Tests of Lorentz Invariance

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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