains unchanged in between successive scatterings. This means that an isotropic radiation field remains isotropic. Since scattering an isotropic radiation field only produces another isotropic field, the flow behaves (to leading order) as if no scatterings took place. Since the radiation field was necessarily isotropic when the flow was optically thick at smaller radii, leading to zero local polarization, it must yield the same (to leading order) when it becomes optically thin, as shown below.

The radiation field is able to accelerate the flow to Γ(*R*) ∝ *R* only if *U*0 *<sup>γ</sup>*/*ρ* 0 *c* <sup>2</sup> <sup>1</sup> and while the matter maintains a small lag, ∆Γ = Γrad − Γmatter, which is initially Γmatter ≡ Γ ≈ Γrad (where these LFs are of the respective local center of momentum frames) corresponding to a relative velocity *βγ*<sup>m</sup> ∼ ∆Γ/Γ ∼ *ρ* 0 *c* <sup>2</sup>/*U*<sup>0</sup> *<sup>γ</sup>* 1, but it gradually increases until it eventually becomes comparable to the two near the saturation radius *R<sup>s</sup>* where *U*0 *<sup>γ</sup>τ<sup>T</sup>* ∼ *ρ* 0 *c* 2 (for *τ<sup>T</sup>* < 1), the point beyond which matter stops accelerating and starts coasting, while the scaling Γrad ∝ *R* remain valid as the radiation-free streams in increasingly more radial directions. In a steady radiation-dominated spherical flow the comoving radiation energy density scales as *U*0 *<sup>γ</sup>* ∝ *V* 0−4/3 ∝ [*R* <sup>2</sup>Γ(*R*)]−4/3, and the rest mass energy density of the particles scale as *ρ* 0 ∝ *n* 0 *<sup>e</sup>* ∝ *V* 0−<sup>1</sup> ∝ [*R* <sup>2</sup>Γ(*R*)]−<sup>1</sup> , where *V* 0 is the comoving volume. This yields *U*0 *<sup>γ</sup>*/*ρ* 0 *c* <sup>2</sup> ∝ [*R* <sup>2</sup>Γ(*R*)]−1/3, which for Γ(*R*) ∝ *R* gives *U*0 *<sup>γ</sup>*/*ρ* 0 *c* <sup>2</sup> ∝ *R* <sup>−</sup><sup>1</sup> and *τ<sup>T</sup>* = *n* 0 *<sup>e</sup>σTR*/Γ(*R*) ∝ [*R*Γ 2 (*R*)]−<sup>1</sup> ∝ *R* −3 . This further yields *U*0 *<sup>γ</sup>τT*/*ρ* 0 *c* <sup>2</sup> = (*U*<sup>0</sup> *<sup>γ</sup>*,ph/*ρ* 0 ph*c* 2 )(*R*/*R*ph) −4 so that *R<sup>s</sup>* ∼ *R*ph(*U*<sup>0</sup> *<sup>γ</sup>*,ph/*ρ* 0 ph*c* 2 ) 1/4 <sup>∼</sup> *<sup>R</sup>*ph*<sup>β</sup>* −1/4 *γ*m,ph and *βγ*<sup>m</sup> ∼ min[1,(*R*/*Rs*) 4 ]. Near *R<sup>s</sup>* the comoving radiation anisotropy becomes significant (*βγ*<sup>m</sup> ∼ 1) and therefore so does the polarization of the radiation scattered at *R* ∼ *R<sup>s</sup>* , but this is only a fraction ∼ *τT*(*Rs*) ∼ (*U*<sup>0</sup> *<sup>γ</sup>*,ph/*ρ* 0 ph*c* 2 ) <sup>−</sup>3/4 <sup>∼</sup> *<sup>β</sup>* 3/4 *<sup>γ</sup>*m,ph 1 of the photons, and therefore the overall local (i.e., from a particular fluid element) polarization is of the same order, i.e., very small.
