arXiv:1905.03413v1 [astro-ph.HE] 9 May 2019 e ( GeV iltos(I)icuevcu iprinadvacuum and dispersion ( vacuum birefringence include (LIV) violations hoispeitta oet ymtymyb bro- scale be energy may of symmetry the Lorentz symmetry gravity at that quantum fundamental ken predict various the However, theories relativity. is Einstein’s invariance Lorentz INTRODUCTION 1 MNRAS otlc´ usl 2011 Russell Kosteleck´y & 2006 Bluhm ZZZ form original in YYY; Received XXX. Accepted hs ffcsaeepce ob eytn tobservable at tiny very be to expected energies are effects these (see systems of range wide a in performed invarianc Lorentz been from have deviations for searches perimental 1991 neeg-eedn pe flgt ec,tearrival- the ⋆ Hence, light. of causes sensi- speed dispersion provide Vacuum energy-dependent symmetry. therefore Lorentz an can of baselines tests tive long measurements the Astronomical involving detectable. become and tances

e osrit nLrnzIvrac ilto with Bursts Violation -Ray Invariance Polarized Lorentz on Constraints New c 2 1 u-i Wei Jun-Jie unx e aoaoyfrRltvsi srpyis Gua Astrophysics, Relativistic for Sciences, Laboratory of Key Academy Guangxi Chinese Observatory, Mountain Purple -al [email protected] (JJW) [email protected] E-mail: 08TeAuthors The 2018 ntepoo etr intrso oet invariance Lorentz of signatures sector, photon the In , 1995 otlc´ aul1989 Samuel Kosteleck´y & 000 ≪ ; , mln-aei ta.1998 al. et Amelino-Camelia ; 1 M – mln-aei 2013 Amelino-Camelia otlc´ ee 2008 Mewes Kosteleck´y & 7 Pl 21)Pern 0My21 oplduigMRSL MNRAS using Compiled 2019 May 10 Preprint (2018) hycnacmlt vrlredis- large over accumulate can they , 1 , 2 ⋆ o compilation). a for general e words: Key b affected greatly GRBs. fro not of data are ranging parameters constraints polarimetric factors spectral our by GRB that photons prove our in we involving to the addition, LIV method of to 60% this sensitivities than Applying existing more conserved. and be suppressed, can severely not is polarization fteplrzto etro iheeg htn ihrsett t rota to respect relative with the photons if energy high that than larger of believed is vector photons is polarization the It linear of of sources. vector polarization astrophysical birefringence, the from vacuum of yield rotation energy-dependent can an (LIV) violations invariance Lorentz ABSTRACT fwo aehg eeto infiac.Floigtemto shown method the Following bur w significance. work, gamma-ray this m detection in In the high polarization large. have Hence, emission too level. whom be prompt observed of not the of should as detections ∆Θ high recent that as implies be polarization not high could and depleted I ffc o ahGB n ofimta,ee f∆ sapproaching is ∆Θ if even that, confirm and GRB, each for effect LIV ( 2016 ; M otlc´ Potting Kosteleck´y & ; ,w ieadtie aclto nteplrzto vlto rsn fro arising evolution polarization the on calculation detailed a give we ), Pl asn2014 Tasson ; .Ee though Even ). ≃ atnl 2005 Mattingly 1 . 22 srpril hsc rvtto oaiain–gmaryburs gamma-ray – polarization – gravitation – physics astroparticle × .Ex- ). 10 π/ giUiest,Nnig500,China 530004, Nanning University, ngxi 19 e ; ,te h e oaiaino h inlwudb significantly be would signal the of polarization net the then 2, ajn 104 China 210034, Nanjing 2015 2008 1998 et.Ti a eudrto ytefc htteformer the factor that a by fact latter the the than by sensitive polariza- more understood are from be can result measure- This dispersion LIV ments. vacuum than on rather measurements limits tion stringent more 2017 cally, Krawczynski & Kislat 2012 al. et Toma 2007 2007 2001 Kozameh & Gleiser 2017 2012 2013 al. et Chang invco falnal oaie ih.Lrnzinvari- Lorentz observa- (e.g., light. polarimetric sources through astrophysical polarized tested of tions be linearly polariza- therefore a can the ance of of vector rotation wavelength-dependent tion a to leads e iutnosyfo natohsclsuc a be can source emit- (e.g., astrophysical invariance energies Lorentz an test different to used from with simultaneously photons ted of differences time ; , ; ; ; ; ; 2013 uioie l 2009 al. et Gubitosi otlc´ ee 2008 Mewes Kosteleck´y & alpuo 2005 Pavlopoulos hn a2015 Ma & Zhang i a2018 Ma & Liu li armts2013 Mavromatos & Ellis ; irfnv2003 Mitrofanov ; ; G t ta.2013 al. et ¨ otz eio ta.2012 al. et Nemiroff .Smlry aumbirefringence vacuum Similarly, ). ; ; ; ; arn ta.2011 al. et Laurent otlc´ ee 2001 Mewes Kosteleck´y & li ta.2006 al. et Ellis ; ; 2017a al. et Wei aosne l 2004 al. et Jacobson remne l 2018 al. et Friedman , ; 2009 abn uln1999 Pullin & Gambini , mln-aei tal. et Amelino-Camelia netite nthe in uncertainties y yplrzdemission polarized ly 2014 ilt&Krawczynski & Kislat ; a flwenergy low of hat inage(∆Θ) angle tion ta polarization itial w otn In ten. to two m boe l 2009 al. et Abdo A ; ; hc eut in results which T t GB) all (GRBs), sts ; to i ta.2016 al. et Lin E ∝ aueetof easurement aieo tal. et Vasileiou , ao Piran & Jacob tl l v3.0 file style X ; b eimprove we , π/ in tce 2011 Stecker 1 ; /ω assemble e ; e Wu & Wei ,tenet the 2, i tal. et Lin a tal. et Fan where , .Typi- ). , the m 2006 t: ω ; ; , ; ; 2 Wei is the frequency of the observed light (Kosteleck´y& Mewes 2 GENERAL FORMULAE 2009). The Lorentz-violating dispersion relation for photon propa- gation takes the generalized form (Myers & Pospelov 2003)

Owing to the high energy polarimetry of prompt emis- 2 2 2ξ 3 sion and their large cosmological distances, gamma-ray E± = p ± p , (1) Mpl bursts (GRBs) have been viewed as promising sources for 19 searching for LIV-induced vacuum birefringence (Mitrofanov where Mpl ≈ 1.22 × 10 GeV is the Planck energy, ± repre- 2003; Jacobson et al. 2004; Kosteleck´y& Mewes 2006, 2007, sents different circular polarization states, and ξ is a dimen- 2013; Fan et al. 2007; Laurent et al. 2011; Stecker 2011; sionless parameter characterizing the order of magnitude of Toma et al. 2012; G¨otz et al. 2013, 2014; Lin et al. 2016). If the LIV effect. If ξ =6 0, then the dispersion relation will lead Lorentz invariance is broken then group velocities of photons to slightly different propagation velocities for different po- with right- and left-handed circular polarizations should dif- larization states. Thus, the polarization vector of a linearly fer slightly, leading to a phase rotation of linear polarization. polarized light will rotate during the propagation. This effect Observations of linear polarization can therefore place strict is known as vacuum birefringence. The rotation angle during limits on the birefringence parameter (ξ). It is believed that the propagation from the source at redshift z to the observer if high polarization is observed, then the difference of ro- is expressed as (Laurent et al. 2011; Toma et al. 2012) tation angles (denoted by ∆Θ) of polarization vectors be- k2 z (1 + z′)dz′ tween the highest and lowest energy photons should not be ∆θ(k)= ξ , (2) ′ 3 too large. Otherwise, the net polarization of the signal will MplH0 Z0 Ωm(1 + z ) + ΩΛ p be significantly depleted and could not be as high as the where k is the energy of the observed light. Here we observed level. Toma et al. (2012) set the upper limit of the use the standard cosmological parameters: H0 = 67.3 relative rotation angle ∆Θ to be π/2, and obtained a severe −1 −1 km s Mpc , Ωm = 0.315, and ΩΛ = 0.685 upper bound on the birefringence parameter ξ in the order of −15 (Planck Collaboration et al. 2014). O(10 ) from the polarimetric data of GRB 110721A. How- Following the process of calculating the polarization ever, GRB 110721A had no direct redshift measurement at evolution in Lin et al. (2016), we assume that there is a that time, they used a distance estimate based on the empir- beam of non-coherent light emitted from a source, in which ical luminosity relation. By using the redshift determination the propagation direction is selected as the z-axis and the (z = 1.33) together with the polarization measurement of polarization direction is in the xy-plane. The intensity of GRB 061122, G¨otz et al. (2013) obtained a much stricter −16 photons with electric vector in the infinitesimal azimuth an- limit, ξ < 3.4 × 10 . The most distant polarized burst, gle interval dθ and with energy in the infinitesimal interval GRB 140206A, which has a confirmed redshift measurement dk is given by of z = 2.739, yielded the deepest limit to date (G¨otz et al. 2014). All these limits were based on the assumption that dj(θ,k)= j0f(θ)kN(k)dθdk , (3) the differential rotation angle ∆Θ is smaller than π/2. How- where j0 denotes a normalized constant, N(k) is the photon ever, Lin et al. (2016) calculated the evolution of GRB po- spectrum, and f(θ) represents a periodic function of θ with larization arising from the LIV effect, and found that more period π. Because of the positive correlation between the than 60% of the initial polarization (depending both on the photon intensity and the square of the electric vector, the photon spectrum and the photon energy range) can still be intensity projected onto the direction of azimuth angle ϕ can conserved at ∆Θ ≤ π/2. It is interesting to constrain the be written as LIV effect under the framework of calculating GRB polar- 2 ization evolution. djϕ(θ,k)= j0f(θ)kN(k) cos (ϕ − θ)dθdk . (4) The initial total intensity of photons polarized along the direction ϕ is then expressed as In this paper, following the calculation proposed in Lin et al. (2016) and using the latest measurements of linear π k2 2 − polarization in GRBs, we update constraints on a possible I(ϕ)= djϕ(θ,k)= dθ dk j0f(θ)kN(k) cos (ϕ θ) , Z Z0 Zk1 deviation from Lorentz invariance through the vacuum bire- (5) fringence effect, and thereby improve existing sensitivities to LIV involving photons by factors ranging from two to ten. where k1

MNRAS 000, 1–7 (2018) Constraints on LIV with Polarized GRBs 3

δ-functions is adopted (Lin et al. 2016): 041219A (Π = 98±33%), but again this result was criticized

+∞ by more detailed analyses (McGlynn et al. 2007; G¨otz et al. f(θ)= δ(θ − nπ) . (7) 2009). n=X−∞ Fortunately, evidence of polarized gamma-ray emission from GRBs has been accumulated since 2011. The gamma- Substituting Equation (7) into Equation (5), the initial pho- ray burst polarimeter (GAP) onboard the Interplanetary ton intensity simplifies to Kite-craft Accelerated by Radiation Of the Sun (IKAROS) k2 2 detected a polarization signal with a polarization degree I(ϕ)= j0 cos ϕ dk kN(k) . (8) of Π = 27 ± 11% from GRB 100826A, and a null po- Zk1 larization degree is ruled out with 2.9σ confidence level It is obvious that Imin = I(π/2) = 0 and Imax = I(0), which (C.L.) (Yonetoku et al. 2011). IKAROS/GAP also detected implies that the initial polarization degree Π is 100%, i.e., gamma-ray polarizations of two other bursts with high sig- the initial polarization state is initially completely polarized. nificance levels, with Π = 70 ± 22% for GRB 110301A, and Vacuum birefringence results in a rotation of the plane +16 Π = 84−28% for GRB 110721A (Yonetoku et al. 2012). The of linear polarization for photons at different energies emit- detection significance is 3.7σ and 3.3σ, respectively. Using ted with the same initial polarization degree. Let ∆Θ ≡ a different instrument, the Imager on Board the Interna- ∆θ(k2) − ∆θ(k1) to be the relative rotation angle of the po- tional Gamma-Ray Astrophysics Laboratory (INTEGRAL) larization vector of high energy photons with respect to that Satellite (IBIS), G¨otz et al. (2013) reported a polarization of low energy photons, then Equation (2) can be rephrased measurement in the prompt emission of GRB 061122. They as put a lower limit on its polarization degree of Π > 60% at k2 68% C.L.. One other highly polarized burst (GRB 140206A) ∆θ(k)=∆Θ 2 2 . (9) with the polarization degree of Π > 48% at 68% C.L. was k2 − k1 also detected by INTEGRAL/IBIS (G¨otz et al. 2014). Re- The received photon intensity can then be derived by replac- cently, Chattopadhyay et al. (2017) presented the polariza- ing f(θ) with f(θ +∆θ(k)) in Equation (5), i.e., tion data for the brightest 11 bursts detected by CZTI on-

π k2 board during the first year of operation. They ′ 2 I (ϕ)= dθ dk j0f (θ +∆θ(k)) kN(k) cos (ϕ − θ) . found that only 7 of these show clear polarization signatures Z Z 0 k1 with ≥ 3σ detection significance for 4 bursts (GRB 160131A, (10) GRB 160802A, GRB 160821A, and GRB 160910A) and ∼ 2.5σ significance for another 3 bursts (GRB 160106A, Substituting Equations (7) and (9) into Equation (10), we 1 have the observed photon intensity GRB 160325A, and GRB 160509A). All of these detections are evidences that the prompt emission of GRBs are highly k2 2 ′ 2 ∆Θk polarized. Therefore, the reliable observations of gamma-ray I (ϕ)= j0 dk kN(k) cos ϕ + 2 2 . (11) Zk1  k2 − k1  linear polarization mentioned here enable us to obtain strict limits on LIV. Accordingly, the observed polarization degree is In assembling the GRB sample, we require the mem- ′ ′ ′ Imax − Imin bers have an independent polarization detection with high Π = ′ ′ , (12) Imax + Imin level of significance. We exclude those bursts whose polariza- ′ ′ tion measurements remain under debate (e.g., GRB 930131, where Imax and Imin are the maximum and minimum values ′ GRB 960924, GRB 021206, GRB 041219A). Lastly, our sam- of I (ϕ), respectively. ple includes 12 GRBs for which strong evidence of gamma- ray linear polarization exists. All of these data are obtained from previously published studies, which are shown in Ta- 3 POLARIZATION MEASUREMENTS OF ble 1. The first eight columns include the following informa- PROMPT GRB EMISSION tion for each GRB: (1) the burst name; (2) the instrumen- The measurement of polarization in the prompt gamma- tation for gamma-ray polarimetry; (3) the energy rang in ray emission of GRBs has always been challenging (see which polarization is observed; (4) the observed polarization Covino & Gotz 2016; McConnell 2017 for reviews). The degree Π (with the corresponding 68% C.L. uncertainty); the first reported highly polarized burst was GRB 021206, spectral parameters (with 90% C.L. uncertainties), including with a linear polarization degree of Π = 80 ± 20% (5) the low-energy photon index α, (6) the high-energy pho- (Coburn & Boggs 2003). However, subsequent re-analyses ton index β, (7) the spectral peak energy Ep or the breaking of the same data could not confirm significant polariza- energy Ec; and (8) the redshift. tion signal (Rutledge & Fox 2004; Wigger et al. 2004). An- Of these 12 GRBs, five GRBs have redshift measure- other early attempt to measure linear polarization during ments, while others have none. The empirical luminosity re- the prompt phase of GRBs was performed by Willis et al. lation (the well-known Amati relation; Amati et al. 2002) is (2005), who reported evidence of high polarization in GRB therefore applied to estimate the redshifts of the other seven 930131 and GRB 960924, Π > 35% and Π > 50%, respec- tively. But unfortunately the authors’ estimation method did not allow them to statistically constrain such results, 1 Note that the polarization measurements of the remaining four but called for further independent measurements to confirm bursts (GRB 151006A, GRB 160607A, GRB 160623A, and GRB whether these two bursts are highly polarized. Kalemci et al. 160703A) with low significance levels are not included in our sam- (2007) reported a high level of polarization for GRB ple.

MNRAS 000, 1–7 (2018) 4 Wei

Table 1. Limits on LIV from GRB polarization measurements

GRB Instrument Energy Π Spectral parameters (90% C.L.) z Refs.b ξ

(keV) (68% C.L.) α β Ep/Ec (68% C.L.) (keV)

+0.04 +20 −17 061122 INTEGRAL/IBIS 250—800 > 60% −1.15−0.04 — 221−20 1.33 1, 1, 1 < 5.2 × 10 +0.06 +0.10 +134 a +1.4 −14 100826A IKAROS/GAP 70—300 27 ± 11% −1.31−0.05 −2.10−0.20 606−109 0.083 2, 3, 4 1.2−0.7 × 10 +0.02 +0.04 +1.85 a +5.4 −15 110301A IKAROS/GAP 70—300 70 ± 22% −0.81−0.02 −2.70−0.05 106.8−1.75 0.082 5, 6, 4 4.3−2.3 × 10 +16 +0.02 +0.02 +26.5 +4.0 −16 110721A IKAROS/GAP 70—300 84−28% −0.94−0.02 −1.77−0.02 372.5−23.6 0.382 5, 7, 8 5.1−5.1 × 10 +0.15 +47 −16 140206A INTEGRAL/IBIS 200—400 > 48% 1.10−0.15 — 114−26 2.739 9, 9, 9 < 1.0 × 10 +0.07 +0.14 +45 a +1.4 −15 160106A AstroSat/CZTI 100—300 68.5 ± 24% −0.53−0.06 −2.31−0.21 400−40 0.091 10, 10, 4 3.4−1.8 × 10 +0.04 +0.07 +518 +2.0 −16 160131A AstroSat/CZTI 100—300 94 ± 31% −1.16−0.04 −1.56−0.10 586−259 0.972 10, 10, 11 1.2−1.2 × 10 +0.07 +0.20 +25 a +1.0 −15 160325A AstroSat/CZTI 100—300 58.75 ± 23.5% −0.71−0.06 −2.26−0.30 238−22 0.161 10, 10, 4 2.3−0.9 × 10 +0.02 +0.03 +12 +2.2 −16 160509A AstroSat/CZTI 100—300 96 ± 40% −0.75−0.02 −2.13−0.03 334−10 1.17 10, 10, 12 0.8−0.8 × 10 +0.04 +0.10 +17 a +1.7 −15 160802A AstroSat/CZTI 100—300 85 ± 29% −0.61−0.04 −2.40−0.13 280−14 0.105 10, 10, 4 2.0−2.0 × 10 +0.01 +0.03 +25 a +1.7 −15 160821A AstroSat/CZTI 100—300 48.7 ± 14.6% −0.97−0.01 −2.25−0.03 866−24 0.047 10, 10, 4 8.9−1.7 × 10 +0.03 +0.05 +13 a +7.6 −16 160910A AstroSat/CZTI 100—300 93.7 ± 30.92% −0.36−0.03 −2.38−0.06 330−13 0.272 10, 10, 4 4.7−4.7 × 10

(a) The redshifts of the seven GRBs are estimated by the luminosity relation. (b) The references appear in the following order: polarization, spectral parameters, and redshift: [1] G¨otz et al. (2013); [2] Yonetoku et al. (2011); [3] Golenetskii et al. (2010); [4] This work; [5] Yonetoku et al. (2012); [6] Foley (2011); [7] Tierney & von Kienlin (2011); [8] Berger (2011); [9] G¨otz et al. (2014); [10] Chattopadhyay et al. (2017); [11] de Ugarte Postigo et al. (2016); [12] Tanvir et al. (2016).

GRBs. We use the observed fluence and Ep of the seven bursts (GRB 100826A: 20–10000 keV fluence 3.0 × 10−4 erg −2 cm and Ep = 606 keV (Golenetskii et al. 2010); GRB 110301A: 10–1000 keV fluence 3.65 × 10−5 erg cm−2 and Ep = 106.8 keV (Foley 2011); GRB 160106A: 100–300 keV −5 −2 160910A fluence 3.5 × 10 erg cm and Ep = 400 keV; GRB −6 −2 160325A: 100–300 keV fluence 7.6×10 erg cm and Ep = ' 238 keV; GRB 160802A: 100–300 keV fluence 2.2 × 10−5 erg −2 cm and Ep = 280 keV; GRB 160821A: 100–300 keV flu- −4 −2 ence 2.0×10 erg cm and Ep = 866 keV; GRB 160910A: −6 −2 100–300 keV fluence 4.2 × 10 erg cm and Ep = 330 keV (Chattopadhyay et al. 2017)) to calculate the intrinsic peak energies and the isotropic gamma-ray energies for dif- ferent redshifts. By requiring that the bursts enter the 2σ region of the relation, we derive z ≥ 0.083 for GRB 100826A, [ ] z ≥ 0.082 for GRB 110301A, z ≥ 0.091 for GRB 160106A, ′ z ≥ 0.161 for GRB 160325A, z ≥ 0.105 for GRB 160802A, Figure 1. Polarization degree Π as a function of the relative z ≥ 0.047 for GRB 160821A, and z ≥ 0.272 for GRB rotation angle ∆Θ for different GRBs with known photon spectra 160910A. Hereafter, the lower limits of redshifts are con- and energy ranges. servatively adopted. We refer the reader to Deng & Zhang (2014) for more details on the redshift estimation. from this plot that although the net polarization degree Π′ decreases rapidly with increasing ∆Θ at ∆Θ ≤ π, more than 4 CONSTRAINTS ON LIV 60% of the initial polarization degree can still be conserved at ∆Θ ≤ π/2. Note that with the chosen initial polarization The photon spectrum of each GRB presented in Table 1 pattern (the sum of δ-functions) and the typical spectral can be well fitted by the Band function (Band et al. 1993) content of GRBs, the polarization measure is a multival- or a power law with an exponential cutoff, i.e., N(k) ∝ ued function for relative rotation angles above π/2. This α k exp(−k/Ec). The third column of Table 1 is the energy is explicitly visualized in Figure 1. Thus, the polarization range in which polarization is observed. With the known measure as a function of ∆Θ can not be treated as an “in- photon spectrum and energy range, for any given relative ro- formative” one in the range ∆Θ > π/2 as being multivalued tation angle ∆Θ, we can numerically compute the maximum function for higher values of ∆Θ. In other words, in practice, and minimum values of I′(ϕ) according to Equation (11), once the relative rotation angle exceeds ∼ π/2 the polariza- and then calculate the polarization degree with Equation tion should be treated as the lost one. Additionally, due to (12). The observed polarization degree Π′ as a function of the differences of spectral parameters and energy ranges, the ∆Θ for different GRBs are shown in Figure 1. One can see polarization evolution of GRBs are somewhat different (see

MNRAS 000, 1–7 (2018) Constraints on LIV with Polarized GRBs 5

GRB 061122 GRB 100826A GRB 110301A ' ' '

-15 -15 -15

[10 ] [10 ] [10 ]

GRB 140206A GRB 160106A GRB 110721A ' ' '

-15 -15 -15

[10 ] [10 ] [10 ]

GRB 160325A GRB 160509A GRB 160131A ' ' '

-15 -15 -15

[10 ] [10 ] [10 ]

GRB 160802A GRB 160821A GRB 160910A ' ' '

-15 -15 -15

[10 ] [10 ] [10 ]

Figure 2. Polarization degree Π′ as a function of the birefringence parameter ξ for different GRBs. The blue shaded areas, estimated by 1000 Monte Carlo simulations, represent the uncertainties due to the observational uncertainties of spectral parameters. The black solid curves represent the average of the Monte Carlo results. The vertical and horizonal dashed lines mark the ξ values that correspond to the observed polarization degrees at 1σ C.L.. The red solid line regime corresponds to the 1σ constrained range of ξ.

Figure 1). That is, the polarization evolution depends both evolution through 1000 Monte Carlo simulations utilizing on the photon spectrum and the energy range (see Lin et al. uncertainties in the spectral parameters. The black solid 2016 for more explanations). curves show the polarization evolution from the average of the Monte Carlo simulations. The blue shaded areas repre- To show the resulting constraints on LIV more clearly, sent the uncertainties taking into account the spread of all we also plot the polarization as a function of the birefrin- the Monte Carlo simulation results, which indicate that the gence parameter ξ for different GRBs in Figure 2. To explore uncertainties of spectral parameters do not have a significant the influence of the observational uncertainties of spectral impact on the polarization evolution. With the observed po- parameters on the polarization evolution and then on the larization degrees, we can obtain strict limits on ξ at 68% test of LIV, we estimate uncertainties in the polarization

MNRAS 000, 1–7 (2018) 6 Wei

C.L. for each burst, which are displayed in column 10 of ACKNOWLEDGEMENTS Table 1. These results represent sensitivities improved by We thank the anonymous referee for constructive sugges- factors of 2 to 10-fold over existing bounds on a deviation tions. We also thank Prof. Xue-Feng Wu for helpful dis- from Lorentz invariance. cussions. This work is partially supported by the National Natural Science Foundation of China (grant Nos. U1831122, 11603076, 11673068, and 11725314), the Youth Innova- tion Promotion Association (2017366), the Key Research Program of Frontier Sciences (QYZDB-SSW-SYS005), the 5 SUMMARY AND DISCUSSION Strategic Priority Research Program “Multi-waveband grav- itational wave Universe” (grant No. XDB23000000) of the Violations of Lorentz invariance can produce vacuum bire- Chinese Academy of Sciences, and the “333 Project” and fringence, which leads to an energy-dependent rotation of the Natural Science Foundation (grant No. BK20161096) of the polarization plane of linearly polarized photons. It has Jiangsu Province. been confirmed that photons in the prompt emission of some GRBs are highly polarized. The polarization measurements of GRBs have been widely used to constrain LIV. Most of previous constraints were based on the assumption that REFERENCES the difference of rotation angles ∆Θ of polarization vectors between the highest and lowest energy photons is smaller Abdo A. 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