A Comprehensive Study of Multiflare GRB Spectral Lag

We first check the multiflare lag distribution and compare it with that of the prompt-emission pulse lag. The multiflare GRB lag distribution is demonstrated in Figure 1 and the lag and related parameters are listed in Table 1. We find from Figure 1 and Table 1 that (1) the lags range from −4.78 ± 0.41 s to 22.9 ± 1.28 s, (2) the distribution is similar to a Gaussian distribution and peaks at ∼5 s, and (3) the corresponding median value is 3.38 s with a mean of 5.03 s. About 90% of these lags (123 flares) are positive, about 10% are negative lags (13 flares), and 1 is zero when fluctuations are counted.

Zoom In Zoom Out Reset image size

It is worth mentioning that the spectral lag in the prompt-emission pulse is mainly concentrated in the range of 10⁻²–10⁻¹ s (Yi et al. 2006; Hakkila et al. 2007; Li et al. 2012b), while the lag of the flare is mainly concentrated in a few to tens of seconds; this shows that the lag of X-ray flares is much longer than that of prompt-emission pulses.

The previous section shows that the flare lag is much greater than that of the prompt-emission pulse. There is a positive correlation between lag and pulse duration in gamma-ray prompt emission as revealed by many authors (e.g., Norris et al. 2005; Peng et al. 2007; Li et al. 2012b). Does the duration of the flare also have a great influence on the spectral lag? Figure 2 shows the correlation between the flare duration and the lag of the X-ray flare. The open circles are the 114 flares that show positive lag (we refer to these 114 flares showing positive time lag as sample 1), and the solid line is the best-fit relationship between the lag and duration of the flare: δ T = 101.81 ± 0.04, lag = 0.54 ± 0.06; the Spearman correlation coefficient is 0.60 with p = 1.19 × 10⁻¹². Previous studies have shown that the GRB prompt-emission pulse duration is also positively correlated with the spectral lag (e.g., Norris et al. 2005). Both the gamma-ray prompt-emission pulse and the X-ray flare have a consistent lag and duration correlation; their spectral lag is positively related to the duration. That is, the longer the duration, the greater the lag. The duration of the prompt-emission pulse is concentrated in a few seconds to tens of seconds, while the average duration of the flare in our sample 1 is 185.5 s. Since the flare has a longer duration, the lag of the flare is also greater.

Zoom In Zoom Out Reset image size

Margutti et al. (2010) showed that X-ray flares evolve with time with a sample including nine single flares. That is, flares become wider as time proceeds, with larger peak lags. Moreover, they found that a single flare has the same width–lag correlation as a prompt-emission pulse; the wider the flare/pulse, the greater the lag value. Employing a much larger multiflare sample we also investigate the two issues.

Figure 3 demonstrates the flare spectral lag versus the flare peak time t_peak for sample 1; the black filled circles are the 114 flares that show positive lags, and the solid line is the best regression line. The best functional form of this relation is log(${t}_{\mathrm{peak}})=A+B\mathrm{log}(\mathrm{lag})$ (A is in units of seconds). A correlation (Spearman correlation coefficient r = 0.34) is identified, with A = 2.26 ± 0.04 and B = 0.26 ± 0.06. Figure 4 plots the flare peak time and the duration of the flare; the solid line is the best-fitting line: t_peak = 101.62 ± 0.13, δ T = 0.37 ± 0.06; the Spearman correlation coefficient is 0.45 with p = 4.13 × 10⁻⁷. These results are consistent with previous studies (e.g., Margutti et al. 2010). But both power-law indices of our sample are much smaller that those of Margutti et al. (2010). Therefore, in the case of multiflare GRBs, the flares also evolve with time, that is, the later the flares appear, the longer the durations and the larger the spectral lags.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

As mentioned above, there are 13 multiflare GRBs whose time lags show opposite signs. In order to study the causes of positive and negative lags, we select GRB 060714 and GRB 060111A to compare their spectral evolution trends since some scholars think that the spectral evolution may be causing the spectral lag (e.g., Kocevski & Liang 2003; Roychoudhury et al.2014). The light curves of the WT models of GRB 060714 and GRB 060111A are shown in the left panel of Figure 5. Both GRB 060714 and GRB 060111A have three obvious flares, which have complete structures and are very bright. We use the DCF to estimate the lags. The first and second flares of GRB 060111A and GRB 060714 show positive lags, while both of the third flares show negative lags (see Table 2). The third flares of these two GRBs have the longest duration and are relatively bright, so we choose these two GRBs to study the effect of spectral evolution on lags.

Zoom In Zoom Out Reset image size

Table 2. Fitting Results of the COMP Model and Band Model to GRB 060714 and GRB 060111A

GRB	Flare	t1 (s)	t2 (s)	CPL				Band					ΔBIC
				α	Ep (keV)	BIC		α	β	Ep (keV)	BIC
060714	1	113	115	$-{0.51}_{-0.21}^{+0.21}$	${8.98}_{-4.23}^{+5.0}$	40.24		$-{0.48}_{-0.2}^{+0.21}$	$-{2.22}_{-0.38}^{+0.38}$	${7.57}_{-1.57}^{+1.59}$	47.58		7.34
060714	1	115	117	$-{0.48}_{-0.38}^{+0.37}$	${4.16}_{-2.06}^{+2.98}$	64.79		$-{0.55}_{-0.35}^{+0.37}$	$-{2.26}_{-0.36}^{+0.36}$	${3.73}_{-1.18}^{+1.31}$	72.69		7.9
060714	1	117	119	$-{0.75}_{-0.31}^{+0.32}$	${6.66}_{-4.09}^{+6.14}$	34.42		$-{0.81}_{-0.32}^{+0.3}$	$-{2.18}_{-0.38}^{+0.38}$	${5.58}_{-2.29}^{+2.42}$	41.09		6.68
060714	1	119	121	$-{0.68}_{-0.19}^{+0.19}$	${6.99}_{-3.39}^{+4.71}$	52.89		$-{0.68}_{-0.2}^{+0.2}$	$-{2.12}_{-0.35}^{+0.36}$	${6.34}_{-1.74}^{+1.81}$	59.83		6.94
060714	2	140	144	${0.08}_{-0.14}^{+0.15}$	${2.27}_{-0.2}^{+0.23}$	95.94		$-{0.05}_{-0.16}^{+0.16}$	$-{2.9}_{-0.27}^{+0.28}$	${2.29}_{-0.19}^{+0.19}$	109.76		13.83
060714	2	144	150	$-{0.33}_{-0.19}^{+0.18}$	${2.47}_{-0.38}^{+0.45}$	104.6		$-{0.39}_{-0.15}^{+0.14}$	$-{2.76}_{-0.28}^{+0.28}$	${2.34}_{-0.19}^{+0.19}$	113.3		8.71
060714	2	150	154	$-{1.07}_{-0.41}^{+0.4}$	${1.54}_{-0.2}^{+0.31}$	32.98		$-{0.9}_{-0.29}^{+0.24}$	$-{3.13}_{-0.27}^{+0.27}$	${0.52}_{-0.09}^{+0.1}$	45.74		12.76
060714	2	154	159	$-{0.44}_{-0.32}^{+0.33}$	${1.39}_{-0.34}^{+0.46}$	56.82		$-{0.63}_{-0.29}^{+0.29}$	$-{2.72}_{-0.27}^{+0.27}$	${1.21}_{-0.17}^{+0.17}$	67.24		10.41
060714	3	175	180	$-{0.61}_{-0.18}^{+0.17}$	${3.62}_{-0.93}^{+1.04}$	84.16		$-{0.39}_{-0.22}^{+0.21}$	$-{2.12}_{-0.18}^{+0.19}$	${2.75}_{-0.39}^{+0.39}$	85.79		1.63
060714	3	180	184	$-{1.41}_{-0.2}^{+0.19}$	${1.74}_{-1.64}^{+1.93}$	72.33		$-{0.99}_{-0.32}^{+0.32}$	$-{2.43}_{-0.2}^{+0.2}$	${1.22}_{-0.22}^{+0.21}$	78.09		5.76
060714	3	184	190	$-{1.22}_{-0.33}^{+0.32}$	${0.89}_{-0.6}^{+0.86}$	45.26		$-{0.96}_{-0.32}^{+0.32}$	$-{2.48}_{-0.17}^{+0.18}$	${0.57}_{-0.13}^{+0.13}$	50.95		5.69
060714	3	190	195	$-{1.01}_{-0.36}^{+0.34}$	${2.14}_{-1.28}^{+1.64}$	33.38		$-{0.95}_{-0.31}^{+0.32}$	$-{2.31}_{-0.25}^{+0.27}$	${1.21}_{-0.64}^{+0.63}$	39.69		6.31
060714	3	195	200	⋯	⋯	⋯		⋯	⋯	⋯	⋯		⋯
060714	3	200	235	$-{0.95}_{-0.32}^{+0.32}$	${1.0}_{-0.47}^{+0.62}$	47.06		$-{0.9}_{-0.32}^{+0.31}$	$-{2.6}_{-0.33}^{+0.33}$	${0.88}_{-0.32}^{+0.29}$	56.07		9.02
060111A	1	99	109	$-{0.73}_{-0.22}^{+0.22}$	${3.15}_{-1.24}^{+1.44}$	78.39		$-{0.53}_{-0.24}^{+0.24}$	$-{2.26}_{-0.26}^{+0.26}$	${2.48}_{-0.35}^{+0.34}$	81.98		3.59
060111A	1	109	118	$-{0.99}_{-0.22}^{+0.2}$	${2.93}_{-1.56}^{+1.75}$	99.3		$-{0.69}_{-0.29}^{+0.3}$	$-{2.18}_{-0.22}^{+0.23}$	${2.03}_{-0.41}^{+0.39}$	103.86		4.56
060111A	1	118	129	$-{0.62}_{-0.24}^{+0.24}$	${1.85}_{-0.44}^{+0.55}$	63.27		$-{0.61}_{-0.22}^{+0.21}$	$-{2.73}_{-0.23}^{+0.23}$	${1.6}_{-0.18}^{+0.18}$	71.58		8.31
060111A	1	129	139	$-{0.62}_{-0.3}^{+0.28}$	${1.74}_{-0.51}^{+0.66}$	67.42		$-{0.74}_{-0.25}^{+0.25}$	$-{2.65}_{-0.27}^{+0.28}$	${1.54}_{-0.2}^{+0.2}$	76.46		9.04
060111A	2	167	175	$-{0.78}_{-0.31}^{+0.3}$	${1.96}_{-0.81}^{+0.94}$	66.38		$-{0.76}_{-0.29}^{+0.28}$	$-{2.52}_{-0.29}^{+0.27}$	${1.6}_{-0.26}^{+0.23}$	74.28		7.91
060111A	2	175	186	⋯	⋯	⋯		⋯	⋯	⋯	⋯		⋯
060111A	2	186	195	$-{1.05}_{-0.33}^{+0.35}$	${2.8}_{-2.38}^{+3.85}$	44.6		$-{0.97}_{-0.31}^{+0.31}$	$-{2.21}_{-0.35}^{+0.34}$	${1.86}_{-0.68}^{+0.59}$	51.19		6.59
060111A	2	195	201	$-{0.83}_{-0.32}^{+0.35}$	${3.07}_{-3.3}^{+5.51}$	24.98		$-{0.85}_{-0.3}^{+0.29}$	$-{2.16}_{-0.37}^{+0.36}$	${2.81}_{-2.16}^{+2.78}$	31.28		6.3
060111A	3	283	302	$-{0.48}_{-0.1}^{+0.1}$	${3.35}_{-0.43}^{+0.48}$	220.69		$-{0.28}_{-0.12}^{+0.11}$	$-{2.27}_{-0.18}^{+0.19}$	${2.84}_{-0.21}^{+0.21}$	221.64		0.95
060111A	3	302	323	$-{0.75}_{-0.1}^{+0.11}$	${3.49}_{-0.71}^{+0.74}$	318.14		$-{0.21}_{-0.09}^{+0.09}$	$-{1.96}_{-0.09}^{+0.09}$	${2.22}_{-0.15}^{+0.15}$	295.71		–22.42
060111A	3	323	342	$-{0.5}_{-0.13}^{+0.13}$	${3.25}_{-0.52}^{+0.59}$	255.67		$-{0.3}_{-0.13}^{+0.12}$	$-{2.25}_{-0.19}^{+0.19}$	${2.71}_{-0.21}^{+0.21}$	255.86		0.19
060111A	3	342	363	$-{1.19}_{-0.12}^{+0.13}$	${4.86}_{-3.44}^{+4.16}$	140.69		$-{0.98}_{-0.3}^{+0.33}$	$-{1.97}_{-0.34}^{+0.28}$	${3.53}_{-1.71}^{+1.67}$	147.49		6.8
060111A	3	363	403	$-{1.27}_{-0.1}^{+0.11}$	${4.7}_{-3.66}^{+4.16}$	98.71		$-{1.13}_{-0.26}^{+0.32}$	$-{1.99}_{-0.38}^{+0.3}$	${3.78}_{-1.95}^{+1.9}$	104.75		6.03
060111A	3	403	432	$-{1.11}_{-0.21}^{+0.22}$	${4.07}_{-3.17}^{+4.39}$	74.75		$-{0.95}_{-0.3}^{+0.3}$	$-{2.08}_{-0.34}^{+0.3}$	${3.0}_{-1.19}^{+1.09}$	81.09		6.34
060111A	3	432	451	$-{0.94}_{-0.29}^{+0.3}$	${3.79}_{-2.64}^{+3.9}$	48.8		$-{0.95}_{-0.29}^{+0.29}$	$-{2.14}_{-0.35}^{+0.36}$	${2.98}_{-1.13}^{+1.08}$	55.44		6.64
060111A	3	451	510	$-{0.92}_{-0.24}^{+0.23}$	${2.97}_{-1.41}^{+1.75}$	101.22		$-{0.71}_{-0.25}^{+0.27}$	$-{2.26}_{-0.23}^{+0.25}$	${2.18}_{-0.43}^{+0.43}$	107.15		5.93

Download table as:  ASCIITypeset image

To examine whether spectral evolution is responsible for the observed spectral lags of GRB 060714 and GRB 060111A, we study the time variation of E_peak (the peak energy in the ν F_ν spectrum) for all flares of GRB 060714 and GRB 060111A under consideration since the two GRBs have the most flares. In the spectral evolution of prompt-emission pulses, E_peak is often used to represent the process of spectral evolution; in X-ray afterglow flares, we also use the trend of E_peak to represent spectral evolution. Several theoretical models have been proposed to explain its wide distribution from several kiloelectronvolts to megaelectronvolts (Sakamoto et al. 2009; Roychoudhury et al. 2014).

We divide the attenuation time of all flares for the two GRBs into several time periods; the XRT data energy range is 0.3–10 keV. We extract the spectra for each time period from the Swift website (https://www.swift.ac.uk/) and then adopt the Multi-mission Maximum Likelihood Framework (3ML; Vianello et al. 2015) to fit the flare spectral data. The 3ML tool adopts the Markov Chain Monte Carlo (MCMC) technique to perform time-resolved spectral fitting. The MCMC technique is based on the Bayesian statistic using the 3ML tool to carry out parameter estimation of data. In order to choose a better model to fit the energy spectrum, we fit the flare spectral data with the BAND model (Band et al. 1993) and the COMP function (Mukherjee et al. 1998) to perform time-resolved spectral analysis and compare the ΔBIC of the BAND model and COMP model. ΔBIC is BIC_BAND − BIC_COMP and ΔBIC greater than zero indicates that the COMP model is better. Then we check all cases and find that all ΔBIC are positive except one. The BAND model does not fit our energy spectrum well and the COMP model is the preferred model, since it systematically has a lower BIC value. Therefore, we mainly adopt the data from the COMP model in addition to one from BAND to analyze E_peak evolution with flare peak time. For the specific definition of the goodness of data fitting by the empirical model, please refer to Yu et al. (2019). The fitting results are shown in Table 2.

The COMP model is a single-power-law model with a high-energy cutoff, and the function form is as follows:

$$\begin{eqnarray}&&N\left(E\right)=A{\left(\displaystyle \frac{E}{100\,\mathrm{keV}}\right)}^{\alpha }\exp \left(-\displaystyle \frac{E}{{E}_{c}}\right).\end{eqnarray} \tag{ 4 }$$

A is the normalization constant of the spectrum, α is the photon spectrum index, E_c is the break energy in the spectrum, and E_peak and E_c have such a relationship: ${E}_{\mathrm{peak}}=\left(2+\alpha \right){E}_{c}$. This function fits all the flares of GRB 060714 and flares 1 and 3 of GRB 060111A very well; the floating point is not enough in the second-flare fitting energy spectrum of GRB 060111A, so we remove it. In GRB 060714, the first flare is a composite flare, which is relatively obvious in the 0.3–1.5 keV energy channel; this may have an impact on the evolution of E_peak.

Some scholars think that the spectral evolution from hard to soft causes the spectral lag (e.g., Kocevski & Liang 2003). Roychoudhury et al. (2014) suggested spectral evolution will cause both positive and negative lags; the evolution from hard to soft causes a positive lag, while the evolution from soft to hard causes a negative lag. A comparison of the time variations of E_peak for GRB 060111A and GRB 060714 is given in Figure 5.

The first and third flares of GRB 060111A show a positive lag and negative lag, respectively. The E_peak evolution trends of these two flares are not the same. The E_peak of the first flare of GRB 060111A has a hard-to-soft trend. However, the E_peak evolution of the third flare of GRB 060111A does not have a clear trend from soft to hard, but has a weak soft-to-hard-to-soft trend near the peak of the flare. This soft-to-hard-to-soft trend may be the cause of the negative lag in the third flare of GRB 060111A. This may be reasonable since Peng et al. (2011) also justified that the spectral evolution trend from soft to hard to soft in prompt-emission pulses will cause a negative lag.

In GRB 060714, a mixed flare appeared at the tail of the first flare. This may be the reason why this flare has a hard-to-soft-to-hard trend. The second flare and the third flare have similar hard-to-soft evolution trends, but their lags show opposite signs: the second flare shows a positive lag, whereas the third flare shows a negative lag. Thus it seems that spectral evolution is not the dominant cause of the spectral lag features of GRB 060714.

Spectral evolution has long been considered as the cause of spectral lag (e.g., Kocevski & Liang 2003). However, many studies have used different considerations to support the observation that spectral evolution may not be the dominant process responsible for the spectral lag of all GRBs (e.g., Ukwatta et al. 2012; Roychoudhury et al. 2014; Chakrabarti et al. 2018). We also think that spectral evolution cannot explain all spectral lags; there may be other physical mechanisms.

We choose 37 multipulse GRBs (including 88 pulses) from Li et al. (2012b) to check if there are similar lag properties between multiflare and multipulse GRBs. The lags of these 88 pulses are obtained by fitting the Gaussian model to the CCF curve, which is similar to the DCF method. The duration δ T of the pulse is also defined by δ T = T_end − T_start; for more details, please refer to Figure 3 in Li et al. (2012b).

Figure 7 is the lag distribution of the 88 pulses: the pulse lags are between 50–100 and 15–25 keV, the red curve is the Gaussian fitting curve, the lags of the 88 pulses have a distribution similar to a Gaussian distribution, and the average value of this Gaussian distribution is about 0.18 s. In order to pick out the positive and negative lags we remove eight pulses with very large errors. Among the 80 remaining pulses, there are 68 positive-lag pulses (85%) and 12 negative-lag pulses (15%). Of the 68 positive-lag pulses, the median value is 0.27 s with a mean of 0.36 s. The pulse lags are much shorter than the flare lags, which may be caused by the fact that the durations of the pulses are much shorter than those of the flares.

Zoom In Zoom Out Reset image size

Figure 8 shows the relationship between the lag and duration of the pulses/flares: the red filled circles on the right axis are the 114 flares, the black filled circles on the left axis are the 68 pulses, and the red solid line with associated slope 0.63 ± 0.08 and the black solid line with associated slope 0.60 ± 0.13 are the best-fit relationships for the flare lag and flare duration and for the pulse lag and pulse duration, respectively. Moreover, the slopes of the two relationships are very similar. From this perspective, multiflare GRBs and multipulse GRBs have similar characteristics (flares are an extension of pulses), which also provides support for X-ray flares and gamma-ray pulses having the same physical origin.

Zoom In Zoom Out Reset image size

There are several phenomena responsible for the observed spectral lags, such as the curvature effect (Ryde & Petrosian 2002), the internal cooling of radiated electrons (Kazanas et al. 1998; Schaefer 2004), the Compton reflection of a medium far away from the radiation source, and the spectral evolution (Kocevski & Liang 2003) during the prompt phase. Mochkovitch et al. (2016) believe that spectral lag depends on all spectral changes including the pulse peak energy and spectral index. Spectral evolution means that as the energy dissipates, the radiation gradually cools down, and the overall trend of the energy spectrum moves toward low energy, which leads to mainly high-energy photons at the beginning, and fewer low-energy photons. After a period of radiation, the photons can fall to the low-energy channel, resulting in a positive lag of high-energy photons arriving first and low-energy photons arriving later. At a certain stage, if the central engine injects energy to accelerate the shell, it causes the energy spectrum to go from low energy to high energy and creates a negative lag.

The curvature effect causes low-latitude photons to arrive first and high-latitude photons to arrive later, and the smaller Doppler factor of high-latitude photons makes these photons fall to a lower-energy range, which causes low-energy photons to arrive later and form a positive lag. Peng et al. (2011) also proposed that spectral lag is a result of the combined actions of the spectral evolution and the curvature effect. The curvature effect always provides the contribution of positive lag, and the spectral evolution provides different contributions of positive and negative lags according to the evolution model of the spectrum. The inherent cooling of radiating electrons means that low-energy radiation will be generated later than high-energy radiation, hence a positive lag; the Compton reflection of a medium follows the same principle.

We also examine if the spectral evolution can explain the spectral lag of a flare. The spectral lag may be related to the peak energy characteristics of the flare. The spectral evolution near the peak of the third flare of GRB 060111A shows a weak soft-to-hard-to-soft trend, which may be the reason for the negative lag of this flare. Both the first and second flares of this GRB have positive lags. But the E_peak evolution models have opposite characteristics: the first flare follows the hard-to-soft evolution mode and the second one shows a soft-to-hard trend. So we suspect that the curvature effect (and the hard-to-soft spectral evolution) and the soft-to-hard-to-soft spectral evolution together affect the GRB, causing the time lags of this GRB to show opposite signs.

In GRB 060714, the second and third flares have similar hard-to-soft spectral evolution trends, but their lags have opposite signs; the spectral evolution cannot explain this phenomenon. This requires a new mechanism to explain the cause of the lags in the GRB. Inverse Compton scattering of low-energy thermal photons by relativistic electrons is one of the feasible schemes for GRB radiation, which may introduce a negative lag (Roychoudhury et al. 2014). The first and the second flares of GRB 060714 have positive lags, which may be affected by curvature effects (and positive spectral evolution), and the third flare has negative lags, which may be caused by inverse Compton scattering of low-energy thermal photons by relativistic electrons. But this is just our guess, and more detailed theoretical research is needed to explain it.

The reason for the two GRBs having opposite-sign lags may be the combined effect of curvature effect, spectral evolution, and inverse Compton effect. The number of negative lags is relatively small after all, and the curvature effect (and hard-to-soft spectral evolution) may be dominant in X-ray flares. More detailed theoretical studies are needed for either the prompt-emission pulses or the X-ray flares. We hope that there will soon be a complete theory to explain the relationship between the spectral lag and spectral evolution in GRBs, so that we can have a deeper understanding of the pulses/flares of GRBs.

Using the observations of the Swift Burst Alert Telescope and the Suzaku wide-area monitor, Roychoudhury et al. (2014) found the multipulse GRB 060814 has a similar phenomenon. They found that the spectral lags of the first two and fourth pulses are positive but the third pulse exhibits a negative lag. However, the time variations of the E_peak of all the pulses show the same trend. The similar phenomenon seems to also support the observation that flares and pulses have the same physical origin.

Most studies of spectral lag have focused on the evolution of the spectrum and the curvature effect. Hakkila et al. (2018b) put forward the theory that the presence of pulse/flare "structures" can explain why some pulses/flares exhibiting hard-to-soft evolution have negative lags while others have positive ones. They demonstrated negative lags can be created by spectrally evolving bumps in GRB pulse light curves (see Figures 18(b) and 19 in their paper) even when the pulses in which they are found exhibit hard-to-soft evolution. Therefore, the presence of evolving pulse structures also supports the observation that the GRB central engine might be responsible, and points to times in the light curve when this might occur.

Hakkila et al. (2018a, 2018b) found that GRB pulse structures also exhibit temporal symmetries. In other words, structures observed during the pulse decay phase match structures in the rise phase, in reverse temporal order. This observation strongly suggests kinematic mechanisms (Hakkila et al. 2018b; Hakkila & Nemiroff 2019), which might be associated with energy fluctuations in the central engine (in the form of impactor waves), but might also result from heterogeneities in the developing jet or from structural fluctuations in the medium through which the jet expands. Therefore, the explained energy injection from the central engine is not the only mechanism capable of forming negative lags.