Exploring Anisotropic Lorentz Invariance Violation from the Spectral-Lag Transitions of Gamma-Ray Bursts_2

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Citation: Wei, J.-N.; Liu, Z.-K.; Wei,

J.-J.; Zhang, B.-B.; Wu, X.-F. Exploring

Anisotropic Lorentz Invariance

Violation from the Spectral-Lag

Transitions of Gamma-Ray Bursts.

Universe 2022,1, 0. https://doi.org/

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4.0/).

universe

Article

Exploring Anisotropic Lorentz Invariance Violation from the

Spectral-Lag Transitions of Gamma-Ray Bursts

Jin-Nan Wei 1,2,3, Zi-Ke Liu 4,5, Jun-Jie Wei 1,2,3,* , Bin-Bin Zhang 4,5 and Xue-Feng Wu 1,2

1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China;

weijn@pmo.ac.cn (J.-N.W.); xfwu@pmo.ac.cn (X.-F.W.)

2School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, China

3Guangxi Key Laboratory for Relativistic Astrophysics, Nanning 530004, China

4School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China;

zkliu@smail.nju.edu.cn (Z.-K.L.); bbzhang@nju.edu.cn (B.-B.Z.)

5Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing

210023, China

*Correspondence: jjwei@pmo.ac.cn

Abstract:

The observed spectral lags of gamma-ray bursts (GRBs) have been widely used to explore

possible violations of Lorentz invariance. However, these studies were generally performed by concen-

trating on the rough time lag of a single highest-energy photon and ignoring the intrinsic time lag at the

source. A new way to test nonbirefringent Lorentz-violating effects has been proposed by analyzing the

multi-photon spectral-lag behavior of a GRB that displays a positive-to-negative transition. This method

gives both a plausible description of the intrinsic energy-dependent time lag and comparatively robust

constraints on Lorentz-violating effects. In this work, we conduct a systematic search for Lorentz-violating

photon dispersion from the spectral-lag transition features of 32 GRBs. By ﬁtting the spectral-lag data of

these 32 GRBs, we place constraints on a variety of isotropic and anisotropic Lorentz-violating coefﬁcients

with mass dimension

6 and 8. While our dispersion constraints are not competitive with existing

bounds, they have the promise to complement the full coefﬁcient space.

Keywords: gamma-ray bursts; astroparticle physics; gravitation; quantum gravity

1. Introduction

Lorentz invariance, the foundational symmetry of Einstein’s relativity, has survived in a wide

range of tests over the past century [

]. However, many quantum gravity models seeking to unify

general relativity and quantum theory predict that Lorentz symmetry may be violated at energies

approaching the Planck scale

EPl =√¯hc5/G'

1.22

GeV [

–

]. While these energies

are unreachable experimentally, tiny deviations from Lorentz invariance at attainable energies

can accumulate to detectable levels over sufficiently large distances. Astrophysical observations

involving long propagation distances can therefore provide precision tests of Lorentz invariance.

In the photon sector, one effect of Lorentz invariance violation (LIV) is an energy-dependent

vacuum dispersion of light, which causes arrival-time delays of photons with different energies

originating from a given astrophysical source [

–

]. LIV models can also lead to vacuum

birefringence, which produces an energy-dependent rotation of the polarization vector of lin-

early polarized light [

–

]. Generally, these effects can be anisotropic, such that arrival-time

differences and polarization rotations possess a direction dependence and require observations

of point sources along many lines of sight or measurements of extended sources such as the

cosmic microwave background to fully explore the LIV model parameter space [15,16,36].

Universe 2022,1, 0. https://doi.org/10.3390/universe1010000 https://www.mdpi.com/journal/universe

arXiv:2210.03897v1 [astro-ph.HE] 8 Oct 2022

Universe 2022,1, 0 2 of 17

The Standard-Model Extension (SME) is an effective ﬁeld theory that characterizes Lorentz

and CPT violations at attainable energies [

–

]. It considers additional Lorentz and/or CPT-

violating terms to the SME Lagrange density, which can be ordered by the mass dimension

of the tensor operator [

]. Photon vacuum dispersion introduced by operators of dimension

(

4) is proportional to

(E/EPl)d−4

. Lorentz-violating operators with even

preserve CPT

symmetry, while those with odd

break CPT. All

(d−

coefﬁcients of odd

produce both

vacuum dispersion and birefringence, whereas for each even

there is a subset of

(d−

nonbirefringent but dispersive Lorentz-violating coefﬁcients. The latter can be well constrained

through vacuum dispersion time-of-ﬂight measurements. It should be stressed that at least

(d−

sources distributed evenly in the sky are needed to fully constrain the anisotropic

Lorentz-violating coefﬁcient space for a given

. In contrast, only one source is required to

fully constrain the corresponding coefﬁcient in the isotropic LIV limit. That is, the restriction to

isotropic LIV disregards d(d−2)possible effects from anisotropic violations at each d.

Thanks to their small variability time scales, large cosmological distances, and very high-

energy photons, gamma-ray bursts (GRBs) are viewed as one of the ideal probes for testing

Lorentz invariance through the dispersion method [

]. Direction-dependent limits on

several combinations of coefﬁcients for Lorentz violation have been placed using vacuum-

dispersion constraints from GRBs. For example, limits on combinations of the 25

nonbirefringent Lorentz-violating coefﬁcients have been derived by studying the dispersion

of light in observations of GRB 021206 [

], GRB 080916C [

], four bright GRBs [

GRB 160625B [

], and GRB 190114C [

]. Bounds on combinations of the 49

8 coefﬁcients

for nonbirefringent vacuum dispersion have been derived from GRB 021206 [

], GRB

080916C [

], GRB 160625B [

], and GRB 190114C [

]. However, most of these studies

limit attention to the time delay induced by nonbirefringent Lorentz-violating effects, while

ignoring possible source-intrinsic time delays. Furthermore, the limits from GRBs are based

on the rough time delay of a single highest-energy photon. To obtain reliable LIV limits, it is

desirable to use the true time lags of high-quality and high-energy light curves in different

energy multi-photon bands.

In two previous papers [

], we derived new direction-dependent limits on combinations

of coefficients for Lorentz-violating vacuum dispersion using the peculiar time-of-flight measure-

ments of GRB 160625B and GRB 190114C, which both have obvious transitions from positive

to negative spectral lags. Spectral lag, which is defined as the arrival-time difference of high-

and low-energy photons, is a ubiquitous feature in GRBs [

–

]. Conventionally, the spectral

lag is considered to be positive when high-energy photons arrive earlier than the low-energy

ones. By fitting the spectral-lag behaviors of GRB 160625B and GRB 190114C, we obtained both a

reasonable formulation of the intrinsic energy-dependent time lag and robust constraints on a

variety of isotropic and anisotropic Lorentz-violating coefficients with mass dimension

6 and

8 [

]. In this work, we analyze the spectral-lag transition features of 32

Fermi

GRBs [

], and

derive limits on photon vacuum dispersion for all of them. We combine these limits with previous

results in order to fully constrain the nonbirefringent Lorentz-violating coefficients with

and 8.

The paper is structured as follows. In Section 2, we brieﬂy describe the theoretical frame-

work of vacuum dispersion in the SME. In Section 3, we introduce our analysis method, and

then present our resulting constraints on the Lorentz-violating coefﬁcients. The physical impli-

cations of our results are discussed in Section 4. Finally, we summarize our main conclusions

in Section 5.

Universe 2022,1, 0 3 of 17

2. Theoretical Framework

In the SME framework, the LIV-induced modiﬁcations to the photon dispersion relation

can be described in the form [15,16]

E(p)'1−ς0±q(ς1)2+ (ς2)2+ (ς3)2p, (1)

where

is the photon momentum. The symbols

ς0

ς1

ς2

, and

ς3

are the combinations of

coefﬁcients for LIV that depend on the momentum and direction of propagation. These four

combinations can be decomposed on a spherical harmonic basis to yield

ς0=∑

djm

pd−40Yjm(ˆn)c(d)

(I)jm,

ς1±iς2=∑

djm

pd−4∓2Yjm(ˆn)k(d)

(E)jm ∓ik(d)

(B)jm,

ς3=∑

djm

pd−40Yjm(ˆn)k(d)

(V)jm,

(2)

where

ˆn

is the direction of the source and

sYjm(ˆn)

represents spin-weighted harmonics of spin

weight

. The coordinates

(θ

φ)

ˆn

are in a Sun-centered celestial-equatorial frame [

such that

θ= (

◦−Dec.)

and

φ=

R.A., where R.A. and Dec. are the right ascension and

declination of the astrophysical source, respectively.

With the above decomposition, all types of LIV for vacuum propagation can be charac-

terized using four sets of spherical coefﬁcients:

c(d)

(I)jm

k(d)

(E)jm

, and

k(d)

(B)jm

for CPT-even effects

and

k(d)

(V)jm

for CPT-odd effects. For each coefﬁcient, the relevant Lorentz-violating operator has

mass dimension

and eigenvalues of total angular momentum written as

. The coefﬁcients

c(d)

(I)jm

are associated with CPT-even operators causing dispersion without leading-order bire-

fringence, while nonzero coefﬁcients

k(d)

(E)jm

k(d)

(B)jm

, and

k(d)

(V)jm

produce birefringence. In the

present work, we focus on the nonbirefringent vacuum dispersion coefﬁcients

c(d)

(I)jm

. Setting

all other coefﬁcients for birefringent propagation to zero, the group-velocity defect including

anisotropies is given by

δvg=−∑

djm

(d−3)Ed−40Yjm(ˆn)c(d)

(I)jm , (3)

in terms of the photon energy

. Note that the factor

(d−

)

refers to the difference between

group and phase velocities [

]. For even

d≥

6, nonzero values of

c(d)

(I)jm

imply an energy-

dependent vacuum dispersion of light, so two photons with different energies (

Eh>El

)

emitted simultaneously from the same astrophysical source at redshift

would be observed at

different times. The arrival-time difference can therefore be derived as [15]

4tLIV =tl−th

≈ −(d−3)Ed−4

h−Ed−4

lZz

(1+z0)d−4

H(z0)dz0∑

0Yjm(ˆn)c(d)

(I)jm ,(4)

where

and

are the arrival times of photons with observed energies

and

, respec-

tively. In the ﬂat

CDM model, the Hubble expansion rate

H(z)

is expressed as

H(z) =

H0Ωm(1+z)3+ΩΛ1/2

, where

H0=

67.36 km s

−1

Mpc

−1

is the Hubble constant,

Ωm=

0.315

Universe 2022,1, 0 4 of 17

is the matter density, and

ΩΛ=

−Ωm

is the cosmological constant energy density [

]. In

the SME case of a direction-dependent LIV, we constrain the combination

∑jm 0Yjm(ˆn)c(d)

(I)jm

a whole. For the limiting case of the vacuum isotropic model (

j=m=

0), all the terms in the

combination become zero except

0Y00 =Y00 =p1/(4π)

. In that case, we constrain a single

c(d)

(I)00 coefﬁcient.

3. Constraints on Anisotropic LIV

By systematically analyzing the spectral lags of 135

Fermi

long GRBs with redshift mea-

surement, Liu et al. [

] identiﬁed 32 of them having well-deﬁned transitions from positive

to negative spectral lags. For each GRB, Liu et al. [

] extracted its multi-band light curves

and calculated the spectral time lags for any pair of light curves between the lowest-energy

band and any other higher-energy bands. In Figure 1, we plot the time lags of each burst as a

function of energy. One can see that all GRBs exhibit a positive-to-negative lag transition. In

this section, we utilize the spectral-lag transitions of these 32 GRBs to pose direction-dependent

constraints on combinations of nonbirefringent Lorentz-violating coefﬁcients c(6)

(I)jm and c(8)

(I)jm.

The 32 GRBs used in our study are listed in Table 1, which includes the following information

for each burst: its name, the right ascension coordinate, the declination coordinate, and the

redshift z.

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Citation:Wei,J.-N.;Liu,Z.-K.;Wei,J.-J.;Zhang,B.-B.;Wu,X.-F.ExploringAnisotropicLorentzInvarianceViolationfromtheSpectral-LagTransitionsofGamma-RayBursts.Universe2022,1,0.https://doi.org/AcademicEditor:Received:Accepted:Published:Publisher'sNote:MDPIstaysneutralwithregardtojurisdictionalclaimsinpubli...

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