**4. Theoretical Models of Prompt GRB Polarization**

The focus of this section is to present polarization predictions for the popular prompt GRB emission mechanisms, as highlighted in Section 2. Since GRBs are cosmological sources (of modest physical size in astrophysical standards), they remain spatially unresolved. Consequently, the measured polarization is the effective average value over the entire image of the burst on the plane of the sky. Therefore, the obtained polarization is affected by several effects, such as the intrinsic level of polarization at every point on the observed part of the outflow; the geometry of the outflow; i.e., the angular profile of the emissivity and bulk Γ; and the observer's LOS. Even though GRBs are intrinsically very luminous, their large distances drastically reduce the observed flux, making them photon starved. This forces observers to integrate either over the entire pulse or large temporal segments of a given emission episode to increase the photon count. This causes additional averaging—time averaging over the instantaneous polarization from the whole source, which in many cases significantly evolves even within a single spike in the prompt GRB lightcurve.

Before presenting the model predictions for time-resolved polarization in Section 4.3, pulse-integrated polarization is discussed first. In the latter, any radial dependence of the flow properties is ignored for simplicity (but without affecting the accuracy of the calculation). As a result, pulse-integrated polarization ultimately amounts to integrating over a single pulse emitted at a fixed radius, where it is not important what that radius is as it does not enter any of the calculations.

Polarization is most conveniently expressed using the Stokes parameters (*I*, *Q*, *U*, *V*), where *I* is the total intensity, *Q* and *U* are the polarized intensities that measure linear polarization, and *V* measures the level of circular polarization. In GRB prompt emission, the circular polarization is typically expected to be negligible compared to the linear polarization (*V* <sup>2</sup> *<sup>Q</sup>*<sup>2</sup> <sup>+</sup> *<sup>U</sup>*<sup>2</sup> ; this is usually expected to hold also for the reverse shock and afterglow emission) and therefore we concentrated here on the linear polarization. The local linear polarization (all local quantities are shown with a "bar") from a given fluid element on the emitting surface of the flow is given by e.g., [220]

$$\text{I\"I\"} = \frac{\sqrt{\vec{\mathbb{Q}}^2 + \vec{\mathbb{U}}^2}}{\vec{I}'} = \frac{\sqrt{\vec{\mathbb{Q}}^2 + \vec{\mathbb{U}}^2}}{\vec{I}} = \text{I\"I,} \tag{5}$$

where

$$\frac{\ddot{\varPi}}{\varPi} = \check{\varPi}\sin 2\theta\_p \, \, \qquad \frac{\not\not\mathbb{Q}}{\varGamma} = \check{\varPi}\cos 2\theta\_p \, \, \, \qquad \theta\_p = \frac{1}{2}\arctan\left(\frac{\tilde{\varPi}}{\tilde{\mathbb{Q}}}\right) \, \, \, \, \tag{6}$$

and ¯*θ<sup>p</sup>* is the local polarization position angle (PA). When moving from the comoving frame of the jet to the observer frame, both the Stokes parameters and the direction of the polarization unit vector (Πˆ¯ <sup>0</sup> = (*n*ˆ <sup>0</sup> <sup>×</sup> *<sup>B</sup>*ˆ<sup>0</sup> )/|*n*ˆ <sup>0</sup> <sup>×</sup> *<sup>B</sup>*ˆ<sup>0</sup> |, where *n*ˆ <sup>0</sup> and *B*ˆ<sup>0</sup> are the unit vectors in the comoving frame pointing along the observer's LOS and direction of the local B-field, respectively) undergo a Lorentz transformation (e.g., Equation (13) of Gill et al. [24]). The degree of polarization (magnitude of the polarization vector), however, remains invariant (since *Q*¯0/*Q*¯ = *U*¯ <sup>0</sup>/*U*¯ = ¯*I* <sup>0</sup>/ ¯*I*). The local polarization is different from the *global* one, Π = p *Q*<sup>2</sup> + *U*2/*I* (all global parameters are denoted without a bar), which is derived from the global Stokes parameters. It is the global polarization that is ultimately measured

for a spatially unresolved source. For an incoherent radiation field, meaning the emission from the different fluid elements is not in phase, which is also true for most astrophysical sources, the Stokes parameters are additive. Therefore, each global Stokes parameter is obtained by integration of the corresponding local Stokes parameter over the image of the GRB jet on the plane of the sky, such that

$$\left\{ \begin{array}{c} \Pi/I \\ \mathbf{Q}/I \end{array} \right\} \rightarrow \left\{ \begin{array}{c} \Pi\_{\mathrm{V}}/I\_{\mathrm{v}} \\ \mathbf{Q}\_{\mathrm{V}}/I\_{\mathrm{v}} \end{array} \right\} = \frac{\int d\Omega \begin{Bmatrix} \Pi\_{\mathrm{V}} = I\_{\mathrm{V}}\Pi\sin(2\tilde{\theta}\_{p}) \\ \tilde{\mathbf{Q}}\_{\mathrm{V}} = I\_{\mathrm{V}}\Pi\cos(2\tilde{\theta}\_{p}) \end{Bmatrix}}{\int d\Omega \mathbf{I}\_{\mathrm{v}}} = \frac{\int d\mathbf{F}\_{\mathrm{V}} \begin{Bmatrix} \Pi\sin(2\tilde{\theta}\_{p}) \\ \Pi\cos(2\tilde{\theta}\_{p}) \end{Bmatrix}}{\int d\mathbf{F}\_{\mathrm{V}}},\tag{7}$$

where *dF<sup>ν</sup>* ∼= *Iνd*Ω = *IνdS*⊥/*d* 2 *A* is the flux contributed by a given fluid element, of observed solid angle *d*Ω and area *dS*<sup>⊥</sup> on the plane of the sky, and *d<sup>A</sup>* is the angular distance to the distant source. We worked with the Stokes parameters per unit frequency for convenience, such as the specific intensity ¯*I<sup>ν</sup>* = *d* ¯*I*/*dν*. For simplicity, we ignored the radial structure of the outflow and assumed that the emission originates from an infinitely "thin shell". This approximation is valid if the time-scale over which particles cool and contribute to the observed radiation is much smaller than the dynamical time. It implies that the emission region is a thin cooling layer of width (in the lab-frame) ∆ *R*/Γ 2 . In this approximation, the differential flux density from each fluid element radiating in the direction *n*ˆ, i.e., the direction of the observer, when the radiating shell is at radius *R* (radial dependence included here for the general expression) can be expressed as [221]

$$dF\_{\nu}(t\_{\rm obs}, \hat{n}, \mathbf{R}) = \frac{(1+z)}{16\pi^2 d\_L^2} \delta\_{\mathbf{D}}^3 L\_{\nu'}^{\prime}(\mathbf{R}) d\mathbf{\hat{D}},\tag{8}$$

where *z* and *d<sup>L</sup>* are the redshift and luminosity distance of the source, respectively; *L* 0 *ν* 0 is the comoving spectral luminosity of the fluid element; and *d*Ω˜ = *dµ*˜*dϕ*˜ is its solid angle; *µ*˜ = cos ˜*θ* with the polar angle ˜*θ* measured from the LOS; and *ϕ*˜ is the azimuthal angle around the LOS. The Doppler factor of the fluid element moving with velocity ~*β* = ~*v*/*c* is given by *<sup>δ</sup>D*(*R*) = [Γ(<sup>1</sup> <sup>−</sup>~*<sup>β</sup>* · *<sup>n</sup>*ˆ)]−<sup>1</sup> = [Γ(<sup>1</sup> <sup>−</sup> *βµ*˜)]−<sup>1</sup> (where the second expression holds for a radial outflow where *β*ˆ = *r*ˆ). In order to calculate the Stokes parameters using the differential flux density, the angular structure of the outflow needs to be specified, as was done next.
