11institutetext: Jun-Jie Wei 22institutetext: Purple Mountain Observatory,
Chinese Academy of Sciences, Nanjing 210023, China 33institutetext: School of
Astronomy and Space Sciences, University of Science and Technology of China,
Hefei 230026, China
33email<EMAIL_ADDRESS>44institutetext: Xue-Feng Wu 55institutetext: Purple
Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China
66institutetext: School of Astronomy and Space Sciences, University of Science
and Technology of China, Hefei 230026, China
66email<EMAIL_ADDRESS>
# Tests of Lorentz Invariance

Jun-Jie Wei and Xue-Feng Wu corresponding authorcorresponding author

###### Abstract

Lorentz invariance is a fundamental symmetry of both Einstein’s theory of
general relativity and quantum field theory. However, deviations from Lorentz
invariance at energies approaching the Planck scale are predicted in many
quantum gravity theories seeking to unify the force of gravity with the other
three fundamental forces of matter. Even though any violations of Lorentz
invariance are expected to be very small at observable energies, they can
increase with energy and accumulate to detectable levels over large distances.
Astrophysical observations involving high-energy emissions and long baselines
can therefore offer exceptionally sensitive tests of Lorentz invariance. With
the extreme features of astrophysical phenomena, it is possible to effectively
search for signatures of Lorentz invariance violations (LIV) in the photon
sector, such as vacuum dispersion, vacuum birefringence, photon decay, and
photon splitting. All of these potential signatures have been studied
carefully using different methods over the past few decades. This chapter
attempts to review the status of our current knowledge and understanding of
LIV, with particular emphasis on a summary of various astrophysical tests that
have been used to set lower limits on the LIV energy scales.

###### Keywords:

Lorentz invariance, Astroparticle physics, Quantum gravity, Relativity and
gravitation

##  1 Introduction

Einstein’s theory of general relativity gives a classical description of
gravity, and the Standard Model of particle physics is a well-tested quantum
theoretical model of particles and all fundamental forces except gravity. They
provide an excellent description of nature together at experimentally
attainable energies. However, at the Planck scale ($E_{\rm Pl}=\sqrt{\hbar
c^{5}/G}\simeq 1.22\times 10^{19}$ GeV), a more fundamental quantum theory of
gravity is required, which would be a giant leap towards unification of all
fundamental forces. A correct and consistent unification scheme has long been
the Holy Grail of modern physics. Several quantum gravity (QG) theories that
attempt to unify general relativity and the standard Model at the Planck scale
predict that Lorentz invariance may be broken at this energy scale, perhaps
due to the underlying quantized nature of spacetime 1989PhRvD..39..683K ;
1991NuPhB.359..545K ; 1995PhRvD..51.3923K ; 2005LRR…..8….5M ;
2005hep.ph….6054B ; 2013LRR….16….5A ; 2014RPPh…77f2901T ; 2021FrPhy..1644300W
. Therefore, the dedicated experimental searches for violations of Lorentz
invariance may help to clear the path to a unification theory. Such
experimental tests obtained from a wide range of systems have been compiled in
Ref. 2011RvMP…83…11K .

Although any deviations from Lorentz invariance are supposed to be very tiny
at attainable energies $\ll E_{\rm Pl}$, they can increase with energy and
over large distances due to cumulative process in particle propagation.
Astrophysical observations involving high energies and long baselines can
therefore provide the most sensitive tests of Lorentz invariance. In the
photon sector, the potential signatures of Lorentz invariance violations (LIV)
include vacuum dispersion, vacuum birefringence, photon decay, photon
splitting, and so on 2008ApJ…689L…1K ; 2001APh….16…97S .

Vacuum dispersion would produce an energy-dependent velocity of light, which
would lead to arrival-time differences between promptly emitted photons with
different energies traveling over astrophysical distances. Astrophysical time-
of-flight measurements can therefore be used to test Lorentz invariance (e.g.,
Refs. 1998Natur.393..763A ; 2006APh….25..402E ; 2008JCAP…01..031J ;
2008ApJ…689L…1K ; 2009PhRvD..80a5020K ; 2009Sci…323.1688A ;
2009Natur.462..331A ; 2012APh….36…47C ; 2012PhRvL.108w1103N ;
2013PhRvD..87l2001V ; 2013APh….43…50E ; 2015PhRvD..92d5016K ;
2015APh….61..108Z ; 2017ApJ…834L..13W ; 2017ApJ…842..115W ; 2017ApJ…851..127W
; 2019PhRvD..99h3009E ; PhysRevLett.125.021301 ; 1999PhRvL..83.2108B ;
Kaaret1999 ). Similarly, the effects of vacuum birefringence would accumulate
over cosmological distances resulting in a measurable rotation of the
polarization plane of linearly polarized photons as a function of energy.
Thus, Lorentz invariance can also be tested with astrophysical polarization
measurements (e.g., Refs. 1990PhRvD..41.1231C ; 1998PhRvD..58k6002C ;
2001PhRvD..64h3007G ; 2001PhRvL..87y1304K ; 2006PhRvL..97n0401K ;
2007PhRvL..99a1601K ; 2013PhRvL.110t1601K ; 2003Natur.426Q.139M ;
2004PhRvL..93b1101J ; 2007MNRAS.376.1857F ; 2009JCAP…08..021G ;
2011PhRvD..83l1301L ; 2011APh….35…95S ; 2012PhRvL.109x1104T ;
2013MNRAS.431.3550G ; 2014MNRAS.444.2776G ; 2016MNRAS.463..375L ;
2017PhRvD..95h3013K ; 2019PhRvD..99c5045F ; 2019MNRAS.485.2401W ). Since time-
of-flight measurements are less sensitive than polarization measurements by a
factor $\propto 1/E$, where $E$ is the energy of the light, time-of-flight
measurements typically place less stringent limits on LIV 2009PhRvD..80a5020K
. However, numerous predicted signals of LIV have no vacuum birefringence, so
constraints from time-of-flight measurements are indispensable in a broad-
based search for nonbirefringent Lorentz-violating effects.

Besides the measurements of time-of-flight and polarization, astrophysical
spectrum measurements can also be used to test Lorentz invariance (e.g., Refs.
2001APh….16…97S ; 2017PhRvD..95f3001M ; 2019JCAP…04..054A ; 2019EPJC…79.1011S
; 2020PhRvL.124m1101A ; 2021arXiv210507927C ; 2021arXiv210612350T ). This is
because superluminal LIV allows photons to decay at high energies. Photons
decay into electron-positron pairs proceed over very short distances
(centimeters or less) once above the energy threshold of the decay process,
which would result in a hard cutoff in the gamma-ray spectra of astrophysical
sources 2017PhRvD..95f3001M . Another superluminal LIV decay process, photon
splitting into multiple photons, also predicts a cutoff at the highest energy
part of the photon spectra of astrophysical sources 2019JCAP…04..054A ;
2019EPJC…79.1011S . Experimental searches for LIV-induced spectral cutoff have
therefore been carried out in the observed spectra of several astrophysical
sources 2017PhRvD..95f3001M ; 2019JCAP…04..054A ; 2019EPJC…79.1011S ;
2020PhRvL.124m1101A ; 2021arXiv210507927C ; 2021arXiv210612350T .

In this chapter, we summarize the current status on astrophysical tests of
Lorentz invariance and attempt to chart the future of the subject. We start by
reviewing the recent achievements in sensitivity of vacuum dispersion time-of-
flight measurements. After that, we overview the progresses on LIV tests
through the effects of vacuum birefringence, photon decay, and photon
splitting. Finally, we present a summary and future prospect.

##  2 Vacuum dispersion

###  2.1 Modified photon dispersion relation

Compared to the standard energy-momentum relationship in Einstein’s special
relativity, the introduction of LIV can induce modifications to the photon
dispersion relation in vacuum, which can be described using a Taylor series
1998Natur.393..763A :

$E^{2}\simeq p^{2}c^{2}\left(1\pm\sum_{n=1}^{\infty}a_{n}\right)\;,$ (1)

where $E$ and $p$ are the energy and momentum of the photon, respectively, $c$
is the Lorentz-invariant speed of light, and $a_{n}=(E/E_{{\rm QG},n})^{n}$ is
a dimensionless expansion coefficient. Also, $E_{{\rm QG},n}$ is the
hypothetical energy scale at which QG effects become significant. At small
energies $E\ll E_{{\rm QG},n}$, the sum is dominated by the lowest-order term
of the series. Considering the sensitivity of current detectors, only the
first two leading terms ($n=1$ or $n=2$) are of interest for independently
experimental searches. They are often referred to as linear and quadratic
energy-dependent modifications, respectively. The energy-dependent photon
group velocity can therefore be derived as

$\upsilon(E)=\frac{\partial E}{\partial p}\approx
c\left[1\pm\frac{1+n}{2}\left(\frac{E}{E_{{\rm QG},n}}\right)^{n}\right]\;,$
(2)

where the sign $\pm$ allows for the “superluminal” ($+$) or “subluminal” ($-$)
scenarios.

Owing to the energy dependence of $\upsilon(E)$, two photons with different
energies radiated simultaneously from the same source would travel at
different speeds, thus arriving at the observer at different times. For
example, in the subluminal case (where high-energy photons are slower), the
photon with a higher energy (denoted by $E_{h}$) would arrive with respect to
the photon with a lower energy ($E_{l}$) by a time delay 2008JCAP…01..031J

$\Delta t_{\rm
LIV}=t_{h}-t_{l}=\frac{(1+n)}{2H_{0}}\frac{E_{h}^{n}-E_{l}^{n}}{E_{{\rm
QG},n}^{n}}\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{\rm
m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}dz^{\prime}\;,$ (3)

where $t_{h}$ and $t_{l}$ represent the observed arrival times of the high-
and low-energy photons, respectively, and $z$ is the redshift of the source.
Here $H_{0}$, $\Omega_{\rm m}$, and $\Omega_{\Lambda}$ are cosmological
parameters of the standard $\Lambda$CDM model.

In principle, Lorentz invariance can be tested using the observed time delays
of photons with different energies arising in astronomical sources. However,
it is difficult to exclude the possibility that the energy-dependent delays
are of pure astrophysical origins. In order to prove the existence of LIV, one
needs to systematically collect numerous delay events and demonstrate that
they all match the prediction of LIV (Eq. (3)).

On the contrary, it is relatively simple to place limits on the non-existence
of LIV at a certain energy scale. A smaller-than-expected observed delay of
high-energy photons with respect to low-energy photons can be used to discard
a given model. More specifically, a conservative lower limit on the QG energy
scale $E_{\rm QG}$ can be placed under the assumption that the observed time
delay is mainly contributed by the LIV effects.

###  2.2 Present constraints from time-of-flight measurements

It is obvious from Eq. (3) that a potential LIV-induced time delay is
predicted to be an increasing function of the photon energy and the source
distance. Additionally, a short timescale of the signal variability provides a
natural reference time to measure the energy-dependent delay more easily. The
greatest sensitivities on $E_{{\rm QG},n}$ can therefore be expected from
those violent astrophysical phenomena with high-energy emissions, large
distances, and short time delays. The first attempt to constrain LIV by
exploiting the rapid variations of gamma-ray emissions from astrophysical
sources at cosmological distances was presented in Ref. 1998Natur.393..763A .
As of now, transient or variable sources such as gamma-ray bursts (GRBs),
active galactic nuclei (AGNs), and pulsars form a group of excellent
candidates for searching for the LIV-induced vacuum dispersion.

#### Gamma-ray bursts

GRBs are among the most distant gamma-ray transients in the Universe and their
prompt emission light curves vary on subsecond timescales. As such, GRBs are
ideal probes that one can use to perform LIV tests. There have been lots of
LIV studies by applying various analysis techniques on the time-of-flight
measurements of GRBs.

Some of the pre-Fermi constraints are those by Ref. Ellis2003  using BATSE
GRBs; by Ref. 2004ApJ…611L..77B  using RHESSI observations of GRB 021206; by
Ref. 2008ApJ…676..532B  using HETE-2 GRBs; by Ref. 2008GReGr..40.1731L  using
INTEGRAL GRBs; by Ref. 2006JCAP…05..017R  using Konus-Wind and Swift
observations of GRB 051221A; by Refs. 2006APh….25..402E ; 2008APh….29..158E
using BATSE, HETE-2, and Swift GRBs; and by Ref. 2017ApJ…851..127W  using
Swift GRBs. Much more stringent constraints on LIV, however, have been
obtained using Fermi observations, entirely thanks to the unprecedented
sensitivity for detecting GRB prompt MeV/GeV emission by the Large Area
Telescope (LAT) onboard the Fermi satellite. These tighter constraints include
those by the Fermi Gamma-Ray Burst Monitor (GBM) and LAT Collaborations using
GRBs 080916C 2009Sci…323.1688A  and 090510 2009Natur.462..331A ; by Ref.
2009PhRvD..80k6005X  using GRB 090510; and by Refs. 2010APh….33..312S ;
2012APh….36…47C ; 2012PhRvL.108w1103N ; 2013PhRvD..87l2001V ;
2019PhRvD..99h3009E  using multiple Fermi GRBs. Remarkably, Ref.
2009Natur.462..331A  studied the time-of-flight effect by analyzing the
arrival time lag between the highest energy (31 GeV) photon and low-energy
photons from GRB 090510. The burst was localized at $z=0.903$. Fermi/LAT
detected this 31 GeV photon 0.829 s after the GBM trigger. Assuming the 31 GeV
photon was emitted at the GBM trigger time, the time delay between the highest
energy photon and low-energy (trigger) photons is then $\Delta t_{\rm
LIV}<829$ ms. This gives the most conservative constraint on the linear LIV
energy scale $E_{{\rm QG},1}>1.19E_{\rm Pl}$. If the 31 GeV photon is assumed
to be associated with the contemporaneous $<1$ MeV spike, one therefore has
$\Delta t_{\rm LIV}<10$ ms, which gives the most radical constraint $E_{{\rm
QG},1}>102E_{\rm Pl}$. These results obviously disfavor the linear LIV models
requiring $E_{{\rm QG},1}\leq E_{\rm Pl}$, and are in good agreement with
Lorentz invariance. Subsequently, Ref. 2013PhRvD..87l2001V  adopted three
different techniques to constrain the degree of dispersion observed in the
data of four Fermi/LAT GRBs. For the subluminal case, their constraints from
GRB 090510, namely $E_{{\rm QG},1}>7.6E_{\rm Pl}$ and $E_{{\rm
QG},2}>1.3\times 10^{11}~{}\mathrm{GeV}$ for linear and quadratic leading-
order LIV-induced vacuum dispersion, further improved previous results by a
factor of $\sim 2$. On 2019 January 14, the Major Atmospheric Gamma Imaging
Cherenkov (MAGIC) telescopes discovered a gamma-ray signal above 0.2 TeV from
GRB 190114C, recording the most energetic photons ever detected from a GRB
2019Natur.575..455M . The MAGIC Collaboration used this unique observation to
test the dependence of the speed of light in vacuum on its energy
PhysRevLett.125.021301 . They obtained competitive lower limits on the
quadratic LIV energy scale, namely $E_{{\rm QG},2}>6.3\times
10^{10}~{}\mathrm{GeV}$ ($E_{{\rm QG},2}>5.6\times 10^{10}~{}\mathrm{GeV}$)
for the subluminal (superluminal) case.

Generally, Lorentz invariance has been tested with high accuracy using the
spectral lags111Spectral lag is defined as the arrival time delay between
correlated photons with different energies (or between light curves in
different energy bands). of GRBs. A key challenge in such time-of-flight
tests, however, is to distinguish an intrinsic time lag at the source from a
delay induced by LIV. Possible source-intrinsic effects caused by the unknown
emission mechanism could cancel-out or enhance the LIV-induced time delay,
which would impact the reliability of the resulting constraints on LIV.

Any effects of LIV would increase with the redshift of the source (see Eq.
(3)), whereas source-intrinsic effects might be independent of the redshift.
Therefore, Refs. Ellis2003 ; 2006APh….25..402E ; 2008APh….29..158E  proposed
working on a statistical sample of GRBs with different redshifts to
disentangle the intrinsic time lag problem. For each GRB, Ref.
2006APh….25..402E  extracted the spectral lag in the light curves recorded in
the selected observer-frame energy bands 25–55 and 115–320 keV. To account for
the unknown intrinsic time lag, they fitted the observed arrival time delays
of 35 GRBs with the inclusion of a term $b_{\rm sf}$ specified in the GRB
source frame. That is, the observed spectral lags are fitted by two
components, $\Delta t_{\rm obs}=\Delta t_{\rm LIV}+b_{\rm sf}(1+z)$,
reflecting the contributions from both the LIV and source-intrinsic effects
2006APh….25..402E . Such parametrization allows one to derive a simple linear
fitting function:

$\frac{\Delta t_{\rm obs}}{1+z}=a_{\rm LV}K+b_{\rm sf}\;,$ (4)

where

$K=\frac{1}{(1+z)}\int_{0}^{z}\frac{(1+z^{\prime})dz^{\prime}}{h(z^{\prime})}$
(5)

is a non-linear redshift function which is related to the specific
cosmological model, the slope $a_{\rm LV}=\Delta E/(H_{0}E_{\rm QG})$ is
connected to the energy scale of LIV, and the intercept $b_{\rm sf}$ stands
for the unknown intrinsic time lag inherited from the sources. Also,
$h(z)=\sqrt{\Omega_{\rm m}(1+z)^{3}+\Omega_{\Lambda}}$ is the dimensionless
expansion rate at $z$, where the standard flat $\Lambda$CDM model with
parameters $\Omega_{\rm m}=0.3$ and $\Omega_{\Lambda}=1-\Omega_{\rm m}$ is
adopted. Note that the linear LIV correction ($n=1$) for the subluminal case
($-$) was considered in this work. Ref. 2008APh….29..158E  compiled the
rescaled time lags $\Delta t_{\rm obs}/(1+z)$ extracted from 35 light curve
pairs as functions of the variable $K$ and performed a linear regression
analysis of the available data. The linear fit corresponds to $\frac{\Delta
t_{\rm obs}}{1+z}=(0.0068\pm 0.0067)K-(0.0065\pm 0.0046)$. With the
corresponding upper limit on the slope parameter $a_{\rm LV}$, as well as the
energy difference between two observer-frame energy bands $\Delta E$, the 95%
confidence-level lower limit on the linear LIV energy scale is $E_{\rm QG}\geq
1.4\times 10^{16}$ GeV 2008APh….29..158E . Going beyond the $\Lambda$CDM
cosmology, Refs. 2009CQGra..26l5007B ; 2015ApJ…808…78P  extended this analysis
to other different cosmological models, finding the result is only weakly
sensitive to the background cosmology. Some cosmology-independent techniques
were furthermore applied to this kind of analysis 2018PhLB..776..284Z ;
2020ApJ…890..169P .

It is worth pointing out that Ref. 2006APh….25..402E  extracted the spectral
lag of each GRB in the light curves between two fixed observer-frame energy
bands. However, due to the fact that different GRBs have different redshift
measurements, these two observer-frame energy bands actually correspond to a
different pair of energy bands in the GRB source frame 2012MNRAS.419..614U ,
thus potentially introducing an artificial energy dependence to the extracted
spectral lag and/or systematic uncertainties to the search for LIV-induced
lags.222For those cosmological sources with various redshifts, their observer-
frame quantities can be quite different than the corresponding source-frame
ones. However, note that if we focus on the observer-frame quantities of
individual cosmological sources, there is no such a problem. This problem can
be solved by choosing two appropriate energy bands fixed in the GRB source
frame and estimating the observed time lag for two projected observer-frame
energy bands by the relation $E_{\rm observer}=E_{\rm source}/(1+z)$, Ref.
2012MNRAS.419..614U  showed that the correlation between observer-frame lags
and source-frame lags for the same GRB sample has a large scatter, supporting
that the source-frame lag can not be directly represented by the observer-
frame lag. Therefore, when analyzing the LIV effects in a large sample of
cosmological sources with different redshifts, it would be good to use the
spectral lags extracted in the source frame. Ref. 2015MNRAS.446.1129B
investigated the source-frame spectral lags of 56 GRBs observed by Swift/Burst
Alert Telescope (BAT). This sample has redshifts ranging from 0.35 (GRB
061021) to 5.47 (GRB 060927). For each GRB, they extracted light curves for
two observer-frame energy bands corresponding to the fixed source-frame energy
bands 100–150 and 200–250 keV. These two specific source-frame energy bands
were chosen so that after transforming to the observer frame (based on the
redshift of each GRB, i.e., $[100-150]/(1+z)$ and $[200-250]/(1+z)$ keV) they
still lie in the detectable energy range of Swift/BAT. Finally, for each
light-curve pairs, Ref. 2015MNRAS.446.1129B  used the improved cross-
correlation function analysis technology to estimate the spectral lag. Note
that the energy difference between the mid-points of the two source-frame
energy bands is fixed at 100 keV, whereas in the observer frame (as expected),
the energy difference varies depending on the redshift of each GRB. For
example, the energy difference is 16 keV in GRB 060927 and 74 keV in GRB
061021. This is in contrast to the spectral lag extractions performed in the
observer frame, where the energy difference is treated as a constant
2012MNRAS.419..614U . Ref. 2017ApJ…851..127W  first took advantage of the
source-frame spectral lags of 56 GRBs presented in Ref. 2015MNRAS.446.1129B
to study the LIV effects. For the subluminal case, the linear LIV-induced time
delay between two observer-frame energy bands with the energy difference
$\Delta E$ is given by

$\Delta t_{\rm LIV}=\frac{\Delta E}{H_{0}E_{\rm
QG}}\int_{0}^{z}\frac{(1+z^{\prime}){\rm d}z^{\prime}}{h(z^{\prime})}\\\
=\frac{\Delta E^{\prime}/(1+z)}{H_{0}E_{\rm
QG}}\int_{0}^{z}\frac{(1+z^{\prime}){\rm d}z^{\prime}}{h(z^{\prime})}\;,$ (6)

where $\Delta E^{\prime}=100~{}\mathrm{keV}$ is the source-frame energy
difference. Similar to the treatment of Ref. 2006APh….25..402E , one can
formulate the intrinsic time delay problem in terms of linear regression:

$\frac{\Delta t_{\rm src}}{1+z}=a^{\prime}_{\rm LV}K^{\prime}+b_{\rm sf}\;,$
(7)

where $\Delta t_{\rm src}$ is the extracted spectral lag for two fixed source-
frame energy bands 100–150 and 200–250 keV,

$K^{\prime}=\frac{1}{(1+z)^{2}}\int_{0}^{z}\frac{(1+z^{\prime})dz^{\prime}}{h(z^{\prime})}$
(8)

is a function of the redshift, and $a^{\prime}_{\rm LV}=\Delta
E^{\prime}/(H_{0}E_{\rm QG})$ is the slope in $K^{\prime}$. In order to probe
the LIV effects, Ref. 2017ApJ…851..127W  performed a linear fit to the $\Delta
t_{\rm src}/(1+z)$ versus $K^{\prime}(z)$ data. With the optimized
$a^{\prime}_{\rm LV}$ and the source-frame energy difference $\Delta
E^{\prime}$, the 95% confidence-level lower limit on the linear LIV energy
scale is $E_{\rm QG}\geq 2.0\times 10^{14}$ GeV 2017ApJ…851..127W . This is a
step forward in the investigation of LIV effects, since all previous studies
used arbitrary observer-frame energy bands.

Another point to note is that in the treatment of Ref. 2006APh….25..402E  an
unknown constant was assumed to be the intrinsic time lag in the linear
fitting function (see Eq. 4). That is, their work based on the assumption that
all GRBs have the same intrinsic time delay. However, since the durations of
GRBs span about five or six orders of magnitude, it is unlikely that high-
energy photons emitted from different GRBs (or from the same GRB) have the
same intrinsic time lag relative to the emission time of low-energy photons
2016ChPhC..40d5102C . As an improvement, Ref. 2012APh….36…47C  estimated the
intrinsic time delay between high- and low-energy photons emitted from each
GRB by using the magnetic jet model. However, the magnetic jet model relies on
some given theoretical parameters, and this introduces uncertainties on the
LIV results.

Figure 1: Multi-band light curves of the second sub-burst of GRB 160625B. The
top thick line represents the full-range (10–20000 keV) light curve. Figure
taken from 2017ApJ…834L..13W .

In 2017, Ref. 2017ApJ…834L..13W  used multiple spectral lags from GRB 160625B
to constrain LIV in the photon sector. GRB 160625B was detected by the Fermi
satellite on 2016 June 25, consisting of three different isolated sub-bursts
with unusually high photon statistics. Since the second sub-burst of GRB
160625B is especially bright, it is easy to extract its high-quality light
curves in different energy bands (see Figure 1). With multi-photon energy
bands, Ref. 2017ApJ…834L..13W  calculated the spectral lags in the light
curves recorded in the lowest-energy band (10–12 keV) relative to any other
light curves with higher-energy bands, finding that the observed spectral lag
$\Delta t_{\rm obs}$ increases at $E\leq 8$ MeV and then gradually decreases
in the energy range $8\,{\rm MeV}\leq E\leq 20$ MeV. The lag behavior is very
peculiar, being an obvious transition from positive to negative spectral lags
discovered within a burst (see Figure 2). Conventionally, a positive spectral
lag means an earlier arrival time for photons with higher energies, whereas a
negative spectral lag corresponds to a delayed arrival of the higher-energy
photons. Ref. 2017ApJ…834L..13W  proposed that the spectral-lag transition of
GRB 160625B provides a great opportunity to distinguish LIV-induced
propagation effects from any source-intrinsic time delay in the emission of
photons at different energy bands. Since the time delay $\Delta t_{\rm LIV}$
caused by LIV may probably be accompanied by an intrinsic energy-dependent
time delay $\Delta t_{\rm int}$ arise from the unknown emission mechanism of
the source, the observed spectral lag should be comprised of two
contributions,

$\Delta t_{\rm obs}=\Delta t_{\rm int}+\Delta t_{\rm LIV}\;.$ (9)

As the observed spectral lags of most GRBs have a positive energy dependence
(i.e., high-energy photons arrive earlier than low-energy ones; see
2017ApJ…844..126S ; 2018ApJ…865..153L ), Ref. 2017ApJ…834L..13W  suggested
that the observer-frame correlation between the intrinsic time lag and the
energy $E$ can be expressed approximately by a power-law with positive
dependence,

$\Delta t_{\rm int}(E)=\tau\left[\left(\frac{E}{\rm
keV}\right)^{\alpha}-\left(\frac{E_{0}}{\rm keV}\right)^{\alpha}\right]\;{\rm
s}\;,$ (10)

where $\tau>0$ and $\alpha>0$, and $E_{0}=11.34$ keV is the median value of
the reference lowest-energy band (10–12 keV). Also, when the subluminal case
($-$) is considered, the LIV effects predict a negative spectral lag (i.e.,
high-energy photons travel slower than low-energy ones). As the Lorentz-
violating term takes the lead at higher energies, the positive correlation of
the lag with energy would gradually become an anti-correlation. The
contributions from both the intrinsic energy-dependent time lag and the LIV-
induced time delay can therefore result in the observed lag behavior with an
apparent transition from positive to negative lags 2017ApJ…834L..13W . By
directly fitting the spectral lag behavior of GRB 160625B, Ref.
2017ApJ…834L..13W  obtained both comparatively robust limits on the QG energy
scales and a reasonable formulation of the intrinsic energy-dependent time
lag. The best-fit theoretical curves for the linear and quadratic LIV models
are shown in Figure 2. The $1\sigma$ confidence-level lower limits on the
linear and quadratic LIV energy scales are $E_{{\rm QG},1}>0.5\times
10^{16}~{}\mathrm{GeV}$ and $E_{{\rm QG},2}>1.4\times 10^{7}~{}\mathrm{GeV}$,
respectively. The multi-photon spectral-lag data of GRB 160625B has also been
used to constrain the Lorentz-violating coefficients of the Standard Model
Extension 2017ApJ…842..115W . Recently, Ref. 2021ApJ…906….8D  showed decisive
evidence that GRB 190114C is an additional burst having a well-defined
transition from positive to negative lags, providing another opportunity not
only to disentangle the intrinsic time lag problem but also to place robust
constraints on LIV. Although the spectral-lag transitions of GRB 160625B and
GRB 190114C do not currently have the best sensitivity to LIV constraints, the
presented method, when applied to future bright short GRBs with similar lag
features, may derive more stringent constraints on LIV.

Figure 2: Energy dependence of the observed spectral lag $\Delta t_{\rm obs}$
of GRB 160625B relative to the softest energy band. The best-fit theoretical
curves correspond to the linear ($n=1$) and quadratic ($n=2$) LIV models,
respectively. Figure taken from 2017ApJ…834L..13W .

#### Active galactic nuclei

Because of their rapid flux variability (ranging from minutes to years), very-
high-energy (VHE, $E\geq 100~{}\mathrm{GeV}$) emissions, and large distances,
flaring AGNs have also been considered as promising sources for LIV studies.
It should be underlined that searching for LIV-induced vacuum dispersions
using both GRBs and VHE flares of AGNs is of great fundamental interest. GRBs
can be detected up to very high redshifts ($z\sim 8$), but with very limited
statistics of photons above a few tens of GeV. Conversely, AGN flares can be
observed by ground-based detectors with large statistics of photons up to a
few tens of TeV. Due to the absorption of VHE gamma-rays by extragalactic
background light, TeV measurements are limited to sources with relatively low
redshifts. Thus, GRBs and AGNs are complementary to each other in testing LIV,
and they enable us to test different energy and redshift ranges.

There have been some competitive constraints on LIV-induced vacuum dispersions
using VHE gamma-ray observations of bright AGN flares, including the Whipple
analysis of the flare of Mrk 421 1999PhRvL..83.2108B , the H.E.S.S. analyses
of the flares of PKS 2155-304 2008PhRvL.101q0402A ; 2011APh….34..738H  and PG
1553+113 2015ApJ…802…65A , and the MAGIC and H.E.S.S. analyses of the flares
of Mrk 501 2008PhLB..668..253M ; 2009APh….31..226M ; 2019ApJ…870…93A . For the
linear LIV effect considering a subluminal case, the limit obtained from the
PKS 2155-304 flare data by H.E.S.S. is the best time-of-flight limit obtained
with an AGN, yielded $E_{{\rm QG},1}>2.1\times 10^{18}~{}\mathrm{GeV}$
2011APh….34..738H . For the quadratic LIV effect, the limits obtained from
H.E.S.S.’s observation of the Mrk 501 flare are the most stringent constraints
ever obtained with an AGN, namely $E_{{\rm QG},2}>8.5\times
10^{10}~{}\mathrm{GeV}$ ($E_{{\rm QG},2}>7.3\times 10^{10}~{}\mathrm{GeV}$)
for the subluminal (superluminal) case 2019ApJ…870…93A .

#### Pulsars

Gamma-ray pulsars are a third class of astrophysical sources that are often
used for LIV time-of-flight tests. There are several reasons why they are a
tempting target for these types of studies. First, due to the stable and
cyclical nature of pulsar emission, sensitivity to LIV can be systematically
planned and improved by observing longer. Second, as the timing of the pulsar
is widely studied throughout the electromagnetic spectrum, energy-dependent
time delays caused by propagation effects can be more easily distinguished
from source-intrinsic effects. Moreover, although being observed several
orders of magnitude closer than GRBs or AGNs, the detection of VHE emission
from pulsars will compensate for a short distance when it comes to testing the
quadratic LIV term.

First limits on LIV using gamma-ray emission from the galactic Crab pulsar
were obtained from the observation of CGRO/EGRET at energies above 2 GeV
Kaaret1999 , and then improved by VERITAS using VHE gamma-rays reaching up to
120 GeV 2011ICRC….7..256O ; 2013ICRC…33.2768Z . Remarkably, Ref.
2017ApJS..232….9M  used the Crab pulsar emission observed up to TeV energies
by MAGIC to put new constraints on the energy scale $E_{\rm QG}$. The 95%
confidence-level lower limits for the subluminal (superluminal) case are
$E_{{\rm QG},1}>5.5\times 10^{17}~{}\mathrm{GeV}$ ($E_{{\rm QG},1}>4.5\times
10^{17}~{}\mathrm{GeV}$) and $E_{{\rm QG},2}>5.9\times 10^{10}~{}\mathrm{GeV}$
($E_{{\rm QG},2}>5.3\times 10^{10}~{}\mathrm{GeV}$) for linear and quadratic
LIV effects, respectively 2017ApJS..232….9M .

Table 1: A census of lower bounds on the energy scale $E_{\rm QG}$ for linear
($n=1$) and quadratic ($n=2$) LIV for the subluminal ($-$) and superluminal
($+$) cases. These limits were produced using the time-of-flight measurements
of various astrophysical sources.

Source(s) | Instrument | Technique |  $E_{{\rm QG},1}$ [GeV] |  $E_{{\rm QG},2}$ [GeV] | Refs.
---|---|---|---|---|---
GRB
9 GRBsa | BATSE+OSSE | Wavelets |  ($-$) $6.9\times 10^{15}$ |  ($-$) $2.9\times 10^{6}$ | Ellis2003
GRB 021206b | RHESSI | Peak times at different energies |  ($-$) $1.8\times 10^{17}$ |  | 2004ApJ…611L..77B
15 GRBs | HETE-2 | Wavelets |  ($-$) $2.0\times 10^{15}$ |  | 2008ApJ…676..532B
11 GRBs | INTEGRAL | Likelihood |  ($-$) $3.2\times 10^{11}$ |  | 2008GReGr..40.1731L
GRB 051221A | Konus-Wind+Swift | Peak times at different energies |  ($-$) $6.6\times 10^{16}$ |  ($-$) $6.2\times 10^{6}$ | 2006JCAP…05..017R
35 GRBsc | BATSE+HETE-2 | Observer-frame spectral lags | ($-$) $1.4\times 10^{16}$ |  | 2006APh….25..402E ; 2008APh….29..158E
+Swift |
56 GRBs | Swift | Source-frame spectral lags |  ($-$) $2.0\times 10^{14}$ |  | 2017ApJ…851..127W
GRB 080916C | Fermi GBM+LAT | Associating a 13.2 GeV photon with | ($-$) $1.3\times 10^{18}$ |  | 2009Sci…323.1688A
the trigger time |
GRB 090510 | Fermi GBM+LAT | Associating a 31 GeV photon with | ($-$) $1.5\times 10^{19}$ |  | 2009Natur.462..331A
the start of the first GBM pulse |
GRB 090510 | Fermi/LAT | PairView+Likelihood |  ($-$) $9.3\times 10^{19}$ |  ($-$) $1.3\times 10^{11}$ | 2013PhRvD..87l2001V
+Sharpness-Maximization Method |  ($+$) $1.3\times 10^{20}$ |  ($+$) $9.4\times 10^{10}$
GRB 190114C | MAGIC | Likelihood |  ($-$) $5.8\times 10^{18}$ |  ($-$) $6.3\times 10^{10}$ | PhysRevLett.125.021301
($+$) $5.5\times 10^{18}$ |  ($+$) $5.6\times 10^{10}$
GRB 160625B | Fermi/GBM | Spectral-lag transition |  ($-$) $0.5\times 10^{16}$ |  ($-$) $1.4\times 10^{7}$ | 2017ApJ…834L..13W
GRB 190114C | Fermi/GBM | Spectral-lag transition |  ($-$) $2.2\times 10^{14}$ |  ($-$) $0.9\times 10^{6}$ | 2021ApJ…906….8D
AGN
Mrk 421 | Whipple | Binning |  ($-$) $4.0\times 10^{16}$ |  | 1999PhRvL..83.2108B
PKS 2155-304 | H.E.S.S. | Modified cross correlation function |  ($-$) $7.2\times 10^{17}$ |  ($-$) $1.4\times 10^{9}$ | 2008PhRvL.101q0402A
PKS 2155-304 | H.E.S.S. | Likelihood |  ($-$) $2.1\times 10^{18}$ |  ($-$) $6.4\times 10^{10}$ | 2011APh….34..738H
PG 1553+113 | H.E.S.S. | Likelihood |  ($-$) $4.1\times 10^{17}$ |  ($-$) $2.1\times 10^{10}$ | 2015ApJ…802…65A
($+$) $2.8\times 10^{17}$ |  ($+$) $1.7\times 10^{10}$
Mrk 501 | MAGIC | Energy cost function |  ($-$) $2.1\times 10^{17}$ |  ($-$) $2.6\times 10^{10}$ | 2008PhLB..668..253M
Mrk 501 | MAGIC | Likelihood |  ($-$) $3.0\times 10^{17}$ |  ($-$) $5.7\times 10^{10}$ | 2009APh….31..226M
Mrk 501 | H.E.S.S. | Likelihood |  ($-$) $3.6\times 10^{17}$ |  ($-$) $8.5\times 10^{10}$ | 2019ApJ…870…93A
($+$) $2.6\times 10^{17}$ |  ($+$) $7.3\times 10^{10}$
Pulsar
Crab | CGRO/EGRET | Pulse arrival times | ($-$) $1.8\times 10^{15}$ |  | Kaaret1999
in different energy bands |
Crab | VERITAS | Likelihood |  ($-$) $3.0\times 10^{17}$ |  ($-$) $7.0\times 10^{9}$ | 2011ICRC….7..256O
Crab | VERITAS | Dispersion Cancellation |  ($-$) $1.9\times 10^{17}$ |  | 2013ICRC…33.2768Z
($+$) $1.7\times 10^{17}$ |
Crab | MAGIC | Likelihood |  ($-$) $5.5\times 10^{17}$ |  ($-$) $5.9\times 10^{10}$ | 2017ApJS..232….9M
($+$) $4.5\times 10^{17}$ |  ($+$) $5.3\times 10^{10}$

aLimit obtained taking no account of the factor $(1+z)$ in the integral of Eq.
(5).
bThe spectral and temporal properties of GRB 021206 were used to estimate the
pseudo redshift.
cThe Limits of Ref. 2006APh….25..402E  were corrected in Ref.
2008APh….29..158E  by taking account of the factor $(1+z)$ in the integral of
Eq. (5).

The most important results obtained to date with vacuum dispersion time-of-
flight measurements of various astrophysical events are listed in Table 1. The
most stringent limits so far on $E_{\rm QG}$ have been obtained with
Fermi/LAT’s observation of GRB 090510. The limits set are $E_{{\rm
QG},1}>9.3\times 10^{19}~{}\mathrm{GeV}$ ($E_{{\rm QG},1}>1.3\times
10^{20}~{}\mathrm{GeV}$) for a linear, and $E_{{\rm QG},2}>1.3\times
10^{11}~{}\mathrm{GeV}$ ($E_{{\rm QG},2}>9.4\times 10^{10}~{}\mathrm{GeV}$)
for a quadratic LIV, for the subluminal (superluminal) case
2013PhRvD..87l2001V . Obviously, these time-of-flight studies using gamma-ray
emissions in the GeV–TeV range provide us at present with the best opportunity
to search for Planck-scale modifications of the photon dispersion relation.
Nevertheless, while they offer meaningful constraints for the linear ($n=1$)
modification, they are still much weaker for deviations that appear at the
quadratic ($n=2$) order.

##  3 Vacuum birefringence

###  3.1 General formulae

The Charge-Parity-Time (CPT) theorem states that the physical laws are
invariant under charge conjugation, parity transformation, and time reversal.
In some QG theories that invoke LIV, the CPT theorem no longer holds. In the
absence of Lorentz symmetry, the CPT invariance, if needed, should be imposed
as an additional assumption of the theory. In the photon sector, these
theories invoke a Lorentz- and CPT-violating dispersion relation of the form
2003PhRvL..90u1601M

$E_{\pm}^{2}=p^{2}c^{2}\pm\frac{2\eta}{E_{\rm pl}}p^{3}c^{3}\;,$ (11)

where the sign $\pm$ corresponds to the helicity, i.e., two different circular
polarization states, and $\eta$ is a dimensionless parameter that needs to be
bound. Note that the parameter $\eta$ exactly vanishes in LIV but CPT
invariant theories. Therefore, in this sense, such tests might be less broad
and general than those tests based on vacuum dispersion.

The linear polarization can be decomposed into right- or left-handed circular
polarization states. If $\eta\neq 0$, then the dispersion relation (11)
indicates that photons with opposite circular polarizations have slightly
different phase velocities and therefore propagate with different speeds,
leading to an energy-dependent rotation of the polarization vector of a
linearly polarized light. This effect is known as vacuum birefringence. The
rotation angle during the propagation from the source at redshift $z$ to the
observer is expressed as 2011PhRvD..83l1301L ; 2012PhRvL.109x1104T

$\Delta\phi_{\rm LIV}(E)\simeq\eta\frac{E^{2}F(z)}{\hbar E_{\rm pl}H_{0}}\;,$
(12)

where $E$ is the observed photon energy, and $F(z)$ is a function related to
the cosmological model, which reads (in the standard flat $\Lambda$CDM model)

$F(z)=\int_{0}^{z}\frac{\left(1+z^{\prime}\right)dz^{\prime}}{\sqrt{\Omega_{\rm
m}\left(1+z^{\prime}\right)^{3}+\Omega_{\Lambda}}}\;.$ (13)

Generally, it is hard to know the intrinsic polarization angles for photons
emitted with different energies from a given astrophysical source. If one
possessed this information, evidence for vacuum birefringence (i.e., an
energy-dependent rotation of the polarization plane of linearly polarized
photons) could be directly obtained by measuring differences between the known
intrinsic polarization angles and the actual observed polarization angles at
different energies. Even in the absence of such knowledge, however, the
birefringent effect can still be tested by polarized sources at arbitrary
redshifts. This is because a large degree of birefringence would add opposite
oriented polarization vectors, effectively washing out most, if not all, of
the net polarization of the signal. Therefore, the polarization detections
with high significance levels can put upper bounds on the energy-dependent
birefringent effect.

###  3.2 Present constraints from polarization measurements

Observations of linear polarization from distant astrophysical sources can be
used to place upper bounds on the birefringent parameter $\eta$. The vacuum
birefringence constraints stem from the fact that if the rotation angles of
photons with different energies (Eq. 12) differ by more than $\pi/2$ over an
energy range ($E_{1}<E<E_{2}$), then the net polarization of the signal would
be significantly depleted and could not be as high as the observed level.
Thus, the detection of highly polarized photons implies that the differential
rotation angle $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ should be smaller than
$\pi/2$, i.e.,

$|\Delta\phi(E_{2})-\Delta\phi(E_{1})|=\eta\frac{\left(E_{2}^{2}-E_{1}^{2}\right)F(z)}{\hbar
E_{\rm pl}H_{0}}\leq\frac{\pi}{2}\;.$ (14)

Previously, Ref. 2001PhRvD..64h3007G  used the observed linear polarization of
ultraviolet light from a galaxy 3C 256 at a distance of around 300 Mpc to set
an upper bound of $\eta<10^{-4}$. Ref. 2008PhRvD..78j3003M  derived a tighter
constraint of $\eta<9\times 10^{-10}$ using the hard X-ray polarization
observation from the Crab Nebula. Much stronger limits of $\eta<10^{-14}$ were
obtained by Refs. 2003Natur.426Q.139M ; 2004PhRvL..93b1101J  using a report of
polarized gamma-rays observed in the prompt emission from GRB 021206
2003Natur.423..415C . However, this claimed polarization detection has been
refuted by subsequent re-analyses of the same data 2004MNRAS.350.1288R ;
2004ApJ…613.1088W . Using the reported detection of polarized soft gamma-ray
emission from GRB 041219A 2007ApJS..169…75K ; 2007A&A…466..895M ;
2009ApJ…695L.208G , Refs. 2011PhRvD..83l1301L ; 2011APh….35…95S  obtained a
stringent upper limit on $\eta$ of $1.1\times 10^{-14}$ and $2.4\times
10^{-15}$, respectively. But again, the previous reports of the gamma-ray
polarimetry for GRB 041219A have been disputed (see 2012PhRvL.109x1104T  for
more explanations), and the arguments for the resulting constraints given by
Refs. 2011PhRvD..83l1301L ; 2011APh….35…95S  are still open to questions.

Contrary to those controversial reports, the detection of gamma-ray linear
polarization by the Gamma-ray burst Polarimeter (GAP) onboard the
Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) is
fairly credible and thus can be used to set a reliable limit on the
birefringent parameter 2012PhRvL.109x1104T . IKAROS/GAP clearly detected
linear polarizations in the prompt gamma-ray emission of three GRBs, with a
polarization degree of $\Pi=27\pm 11\%$ for GRB 100826A 2011ApJ…743L..30Y ,
$\Pi=70\pm 22\%$ for GRB 110301A 2012ApJ…758L…1Y , and $\Pi=84^{+16}_{-28}\%$
for GRB 110721A 2012ApJ…758L…1Y . The detection significance is $2.9\sigma$,
$3.7\sigma$, and $3.3\sigma$, respectively. With the assumption that
$|\Delta\phi(E_{2})-\Delta\phi(E_{1})|\leq\pi/2$, Ref. 2012PhRvL.109x1104T
applied the polarization data of these three GRBs to derive a severe upper
bound on $\eta$ in the order of $\mathcal{O}(10^{-15})$. Note that since the
redshifts of these GRBs were not measured, Ref. 2012PhRvL.109x1104T  used a
luminosity indicator for GRBs to estimate the possible redshifts. Utilizing
the real redshift determination ($z=1.33$) together with the polarimetric data
of GRB 061122, Ref. 2013MNRAS.431.3550G  derived a more stringent limit
($\eta<3.4\times 10^{-16}$) on the possibility of LIV based on the vacuum
birefringent effect. The current deepest limit of $\eta<1.0\times 10^{-16}$
was obtained by Ref. 2014MNRAS.444.2776G  using the most distant polarized
burst GRB 140206A at redshift $z=2.739$.

All the polarization constraints presented above were based on the assumption
that the observed polarization degree would be severely suppressed for a given
energy range if the differential rotation angle
$|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ is too large, regardless of the
intrinsic polarization fraction at the corresponding rest-frame energy range.
However, Ref. 2016MNRAS.463..375L  gave a detailed calculation on the
evolution of GRB polarization induced by the vacuum birefringent effect, and
showed that, even if $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ is as large as
$\pi/2$, more than 60% of the initial polarization degree can be conserved.
This is in conflict with the general perception that
$|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ could not be larger than $\pi/2$ when
high polarization is observed. Thus, Ref. 2016MNRAS.463..375L  suggested that
it is inappropriate to simply use $\pi/2$ as the upper limit of
$|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ to constrain the vacuum birefringent
effect. Applying their formulae for calculating the polarization evolution to
two true GRB events, Ref. 2016MNRAS.463..375L  obtained the most conservative
limits on the birefringent parameter from the polarimetric data of GRB 061122
and GRB 110721A as $\eta<0.5\times 10^{-16}$ and $\eta<4.0\times 10^{-16}$,
respectively. Following the analysis method proposed in Ref.
2016MNRAS.463..375L  and using the latest detections of prompt emission
polarization in GRBs, Ref. 2019MNRAS.485.2401W  improved existing bounds on a
deviation from Lorentz invariance through the vacuum birefringent effect by
factors ranging from two to ten.

Instead of requiring the more complicated and indirect assumption that
$|\Delta\phi(E_{2})-\Delta\phi(E_{1})|\leq\pi/2$, some authors simply assume
that all photons in the observed energy band are emitted with the same
(unknown) intrinsic polarization angle 2007MNRAS.376.1857F ;
2020EPJP..135..527W ; 2021Galax…9…44Z . If the rotation angle of the linear
polarization plane arising from the birefringent effect $\Delta\phi_{\rm
LIV}(E)$ is considered here, the observed linear polarization angle at a
certain $E$ with an intrinsic polarization angle $\phi_{0}$ should be

$\phi_{\rm obs}=\phi_{0}+\Delta\phi_{\rm LIV}\left(E\right)\;.$ (15)

Since $\phi_{0}$ is assumed to be an unknown constant, we expect to observe
the birefringent effect as an energy-dependent linear polarization vector.
Such an observation could give a robust limit on the birefringent parameter
$\eta$. Ref. 2007MNRAS.376.1857F  searched for the energy-dependent change of
the polarization angle in the spectropolarimetric observations of the optical
afterglows of GRB 020813 and GRB 021004. By fitting the multiband polarimetric
data of these two GRBs with Eq. (15), Ref. 2007MNRAS.376.1857F  obtained
constraints on both $\phi_{0}$ and $\eta$ (see Figure 3). At the $3\sigma$
confidence level, the joint constraint on $\eta$ from two GRBs is $-2\times
10^{-7}<\eta<1.4\times 10^{-7}$ (see also 2020EPJP..135..527W ). Ref.
2021Galax…9…44Z  applied the same analysis method to multiband optical
polarimetry of five blazars and obtained $\eta$ constraints with similar
accuracy. It is clear from Eq. (12) that the larger the distance of the
polarized source, and the higher the energy band of the polarization
observation, the greater the sensitivity to small values of $\eta$. As
expected, less stringent constraints on $\eta$ were obtained from the optical
polarization data 2007MNRAS.376.1857F ; 2020EPJP..135..527W ; 2021Galax…9…44Z
.

Figure 3: Fit to multiband polarization observations of the optical afterglows
of GRB 020813 and GRB 021004. Left panels: linear fits of the
spectropolarimetric data. Right panels: 1-3$\sigma$ confidence levels in the
$\phi_{0}$-$\eta$ plane. Reproduced from 2020EPJP..135..527W . Table 2: A
census of limits on the vacuum birefringent parameter $\eta$. These limits
were obtained from the linear polarization measurements of various
astrophysical sources.

Source | Distance | Polarimeter |  Energy Ranga | $\Pi$ | $\eta$ | Refs.
---|---|---|---|---|---|---
Galaxy 3C 256 |  $d=300$ Mpc | Spectropolarimeter | Ultraviolet | $16.4\pm 2.2\%$ | $<10^{-4}$ | 2001PhRvD..64h3007G
Crab Nebula |  $d=1.9$ kpc | INTEGRAL/SPI | 100–1000 keV | $46\pm 10\%$ | $<9\times 10^{-10}$ | 2008PhRvD..78j3003M
GRB
GRB 021206 |  $d=10^{10}$ light yearsc | RHESSI | 150–2000 keV |  $80\pm 20\%$b | $<10^{-14}$ | 2003Natur.426Q.139M
GRB 021206 |  $d=0.5$ Gpcd | RHESSI | 150–2000 keV |  $80\pm 20\%$b | $<0.5\times 10^{-14}$ | 2004PhRvL..93b1101J
GRB 041219A |  $z=0.02$e | INTEGRAL/IBIS | 200–800 keV |  $43\pm 25\%$b | $<1.1\times 10^{-14}$ | 2011PhRvD..83l1301L
GRB 041219A |  $z=0.2$f | INTEGRAL/SPI | 100–350 keV |  $96\pm 40\%$b | $<2.4\times 10^{-15}$ | 2011APh….35…95S
GRB 100826A |  $z=0.71$f | IKAROS/GAP | 70–300 keV | $27\pm 11\%$ | $<2.0\times 10^{-14}$ | 2012PhRvL.109x1104T
GRB 110301A |  $z=0.21$f | IKAROS/GAP | 70–300 keV | $70\pm 22\%$ | $<1.0\times 10^{-14}$ | 2012PhRvL.109x1104T
GRB 110721A |  $z=0.45$f | IKAROS/GAP | 70–300 keV | $84^{+16}_{-28}\%$ | $<2.0\times 10^{-15}$ | 2012PhRvL.109x1104T
GRB 061122 | $z=1.33$ | INTEGRAL/IBIS | 250–800 keV | $>60\%$ | $<3.4\times 10^{-16}$ | 2013MNRAS.431.3550G
GRB 140206A | $z=2.739$ | INTEGRAL/IBIS | 200–400 keV | $>48\%$ | $<1.0\times 10^{-16}$ | 2014MNRAS.444.2776G
GRB 061122 | $z=1.33$ | INTEGRAL/IBIS | 250–800 keV | $>60\%$ | $<0.5\times 10^{-16}$ | 2016MNRAS.463..375L
GRB 110721A |  $z=0.45$f | IKAROS/GAP | 70–300 keV | $84^{+16}_{-28}\%$ | $<4.0\times 10^{-16}$ | 2016MNRAS.463..375L
GRB 061122 | $z=1.33$ | INTEGRAL/IBIS | 250–800 keV | $>60\%$ | $<0.5\times 10^{-16}$ | 2019MNRAS.485.2401W
GRB 100826A |  $z=0.083$f | IKAROS/GAP | 70–300 keV | $27\pm 11\%$ | $1.2^{+1.4}_{-0.7}\times 10^{-14}$ | 2019MNRAS.485.2401W
GRB 110301A |  $z=0.082$f | IKAROS/GAP | 70–300 keV | $70\pm 22\%$ | $4.3^{+5.4}_{-2.3}\times 10^{-15}$ | 2019MNRAS.485.2401W
GRB 110721A | $z=0.382$ | IKAROS/GAP | 70–300 keV | $84^{+16}_{-28}\%$ | $5.1^{+4.0}_{-5.1}\times 10^{-16}$ | 2019MNRAS.485.2401W
GRB 140206A | $z=2.739$ | INTEGRAL/IBIS | 200–400 keV | $>48\%$ | $<1.0\times 10^{-16}$ | 2019MNRAS.485.2401W
GRB 160106A |  $z=0.091$f | AstroSat/CZTI | 100–300 keV | $68.5\pm 24\%$ | $3.4^{+1.4}_{-1.8}\times 10^{-15}$ | 2019MNRAS.485.2401W
GRB 160131A | $z=0.972$ | AstroSat/CZTI | 100–300 keV | $94\pm 31\%$ | $1.2^{+2.0}_{-1.2}\times 10^{-16}$ | 2019MNRAS.485.2401W
GRB 160325A |  $z=0.161$f | AstroSat/CZTI | 100–300 keV | $58.75\pm 23.5\%$ | $2.3^{+1.0}_{-0.9}\times 10^{-15}$ | 2019MNRAS.485.2401W
GRB 160509A | $z=1.17$ | AstroSat/CZTI | 100–300 keV | $96\pm 40\%$ | $0.8^{+2.2}_{-0.8}\times 10^{-16}$ | 2019MNRAS.485.2401W
GRB 160802A |  $z=0.105$f | AstroSat/CZTI | 100–300 keV | $85\pm 29\%$ | $2.0^{+1.7}_{-2.0}\times 10^{-15}$ | 2019MNRAS.485.2401W
GRB 160821A |  $z=0.047$f | AstroSat/CZTI | 100–300 keV | $48.7\pm 14.6\%$ | $8.9^{+1.7}_{-1.7}\times 10^{-15}$ | 2019MNRAS.485.2401W
GRB 160910A |  $z=0.272$f | AstroSat/CZTI | 100–300 keV | $93.7\pm 30.92\%$ | $4.7^{+7.6}_{-4.7}\times 10^{-16}$ | 2019MNRAS.485.2401W
GRB 020813 | $z=1.255$ | LRISp | 3500–8800 Å | 1.8%–2.4% |  ($-2.0$–1.4)$\times 10^{-7}$ | 2007MNRAS.376.1857F
GRB 021004 | $z=2.328$ | VLT | 3500–8600 Å | $1.88\pm 0.05\%$ |  ($-2.0$–1.4)$\times 10^{-7}$ | 2007MNRAS.376.1857F
GRB 020813 | $z=1.255$ | LRISp | 3500–8800 Å | 1.8%–2.4% |  ($-1.8$–1.1)$\times 10^{-7}$ | 2020EPJP..135..527W
GRB 021004 | $z=2.328$ | VLT | 3500–8600 Å | $1.88\pm 0.05\%$ |  ($-1.8$–1.1)$\times 10^{-7}$ | 2020EPJP..135..527W
Blazar
3C 66A | $z=0.444$ | Photopolarimeter | $UBVRI$ | 14.1%–33.7% | $(3.4\pm 1.2)\times 10^{-7}$ | 2021Galax…9…44Z
S5 0716+714 | $z=0.310$ | Photopolarimeter | $UBVRI$ | 15.1%–20.1% | $(-1.0\pm 4.6)\times 10^{-7}$ | 2021Galax…9…44Z
OJ 287 | $z=0.306$ | Photopolarimeter | $UBVRI$ | 5.2%–22.8% | $(6.2\pm 7.2)\times 10^{-7}$ | 2021Galax…9…44Z
MK 421 | $z=0.031$ | Photopolarimeter | $UBVRI$ | 1.7%–3.5% | $(6.5\pm 22.8)\times 10^{-6}$ | 2021Galax…9…44Z
PKS 2155-304 | $z=0.116$ | Photopolarimeter | $UBVRI$ | 3%–7% | $(-5.3\pm 3.6)\times 10^{-7}$ | 2021Galax…9…44Z

aThe energy range in which polarization is observed.
bThe claimed polarization detections have been refuted.
cThe distance of GRB 021206 was assumed to be $10^{10}$ light years.
dThe distance of GRB 021206 was assumed to be 0.5 Gpc.
eThe lower limit to the photometric redshift of GRB 041219A was adopted.
fThe redshifts of these GRBs were estimated by the empirical luminosity
relation.

Table 2 presents a summary of astrophysical polarization constraints on a
possible LIV through the vacuum birefringent effect. The current best limits
on the birefringent parameter, $\eta<\mathcal{O}(10^{-16})$, have been
obtained by the detections of gamma-ray linear polarization of GRBs
2013MNRAS.431.3550G ; 2014MNRAS.444.2776G ; 2016MNRAS.463..375L ;
2019MNRAS.485.2401W .

##  4 Photon decay and photon splitting

There are various forms of modified dispersion relation for different
particles and underlying QG theories. One of the Lorentz-violating dispersion
relations for photons takes the generalized form 1998Natur.393..763A

$E^{2}-p^{2}=\pm|\alpha_{n}|p^{n+2}\;,$ (16)

where $n$ is the leading order of the modification from the underlying QG
theory, $\alpha_{n}$ is the $n$th order LIV parameter, and the sign $\pm$
refers to the superluminal ($+$) and subluminal ($-$) cases. For $n>0$, limits
on $\alpha_{n}$ can be interpreted in terms of the QG energy scale,

$E_{{\rm QG},n}=\alpha_{n}^{-1/n}\;.$ (17)

###  4.1 Photon decay

In the superluminal LIV case, photons can decay into electron-positron pairs,
$\gamma\rightarrow e^{+}e^{-}$. The resulting decay rates
$\Gamma_{\gamma\rightarrow e^{+}e^{-}}$ are rapid and effective at energies
where the process is allowed. Once above the energy threshold, this photon
decay process would lead to a hard cutoff in the gamma-ray spectrum without
any high-energy photons being observed 2017PhRvD..95f3001M . The threshold for
any order $n$ is given by

$\alpha_{n}\leq\frac{4m_{e}^{2}}{E^{n}\left(E^{2}-4m_{e}^{2}\right)}\;,$ (18)

where $m_{e}$ represents the electron mass 2017PhRvD..95f3001M . One can see
from Eqs. (17) and (18) that the upper limits on $\alpha_{n}$ (lower limits on
$E_{{\rm QG},n}$) become tighter with the increase in the observed photon
energy by a factor of $E^{-(n+2)}$ ($E^{1+2/n}$ for lower limits on $E_{{\rm
QG},n}$).

Following Eqs. (17) and (18), one then has $E_{{\rm QG},n}$ for $n=1$ and $2$,

$E_{{\rm QG},1}\geq 9.57\times 10^{23}\;{\rm eV}\left(\frac{E}{\rm
TeV}\right)^{3}\;,$ (19) $E_{{\rm QG},2}\geq 9.78\times 10^{17}\;{\rm
eV}\left(\frac{E}{\rm TeV}\right)^{2}\;.$ (20)

Therefore, a lower limit for $E_{{\rm QG},n}$ in the photon sector can be
obtained directly from any observed high-energy photon event.

###  4.2 Photon splitting

A second superluminal LIV decay process is photon splitting to multiple
photons, $\gamma\rightarrow N\gamma$. The dominant splitting process with
quadratic modification ($n=2$) is the photon decay into three photons
($\gamma\rightarrow 3\gamma$) 2019JCAP…04..054A . The decay rate of photon
splitting (in units of energy) is 2019JCAP…04..054A ; 2019EPJC…79.1011S

$\Gamma_{\gamma\rightarrow 3\gamma}=5\times
10^{-14}\frac{E^{19}}{m_{e}^{8}E_{{\rm QG},2}^{10}}\;,$ (21)

which is much smaller than the photon decay rate $\Gamma_{\gamma\rightarrow
e^{+}e^{-}}$. Despite the lack of an energy threshold, this process is
kinematically allowed whenever $E^{2}>p^{2}$ (i.e., superluminal LIV). It
becomes significant when photons travel through astrophysical distances and
also predicts a hard cutoff at high photon energies in astrophysical spectra.

If we find the evidence of a cutoff in the spectrum of an energetic
astrophysical source at distance $d$, we can equate the mean free path of a
photon to the source distance. That is, we take
$d\times\Gamma_{\gamma\rightarrow 3\gamma}=1$, with $\Gamma_{\gamma\rightarrow
3\gamma}$ translated to units of $\rm kpc^{-1}$. The corresponding LIV limit
is therefore given by

$E_{{\rm QG},2}\geq 3.33\times 10^{19}\;{\rm eV}\left(\frac{d}{\rm
kpc}\right)^{0.1}\left(\frac{E}{\rm TeV}\right)^{1.9}\;,$ (22)

which is a function of the highest photon energy. Once again, this photon
splitting in flight from the source provides a direct way to constrain the
quadratic LIV energy scale that mainly relies on the highest energy photons
observed.

###  4.3 Present constraints from spectral cutoff

If there is no sign of clear cutoff at the highest energy end of the photon
spectra of astrophysical sources, then stringent lower limits on the LIV
energy scale can be derived. There have been some resulting constraints on
LIV-induced spectral cutoff using sub-PeV measurements of gamma-ray spectra
from astrophysical sources, including the HEGRA observations of the Crab
Nebula spectrum 2017PhRvD..95f3001M ; 2019JCAP…04..054A , the Tibet
observations of the Crab Nebula spectrum 2019EPJC…79.1011S , the HAWC
observations of the spectra of the Crab and three other sources
2020PhRvL.124m1101A , and the LHAASO observations of the spectra of the Crab
and four other sources 2021arXiv210507927C ; 2021arXiv210612350T .

For instance, the LHAASO Collaboration studied the superluminal LIV effect
using the observations of unprecedentedly high-energy gamma rays, with a
rigorous statistical technique and a careful assessment of the systematic
uncertainties 2021arXiv210612350T . Recently, 12 ultra-high-energy gamma-ray
Galactic sources above 100 TeV have been detected by LHAASO 2021Natur.594…33C
. The highest energy photon-like event from the source LHAASO J2032+4102 is
about 1.4 PeV. The second highest energy photon-like event from LHAASO
J0534+2202 (Crab Nebula) is about 0.88 PeV. Since higher energy photons may
provide better limits on the LIV energy scale, the LHAASO Collaboration
adopted the two sources J2032+4102 and J0534+2202 for their purpose
2021arXiv210612350T . Two general spectral forms are assumed for both sources,
i.e., a log-parabolic form and a power-law form. Both the photon decay
($\gamma\rightarrow e^{+}e^{-}$) and photon splitting ($\gamma\rightarrow
3\gamma$) processes predict that the energy spectrum of a source has a quasi-
sharp cutoff at the highest energy part. If LIV does exist, the expected
spectrum of the source can then be expressed as 2021arXiv210612350T

$f(E)=A_{0}\left(\frac{E}{E_{0}}\right)^{-\alpha-\beta\ln\left(E/E_{0}\right)}H(E-E_{\rm
cut})\;,$ (23)

where $A_{0}$ is the flux normalization, $\alpha$ and $\beta$ are the spectral
indices, $E_{0}=20$ TeV is a reference energy, $H(E-E_{\rm cut})$ stands for
the Heaviside step function, and $E_{\rm cut}$ is the cutoff energy. The above
expression is for the log-parabolic spectrum. When $\beta\equiv 0$, the log-
parabolic spectrum reduces to the power-law spectrum. The spectral parameters
can be optimized by maximizing the likelihood function, i.e.,

$\mathcal{L}(A_{0},\alpha,\beta,E_{\rm cut})=\prod_{i}^{n}{\rm
Poisson}\left(N_{{\rm obs},i},\;N_{{\rm exp},i}(A_{0},\alpha,\beta,E_{\rm
cut})+N_{{\rm bkg},i}\right)\;,$ (24)

where $i$ represents the $i$-th energy bin, $N_{{\rm obs},i}$ is the number of
observed events from the source, $N_{{\rm exp},i}$ is the expected signal from
the chosen energy spectrum, and $N_{{\rm bkg},i}$ is the estimated background.
For each $E_{\rm cut}$, Ref. 2021arXiv210612350T  obtained the best-fit
spectral parameters ($A_{0}$, $\alpha$, and $\beta$) and the corresponding
likelihood value. The statistical significance of the existence of such a
cutoff was calculated using a test statistic (TS) variable, which is the log-
likelihood ratio of the fit with a cutoff and the fit with no cutoff,

${\rm TS}(E_{\rm cut})=-\sum_{\rm
bin}2\ln\left(\frac{\mathcal{L}_{1}(\hat{\hat{A_{0}}},\hat{\hat{\alpha}},\hat{\hat{\beta}},E_{\rm
cut})}{\mathcal{L}_{0}(\hat{A_{0}},\hat{\alpha},\hat{\beta},E_{\rm
cut}\rightarrow\infty)}\right)\;,$ (25)

where the null hypothesis (without the LIV effect) is the case with $E_{\rm
cut}\rightarrow\infty$ and the alternative hypothesis (with the LIV effect)
corresponds to a finite $E_{\rm cut}$. Because the spectral fits do not
indicate a significant preference for $E_{\rm cut}<\infty$, a lower limit on
$E_{\rm cut}$ was proceed to set, below which there is weak or no evidence for
the photon decay. This lower limit on $E_{\rm cut}$ also serves as an upper
limit on the observed photon energy $E$. Then Eqs. (19), (20), and (22)
directly lead to lower limits on $E_{{\rm QG},1}$ and $E_{{\rm QG},2}$.

Table 3: Summary of 95% confidence-level lower limits on the cutoff energy
$E_{\rm cut}$ and the linear and quadratic energy scales of the superluminal
LIV as obtained by photon decay ($e^{+}e^{-}$) and photon splitting
($3\gamma$).

Experiment | Source | Distance | $E_{\rm cut}$ |  $E_{{\rm QG},1}$ ${(e^{+}e^{-})}$ |  $E_{{\rm QG},2}$ ${(e^{+}e^{-})}$ |  $E_{{\rm QG},2}$ ${(3\gamma)}$ | Refs.
---|---|---|---|---|---|---|---
|  | [kpc] | [TeV] | [GeV] | [GeV] | [GeV] |
HEGRA | Crab Nebula | 2 | 56 | $1.5\times 10^{20}$ | $2.8\times 10^{12}$ | — | 2017PhRvD..95f3001M
Crab Nebula | 2 | 75 | — | — | $1.3\times 10^{14}$ | 2019JCAP…04..054A
Tibet | Crab Nebula | 2 | 140 | — | — | $4.1\times 10^{14}$ | 2019EPJC…79.1011S
HAWC | J1825–134 | 1.55 | 244 | $1.4\times 10^{22}$ | $5.8\times 10^{13}$ | $1.2\times 10^{15}$ | 2020PhRvL.124m1101A
J1907+063 | 2.37 | 218 | $1.0\times 10^{22}$ | $4.7\times 10^{13}$ | $1.0\times 10^{15}$
Crab Nebula | 2 | 152 | $3.4\times 10^{21}$ | $2.3\times 10^{13}$ | $5.0\times 10^{14}$
J2019+368 | 1.8 | 120 | $1.7\times 10^{21}$ | $1.4\times 10^{13}$ | $3.2\times 10^{14}$
LHAASO | J2226+6057 | $0.8^{a}$ | 280.7 | $2.1\times 10^{22}$ | $7.7\times 10^{13}$ | $1.5\times 10^{15}$ | 2021arXiv210507927C
J1908+0621 | 2.37 | 370.5 | $4.9\times 10^{22}$ | $1.3\times 10^{14}$ | $2.8\times 10^{15}$ | 2021arXiv210507927C
J1825–1326 | 1.55 | 169.9 | $4.7\times 10^{21}$ | $2.8\times 10^{13}$ | $0.6\times 10^{15}$ | 2021arXiv210507927C
Crab Nebula | 2 | 750 | $4.0\times 10^{23}$ | $5.5\times 10^{14}$ | $1.0\times 10^{16}$ | 2021arXiv210612350T
J2032+4102 | 1.4 | 1140 | $1.4\times 10^{24}$ | $1.3\times 10^{15}$ | $2.2\times 10^{16}$ | 2021arXiv210612350T

aThe distance of possible celestial objects associated with J2226+6057 was
given.

Table 3 lists a summary of 95% confidence-level lower limits on $E_{\rm cut}$
and the inferred LIV energy scales obtained through photon decay and photon
splitting. As illustrated in Table 3, the highest-energy source LHAASO
J2032+4102 provides the strongest constraints on the energy scales of the
superluminal LIV among experimental results with the same method. The linear
LIV energy scale is constrained to be higher than about $10^{5}$ times the
Planck energy scale $E_{\rm Pl}$, and the quadratic LIV energy scales is over
$10^{-3}E_{\rm Pl}$.

###  4.4 Comparison with different methods

The comparison with the resulting constraints on the superluminal LIV energy
scales obtained from different experiments is shown in Figure 4. We show
strong limits on photon decay and photon splitting using ultra-high-energy
gamma-ray observations from HEGRA 2017PhRvD..95f3001M ; 2019JCAP…04..054A ,
Tibet 2019EPJC…79.1011S , HAWC 2020PhRvL.124m1101A , and LHAASO
2021arXiv210612350T . We also show limits due to LIV energy-dependent time
delay searches with the Fermi/LAT 2013PhRvD..87l2001V . Obviously, the limits
on $E_{{\rm QG},1}$ and $E_{{\rm QG},2}$ from photon decay and photon
splitting are at least four orders of magnitude stronger than those from the
time delay method.

Additionally, observations of linear polarization from astrophysical sources
have been widely used to place limits on the vacuum birefringent parameter
$\eta$. By comparing Eqs. (1) and (11), we can derive the conversion from
$\eta$ to the limit on the linear LIV energy scale, i.e., $E_{{\rm
QG},1}=\frac{E_{\rm pl}}{2\eta}$. At this point, it is interesting to make a
comparison of recent achievements in sensitivity of polarization measurements
versus photon decay measurements. The detection of linear polarization in the
prompt gamma-ray emission of GRB 061122 yielded the strictest limit on the
birefringent parameter, $\eta<0.5\times 10^{-16}$, which corresponds to
$E_{{\rm QG},1}>1.2\times 10^{35}~{}\mathrm{GeV}$ 2016MNRAS.463..375L ;
2019MNRAS.485.2401W . Obviously, this polarization constraint is about 11
orders of magnitude stronger than the constraint from photon decay ($E_{{\rm
QG},1}>1.4\times 10^{24}~{}\mathrm{GeV}$) 2021arXiv210612350T . However, both
time-of-flight constraints and photon decay constraints are essential in an
extensive search for nonbirefringent Lorentz-violating effects.

Figure 4: Comparison of the constraints on the superluminal LIV energy scales
derived from different experiments with different methods. We show strong
constraints due to photon decay ($e^{+}e^{-}$), photon splitting ($3\gamma$),
and energy-dependent time delay ($\Delta t$). The vertical dashed lines
correspond to the Planck energy scale $E_{\rm Pl}$.

##  5 Summary and outlook

Many tight constraints on violations of Lorentz invariance have been achieved
by observing tiny changes in light that has propagated over astrophysical
distances. Some of these search for a frequency-dependent velocity resulting
from vacuum dispersion. Some works seek a change in polarization arising from
vacuum birefringence. Others look for a hard cutoff in the gamma-ray spectra
due to photon decay and photon splitting.

For vacuum dispersion studies, in order to tightly constrain the LIV effects
one should choose those explosive astrophysical sources with shorter spectral
lags, higher energy emissions, and longer propagation distances. As the most
energetic bursting events occurring at cosmological sources, GRBs are
excellent astrophysical probes for LIV constraints in the dispersive photon
sector. In the past, emission from GRBs had been detected only at energies
below 100 GeV, and LIV limits from time-of-flight measurements of GRBs had
also been restricted in the relatively low energy range. Now that photons of
energies above 100 GeV have been detected from two bright GRBs (GRB 190114C
2019Natur.575..455M  and GRB 180720B 2019Natur.575..464A ), opening a new
spectral window in GRB research. It is expected that such detections will
become routine in the future 2019Natur.575..448Z , especially with the
operations of facilities such as HAWC, LHAASO, and the future international
Cherenkov Telescope Array. Vacuum dispersion constraints will greatly benefit
from the observations of extremely high-energy emission of more GRBs. For
vacuum birefringence studies, in order to obtain tighter constraints on the
LIV effects one should choose those astrophysical sources with longer
distances and polarization measurements at higher energies. In this respect,
the ideal sources to seek LIV-induced vacuum birefringence would be polarized
gamma-rays from bright GRBs at deep cosmological distances. The technology for
measuring polarization in the 5 to 100 MeV energy range has been constantly
improved and developed. As more and more gamma-ray polarimeters (such as
POLAR-II, TSUBAME, NCT/COSI, GRAPE; see 2017NewAR..76….1M  for a review) enter
service, more polarized astrophysical sources at higher gamma-ray energies are
expected to be detected. Polarization measurements of cosmological sources
such as GRBs at such energies will further improve LIV constraints through the
vacuum birefringent effect. For photon decay studies, in order to obtain more
stringent bounds on the LIV effects one should choose those astrophysical
sources with abundant ultra-high-energy photons. High energy resolution of
spectrum measurements are desired to determine whether there is a hard cutoff
consistent with the prediction of photon decay in the observed spectra of
astrophysical sources. When using ultra-high-energy photons for LIV limits, we
should be careful with the uncertainty for failure to reject a cosmic-ray
event. Owing to its unprecedented capability of cosmic ray background
rejection and high energy resolution 2021Natur.594…33C , LHAASO has
competitive advantage in constraining LIV-induced photon decay.

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