Lorentz Invariance Violation Limits from the Spectral Lag Transition of GRB 190114C

Spectral lag, referring to the different arrival times for the photons with different energies, is a common feature in gamma-ray burst (GRB) pulses observed during the prompt emission (Norris et al., 1986, 2000; Cheng et al., 1995; Band, 1997; Chen et al., 2005; Peng et al., 2007). Observationally, most of the GRB pulses are dominated by positive lags (i.e., the soft photons lag behind the hard photons), and a small fraction of pulses show zero and even negative lags (e.g, Norris et al. 2000; Yi et al. 2006; Wei et al. 2017a). It was suggested that the high-latitude emission “curvature effect” of a relativistic jet can plausibly account for the intrinsic origin of (positive) spectral lags (e.g., Ioka & Nakamura 2001; Shen et al. 2005; Lu et al. 2006; Shenoy et al. 2013). Zhang et al. (2009) pointed out that the main difficulty of the curvature effect scenario is that the flux levels at different energy bands are significantly lower than those observed. Uhm & Zhang (2016) further showed that there would be essentially unnoticeable time lags given rise from the high-latitude emission of a spherical fireball and the reproduced properties of light curves are not compatible with the observations. Instead, they established a physical model invoking a spherical shell rapidly expands in the bulk-accelerating region, which is suggested to account for the spectral lags more reasonably (Wei et al., 2017a; Shao et al., 2017; Lu et al., 2018; Uhm et al., 2018). On the other hand, Lu et al. (2018) found that the spectral lag is closely related to the spectral evolution within 92 pulses from 84 GRBs observed by the Fermi Gamma-ray Burst Monitor (GBM). These researches motivate the work by Du et al. (2019), who investigated the detailed relationship of the spectral lag with spectral evolution by using a phenomenological model with different spectral evolution patterns. Their results can naturally reproduce the negative lags observed in GRB 160625B for more generic physical models.

The candidate quantum-gravitational (QG) effect in the particle-physics field provides an explanation of the negative lags. One such effect is the so-called Lorentz invariance violation (LIV). There is great interest in the expectation that the Lorentz invariance breaks presumably at the energy on the Planck scale (i.e., $E_{\rm QG}\approx E_{\rm pl}=\sqrt{hc^{5}/2\pi G}\simeq 1.22\times 10^{19}$ GeV; Amelino-Camelia et al. 1998; Mattingly 2005; Amelino-Camelia 2013). Such type of QG effects may appear as the high-energy photons interact with the ‘foam’ on short time and small scales (Amelino-Camelia et al., 1997), resulting in the in vacuo photon dispersion (i.e., an energy-dependent speed of light). In such scenarios, the high-energy photons propagate more slowly than the low-energy photons in vacuum (Amelino-Camelia et al., 1998). Thus, the discrepant arrival time of photons with different energies emitted from the same astrophysical source allows us to constrain the quantum-gravity energy scale $E_{\rm QG}$ (Amelino-Camelia et al., 1998; Mattingly, 2005; Ellis & Mavromatos, 2013).

GRBs are characterized by long distances, short spectral lags, and very energetic emissions, which make them the current most powerful probes to the energy scales of LIV in the dispersive photon sector (Mattingly, 2005; Abdo et al., 2009a, b; Wei et al., 2017a; Ellis et al., 2019). The bounds on LIV effects have been obtained by analyzing the rough time lag of a single high-energy photon detected in some bursts or the arrival times of sharp features of intensity profile with statistical samples of GRBs (Amelino-Camelia et al. 1998; Schaefer 1999; Ellis et al. 2003, 2006; Boggs et al. 2004; Abdo et al. 2009a, b; Xiao & Ma 2009; Chang et al. 2012; Nemiroff et al. 2012; Zhang & Ma 2015). The most stringent lower limit on the linear LIV leading term of vacuum dispersion with Taylor expansion comes from the studies of the effects on the time of flight by analyzing the arrival time delay between a 31 GeV photon and low-energy photons from short GRB 090510, with $E_{\rm QG,1}>(1-10)\times E_{\rm pl}$ (Abdo et al., 2009a; Vasileiou et al., 2013). For the quadratic LIV, the strictest lower limit has resulted from the observations of active galactic nuclei Markarian 501 (Mrk 501) by the H.E.S.S. telescope, i.e., $E_{\rm QG,2}>8.5\times 10^{10}$ GeV ($E_{\rm QG,2}>7.8\times 10^{11}$ GeV) from the arrival time delays of multi-TeV photons (from the modifications to the interactions between very high energy gamma-rays and extragalactic background light photons; Abdalla et al. 2019). Recently, the TeV gamma-rays detected from GRB 190114C by the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescopes yield a competitive lower limit $E_{\rm QG,2}>6.3\times 10^{10}$ GeV ($E_{\rm QG,2}>5.6\times 10^{10}$ GeV) in the subluminal (superluminal) scenario (Acciari et al., 2020). Actually, these lower constraints correspond to no significant LIV effects but suffer from some conservative assumptions. For instance, in Acciari et al. (2020), it is not a multi-wavelength fit in investigating effects on the time of flight; there is no comparison in investigating the likelihood of synchrotron-self-Compton versus a base synchrotron explanation; the parameters of intrinsic spectral and temporal emissions are assumed. Furthermore, distinguishing an intrinsic time delay at the source from a delay caused by the propagation effects in a vacuum is a critical obstacle. Ellis et al. (2006) first considered the possible intrinsic time lag ($\Delta t_{\rm int}$) by fitting the spectral lags extracted from broad light curves observed in different energy bands with a linear regression model in which the slope is related to the LIV effects and the intercept corresponds to the $\Delta t_{\rm int}$ in the observer. This approach directs to weak evidence for LIV by assuming that all GRBs have the same intrinsic time lag. Thus far, it is necessary to constrain LIV by using the detailed spectral lag measurements from high-quality and high-energy light curves from GRBs.

Recently, Wei et al. (2017a) used the spectral lag data between the light curves observed in different energy bands from GRB 160625B to put robust constraints on the LIV effects (see also Pan et al. 2020). This method, for the first time, gives a reasonable formulation for the intrinsic (positive) time lags, and the $E_{\rm QG}$ can be constrained if a transition from positive lags to negative lags appears in high-energy range¹¹1The potential LIV effect that is expected to appear on QG scales could lead to the energy-dependent time delay in the arrival times of observed photons, it would influence the observed temporal and spectral features (e.g., the spectral evolution) of GRBs and contribute to the entire observed spectral lags. As shown by Du et al. (2019), different spectral evolution patterns can produce different evolving behaviors of spectral lags in GRB pulses, which would inherently include the contributions of the LIV effects if which exist.. In this way, the observed arrival time delay between light curves in different energy bands are comprised of two contributions²²2In fact, for cosmic transient sources (e.g., GRB) there are three other effects attributed to the time delay between photons with different energies (Gao et al., 2015): (1) the potential time delay from the special-relativistic effects in the case that photons have non-zero rest mass; (2) the time delay from the dispersion by the free electrons in the line of sight and (3) the gravitational potential integrated from the emission site to the Earth if Einstein’s equivalence principle is violated. The contributions of those three effects can be negligible for the photons emitted from GRBs, as shown by (Gao et al., 2015) and (Wei et al., 2015).

where $\Delta t_{\rm LIV}$ is the observed time lags induced by the LIV effects and $\Delta t_{\rm int,z}=\Delta t_{\rm int}(1+z)$ is the intrinsic time lags corrected by the cosmological time dilation. In this work, we show that, in GRB 190114C, the spectral lag monotonically increases with the photon energy and approximately saturates at a certain photon energy. In particular, we show decisive evidence that there is a transition from positive lags to negative lags well measured at $\sim 0.7$ MeV, lower than that in GRB 160625B with one order of magnitude ($\sim 8$ MeV, Wei et al. 2017a). Thus, GRB 190114C is another case that allows us to constrain the quantum-gravity energy scale by directly fitting its observed spectral lag data.

The contents of this paper are arranged as follows. In Section 2, we describe the observed properties of GRB 190114C. We present our method and constraint results in Section 3. Our conclusion and discussions are presented in Section 4. The flat $\Lambda$CDM cosmology (radiation density is ignored) with parameters $\Omega_{\rm m,0}=0.315$ ($\Omega_{\Lambda,0}=1-\Omega_{\rm m,0}$) and $H_{0}=67.36$ km$\cdot$s${}^{-1}\cdot$Mpc^-1 (Planck Collaboration et al., 2018) is adopted throughout this work.

On 14 January 2019 at 20:57:03 universal time, GRB 190114C was first triggered by the GBM onboard the Fermi Gamma-Ray Space Telescope (Hamburg et al., 2019), and the Burst Alert Telescope (BAT) onboard the Neil Gehrels Swift Observatory located this burst with coordinates (J2000) right ascension, ${\rm R.A.}=54.5^{\circ}$, and declination, ${\rm Dec}=-26.9^{\circ}$ (Gropp et al., 2019). BAT and GBM measured the duration time of this burst in terms of $T_{90}$ (the time interval containing 90% of the total photon counts) to be $\sim 362$ s (in 15$-$350 keV band) and $\sim 116$ s (in 50$-$300 keV band), respectively. Spectroscopic observations reveal absorption features consistent with Ca H&K, Na ID, Mg II and Mg I at a redshift of $z=0.4245\pm 0.0005$ (Selsing et al., 2019; Kann et al., 2019). Remarkably, MAGIC detected the very-high-energy (VHE) emissions up to at least 1 TeV from GRB 190114C about 50 s after the Swift trigger (MAGIC Collaboration et al., 2019). Fermi Large Area Telescope (LAT) also detected this source, lasting until $\sim$150 s after GBM trigger (Kocevski et al., 2020).

Figure 1 shows the Fermi-GBM light curves of GRB 190114C during $-1-14$ s (the main pulses) observed in different energy bands. With the high photon statistics for the bright light curves of GRB 190114C in amply populated energy bands shown in Figure 1, the time lags between the light curves observed in different energy bands are calculated by employing the cross-correlation function (CCF) method. We direct the interested readers to Ukwatta et al. (2010) and Zhang et al. (2012) for more details of the CCF method. We adopt the arrival time of the time series in the lowest energy band ($10-15$ keV) as a reference, and the time lags between this energy band and any other high-energy bands are shown in Figure 2 and tabulated in Table 1. One can see that the spectral lag monotonically increases with energy and saturates at about 60 keV. In particular, there is an obvious transition from the positive lags to negative lags at around 0.7 MeV, which is analogous to (but the transition point is about one order of magnitude smaller than) that observed in GRB 160626B ($\sim$8 MeV) (Wei et al., 2017a). Considering the negative lags that the LIV involves, this behavior makes GRB 190114C the another case that allows us to constrain the energy scales of LIV effects by directly fitting its spectral lag data.

By analyzing the arrival time delays between broad light curves in different energy bands observed in GRB 190114C, we find a well-observed transition from positive lags to negative lags at around 0.7 MeV, analogous to that from GRB 160625B. This spectral lag behavior allows us to perform a robust test for the LIV effects’ possible signatures by directly fitting the detailed spectral lag measurements observed in this astrophysical source. Using the dynamic nested sampling method based on the Bayes’ Theorem, the outcomes of model fitting give a suitable formulation for the intrinsic energy-dependent time lag and lower limits on the quantum-gravity energy scales, i.e., $E_{\rm QG,1}\geq 2.23\times 10^{14}$ GeV and $E_{\rm QG,2}\geq 0.87\times 10^{6}$ GeV in linear and quadratic ($n=2$) LIV cases at 2$\sigma$ confidence levels, respectively. Considering a negative spectral lag due to LIV in the SME framework, we obtain $\sum_{jm}{}_{0}Y_{jm}(\theta,\phi)c_{(I)jm}^{(d)}\leq 6.77\times 10^{-13}$ GeV^-2 and $1.56\times 10^{-7}$ GeV^-4 (2$\sigma)$ for LIV effects with mass dimensions $d=6$ and 8, respectively. The Bayes factors obtained in both the LIV models with Taylor expansion and SME and the null hypothesis of no LIV-induced time delay indicate decisive evidence for the existence of spectral-lag transition.

Because the negative lags exist in the softer energy bands and the lower redshift, the spectral lags of GRB 190114C are not sensitive enough to the LIV constraints compared to GRB 160625B ($z=1.41$). The limits of $E_{\rm QG}$ we obtained from GRB 190114C are below that from GRB 160625B with about one order of magnitude in linear and quadratic LIV models. The lower limit in the linear LIV model is four orders of magnitude below the Planck scale, one order of magnitude below obtained by Ellis et al. (2006) who considered the intrinsic time lag as an unknown constant. The result for the quadratic LIV case is five orders of magnitude below the limit of multi-TeV gamma-ray flare from Mrk 501. Our constraints on the SME coefficients are somewhat weaker than existing bounds, but they can be viewed as comparatively robust and have the promise to complement existing LIV constraints.

Despite the current relatively lower constraints, GRBs are advantageous for their detectability at high redshifts with long-lasting VHE emissions and short spectral lags compared to such competitive strong gamma-ray emitters like AGNs. We are desperate for such kinds of GRB sources emerging at high redshifts with higher energy emissions and higher temporal resolutions for more stringent LIV constraints. To date, the sub-TeV and even TeV photons have been detected in GRB 180720B ($z=0.653$; Abdalla et al. 2019) and GRB 190114C (MAGIC Collaboration et al., 2019), such detections are expected to become routine (see discussions by Zhang 2019). The performance of observatories for the TeV astronomy, e.g., the future Cherenkov Telescope Array (CTA, $\sim$30 GeV – 100 TeV) and the Large High Altitude Air Shower Observatory (LHAASO, $\sim$100 GeV – 1 PeV) in China, could be rewarding for conducting the LIV constraints from VHE GRBs in the next decades. If only the VHE GRB events are detected, the spectral lag transition may be determined. Our method can release the systematic effects from the assumptions of intrinsic spectral and temporal distributions of VHE emissions beyond the detector’s sensitivity and the extragalactic background light model, which severely bias the constraints by searching for an energy dependence in the arrival time of VHE photons (Acciari et al., 2020). On the other hand, the physical origin of the intrinsic time lag is still an issue as the emission mechanism of GRBs remains largely unknown. Du et al. (2019) found that the spectral lag is strongly related to the spectral evolution. This gives a possibility to investigate the physical origin of intrinsic time lag with the detailed temporal and spectral features of any GRB, whilst exploring the emission mechanism of the prompt gamma-rays.