*4.3. Temporal Evolution of Polarization*

The earlier sections only discuss the pulse-integrated polarization, which is relevant for most GRBs that are not bright enough to be able to yield any time-resolved polarimetric results. However, with the upcoming more sensitive gamma-ray polarimeters in the next decade time-resolved polarimetry of prompt GRB emission will become possible. Therefore, in anticipation of such a development, it is prudent to also construct accurate theoretical model predictions to compare with time-resolved polarization measurements.

When discussing time-resolved polarization it becomes important to include the radial dependence of the flow properties, which were ignored for the pulse-integrated

discussion. We first describe a simple and very general pulse model of an accelerating, coasting, or decelerating flow (see, e.g., [227,228]), which is then used to calculate the time-resolved polarization. Consider a thin ultra-relativistic shell that starts to emit prompt GRB photons at radius *R* = *R*0. The emission continues over a radial extent ∆*R* and terminates at *R<sup>f</sup>* = *R*<sup>0</sup> + ∆*R*. During this time, the comoving spectrum, with *ν* 0*L* 0 *ν* <sup>0</sup> spectral peak frequency *ν* 0 pk, and spectral luminosity evolve as a power law with radius,

$$L'\_{\nu'}(R,\theta) = L'\_0 \left(\frac{R}{R\_0}\right)^a S\left(\frac{\nu'}{\nu'\_{\rm pk}}\right) f(\theta) \qquad \text{with} \qquad \nu'\_{\rm pk} = \nu'\_0 \left(\frac{R}{R\_0}\right)^d \tag{16}$$

where *L* 0 <sup>0</sup> = *L* 0 *ν* <sup>0</sup>(*R*0) and *ν* 0 <sup>0</sup> = *ν* 0 pk(*R*0) are the normalizations. The factor *f*(*θ*) describes the angular profile of *L* 0 *ν* <sup>0</sup> where it is normalized to unity at the jet-symmetry axis with *f*(0) = 1, for a uniform spherical flow *f*(*θ*) = 1 and for a top-hat jet *f*(*θ*) = H(*θ<sup>j</sup>* − *θ*) with H being the Heaviside function and *θ<sup>j</sup>* the jet half-opening angle. The comoving spectrum is described by the function *S*(*x*), which is considered here to be the Band function, where *x* = *ν* 0/*ν* 0 pk. The dynamics of the thin shell are given by the radial profile of the bulk-Γ, such that Γ 2 (*R*) = Γ 2 0 (*R*/*R*0) <sup>−</sup>*<sup>m</sup>* where Γ<sup>0</sup> = Γ(*R*0). The shell is coasting when *m* = 0 and accelerating (decelerating) for *m* < 0 (*m* > 0). Once the power law indices *a* and *d* for *L* 0 *ν* <sup>0</sup> are provided, one has complete information of the temporal evolution of the pulse. These indices depend on the details of the underlying prompt GRB model, e.g., on the composition and dissipation mechanism. If the prompt GRB spectrum is assumed to be of synchrotron origin, then it can be shown [84] that for a KED flow, where energy is dissipated at internal shocks (*m* = 0), *a* = 1 and *d* = −1. Alternatively, if the flow is PFD with a striped wind B-field structure and energy is dissipated due to magnetic reconnection, which also accelerates the flow with *m* = −2/3, then it is found that *a* = 4/3 and *d* = −2.

The pulse profile and temporal evolution of polarization for a KED flow coasting at Γ<sup>0</sup> 1 is shown in Figure 11 for an ordered B-field (*B*ord). The different curves are shown for observed frequency *ν* = *x*0*ν*0, which is a fraction *x*<sup>0</sup> of the peak frequency *ν*<sup>0</sup> = 2Γ0*ν* 0 0 of the first photons emitted along the LOS at radius *R*0. The apparent arrival time of these first photons is given by *t*0,*<sup>z</sup>* ≡ *t*0/(1 + *z*) = *R*0/2(1 + *m*)Γ 2 0 *c*, which is the characteristic radial delay time between the shell to arrive at radius *R*<sup>0</sup> and the hypothetical photon that was emitted by the engine at the same time as the shell. For *m* = 0, this is also the angular time over which radiation from within the beaming cone around the LOS arrives at the observer. Depending on *x*0, the pulse profile changes and shows a peak at different times with the latest peak occurring at ˜*t<sup>f</sup>* <sup>≡</sup> *<sup>t</sup><sup>f</sup>* /*t*<sup>0</sup> <sup>=</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>1</sup>+*<sup>m</sup> <sup>f</sup>* = (*R<sup>f</sup>* /*R*0) <sup>1</sup>+*<sup>m</sup>* = (1 + ∆*R*/*R*0) <sup>1</sup>+*m*, the arrival time of last photons emitted along the LOS from radius *R<sup>f</sup>* . At ˜*t* > ˜*t<sup>f</sup>* , the flux density declines rapidly, and the pulse becomes dominated by high-latitude emission that originates from outside of the beaming cone, i.e., from angles larger than 1/Γ<sup>0</sup> from the LOS.

**Figure 11.** Pulse profile (**left**) and temporal evolution of polarization (**right**) for a coasting (*m* = 0) ultrarelativistic (Γ<sup>0</sup> 1) thin spherical shell with an ordered field (*B*ord). Here, energy is dissipated in internal shocks in a KED flow and the emission is synchrotron, which is modeled using a Band function with asymptotic spectral indices *b*<sup>1</sup> and *b*2. The shell starts to radiate at *R* = *R*<sup>0</sup> and terminates at radius *R<sup>f</sup>* = *R*0(1 + ∆*R*/*R*0). The comoving spectral luminosity and spectral peak evolve as a power law in radius with indices *a* and *d*, respectively (see Equation (16)). The different curves show the trend at the observed frequency *ν* = *x*0*ν*<sup>0</sup> where *ν*<sup>0</sup> is the *νFν*-peak frequency of the first photons emitted along the observer's LOS from radius *R*0, which then arrive at the apparent time *t* = *t*0. The emission is assumed to have a Band function spectrum with asymptotic power-law spectral indices *b*<sup>1</sup> and *b*<sup>2</sup> below and above the spectral peak energy, respectively. Figure adapted from [84].

The polarization curves show maximal polarization initially, corresponding to Πmax(*α*) depending on the local value of the spectral index *α* for the Band function as set by *x*0. For ˜*t* < ˜*t<sup>f</sup>* , the polarization first declines and then saturates, which reflects the averaging of local polarization over the beaming cone as seen on the plane of the sky, which tends to yield a net polarization lower than Πmax. For ˜*t* > ˜*t<sup>f</sup>* , like the pulse profile, the polarization also declines rapidly when high-latitude emission becomes dominant. The polarization curves at different *x*<sup>0</sup> merge at ˜*t* = ˜*t*cross(*x*0), the crossing time of the break frequency across the observed frequency as the entire spectrum drifts towards softer energies over time. The merging of the polarization curves occurs due to the fact that after time ˜*t*cross all photons at the observed frequency *ν* are harder than the Band-function break frequency beyond which the Band function features a strict power law with a given spectral index. Therefore, the level of polarization for all photons sampling the power law is also the same as dictated by Πmax(*α*).

The polarization is not always maximal at the start of the pulse if the magnetic field is not ordered. This is demonstrated in Figure 12 that shows the pulse profile and temporal evolution of synchrotron polarization for different B-field configurations in a top-hat jet. As argued earlier, in B-field configurations, e.g., *<sup>B</sup>*<sup>⊥</sup> and *<sup>B</sup>*<sup>k</sup> , that produce axisymmetric polarization maps around the LOS the net polarization vanishes. This symmetry is only broken when the observer becomes aware of the jet edge, e.g., in a top-hat jet. It is at that instant the magnitude of polarization begins to grow above zero. The polarization curves for the three B-field configurations also show a change in the PA by ∆*θ<sup>p</sup>* = 90◦ when the curves cross zero. Interestingly, this happens more than once for *B*tor. The reason for this can be understood from the polarization maps shown in Figure 6 where the 90◦ change in the PA occurs when the net polarization begins to be dominated by emission polarized along the line connecting the jet symmetry axis and the observer's LOS over that polarized in the transverse direction or vice versa. At late times, the observed emission vanishes after the arrival time of the last photons from the edge of the jet furthest from the LOS. Since the flux declines very rapidly at ˜*t* > ˜*t<sup>f</sup>* , the changes in the PA are challenging to detect in practice.

**Figure 12.** Pulse profile (black) and temporal evolution of synchrotron polarization in a top-hat jet (THJ, with *ξ<sup>j</sup>* = (Γ*θ<sup>j</sup>* ) <sup>2</sup> and *q* = *θ*obs/*θ<sup>j</sup>* ) for different B-field configurations. See caption of Figure 11 for explanation of different symbols and parameters. Figure adapted from [84].
