Lorentz Invariance is a cornerstone of modern physics and an essential symmetry of quantum field theory. It is a requirement of special relativity, which postulates that inertial frames are equivalent. However, recently some quantum field theories have suggested that well-established concepts like Lorentz Invariance may be violated at very high energies that approach the Planck energy scale, about 1.2 × 10¹⁹ GeV [1] . Of course, such energies cannot currently be achieved through experiments on Earth, but energetic photons traveling over huge cosmological distances may provide hints or evidence to a possible violation of Lorentz Invariance [2] .

Physically, Lorentz Invariance Violation (LIV) can be understood in analogy to how light behaves when it enters a dispersive medium. The speed of light in a dispersive medium is not constant but depends on its wavelength because light waves are sensitive to the specific structure of the dispersive medium. Likewise, according to some quantum gravity theories, exceedingly high-energy photons may be sensitive to the structure of spacetime implying that the speed of light is not constant in vacuum, thus violating Lorentz Invariance [2] [3] . One way to measure this effect is to compare the arrival time of energetic photons, but of different energies, that emanate from the same source, for example a gamma-ray burst (GRB) [4] [5] [6] [7] [8] . A recent study in 2023, by Finke and Razzaque [9] , finds possible evidence for LIV in the gamma-ray burst GRB221009A.

In this paper, we use a recent data sample of short GRBs to place constraints on LIV. Short GRBs are bursts with an observed duration less than 2 s, and they are believed to emerge from the coalescence of two compact objects, like two neutron stars or a neutron star and a black hole [10] . In investigating LIV, short GRBs are preferred to long GRBs because the latter have larger and more varied intrinsic spectral lags, which complicates their utilization as probes of LIV [11] .

In Section 2, we provide a brief review of the relevant quantum gravity equations that pertain to LIV. In Section 3, we introduce the data sample of short GRBs that we use and then present our analysis and results. Finally, in Section 4 we give our conclusions and future prospects.

According to some quantum gravity models [12] , the speed of light in vacuum, typically labeled c, is not constant, and thus the relationship between a photon’s energy, E, and its momentum, p, is not given by E = pc, but rather by the following dispersion equation, which is obtained through a Taylor expansion:

E 2 ≈ p 2 c 2 { 1 − s n [ p c E Q G ] n } (1)

where E_QG is the quantum gravity energy, s_n is a sign factor which is equal to +1 or −1 depending on whether higher energy photons move slower or faster than lower energy photons, respectively, and n refers to the dependence order of the energy [13] .

The speed of propagation of light, v_γ, is simply the group velocity and can be calculated by differentiating E with respect to p:

v γ = ∂ E ∂ p ≈ c { 1 − s n [ n + 1 2 ] [ E E Q G ] } (2)

According to Equation (2), the speed of propagation of light in vacuum is energy dependent and not constant, which violates Lorentz Invariance and implies that photons of different energies emanating from the same source, for example a GRB, would exhibit a spectral time delay, typically called a spectral lag, Δ t L I V , which can be expressed as [11] [14] :

Δ t L I V = − s n [ 1 + n 2 H 0 ] [ E h n − E l n ( 1 + z ) n E Q G n ] ∫ 0 z ( 1 + z ) n h ( z ) d z (3)

where H₀ is the Hubble constant, E_h and E_l are the high and low energies in the GRB’s rest frame, respectively, and h(z) is the Hubble expansion rate, which is given by [11] :

h ( z ) = Ω m ( 1 + z ) 3 + Ω λ (4)

where z is the GRB’s redshift, and Ω_m and Ω_λ are the matter and curvature density parameters, respectively. To simplify the notation, Equation (3) can be rewritten as [11] :

Δ t L I V = a L I V ⋅ K ( z ) ⋅ ( 1 + z ) (5)

where K(z) includes the terms that depend on z and a L I V includes the other parameters. In the rest frame of the burst, the spectral lag due to LIV is simply:

Δ t L I V 1 + z = a L I V ⋅ K ( z ) (6)

It is important to keep in mind that GRBs may have intrinsic spectral lags, due to their prompt and delayed gamma-ray emissions, that have nothing to do with LIV, and this intrinsic spectral lag must be factored in and included. In this study, we assume the intrinsic spectral lag is constant for the short GRBs that we use and is given, in the rest frame of the burst, by:

Δ t i n t 1 + z = b (7)

where b is a constant. Now, we can combine the intrinsic spectral lag with that due to LIV to obtain the total observed spectral lag:

Δ t = Δ t L I V + Δ t i n t = a L I V ⋅ K ( z ) ⋅ ( 1 + z ) + b ⋅ ( 1 + z ) (8)

Thus, if there is no LIV, then ∆t should be directly proportional to (1 + z), and a plot of ∆t versus (1 + z) should exhibit linear behavior with a slope b.

The data sample that we use consists of 46 short GRBs, which we took from the recent study by Xiao et al. [11] . The bursts in the sample consist of GRBs observed by either the Fermi satellite or the Swift satellite and have a redshift range from 0.0098 to 2.6. To extract the spectral lag, [11] selected the energy bands in the rest frame of the burst to be 15 - 70 keV and 120 - 250 keV. They then applied a new technique, the improved Li-CCF method, to extract the observed spectral lags, ∆t.

Our first step involved plotting ∆t as a function of (1 + z) and checking whether we obtain a good linear fit, and if so, extracting the slope, b_o. As Figure 1 shows, we obtained a good linear fit with a best-fit value of b_o = 1.41 ± 0.41 ms and a reduced chi-square of 1.05 with a p-value of 0.38 for the chi-square distribution. Note that if the Swift and Fermi satellites gave different values of ∆t for the same burst, then we took the mean value.

Our next step involved binning the data in redshift while keeping, as much as possible, the number of bursts per bin the same. We then applied a linear fit similar to what we did in Figure 1 and extracted the value of the slope, b, for each bin, and then compared these values of b with that for the entire sample – namely, b_o. The point is that we wanted to check whether the values of b showed any evolution or dependence on redshift, which would hint at a possible LIV contribution, or whether they were consistent with a constant value. We repeated our analysis with a different number of bins to check whether binning introduced any biases. Our results are presented in Figure 2, and the different panels show our results for the different bin numbers that we used.

Our results, shown in Figure 2, indicate that when 3 bins were used, b showed no redshift dependence or evolution, but when a higher number of bins was used, there were hints of small deviations from what is expected for Lorentz Invariance, but we caution that these deviations are not statistically significant, and our results are consistent with no LIV at the 1σ level.

Lorentz Invariance is one of the fundamental principles in modern physics, and checking its validity is an important task since any violation of Lorentz Invariance would have significant consequences regarding key concepts in physics. In this study, we employed a recent sample of short GRBs and used their observed spectral lags to check for any deviation from Lorentz Invariance. To do that, we binned the data in redshift and then checked whether the extracted intrinsic spectral lag for each bin, b, is redshift dependent or whether it is constant and consistent with its no LIV value, b_o. Our results indicate that although there are slight deviations from what is expected for Lorentz Invariance, these deviations are not statistically significant. In the future, it would be important to reexamine this issue using an expanded data sample that includes not just short GRBs, but also long GRBs, assuming that in the future we will have a better understanding of the mechanisms that contribute to the relatively large values and scatter in the spectral lags of long GRBs.

The authors declare no conflicts of interest regarding the publication of this paper.

Azzam, W.J. and Hasan, A.M. (2023) Constraining Lorentz Invariance Violation Using Short Gamma-Ray Bursts. Journal of Applied Mathematics and Physics, 11, 2179-2184. https://doi.org/10.4236/jamp.2023.118139