tiny [table]font=tiny

# Relativistic Effects and GRB Polarization in Power-Law Evolution

###### Abstract

Despite decades of polarization observations and high-significance polarized
$\gamma$-ray, X-ray, optical, and radio emissions in gamma-ray bursts (GRBs)
have been accumulating in dozens of cases[3, 4, 5, 6, 7, 8, 9], people has yet
to find a consistent scenario for understanding the globally observed timing
properties of GRB polarization to date. Here, we report that the observed
properties of GRB polarization exhibit a four-segment timing evolution at the
cosmological distance: (I) an initial hump early on (within the first few
seconds); (II) a later on power-law decay (from $\sim$101 to $\sim$104 s),
which takes the form of $\pi_{\rm obs}\propto t^{-0.50\pm 0.02}$; (III)
afterwards a late-time rebrightening hump (from $\sim$104 to $\sim$105 s); and
(IV) finally a flatting power-law decay (from $\sim$ 105 to $\sim$ 107 s),
with the the form of $\pi_{\rm obs}\propto t^{-0.21\pm 0.08}$. These finding
may present a challenge to the mainstream of polarization models that assume
the polarization time evolution change in different emission regions. We show
that these results can be explained by relativistic and geometric effects[10]
of a highly relativistic and magnetized jet generated by central engine, and
“magnetic patches”[11, 12, 13, 14] distributed as a globally random but
locally coherent form. The long-term timing evolution of observed GRB
polarization follows a scaling law $\pi_{\rm obs}\propto 1/S_{\rm obs}$,
dominantly determined by how “magnetic patches” are randomly distributed in
observed emission region $S_{\rm obs}$ on the jet plane of $1/\Gamma$ cone. It
predicts the polarization hump and tail form in accordance with the luminosity
jet break phenomenon. Our analysis suggests that there is a single dominant
mechanism (relativistic and geometric effects) that might account for the
global observational properties of GRB polarization, and other emission
mechanisms and effects might play a role in spatially local and temporally
short effects on GRB polarization.

Liang Li${}^{\ref{ICRANet},\ref{INAF},\ref{ICRA}}$, She-Sheng
Xue${}^{\ref{ICRANet},\ref{INAF},\ref{ICRA}}$, and Zi-Gao
Dai${}^{\ref{USTC},\ref{NU}}$

ICRANet, Piazza della Repubblica 10, 65122 Pescara, Italy

INAF – Osservatorio Astronomico d’Abruzzo, Via M. Maggini snc, I-64100,
Teramo, Italy

Dip. di Fisica and ICRA, Sapienza Universita di Roma, Piazzale Aldo Moro 5,
I-00185 Rome, Italy

Department of Astronomy, University of Science and Technology of China, Hefei
230026, China

School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China

Measuring the polarization from GRB emission play a crucial role in
understanding the physics of relativistic jets–in particular–providing a
unique probe of the magnetic fields in the collimated jets[15, 16, 17, 18, 19,
20]. After more than two decades of polarimetric observations, GRB
polarimetric measurements have accumulated in dozens of cases[4, 6, 7], and
have been observed in both early and late emission phases, spanning from a few
seconds to a few months (covering six to seven orders of magnitude in time)
after the burst trigger, making it possible to study GRB polarization
properties in a global manner.

Despite decades of multi-wavelength observations and considerable theoretical
efforts, there is still no a consistent scenario for explaining the globally
observed timing properties of GRB polarization to date. A better understanding
of the physics of GRB polarization properties is required to have a more
complete picture of the long-timescale evolutionary connections from burst to
burst. Therefore, we searched the literature for published and archival
observations that would allow us to provide a complete picture of polarization
evolution. With this dedicated search, the complete GRB polarization sample,
which consists of 73 bursts (39 bursts with known redshifts) and covers a
broad wavelength range (from radio to $\gamma$-ray) of polarization
measurements, is provided (see Table Extended Data).

In order to investigate the temporal properties of GRB polarization, a
corresponding epoch of the polarization measurement for each burst is
required. In order to explore the intrinsic properties of GRB polarization at
the cosmological distance, a time-dilation correction factor 1/(1+$z$) is
applied. The degree of polarization ($\pi_{\rm obs}$) as a function of time
after the burst plotted in the cosmological rest-frame is shown in Figure 1
for the full polarization sample (with known redshift, see Table Extended
Data). By modeling the data, we discover that the globally time-sampled
evolutionary properties of GRB polarization degrees $\pi_{\rm obs}$
(percentages, ranging from 0% to 100%) consist of four segments (Figure 1):
(I) an initial hump early on (within the first few seconds); (II) a power-law
decay over a long-term timescale period (from $\sim$101 to $\sim$104 s) later
on, which takes the form of $\pi_{\rm obs}\propto t^{-0.50\pm 0.02}$ with a
decay index around at -0.50 (see Methods); (III) a rebrightening hump occurs
at a later time (from $\sim$104 to $\sim$105 s); and (IV) finally, a new
power-law decay (from $\sim$ 105 to $\sim$ 107 s), which takes the form of
$\pi_{\rm obs}\propto t^{-0.21\pm 0.08}$. These observational properties are
naturally in agreement with theoretical previsions based on the scenario of
relativistic geometric effects and outflow dynamics (Figure 2). In the
standard GRB fireball “internal-external” shock model[21, 22, 23, 24, 25, 26],
following the collimated relativistic outflows (jets) launched from the
central engine: the early and short-lived prompt $\gamma$-ray emissions
generated by an internal shock, where two consecutive shells interact and
collide, occur at a close distance from the progenitor; while the late and
long-lasting, X-ray, optical, and radio wavelengths afterglow emissions
originated from an external shock, where outflows interact with ambient
medium, occur at large distances from progenitors. Within this scenario,
relativistic magnetized jets are also expected to be generated by the central
engines and launched along with the relativistic collimated jets, either by
hyper-accreting black holes or by rapidly spinning magnetars[1, 15, 2, 16, 3,
4, 17, 5, 18, 6, 19, 20, 7, 8], very likely distributed as globally random,
but locally coherent magnetic fields. A local coherent magnetic domain of
typical area dimension $S_{\rm patch}$ is named as a “magnetic patch”[11, 12,
13, 14], where magnetic field $H_{\rm patch}$ orientation is in order. It
emits polarized photon about $\sim\Pi_{\rm max}$ (Eq.4). The orientations of
magnetic fields and photon polarization vary from one magnetic patch to
another. These magnetic patches and fields randomly distribute in a outflow
due to magnetized fluid dynamics.

In early emission (within the first few seconds), starting at an bulk Lorentz
factor $\Gamma\gg 1$, due to relativistic beaming, only the emission inside
the 1/$\Gamma$ cone can be contributed to the observed flux[9], so as to the
observable area $S_{\rm obs}$ is a small portion of outflow. When only one
magnetic patch in outflow is seen ($S_{\rm obs}\lesssim S_{\rm patch}$), the
observed polarization $\pi_{\rm obs}$ is about $\Pi_{\rm max}$, associating to
magnetic patch field $H_{\rm patch}$ and orientation. Afterwards, $\Gamma$
decreases due to energy dissipation, and $S_{\rm obs}$ increases. More and
more magnetic patches in outflow are seen. The observed net polarization from
all magnetic patches seen should be smaller than $\Pi_{\rm max}$, attributed
to globally and randomly disorder distributions of magnetic patch fields and
orientation in a outflow. The more magnetic patches are seen and contribute to
the observed flux, the smaller measured net polarization is, owing to an
increase of randomness degree of magnetic patches distributing over the
observed area $S_{\rm obs}$. Considering that the local synchrotron radiation
and magnetized fluid dynamics timescales are much smaller than the period of
observed polarization data collected and analyzed, we propose that outflow
relativistic and geometric effects play an essential role in explaining
observed polarization properties in a long-time duration. The observed
polarization $\pi_{\rm obs}$ should correspond to the net polarization
averaged over magnetic patches distribution and their fluid and radiation
dynamics effects. As a consequence, the $\pi_{\rm obs}$ long-time evolution
mainly follows a scaling law $\pi_{\rm obs}\propto 1/S_{\rm obs}$ ($S_{\rm
patch}\lesssim S_{\rm obs}\lesssim S_{\rm jet}$ and from $\sim 10^{1}$ to
$\sim 10^{4}$ s) until a jet-break time (from $\sim 10^{4}$ to $\sim 10^{5}$
s) is reached when $S_{\rm obs}\approx S_{\rm jet}$ (see Methods). The scaling
law seems in agreement with data. At the moment of jet-break time, an Earth
based observer naturally starts to see non-axial symmetric or irregular
boundaries of outflows. Such an irregular or fractal geometry of outflow
boundary decrease the randomness of spatial distributions of magnetic patches,
and should in principle result in net polarization increases. Therefore the
scenario a priori previews that at jet-break time the observed polarization
has an increment correlating known afterglow jet break phenomena, and change
its decay scaling law from $\pi_{\rm obs}\propto 1/S_{\rm obs}$ to a shallower
one, since total net polarization of entire jet is seen and the observable
area $S_{\rm obs}>S_{\rm jet}$ becomes irrelevant in net polarization time
evolution. In four GRBs polarization data (see Table Extended Data), we indeed
find around $10^{4}$ to $10^{5}$ seconds bit-increments appear correlating
light-curve jet breaks, and a shallower decaying law follows after $10^{5}$
seconds, the typical epochs and temporal slopes for jet breaks that are
observed for these bursts (see Methods). We also constrain shallower decaying
law in polarization data after the jet-break time ($S_{\rm obs}>S_{\rm jet}$
and $\gtrsim 10^{5}$ seconds), see $\Gamma\gtrsim 1$ in Figure 2, the observed
net polarization decaying law clearly deviates from the scaling power law
$\pi_{\rm obs}\propto 1/S_{\rm obs}$ and becomes shallower one. It may remain
as a constant soon after the jet-break time, then decreases slowly, due to the
time variations of magnetic patch fields and sizes, as well as magnetized
outflow itself after jet break.

There are two remarks in order. First, in the first few seconds (Segment I,
see Figure 1) for the early polarization evolution, the observed area should
be smaller than a magnetic patch area $S_{\rm obs}<S_{\rm patch}$ because of a
giant $\Gamma$, and the observed polarization $\pi_{\rm obs}\sim\Pi_{\rm
max}(H_{\rm patch})$ should follow the variations of magnetic patch field
$H_{\rm patch}$ and electron distribution. We speculate $\pi_{\rm
obs}\sim\Pi_{\rm max}$ slightly increases, reaching its maximal value when
$S_{\rm obs}\approx S_{\rm patch}$, for the possible reasons that $\Gamma$
decreases, $S_{\rm obs}$ increases and more electrons (involved in) emitting
radiation are observed in time. Unfortunately, current observations do not
contain high-resolution polarization data for this earliest period, and we
have not been able to determine the polarization evolution in time. Second, in
the period after the jet break $>10^{5}$ sec and $S_{\rm obs}>S_{\rm jet}$
(Segment IV, see Figure 2) for the later polarization evolution, we are lack
of better-established polarization data. Therefore the power-law decay index
during Segment IV is poorly constrained as determined by the current sample.
To gain a deeper insight into the polarization time evolution in early
(Segment I) and later (Segment IV) periods, we look forward to high-
sensitivity polarimetric observations from upcoming instruments[10] to
establish well-sampled polarization light curves in these two periods. It will
be subjects for future studies.

In addition to the time-sampled polarization lightcurve from observed data, we
also discuss other observational evidences supporting the scaling law in
Segment II and bit-increment in Segment III. In Segment II ($S_{\rm
patch}\lesssim S_{\rm obs}<S_{\rm jet}$), the observed net polarization mainly
follows a scaling law $\pi_{\rm obs}\propto S^{-1}_{\rm obs}$ and $S_{\rm
jet}/S_{\rm obs}=\theta^{2}_{j}/\theta^{2}_{e}$ (see Methods). Due to
relativistic beaming, only the emission inside the 1/$\Gamma$ cone contributes
to the observed flux, one has $\theta_{e}\simeq\Gamma^{-1}$. One therefore has
$\pi_{\rm obs}\propto\Gamma^{2}$. A possible test can be made by investigating
the relation between the polarization data $\pi_{\rm obs}$ and Lorentz factor
$\Gamma(\pi)$ and the latter is measured in the same epoch of polarization
data (see Methods). Since $\Gamma(\pi)$ cannot be precisely estimated for
certain bursts, one can use $\Gamma_{0}$ as a proxy. An empirical relation[11,
12] ($\Gamma_{0}\simeq 249L^{0.30}_{\gamma,\rm iso,52}$) is used to estimate
the values of $\Gamma_{0}$, as long as their energy flux $F_{\gamma}$ and
redshift are known (see Methods). Extended Data Figure 6 shows $\pi_{\rm obs}$
as functions of $\Gamma_{0}$. We find the data points indeed cluster around
the scaling law $\pi_{\rm obs}\propto\Gamma^{2}$ (see the solid lines of
Figure 6), where various external shock models are considered. Moreover, in
the epoch of jet-break (Segment III, $S_{\rm obs}\approx S_{\rm jet}$), we
find five bursts (GRB 990510, GRB 010222, GRB 020405, GRB 030328, and GRB
080928) that exhibit the bit-increment polarization data. It is further shown
that in these bursts the occurrences of polarization bit-increments are in
agreement with the jet-break analysis using their afterglow data, where three
(or possibly four) bursts may observe a jet break (see Methods).

The intrinsic mechanisms and effects that account for GRB polarization are
complicated. The observed net polarization should come from the different
microscopic radiation mechanisms and macroscopic effects of jet geometries,
viewing angles, magnetic field configurations, and degree of magnetized jet,
etc. Our data and theoretical analysis suggest that the relativistic
geometrical effect is the dominant one across various emission regions,
naturally explaining the observed power-law evolution of GRB polarization in a
long time scale ${\mathcal{O}}(10^{6})$ seconds. This single power-law over
long time scales across multiple emission regions challenges the current
mainstream of polarization models that assume different polarization
properties in distinct emission regions[13, 14, 15, 19], as described by these
models, the temporal evolution of GRB polarization may display a “three-stage
jump” pattern (see Figure 3) from early prompt emission to later afterglow
“reverse-forward” shock emission regions. These results suggest that other
possible mechanisms and effects (e.g., the internal/external shock model[21],
the dissipative photosphere model[16, 17], the ICMART model[19]) give
contributions to GRB polarization in small time scales, that result in
observational data scattering up and down around the power-law evolution.

We find four dramatically different segments spanning from a few seconds to a
few months, covering six to seven orders of magnitude in time. They
characterize a complete picture of the global timing properties of GRB
polarization for the first time. Our results show that the time evolution of
GRB polarization follows the power-law decay with an index $\alpha\simeq-0.50$
until the jet-break time. It may provide an empirical estimate of $\pi_{\rm
obs}$ at a given time. The estimated $\pi_{\rm obs}$ value can be used for
further constraints on physical models and their relevant parameters. For
instance, the scaling law $\pi_{\rm obs}\propto S^{-1}_{\rm
obs}\propto\Gamma^{2}$ gives an insight into the traditional GRB afterglow
models, which describe the environment surrounding GRBs (see Figure 7 and
Methods). Our result supports the afterglow scenarios of constant energy and
density stratification (wind) environment[24, 26]. It may disfavor the case of
a large-scale ordered magnetic field advocated from the central engine with an
on-axis observation, because an rising scaling law of GRB polarization is
predicted by the case; but favor the case of a large-scale ordered magnetic
field with an off-axis observation. Moreover, analogously to the Amati
relation[18], the GRB polairzation power law evolution in time shows though
observed GRB phenomena are individually different (e.g., total energy,
lightcurve, and spectrum), they have some common features in central engine
and relativistic magnetized jet evolution at large scales in time and space.

References

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————————————————————–

## Author contributions

LL led the data analysis, and contributed to part of the physical
explanations. SSX proposed the theoretical model for explanations and
previsions of the observational data in four-time segments. LL and SSX wrote
the article. ZGD participated in discussions. All authors have reviewed,
discussed, and commented on the present results and on the manuscript.

## Acknowledgements

LL thank Yu Wang, Felix Ryde, Bing Zhang, Xue-Feng Wu, Jin-Jun Geng, Soroush
Shakeri, Shuang-Nan Zhang, Remo Ruffini, and the ICRANet members for many
helpful discussions on GRB physics and phenomena.

## Author information

Correspondence and requests for materials should be addressed to LL
(liang.li@icranet.org), SSX (xue@icra.it), and ZGD (dzg@nju.edu.cn).

## Competing Interests

The authors declare that they have no competing financial interests.

Figure 1: The degree of polarization ($\pi_{\rm obs}$, percentages ranging
from 0% to 100%) for the full sample with known redshift (see Table Extended
Data) is plotted as a function of time after the burst at the cosmological
distance. Data points with solid colors represent the polarization data in
different energy band. Green: $\gamma$-rays; yellow: hard X-rays; cyan:
optical; orange: radio; blue-magenta: the polarization data placed at the
upper limits on the measurements; red-green: the polarization data placed at
the lower limits on the measurements. Segment I: an initially slight hump
caused by the onset effect ($S_{\rm obs}\lesssim S_{\rm patch}$) within the
first few seconds. Segment II: a power-law decay ($S_{\rm patch}\lesssim
S_{\rm obs}\lesssim S_{\rm jet}$) over a long timescale period (from $\sim$101
to $\sim$104 s), which takes the form of $\pi_{\rm obs}\propto t^{-0.50\pm
0.02}$. Segment III: a late-rebrightening hump caused by the “jet break”
boundary effect ($S_{\rm obs}\approx S_{\rm jet}$) occurs at a later time
(from $\sim$104 to $\sim$105 s). Segment IV: a flatting power-law decay
($S_{\rm obs}>S_{\rm jet}$) after the “jet break” (from $\sim$ 105 to $\sim$
107 s), which takes the form of $\pi_{\rm obs}\propto t^{-0.21\pm 0.08}$. The
solid line is the best fits using the power-law model and with $2\sigma$ (95%
confidence interval) error shadow region.

Figure 2: A cartoon picture for depicting the timing evolution of the GRB
polarization degree based on relativistic geometrical effects. For simplicity,
we assume that the jet opening angle $\theta_{j}$ remains unchanged throughout
jet evolution, corresponding to an area projected onto the jet plane as
$S_{\rm jet}$ (see white circle symbol in the right panel). Due to
relativistic beaming, only the emission inside the 1/$\Gamma$ cone contributes
to the observed flux, corresponding to an observed area projected onto the jet
plane as $S_{\rm obs}$ (see grey circle symbol in the right panel). A
“magnetic patch” corresponds to an area projected onto the jet plane as
$S_{\rm patch}$ (see orange circle symbol in the right panel). (a) The onset
effect (Segment I) occurs at an early emission (within the first few seconds),
corresponding to $S_{\rm obs}\lesssim S_{\rm patch}$ with a huge $\Gamma\gg 1$
(e.g., $\Gamma$=300). (b) The main power-law decay phase (Segment II) occur
over a long timescale period (from $\sim$ 101 to $\sim$ 104 s), corresponding
to $S_{\rm patch}\lesssim S_{\rm obs}\lesssim S_{\rm jet}$ with a moderate
$\Gamma$ (e.g., $\Gamma$=50). (c) The “jet break” boundary effect (Segment
III) occurs at a later emission when the “jet break” is reached (from $\sim$
104 to $\sim$ 105 s), corresponding to $S_{\rm obs}\approx S_{\rm jet}$ with a
relatively low $\Gamma$ (e.g., $\Gamma$=10). (d) Finally the new power-law
decay phase (Segment IV) occurs after the “jet break” (from $\sim$ 105 to
$\sim$ 107 s), corresponding to $S_{\rm obs}>S_{\rm jet}$ with a small
$\Gamma$ (e.g., $\Gamma$=1).

Figure 3: The contrast between the observed polarization degree and the
theoretical predictions by different radiation mechanisms[19]. In the prompt
emission (PE) phase, we consider the synchrotron radiation from different
magnetic field structures: (a) a random field normal to the radial direction
$B_{\perp}$ ($\pi\sim$ 25%-45%); (b) an ordered field radial $B_{\parallel}$
($\pi\sim$ 10%-56%); (c) a toroidal $B_{\rm tor}$ ($\pi\sim$ 50%-60%), as well
as (d) photosphere emission model ($\pi\sim$ 0%-15%); (e) Compton drag
model[20] ($\pi\sim$ 60%-100%). In the afterglow emission phase, we consider
the highly polarized reverse-shock (RS,[21, 22]) emission ($\pi\sim$ 10%-20%),
and forward-shock (FS,[11]) emission ($\pi\sim$ 1%-10%).

Methods

###  0.1 The GRB polarimetric sample

We conducted an extensive search of the literature for published and archival
polarimetric observations, and attempted to include all the bursts that had
polarization measurements to date. With this dedicated search, the complete
GRB polarization sample, which consists of 73 bursts (39 bursts with known
redshifts, see Figure 1) and covers a broad wavelength range (from radio to
$\gamma$-ray emission) of polarization measurements, is provided (see Table
Extended Data).

These observed polarization data are categorized into three types of data
sets: i) Time-integrated polarization data, which represent average
polarimetric properties, for the entire emission period are treated as a
single time event, resembling the approach used in GRB spectral analyses; ii)
Time-resolved polarization data, in which the whole emission period is divided
into multiple timing events, and polarimetric measurements, are therefore
performed on each event individually; iii) Polarimetric measurements are
placed on an upper/lower limit. We note that (1) For a given burst, time-
integrated polarization data can in principle include as many numbers of
detected photons as possible, thereby reducing the uncertainties and avoiding
unreliable fluctuations introduced by the time-resolved data; (2) The
polarization data for a good fraction of the bursts have placed an upper/lower
limit on the measurements, and therefore, this data set may not be as tightly
constrained (has a loose constraint) to the model as on the other data sets.

###  0.2 Modeling the multi-wavelength time-sampled polarization data

In order to explore the timing properties of GRB polarization, a corresponding
epoch of the polarization measurement for each burst is required. To
investigate the intrinsic properties of GRB polarization, we select those
bursts whose redshift measurements are also reported. Extended Data Figure 1
shows that the time-integrated polarization degree $\pi_{\rm obs}$
(percentage, ranges from 0% to 100%) as a function of its corresponding epoch
in the cosmological rest-frame, plotting in logarithmic-logarithmic space. As
an initial result, we discovered that $\pi_{\rm obs}$ tends to be smaller over
time. Figure 1 shows the polarization degree $\pi_{\rm obs}$ as a function of
time $t/(1+z)$ after the burst at the cosmological distance for the full
sample, where $t$ =($T_{\rm stop}$+$T_{\rm start}$)/2), and $T_{\rm start}$
and $T_{\rm stop}$ are the start time and stop time of the related epoch
listed in the Table Extended Data, and errors $\sigma_{\rm t}$ can be
calculated by $\sigma_{\rm t}$=($T_{\rm stop}$-$T_{\rm start}$)/2, and $z$ is
reshift. Using this full sample, we find that the polarization lightcurve can
be best fitted by the power-law model ($\pi=At^{-\alpha}$), the best-fit
function gives

$\displaystyle\rm log_{10}(\pi_{\rm obs})/(\%)=(4.40\pm 0.36)+(-0.498\pm
0.020)\times log_{10}[{\it t}/(1+z)]/(s),$ (1)

with the number of data points $N$=115, the Spearman’s rank correlation
coefficient of $R$=-0.86, and a chance probability $p<10^{-9}$. These results
indicate that the time-sampled polarization evolution over long-timescale
periods decays as a power-law111It’s worth noting that the “$t_{0}$” effect
might have a significant impact on how early polarization evolves. This
happens often in multi-pulse bursts in which adjacent pulses are well-
separated by a long quiescent period. The zero time on polarization
measurements may need to be corrected to be physical due to the GRB trigger
time being no longer special in such cases. at cosmological distances, which
takes the form of $\pi_{\rm obs}\propto[t/(1+z)]^{-0.50\pm 0.02}$.

###  0.3 Correlation coefficient analysis from the Markov chain Monte Carlo
algorithm

Our analysis results were also double-checked using a correlation coefficient
analysis from the Markov chain Monte Carlo algorithm. The correlation
coefficient can be used to assess the reliability of the correlation between
the cosmological rest-frame time $t/(1+z)$ and the GRB observational
polarization ($\pi$) for our samples. By implementing the Python package
PyMC3, ref.[23], we use a normal-LKJ correlation prior distribution and the
Markov chain Monte Carlo (MCMC) algorithm to obtain the covariance matrix of
the multivariate normal distribution, and the correlation coefficient between
the parameters is obtained as a result[24] by iterating 105 times and burning
the first 104 times of the MCMC samples. Our results are summarized in Table
0.3, including the expected values ($\mu_{1}$) and standard deviation
($\sigma_{1}$) of the normal distribution of the analyzed parameters, and
their correlation coefficients, as well as the related Highest Density
Interval (HDI) of the posterior distributions, ranging from 3% to 97%. In
Extended Data Figure 4, we show an MCMC iteration for the mean values and
standard deviation of the $t_{\rm z}$ and $\pi$, and their correlation
coefficient, including the value of 105 iterations (right) and their
distribution (left).

Results of correlation coefficient of Markov chain Monte Carlo algorithm
Sample $\mu_{1}$ $\sigma_{1}$ $\mu_{2}$ $\sigma_{2}$ Correlation Coefficient
hdi interval ($t_{\rm z}$) ($t_{\rm z}$) ($\pi$) ($\pi$) [3% to 97%] Time-
integrated GRB polarization data 2.112$\pm$0.253 0.941$\pm$0.049
1.044$\pm$0.145 0.537$\pm$0.042 -0.950$\pm$0.028 [-0.985 to -0.902]

Figure 4: MCMC iteration for the mean values and standard deviation of the
$t_{\rm z}$ and $\pi$, and their correlation coefficient for our sample,
including the value of 105 iterations (right) and their distribution (left).

###  0.4 “Magnetic patch” polarization properties and the $\Pi_{\rm max}$
point

After a relativistic jet is emitted, the outflow is likely to carry a fraction
of the magnetic field energy in the form of a random magnetic field, known as
relativistic magnetized jets. In this random magnetic field, the degree of
order of the magnetic field is related to the observed area. The smaller the
area is, the higher the degree of order it contains. Naturally, there is a
critical area, $S_{\rm patch}$, know as “magnetic patch[11, 12, 13, 14]”,
where the magnetic field can be considered to be completely ordered. As shown
a yellow arrowed spot in sketch Figure 2, we indicate the “magnetic patch”,
where magnetic field orientations are in order, producing polarized
photons222Here, instead of specifying how the magnetic domain is form, we
refer such patch to the region where polarized photons are produced in
general.. The magnetic patch area is $S_{\rm patch}$ and its contribution to
the observed polarization is about $\Pi_{\rm max}$, where $\Pi_{\rm max}$ is
the possibly maximal polarization.

In astrophysical observations, the photon polarization is measured by using
the percentages (ranging from 0% to 100% percent) of polarized and non-
polarized photons to represent the degree of GRB polarization, namely the
measured polarization degree is given by

$P_{\rm obs}=\pi_{\rm obs}=\frac{I^{\rm p}}{I^{\rm p}+I^{\rm n}},\\\ $ (2)

where $I^{\rm n}$ and $I^{\rm p}$ respectively refer to the intensities of the
natural and fully polarized lights from the source. There are two components:
fully polarized photons (e.g., elliptical, circular, and linear polarized
light) and non-polarization photons (natural light). The former is completely
polarized with $\pi=1$, and the latter is completely non-polarized with
$\pi=0$. We illustrate in Figure 1 the observed $\pi_{\rm obs}$ data and their
uncertainties.

The measured $\pi_{\rm obs}$ receives all contributions from the area $S_{\rm
obs}$ on the jet plane visible to the observer. As discussed soon, the initial
$S_{\rm obs}$ is smaller than a magnetic patch area $S_{\rm patch}$. First we
introduce the polarization degree of photons emitted from a “magnetic patch”,
correspondingly to the observation one (Eq.2),

$\pi_{\rm patch}=\frac{I^{\rm p}_{\rm patch}}{I_{\rm patch}},\\\ $ (3)

where $I_{\rm patch}=I^{\rm p}_{\rm patch}+I^{\rm n}_{\rm patch}$, and $I^{\rm
p}_{\rm patch}$ and $I^{\rm n}_{\rm patch}$ correspond to the intensities of
natural and fully polarized lights from a “magnetic patch”. It can be seen
from Figure 1 that the time-integrated polarization lightcurve at $S_{\rm
obs}\approx S_{\rm patch}$ is a peak, from which we define as $\Pi_{\rm max}$,

$\Pi_{\rm max}=\pi_{\rm obs}\mid_{S_{\rm patch}\approx S_{\rm
obs}}\approx\pi_{\rm patch}.$ (4)

Based on observations (within the uncertainties), the value of $\Pi_{\rm max}$
could be estimated to be between 70 and 100 percent (see Figure 1). Given a
GRB source and its surrounding, we assume that magnetic patches have the
similar structure of locally coherent domain, thus the $\Pi_{\rm max}$ of
different patches are considered to be approximately the same. The reasons are
given below.

When the observed area $S_{\rm obs}$ becomes larger than many magnetic
patches, the observed polarization degree $\pi_{\rm obs}$ contributed by
photons coming from different magnetic patches. Moreover, all magnetic patches
are randomly distributed in entire observed area $S_{\rm obs}$. The net
polarization degree $\pi_{\rm obs}$ is expected to be smaller than $\Pi_{\rm
max}$ and monochromatically decrease in time, as the number of observed
magnetic patches increases[11, 12, 13, 14]. Thus the $\Pi_{\rm max}$ (Eq.4)
represents a maximal polarization degree in observation. While the net
polarization degree $P_{\rm obs}$ contributed by polarised photons from all
visible patches will be defined in due course.

###  0.5 Magnetic field orderliness and relativistic geometrical effects in
GRB polarization

We follow the discussions in jet-break phenomenon, the observed area $S_{\rm
obs}$ (see grey circular area in Figure 2) varies in time depending on the
bulk Lorentz factor $\Gamma$ of the outflow. The evolution of such
relativistic effects on observed polarization may consist of four possible
segments depending on whether the observed area ($S_{\rm obs}$) is greater (or
less) than the magnetic patch area ($S_{\rm patch}$) and the area ($S_{\rm
jet}$) at the break time.

When the Lorentz factor $\Gamma\gg 1$ (early emission phase, such as when the
outflow becomes optically transparent), due to a huge $\Gamma$, the observed
cone ($\sim 1/\Gamma$) is very small. Only one or a few magnetic patches are
seen, depending on magnetic patch size. If only one magnetic patch is seen,
the observed area $S_{\rm obs}$ is smaller than and increasing to the magnetic
patch area $S_{\rm patch}$ (Segment I, see $\Gamma\sim 300$ in Figure 2), we
therefore obtain $\pi_{\rm obs}\approx\Pi_{\rm max}$ for $S_{\rm obs}\lesssim
S_{\rm patch}$ from (Eqs.5-8). Indeed, as shown by three data points in the
Segment I, the observed photon polarization $\pi_{\rm obs}$ very slightly
increases and reach its maximal value. However, detail analysis and data need
to consider also the time evolution of random magnetic fields in their sizes
and strengths for slightly increasing $\Pi_{\rm max}$ value in the Segment I.
This is a subject of future study. Nevertheless, using the data at initial
time (around 0-10 s), we can infer the maximal value of polarization degree
$\Pi_{\rm max}S_{\rm patch}/S_{\rm obs}$ for $S_{\rm patch}\approx S_{\rm
obs}$ in $\Gamma\sim 300$ in Figure 2.

When the the observed area $S_{\rm obs}$ is larger than $S_{\rm patch}$
(Segment II, see $\Gamma\sim 50$ in Figure 2), it covers more than one
magnetic patch. The orientations of magnetic fields and photon polarizations
vary from one patch to another, see right illustration in Figure 2. Suppose
that the magnetic patch $S_{\rm patch}$ emits the polarized radiation of total
intensity $I_{\rm patch}$ and polarized intensity ${\bf I}^{\rm p}_{\rm
patch}$. Here the bold ${\bf I}^{\rm p}_{\rm patch}$ represents the orientated
polarization intensity of the value $I^{\rm p}_{\rm patch}=|{\bf I}^{\rm
p}_{\rm patch}|$. The intensities $I_{\rm patch}$ and $I^{\rm p}_{\rm patch}$
are assumed to be approximately equal for all magnetic patches. The observed
area $S_{\rm obs}\approx\sum_{S_{\rm patch}}S_{\rm patch}$ is smaller than
entire emission region. An Earth based observer see the total intensity and
net polarization intensity of radiation emitted from all magnetic patches
within the observed area $S_{\rm obs}$. Their values can be approximately
expressed as

$\displaystyle I^{\rm p}$ $\displaystyle=$ $\displaystyle\sum_{S_{\rm
patch}\in S_{\rm obs}}{\bf I}^{\rm p}_{\rm patch}S_{\rm patch}\approx I^{\rm
p}_{\rm patch}S_{\rm patch}^{\rm eff},$ (5) $\displaystyle I$
$\displaystyle\approx$ $\displaystyle\sum_{S_{\rm patch}\in S_{\rm obs}}I_{\rm
patch}S_{\rm patch}\approx I_{\rm patch}S_{\rm obs}.$ (6)

Here we introduce an effective patch area $S_{\rm patch}^{\rm eff}$ to
represent the net polarization observed from different patches.

$\displaystyle S_{\rm patch}^{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{S_{\rm patch}\in S_{\rm obs}}(-)^{p}S_{\rm patch}<S_{\rm
obs},$ (7)

where the notation $(-)^{p}$ symbolically indicates the randomness of
polarization orientations from patches, indicating the destructive (rather
than constructive) phenomenon of superposition of light polarizations from
different patches. Due to the relativistic beaming effect, the observed area
$S_{\rm obs}\propto\Gamma^{-2}$. The observational polarization degree
$\pi_{\rm obs}$ (Eq.2) is given by

$\begin{split}\pi_{\rm obs}=\frac{I^{\rm p}}{I}\approx\Pi_{\rm
max}(\frac{S_{\rm patch}^{\rm eff}}{S_{\rm
obs}})\propto\Gamma^{2},\end{split}$ (8)

where $\Pi_{\rm max}=I^{\rm p}_{\rm patch}/I_{\rm patch}$ is the patch maximal
polarization degree (Eq.4). We will discuss that in long time scale when
$S_{\rm obs}\gg S_{\rm patch}$, the factor $\Pi_{\rm max}S_{\rm patch}^{\rm
eff}$ slowly varies compared with $S_{\rm obs}$ variation, leading to
$\pi_{\rm obs}\propto S^{-1}_{\rm obs}\propto\Gamma^{2}$. It is one of the
basic formulae of our theoretical model to study the observed polarization
degree of GRBs.

Figure 5: A cartoon picture for “magnetic patches” on the jet plane
represented by many small circles with random red arrows; the observed area
$S_{\rm obs}\propto 1/\Gamma^{2}$ increase in time represented by four hollow
circles corresponding to the hollow circles in Fig.2; the irregular jet
boundary represented by a waved circle.

Let us consider a GRB jet model[25, 26] to illustrate the formula (Eq.8) and
its time evolution. According to the jet geometry, at the distance $R_{\rm
GRB}\approx ct_{\rm GRB}$ from the center engine and a fixed opening angle
$\theta_{j}$, the jet plane area $S_{\rm jet}\approx\hat{\pi}(\theta_{j}R_{\rm
GRB}/2)^{2}$ increasing with time. The magnetic patch area on the jet plane is
$S_{\rm patch}\approx\hat{\pi}(\theta_{\rm patch}R_{\rm GRB}/2)^{2}$. We
assume that (i) the areas of magnetic patches are approximately equal and (ii)
following the dynamics of jet evolution, the magnetic patches are at anchor in
the jet flow. Namely, the patch opening angle $\theta_{\rm patch}$ is fixed
and the total patch number on the jet plane is approximately constant in time,

$\displaystyle N_{\rm patch}$ $\displaystyle\equiv$ $\displaystyle\frac{S_{\rm
jet}}{S_{\rm patch}}=\frac{\theta^{2}_{j}}{\theta^{2}_{\rm patch}}\cong{\rm
const.}\quad\theta_{j}>\theta_{\rm patch},$ (9)

as well as the patch distribution on the jet plane are assumed to weakly
depend on the jet evolution. The observed polarization $\pi_{\rm
obs}\approx\Pi_{\rm max}$ is maximal, when $S_{\rm obs}\approx S_{\rm patch}$.
When $S_{\rm obs}>S_{\rm patch}$, due to the fact that polarization
orientations are randomly different among magnetic patches, the net
polarization observed is smaller than $\Pi_{\rm max}$. Suppose that the
observer is along the line-of-sight of the jet axis, i.e., approximately axial
symmetric, and patch magnetic field orientations randomly distribute in the
observed area $S_{\rm obs}$ (see Fig. 2 and Fig. 5). The polarizations of
radiation from all magnetic patches within the observed area $S_{\rm obs}$
should almost, but not exactly, cancel each others. The reason is that the
number of observed magnetic patches is finite and their orientations cannot be
completely random. To represent the observer relevant net polarization of
radiation from different magnetic patches, we introduce an effective patch
area $S_{\rm patch}^{\rm eff}$ (Eq.7). Its value depend on the magnetic patch
numbers and their random distribution inside $S_{\rm obs}$. In general, the
larger $S_{\rm obs}$ is, the more $S_{\rm patch}$ are seen, the higher degree
of their random distribution is, the smaller observed net polarization should
be. Therefore, as increasing $S_{\rm obs}$, more and more magnetic patches are
observed, the randomness of their magnetic field orientations increases, the
effective patch area $S_{\rm patch}^{\rm eff}$ decreases. As a result, the net
observed polarization of photons from different magnetic patches decreases.
However, as $S_{\rm obs}$ becomes much larger than $S_{\rm patch}$, the
polarization contributions from the patches $S_{\rm patch}$ distributing in
the $S_{\rm obs}$ outer area tend to be destructive each other, due to the
axial symmetry of their random distribution, as illustrated in sketch Figure
5. Namely the observed net polarization contributions mainly come from the
patches $S_{\rm patch}$ distributing in the $S_{\rm obs}$ inner area. We
formally express the $S_{\rm patch}^{\rm eff}$ (Eq.7) as

$\displaystyle S_{\rm patch}^{\rm eff}\approx\sum_{\rm inner}(-)^{p}S_{\rm
patch}+\sum_{\rm outer}(-)^{p}S_{\rm patch}\Rightarrow\sum_{\rm
inner}(-)^{p}S_{\rm patch}.$ (10)

Its approximate constancy in later time is due to the patch distribution on
the jet plane is weakly time dependent, the aforementioned properties of
magnetic patches on the jet plane. The discussed time evolution of $S_{\rm
patch}^{\rm eff}$ is qualitatively sketched in Fig. 5.

These assumptions are on the basis that the $S_{\rm patch}$ time-varying scale
$\tau_{\rm patch}\approx S^{1/2}_{\rm patch}/c$ should be smaller than
$S^{1/2}_{\rm obs}/c$. In addition, the maximal polarization $\Pi_{\rm max}$
time-varying scale is smaller than $\tau_{\rm patch}$. Therefore, due to the
local patch magnetic field dynamics and orientation randomness, we expect in
the net polarization the factor $\Pi_{\rm max}S_{\rm patch}^{\rm eff}$ has
short-time $\tau_{\rm patch}$ fluctuating features and its mean value weakly
depends on time at a much long time scale ($\sim$105 seconds). Thus we
parametrize $\Pi_{\rm max}S_{\rm patch}^{\rm eff}\propto t^{-\alpha}$
($\alpha\ll 1)$, which varies much slowly than $S_{\rm obs}$ in time. As a
result, in a long time period of data collection and analysis (Figure 1), the
time dependence of observed net polarization should be dominantly determined
by the time dependence of the observed area $S_{\rm obs}$, whose evolution in
time is determined by relativistic effect and kinetic motion of emitting area.
All above discussions can be generalized to the case that the observer line
sign is not exactly along the axial symmetry line of emitting region.

The observed area $S_{\rm obs}$ on the jet plan area $S_{\rm jet}$ evolves in
time. In the period of $S_{\rm patch}\lesssim S_{\rm obs}\lesssim S_{\rm jet}$
before the jet-break time scale ($\sim 10^{5}$ seconds), the observed net
polarization mainly follow scaling law $\pi_{\rm obs}\propto 1/S_{\rm obs}$
(Eq.8) that explains in the numerical data by the following Section (see
Methods). We should point out that the local and short-time fluctuating
features of the factor $\Pi_{\rm max}S^{\rm eff}_{\rm patch}$ lead to the
observed polarization data from GRB sources scattering around the scaling law
$\pi_{\rm obs}\propto 1/S_{\rm obs}$, see Figure 2. Note that our attention is
to explain such a scaling law behavior observed in jet evolution over a long
time scale ($\sim 10^{5}$ seconds). Therefore, we ignore magnetic field and
patch fluctuation in small region and short-time scales caused by local and
fast varying dynamics, whose averages in a long time and large spatial scales
vanish. Moreover, in observed photon polarization samples, these detail
features should be also washed away by the randomness of GRB sources in their
engines, surroundings and evolution histories. Only GRB common features
attributed to the relativistic effect and jet geometry remain, resulting in
photon polarization data scattering around the power law in time scale of
$10^{5}$ sec, see Fig. 1.

When the jet-break $S_{\rm obs}\approx S_{\rm jet}$ is reached (Segment III,
see $\Gamma\sim 10$ in Figure 2), we can obtain the value $\pi_{\rm
obs}\approx\Pi_{\rm max}S^{\rm eff}_{\rm patch}/S_{\rm jet}$ from data. At
this moment, the non-axially symmetric jet boundary is seen, and $S^{\rm
eff}_{\rm patch}$ shows a small hump in Fig.5. Such an irregular jet boundary
should in principle results in net polarization increases. This explains a
slight increasing data in sketch plot in Figure 2.

After the jet-break $S_{\rm obs}>S_{\rm jet}$ (Segment IV, see $\Gamma\gtrsim
1$ in Figure 2), the observed net polarization is $\Pi_{\rm max}S^{\rm
eff}_{\rm patch}/S_{\rm jet}\propto t^{-\alpha}$, which clearly deviates from
the power law. It should remain as an approximate constant soon after jet
break, due to the total net polarization of entire jet is observed. Later on,
it is expected to slowly decrease in time, namely $\pi_{\rm obs}\propto
t^{-\alpha}$, due to the slowly time variations of maximal polarization
$\Pi_{\rm max}$, magnetic patch $S_{\rm patch}$ area and distribution, as well
as jet dynamics. These features are illustrated by sketches in Figure 2. We
show in Figure 1 $\pi_{\rm obs}\propto t^{-0.21\pm 0.08}$. We note that the
index $\alpha\approx 0.21$ is likely to be an upper bound due to only having
only one data point after $10^{5}$ seconds of data being used, and need to
pending the availability of a sufficiently comprehensive data set. More data
analysis at this point are needed, yielding a more definite result.
Nevertheless, the index $\alpha\approx 0.30$ being smaller than 0.5 shows that
the presented scenario for understanding polarization data is self consistent
and contained. In fact, four GRBs (GRB990510, GRB010222, GRB030328, and
GRB080928) exhibit the correlation between flux decaying and polarization
rising at the same jet break time, which provide evidences for the present
theoretical scenario.

###  0.6 Correlation between GRB polarization and bulk Lorentz factor

Consider a conical jet with line-of-sight aligned with the jet axis and with a
jet opening angle $\theta_{j}$, launched from the GRB central engine (see A in
Extended Data Figure 2). For simplicity, we assume that the jet opening angle
$\theta_{j}$ remains unchanged during the epoch when the jet break happens,
corresponding to an area projected onto the plane as $S_{\rm jet}$ (see white
circle symbol in the right panel of Figure 2), with a radius of $R_{\rm jet}$.
Due to relativistic beaming, only the emission inside the 1/$\Gamma$ cone
contributes to the observed flux. It projects onto the jet plane an observed
area $S_{\rm obs}$ of radius $R_{\rm obs}$ (see orange circle symbol in the
right panel of Figure 2).

A sketch plot is shown in Extended Data Figure 2, where A is the GRB central
engine, B is the observer, O is the center point of the plane, C is the
highest latitude photon on the conical jet that the relativistic emission can
be observed by the observer, and D is the boundary of the conical jet on the
plane. We set

$\left\\{\begin{array}[]{ll}\angle\rm OAC=\alpha,\\\ \angle\rm OBC=\beta,\\\
\angle\rm BCF=\theta_{e},\\\ \angle\rm BCE=\varphi=\frac{\theta_{e}}{2},\\\
\angle\rm OAD=\tau=\frac{\theta_{j}}{2},\\\ {\rm OC}=R_{\rm obs},\\\ {\rm
OD}=R_{\rm jet},\\\ {\rm OA}=R_{\rm GRB},\\\ {\rm OB}=L_{\rm dis},\\\
\end{array}\right.$ (11)

where $R_{\rm GRB}$ and $L_{\rm dis}$ are the GRB emission radius and
luminosity distance, respectively.

Due to relativistic beaming, only the emission inside the 1/$\Gamma$ cone can
be attributed to the observed flux. For instance, the relativistic emission
produced by the photon C can be observed only inside the cone $\angle$ BCF
($\theta_{e}$). One therefore has

$\theta_{e}\simeq\frac{1}{\Gamma}.$ (12)

Through geometric correlations, one has

$\left\\{\begin{array}[]{ll}\alpha=\frac{R_{\rm obs}}{R_{\rm GRB}},\\\
\beta=\frac{R_{\rm obs}}{L_{\rm dis}}.\end{array}\right.$ (13)

On the other hand, due to $L_{\rm dis}\gg R_{\rm GRB}$,

$\alpha\gg\beta,$ (14)

therefore,

$\varphi=\alpha+\beta\simeq\alpha,$ (15)

and

$\theta_{e}=2\varphi=2\alpha$ (16) $\left\\{\begin{array}[]{ll}R_{\rm
jet}=\tau\times R_{\rm GRB}\approx\tau\times(ct_{\rm GRB}),\\\ R_{\rm
obs}=\alpha\times R_{\rm GRB}\approx\alpha\times(ct_{\rm GRB})\\\
\end{array}\right.$ (17)

where $t_{\rm GRB}$ is the relativistic ejecta emission time measured in the
cosmological rest frame, relating to the observed time $t_{\rm obs}$ by
$t_{\rm GRB}\simeq 2\Gamma^{2}t_{\rm obs}$, ref.[27].

As shown in the sketch plot in Figure 2, the three characteristic areas are
expressed as

$\left\\{\begin{array}[]{ll}S_{\rm jet}=\hat{\pi}(R_{\rm
jet})^{2}=\hat{\pi}(\tau\times R_{\rm
GRB})^{2}=\hat{\pi}(\frac{\theta_{j}}{2}R_{\rm
GRB})^{2}=\hat{\pi}\frac{\theta^{2}_{j}}{4}c^{2}t^{2}_{\rm
GRB}=\hat{\pi}(\theta_{j}c\Gamma^{2}t_{\rm obs})^{2},\\\ S_{\rm
obs}=\hat{\pi}(R_{\rm obs})^{2}=\hat{\pi}(\alpha\times R_{\rm
GRB})^{2}=\hat{\pi}(\frac{\theta_{e}}{2}R_{\rm
GRB})^{2}=\hat{\pi}(\frac{R_{\rm
GRB}}{2\Gamma})^{2}=\hat{\pi}\frac{c^{2}}{4\Gamma^{2}}t^{2}_{\rm
GRB}=\hat{\pi}(c\Gamma t_{\rm obs})^{2},\\\ S_{\rm
patch}=\hat{\pi}(\frac{\theta_{\rm patch}}{2}R_{\rm
GRB})^{2}=\hat{\pi}\frac{\theta^{2}_{\rm patch}}{4}c^{2}t^{2}_{\rm
GRB}=\hat{\pi}(\theta_{\rm patch}c\Gamma^{2}t_{\rm
obs})^{2}.\end{array}\right.$ (18)

In addition to the constant ratio $N_{\rm patch}\equiv S_{\rm jet}/S_{\rm
patch}$ (Eq.9), we examine below the ratios that are relevant for our
description of timing properties of GRB polarization,

$\displaystyle N_{\rm p}$ $\displaystyle\equiv$ $\displaystyle\frac{S_{\rm
obs}}{S_{\rm patch}}\approx(\theta_{\rm patch}\Gamma)^{-2},$ (19)
$\displaystyle N_{\rm obs}$ $\displaystyle\equiv$ $\displaystyle\frac{S_{\rm
jet}}{S_{\rm
obs}}=\frac{\theta^{2}_{j}}{\theta^{2}_{e}}\approx(\theta_{j}\Gamma)^{2},$
(20)

where $N_{\rm p}$ is the number of patches on the observed area $S_{\rm obs}$
and $N_{\rm obs}$ represents a fraction of $S_{\rm obs}$ over $S_{\rm jet}$.
Both $N_{\rm p}$ and $N_{\rm obs}$ are $\Gamma$ and time dependent. In terms
of the ratio $N_{\rm p}$, the observed polarization degree $\pi_{\rm obs}$
(Eq.8) can be rewritten as

$\displaystyle\pi_{\rm obs}=\Pi_{\rm max}(\frac{S_{\rm patch}^{\rm eff}/S_{\rm
patch}}{S_{\rm obs}/S_{\rm patch}})=\Pi_{\rm max}(\frac{S_{\rm patch}^{\rm
eff}/S_{\rm patch}}{N_{\rm p}}).$ (21)

In the initial phase of the Segment II, when $S_{\rm obs}>S_{\rm patch}$ and
$S_{\rm obs}$ covers the centre part of the jet plane, the contribution
$S_{\rm patch}^{\rm eff}/S_{\rm patch}\propto\sqrt{N_{\rm p}}$ following to
the probability of random distribution. Thus, the polarization degree
$\pi_{\rm obs}\propto 1/\sqrt{N_{\rm p}}$, in consistent with the study of
Ref.[11]. Then from Eq. (19), we obtain its $\Gamma$-dependence,

$\displaystyle\pi_{\rm obs}\propto\frac{1}{\sqrt{N_{\rm p}}}\propto\Gamma,$
(22)

for large polarization degree in the initial and short period. In the later
phase of the Segment II, when $S_{\rm obs}\gg S_{\rm patch}$ and $N_{\rm p}$
increases, the total contribution comes from the inner and outer parts, see
(Eq.10). The outer part contributions cancel among themselves due to an axial
symmetry. The inner part contribution remains approximately equal due to the
structure and dynamics of the jet and magnetic patch (Eq.9). As a result, the
$N_{\rm p}$\- and $\Gamma$-dependence of the polarization degree $\pi_{\rm
obs}$ becomes

$\displaystyle\pi_{\rm obs}\propto\frac{1}{N_{\rm p}}\propto\Gamma^{2},$ (23)

for small polarization degree in the later and long period. The
$\Gamma$-dependence of (Eq.22) and (Eq.23) is distinctly different. Using
observed data we examine them in Extended Data Figure 6. It indeed shows that
the large polarization degree follows (Eq.22) (see yellow line in Figure 6)
and the small polarization follows (Eq.23) (see cyan line in Figure 6). This
supports our theoretical scenario of the time evolution of polarization degree
$\pi_{\rm obs}$, described in the previous Section. This is also in accordance
with our explanations that the weakly $\Gamma$-depending in the initial
Segment I and in the final Segment IV, as well as $\pi_{\rm obs}$ has an
increment in the Segment III at the jet break time.

###  0.7 Numerical results (or Self-consistency checking results)

Using a typical jet opening angle $\theta_{j}$=0.1, we show the ratios
(numbers) $N_{\rm patch}$, $N_{\rm p}$ and $N_{\rm obs}$ as functions of time
and Lorentz factor $\Gamma$ in Figs. 3 and 4 respectively. The observed
polarization lightcurve shows a hump at initial times 0–10 s (see Segment I in
Figure 1). The peak time occurs at the point $\pi_{\rm obs}\approx\Pi_{\rm
max}$ when $S_{\rm obs}\approx S_{\rm patch}$. On the other hand, we have
$S_{\rm obs}\approx S_{\rm jet}$ at the jet break time $t_{\rm obs}\simeq$
3$\times$ 104 seconds (see Segment III in Figure 1). At the peak time and jet
break time, we numerically calculate the following quantities for consistency
checks.

At the initial time $t_{\rm obs}\simeq 3$ seconds and $\Gamma\simeq 300$ for a
hump $\pi_{\rm obs}\approx\Pi_{\rm max}$, $S_{\rm obs}\approx S_{\rm patch}$,
we obtain

  * •

$R_{\rm GRB}\simeq ct_{\rm GRB}=c(2\Gamma^{2}t_{\rm obs})\simeq(3\times
10^{10}{\rm cm/sec})\times 2\times(300)^{2}\times 3{\rm sec}=1.62\times
10^{16}$ cm.

  * •

$S_{\rm obs}\approx S_{\rm patch}=\hat{\pi}(c\Gamma t_{\rm obs})^{2}\simeq
3.14\times(3\times 10^{10}{\rm cm/sec})^{2}\times(300)^{2}\times(3{\rm
sec})^{2}=2.5\times 10^{26}{\rm cm^{2}}$.

  * •

$S_{\rm jet}=\hat{\pi}(\theta_{j}c\Gamma^{2}t_{\rm obs})^{2}\simeq 3.14\times
0.1^{2}\times(3\times 10^{10}{\rm cm/sec})^{2}\times(300)^{4}\times(3{\rm
sec})^{2}=2.25\times 10^{28}{\rm cm^{2}}$.

  * •

$N_{\rm p}\equiv S_{\rm obs}/S_{\rm patch}\approx 1$.

  * •

$N_{\rm obs}\equiv$ $S_{\rm jet}/S_{\rm
obs}\approx\theta^{2}_{j}\Gamma^{2}=(0.1)^{2}\times(300)^{2}=900$.

  * •

$N_{\rm patch}\equiv S_{\rm jet}/S_{\rm patch}\approx S_{\rm jet}/S_{\rm
obs}=900$.

At the jet break time $t_{\rm obs}\simeq$ 3$\times$ 104 seconds and
$\Gamma\simeq 10$, we obtain

  * •

$R_{\rm GRB}\simeq ct_{\rm GRB}=c(2\Gamma^{2}t_{\rm obs})\simeq(3\times
10^{10}{\rm cm/sec})\times 2\times(10)^{2}\times 3\times 10^{4}{\rm
sec})=1.8\times 10^{17}$ cm.

  * •

$S_{\rm obs}\approx S_{\rm jet}=\hat{\pi}(c\Gamma t_{\rm obs})^{2}\simeq
3.14\times(3\times 10^{10}{\rm cm/sec})^{2}\times(10)^{2}\times(3\times
10^{4}{\rm sec})^{2}=2.54\times 10^{32}{\rm cm^{2}}$.

  * •

$S_{\rm jet}=\hat{\pi}(\theta_{j}c\Gamma^{2}t_{\rm obs})^{2}\simeq
3.14\times(0.1)^{2}\times(3\times 10^{10}{\rm
cm/sec})^{2}\times(10)^{4}\times(3\times 10^{4}{\rm sec})^{2}=2.54\times
10^{32}{\rm cm^{2}}$.

  * •

$N_{\rm p}\equiv S_{\rm obs}/S_{\rm patch}\approx 900$.

  * •

$N_{\rm obs}\equiv S_{\rm jet}/S_{\rm
obs}\approx\theta^{2}_{j}\Gamma^{2}=(0.1)^{2}\times(10)^{2}=1$.

  * •

$N_{\rm patch}\equiv S_{\rm jet}/S_{\rm
patch}\approx\frac{\theta^{2}_{j}}{\theta^{2}_{\rm patch}}=900$.

  * •

$\theta_{\rm patch}=0.1/30$.

The results are summarized in Table Extended Data.

###  0.8 Testing the $\pi_{\rm obs}$-$\Gamma$ and $\pi_{\rm obs}$-[t/(1+z)]
relations with observational data (Segment II)

In the major power-law phase (Segment II, $S_{\rm patch}\lesssim S_{\rm
obs}\lesssim S_{\rm jet}$), we have obtained two important scaling laws: the
observed net polarization mainly follows a scaling law $\pi_{\rm obs}\propto
S^{-1}_{\rm obs}\propto\Gamma^{2}$ (Eq.8), and $S_{\rm jet}/S_{\rm
obs}=\theta^{2}_{j}/\theta^{2}_{e}\propto\Gamma^{2}$ (Eq.20), due to
relativistic beaming effect. In Figure 6, we show that the scaling law
$\pi_{\rm obs}\propto\Gamma^{2}$ is consistent with GRB data.

There are several external-shock models widely discussed in the literature.
The simplest model invokes a decelerating blastwave that enters the self-
similar regime with constant energy (adiabatic and no energy injection), and
constant medium density (ISM)[22, 23]. Several other more complicated cases,
such as the dynamics of a blastwave with a varying ambient medium density
(e.g., stellar wind[24, 28]), or varying total energy in a blastwave (e.g.,
with radiative loss[27] or with energy injection[29]) have also been
discussed. These traditional external-shock models give

$\left\\{\begin{array}[]{ll}\Gamma\propto R_{\rm GRB}^{-3/2}\propto
t^{-3/8},R_{\rm GRB}\propto t^{1/4};\quad\rm ISM\\\ \Gamma\propto
R^{-1/2}_{\rm GRB}\propto t^{-1/4},R_{\rm GRB}\propto t^{1/2};\quad\rm
wind,\\\ \Gamma\propto R^{-3}_{\rm GRB}\propto t^{-3/7},R_{\rm GRB}\propto
t^{1/7};\quad\rm Radiative\,fireball,\\\ \Gamma\propto R^{-\frac{5}{6}}_{\rm
GRB}\propto t^{-\frac{5}{16}},R_{\rm GRB}\propto t^{\frac{3}{8}};\quad\rm
Energy\,injection\end{array}\right.$ (24)

Note that the energy injection case has taken a typical value of $q$=0.5.

Relativistic effects gives

$S_{\rm obs}\propto\theta^{2}_{e}\propto\Gamma^{-2}$ (25)

The geometric correlation gives tan($\theta_{j}$/2)=$R_{\rm jet}/R_{\rm GRB}$,
and combining Eq. (24), therefore,

$S_{\rm jet}\propto R_{\rm
GRB}^{2}\propto\left\\{\begin{array}[]{ll}(\Gamma^{-2/3})^{2}\propto\Gamma^{-4/3}\propto
t^{1/2},\quad\rm ISM\\\ (\Gamma^{-2})^{2}\propto\Gamma^{-4}\propto
t^{1},\quad\rm wind,\\\ (\Gamma^{-1/3})^{2}\propto\Gamma^{-2/3}\propto
t^{2/7},\quad\rm Radiative\,fireball,\\\
(\Gamma^{-6/5})^{2}\propto\Gamma^{-12/5}\propto t^{3/4},\quad\rm
Energy\,injection\end{array}\right.$ (26)

Before the jet break effect $t\lesssim t_{\rm jet}$, the observed area is
$S_{\rm obs}$ on the jet plane and $\pi_{\rm obs}\propto S^{-1}_{\rm obs}$,
one therefore has

$\pi_{\rm obs}\propto S^{-1}_{\rm
obs}\propto\Gamma^{2}\propto\left\\{\begin{array}[]{ll}t^{-3/4},\quad\rm
ISM\\\ t^{-1/2},\quad\rm wind,\\\ t^{-6/7},\quad\rm Radiative\,fireball,\\\
t^{-5/8},\quad\rm Energy\,injection\end{array}\right.$ (27)

After the jet break effect $t\gtrsim t_{\rm jet}$, the observed area $S_{\rm
obs}$ is larger than the total emission area $S_{\rm jet}$. The total net
polarization degree $\pi_{\rm obs}$ has been accounted, thus approximately
remains as a constant or slowly decreases in time.

A possible test can be made by investigating the $\pi_{\rm obs}$-$\Gamma(\pi)$
relation via observational data, where $\Gamma(\pi)$ is the bulk Lorentz
factor measured during the epoch of polarization. Two main methods have been
proposed to estimate $\Gamma$ of a GRB fireball from the literature[30, 11],
either using a thermal component from prompt emission spectra[30] or using the
early afterglow light curves that show the signal of fireball
deceleration[11]. The former one may not be universally applicable since
thermal component spectra have only been observed in a handful of Fermi-
detected GRBs to date. We therefore focus on the latter one. Additionally,
$\Gamma(\pi)$ cannot be precisely estimated in many bursts due to their
polarization being measured in various emission regions (some in early prompt
emission, some others in late after emission). As a proxy of $\Gamma(\pi)$, we
use the initial Lorentz factor $\Gamma_{0}$ (at the deceleration radius of the
GRBs), which can be estimated by using an empirical relation[11]
($\Gamma_{0}\simeq 249L^{0.30}_{\gamma,\rm iso,52}$), where $L_{\gamma,\rm
iso}$ is the average isotropic-equivalent luminosity during prompt emission
phase. In order to obtain $L_{\gamma,\rm iso}$, we perform time-integrated
spectral analysis of the prompt emission for each burst, following the
standard practices[31, 32] used by the full Python package, namely, the Multi-
Mission Maximum Likelihood Framework (3ML[33]). The energy flux $F_{\gamma}$
(erg cm-2s-1) can therefore be obtained from the spectral fit, with a
$k$-correction ($k_{c}$) applied. With $F_{\gamma}$ and redshift measurements,
one can estimate the average isotropic-equivalent luminosity $L_{\gamma,\rm
iso}=4\hat{\pi}d^{2}_{L}F_{\gamma}k_{c}$.

Extended Data Figure 6 shows the $\pi_{\rm obs}$-$\Gamma_{0}$ correlation. We
note that for those bursts in which the polarization is measured during the
prompt emission phase, one has $\Gamma_{0}<\Gamma(\pi)$ due to
$t_{\Gamma_{0}}<t_{\Gamma(\pi)}$ (see the data points in Extended Data Figure
6 with rightward arrows). While the polarization for the remaining GRBs is
measured in the optical band during afterglow emission and has
$t_{\Gamma_{0}}>t_{\Gamma(\pi)}$, one therefore has $\Gamma_{0}>\Gamma(\pi)$
(see the data points in Figure 6 with leftward arrows). In summary, the data
points are clustered around the solid line defined by the function $\pi_{\rm
obs}\propto\Gamma^{2}$ (see the solid line in Figure 6), which strongly
supports our explanations.

We show that the GRB polarization degrees ($\pi_{\rm obs}$) decay as a power-
law over time, which takes the form of $\pi_{\rm obs}\propto t^{-0.50\pm
0.02}$ (Figure 1) and the theoretical explanation $\pi_{\rm obs}\propto
S^{-1}_{\rm obs}\propto\Gamma^{2}$ (Eq.27), which is verified by the $\pi_{\rm
obs}-\Gamma$ relation (Figure 6) from data. As a result, using afterglow
models of time-decaying Lorentz factors $\Gamma\propto t^{-\alpha/2}$ (Eq.24),
we obtain the polarization scaling law $\pi_{\rm obs}\propto t^{-\alpha}$.
Therefore, the obtained power-law index of polarization ($\alpha_{\rm
obs}$=-0.50 $\pm$ 0.02) from the time-integrated polarization sample can be
used as a probe for further information constraints of the afterglow models
and environment surrounding GRBs (Extended Data Figure 7).

###  0.9 Searching for a jet break in the afterglow lightcurve (Segment III)

A bit-increment polarization data (deviating from the main power-law function)
around $10^{4}$ to $10^{5}$ seconds were observed in 5 GRBs (GRB 990510, GRB
010222, GRB 020405, GRB 030328, and GRB 080928, see Figure 1). In the
presented scenario, these observed properties are naturally explained as the
“jet break” boundary effects that non-axial symmetric irregularities on jet-
plan boundary are seen and effective magnetic patch area $S^{\rm eff}_{\rm
patch}$ increases in Eq.(5). To verify such an explanation, we further analyse
their afterglow lightcurves. This is because the afterglow lightcurves are
usually characterized by a post-jet break when the jet boundary is observed,
with a steeper temporal decay slope ($\alpha\sim p>1.5$, Ref.[25]) than normal
decay, where $p$ is the electron spectral distribution index. As a result
shown in the left panel of Extended Data Figure 8, it is clearly evident that
in these four GRB events, net ploarization increment and afterglow lightcurve
decrement are indeed correlated at jet break time.

Extended Data Figure 8 shows the optical afterglow lightcurves for these five
bursts and their polarization data. We fit these lightcurves using a BKPL/SPL
model, and pay special attention to their break time (if any) and post-break
slope. The best fit results are summarized in Table Extended Data. We find
that four out of five bursts have post-break slopes larger than 1.5, which can
be accounted for by the jet-break phenomenon. In addition, the post-break
times measured from the fitting (with uncertainties) of three bursts (GRB
990510, GRB 010222, and GRB 030328) overlap with the epochs of their
polarization observations.

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  * [96] Gupta, R. _et al._ Probing into emission mechanisms of GRB 190530A using time-resolved spectra and polarization studies: Synchrotron Origin?  _MNRAS_ (2022).  2201.01167.
  * [97] Shrestha, M. _et al._ GRB 191016A: A highly collimated gamma-ray burst jet with magnetised energy injection.  _arXiv e-prints_ arXiv:2111.09123 (2021).  2111.09123.

## Extended Data

  1. 1.

Extended Data Figure 1: The degree of polarization ($\pi_{\rm obs}$) in a
time-integrated manner is plotted as a function of their time intervals after
the burst at the cosmological distance.

  2. 2.

Extended Data Figure 2: Upper panel (a): a sketch plot for delineating the
correlation between GRB polarization and the Lorentz factor. We consider a
conical jet with line-of-sight aligned with the jet axis with a jet opening
angle $\theta_{j}$(=$\angle$HAD). Assuming that A is the GRB central engine, B
is the observer, O is the center point of the plane, C(G) are the highest
latitude photons on the conical jet that the relativistic emission can be
observed by the observer, and D (H) are the boundary of the conical jet on the
plane. Due to relativistic beaming, the relativistic emission produced by the
photon C can be observed only inside the cone $\theta_{e}$ ($\angle$ BCF).
Lower panel (b): same as (a) but considering a conical jet with line-of-sight
unaligned with the jet axis.

  3. 3.

Extended Data Figure 5: A cartoon picture for “magnetic patches” on the jet
plane represented by many small circles with random red arrows; the observed
area $S_{\rm obs}\propto 1/\Gamma^{2}$ increase in time represented by four
hollow circles corresponding to the hollow circles in Fig.2; the irregular jet
boundary represented by a waved circle.

  4. 4.

Extended Data Figure 3: The numbers $N_{\rm p}$(=$S_{\rm obs}/S_{\rm
patch}$:cyan line), $N_{\rm patch}$(=$S_{\rm jet}/S_{\rm patch}$:yellow line),
and $N_{\rm obs}$(=$S_{\rm jet}/S_{\rm obs}$:grey line) as functions of rest-
frame time, as well as their typical values (indicated by different colored
data points). Data points colored by green and magenta indicate $t_{\rm
obs}=t_{\rm patch}$ and $t_{\rm obs}=t_{\rm jet}$, respectively. $N_{\rm p}$:
when $t_{\rm obs}\lesssim t_{\rm patch}$, one has $S_{\rm obs}\lesssim S_{\rm
patch}$, and $N_{\rm p}\lesssim 1$; when $t_{\rm patch}\lesssim t_{\rm
obs}\lesssim t_{\rm jet}$, one has $S_{\rm patch}\lesssim S_{\rm obs}\lesssim
S_{\rm jet}$, and $N_{\rm p}\gtrsim 1$ follows a rapidly rising scaling law
over time (cyan line); when $t_{\rm obs}\gtrsim t_{\rm jet}$, one has $S_{\rm
obs}\gtrsim S_{\rm jet}$ and $N_{\rm p}$ follows a slowly rising scaling law
over time (yellow). $N_{\rm patch}$: compared with $N_{\rm p}$, $N_{\rm
patch}$ follows a slowly rising scaling law (note that we have assumed typical
observed parameters: $N_{\rm patch}$=600 at the $\Pi_{\rm max}$ point and
$N_{\rm patch}$=900 at the jet break time) over time when $t_{\rm obs}\lesssim
t_{\rm jet}$. However, $N_{\rm p}$ and $N_{\rm patch}$ will share the same
scaling law, when $t_{\rm obs}\gtrsim t_{\rm jet}$. $N_{\rm obs}$: given the
typical initial observed parameters (e.g., $N_{\rm obs}$=900, see Methods),
$N_{\rm obs}$ as a function of time can be described by the orange line. Note
that: one has $N_{\rm obs}\approx 1$, when $t\gtrsim t_{\rm jet}$. Summarized
information is given in Table Extended Data.

  5. 5.

Extended Data Figure 4: The numbers $N_{\rm p}$, $N_{\rm patch}$, and $N_{\rm
obs}$ depend on $\Gamma^{2}$. By assuming typical values (e.g., $N_{\rm
patch}$=900 when $\Gamma$=300) and a typical jet opening angle
($\theta_{j}$=0.1) measured from GRBs, $N_{\rm p}$, $N_{\rm patch}$, and
$N_{\rm obs}$ as functions of $\Gamma$ can be described by the cyan (Eq.19),
grey (Eq.20), and yellow (Eq.9) lines. The data points with different colors
represent several typical $\Gamma$ values and their corresponding $N$ values.
For example, when $\Gamma=300$ (at $\Pi_{\rm max}$ point), one has $N_{\rm
p}=1$, $N_{\rm patch}=900$, and $N_{\rm obs}=900$; when $\Gamma=10$ (at jet
break time); $N_{\rm p}=900$, $N_{\rm patch}=900$ and $N_{\rm obs}=1$; and
when $\Gamma=1$, one has $N_{\rm p}=900$, $N_{\rm patch}=900$, and $N_{\rm
obs}=1$. Note that (i) when $\Gamma=1$, $N_{\rm obs}=1$, rather than $N_{\rm
obs}=0.01$ due to the “jet break” boundary effect; (ii) when $\Gamma\lesssim
10$ ($t_{\rm obs}\gtrsim t_{\rm jet}$), $N_{\rm p}$ and $N_{\rm patch}$ share
the same function (dashed line). Summarized information is given in Table
Extended Data.

  6. 6.

Extended Data Figure 5: The solid line is qualitatively illustrated by the
time evolution of $S_{\rm patch}^{\rm eff}/S_{\rm patch}$, while the data
points with purple and orange colors indicate the $\Pi_{\rm max}$ point and
the typical jet break time.

  7. 7.

Extended Data Figure 6: Time-integrated degrees of polarization $\pi_{\rm
obs}$ as a function of the initial Lorentz factor $\Gamma_{0}$ at the
deceleration radius, where $\Gamma_{0}$ is estimated by the empirical relation
($\Gamma_{0}\simeq 249L^{0.30}_{\gamma,\rm iso,52}$). The solid lines
represent the functions of $\pi_{\rm obs}\propto\Gamma^{2}$. The rightward
($\Gamma_{0}<\Gamma_{\pi}$: prompt emission phase) and leftward
($\Gamma_{0}>\Gamma_{\pi}$: afterglow emission phase) arrows represent the
region in which the bulk Lorentz factor is measured during the epoch of
polarization.

  8. 8.

Extended Data Figure 7: Comparing various afterglow models with observed
results in the $\pi_{\rm obs}$-[t/(1+z)] plane. The solid line is the best
fits using the power-law model from the observational data and with $2\sigma$
(95% confidence interval) error shadow region. The solid lines indicated by
different colors represent different afterglow models.

  9. 9.

Extended Data Figure 8: Optical ($R$-band) afterglow lightcurves (dotted
lines) (left panel) and the degree of polarization (right panel) for the five
bursts that have a jet break around the polarization epoch. (GRB 990510, GRB
010222, GRB 020405, GRB 030328, and GRB 080928). The dashed lines in left
panel represent the best fits using a BKPL/SPL model.

  10. 10.

Extended Data Table Extended Data: A full catalog of GRB polarimetric
observations.

  11. 11.

Extended Data Table Extended Data: Fit Results for the Optical Afterglow
Lightcurve (Jet Breaks).

  12. 12.

Extended Data Table Extended Data: Results of Inferred $\Gamma_{0}$.

Figure 1: The degree of polarization ($\pi_{\rm obs}$) in a time-integrated
manner is plotted as a function of their time intervals after the burst at the
cosmological distance.

Figure 2: Upper panel (a): a sketch plot for delineating the correlation
between GRB polarization and the Lorentz factor. We consider a conical jet
with line-of-sight aligned with the jet axis with a jet opening angle
$\theta_{j}$(=$\angle$HAD). Assuming that A is the GRB central engine, B is
the observer, O is the center point of the plane, C(G) are the highest
latitude photons on the conical jet that the relativistic emission can be
observed by the observer, and D (H) are the boundary of the conical jet on the
plane. Due to relativistic beaming, the relativistic emission produced by the
photon C can be observed only inside the cone $\theta_{e}$ ($\angle$ BCF).
Lower panel (b): same as (a) but considering a conical jet with line-of-sight
unaligned with the jet axis.

Figure 3: The numbers $N_{\rm p}$(=$S_{\rm obs}/S_{\rm patch}$:cyan line),
$N_{\rm patch}$(=$S_{\rm jet}/S_{\rm patch}$:yellow line), and $N_{\rm
obs}$(=$S_{\rm jet}/S_{\rm obs}$:grey line) as functions of rest-frame time,
as well as their typical values (indicated by different colored data points).
Data points colored by green and magenta indicate $t_{\rm obs}=t_{\rm patch}$
and $t_{\rm obs}=t_{\rm jet}$, respectively. $N_{\rm p}$: when $t_{\rm
obs}\lesssim t_{\rm patch}$, one has $S_{\rm obs}\lesssim S_{\rm patch}$, and
$N_{\rm p}\lesssim 1$; when $t_{\rm patch}\lesssim t_{\rm obs}\lesssim t_{\rm
jet}$, one has $S_{\rm patch}\lesssim S_{\rm obs}\lesssim S_{\rm jet}$, and
$N_{\rm p}\gtrsim 1$ follows a rapidly rising scaling law over time (cyan
line); when $t_{\rm obs}\gtrsim t_{\rm jet}$, one has $S_{\rm obs}\gtrsim
S_{\rm jet}$ and $N_{\rm p}$ follows a slowly rising scaling law over time
(yellow). $N_{\rm patch}$: compared with $N_{\rm p}$, $N_{\rm patch}$ follows
a slowly rising scaling law (note that we have assumed typical observed
parameters: $N_{\rm patch}$=900 at the $\Pi_{\rm max}$ point and $N_{\rm
patch}$=900 at the jet break time) over time when $t_{\rm obs}\lesssim t_{\rm
jet}$. However, $N_{\rm p}$ and $N_{\rm patch}$ will share the same scaling
law, when $t_{\rm obs}\gtrsim t_{\rm jet}$. $N_{\rm obs}$: given the typical
initial observed parameters (e.g., $N_{\rm obs}$=900, see Methods), $N_{\rm
obs}$ as a function of time can be described by the orange line. Note that:
one has $N_{\rm obs}\approx 1$, when $t\gtrsim t_{\rm jet}$. Summarized
information is given in Table Extended Data.

Figure 4: The numbers $N_{\rm p}$, $N_{\rm patch}$, and $N_{\rm obs}$ depend
on $\Gamma^{2}$. By assuming typical values (e.g., $N_{\rm patch}$=900 when
$\Gamma$=300) and a typical jet opening angle ($\theta_{j}$=0.1) measured from
GRBs, $N_{\rm p}$, $N_{\rm patch}$, and $N_{\rm obs}$ as functions of $\Gamma$
can be described by the cyan (Eq.19), grey (Eq.20), and yellow (Eq.9) lines.
The data points with different colors represent several typical $\Gamma$
values and their corresponding $N$ values. For example, when $\Gamma=300$ (at
$\Pi_{\rm max}$ point), one has $N_{\rm p}=1$, $N_{\rm patch}=900$, and
$N_{\rm obs}=900$; when $\Gamma=10$ (at jet break time); $N_{\rm p}=900$,
$N_{\rm patch}=900$ and $N_{\rm obs}=1$; and when $\Gamma=1$, one has $N_{\rm
p}=900$, $N_{\rm patch}=900$, and $N_{\rm obs}=1$. Note that (i) when
$\Gamma=1$, $N_{\rm obs}=1$, rather than $N_{\rm obs}=0.01$ due to the “jet
break” boundary effect; (ii) when $\Gamma\lesssim 10$ ($t_{\rm obs}\gtrsim
t_{\rm jet}$), $N_{\rm p}$ and $N_{\rm patch}$ share the same function (dashed
line). Summarized information is given in Table Extended Data. Figure 5: The
solid line is qualitatively illustrated by the time evolution of $S_{\rm
patch}^{\rm eff}/S_{\rm patch}$, while the data points with purple and orange
colors indicate the $\Pi_{\rm max}$ point and the typical jet break time.

Figure 6: Time-integrated degrees of polarization $\pi_{\rm obs}$ as a
function of the initial Lorentz factor $\Gamma_{0}$ at the deceleration
radius, where $\Gamma_{0}$ is estimated by the empirical relation
($\Gamma_{0}\simeq 249L^{0.30}_{\gamma,\rm iso,52}$). The cyan solid line
represents the function of $\pi_{\rm obs}\propto\Gamma$ (Eq.22) for small
polarization degree while the yellow solid line represents the function of
$\pi_{\rm obs}\propto\Gamma^{2}$ (Eq.23) for large polarization degree. The
rightward ($\Gamma_{0}<\Gamma(\pi)$: prompt emission phase) and leftward
($\Gamma_{0}>\Gamma(\pi)$: afterglow emission phase) arrows represent the
region in which the bulk Lorentz factor is measured during the epoch of
polarization.

Figure 7: Comparing various afterglow models with observed results in the
$\pi_{\rm obs}$-[t/(1+z)] plane. The solid line is the best fits using the
power-law model from the observational data and with $2\sigma$ (95% confidence
interval) error shadow region. The solid lines indicated by different colors
represent different afterglow models.

Figure 8: Optical ($R$-band) afterglow lightcurves (dotted lines) (left panel)
and the degree of polarization (right panel) for the five bursts that have a
jet break around the polarization epoch. (GRB 990510, GRB 010222, GRB 020405,
GRB 030328, and GRB 080928). The dashed lines in left panel represent the best
fits using a BKPL/SPL model.

Table 1 A full catalog of GRB polarimetric observations GRB PD PA $\nu$/keV
Energy/wave band Time Significance Instrument Ref. z ($\pi_{\rm obs}\%$) (∘)
($t-t_{0}$) 930131 $>$35 5 20 keV-1 MeV $\gamma$-ray NA NA BATSE-CGRO [34] NA
960924 $>$50 47 20 keV-1 MeV $\gamma$-ray NA NA BATSE-CGRO [34] NA 980329
$<$21 NA 8.3$\times$109 Radio 500h 2$\sigma$ VLA [35] 3.5pha 980425 0.6 80
4000-7000 $\mathring{A}$ Optical 8days NA NA [36] 0.0085 980425 0.4 67
4000-7000 $\mathring{A}$ Optical 25days NA NA [36] 980425 0.53$\pm$0.08
49$\pm$3 390-750 nm Optical 42days NA CTIO 4-m telescope [37] 980703 $<$8 NA
4.86$\times$109 Radio 100h 3$\sigma$ VLA [3] 0.966 980703 $<$8 NA
8.46$\times$109 Radio 100h 3$\sigma$ VLA [3] 990123 $<$2.3 NA 4.55$\times$1014
Optical(R) 18.25h 2$\sigma$ NOT [38] 1.60 990123 $<$23l,$<$32c NA
8.46$\times$109 Radio 30h 3$\sigma$ VLA [39] 990510 1.7$\pm$0.2 101$\pm$3
4.55$\times$1014 Optical(R) 18.5h NA VLT [4] 1.619 990510 $1.6\pm 0.2$
96$\pm$4 4.55$\times$1014 Optical(R) 20.6h NA VLT [40] 990510 $<$3.9 NA
4.55$\times$1014 Optical(R) 43h NA VLT [40] 990510 2.2${}^{+1.1}_{-0.9}$
112${}^{+17}_{-15}$ 4.55$\times$1014 Optical(R) 43.4h NA VLT [40] 990712
$2.9\pm 0.4$ 121.1$\pm$3.5 4.55$\times$1014 Optical(R) 10.56h NA VLT [41]
0.434 990712 $1.2\pm 0.4$ 116.2$\pm$10.1 4.55$\times$1014 Optical(R) 16.8h NA
VLT [41] 990712 $2.2\pm 0.7$ 139.1$\pm$10.4 4.55$\times$1014 Optical(R) 34.8h
NA VLT [41] 991216 $<$2.7 NA 5.45$\times$1014 Optical(V) 35.0h 2$\sigma$ VLT
[42] 1.02? 991216 $<$5 NA 5.45$\times$1014 Optical(V) 60.0h 2$\sigma$ VLT [42]
991216 $<$11l,$<$17c NA 8.46$\times$109 Radio 35.8h 3$\sigma$ VLA [14] 991216
$<$9l,$<$15c NA 8.46$\times$109 Radio 64.3h 3$\sigma$ VLA [14] 000301C $<$30
NA 4.55$\times$1014 Optical(R) 43.2h NA VLT [43] 2.03 010222 $1.36\pm 0.64$ NA
5.45$\times$1014 Optical(V) 22.65h NA NOT [44] 1.477 011211 $<$2.7 NA
4.55$\times$1014 Optical(R) 35h 3$\sigma$ VLT [45] 2.14 020405 $<$11l,$<$19c
NA 8.46$\times$109 Radio 28.6h 3$\sigma$ VLA [14] 0.69 020405 1.50$\pm$0.40
172$\pm$8 4.55$\times$1014 Optical(R) 29.5h NA VLT [46] 020405 9.89$\pm$1.3
179.9$\pm$3.8 5.45$\times$1014 Optical(V) 31.7h NA MMT [47] 020405
1.96$\pm$0.33 154$\pm$5 5.45$\times$1014 Optical(V) 52.0h NA VLT [48] 020405
1.47$\pm$0.43 168$\pm$9 5.45$\times$1014 Optical(V) 76.2h NA VLT [48] 020813
1.8-2.4 153-162 3.26-9.37$\times$1014 Optical 4.7-7.9h NA Keck [49] 1.25
020813 0.80$\pm$0.16 144$\pm$6 NA NA 13.92h NA NA [50] 020813 1.07$\pm$0.22
154.3$\pm$5.9 5.45$\times$1014 Optical(V) 21.55h NA VLT [51] 020813
1.42$\pm$0.25 137.0$\pm$4.4 5.45$\times$1014 Optical(V) 22.5h NA VLT [51]
020813 1.11$\pm$0.22 150.5$\pm$5.5 5.45$\times$1014 Optical(V) 23.41h NA VLT
[51] 020813 1.05$\pm$0.23 146.4$\pm$6.2 5.45$\times$1014 Optical(V) 24.39h NA
VLT [51] 020813 1.43$\pm$0.44 155.8$\pm$8.5 5.45$\times$1014 Optical(V) 26.80h
NA VLT [51] 020813 1.07$\pm$0.53 163.0$\pm$14.6 5.45$\times$1014 Optical(V)
27.34h NA VLT [51] 020813 1.37$\pm$0.49 142.1$\pm$8.9 5.45$\times$1014
Optical(V) 27.78h NA VLT [51] 020813 1.26$\pm$0.34 164.7$\pm$7.4
5.45$\times$1014 Optical(V) 47.51h NA VLT [51] 020813 0.58$\pm$1.08
13.7$\pm$24.4 5.45$\times$1014 Optical(V) 97.29h NA VLT [51] 021004
1.88$\pm$0.46 189$\pm$7 4.55$\times$1014 Optical(R) 8.88h NA NOT [52] 2.33
021004 2.24$\pm$0.51 175$\pm$6 4.55$\times$1014 Optical(R) 9.12h NA NOT [52]
021004 $<$0.60 NA 4.55$\times$1014 Optical(R) 9.60h NA NOT [52] 021004
0.71$\pm$0.13 140$\pm$5 5.45$\times$1014 Optical(V) 16.08h NA VLT [52] 021004
$<$5.0 NA 2.43$\times$1014 Optical(J) 10.76h 2$\sigma$ TNG [53] 021004
0.51$\pm$0.10 126$\pm$5 5.45$\times$1014 Optical(V) 14.62h NA VLT [53] 021004
0.8-1.7 100-147 3.49-8.57$\times$1014 Optical 18.83h NA VLT [53] 021004
0.43$\pm$0.20 45$\pm$12 5.45$\times$1014 Optical(V) 90.7h NA VLT [53] 021206
$80\pm 20$ NA 150-2000 keV $\gamma$-ray NA $\geq$5.7$\sigma$ RHESSI [5] NA
021206 0 NA NA NA NA NA NA [54] 021206 $<$4.1 NA 150-2000 keV $\gamma$-ray NA
NA NA [54] 021206 $41^{+57}_{-44}$ NA 150-2000 keV $\gamma$-ray NA NA RHESSI
[55] 030226 $<$1.1 NA 4.55$\times$1014 Optical 25.39h 2$\sigma$ VLT [56] 1.98
030328 2.4$\pm$0.6 170$\pm$7 5.45$\times$1014 Optical 18.5h NA VLT [57] 1.52

Table 2: — continued GRB | PD | PA |  $\nu$/keV | Energy band | Time | Significance | Instrument | Ref. | z
---|---|---|---|---|---|---|---|---|---
| ($\pi_{\rm obs}\%$) | (∘) |  |  | ($t-t_{0}$) |  |  |  |
030329 | 0.92$\pm$0.10 | 86.13$\pm$2.43 | 4.55$\times$1014 | Optical(R) | 12.77h | NA | VLT | [58] | 0.168
030329 | 0.86$\pm$0.09 | 86.74$\pm$2.40 | 4.55$\times$1014 | Optical(R) | 13.18h | NA | VLT | [58] |
030329 | 0.87$\pm$0.09 | 88.60$\pm$2.64 | 4.55$\times$1014 | Optical(R) | 13.61h | NA | VLT | [58] |
030329 | 0.80$\pm$0.09 | 91.12$\pm$2.88 | 3.53-4.55$\times$1014 | Optical | 14.04h | NA | VLT | [58] |
030329 | 0.66$\pm$0.07 | 78.52$\pm$2.94 | 4.55$\times$1014 | Optical(R) | 16.61h | NA | VLT | [58] |
030329 | 0.66$\pm$0.07 | 76.69$\pm$2.89 | 4.55$\times$1014 | Optical(R) | 17.11h | NA | VLT | [58] |
030329 | 0.56$\pm$0.05 | 74.37$\pm$3.11 | 4.55$\times$1014 | Optical(R) | 17.62h | NA | VLT | [58] |
030329 | 1.10$\pm$0.40 | 70$\pm$11 | 4.55$\times$1014 | Optical(R) | 36.72h | NA | CAHA | [58] |
030329 | 1.37$\pm$0.11 | 61.65$\pm$2.38 | 4.55$\times$1014 | Optical(R) | 37.20h | NA | VLT | [58] |
030329 | 1.50$\pm$0.12 | 62.29$\pm$2.44 | 4.55$\times$1014 | Optical(R) | 37.92h | NA | VLT | [58] |
030329 | 1.07$\pm$0.09 | 59.41$\pm$2.51 | 3.53-4.55$\times$1014 | Optical | 40.08h | NA | VLT | [58] |
030329 | 1.09$\pm$0.08 | 66.07$\pm$2.45 | 4.55$\times$1014 | Optical(R) | 40.80h | NA | VLT | [58] |
030329 | 1.02$\pm$0.08 | 67.05$\pm$2.60 | 4.55$\times$1014 | Optical(R) | 41.28h | NA | VLT | [58] |
030329 | 1.13$\pm$0.08 | 70.56$\pm$2.51 | 4.55$\times$1014 | Optical(R) | 41.76h | NA | VLT | [58] |
030329 | 0.52$\pm$0.06 | 30.76$\pm$5.04 | 3.53-4.55$\times$1014 | Optical | 64.32h | NA | VLT | [58] |
030329 | 0.52$\pm$0.12 | 12.55$\pm$4.63 | 4.55$\times$1014 | Optical(R) | 64.80h | NA | VLT | [58] |
030329 | 0.31$\pm$0.07 | 24.50$\pm$6.94 | 4.55$\times$1014 | Optical(R) | 65.28h | NA | VLT | [58] |
030329 | 0.57$\pm$0.09 | 52.85$\pm$4.08 | 4.55$\times$1014 | Optical(R) | 84.96h | NA | VLT | [58] |
030329 | 0.53$\pm$0.08 | 57.08$\pm$4.06 | 4.55$\times$1014 | Optical(R) | 85.44h | NA | VLT | [58] |
030329 | 0.42$\pm$0.10 | 62.21$\pm$6.10 | 4.55$\times$1014 | Optical(R) | 85.92h | NA | VLT | [58] |
030329 | 1.68$\pm$0.18 | 66.32$\pm$3.38 | 4.55$\times$1014 | Optical(R) | 135.84h | NA | NOT | [58] |
030329 | 2.22$\pm$0.28 | 75.16$\pm$3.32 | 4.55$\times$1014 | Optical(R) | 183.36h | NA | VLT | [58] |
030329 | 1.33$\pm$0.14 | 70.91$\pm$3.31 | 4.55$\times$1014 | Optical(R) | 230.16h | NA | VLT | [58] |
030329 | 2.04$\pm$0.57 | 1.16$\pm$7.64 | 4.55$\times$1014 | Optical(R) | 326.40h | NA | VLT | [58] |
030329 | 0.58$\pm$0.10 | 42.70$\pm$9.26 | 4.55$\times$1014 | Optical(R) | 540.00h | NA | VLT | [58] |
030329 | 1.49$\pm$0.56 | 99.71$\pm$6.60 | 4.55$\times$1014 | Optical(R) | 696.00h | NA | VLT | [58] |
030329 | 1.48$\pm$0.48 | 25.42$\pm$9.41 | 4.55$\times$1014 | Optical(R) | 900.00h | NA | VLT | [58] |
030329 | 2.10$\pm$1.20 | 54.1$\pm$10.4 | 4.55$\times$1014 | Optical(R) | 35.19h | NA | CAHA | [59] |
030329 | 1.97$\pm$0.48 | 83.2 | 4.55$\times$1014 | Optical(R) | 36.49h | NA | IAG-USP | [60] |
030329 |  $<$1 | NA | 8.4$\times$109 | Radio | 185.04h | 3$\sigma$ | VLBA | [61] |
030329 |  $<$1.8 | NA | 8.4$\times$109 | Radio | 1992h | 3$\sigma$ | VLBA | [62] |
030329 |  $<$4.7 | NA | 8.4$\times$109 | Radio | 5208h | 3$\sigma$ | VLBA | [62] |
041219A | 98$\pm$33 | NA | 100-350 keV |  $\gamma$-ray | NA |  $\sim$2.3$\sigma$ | INTEGRAL-SPI | [63] | 0.3
041219A | 63${}^{+31}_{-30}$,96$\pm$40 | 70${}^{+14}_{-11}$ | 100-350 keV |  $\gamma$-ray | NA |  $\sim$2$\sigma$ | INTEGRAL-SPI | [64] |
041219A |  $\leq 4$ and 43$\pm$25 | 38$\pm$16 | 200-800 keV |  $\gamma$-ray | first and second peak |  $<$2$\sigma$ | INTEGRAL-IBIS | [65] |
060418 |  $<$8 | NA | 5.08$\times$1014 | Optical | 0.057h | 2$\sigma$ | LT | [6] | 1.489
061122 | 29${}^{+25}_{-26}$,$<60$ | NA | 100-1000 keV |  $\gamma$-ray | NA | 1$\sigma$ | INTEGRAL-SPI | [66] | NA
061122 |  $>$33 | 160$\pm$20 | 250-800 keV |  $\gamma$-ray | NA | 90% confidence | INTEGRAL-IBIS | [67] |
071010 |  $<$1.3 | NA | 4.55$\times$1014 | Optical | 21.51h | 3$\sigma$ | VLT | [68] | 0.98
080310 | $<2.5$ | NA | 5.45$\times$1014 | Optical(V) | 87171s | 2$\sigma$ | VLT | [69] | 2.42
080310 | $<2.5$ | NA | 5.45$\times$1014 | Optical(V) | 169501s | 2$\sigma$ | VLT | [69] |
080310 | $<2.6$ | NA | 5.45$\times$1014 | Optical(V) | 253724s | 2$\sigma$ | VLT | [69] |
080928 | $2.5\pm 0.5$ | 27$\pm$3 | 4.25-6.00$\times$1014 | Optical | 15.2h | NA | VLT | [70] | 1.692
090102 | $10.1\pm 1.3$ | NA | 5.08$\times$1014 | Optical | 160.8-220.8s |  $>$3$\sigma$ | LT-RINGO | [71] | 1.547
091018 |  $<$0.15c | NA | 4.55$\times$1014 | Optical(R) | 3.74h | 2$\sigma$ | VLT | [72] | 0.971
091018 |  $<$0.32 | NA | 4.55$\times$1014 | Optical(R) | 3.17h | 1$\sigma$ | VLT | [72] |
091018 | 0.21$\pm$0.31 | 177.0$\pm$47.5 | 4.55$\times$1014 | Optical(R) | 4.33h | NA | VLT | [72] |
091018 | 0.56$\pm$0.27 | 37.7$\pm$24.7 | 4.55$\times$1014 | Optical(R) | 4.73h | NA | VLT | [72] |
091018 | 0.26$\pm$0.31 | 9.2$\pm$43.7 | 4.55$\times$1014 | Optical(R) | 5.11h | NA | VLT | [72] |
091018 |  $<$0.32 | NA | 4.55$\times$1014 | Optical(R) | 5.53h | NA | VLT | [72] |
091018 | 1.07$\pm$0.30 | 179.2$\pm$16.1 | 4.55$\times$1014 | Optical(R) | 5.91h | NA | VLT | [72] |
091018 | 0.78$\pm$0.31 | 3.9$\pm$21.2 | 4.55$\times$1014 | Optical(R) | 6.33h | NA | VLT | [72] |
091018 | 0.84$\pm$0.30 | 171.1$\pm$19.9 | 4.55$\times$1014 | Optical(R) | 6.70h | NA | VLT | [72] |
091018 | 2.0$\pm$0.8 | 10.9$\pm$20.9 | 1.39$\times$1014 | Optical(R) | 10.34h | NA | VLT | [72] |
091018 | 1.44$\pm$0.32 | 2.2$\pm$12.6 | 4.55$\times$1014 | Optical(R) | 10.92h | NA | VLT | [72] |
091018 | 0.94$\pm$0.32 | 8.0$\pm$18.6 | 4.55$\times$1014 | Optical(R) | 11.30h | NA | VLT | [72] |
091018 | 1.73$\pm$0.36 | 69.8$\pm$11.7 | 4.55$\times$1014 | Optical(R) | 27.35h | NA | VLT | [72] |
091018 | 3.25$\pm$0.35 | 57.6$\pm$6.1 | 4.55$\times$1014 | Optical(R) | 27.73h | NA | VLT | [72] |
091018 | 1.99$\pm$0.35 | 27.6$\pm$10.0 | 4.55$\times$1014 | Optical(R) | 28.16h | NA | VLT | [72] |
091018 | 1.42$\pm$0.36 | 114.6$\pm$14.0 | 4.55$\times$1014 | Optical(R) | 28.54h | NA | VLT | [72] |
091018 | 0.27$\pm$0.38 | 101.6$\pm$47.1 | 4.55$\times$1014 | Optical(R) | 28.97h | NA | VLT | [72] |
091018 | 1.00$\pm$0.38 | 102.2$\pm$20.2 | 4.55$\times$1014 | Optical(R) | 29.35h | NA | VLT | [72] |
091018 | 0.41$\pm$0.34 | 168.8$\pm$36.6 | 4.55$\times$1014 | Optical(R) | 32.64h | NA | VLT | [72] |
091018 | 0.97$\pm$0.32 | 32.8$\pm$17.8 | 4.55$\times$1014 | Optical(R) | 33.40h | NA | VLT | [72] |
091018 | 1.08$\pm$0.35 | 88.7$\pm$17.9 | 4.55$\times$1014 | Optical(R) | 34.78h | NA | VLT | [72] |
091018 | 1.45$\pm$0.37 | 169.0$\pm$14.3 | 4.55$\times$1014 | Optical(R) | 55.37h | NA | VLT | [72] |
091208B | $10.4\pm 2.5$ | 92$\pm$6 | 4.55$\times$1014 | Optical(R) | 149-706s | NA | Kanata | [73] | 1.063
Table 2: — continued GRB | PD | PA |  $\nu$/keV | Energy band | Time | Significance | Instrument | Ref. | z
---|---|---|---|---|---|---|---|---|---
| ($\pi_{\rm obs}\%$) | (∘) |  |  | ($t-t_{0}$) |  |  |  |
100826A | $27\pm 11$ | 159$\pm$18, 75$\pm$20 | 20 keV-10 MeV |  $\gamma$-ray | 0-100s | 2.9$\sigma$ | IKAROS-GAP | [74] | NA
100906A |  $<$10 | NA |  $\sim$5$\times$1014 | Optical |  $\sim$0.5h | 60% | MASTER | [75] | 1.727
110205A | 1.4 | NA | 4.55$\times$1014 | Optical(R) | 2.73-4.33hours | NA | CAHA | [76] | 2.22
110205A | $<16$ | NA | 5.08$\times$1014 | Optical | 243s | 3$\sigma$ | LT-RINGO2 | [77] |
110205A | $<6.2$ | NA | 5.08$\times$1014 | Optical | 0.93h | 2$\sigma$ | LT-RINGO2 | [77] |
110301A | $70\pm 22$ | 73$\pm$11 | 10 keV-1 MeV |  $\gamma$-ray | 0-7s | 3.7$\sigma$ | IKAROS-GAP | [78] | NA
110721A | $84^{+16}_{-28}$ | 160$\pm$11 | 10 keV-1 MeV |  $\gamma$-ray | 0-11s | 3.3$\sigma$ | IKAROS-GAP | [78] | 0.382
120308A |  $28\pm 4$ to $16^{+5}_{-4}$ | 34$\pm$4 | 5.08$\times$1014 | Optical |  $\sim$200-700s | 3$\sigma$ | LT-RINGO2 | [22] |  $<$4.5
120308A | $23\pm 4$ | 44$\pm$6 | 5.08$\times$1014 | Optical | 0.0594h | 3$\sigma$ | LT-RINGO2 | [22] |
120308A | $17^{+5}_{-4}$ | 51$\pm$9 | 5.08$\times$1014 | Optical | 0.0892h | 3$\sigma$ | LT-RINGO2 | [22] |
120308A | $16^{+7}_{-4}$ | 40$\pm$10 | 5.08$\times$1014 | Optical | 0.1189h | 3$\sigma$ | LT-RINGO2 | [22] |
120308A | $16^{+5}_{-4}$ | 55$\pm$9 | 5.08$\times$1014 | Optical | 0.1792h | 3$\sigma$ | LT-RINGO2 | [22] |
121011A |  $<$15 | NA |  $\sim$5$\times$1014 | Optical |  $\sim$0.5h | 60% confidence | MASTER | [79] | NA
121024A | 0.61$\pm$0.13c | NA | 4.55$\times$1014 | Optical(R) | 0.15days | NA | VLT | [80] | 2.298
121024A | 4.09$\pm$0.20 | 163.7$\pm$2.8 | 4.55$\times$1014 | Optical(R) | 2.69h | NA | VLT | [80] |
121024A | 4.83$\pm$0.20 | 160.3$\pm$2.3 | 4.55$\times$1014 | Optical(R) | 2.96h | NA | VLT | [80] |
121024A | 3.82$\pm$0.20 | 182.7$\pm$3.0 | 4.55$\times$1014 | Optical(R) | 4.11h | NA | VLT | [80] |
121024A | 3.12$\pm$0.19 | 175.3$\pm$3.5 | 4.55$\times$1014 | Optical(R) | 4.46h | NA | VLT | [80] |
121024A | 3.39$\pm$0.18 | 178.0$\pm$2.9 | 4.55$\times$1014 | Optical(R) | 4.84h | NA | VLT | [80] |
121024A | 3.49$\pm$0.18 | 180.3$\pm$3.0 | 4.55$\times$1014 | Optical(R) | 5.23h | NA | VLT | [80] |
121024A | 3.20$\pm$0.18 | 174.5$\pm$3.3 | 4.55$\times$1014 | Optical(R) | 5.62h | NA | VLT | [80] |
121024A | 0.34$\pm$1.09 | 51.9$\pm$67.5 | 4.55$\times$1014 | Optical(R) | 23.45h | NA | VLT | [80] |
121024A | 1.49$\pm$0.78 | 93.1$\pm$26.6 | 4.55$\times$1014 | Optical(R) | 26.62h | NA | VLT | [80] |
121024A | 2.66$\pm$0.60 | 83.0$\pm$12.6 | 4.55$\times$1014 | Optical(R) | 28.62h | NA | VLT | [80] |
121024A | 0.86$\pm$0.72 | 101.3$\pm$36.4 | 4.55$\times$1014 | Optical(R) | 29.39h | NA | VLT | [80] |
130427A |  $<$3.9l,$<$2.7c | NA | 4.8$\times$109 | Radio | 36h | 3$\sigma$ | EVN | [81] | 0.3399
130427A |  $<$7.5l,$<$5.7c | NA | 4.8$\times$109 | Radio | 60h | 3$\sigma$ | EVN | [81] |
130427A |  $<$21l,$<$15c | NA | 4.8$\times$109 | Radio | 110h | 3$\sigma$ | EVN | [81] |
131030A | $2.1\pm 1.6$ | 27$\pm$22 | 4.55$\times$1014 | Optical(R) | 665s-2h | NA | Skinakas 1.3m | [82] | 1.295
140206A |  $>28$, $>48$? | 80$\pm$15 | 15-350 keV |  $\gamma$-ray | Second peak($\sim$4-26s) | 90% confidence | INTEGRAL-IBIS | [83] | 2.73
140430A | $<22$ | NA | 5.08$\times$1014 | Optical | 124-424s | 3$\sigma$ | LT-RINGO3 | [84] | NA
150301B | $>7.6$ | NA | 5$\times$1014 | Optical | 79-89s | 3.2$\sigma$ | MASTER | [85] | 1.5169
151006A | $<84$ | NA | 100-300 keV | hard X-rays | NA | NA |  $AstroSat$-CZTI | [9] | NA
160106A | $68.5\pm 24$ | $-22.5\pm 12$ | 100-400 keV | hard X-rays | NA |  $\gtrsim$3$\sigma$ |  $AstroSat$-CZTI | [9] | NA
160131A | $94\pm 31$ | $41.2\pm 5$ | 100-300 keV | hard X-rays | NA |  $\gtrsim$3$\sigma$ |  $AstroSat$-CZTI | [9] | 0.972
160325A | $58.75\pm 23.5$ | $10.9\pm 17$ | 100-300 keV | hard X-rays | NA |  $\sim$2.2$\sigma$ |  $AstroSat$-CZTI | [9] | NA
160509A |  $<92$,$96\pm 40$ | $-28.6\pm 11$ | 100-300 keV | hard X-rays | NA |  $\sim$2.5$\sigma$ |  $AstroSat$-CZTI | [9] | 1.17
160530A | $<46$ | NA | 0.2-5 MeV | soft $\gamma$-ray | NA | 90% confidence | COSI | [86] | NA
160607A | $<77$ | NA | 100-300 keV | hard X-rays | NA | NA |  $AstroSat$-CZTI | [9] | NA
160623A | $<46$ | NA | 100-300 keV | hard X-rays | NA | NA |  $AstroSat$-CZTI | [9] | 0.367
160625B | 8.0$\pm$0.5 | NA | NA | Optical | 95-360s | 3$\sigma$ | MASTER-IAC | [87] | 1.406
160703A | $<55$ | NA | 100-300 keV | hard X-rays | NA | NA |  $AstroSat$-CZTI | [9] |  $<$1.5ph
160802A | 85$\pm$29 | -36.1$\pm$4.6 | 200-300 keV | hard X-rays | NA |  $\geq$3$\sigma$ |  $AstroSat$-CZTI | [9] | NA
160802A | 85$\pm$29 |  $\sim$-32 | 200-300 keV | hard X-rays | 0-20.34s |  $\sim$3$\sigma$ |  $AstroSat$-CZTI | [88] |
160821A | $48.7\pm 14.6$ | $-34.0\pm 5.0$ | 100-300 keV | hard X-rays | NA |  $\geq$3$\sigma$ |  $AstroSat$-CZTI | [9] | NA
160821A | $21^{+24}_{-19}$ | NA | 100-300 keV | hard X-rays | 115-155s | 2.2$\sigma$ |  $AstroSat$-CZTI | [89] |
160821A | $66^{+26}_{-27}$ | NA | 100-300 keV | hard X-rays | NA |  $\sim$5.3$\sigma$ |  $AstroSat$-CZTI | [89] |
160821A | $71^{+29}_{-41}$ | $110^{+14}_{-15}$ | 100-300 keV | hard X-rays | 115-129s | 3.5$\sigma$ |  $AstroSat$-CZTI | [89] |
160821A | $58^{+29}_{-30}$ | $31^{+12}_{-10}$ | 100-300 keV | hard X-rays | 131-139s | 4$\sigma$ |  $AstroSat$-CZTI | [89] |
160821A | $61^{+39}_{-46}$ | $110^{+25}_{-26}$ | 100-300 keV | hard X-rays | 142-155s | 3.1$\sigma$ |  $AstroSat$-CZTI | [89] |
160821A | $54\pm 21$ | -39$\pm$4 | 100-300 keV | hard X-rays | 130-149s |  $\geq$3$\sigma$ |  $AstroSat$-CZTI | [90] |
160910A | $93.7\pm 30.92$ | $43.5\pm 4.0$ | 100-300 keV | hard X-rays | NA |  $\geq$3$\sigma$ |  $AstroSat$-CZTI | [9] | NA
161203A | $16^{+29}_{-15}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
161217C | $21^{+30}_{-16}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
161218A | $7.0^{+10.7}_{-7.0}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
161218A |  $<$41% | NA | 50-500 keV | hard X-rays | NA | 99% confidence | POLAR | [8] |
161218B | $13^{+28}_{-13}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
161229A | $17^{+24}_{-13}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170101A | $6.3^{+10.8}_{-6.3}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170101A |  $<$30% | NA | 50-500 keV | hard X-rays | NA | 99% confidence | POLAR | [8] |
170101B | $60^{+24}_{-36}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170114A | $10.1^{+10.5}_{-7.4}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170114A | 28$\pm$9c | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [8] |
170114A |  $<$28% | NA | 50-500 keV | hard X-rays | NA | 99% confidence | POLAR | [8] |
Table 2: — continued

GRB | PD | PA |  $\nu$/keV | Energy band | Time | Significance | Instrument | Ref. | z
---|---|---|---|---|---|---|---|---|---
| ($\pi_{\rm obs}\%$) | (∘) |  |  | ($t-t_{0}$) |  |  |  |
170127C | $9.9^{+19.3}_{-8.4}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170127C |  $<$68% | NA | 50-500 keV | hard X-rays | NA | 99% confidence | POLAR | [8] |
170206A | $13.5^{+7.4}_{-8.6}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170206A |  $<$31% | NA | 50-500 keV | hard X-rays | NA | 99% confidence | POLAR | [8] |
170207A | $5.9^{+9.6}_{-5.9}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170210A | $11.4^{+35.7}_{-9.7}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170305A | $40^{+25}_{-25}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170320A | $18^{+32}_{-18}$ | NA | 50-500 keV | hard X-rays | NA | NA | POLAR | [90] | NA
170817A |  $<$12 | NA | 2.8 GHz | Radio | 244days | 99% confidence | VLA | [91] | 0.0093
171010A |  $\sim$40 | variable | 100-300keV | hard X-rays | NA | NA |  $AstroSat$-CZTI | [92] | 0.3285
171205A | 0.27$\pm$0.04 | -71.3$\pm$3.3 | 97.5GHz | Radio | 5.187 days | 5$\sigma$ | ALMA1 | [93] | 0.0368
171205A | 0.30$\pm$0.06 | -67.9$\pm$4.7 | 90.5GHz | Radio | 5.187 days | 5$\sigma$ | ALMA | [93] |
171205A |  $<$0.32 |  $<$-78.1 | 92.4GHz | Radio | 5.187 days | 5$\sigma$ | ALMA | [93] |
171205A | 0.35$\pm$0.08 | -71.3$\pm$5.5 | 102.5GHz | Radio | 5.187 days | 5$\sigma$ | ALMA | [93] |
171205A | 0.31$\pm$0.06 | -58.0$\pm$4.9 | 104.5GHz | Radio | 5.187 days | 5$\sigma$ | ALMA | [93] |
190114C | $0.87\pm 0.13$ | 10$\pm$5 | 97.5$\times$109 | Radio | 2.2-5.2h |  $\geq$5$\sigma$ | ALMA | [94] | 0.425
190114C | $0.60\pm 0.19$ | -44$\pm$12 | 97.5$\times$109 | Radio | 2.2-5.2h |  $\geq$5$\sigma$ | ALMA | [94] |
190114C |  $7.7\pm 1.1$ to 2-4 | NA | NA | Optical | 52-109s | NA | RINGO3 | [95] |
190530A | $55.43\pm 21.30$ | 46.74$\pm$4.0 | 100-300keV | hard X-rays | 0-25s | $\leq 3\sigma$ | AstroSat-CZTI | [96] | 0.9386
190530A | $<64.40$ | NA | 100-300keV | hard X-rays | 7.75-12.25s | 95% confidence | AstroSat-CZTI | [96] |
190530A | $53.95\pm 24.13$ | 48.17$\pm$6.0 | 100-300keV | hard X-rays | 12.25-25.0s | $\leq 3\sigma$ | AstroSat-CZTI | [96] |
190530A | $49.99\pm 21.80$ | 49.61$\pm$6.0 | 100-300keV | hard X-rays | 7.75-25.0s | $\leq 3\sigma$ | AstroSat-CZTI | [96] |
190530A | $<65.29$ | NA | 100-300keV | hard X-rays | 15.0-19.5s | 95% confidence | AstroSat-CZTI | [96] |
191016A | $<13.4$ | NA | V | Optical | 3987-4587s | NA | LT-RINGO3 | [97] | NA
191016A | 5.7$\pm$5.6 | 82$\pm$26.4 | V | Optical | 4587-5187s | NA | LT-RINGO3 | [97] |
191016A | $<10.8$ | NA | V | Optical | 5187-5787s | NA | LT-RINGO3 | [97] |
191016A | $<9.2$ | NA | V | Optical | 5787-6387s | NA | LT-RINGO3 | [97] |
191016A | $<13.5$ | NA | V | Optical | 6387-6987s | NA | LT-RINGO3 | [97] |
191016A | $<16.8$ | NA | V | Optical | 6987-7587s | NA | LT-RINGO3 | [97] |
191016A | $<9.1$ | NA | R | Optical | 3987-4587s | NA | LT-RINGO3 | [97] |
191016A | 11.2$\pm$6.6 | 90.1$\pm$15.4 | R | Optical | 4587-5187s | NA | LT-RINGO3 | [97] |
191016A | $<5.5$ | NA | R | Optical | 5187-5787s | NA | LT-RINGO3 | [97] |
191016A | 6.1$\pm$6.0 | 89.8$\pm$30.5 | R | Optical | 5787-6387s | NA | LT-RINGO3 | [97] |
191016A | $<12.0$ | NA | R | Optical | 6387-6987s | NA | LT-RINGO3 | [97] |
191016A | $<11.0$ | NA | R | Optical | 6987-7587s | NA | LT-RINGO3 | [97] |
191016A | 4.7$\pm$4.1 | 93.16$\pm$22.6 | I | Optical | 3987-4587s | NA | LT-RINGO3 | [97] |
191016A | $<5.2$ | NA | I | Optical | 4587-5187s | NA | LT-RINGO3 | [97] |
191016A | $<14.0$ | NA | I | Optical | 5187-5787s | NA | LT-RINGO3 | [97] |
191016A | 14.6$\pm$7.2 | 100$\pm$12.4 | I | Optical | 5787-6387s | NA | LT-RINGO3 | [97] |
191016A | $<10.7$ | NA | I | Optical | 6387-6987s | NA | LT-RINGO3 | [97] |
191016A | $<17.6$ | NA | I | Optical | 6987-7587s | NA | LT-RINGO3 | [97] |

The “Note”.
(a): A photometric redshift is an estimate for the recession velocity of an
astronomical object such as a galaxy or quasar, made without measuring its
spectrum.
(l): Linear polarization measurements.
(c): Circular polarization measurements.
(1): The Atacama Large Millimeter/submillimeter Array.
(d): The Gamma-Ray Burst Polarimeter (GAP) on board the small solar-power-sail
demonstrator IKAROS.
(e): The purpose-built RINGO2 polarimeter20 on the Liverpool Telescope.
(f): Compton Spectrometer and Imager.

Extended Data Table 2 Polarimetric Observations in Different Segments Segment
observed time $t_{\rm obs}$(s) observed area ($S_{\rm obs}$) $\Gamma$
$\pi_{\rm obs}$ $\pi_{\rm obs}$(t) $\pi_{\rm obs}$($S_{\rm obs},\Gamma$)
$N_{\rm p}$ $N_{\rm patch}$ $N_{\rm obs}$ (in the rest-frame) (on the jet
plane) ($S_{\rm obs}/S_{\rm patch}$) ($S_{\rm jet}/S_{\rm patch}$) ($S_{\rm
jet}/S_{\rm obs}$) I $\lesssim t_{\rm patch}$($0\sim 3$) $\lesssim S_{\rm
patch}$ $>300$ $\lesssim\Pi_{\rm max}$ increase $\propto 1/S_{\rm
obs},\propto\Gamma^{2}$ $\lesssim$1 $\approx$900 $>$900 $\Pi_{\rm max}$
$\approx t_{\rm patch}$($\sim$3) $\approx S_{\rm patch}$ $\approx 300$
$\approx\Pi_{\rm max}$ $\propto t^{-0.50\pm 0.02}$ $\propto 1/S_{\rm
obs},\propto\Gamma^{2}$ $\approx$ 1 $\approx$900 $\approx$900 II $t_{\rm
patch}\lesssim t_{\rm obs}\lesssim t_{\rm jet}$ ($3\sim 3\times 10^{4}$)
$S_{\rm patch}\lesssim S_{\rm obs}\lesssim S_{\rm jet}$ $\approx 50$
$<\Pi_{\rm max}$ $\propto t^{-0.50\pm 0.02}$ $\propto 1/S_{\rm
obs},\propto\Gamma^{2}$ $>$1 $\approx$900 $\approx$25 III $\approx t_{\rm
jet}$($10^{4}\sim 10^{5}$) $\approx S_{\rm jet}$ $\approx 10$ $<\Pi_{\rm max}$
increase $\propto 1/S_{\rm obs},\propto\Gamma^{2}$ $\approx$900 $\approx$900
$\approx$1 IV $>t_{\rm jet}$($>10^{5}$) $>S_{\rm jet}$ $\approx 1$ $<\Pi_{\rm
max}$ $\propto t^{-0.21\pm 0.08}$ $\propto 1/S_{\rm obs},\propto\Gamma^{2}$
$\approx$900 $\approx$900 $\approx$1

Extended Data Table 3 Fit Results for the Optical Afterglow Lightcurve (Jet
Breaks)
GRB $t_{\rm start}$$\sim$$t_{\rm stop}$ Model $\alpha_{1}$ $t_{\rm break}$
$\alpha_{2}$ Is a jet break? ($10^{4}$ s) ($10^{4}$ s) 990510(R)
1.24$\sim$34.02 BKPL 0.88$\pm$0.02 $11.4\pm 1.3$ 2.11$\pm$0.30 Yes 010222(R)
1.31$\sim$212.48 BKPL 0.67$\pm$0.15 $12.3\pm 9.3$ 1.86$\pm$0.48 Likely
020405(R) 8.50$\sim$88.26 SPL … … 1.30$\pm$0.05 No 030328(R) 0.49$\sim$22.75
BKPL 0.54$\pm$0.06 4.81$\pm$0.89 2.20$\pm$0.31 Yes 080928($R_{\rm c}$)
0.02$\sim$23.30 SPL … … 2.21$\pm$0.06 (Late part) Yes

Extended Data Table 4 Results of Inferred $\Gamma_{0}$ GRB
$t_{1}$$\sim$$t_{2}$ $S$ Detectors $\Delta T_{\rm src}$ $[\Delta
T_{\rm(bkg,1)},\Delta T_{\rm(bkg,2)}]$ $F^{\rm obs}_{\gamma}$ $L_{\gamma}$
$\Gamma_{0}$ (s) (s) (s) (erg cm-2 s-1) (erg s-1) 080928 -1$\sim$14.336 11.67
(n3)n6n7b0 (-1 to 14.336) (-20 to -10, 40 to 60) $(6.13\pm 3.98)\times
10^{-8}$ $(1.48\pm 0.96)\times 10^{51}$ 140 090102 -1$\sim$26.624 44.69
n9na(nb)b1 (-1 to 26.624) (-20 to -10, 180 to 200) $(1.34\pm 0.08)\times
10^{-6}$ $(2.19\pm 0.13)\times 10^{52}$ 315 091208B -1$\sim$12.480 32.93
n9(na)b0 (-1 to 12.48) (-20 to -10, 40 to 60) $(6.98\pm 0.17)\times 10^{-7}$
$(4.39\pm 1.07)\times 10^{51}$ 195 100826A -1$\sim$82 11.67 n7(n8)b1 (-1 to
82) (-20 to -10, 200 to 250) $(3.92\pm 0.25)\times 10^{-6}$ $(1.37\pm
0.09)\times 10^{53}$ 546 110301A -1$\sim$5.693 269.92 n7(n8)nbb1 (-1 to 5.693)
(-20 to -10, 40 to 60) $(6.11\pm 0.30)\times 10^{-6}$ $(2.96\pm 0.14)\times
10^{51}$ 173 110721A -1$\sim$21.822 84.67 (n6)n7n9b1 (-1 to 21.822) (-20 to
-10, 40 to 60) $(3.76\pm 0.40)\times 10^{-6}$ $(1.51\pm 0.16)\times 10^{51}$
141 160625B -1$\sim$300 156.58 (n6)n7n9b1 (-1 to 300) (-50 to -20, 80 to 100)
$(3.43\pm 0.35)\times 10^{-6}$ $(2.54\pm 0.26)\times 10^{52}$ 329 160802A
-1$\sim$20.34 151.65 (n2)b0 (-1 to 20.34) (-20 to -10, 60 to 80) $(4.87\pm
0.49)\times 10^{-6}$ $(1.71\pm 0.17)\times 10^{52}$ 293 190114C
-1$\sim$116.354 189.04 n3(n4)b0 (-1 to 116.354) (-20 to -10, 180 to 200)
$(7.52\pm 0.33)\times 10^{-6}$ $(4.96\pm 0.22)\times 10^{51}$ 202