Testing Fundamental Physics with Astrophysical Transients
aa r X i v : . [ a s t r o - ph . H E ] F e b Front. Phys. 16(4), 44300 (2021)DOI https://doi.org/10.1007/s11467-020-1049-x
Review article
Testing Fundamental Physics with Astrophysical Transients
Jun-Jie Wei , , ∗ , Xue-Feng Wu , , †
1. Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China2. School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, ChinaCorresponding authors. E-mail: ∗ [email protected], † [email protected] Explosive astrophysical transients at cosmological distances can be used to place precision tests of thebasic assumptions of relativity theory, such as Lorentz invariance, the photon zero-mass hypothesis,and the weak equivalence principle (WEP). Signatures of Lorentz invariance violations (LIV) includevacuum dispersion and vacuum birefringence. Sensitive searches for LIV using astrophysical sourcessuch as gamma-ray bursts, active galactic nuclei, and pulsars are discussed. The most direct conse-quence of a nonzero photon rest mass is a frequency dependence in the velocity of light propagating invacuum. A detailed representation of how to obtain a combined severe limit on the photon mass usingfast radio bursts at different redshifts through the dispersion method is presented. The accuracy ofthe WEP has been well tested based on the Shapiro time delay of astrophysical messengers travelingthrough a gravitational field. Some caveats of Shapiro delay tests are discussed. In this article, wereview and update the status of astrophysical tests of fundamental physics.
Keywords
Astroparticle physics, Gravitation, Astrophysical transients
PACS numbers
Contents
Einstein’s theory of special and general relativity is amajor pillar of modern physics, with a wide applicationin astrophysics. It is, therefore, of great scientific sig- nificance to test the validity of the basic assumptions ofrelativity theory, such as Lorentz invariance, the photonzero-mass hypothesis, and the weak equivalence principle(WEP). When the terrestrial conditions eventually im-pose some limitations, the extreme features of astrophys-ical phenomena afford the ideal testbeds for obtaininghigher precision tests of the fundamental laws of physics.Lorentz invariance, which is the fundamental symme-try of Einstein’s special relativity, says that the relevantphysical laws of a non-accelerating physical system areinvariant under Lorentz transformation. However, devi-ations from Lorentz invariance at a natural energy scaleare suggested in many quantum gravity (QG) theoriesseeking to unify general relativity and quantum mechan-ics, such as loop quantum gravity [1, 2], double specialrelativity [3–6], and superstring theory [7]. This nat-ural scale, referred to as the “QG energy scale” E QG ,is generally supposed to be around the Planck energy E Pl = p ¯ hc /G ≃ . × GeV [7–13]. The dedicatedexperimental tests of Lorentz invariance may thereforehelp to clear the path to a grand unified theory. A compi-lation of various recent experimental tests may be foundin Ref. [14]. Although any violations of Lorentz symme-try are predicted to be tiny at observable energies ≪ E Pl ,they can accumulate to measurable levels over large dis-tances. Therefore, astrophysical measurements involvinglong baselines can provide sensitive probes of Lorentz in-variance violation (LIV). In the photon sector, the po-tential signatures of LIV include vacuum dispersion and © Higher Education Press and Springer-Verlag Berlin Heidelberg 2021 eview article vacuum birefringence [15]. Vacuum dispersion wouldproduce an energy-dependent speed of light that in turnwould translate into differences in the arrival time of pho-tons with different energies traveling over cosmologicaldistances. Lorentz invariance can therefore be tested us-ing astrophysical time-of-flight measurements (e.g., Refs.[15–35]). Similarly, the effect of vacuum birefringencewould accumulate over astrophysical distances resultingin a detectable rotation of the polarization vector of lin-early polarized emission as a function of energy. Thus,Lorentz invariance can also be probed with astrophysicalpolarization measurements (e.g., Refs. [36–55]). Typ-ically, polarization measurements place more stringentconstraints on LIV. This can be understood by the factthat polarization measurements are more sensitive thantime-of-flight measurements by a factor ∝ /E , where E is the energy of the photon [20]. However, numer-ous predicted Lorentz-violating signals have no vacuumbirefringence, so constraints from time-of-flight measure-ments are indispensable in an extensive search for non-birefringent effects.The postulate that all electromagnetic waves travel invacuum at the constant speed c is one of the foundationsof Maxwell’s electromagnetism and Einstein’s special rel-ativity. The constancy of light speed implies that thequantum of light, or photon, should be massless. The va-lidity of this postulate can therefore be tested by search-ing for a rest mass of the photon. However, none of theexperiments so far could confirm that the photon restmass is absolutely zero. Based on the uncertainty prin-ciple, when using the age of the universe ( T ∼ yr),there is an ultimate upper limit on the photon rest mass,i.e., m γ ≤ ¯ h/T c ≈ − kg [56, 57]. The best one canhope to do is to set ever tighter limits on m γ and pushthe experimental results even closer to the ultimate up-per limit. In theory, a nonzero photon rest mass can beaccommodated into electromagnetism through the Procaequations. Using them, some possible visible effects as-sociated with a massive photon have been carefully stud-ied, which open the door to useful approaches for terres-trial experiments or astrophysical observations aimed atplacing upper limits on the photon rest mass (see Refs.[56–60] for reviews). To date, the experimental methodsfor constraining m γ include the frequency dependenceof the speed of light ( m γ ≤ . × − kg ) [61–72],Coulomb’s inverse square law ( m γ ≤ . × − kg ) [73],Amp `e re’s law ( m γ ≤ (8 . ± . × − kg ) [74], torsionbalance ( m γ ≤ . × − kg ) [75–78], gravitationaldeflection of electromagnetic waves ( m γ ≤ − kg )[79, 80], Jupiter’s magnetic field ( m γ ≤ × − kg )[81], magnetohydrodynamic phenomena of the solar wind( m γ ≤ . × − − . × − kg ) [82–84], cosmic mag-netic fields ( m γ ≤ − kg ) [85–87], supermassive black-hole spin ( m γ ≤ − − − kg ) [88], spindown of a white-dwarf pulsar ( m γ ≤ (6 . − . × − kg ) [89],and so on. Among these methods, the most direct androbust one is to detect a possible frequency dependencein the velocity of light. In this article, we will review thephoton mass limits from the dispersion of electromag-netic waves of astrophysical sources.Einstein’s WEP states that any freely falling, un-charged test body will follow a trajectory, independentof its internal structure and composition [90, 91]. It isthe basic ingredient of general relativity and other metrictheories of gravity. The most famous tests of the WEPare the E ¨o tv ¨o s-type experiments, which compare the ac-celerations of two laboratory-sized objects consisted ofdifferent composition in a known gravitational field (seeRef. [91] for a review). For laboratory-sized objectswith macroscale masses, the accuracy of the WEP canbe tested in a Newtonian context. However, the motionof test particles (like photons or neutrinos) in a grav-itational field is not precisely described by Newtoniandynamics. As such, the parameterized post-Newtonian(PPN) formalism has been developed to describe exactlytheir motion. Each theory of gravitation satisfying theWEP is specified by a set of PPN parameters. TheWEP can then also be tested by massless (or negligi-ble rest-mass) particles in the context of the PPN for-malism and any possible deviation from WEP is charac-terized by the PPN parameter, γ , of a particular grav-ity theory. Here γ reflects the level of space curved byunit rest mass [90, 91]. In general relativity, γ is pre-dicted to be strictly 1. The determination of the abso-lute γ value has reached high precision. The light de-flection measurements through very-long-baseline radiointerferometry yielded an agreement with general rela-tivity to 0.01 percent, i.e., γ − − . ± . × − [92, 93]. The radar time-delay measurement from theCassini spacecraft obtained a more stringent constraint γ − . ± . × − [94]. Regardless of the absolutevalue of γ , all metric theories of gravity incorporating theWEP predict that different species of messenger particles(photons, neutrinos, and gravitational waves (GWs)), orthe same species of particles but with different internalproperties (e.g, energies or polarization states), travel-ing through the same gravitational fields, must followidentical trajectories and undergo the same γ -dependentShapiro delay [95]. To test the WEP, therefore, the issueis not whether the value of γ is very nearly unity, butwhether it is the same for all test particles. The Shapirotime delay of test particles emitted from the same astro-physical sources has been widely applied to constrain apossible violation of the WEP through the relative dif-ferential variations of the γ values (e.g., [96–102]).In this review, we summarize the current status on as-trophysical tests of fundamental physics and attempt tochart the future of the subject. In Section 2, we review Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article two common astrophysical approaches to testing the LIVeffects. Section 3 is dedicated to discuss astrophysicalbounds on the photon rest mass through the measure-ment of the frequency dependence of the speed of light.In Section 4, we focus on tests of the WEP through theShapiro (gravitational) time delay effect. Finally, a sum-mary and future prospect are presented in Section 5.
Various QG theories that extend beyond the stan-dard model predict violations of Lorentz invariance atenergies approaching the Planck scale [7–13]. The ex-istence of LIV can produces an energy-dependent vac-uum dispersion of light, which leads to time delays be-tween promptly emitted photons with different energies,as well as an energy-dependent rotation of the polariza-tion plane of linearly polarized photons resulting fromvacuum birefringence. In this section, we first reviewthe recent achievements in sensitivity of vacuum disper-sion time-of-flight measurements. Then the progress onLIV limits from astrophysical polarization measurementsis discussed.2.1 Vacuum dispersion from LIV
Several QG theories proposed to describe quantumspacetime, predict the granularity of space and time.For example, string theory that remove the point-likenature of the particles by introducing to each of thema (mono)-dimensional extension: the string (see, e.g.,Smolin [103] for reviews). Loop quantum gravity thatquestion the continuousness and smoothness of space-time quantizing it into discrete energy levels like thoseobserved in the classical quantum-mechanical systemsto construct a complex geometric structure called spinnetworks (see, e.g., Rovelli [104] for reviews). In bothproposed theories emerge a minimal length for physicalspace (and time), although with different and somewhatopposite theoretical approaches [105]. The minimal spa-tial and temporal scales associated to the quantizationare in terms of standard units: l P ∼ p ¯ hG/c ∼ − cm and t P ∼ p ¯ hG/c ∼ − s for the Planck lengthand time, respectively. A significant class of QG theo-ries predict that the propagation of photons through thisdiscrete spacetime might exhibit a non-trivial dispersionrelation in vacuum [16], corresponding to the possible vi-olation or break of the Lorentz invariance via an energy-dependent speed of light. In vacuum, the LIV-inducedmodifications to the dispersion relation of photons can be described using a Taylor series expansion: E ≃ p c " − ∞ X n =1 s ± (cid:18) EE QG ,n (cid:19) n , (1)where p is the photon momentum, E QG is the hypothet-ical energy scale at which QG effects would become sig-nificant, and s ± = ± is a theory-dependent factor. For E ≪ E QG , the sum is dominated by the lowest-orderterm in the series. Considering only the lowest-orderdominant term, the photon group velocity can thereforebe expressed by v ( E ) = ∂E∂p ≈ c (cid:20) − s ± n + 12 (cid:18) EE QG ,n (cid:19) n (cid:21) , (2)where n = 1 and n = 2 correspond to the linear andquadratic LIV, respectively. The coefficient s ± = +1 ( s ± = − ) stands for a decrease (increase) in the photonspeed along with increasing photon energy (also refers tothe “subluminal” and “superluminal” scenarios).Because of the energy dependence of v ( E ) , two pho-tons with different observer-frame energies ( E h > E l )emitted simultaneously from the same astrophysicalsource at redshift z would arrive at the observer at differ-ent times. Taking account of the cosmological expansion,we derive the expression of the LIV-induced time delay[19]: ∆ t LIV = t h − t l = s ± n H E nh − E nl E n QG ,n Z z (1 + z ′ ) n dz ′ p Ω m (1 + z ′ ) + Ω Λ , (3)where t h and t l , respectively, denote the arrival timesof the high-energy and low-energy photons, H is theHubble constant, and Ω m and Ω Λ are the matter en-ergy density and the vacuum energy density (param-eters of the flat Λ CDM model). Since the adoptedenergy range generally spans several orders of magni-tude, one can approximate ( E nh − E nl ) ≈ E nh . Adopt-ing z = 20 as a firm upper limit for the redshift of anysource, we find the LIV-induced time delay is | ∆ t LIV | ≤ . E h GeV /ζ ) s for the linear ( n = 1 ) LIV case and | ∆ t LIV | ≤ . × − ( E h GeV /ζ ) s for the quadratic( n = 2 ) LIV case, where E h GeV = E h / (1 GeV) and ζ = E QG /E Pl . These indicate that first order ( n = 1 )effects would lead to potentially observable delays, whilesecond order ( n = 2 ) effects are so tiny that it wouldbe impossible to observe them with this time-of-flighttechnique. Eq. (3) shows that the LIV-induced time delay in-creases with the energy of the photons and the distance of
Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
Fig. 1
Left panel: the rescaled time lags between two selected observer-frame energy bands from the full set of 35 GRBswith known redshifts observed until 2006 by BATSE (closed circles), HETE-2 (open circles) and Swift (triangles). Rightpanel: the likelihood function for the slope parameter a LV . Reproduced from Ref. [106]. the source. A short timescale of the signal variability eas-ily provides a reference time to measure the time delay.The higher the photon energy, the longer the source dis-tance, and the shorter the time delay, the more stringentlimits on the QG energy scale one can reach. Exploit-ing the rapid variations of gamma-ray emissions fromastrophysical sources at large distances to constrain LIVwas first proposed by Amelino-Camelia et al. [16]. Atpresent, tests of LIV have been made using the observa-tions of gamma-ray bursts (GRBs), active galactic nuclei(AGNs), and pulsars.• Gamma-ray burstsGRBs are among the most distant gamma-ray sourcesand 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 studieson the time-of-flight measurements of GRBs. Some ofthe pre-Fermi studies are those by Ellis et al. [107] us-ing BATSE GRBs; by Boggs et al. [108] using RHESSIobservations 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 obser-vations of GRB 051221A; by Bolmont et al. [110] usingHETE-2 GRBs; and by Lamon et al. [111] using INTE-GRAL GRBs. Because of the unprecedented sensitiv-ity 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 beenobtained using Fermi observations. These constraintsinclude those by the Fermi Collaboration using GRBs080916C [21] and 090510 [22]; by Xiao & Ma [112] usingGRB 090510; and by Refs. [23–25, 28, 32, 113–117] us-ing multiple Fermi GRBs. Particularly, Abdo et al. [22] used the highest energy (31 GeV) photon of the shortGRB 090510 detected by Fermi/LAT to constrain thelinear LIV energy scale ( E QG , ). The burst has a redshift z = 0 . . This 31 GeV photon was detected 0.829 s afterthe Fermi Gamma-Ray Burst Monitor (GBM) trigger. Ifthe 31 GeV photon was emitted at the beginning of thefirst GBM pulse, one has ∆ t LIV < ms, which givesthe most conservative constraint E QG , > . E Pl . If the31 GeV photon is associated with the contemporaneous < MeV spike, one has ∆ t LIV < ms, which gives theleast conservative constraint E QG , > E Pl . We cansee that the linear LIV models requiring E QG , ≤ E Pl aredisfavored by these results. Subsequently, Vasileiou et al.[25] used three statistical techniques to constrain the to-tal degree of dispersion in the data of four LAT-detectedGRBs. For the subluminal case, their most stringent lim-its are derived from GRB 090510 and are E QG , > . E Pl and E QG , > . × GeV for linear and quadratic LIV,respectively. These limits improve previous constraintsby a factor of ∼ . Recently, the Major AtmosphericGamma Imaging Cherenkov (MAGIC) telescopes firstdetected GRB 190114C in the sub-TeV energy domain(i.e., 0.2—1 TeV), recording the highest energy photonsever observed from a GRB [118]. Using conservative as-sumptions on the possible intrinsic spectral and temporalemission properties, the MAGIC Collaboration searchedfor an energy dependence in the arrival time of the mostenergetic photons and presented competitive limits onthe quadratic leading-order LIV-induced vacuum disper-sion [33]. The resulting constraints from GRB 190114Care E QG , > . × GeV and E QG , > . × 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
Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article lags) from GRBs, Lorentz invariance has been testedwith unprecedented accuracy. Such time-of-flight tests,however, are subject to a bias related to a possible intrin-sic time lag introduced from the unknown emission mech-anism of the sources, which would enhance or cancel-outthe delay caused by the LIV effects. That is, the methodof the arrival-time difference is tempered by our igno-rance concerning potential source-intrinsic effects. Thefirst attempt to mitigate the intrinsic time lag problemwas proposed by Ellis et al. [18, 106, 107], who suggestedworking on a large sample of GRBs with different red-shifts. For each GRB, Ellis et al. [18] looked for thespectral lag in the light curves recorded in the chosenobserver-frame energy bands 115–320 and 25–55 keV. Toaccount for the poorly known intrinsic time lag, theyfitted the observed spectral lags of 35 GRBs with theinclusion of a constant b sf specified in the rest frame ofthe source. The observed arrival time delays thus havetwo contributions ∆ t obs = ∆ t LIV + b sf (1 + z ) , reflectingthe possible LIV effects and source-intrinsic effects [18].Rescaling ∆ t obs by a factor (1+ z ) , then one has a simplelinear fitting function ∆ t obs z = a LV K + b sf , (4)where K = 1(1 + z ) Z z (1 + z ′ ) dz ′ h ( z ′ ) (5)is a function of redshift which depends on the cosmolog-ical model, the slope a LV = ∆ E/ ( H E QG ) is related tothe QG energy scale, and the intercept b sf denotes thepossible unknown intrinsic time lag. In the standard flat Λ CDM model, the dimensionless expansion rate h ( z ) isexpressed as h ( z ) = p Ω m (1 + z ) + Ω Λ . Note that thelinear LIV ( n = 1 ) in the subluminal case ( s ± = +1 )was considered in this work. Within a frame work of theconcordance Λ CDM model, a linear fit to the rescaledtime lags extracted from 35 light curve pairs is shownin the left panel of Figure . The best-fit line corre-sponds to ∆ t obs z = (0 . ± . K − (0 . ± . and the likelihood function for the slope parameter a LV is presented in the right panel of Figure . The 95%confidence-level lower limit on the linear LIV energy scalederived from the likelihood function of a LV is E QG ≥ . × GeV [106]. Going beyond the Λ CDM cos-mology, Refs. [119, 120] extended this analysis to dif-ferent cosmological models and showed that the result isinsensitive to the adopted background cosmology. Sub-sequently, some cosmology-independent approaches wereapplied to probe the possible LIV effects [121, 122].It should be noted that there are two limitations in thetreatment of Ellis et al. [18]. First, they extracted spec-tral lags in the light curves between two fixed observer-frame energy bands. However, because different GRBs have different redshift measurements, these two energybands correspond to a different pair of energy bands inthe source frame [123], thus potentially causing an ar-tificial energy dependence to the extracted spectral lagand/or a systematic uncertainty to the search for LIVlags. Ukwatta et al. [123] found that there is alarge scatter in the correlation between observer-framelags and source-frame lags for the same GRB sample, in-dicating that the observer-frame lag does not faithfullyrepresent the source-frame lag. The first limitation ofEllis et al. treatment can be resolved by selecting twoappropriate energy bands fixed in the rest frame and cal-culating the time lag for two projected observer-frameenergy bands by the relation E observer = E source / (1 + z ) .Bernardini et al. [124] studied the source-frame spec-tral lags of 56 GRBs detected by Swift/Burst Alert Tele-scope (BAT). For each GRB, they extracted light curvesfor two observer-frame energy bands corresponding tothe fixed energy bands in the source frame, i.e., 100–150and 200-250 keV. These two particular source-frame en-ergy bands were selected to ensure that the projectedobserver-frame energy bands (i.e., [100 − / (1 + z ) and [200 − / (1 + z ) keV) are within the detectableenergy range of the BAT instrument (see Figure ). Foreach extracted light-curve pairs, they used the discretecross-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 at100 keV, whereas in the observer frame, the energy differ-ence varies depending on the redshift of each burst. Thisis in contrast to the spectral lag extractions achieved inthe observer frame, where the energy difference is treatedas a constant [123]. Wei & Wu [31] first took advantageof the source-frame spectral lags of 56 Swift GRBs pre-sented in Bernardini et al. [124] to investigate the LIVeffects. For the subluminal case, the arrival-time differ-ence of two photons with observer-frame energy differ-ence ∆ E that induced by the linear LIV reads: ∆ t LIV = ∆ EH E QG Z z (1 + z ′ )d z ′ h ( z ′ )= ∆ E ′ / (1 + z ) H E QG Z z (1 + z ′ )d z ′ h ( z ′ ) , (6)where ∆ E ′ = 100 keV is the rest-frame energy difference.Similar to Ellis et al. [18], one can formulate the intrinsic Due to the redshift dependence of cosmological sources, ob-server frame quantities can be quite different than source frameones. In principle, there is a similar problem associated with theenergy dependence of the observer-frame quantities for vacuumbirefringence or WEP bounds (more on this below) when analyz-ing a large sample of cosmological sources with different redshifts.However, note that if we work on the observer-frame quantities ofindividual cosmological sources, there is no such a problem.
Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
15 keV 200 keV
Variable Energy Bands at the Observer FrameFixed Energy Band at the GRB Source Frame - k e V - k e V - k e V For a GRB with redshift of 3.0 - k e V For a GRB with redshift of 2.0For a GRB with redshift of 1.0 - k e V - k e V - k e V - k e V Swift BAT Energy Band15 keV 200 keV
Variable Energy Bands at the Observer FrameFixed Energy Band at the GRB Source Frame - k e V - k e V - k e V For a GRB with redshift of 3.0 - k e V For a GRB with redshift of 2.0For a GRB with redshift of 1.0 - k e V - k e V - k e V - k e V Swift BAT Energy Band
Fig. 2
Fixed energy bands in the GRB source frame are transformed to various energy bands in the observer frame,depending on the redshift. Reproduced from Ref. [123]. time lag problem in terms of linear regression: ∆ t src z = a ′ LV K ′ + b sf , (7)where ∆ t src is the extracted spectral lag for the source-frame energy bands 100–150 and 200-250 keV, K ′ = 1(1 + z ) Z z (1 + z ′ ) dz ′ h ( z ′ ) (8)is a dimensionless redshift function, and a ′ LV =∆ E ′ / ( H E QG ) is the slope in K ′ . Using the sample of56 GRBs with known redshifts, Wei & Wu [31] obtainedrobust limits on the slope a ′ LV and the intercept b sf byfitting their source-frame spectral lag data. The 95%confidence-level lower limit on the QG energy scale de-rived from a ′ LV is E QG ≥ . × GeV [31]. This isa step forward in the study of LIV effects, since all pre-vious investigations used spectral lags extracted in theobserver frame only.The second limitation of Ellis et al. treatment is thatan unknown constant was supposed to be the intrinsictime delay in the linear fitting function (see Eq. 4), whichis tantamount to suggesting that all GRBs have the sameintrinsic time lag. However, since the time durations ofGRBs span nearly six orders of magnitude, it is not pos-sible that high-energy photons radiated from differentGRBs (or from the same burst) have the same intrinsictime lag relative to the radiation time of low-energy pho-tons [125]. As an improvement, Zhang & Ma [28] fittedthe data of the high-energy photons from GRBs on sev-eral straight lines with the same slope as /E n QG ,n butwith different intercepts (i.e., different intrinsic emission times; see also Refs. [114–116]). However, photons fromdifferent GRBs fall on the same line, which still impliesthat the intrinsic time lags between the high-energy pho-tons and the low-energy (trigger) photons are much thesame for these GRBs. Chang et al. [23] made use ofthe magnetic jet model to estimate the intrinsic emis-sion time delay between high- and low-energy photonsfrom GRBs. However, the magnetic jet model dependson some specific theoretical parameters, and thus intro-duces uncertainties on the LIV results. In 2017, Weiet al. [29] first proposed that GRB 160625B, the bursthaving a well-defined transition from positive to negativespectral lags (see Figure ), provides a good opportunitynot only to disentangle the intrinsic time lag problembut also to put new constraints on LIV. The spectrallag is conventionally defined positive when high-energyphotons arrive earlier than low-energy photons, while anegative lag corresponds to a delayed arrival of high-energy photons. As discussed above, the LIV-inducedtime delay ∆ t LIV is likely to be accompanied by a po-tential intrinsic energy-dependent time lag ∆ t int due tounknown properties of the source. Therefore, the ob-served time lag between two different energy bands of aGRB should consist of two parts, ∆ t obs = ∆ t int + ∆ t LIV . (9)Since the observed time lags of most GRBs have a pos-itive energy dependence (e.g., Refs. [126, 127]), Wei etal. [29] approximated the observer-frame relation of theintrinsic time lag and the energy E as a power law with Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article n=1 n=2 t ob s ( s ) E (keV)
Fig. 3
Energy dependence of the observed time lag ∆ t obs between the lowest-energy band and any other high-energybands of GRB 160625B. The solid and dashed curves corre-spond to the best-fit linear ( n = 1 ) and quadratic ( n = 2 )LIV models, respectively. Reproduced from Ref. [29]. positive dependence, ∆ t int ( E ) = τ (cid:20)(cid:18) E keV (cid:19) α − (cid:18) E keV (cid:19) α (cid:21) s , (10)where τ > and α > , and where E = 11 . keV is themedian value of the lowest reference energy band (10–12keV). We emphasize that the intrinsic positive time lagimplies an earlier arrival time for the higher energy pho-tons. Also, when the subluminal case ( s ± = +1 ) is con-sidered, high-energy photons would arrive on Earth afterlow-energy ones, implying a negative spectral lag due tothe LIV effects. As the Lorentz-violating term becomesdominant at higher energy scales, the positive correla-tion between the time lag and the energy would grad-ually trend in an opposite way. The combined contri-butions from the intrinsic time lag and the LIV-inducedtime lag can therefore produce the observed lag behaviorwith a turnover from positive to negative lags [29]. Byfitting the spectral lag behavior of GRB 160625B, Weiet al. [29, 30] obtained both a reasonable formulationof the intrinsic energy-dependent time lag and compar-atively robust limits on the QG energy scale and thecoefficients of the Standard Model Extension. The σ confidence-level lower limits are E QG , > . × GeV and E QG , > . × GeV for linear and quadratic LIV,respectively. The spectral lag data of GRB 160625B doesnot have the current best sensitivity to LIV constraints,but the analysis method, when applied to future brightshort GRBs with similar spectral-lag transitions, mayresult in more stringent constraints on LIV.• Active galactic nucleiThanks to their fast flux variations (hours to years), cosmological distances, and very high energy (VHE, E ≥
100 GeV ) gamma-rays, TeV flares of AGNs havealso been viewed as very effective probes for searching forthe LIV-induced vacuum dispersions. It is worth point-ing out that testing LIV with both GRBs and flaringAGNs is of great fundamental interest. GRBs can bedetected at very large distances (up to z ∼ ), but withvery limited high-energy ( E > tens of GeV) photons. Onthe contrary, AGN flares can be well observed with largestatistics of photons up to a few tens of TeV. But due toextinction of high-energy photons by extragalactic back-ground light, TeV detections are limited to those sourceswith relatively low redshifts z ≤ . . Hence, GRBsand flaring AGNs are mutually complementary in testingLIV, and they allow to test different redshift and energyranges. There have been some resulting constraints onLIV using TeV observations of bright AGN flares, includ-ing the Whipple analysis of the flare of Mrk 421 [34], theMAGIC and H.E.S.S. analyses of the flares of Mrk 501[128–130], and the H.E.S.S. analysis of the flare of PKS2155-304 [131, 132]. The current best limit for the linearcase considering a subluminal LIV effect obtained withAGNs is from the H.E.S.S. analysis of the PKS 2155-304flare data, namely E QG , > . × GeV [132]. For thequadratic LIV, the best limits derived from AGNs havebeen set by H.E.S.S.’s observation of the TeV flare of Mrk501. The reported limits are E QG , > . × GeV ( E QG , > . × GeV ) for the subluminal (superlu-minal) case [130].• PulsarsA third class of astrophysical sources used for the time-of-flight tests on gamma-rays are pulsars. When it comesto testing the quadratic LIV term, having a VHE emis-sion will compensate for a short of distance. Gamma-raypulsars, albeit being detected many orders of magnitudecloser than GRBs or AGNs, have the advantage of pre-cisely periodic flux variation, as well as the fact thatthey are the only stable candidate astrophysical sourcesfor such time-of-flight studies. Sensitivity to LIV cantherefore be improved by simply observing longer. Ad-ditionally, since the timing of the pulsar is carefully stud-ied throughout the electromagnetic spectrum, energy-dependent time delays induced by propagation effectscan be more easily distinguished from intrinsic delays.First limits on LIV using gamma-ray radiation from thegalactic Crab pulsar were obtained from the observa-tion of the Energetic Gamma-Ray Experiment Telescope(EGRET) onboard the Compton Gamma-Ray Obser-vatory (CGRO) at energies above 2 GeV [35], and im-proved by the VERITAS data above 120 GeV [133, 134].Recently, the MAGIC collaboration presented the bestlimits derived from pulsars by studying the Crab pulsaremission observed up to TeV energies, yielding E QG , > Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
Table 1
A selection of lower limits on E QG ,n for linear ( n = 1 ) and quadratic ( n = 2 ) LIV for the subluminal ( s ± = +1 ) andsuperluminal ( s ± = − ) cases. These limits were obtained from vacuum dispersion time-of-flight measurements of variousastrophysical sources. Source(s) Instrument Technique E QG , (GeV) E QG , (GeV) Refs. s ± = +1 s ± = − s ± = +1 s ± = − a BATSE+OSSE Wavelets . × — . × — [107]GRB 021206 b RHESSI Peak times at different energies . × — — — [108]35 GRBs c BATSE+HETE-2 Wavelets . × — — — [18, 106]+SwiftGRB 051221A Konus-Wind+Swift Peak times of the light curves . × — . × — [109]in different energy bands15 GRBs HETE-2 Wavelets . × — — — [110]11 GRBs INTEGRAL Likelihood . × — — — [111]GRB 080916C Fermi GBM+LAT Associating a 13.2 GeV photon with . × — — — [21]the trigger timeGRB 090510 Fermi GBM+LAT Associating a 31 GeV photon with . × — — — [22]the start of the first GBM pulseFermi/LAT PairView+Likelihood . × . × . × . × [25]+Sharpness-Maximization MethodGRB 160625B Fermi/GBM Spectral lag transition . × — . × — [29]56 GRBs Swift Rest-frame spectral lags . × — — — [31]GRB 190114C MAGIC Likelihood . × . × . × . × [33]Mrk 421 Whipple Binning . × — — — [34]Mrk 501 MAGIC Energy cost function . × — . × — [128]Likelihood . × — . × — [129]H.E.S.S. Likelihood . × . × . × . × [130]PKS 2155-304 H.E.S.S. Modified cross correlation function . × — . × — [131]Likelihood . × — . × — [132]Crab pulsar CGRO/EGRET Pulse arrival times . × — — — [35]in different energy bandsVERITAS Likelihood . × — . × — [133]Dispersion Cancellation . × . × — — [134]MAGIC Likelihood . × . × . × . × [135] a Limits obtained not taking into account the factor (1 + z ) in the intergrand of Eq. (3). b The pseudo redshift was estimated from the spectral and temporal properties of GRB 021206. c The Limits of Ellis et al. [18] were corrected in Ellis et al. [106] taking into account the factor (1 + z ) in the intergrand of Eq. (3). . × GeV ( E QG , > . × GeV ) for a linear,and E QG , > . × GeV ( E QG , > . × GeV )for a quadratic LIV, for the subluminal (superluminal)case, respectively [135].The most important constraints obtained so far withvacuum dispersion time-of-flight measurements of var-ious astrophysical sources are summarized in Table .The most stringent lower limits to date on the linear andquadratic LIV energy scales were set by the observationof GRB 090510 with Fermi/LAT. The values for the sub-luminal (superluminal) case are E QG , > . × GeV ( E QG , > . × GeV ) and E QG , > . × GeV ( E QG , > . × GeV ) [25]. Clearly, these vacuumdispersion studies using gamma rays in the GeV–TeVrange offer us at present with the best opportunity tosearch for Planck-scale modifications of the dispersionrelation. Unfortunately, while they provide meaningfulbounds for the linear ( n = 1 ) modification, they are muchweaker for deviations that arise at the quadratic ( n = 2 )order.2.2 Vacuum birefringence from LIV In QG theories that invoke LIV, the Charge-Parity-Time (CPT) theorem, i.e., the invariance of the laws ofphysics under charge conjugation, parity transformation,and time reversal, no longer holds. Note that the factCPT does not hold does not imply that it should beviolated. In the absence of Lorentz invariance, the CPTinvariance, if needed, should be imposed as an additionalassumption. In the effective field theory approach [136],the Lorentz- and CPT-violating dispersion relation forphoton propagation can be parameterized as E ± = p c ± ηE pl p c , (11)where ± represents the left- or right-handed circular po-larization states of the photon, and η is a dimensionlessparameter that needs to be constrained. In LIV but CPTinvariant theories, the parameter η exactly vanishes. Inthis sense, such tests might be less general than the onesbased on vacuum dispersion. The linear polarization canbe decomposed into left- and right-handed circular po-larization states. For η = 0 , photons with opposite circu-lar polarizations have slightly different group velocities,which leads to a rotation of the polarization vector of a Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article linearly polarized wave. This effect is known as vacuumbirefringence. The rotation angle propagating from thesource at redshift z to the observer can be derived as[47, 49] ∆ φ LIV ( E ) ≃ η E F ( z )¯ hE pl H , (12)where E is the observed photon energy, and F ( z ) = Z z (1 + z ′ ) dz ′ q Ω m (1 + z ′ ) + Ω Λ . (13)Generally speaking, it is impossible to know the in-trinsic polarization angles in the emission of photonsof different energies from a given source. If one hadthis information, evidence for vacuum birefringence (i.e.,an energy-dependent rotation of the polarization plane)can be examined by measuring differences between theknown intrinsic polarization angle and the observed po-larization angles at different energies. However, evenwithout such knowledge, the birefringent effect can stillbe constrained for polarized sources at arbitrary cosmo-logical distances, because the differential rotation act-ing on the polarization angle as a function of energywould add opposite oriented polarization vectors, effec-tively erasing most, if not all, of the observed polarizationsignal. Therefore, the detection of the polarization signalcan put an upper bound on such a possible violation. Observations of linear polarization from distantsources have been widely used to place upper limitson the birefringent parameter η . The vacuum birefrin-gence constraints arise from the fact if the rotation angle(Eq. 12) differs by more than π/ over an energy range( E < E < E ), then the net polarization of the sig-nal would be severely suppressed and well below any ob-served value. Hence, the measurement of polarization ina given energy band implies that the differential rotationangle | ∆ φ ( E ) − ∆ φ ( E ) | should not be larger than π/ .Previously, Gleiser & Kozameh [38] set an upperbound of η < − by analyzing the linearly polar-ized ultraviolet light from the distant radio galaxy 3C256. Much more stringent limits, η < − , have beenobtained by using the linear polarization detection inthe gamma-ray emission of GRB 021206 [43, 44]. How-ever, the originally reported detection of high polariza-tion from GRB 021206 [137] has been refuted by there-analyses of the same data [138, 139]. Maccione et al.[140] used the hard X-ray polarization measurement ofthe Crab Nebula to get a constraint of η < × − .Laurent et al. [47] used a report of polarized soft gamma-ray emission from GRB 041219A to derive a strongerlimit of η < . × − (see also Ref. [48]). But again, this claimed polarization detection [141–143] has beendisputed (see the explanations in Ref. [49]). That is, theprevious reports of the gamma-ray polarimetry for GRB041219A are controversial and, thus, the arguments forthe limits on η given by Refs. [47, 48] are still open toquestions.Contrary to those disputed reports, the evidences oflinearly polarized gamma-ray emission detected by thegamma-ray burst polarimeter (GAP) onboard the In-terplanetary Kite-craft Accelerated by Radiation Of theSun (IKAROS) are convincing, and thus these detectionscan be used to set more reliable limits on the birefringentparameter [49]. IKAROS/GAP detected gamma-ray po-larizations of three GRBs with high significance levels,with a linear polarization degree of Π = 27 ± forGRB 100826A [144], Π = 70 ± for GRB 110301A,and Π = 84 +16 − % for GRB 110721A [145]. The detectionsignificance are . σ , . σ , and . σ , respectively. Tomaet al. [49] set the upper limit of the differential rotationangle | ∆ φ ( E ) − ∆ φ ( E ) | to be π/ , and obtained a se-vere upper limit on η in the order of O (10 − ) from thereliable polarimetric data of these three GRBs. However,the GRBs had no direct redshift measurement, they useda redshift estimate based on an empirical luminosity rela-tion. Utilizing the real redshift measurement ( z = 1 . )together with the polarization data of GRB 061122, Götzet al. [50] obtained a stricter limit ( η < . × − ) ona possible LIV. Götz et al. [51] used the most distantpolarized burst (up to z = 2 . ), GRB 140206A, to ob-tain the deepest limit to date ( η < . × − ) on thepossibility of LIV.It is worth noting that most of previous polarizationconstraints were derived under the assumption that thedifferential rotation angle | ∆ φ ( E ) − ∆ φ ( E ) | is smallerthan π/ . However, Lin et al. [52] gave a detailed analy-sis on the evolution of GRB polarization arising from thevacuum birefringent effect, and showed that a consider-able amount of the initial polarization degree (dependingboth on the photon energy band and the photon spec-trum) can be conserved even if | ∆ φ ( E ) − ∆ φ ( E ) | is ap-proaching to π/ . This is incompatible with the commonbelief that | ∆ φ ( E ) − ∆ φ ( E ) | should not be too largewhen high polarization is detected. Therefore, Lin et al.[52] suggested that it is unsuitable to constrain the bire-fringent effect by simply setting π/ as the upper limitof | ∆ φ ( E ) − ∆ φ ( E ) | . Lin et al. [52] applied their for-mulae for the polarization evolution to some true GRBevents, and obtained the most stringent upper limit todate on the birefringent parameter from the polarimetricdata of GRB 061122, i.e., η < . × − . Following theanalysis method presented in Ref. [52] and utilizing therecent measurements of gamma-ray linear polarization ofGRBs, Wei [55] updated constraints on a possible LIVthrough the vacuum birefringent effect, and thereby im- Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article P o l a r i z a t i on A ng l e (r ad ) -5.0x10 -7 -7 -6 -6 GRB020813 =0.47/3 P o l a r i z a t i on A ng l e (r ad ) V /10
29 -4.0x10 -7 -3.0x10 -7 -2.0x10 -7 -1.0x10 -7 -7 -7 GRB021004 =11.3/13
Fig. 4
Fit to the multiwavelength polarimetric data of the optical afterglows of GRB 020813 and GRB 021004. Left panels:linear fits of the observed linear polarization angles versus quantities of frequency squared. Right panels: 1-3 σ confidencelevels in the φ - η plane. Reproduced from Ref. [45]. proving previous results by factors ranging from two toten.If both the intrinsic polarization angle φ and the ro-tation angle induced by the birefringent effect ∆ φ LIV ( E ) are considered here, the observed linear polarization an-gle at a certain E from a source should be φ obs = φ + ∆ φ LIV ( E ) . (14)Assuming that all photons in the observed energy band-pass are emitted with the same (unknown) intrinsic po-larization angle, we then expect to observe the birefrin-gent effect as an energy-dependent linear polarizationvector. To constrain the birefringent parameter η , Fanet al. [45] looked for a similar energy-dependent trendin the multiwavelength polarization observations of theoptical afterglows of GRB 020813 and GRB 021004. Byfitting the spectropolarimetric data of these two GRBs,Fan et al. [45] obtained constraints on both φ and η (seeFigure ). At the σ confidence level, the combined limiton η from two GRBs is − × − < η < . × − . Itis clear from Eq. (12) that the higher energy band of thepolarization observation and the larger distance of thepolarized source, the greater sensitivity to small valuesof η . As expected, the optical polarization data of GRBafterglow obtained a less stringent constraint on η [45]. Table presents a summary of the corresponding lim-its on LIV from the polarization measurements of astro- physical sources. The hitherto most stringent constraintson the birefringent parameter, η < O (10 − ) , have beenobtained by the detections of linear polarization in theprompt gamma-ray emission of GRBs [50–52, 55].By comparing Eqs. (1) and (11), we can derive theconversion from η to the Limit on the linear LIV en-ergy scale, i.e., E QG , = E pl η . With the data pre-sented in Tables and , it is easy to compare therecent achievements in sensitivity of time-of-flight mea-surements versus polarization measurements. For thesubluminal case, the time-of-flight analysis of multi-GeVphotons from GRB 090510A detected by Fermi/LATyielded the strictest limit on the linear LIV energy scale, E QG , > . × GeV , which corresponds to η < . [25]. Obviously, this time-of-flight constraint is manyorders of magnitude weaker than the best polarizationconstraint. However, time-of-flight constraints are essen-tial in a broad-based search for nonbirefringent Lorentz-violating effects. m γ = 0 ), the energyof the photon can be written as E = q p c + m γ c . (15) Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
Table 2
A selection of limits on the vacuum birefringent parameter η from the linear polarization measurements ofastrophysical sources. Author (year) Source Polarimeter Energy band a Π η Refs.Gleiser and Kozameh (2001) 3C 256 Spectropolarimeter Ultraviolet . ± . < − [38]Mitrofanov (2003) GRB 021206 c RHESSI 150–2000 keV ± b < − [43]Jacobson et al. (2004) GRB 021206 d RHESSI 150–2000 keV ± b < . × − [44]Fan et al. (2007) GRB 020813 LRISp 3500–8800 Å 1.8%–2.4% ( − . –1.4) × − [45]GRB 021004 VLT 3500–8600 Å . ± . ( − . –1.4) × − Maccione et al. (2008) Crab Nebula INTEGRAL/SPI 100–1000 keV ± < × − [140]Laurent et al. (2011) GRB 041219A e INTEGRAL/IBIS 200–800 keV ± b < . × − [47]Stecker (2011) GRB 041219A f INTEGRAL/SPI 100–350 keV ± b < . × − [48]Toma et al. (2012) GRB 100826A f IKAROS/GAP 70–300 keV ± < . × − [49]GRB 110301A f IKAROS/GAP 70–300 keV ± < . × − GRB 110721A f IKAROS/GAP 70–300 keV +16 − % < . × − Götz et al. (2013) GRB 061122 INTEGRAL/IBIS 250–800 keV > < . × − [50]Götz et al. (2014) GRB 140206A INTEGRAL/IBIS 200–400 keV > < . × − [51]Lin et al. (2016) GRB 061122 INTEGRAL/IBIS 250–800 keV > < . × − [52]GRB 110721A f IKAROS/GAP 70–300 keV +16 − % < . × − Wei (2019) GRB 061122 INTEGRAL/IBIS 250–800 keV > < . × − [55]GRB 100826A f IKAROS/GAP 70–300 keV ±
11% 1 . +1 . − . × − GRB 110301A f IKAROS/GAP 70–300 keV ±
22% 4 . +5 . − . × − GRB 110721A IKAROS/GAP 70–300 keV +16 − % 5 . +4 . − . × − GRB 140206A INTEGRAL/IBIS 200–400 keV > < . × − GRB 160106A f AstroSat/CZTI 100–300 keV . ±
24% 3 . +1 . − . × − GRB 160131A AstroSat/CZTI 100–300 keV ±
31% 1 . +2 . − . × − GRB 160325A f AstroSat/CZTI 100–300 keV . ± .
5% 2 . +1 . − . × − GRB 160509A AstroSat/CZTI 100–300 keV ±
40% 0 . +2 . − . × − GRB 160802A f AstroSat/CZTI 100–300 keV ±
29% 2 . +1 . − . × − GRB 160821A f AstroSat/CZTI 100–300 keV . ± .
6% 8 . +1 . − . × − GRB 160910A f AstroSat/CZTI 100–300 keV . ± .
92% 4 . +7 . − . × − a The energy band in which polarization is observed. b The claimed polarization detections have been refuted. c The distance of GRB 021206 was taken to be light years. d The distance of GRB 021206 was taken to be 0.5 Gpc. e The lower limit to the photometric redshift of GRB 041219A ( z = 0 . ) was adopted. f The redshifts of these GRBs were estimated by the empirical luminosity relation.
Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
Then the massive photon group velocity υ in vacuumis no longer a constant c , but depends on the photonfrequency ν . That is, υ = ∂E∂p = c r − m γ c E ≈ c − m γ c h ν ! , (16)where the last approximation is valid when m γ ≪ hν/c ≃ × − (cid:0) ν GHz (cid:1) kg . Eq. (16) implies thatthe lower frequency, the slower the photon travels invacuum. Two massive photons with different frequen-cies ( ν l < ν h ), if emitted simultaneously from a samesource, would be received at different times by the ob-server. For a cosmic source, the arrival time differencedue to a nonzero photon mass is given by ∆ t m γ = m γ c h H (cid:0) ν − l − ν − h (cid:1) H γ ( z ) , (17)where H γ ( z ) is a dimensionless function of the sourceredshift z , H γ ( z ) = Z z (1 + z ′ ) − dz ′ p Ω m (1 + z ′ ) + Ω Λ . (18)It is obvious from Eq. (17) that observations of shortertime structures at lower frequencies from sources at cos-mological distances are particularly powerful for con-straining the photon mass. In contrast, observations ofhigh-energy photons from cosmological sources are bet-ter suited for probing LIV [16].The observed time delays between different energybands from astrophysical sources have been used to con-strain the photon mass. For instance, Lovell et al. [61]analysed the delay between the optical and radio emis-sion from flare stars and concluded that the relative ve-locity of light and radio waves was constrained to be × − over a wavelength range from 0.54 µ m to 1.2m, which implied an upper limit on the photon mass of m γ ≤ . × − kg . With the arrival time delay ofoptical pulses from the Crab Nebula pulsar over a wave-length range of 0.35–0.55 µ m, Warner & Nather [62] seta stringent limit on the possible frequency-dependenceof the speed of light, which led to an upper limit of m γ ≤ . × − kg . By analyzing the arrival time de-lay between the radio afterglow and the prompt gamma-ray emission from GRB 980703, Schaefer [63] obtained astricter upper limit of m γ ≤ . × − kg . Using GRBearly-time radio detections as well as multi-band radioafterglow peaks, Zhang et al. [64] improved the resultsof Schaefer [63] by nearly half an order of magnitude.Although the optical emissions of the Crab Nebula pul-sar have been used to constrain the photon mass, Weiet al. [65] showed that much more severe limits on thephoton mass can be obtained with radio observations of pulsars in the Large and Small Magellanic Clouds (LMCand SMC). The photon mass limits can be as low as m γ ≤ . × − kg for the radio pulsar PSR J0451-67in the LMC and m γ ≤ . × − kg for PSR J0045-7042in the SMC [65]. Owing to their fine time structures, lowfrequency emissions, and large cosmological distances,extragalactic fast radio bursts (FRBs) have been viewedas the most promising celestial laboratory so far for test-ing the photon mass [66–69, 71, 72]. The first attemptsto constrain the photon mass using FRBs were presentedin Wu et al. [66] and Bonetti et al. [67]. Adopting thepossible redshift z = 0 . for FRB 150418, and assum-ing the dispersive delay was caused by the nonzero pho-ton mass effect, Wu et al. [66] improved the limit of thephoton mass to be m γ ≤ . × − kg (see also [67]).However, the identification of a radio transient, whichprovided the redshift measurement to FRB 150418, waschallenged with a common AGN variability [146, 147].Now this redshift measurement is generally thought to beunreliable [148]. Subsequently, Bonetti et al. [68] usedthe confirmed redshift measurement of FRB 121102 toobtain a similar result of m γ ≤ . × − kg . Aftercorrecting for dispersive delay, Hessels et al. [149] foundthat the subbursts of FRB 121102 still show a time-frequency downward drifting pattern. The frequency-dependent time delay between subbursts is much smallerthan the dispersive delay, resulting in a tighter upperlimit on the photon mass of m γ ≤ . × − kg [71].The current astrophysical constraints on the photon restmass derived through the dispersion method are shownin Figure . Most of these results were based on a singlesource, in which the observed time delay was assumed tobe due to the nonzero photon mass and the dispersionfrom the plasma effect (see below) was ignored.3.2 Dispersion from the plasma effectDue to the dispersive nature of plasma, radio waveswith lower frequencies would travel through the ionizedmedian slower than those with higher frequencies [150].That is, the group velocity of electromagnetic wavespropagating through a plasma has a frequency depen-dence, i.e., υ p = c (cid:20) − (cid:16) ν p ν (cid:17) (cid:21) / , (19)where the plasma frequency ν p = [ n e e / (4 π m e ǫ )] / with n e the average electron number density along theline of sight, e and m e the charge and mass of an electron,respectively, and ǫ the permittivity of vacuum. The ar-rival time delay between two wave packets with differentfrequencies, which caused by the plasma effect, can then Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article -51 -50 -49 -48 -47 -46 -45 -44 -43 FRB 121102FRB 121102 subbursts
Extragalactic radio pulsar m g ( kg ) YEAR OF EXPERIMENT
Flare starsCrab Nebula pulsarGRB 980703FRB 150418
Fig. 5
Astrophysical limits on the photon rest mass throughthe dispersion method, including the strict upper boundsfrom flare stars [61], Crab nebula pulsar [62], GRB 980703[63], extragalactic radio pulsar [65], FRB 150418 [66, 67],FRB 121102 [68], FRB 121102 subbursts [71], and nine lo-calized FRBs [72]. be expressed as ∆ t DM = Z d lc ν p (cid:0) ν − l − ν − h (cid:1) = e π m e ǫ c (cid:0) ν − l − ν − h (cid:1) DM astro . (20)Here the dispersion measure (DM) is defined as the in-tegrated electron number density along the propagationpath, DM astro ≡ R n e d l . In a cosmological context, themeasured DM astro by an earth observer is DM astro ≡ R n e,z (1 + z ) − d l , where n e,z is the rest-frame numberdensity of free electrons [151]. For a cosmological sourceat redshift z , we expect DM astro to separate into fourcomponents: DM astro = DM MW + DM MWhalo + DM
IGM + DM host z (21)with DM MW the contribution from the Milky Way ion-ized interstellar medium, DM MWhalo the contributionfrom the Milky Way halo, DM IGM the contribution fromthe intergalactic medium (IGM), and DM host the contri-bution from the host galaxy and source environment inthe cosmological rest frame of the source. The measuredvalue of DM host is smaller by a factor of (1 + z ) [151].It is evident from Eq. (20) that radio waves propa-gating through a plasma are expected to arrive witha frequency-dependent dispersion in time of the /ν behavior. However, a similar dispersion ∝ m γ /ν [seeEq. (17)] could also be caused by a nonzero photon mass.The dispersion method used for constraining the photon mass is, therefore, hindered by the similar frequency de-pendences of the dispersions arise from the plasma andnonzero photon mass effects.3.3 Combined limits on the photon massIn order to diagnose an effect as radical as a finitephoton mass, statistical and possible systematic uncer-tainties must be carefully handled. One can not relysolely on a single source, for which it would be hard todistinguish the dispersions from the plasma and photonmass. For this reason, Shao & Zhang [69] constructeda Bayesian framework to derive a combined constraintof m γ ≤ . × − kg from a sample of 21 FRBs (in-cluding 20 FRBs without measured redshift, and one,FRB 121102, with a known redshift), where an uninfor-mative prior was used for the unknown redshift. Wei& Wu [70] also developed a statistical approach to studythis problem by analyzing a catalog of radio sources withmeasured DMs. This technique has the advantage thatit can both give a combined limit of the photon mass andestimate an average DM astro contributed by the plasmaeffect. Using the measured DMs from two statisticalsamples of extragalactic radio pulsars, Wei & Wu [70]placed combined limits on the photon mass at 68% con-fidence level, i.e., m γ ≤ . × − kg for the sample of22 LMC pulsars and m γ ≤ . × − kg for the othersample of 5 SMC pulsars.Since the dispersions from the plasma and photonmass have different redshift dependences, Refs. [67, 68,150] suggested that they could in principle be distin-guished by a statistical sample of FRBs at a range ofdifferent redshifts, thereby improving the sensitivity to m γ . Recently, nine FRBs with different redshift mea-surements (FRB 121102: z = 0 . [148, 152, 153];FRB 180916.J0158+65: z = 0 . [154]; FRB 180924: z = 0 . [155]; FRB 181112: z = 0 . [156]; FRB190523: z = 0 . [157]; FRB 190102: z = 0 . ; FRB190608 z = 0 . ; FRB 190611: z = 0 . ; FRB190711: z = 0 . [158]) have been reported. Wei & Wu[72] applied the idea suggested by Refs. [67, 68, 150] tothese nine localized FRBs to give a combined constrainton the photon mass. From observations, the radio signalsof all FRBs exhibit an apparent ν − -dependent time de-lay, which is expected from both the free electron contentalong the line of the sight and nonzero mass effects onphoton propagation (see Eqs. 17 and 20). Thus, Wei &Wu [72] attributed the observed time delay to two terms: ∆ t obs = ∆ t DM + ∆ t m γ . (22)In practice, the observed DM of each FRB, DM obs , isdirectly determined by fitting the ν − behavior of itsobserved time delay. This implies that both the line-of-sight free electron density and a massive photon (if it Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article H e (z) H (z) z Fig. 6
Dependence of functions H γ ( z ) and H e ( z ) on theredshift z . Reproduced from Ref. [72]. exists) determine the same DM obs , i.e., DM obs = DM astro + DM γ , (23)where DM astro is given in Eq. (21) and DM γ denotes the“effective DM” due to a nonzero photon mass [69], DM γ ≡ π m e ǫ c h e H γ ( z ) H m γ . (24)To investigate the possible DM γ , we need to figureout different DM contributions in Eq. (21). For a lo-calized FRB, the DM MW term can be well estimatedfrom a model of our Galactic electron distribution. The DM MWhalo term is not well modeled, but is expectedto contribute about 50–80 pc cm − [159]. Wei & Wu[72] adopted DM MWhalo = 50 pc cm − . The DM host term is highly uncertain, due to the dependences onthe type of the FRB host galaxy, the relative orien-tations of the galaxy disk and source, and the near-source plasma [160]. Wei & Wu [72] assumed that therest-frame DM host scales with the star formation rate(SFR; see Ref. [161] for more details), DM host ( z ) =DM host , p SFR( z ) / SFR(0) , where DM host , representsthe present value of DM host ( z = 0) and SFR( z ) = . . z z/ . . M ⊙ yr − Mpc − is the empirical form ofthe star formation history [162, 163]. In the analysis ofWei & Wu [72], DM host , was treated as a free parameter.The IGM contribution to the DM of an FRB at redshift z is related to the ionization fractions of hydrogen andhelium in the universe. Since both hydrogen and heliumare fully ionized at z < , one then has [151] DM IGM ( z ) = 21 cH Ω b f IGM πGm p H e ( z ) , (25)where m p is the proton mass, Ω b = 0 . is the baryonicmatter energy density [164], f IGM ≃ . is the fractionof baryon mass in the IGM [165], and H e ( z ) is the di-mensionless redshift function, H e ( z ) = Z z (1 + z ′ ) dz ′ p Ω m (1 + z ′ ) + Ω Λ . (26) Excluded at 95% CL
68% confidence level C u m u l a ti v e P r ob a b ilit y m [10 -50 kg] m [10 -15 eV/c ]
95% confidence level
Fig. 7
Cumulative posterior probability distribution on thephoton rest mass m γ derived from nine localized FRBs. Theexcluded values for m γ at 68% and 95% confidence levels aredisplayed with shadowed areas. Reproduced from Ref. [72]. The two dimensionless quantities ( H e and H γ ) as a func-tion of the redshift z are displayed in Figure . One cansee from this plot that the IGM and a possible photonmass contributions to DMs have different redshift depen-dences. As already commented, Bonetti et al. [67, 68] in-dicated that the different redshift dependences might notonly be able to break dispersion degeneracy but also im-prove the sensitivity to m γ at the point when a few red-shift measurements of FRBs become available (see alsoRefs. [69, 150]).With the redshift measurements of nine FRBs, Wei &Wu [72] maximized the likelihood function, L = Y i √ πσ tot ,i × exp ( − [DM obs ,i − DM astro ,i − DM γ ( m γ , z i )] σ ,i ) , (27)to derive a combined limit on the photon mass m γ . Herethe total variance on each FRB is given by σ = σ + σ + σ + σ , (28)where σ obs , σ MW , and σ MWhalo correspond to the uncer-tainties of DM obs , DM MW , and DM MWhalo , respectively,and σ int represents the global intrinsic scatter that mightoriginate from the diversity of host galaxy contributionand the large IGM fluctuation. The marginalized cumu-lative posterior distribution of the photon mass m γ isshown in Figure . The 68% confidence-level upper limiton m γ from nine localized FRBs is m γ ≤ . × − kg [72], which is comparable with or represents a factor of 7improvement over existing photon mass limits from theindividual FRBs [66–68, 71]. Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
Table 3
Summary of astrophysical upper bounds on the photon rest mass as obtained by the dispersion method.Wavelength (energy orAuthor (year) Source(s) frequency) range m γ (kg) Refs.Lovell et al. (1964) flare stats 0.54 µ m–1.2 m . × − [61]Warner and Nather (1969) Crab Nebula pulsar 0.35–0.55 µ m . × − [62]Schaefer (1999) GRB 980703 . × – . × Hz . × − [63]Zhang et al. (2016) GRB 050416A 8.46 GHz–15 keV . × − [64]Wei et al. (2017) Extragalactic radio pulsar ∼ . GHz . × − [65](PSR J0451-67)Extragalactic radio pulsar ∼ . GHz . × − (PSR J0045-7042)Wei and Wu (2018) 22 radio pulsars in the LMC ∼ . GHz . × − [70]5 radio pulsars in the SMC ∼ . GHz . × − Wu et al. (2016) FRB 150418 1.2–1.5 GHz . × − [66]Bonetti et al. (2016) FRB 150418 1.2–1.5 GHz . × − [67]Bonetti et al. (2017) FRB 121102 1.1–1.7 GHz . × − [68]Shao and Zhang (2017) 21 FRBs (20 of them without ∼ GHz . × − [69]redshift measurement)Xing et al. (2019) FRB 121102 subbursts 1.34–1.37 GHz . × − [71]Wei and Wu (2020) 9 localized FRBs ∼ GHz . × − [72] Table presents a summary of astrophysical upperbounds on the photon mass m γ obtained through the dis-persion method. As illustrated in Figure and Table ,the current best m γ limits were made by using the obser-vations of FRB 121102 subbursts ( m γ ≤ . × − kg )[71] and nine localized FRBs ( m γ ≤ . × − kg ) [72]. The WEP states that inertial and gravitational massesare identical. An alternative statement is that the tra-jectory of a freely falling, uncharged test body is inde-pendent of its internal structure and composition. Astro-physical observations provide a unique test of WEP bytesting if different massless particles experience gravitydifferently. In this section, we present the field-standardtest method through the Shapiro time delay effect.Particles which traverse a gravitational field would ex-perience a time delay (named the Shapiro delay) due tothe warping of spacetime. Adopting the PPN formalism,the γ -dependent Shapiro delay is given by [95] t gra = − γc Z r o r e U ( r ) dr , (29)where r e and r o denote the locations of the emittingsource and observer, respectively, and U ( r ) is the grav-itational potential along the propagation path. A pos-sible violation of the WEP implies that, if two particlesfollow the same path through a gravitational potential,then they would undergo different Shapiro delays. In thiscase, two test particles emitted simultaneously from thesource would reach our Earth with a arrival-time differ- ence (see Figure ) ∆ t gra = ∆ γc Z r o r e U ( r ) dr , (30)where ∆ γ = | γ − γ | represents the difference of the γ values for different particles, which can be used as mea-sure of a possible deviation from the WEP.To estimate the relative Shapiro delay ∆ t gra withEq. (30), one needs to figure out the gravitational po-tential U ( r ) . For sources at cosmological distances, U ( r ) should have contributions from the local gravitationalpotential U local ( r ) , the intergalactic potential U IG ( r ) ,and the host galaxy potential U host ( r ) . Since the poten-tial function for U IG ( r ) and U host ( r ) is hard to model, forthe purposes of obtaining lower limits, it is reasonable toextend the local potential U local ( r ) to the distance of thesource. In previous articles, the gravitational potential ofthe Milky Way, Virgo Cluster, or the Laniakea superclus-ter [166] has been adopted as the local potential, whichcan be modeled as a Keplerian potential U ( r ) = − GM/r .We thus have [96, 100] ∆ t gra = ∆ γ GMc × (31) ln h d + (cid:0) d − b (cid:1) / i h r L + s n (cid:0) r L − b (cid:1) / i b , where M is the mass of the gravitational field source, d is the distance from the particle source to the centerof gravitational field, b is the impact parameter of theparticle paths relative to the center, r L is the distance ofthe center, and s n = +1 (or − ) corresponds to the casewhere the particle source is located in the same (or oppo-site) direction with respect to the center of gravitational Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article
Fig. 8
A cartoon picture shows how two photons, one ata low frequency ( ν l ) and another at a high frequency ( ν h ),travel in curved space-time from their origin in a distant ex-tragalactic transient until reaching our Earth. A lower-limitestimate of the gravitational pull that the photons experiencealong their way is given by the mass in the center of the MilkyWay. field. For a particle source at the coordinates (R.A.= β s ,Dec.= δ s ), the impact parameter b can be estimated as b = r G p − (sin δ s sin δ L + cos δ s cos δ L cos( β s − β L )) , (32)where β L and δ L represent the coordinates of the centerof gravitational field.4.1 Arrival time tests of the WEPIf one assumes that the observed time delay ( ∆ t obs )between messengers or within messengers is mainlycontributed by the relative Shapiro delay ( ∆ t gra ) andwe know that the intrinsic (astrophysical) time delay ∆ t int > , a conservative upper limit on the WEP vi-olation could be placed by ∆ γ < ∆ t obs (cid:18) GMc (cid:19) − × (33) ln − h d + (cid:0) d − b (cid:1) / i h r L + s n (cid:0) r L − b (cid:1) / i b . The first multimessenger test of the WEP was be-tween photons and neutrinos from supernova SN 1987A. Minazzoli et al. [167] discussed the shortcomings of standardShapiro delay-based tests of the WEP. Because such tests are basedon the estimation for different messenger particles of the one-waypropagation time between the emitting source and the observerthat is not an observable per se. As a consequence, Minazzoli et al.[167] suggested that such tests are extremely model dependent andcan not provide reliable quantitative limits on the WEP violation.
Logo [97] and Krauss & Tremaine [96] used the ob-served time delay between photons and MeV neutrinosfrom SN 1987A to prove that the γ values for photonsand neutrinos are identical to an accuracy of approxi-mately 0.3–0.5%. These constraints can be further im-proved using the likely associations of high-energy neu-trinos with the flaring blazars [168–171] or GRBs [172].Besides the neutrino-photon sectors, the coincident de-tections of GW events with electromagnetic counterpartsalso provided multimessenger tests of the WEP, extend-ing the WEP tests with GWs and photons [100, 173–180]. For example, with the assumption that the ar-rival time delay between GW170817 and GRB 170817A( ∼ . s) from a binary neutron star merger is mainlydue to the gravitational potential of the Milky Way out-side a sphere of 100 kpc, Abbott et al. [174] derived − . × − ≤ γ g − γ γ ≤ . × − . A more severeconstraint of | γ g − γ γ | < . × − can be achieved forGW170817/GRB 170817A when the gravitational poten-tial of the Virgo Cluster is considered [177].WEP tests have also been performed within the samespecies of messenger particles (neutrinos, photons, orGWs) but with varying energies (e.g., [71, 96, 98–100, 181–194]). In the neutrino sector, Longo [96] usedthe observed delay between 7.5 MeV and 40 MeV neutri-nos from SN 1987A to set | γ ν (40 MeV) − γ ν (7 . | < . × − . Wei et al. [171] adopted the delay forneutrinos ranging in energy from about 0.1 to 20 Tevfrom the direction of the blazar TXS 0506+056 to ob-tain | γ ν (20 TeV) − γ ν (0 . | < . × − . In thephoton sector, Gao et al. [98] (see also Sivaram [181])suggested that one can use the time delays of photonsof different energies from cosmological transients to testthe WEP. They applied the similar approach to GRBsand derived | γ γ (eV) − γ γ (MeV) | < . × − for GRB080319B and | γ γ (GeV) − γ γ (MeV) | < . × − for GRB090510. Recently, such a test has also been applied todifferent-energy photons in other transient sources, in-cluding FRBs [71, 99, 185–187], TeV blazars [188], andthe Crab pulsar [189–192]. In the GW sector, Wu et al.[100] suggested that one can treat the GW signals withdifferent frequencies as different gravitons to test theWEP. They used the delay for the GW signals rangingin frequency from about 35 to 150 Hz from GW150914to set | γ g (35 Hz) − γ g (150 Hz) | < − (see also [193]).Very recently, Yang et al. [194] obtained new constraintsof ∆ γ by using the GW data of binary black holes merg-ers in the LIGO-Virgo catalogue GWTC-1. The bestconstraints came from GW170104 and GW170823, i.e., ∆ γ < − .Yu & Wang [195] and Minazzoli [196] proposed a newmultimessenger test of the WEP using strongly lensedcosmic transients. By measuring the time delays betweenlensed images seen in different messengers, one can ob- Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article =0.89/3
GRB 020813 ob s (r a d ) -1 0 1 2 3 4 2.32.42.52.62.7 (r a d ) =11.32/13 GRB 021004 ob s (r a d ) E (eV) -4 -3 -2 -1 0 1 2 2.02.22.42.6 (r a d ) (10 -24 ) Fig. 9
Fit to the afterglow polarimetry data of GRB 020813 and GRB 021004. Left panels: linear fits of the spectropolari-metric data. Right panels: 1-3 σ confidence levels in the φ - ∆ γ plane. Reproduced from Ref. [198]. tain robust constraints on the differences of the γ values.4.2 Polarization tests of the WEPThe fact that the trajectory of any freely falling testbody does not depend on its internal structure is one ofthe consequences of the WEP. Since the polarization isviewed as a basic component of the internal structure ofphotons, Wu et al. [197] proposed that multiband elec-tromagnetic emissions exploiting different polarizationsare an essential tool for testing the accuracy of the WEP.For a linearly polarized light, it is the combinationof two monochromatic waves with opposite circular po-larizations (labeled with ‘ l ’ and ‘ r ’). If the WEP fails,then the two circularly polarized beams travel throughthe same gravitational field with different Shapiro delays.The relative Shapiro delay ∆ t gra of these two beams isthe same as Eq. (30), except now ∆ γ = | γ l − γ r | cor-responds to the difference of the γ values for the left-and right-handed circular polarization states. Because ofthe arrival-time difference between the two circular com-ponents, the polarization plane of a linearly polarizedlight would be subject to a rotation along the photons’propagation path. The rotation angle due to the WEPviolation is given by [101, 102] ∆ φ WEP ( E ) = ∆ t gra πcλ = ∆ t gra E ¯ h , (34) where E is the observed energy.Since the initial angle of the linearly polarized lightis not available, the exact value of ∆ φ WEP is unknown.Yet, we can set an upper limit for the differential rotationangle | ∆ φ ( E ) − ∆ φ ( E ) | , because the polarization signalwould be erased as the path difference goes beyond thecoherence length. Therefore, the detection of liner polar-ization indicates that | ∆ φ ( E ) − ∆ φ ( E ) | should not betoo large. By considering the Shapiro delay of the MilkyWay’s gravitational potential, and setting the upper limitof | ∆ φ ( E ) − ∆ φ ( E ) | to be π , Yang et al. [101] obtainedan upper limit on the γ discrepancy of ∆ γ < . × − from the polarization measurement of GRB 110721A. Through a detailed calculation on the evolution of GRBpolarization arising from the WEP violation, Wei & Wu[102] proved that the initial polarization signal is not sig-nificantly suppressed even if | ∆ φ ( E ) − ∆ φ ( E ) | is largerthan π/ . Applying their formulae to the GRB polari-metric data, Wei & Wu [102] placed the most stringentlimits so far on a deviation from the WEP for two cases: ∆ γ < . × − for GRB 061122 and ∆ γ < . × − for GRB 110721A.If the rotation angle induced by the WEP violationis considered here, the observed linear polarization an- Similar LIV constraints have been obtained by astrophysicalpolarization measurements [45, 49]. Here, the LIV effect was sup-posed not to work simultaneously to accidentally enhance or cancelthe effect from the WEP violation.
Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article gle for photons emitted at a certain energy E with anintrinsic polarization angle φ should be φ obs = φ + ∆ φ WEP ( E ) . (35)Instead of requiring the argument that the rotation ofthe polarization plane would severely reduce polariza-tion over a broad bandwidth, Wei & Wu [198] simplyassumed that φ is an unknown constant. One can thenconstrain the WEP violation by measuring the energy-dependent change of the polarization angle. Wei & Wu[198] showed that it is possible to give constraints onboth ∆ γ and φ through direct fitting of the multiwave-length polarimetric data of the optical afterglows of GRB020813 and GRB 021004 (see Figure ). At the σ con-fidence level, the joint constraint on ∆ γ from two GRBsis − . × − < ∆ γ < . × − . Yi et al. [199, 200]applied the same method to radio polarimetry data ofthe blazar 3C 279 and extragalactic radio sources, andobtained less stringent constraints on ∆ γ .Shapiro delay-based tests of the WEP are summarizedin Table for comparison. When the test particles aredifferent messengers, the best multimessenger constraintis ∆ γ < . × − for keV photons and the TeV neutrinofrom GRB 110521B [172]. When the test particles arethe same messengers but with different energies, the bestconstraints are ∆ γ < . × − for 1.344–1.374 GHzphotons from FRB 121102 subbursts [71] and ∆ γ < . × − for ∼ Hz GW signals from GW170104 [194].When the test particles are the same messengers but withdifferent polarization states, the best constraint is ∆ γ < . × − for polarized gamma-ray photons from GRB061122 [102]. In this paper, we have reviewed the status of the high-precision tests of fundamental physics with astrophysicaltransients. A few solid conclusions and future prospectcan be summarized:(i) Violations of Lorentz invariance can be tested byseeking a frequency-dependent velocity resulting fromvacuum dispersion and by searching for a change in po-larization arising from vacuum birefringence.For vacuum dispersion studies, in order to obtaintighter limits on the LIV effects one should choose thewaves of higher frequencies that propagate over longerdistances. GRBs, with their short spectral lags, cosmo-logical distances, and gamma-ray emissions, are the mostpowerful probes so far for LIV constraints in the disper-sive photon sector. In the past, emission from GRBs hasbeen observed only at energies below 100 GeV. Recently,VHE photons above 100 GeV have been detected fromGRB 190114C [118] and GRB 180720B [201], opening a new window to study GRBs in sub-TeV gamma-rays.Such detections are expected to become routine in the fu-ture [202], especially with the operations of facilities suchas the international Cherenkov Telescope Array (CTA)and the Large High Altitude Air Shower Observatory(LHAASO) in China. Observations of extremely high-energy emission from more GRBs will further improveconstraints on LIV. For vacuum birefringence studies,in order to tightly constrain the LIV effects one shouldchoose those astrophysical sources with larger distancesand polarization observations in a higher energy band.Thanks to their polarized gamma-ray emissions and largecosmological distances, GRBs are promising sources forseeking LIV-induced vacuum birefringence. As moreand more GRB polarimeters (such as POLAR-II, TSUB-AME, NCT/COSI, GRAPE; see McConnell [203] for areview) enter service, it is reasonable to expect that theGRB polarimetric data will be significantly enlarged.Limits on the vacuum birefringent effect can be furtherimproved with larger number of GRBs with higher en-ergy polarimetry and higher redshifts.(ii) The most direct and robust method for constrain-ing the photon rest mass is to measure the frequency-dependent dispersion of light.Theoretically speaking, to obtain more stringentbounds on the photon mass one should choose the wavesof lower frequencies that propagate over longer distances.Cosmological FRBs, with their short time durations, lowfrequency emissions, and long propagation distances, arethe most excellent astrophysical probes so far for con-straining the photon mass. However, the derivation ofa photon mass limit through the dispersion method iscomplicated by the similar frequency dependences of thedispersions due to the plasma and nonzero photon masseffects. If in the future more FRB redshifts are measured,the different redshift dependences of the plasma and pho-ton mass contributions to the DM can be used to breakdispersion degeneracy and to improve the sensitivity tothe photon mass [67–69, 72, 150]. It is encouraging thatradio telescopes, such as the Canadian Hydrogen Inten-sity Mapping Experiment (CHIME) [154], the AustraliaSquare Kilometer Array Pathfinder (ASKAP) [155], andthe Deep Synoptic Array 10-dish prototype (DSA-10)[157], have led to direct localizations of FRBs. The fieldwill also greatly benefit from the operations of facilities,such as the Five Hundred Meter Aperture Spherical ra-dio Telescope (FAST), the DSA-2000, and the SquareKilometer Array (SKA). The rapid progress in localizingFRBs will further improve the constraints on the photonmass. In the future, a swarm of nano-satellites orbitingthe Moon will open a new window at very low frequen-cies in the KHz–MHz range [150]. These satellites areexpected to function as a distributed low frequency ar-ray far away from the blocking ionosphere and terrestrial
Jun-Jie Wei and Xue-Feng Wu, Front. Phys. 16(4), 44300 (2021) eview article radio frequency interference, which will have stable con-ditions for observing the cosmic signals. Such low fre-quency observations would offer a more sensitive probeof any delays expected from a nonzero photon mass.(iii) Shapiro delay-based tests of the WEP havereached high precision.Pioneered by SN 1987A tests [96, 97], the Shapiro de-lay experienced by an astrophysical messenger travelingthrough a gravitational field has been intensively em-ployed to constrain possible violations of the WEP (e.g.,[98–102]). However, there are some uncertainties andcaveats involved in such tests. The large uncertaintyis from the estimation of the local gravitational poten-tial U ( r ) . The exact gravitational potential function isnot well known. More accurate models of the function U ( r ) could improve the constraints on the WEP viola-tion, but the improvement should be very limited. Mostoften the Shapiro delay terms caused by the host galaxyand the intergalactic gravitational potential are ignored.In principle, these terms might be much larger than thatcaused by the local gravitational potential. With thebetter understanding of the potential functions for theseterms, the WEP tests might be improved by orders ofmagnitude. As explained in Gao & Wald [204], Eq. (29)is gauge dependent, i.e., depending on the coordinatechoice one can obtain both positive and negative valuesof the Shapiro delay. Additionally, Eq. (29) implicitlyassumes that the trajectory is short enough that one cantreat the cosmological spacetime as Minkowski plus alinear perturbation. Minazzoli et al. [167] showed thatthis assumption is well justified for sufficiently nearbysources like GW170817, but not for sources at cosmologi-cal distances such as GRBs or FRBs with redshifts z ≥ .While Nusser [186] has provided a formulation that, inprinciple, can be applied to more distant sources, mostof the previously cited works use the standard expres-sion for the Shapiro delay (Eq. 29). This is an additionalsource of uncertainty involved in such WEP tests. Thestandard use of Eq. (29) also suffers from an assumptionthat there exists a coordinate time such that the poten-tial and its derivative vanishes at infinity. Minazzoli etal. [167] showed that such an assumption results in aspurious divergence of the Shapiro delay for increasinglyremote sources. With the use of an adequate coordi-nate time, Minazzoli et al. [167] found that the Shapirodelay is no longer monotonic with the number of thesources of the gravitational field. Hence, without fur-ther assumptions and/or observational input, one cannot obtain a conservative lower limit on the Shapiro de-lay from a subset of the gravitational sources based onEq. (29). It might be possible to constrain the gravita-tional potential along the line of sight using cosmologicalobservations (e.g., of galaxy peculiar velocities, as in Ref.[205]), and thus avoid the need to use Eq. (29). Minaz- zoli et al. [167] suggested that any further developmentsof this Shapiro delay-based test should be inspired bya fundamental theory to avoid the gauge dependence ofthe current expression of the test. Acknowledgements
We are grateful to the anonymous refer-ees for insightful comments. This work is partially supported bythe National Natural Science Foundation of China (grant Nos.11673068, 11725314, U1831122, and 12041306), the Youth Innova-tion Promotion Association (2017366), the Key Research Programof Frontier Sciences (grant Nos. QYZDB-SSW-SYS005 and ZDBS-LY-7014), and the Strategic Priority Research Program “Multi-waveband gravitational wave universe” (grant No. XDB23000000)of Chinese Academy of Sciences.
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Upper constraints on the differences of the γ values through the Shapiro (gravitational) time delay effect. Categorization Author (year) Source Messengers Gravitational Field ∆ γ Refs.Longo (1988) SN 1987A eV photons and MeV neutrinos Milky Way . × − [96]Krauss and Tremaine (1988) SN 1987A eV photons and MeV neutrinos Milky Way . × − [97]Wang et al. (2016) PKS B1424-418 MeV photons and PeV neutrino Virgo Cluster . × − [168]PKS B1424-418 MeV photons and PeV neutrino Great Attractor . × − Boran et al. (2019) TXS 0506+056 GeV photons and TeV neutrino Milky Way . × − [169]Laha (2019); Wei et al. (2019) TXS 0506+056 GeV photons and TeV neutrino Laniakea supercluster of galaxies − – − [170, 171]Different messengers Wei et al. (2016a) GRB 110521B keV photons and TeV neutrino Laniakea supercluster of galaxies . × − [172]Abbott et al. (2017) GW170817 MeV photons and GW signals Milky Way -2.6 × − —1.2 × − [174]Wei et al. (2017) GW170817 MeV photons and GW signals Virgo Cluster 9.2 × − [177]GW170817 eV photons and GW signals Virgo Cluster 2.1 × − Shoemaker and Murase (2018) GW170817 MeV photons and GW signals Milky Way 7.4 × − [178]Boran et al. (2018) GW170817 MeV photons and GW signals Milky Way 9.8 × − [179]Yao et al. (2020) GW170817 MeV photons and GW signals Milky Way + host galaxy ∼ − [180]GW170817 eV photons and GW signals Milky Way + host galaxy ∼ − Longo (1988) SN 1987A 7.5–40 MeV neutrinos Milky Way . × − [96]Wei et al. (2019) TXS 0506+056 0.1–20 TeV neutrinos Laniakea supercluster of galaxies . × − [171]Sivaram (1999) GRB 990123 eV–MeV photons Milky Way . × − [181]Gao et al. (2015) GRB 090510 MeV–GeV photons Milky Way . × − [98]GRB 080319B eV–MeV photons Milky Way . × − Wei et al. (2015) FRB 110220 1.2–1.5 GHz photons Milky Way . × − [99]FRB/GRB 100704A 1.23–1.45 GHz photons Milky Way . × − Same messengers Tingay and Kaplan (2016) FRB 150418 1.2–1.5 GHz photons Milky Way (1–2) × − [185]with Nusser (2016) FRB 150418 1.2–1.5 GHz photons Large-scale structure − – − [186]different energies Xing et al. (2019) FRB 121102 subbursts 1.344–1.374 GHz photons Laniakea supercluster of galaxies . × − [71]Wei et al. (2016b) Mrk 421 keV–TeV photons Milky Way . × − [188]PKS 2155-304 sub TeV–TeV photons Milky Way . × − Yang and Zhang (2016) Crab pulsar 8.15–10.35 GHz photons Milky Way (0.6–1.8) × − [189]Zhang and Gong (2017) Crab pulsar eV–MeV photons Milky Way 3.0 × − [190]Desai and Kahya (2018) Crab pulsar 8.15–10.35 GHz photons Milky Way 2.4 × − [191]Leung et al. (2018) Crab pulsar 1.52–2.12 eV photons Milky Way 1.1 × − [192]Wu et al. (2016) GW150914 35–150 Hz GW signals Milky Way ∼ − [100]Kahya and Desai (2016) GW150914 35–250 Hz GW signals Milky Way 2.6 × − [193]Yang et al. (2020) GW170104 ∼ Hz GW signals Milky Way 6.2 × − [194]GW170823 ∼ Hz GW signals Milky Way 1.0 × − Wu et al. (2017) FRB 150807 Polarized radio photons Laniakea supercluster of galaxies . × − [197]GRB 120308A Polarized optical photons Laniakea supercluster of galaxies . × − Same messengers GRB 100826A Polarized gamma-ray photons Laniakea supercluster of galaxies . × − with different Yang et al. (2017) GRB 110721A Polarized gamma-ray photons Milky Way . × − [101]polarization states Wei and Wu (2019) GRB 061122 Polarized gamma-ray photons Laniakea supercluster of galaxies . × − [102]GRB 110721A Polarized gamma-ray photons Laniakea supercluster of galaxies . × − Wei and Wu (2020) GRB 020813 and Polarized optical photons Milky Way -2.7 × − —3.1 × − [198]GRB 021004 - J u n - J i e W e i a n d X u e - F e n g W u , F r o n t . P h y s . ( ) , (2021