Testing Fundamental Physics with Astrophysical Transients

Several QG theories proposed to describe quantum spacetime, predict the granularity of space and time. For example, string theory that remove the point-like nature of the particles by introducing to each of them a (mono)-dimensional extension: the string (see, e.g., Smolin [103] for reviews). Loop quantum gravity that question the continuousness and smoothness of spacetime quantizing it into discrete energy levels like those observed in the classical quantum-mechanical systems to construct a complex geometric structure called spin networks (see, e.g., Rovelli [104] for reviews). In both proposed theories emerge a minimal length for physical space (and time), although with different and somewhat opposite theoretical approaches [105]. The minimal spatial and temporal scales associated to the quantization are in terms of standard units: $\textit{l}_{\rm P}\sim\sqrt{\hbar G/c^{3}}\sim 10^{-33}$ cm and $t_{\rm P}\sim\sqrt{\hbar G/c^{5}}\sim 10^{-43}$ s for the Planck length and time, respectively. A significant class of QG theories predict that the propagation of photons through this discrete spacetime might exhibit a non-trivial dispersion relation in vacuum [16], corresponding to the possible violation or break of the Lorentz invariance via an energy-dependent speed of light. In vacuum, the LIV-induced modifications to the dispersion relation of photons can be described using a Taylor series expansion:

where $p$ is the photon momentum, $E_{\rm QG}$ is the hypothetical energy scale at which QG effects would become significant, and $s_{\pm}=\pm 1$ is a theory-dependent factor. For $E\ll E_{\rm QG}$, the sum is dominated by the lowest-order term in the series. Considering only the lowest-order dominant term, the photon group velocity can therefore be expressed by

where $n=1$ and $n=2$ correspond to the linear and quadratic LIV, respectively. The coefficient $s_{\pm}=+1$ ($s_{\pm}=-1$) stands for a decrease (increase) in the photon speed along with increasing photon energy (also refers to the “subluminal” and “superluminal” scenarios).

Because of the energy dependence of $v(E)$, two photons with different observer-frame energies ($E_{h}>E_{l}$) emitted simultaneously from the same astrophysical source at redshift $z$ would arrive at the observer at different times. Taking account of the cosmological expansion, we derive the expression of the LIV-induced time delay [19]:

where $t_{h}$ and $t_{l}$, respectively, denote the arrival times of the high-energy and low-energy photons, $H_{0}$ is the Hubble constant, and $\Omega_{\rm m}$ and $\Omega_{\Lambda}$ are the matter energy density and the vacuum energy density (parameters of the flat $\Lambda$CDM model). Since the adopted energy range generally spans several orders of magnitude, one can approximate $(E_{h}^{n}-E_{l}^{n})\approx E_{h}^{n}$. Adopting $z=20$ as a firm upper limit for the redshift of any source, we find the LIV-induced time delay is $|\Delta t_{\rm LIV}|\leq 0.5(E_{h\;{\rm GeV}}/\zeta)$ s for the linear ($n=1$) LIV case and $|\Delta t_{\rm LIV}|\leq 5.2\times 10^{-19}(E_{h\;{\rm GeV}}/\zeta)^{2}$ s for the quadratic ($n=2$) LIV case, where $E_{h\;{\rm GeV}}=E_{h}/$(1 GeV) and $\zeta=E_{\rm QG}/E_{\rm Pl}$. These indicate that first order ($n=1$) effects would lead to potentially observable delays, while second order ($n=2$) effects are so tiny that it would be impossible to observe them with this time-of-flight technique.

Eq. (3) shows that the LIV-induced time delay increases with the energy of the photons and the distance of the source. A short timescale of the signal variability easily provides a reference time to measure the time delay. The higher the photon energy, the longer the source distance, and the shorter the time delay, the more stringent limits on the QG energy scale one can reach. Exploiting the rapid variations of gamma-ray emissions from astrophysical sources at large distances to constrain LIV was first proposed by Amelino-Camelia et al. [16]. At present, tests of LIV have been made using the observations of gamma-ray bursts (GRBs), active galactic nuclei (AGNs), and pulsars.

• •

  Gamma-ray bursts

GRBs are among the most distant gamma-ray sources and their signals vary on subsecond timescales. As such, GRBs are identified as promising sources for LIV studies. There have been many searches for LIV through studies on the time-of-flight measurements of GRBs. Some of the pre-Fermi studies are those by Ellis et al. [107] using BATSE GRBs; by Boggs et al. [108] using RHESSI observations of GRB 021206; by Ellis et al. [18, 106] using BATSE, HETE-2, and Swift GRBs; by Rodríguez-Martínez et al. [109] using Konus-Wind and Swift observations of GRB 051221A; by Bolmont et al. [110] using HETE-2 GRBs; and by Lamon et al. [111] using INTEGRAL GRBs. Because of the unprecedented sensitivity for detecting the prompt high-energy GRB emission (up to tens of GeV) by the Fermi Large Area Telescope (LAT), more stringent constraints on LIV have been obtained using Fermi observations. These constraints include those by the Fermi Collaboration using GRBs 080916C [21] and 090510 [22]; by Xiao & Ma [112] using GRB 090510; and by Refs. [113, 23, 24, 25, 28, 114, 115, 116, 117, 32] using multiple Fermi GRBs. Particularly, Abdo et al. [22] used the highest energy (31 GeV) photon of the short GRB 090510 detected by Fermi/LAT to constrain the linear LIV energy scale ($E_{{\rm QG},1}$). The burst has a redshift $z=0.903$. This 31 GeV photon was detected 0.829 s after the Fermi Gamma-Ray Burst Monitor (GBM) trigger. If the 31 GeV photon was emitted at the beginning of the first GBM pulse, one has $\Delta t_{\rm LIV}<859$ ms, which gives the most conservative constraint $E_{{\rm QG},1}>1.19E_{\rm Pl}$. If the 31 GeV photon is associated with the contemporaneous $<1$ MeV spike, one has $\Delta t_{\rm LIV}<10$ ms, which gives the least conservative constraint $E_{{\rm QG},1}>102E_{\rm Pl}$. We can see that the linear LIV models requiring $E_{{\rm QG},1}\leq E_{\rm Pl}$ are disfavored by these results. Subsequently, Vasileiou et al. [25] used three statistical techniques to constrain the total degree of dispersion in the data of four LAT-detected GRBs. For the subluminal case, their most stringent limits are derived from GRB 090510 and are $E_{{\rm QG},1}>7.6E_{\rm Pl}$ and $E_{{\rm QG},2}>1.3\times 10^{11}~{}\mathrm{GeV}$ for linear and quadratic LIV, respectively. These limits improve previous constraints by a factor of $\sim 2$. Recently, the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescopes first detected GRB 190114C in the sub-TeV energy domain (i.e., 0.2—1 TeV), recording the highest energy photons ever observed from a GRB [118]. Using conservative assumptions on the possible intrinsic spectral and temporal emission properties, the MAGIC Collaboration searched for an energy dependence in the arrival time of the most energetic photons and presented competitive limits on the quadratic leading-order LIV-induced vacuum dispersion [33]. The resulting constraints from GRB 190114C are $E_{{\rm QG},2}>6.3\times 10^{10}~{}\mathrm{GeV}$ and $E_{{\rm QG},2}>5.6\times 10^{10}~{}\mathrm{GeV}$ for the subluminal and superluminal cases, respectively.

By using the arrival-time differences between high-energy and low-energy photons (the so-called spectral lags) from GRBs, Lorentz invariance has been tested with unprecedented accuracy. Such time-of-flight tests, however, are subject to a bias related to a possible intrinsic time lag introduced from the unknown emission mechanism of the sources, which would enhance or cancel-out the delay caused by the LIV effects. That is, the method of the arrival-time difference is tempered by our ignorance concerning potential source-intrinsic effects. The first attempt to mitigate the intrinsic time lag problem was proposed by Ellis et al. [107, 18, 106], who suggested working on a large sample of GRBs with different redshifts. For each GRB, Ellis et al. [18] looked for the spectral lag in the light curves recorded in the chosen observer-frame energy bands 115–320 and 25–55 keV. To account for the poorly known intrinsic time lag, they fitted the observed spectral lags of 35 GRBs with the inclusion of a constant $b_{\rm sf}$ specified in the rest frame of the source. The observed arrival time delays thus have two contributions $\Delta t_{\rm obs}=\Delta t_{\rm LIV}+b_{\rm sf}(1+z)$, reflecting the possible LIV effects and source-intrinsic effects [18]. Rescaling $\Delta t_{\rm obs}$ by a factor $(1+z)$, then one has a simple linear fitting function

where

is a function of redshift which depends on the cosmological model, the slope $a_{\rm LV}=\Delta E/(H_{0}E_{\rm QG})$ is related to the QG energy scale, and the intercept $b_{\rm sf}$ denotes the possible unknown intrinsic time lag. In the standard flat $\Lambda$CDM model, the dimensionless expansion rate $h(z)$ is expressed as $h(z)=\sqrt{\Omega_{\rm m}(1+z)^{3}+\Omega_{\Lambda}}$. Note that the linear LIV ($n=1$) in the subluminal case ($s_{\pm}=+1$) was considered in this work. Within a frame work of the concordance $\Lambda$CDM model, a linear fit to the rescaled time lags extracted from 35 light curve pairs is shown in the left panel of Figure 1. The best-fit line corresponds to $\frac{\Delta t_{\rm obs}}{1+z}=(0.0068\pm 0.0067)K-(0.0065\pm 0.0046)$ and the likelihood function for the slope parameter $a_{\rm LV}$ is presented in the right panel of Figure 1. The 95% confidence-level lower limit on the linear LIV energy scale derived from the likelihood function of $a_{\rm LV}$ is $E_{\rm QG}\geq 1.4\times 10^{16}$ GeV [106]. Going beyond the $\Lambda$CDM cosmology, Refs. [119, 120] extended this analysis to different cosmological models and showed that the result is insensitive to the adopted background cosmology. Subsequently, some cosmology-independent approaches were applied to probe the possible LIV effects [121, 122].

It should be noted that there are two limitations in the treatment of Ellis et al. [18]. First, they extracted spectral lags in the light curves between two fixed observer-frame energy bands. However, because different GRBs have different redshift measurements, these two energy bands correspond to a different pair of energy bands in the source frame [123], thus potentially causing an artificial energy dependence to the extracted spectral lag and/or a systematic uncertainty to the search for LIV lags. ¹¹1Due to the redshift dependence of cosmological sources, observer frame quantities can be quite different than source frame ones. In principle, there is a similar problem associated with the energy dependence of the observer-frame quantities for vacuum birefringence or WEP bounds (more on this below) when analyzing a large sample of cosmological sources with different redshifts. However, note that if we work on the observer-frame quantities of individual cosmological sources, there is no such a problem. Ukwatta et al. [123] found that there is a large scatter in the correlation between observer-frame lags and source-frame lags for the same GRB sample, indicating that the observer-frame lag does not faithfully represent the source-frame lag. The first limitation of Ellis et al. treatment can be resolved by selecting two appropriate energy bands fixed in the rest frame and calculating the time lag for two projected observer-frame energy bands by the relation $E_{\rm observer}=E_{\rm source}/(1+z)$. Bernardini et al. [124] studied the source-frame spectral lags of 56 GRBs detected by Swift/Burst Alert Telescope (BAT). For each GRB, they extracted light curves for two observer-frame energy bands corresponding to the fixed energy bands in the source frame, i.e., 100–150 and 200-250 keV. These two particular source-frame energy bands were selected to ensure that the projected observer-frame energy bands (i.e., $[100-150]/(1+z)$ and $[200-250]/(1+z)$ keV) are within the detectable energy range of the BAT instrument (see Figure 2). For each extracted light-curve pairs, they used the discrete cross-correlation function to calculate the spectral lag. Note that the energy difference between the median-values of the two source-frame energy bands is fixed at 100 keV, whereas in the observer frame, the energy difference varies depending on the redshift of each burst. This is in contrast to the spectral lag extractions achieved in the observer frame, where the energy difference is treated as a constant [123]. Wei & Wu [31] first took advantage of the source-frame spectral lags of 56 Swift GRBs presented in Bernardini et al. [124] to investigate the LIV effects. For the subluminal case, the arrival-time difference of two photons with observer-frame energy difference $\Delta E$ that induced by the linear LIV reads:

where $\Delta E^{\prime}=100~{}\mathrm{keV}$ is the rest-frame energy difference. Similar to Ellis et al. [18], one can formulate the intrinsic time lag problem in terms of linear regression:

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

is a dimensionless redshift function, and $a^{\prime}_{\rm LV}=\Delta E^{\prime}/(H_{0}E_{\rm QG})$ is the slope in $K^{\prime}$. Using the sample of 56 GRBs with known redshifts, Wei & Wu [31] obtained robust limits on the slope $a^{\prime}_{\rm LV}$ and the intercept $b_{\rm sf}$ by fitting their source-frame spectral lag data. The 95% confidence-level lower limit on the QG energy scale derived from $a^{\prime}_{\rm LV}$ is $E_{\rm QG}\geq 2.0\times 10^{14}$ GeV [31]. This is a step forward in the study of LIV effects, since all previous investigations used spectral lags extracted in the observer frame only.

The second limitation of Ellis et al. treatment is that an unknown constant was supposed to be the intrinsic time delay in the linear fitting function (see Eq. 4), which is tantamount to suggesting that all GRBs have the same intrinsic time lag. However, since the time durations of GRBs span nearly six orders of magnitude, it is not possible that high-energy photons radiated from different GRBs (or from the same burst) have the same intrinsic time lag relative to the radiation time of low-energy photons [125]. As an improvement, Zhang & Ma [28] fitted the data of the high-energy photons from GRBs on several straight lines with the same slope as $1/E_{{\rm QG},n}^{n}$ but with different intercepts (i.e., different intrinsic emission times; see also Refs. [114, 115, 116]). However, photons from different GRBs fall on the same line, which still implies that the intrinsic time lags between the high-energy photons and the low-energy (trigger) photons are much the same for these GRBs. Chang et al. [23] made use of the magnetic jet model to estimate the intrinsic emission time delay between high- and low-energy photons from GRBs. However, the magnetic jet model depends on some specific theoretical parameters, and thus introduces uncertainties on the LIV results. In 2017, Wei et al. [29] first proposed that GRB 160625B, the burst having a well-defined transition from positive to negative spectral lags (see Figure 3), provides a good opportunity not only to disentangle the intrinsic time lag problem but also to put new constraints on LIV. The spectral lag is conventionally defined positive when high-energy photons arrive earlier than low-energy photons, while a negative lag corresponds to a delayed arrival of high-energy photons. As discussed above, the LIV-induced time delay $\Delta t_{\rm LIV}$ is likely to be accompanied by a potential intrinsic energy-dependent time lag $\Delta t_{\rm int}$ due to unknown properties of the source. Therefore, the observed time lag between two different energy bands of a GRB should consist of two parts,

Since the observed time lags of most GRBs have a positive energy dependence (e.g., Refs. [126, 127]), Wei et al. [29] approximated the observer-frame relation of the intrinsic time lag and the energy $E$ as a power law with positive dependence,

where $\tau>0$ and $\alpha>0$, and where $E_{0}=11.34$ keV is the median value of the lowest reference energy band (10–12 keV). We emphasize that the intrinsic positive time lag implies an earlier arrival time for the higher energy photons. Also, when the subluminal case ($s_{\pm}=+1$) is considered, high-energy photons would arrive on Earth after low-energy ones, implying a negative spectral lag due to the LIV effects. As the Lorentz-violating term becomes dominant at higher energy scales, the positive correlation between the time lag and the energy would gradually trend in an opposite way. The combined contributions from the intrinsic time lag and the LIV-induced time lag can therefore produce the observed lag behavior with a turnover from positive to negative lags [29]. By fitting the spectral lag behavior of GRB 160625B, Wei et al. [29, 30] obtained both a reasonable formulation of the intrinsic energy-dependent time lag and comparatively robust limits on the QG energy scale and the coefficients of the Standard Model Extension. The $1\sigma$ confidence-level lower limits are $E_{{\rm QG},1}>0.5\times 10^{16}~{}\mathrm{GeV}$ and $E_{{\rm QG},2}>1.4\times 10^{7}~{}\mathrm{GeV}$ for linear and quadratic LIV, respectively. The spectral lag data of GRB 160625B does not have the current best sensitivity to LIV constraints, but the analysis method, when applied to future bright short GRBs with similar spectral-lag transitions, may result in more stringent constraints on LIV.

• •

  Active galactic nuclei

Thanks to their fast flux variations (hours to years), cosmological distances, and very high energy (VHE, $E\geq 100~{}\mathrm{GeV}$) gamma-rays, TeV flares of AGNs have also been viewed as very effective probes for searching for the LIV-induced vacuum dispersions. It is worth pointing out that testing LIV with both GRBs and flaring AGNs is of great fundamental interest. GRBs can be detected at very large distances (up to $z\sim 8$), but with very limited high-energy ($E>$ tens of GeV) photons. On the contrary, AGN flares can be well observed with large statistics of photons up to a few tens of TeV. But due to extinction of high-energy photons by extragalactic background light, TeV detections are limited to those sources with relatively low redshifts $z\leq 0.5$. Hence, GRBs and flaring AGNs are mutually complementary in testing LIV, and they allow to test different redshift and energy ranges. There have been some resulting constraints on LIV using TeV observations of bright AGN flares, including the Whipple analysis of the flare of Mrk 421 [34], the MAGIC and H.E.S.S. analyses of the flares of Mrk 501 [128, 129, 130], and the H.E.S.S. analysis of the flare of PKS 2155-304 [131, 132]. The current best limit for the linear case considering a subluminal LIV effect obtained with AGNs is from the H.E.S.S. analysis of the PKS 2155-304 flare data, namely $E_{{\rm QG},1}>2.1\times 10^{18}~{}\mathrm{GeV}$ [132]. For the quadratic LIV, the best limits derived from AGNs have been set by H.E.S.S.’s observation of the TeV flare of Mrk 501. The reported limits are $E_{{\rm QG},2}>8.5\times 10^{10}~{}\mathrm{GeV}$ ($E_{{\rm QG},2}>7.3\times 10^{10}~{}\mathrm{GeV}$) for the subluminal (superluminal) case [130].

• •

  Pulsars

A third class of astrophysical sources used for the time-of-flight tests on gamma-rays are pulsars. When it comes to testing the quadratic LIV term, having a VHE emission will compensate for a short of distance. Gamma-ray pulsars, albeit being detected many orders of magnitude closer than GRBs or AGNs, have the advantage of precisely periodic flux variation, as well as the fact that they are the only stable candidate astrophysical sources for such time-of-flight studies. Sensitivity to LIV can therefore be improved by simply observing longer. Additionally, since the timing of the pulsar is carefully studied throughout the electromagnetic spectrum, energy-dependent time delays induced by propagation effects can be more easily distinguished from intrinsic delays. First limits on LIV using gamma-ray radiation from the galactic Crab pulsar were obtained from the observation of the Energetic Gamma-Ray Experiment Telescope (EGRET) onboard the Compton Gamma-Ray Observatory (CGRO) at energies above 2 GeV [35], and improved by the VERITAS data above 120 GeV [133, 134]. Recently, the MAGIC collaboration presented the best limits derived from pulsars by studying the Crab pulsar emission observed up to TeV energies, yielding $E_{{\rm QG},1}>5.5\times 10^{17}~{}\mathrm{GeV}$ ($E_{{\rm QG},1}>4.5\times 10^{17}~{}\mathrm{GeV}$) for a linear, and $E_{{\rm QG},2}>5.9\times 10^{10}~{}\mathrm{GeV}$ ($E_{{\rm QG},2}>5.3\times 10^{10}~{}\mathrm{GeV}$) for a quadratic LIV, for the subluminal (superluminal) case, respectively [135].

The most important constraints obtained so far with vacuum dispersion time-of-flight measurements of various astrophysical sources are summarized in Table 1. The most stringent lower limits to date on the linear and quadratic LIV energy scales were set by the observation of GRB 090510 with Fermi/LAT. The values for the subluminal (superluminal) case are $E_{{\rm QG},1}>9.3\times 10^{19}~{}\mathrm{GeV}$ ($E_{{\rm QG},1}>1.3\times 10^{20}~{}\mathrm{GeV}$) and $E_{{\rm QG},2}>1.3\times 10^{11}~{}\mathrm{GeV}$ ($E_{{\rm QG},2}>9.4\times 10^{10}~{}\mathrm{GeV}$) [25]. Clearly, these vacuum dispersion studies using gamma rays in the GeV–TeV range offer us at present with the best opportunity to search for Planck-scale modifications of the dispersion relation. Unfortunately, while they provide meaningful bounds for the linear ($n=1$) modification, they are much weaker for deviations that arise at the quadratic ($n=2$) order.

In QG theories that invoke LIV, the Charge-Parity-Time (CPT) theorem, i.e., the invariance of the laws of physics under charge conjugation, parity transformation, and time reversal, no longer holds. Note that the fact CPT does not hold does not imply that it should be violated. In the absence of Lorentz invariance, the CPT invariance, if needed, should be imposed as an additional assumption. In the effective field theory approach [136], the Lorentz- and CPT-violating dispersion relation for photon propagation can be parameterized as

where $\pm$ represents the left- or right-handed circular polarization states of the photon, and $\eta$ is a dimensionless parameter that needs to be constrained. In LIV but CPT invariant theories, the parameter $\eta$ exactly vanishes. In this sense, such tests might be less general than the ones based on vacuum dispersion. The linear polarization can be decomposed into left- and right-handed circular polarization states. For $\eta\neq 0$, photons with opposite circular polarizations have slightly different group velocities, which leads to a rotation of the polarization vector of a linearly polarized wave. This effect is known as vacuum birefringence. The rotation angle propagating from the source at redshift $z$ to the observer can be derived as [47, 49]

where $E$ is the observed photon energy, and

Generally speaking, it is impossible to know the intrinsic polarization angles in the emission of photons of different energies from a given source. If one had this information, evidence for vacuum birefringence (i.e., an energy-dependent rotation of the polarization plane) can be examined by measuring differences between the known intrinsic polarization angle and the observed polarization angles at different energies. However, even without such knowledge, the birefringent effect can still be constrained for polarized sources at arbitrary cosmological distances, because the differential rotation acting on the polarization angle as a function of energy would add opposite oriented polarization vectors, effectively erasing most, if not all, of the observed polarization signal. Therefore, the detection of the polarization signal can put an upper bound on such a possible violation.

Observations of linear polarization from distant sources have been widely used to place upper limits on the birefringent parameter $\eta$. The vacuum birefringence constraints arise from the fact if the rotation angle (Eq. 12) differs by more than $\pi/2$ over an energy range ($E_{1}<E<E_{2}$), then the net polarization of the signal would be severely suppressed and well below any observed value. Hence, the measurement of polarization in a given energy band implies that the differential rotation angle $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ should not be larger than $\pi/2$.

Previously, Gleiser & Kozameh [38] set an upper bound of $\eta<10^{-4}$ by analyzing the linearly polarized ultraviolet light from the distant radio galaxy 3C 256. Much more stringent limits, $\eta<10^{-14}$, have been obtained by using the linear polarization detection in the gamma-ray emission of GRB 021206 [43, 44]. However, the originally reported detection of high polarization from GRB 021206 [137] has been refuted by the re-analyses of the same data [138, 139]. Maccione et al. [140] used the hard X-ray polarization measurement of the Crab Nebula to get a constraint of $\eta<9\times 10^{-10}$. Laurent et al. [47] used a report of polarized soft gamma-ray emission from GRB 041219A to derive a stronger limit of $\eta<1.1\times 10^{-14}$ (see also Ref. [48]). But again, this claimed polarization detection [141, 142, 143] has been disputed (see the explanations in Ref. [49]). That is, the previous reports of the gamma-ray polarimetry for GRB 041219A are controversial and, thus, the arguments for the limits on $\eta$ given by Refs. [47, 48] are still open to questions.

Contrary to those disputed reports, the evidences of linearly polarized gamma-ray emission detected by the gamma-ray burst polarimeter (GAP) onboard the Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) are convincing, and thus these detections can be used to set more reliable limits on the birefringent parameter [49]. IKAROS/GAP detected gamma-ray polarizations of three GRBs with high significance levels, with a linear polarization degree of $\Pi=27\pm 11\%$ for GRB 100826A [144], $\Pi=70\pm 22\%$ for GRB 110301A, and $\Pi=84^{+16}_{-28}\%$ for GRB 110721A [145]. The detection significance are $2.9\sigma$, $3.7\sigma$, and $3.3\sigma$, respectively. Toma et al. [49] set the upper limit of the differential rotation angle $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ to be $\pi/2$, and obtained a severe upper limit on $\eta$ in the order of $\mathcal{O}(10^{-15})$ from the reliable polarimetric data of these three GRBs. However, the GRBs had no direct redshift measurement, they used a redshift estimate based on an empirical luminosity relation. Utilizing the real redshift measurement ($z=1.33$) together with the polarization data of GRB 061122, Götz et al. [50] obtained a stricter limit ($\eta<3.4\times 10^{-16}$) on a possible LIV. Götz et al. [51] used the most distant polarized burst (up to $z=2.739$), GRB 140206A, to obtain the deepest limit to date ($\eta<1.0\times 10^{-16}$) on the possibility of LIV.

It is worth noting that most of previous polarization constraints were derived under the assumption that the differential rotation angle $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ is smaller than $\pi/2$. However, Lin et al. [52] gave a detailed analysis on the evolution of GRB polarization arising from the vacuum birefringent effect, and showed that a considerable amount of the initial polarization degree (depending both on the photon energy band and the photon spectrum) can be conserved even if $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ is approaching to $\pi/2$. This is incompatible with the common belief that $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$ should not be too large when high polarization is detected. Therefore, Lin et al. [52] suggested that it is unsuitable to constrain the birefringent effect by simply setting $\pi/2$ as the upper limit of $|\Delta\phi(E_{2})-\Delta\phi(E_{1})|$. Lin et al. [52] applied their formulae for the polarization evolution to some true GRB events, and obtained the most stringent upper limit to date on the birefringent parameter from the polarimetric data of GRB 061122, i.e., $\eta<0.5\times 10^{-16}$. Following the analysis method presented in Ref. [52] and utilizing the recent measurements of gamma-ray linear polarization of GRBs, Wei [55] updated constraints on a possible LIV through the vacuum birefringent effect, and thereby improving previous results by factors ranging from two to ten.

If both the intrinsic polarization angle $\phi_{0}$ and the rotation angle induced by the birefringent effect $\Delta\phi_{\rm LIV}(E)$ are considered here, the observed linear polarization angle at a certain $E$ from a source should be

Assuming that all photons in the observed energy bandpass are emitted with the same (unknown) intrinsic polarization angle, we then expect to observe the birefringent effect as an energy-dependent linear polarization vector. To constrain the birefringent parameter $\eta$, Fan et al. [45] looked for a similar energy-dependent trend in the multiwavelength polarization observations of the optical afterglows of GRB 020813 and GRB 021004. By fitting the spectropolarimetric data of these two GRBs, Fan et al. [45] obtained constraints on both $\phi_{0}$ and $\eta$ (see Figure 4). At the $3\sigma$ confidence level, the combined limit on $\eta$ from two GRBs is $-2\times 10^{-7}<\eta<1.4\times 10^{-7}$. It is clear from Eq. (12) that the higher energy band of the polarization observation and the larger distance of the polarized source, the greater sensitivity to small values of $\eta$. As expected, the optical polarization data of GRB afterglow obtained a less stringent constraint on $\eta$ [45].

Table 2 presents a summary of the corresponding limits on LIV from the polarization measurements of astrophysical sources. The hitherto most stringent constraints on the birefringent parameter, $\eta<\mathcal{O}(10^{-16})$, have been obtained by the detections of linear polarization in the prompt gamma-ray emission of GRBs [50, 51, 52, 55].

By comparing Eqs. (1) and (11), we can derive the conversion from $\eta$ to the Limit on the linear LIV energy scale, i.e., $E_{{\rm QG},1}=\frac{E_{\rm pl}}{2\eta}$. With the data presented in Tables 1 and 2, it is easy to compare the recent achievements in sensitivity of time-of-flight measurements versus polarization measurements. For the subluminal case, the time-of-flight analysis of multi-GeV photons from GRB 090510A detected by Fermi/LAT yielded the strictest limit on the linear LIV energy scale, $E_{{\rm QG},1}>9.3\times 10^{19}~{}\mathrm{GeV}$, which corresponds to $\eta<0.07$ [25]. Obviously, this time-of-flight constraint is many orders of magnitude weaker than the best polarization constraint. However, time-of-flight constraints are essential in a broad-based search for nonbirefringent Lorentz-violating effects.