Searching for Gravitationally Lensed Gamma-ray Bursts with Their Afterglows

As one of the most violent explosions in the Universe, GRBs and their afterglows are bright enough to be detected in high-redshift range up to at least $z\sim 10$ (Tanvir et al., 2009; Salvaterra et al., 2009). In view of their high redshift, the gravitational lensing effect on GRBs has long been discussed ever since the cosmological origin for GRBs was first proposed (Paczyński, 1986). With detailed study on the probability distributions of time delay in gravitational lensing by point masses and isolated galaxies (modeled as singular isothermal spheres), Mao (1992) predicted that the probability of multiple GRB images obtained by galactic lensing event could be between $0.05\%$ and $0.4\%$. And by assuming that the GRB luminosity function has no cosmological evolution and the lens is modeled as singular isothermal spheres, Hurley et al. (2019) estimated that the detection rate of both lensed GRB images is roughly 0.05 times per year.

Thanks to the successful operation of several dedicated detectors, e.g., the Burst And Transient Source Experiment (BATSE) on Compton Gamma Ray Observatory (CGRO) (Meegan et al., 1992), the Burst Alert Telescope (BAT) on the Neil Gehrels Swift Observatory (Gehrels et al., 2004; Barthelmy et al., 2005) and the Gamma-Ray Burst Monitor (GBM) on the Fermi Observatory (Meegan et al., 2009), $\sim 10^{4}$ GRBs have been detected. The searches for gravitational lensing of GRBs have been widely carried out, for instance, Li & Li (2014) searched $\sim$2100 GRBs observed by BATSE, Hurley et al. (2019) searched $\sim$2300 GRBs observed by Konus-Wind, and Ahlgren & Larsson (2020) searched $\sim$ 2700 GRBs observed by GBM. In addition, Veres et al. (2009) and Davidson et al. (2011) also failed to search for lensing events with two smaller samples of GRBs observed in the first years of Fermi-GBM.

Unfortunately, all searches for galaxy lensing events (with time delay in order of days) have yielded null results. Most recently, Paynter et al. (2021) claimed that they had found one statistically significant lensing event in the lightcurve of GRB 950830 (BATSE trigger 3770) with sub-second time delay. The inferred lens mass, although depending on the unknown lens redshift, falls into the mass range of intermediate-mass black holes (e.g. $\sim 10^{4}-10^{5}M_{\odot}$). The mismatch between theoretical predictions concerning lensing rates and the data search results might be due to the uncertainty of the theoretical model, or it may also be due to the fact that current gamma-ray detectors do not yet have all-sky monitoring (Li & Li, 2014). Nevertheless, it is also possible that some real lensing events are missed in the data search process, considering that the current searches are mainly based on the comparison of gamma-ray lightcurves and the coincidence degree of positions, but 1) the radiation time scale of GRBs in gamma-ray bands is relatively short, 2) the shape of lightcurve is greatly affected by the background, and 3) the positioning error of gamma-ray detectors is very large.

Here we propose that in addition to looking for lensed GRBs in the gamma-ray band, we can also search them with GRB’s afterglow observations, taking into account that the afterglow’s detectable timescale is much longer (up to years), their lightcurve shape is relatively simple, their physical model is much more clear, and they can be monitored by either satellites or ground based multi-band detectors in order to enhance the probability of capturing the lensing effect (Zhang, 2018, for a review). For strongly gravitational lensing events with lens mass $M_{l}$, the typical angular separation between multiple images could be roughly estimated as $0.1^{\prime\prime}{({M_{l}}/{10^{10}}{M_{\odot}})^{1/2}}{({D_{l}}{D_{s}}/{D_{ls}}/{\rm{Gpc}})^{-1/2}}$ (Meneghetti, 2016), which could hardly be resolved by facilities that are commonly used to observe GRB afterglows, unless the lens galaxy is extremely massive (e.g. $M_{l}>10^{12}M_{\odot}$). For instance, the angular resolution of the Swift X-ray Telescope (XRT) is only $18^{\prime\prime}$ (Burrows et al., 2005) and the resolution $1$ meter-class optical telescopes is $\sim 1^{\prime\prime}$ (Oguri, 2010). On the other hand, by studying the broad time delay probability distribution of the point lens, Mao (1992) found that the peak of the time delay probability appeared at $50\hskip 1.13791pt{\rm{s}}\;({M_{l}}/{10^{6}}{M_{\odot}})$. Unlike the short-term gamma-ray emission, the detectable timescale for afterglows is typically much longer than the time delay timescale. In this case, if the multiple images are unresolved, the signals of multiple images will be superimposed to disguise as one signal.

In this paper, we first use the standard GRB afterglow and gravitational lensing models to calculate the lightcurve of such superimposed signal in different bands. Based on the characteristics of the calculated lightcurves, we then searched the current GRB afterglow observation data in the optical band. One potential candidate, GRB 130831A is found, and the possible parameter space is investigated in order to check if it is indeed a lensed event. Future prospects are discussed in the final section. Throughout this paper, we assume the Planck cosmology (Planck Collaboration et al., 2016) as a fiducial model, with ${\Omega_{\rm{m}}}=0.307$, ${\Omega_{\Lambda}}=0.693$ and ${H_{0}}=67$ ${\rm{km}}$ ${{\rm{s}}^{-1}}$ ${\rm{Mp}}{{\rm{c}}^{-1}}$.

Based on the predicted characteristics in Section 2.3, we analyze optical band by searching for the presence of additional signal due to gravitational lensing of GRBs. We extensively search for the optical data from published papers or from the Gamma-ray Coordinates Network (GCN) Circulars if no published paper is available. We totally found 358 GRBs with optical observations being reported from 1997 February to 2020 December, including 308 GRBs having well-sampled optical light curves, which contain at least 3 data points, excluding upper limits ²²2Part of our sample have been published in the literature (e.g. Li et al., 2012; Wang et al., 2015).. By visual inspection, we systematically searched through our sample and identify 3 bursts that show a clear onset peak, followed by a steep rebrightening. These are GRB 100621A, GRB 100901A and GRB 130831A. It is worth noting that some sources are also found to contain a steep rebrightening signature, but their early onset peak has not been observed (some examples are collected in de Ugarte Postigo et al. (2018)). These sources together with GRB 100621A and GRB 100901A are further excluded because their rebrightening peak is too wide to be explained by lensing effect, but more like a giant bump caused by fallback accretion process (Wu et al., 2013; Zhao et al., 2021). Overall, only one possible candidate is left: GRB 130831A with a redshift of 0.479 (Cucchiara & Perley, 2013). De Pasquale et al. (2016) provided a comprehensive analysis for the ultraviolet, optical and infrared photometry of GRB 130831A, whose combined optical light-curves show a rapid temporal evolution between 400 and 800 s after the trigger (the optical flux suddenly increased by a factor of $\sim$5). The sharp rise feature is very difficult to be explained within the external shock framework, unless invoking some complicate energy injection process, such as a continuously refreshed shock following the prompt emission phase (Zhang et al., 2016). De Pasquale et al. (2016) proposed that the sharp rising should be connected to internal dissipation processes which occurred in the outflow, when the Lorentz factor is very high and relativistic effects cause rapid variations of the observed flux. We are interested here in testing whether the sharp rise feature is a signature of lensing effect. Considering that the UV/optical/NIR data collected in De Pasquale et al. (2016) have not been corrected for Galactic and host galaxy extinction, we use the Rc band data of GRB 130831A provided by de Ugarte Postigo et al. (2018), which has been calibrated to the Vega system, corrected for Galactic extinction. In addition, the host-galaxy and supernova have been subtracted ³³3Note that GRB 130831A is known to be associated with SN 2013fu (Cano et al., 2014). The Rc band data we used here has subtracted the supernova contribution.. We apply the model described in Section 2 to fit the data of GRB 130831A, by performing the $\chi^{2}$ objective function minimization procedure with using Markov Chain Monte Carlo (MCMC) method through the emcee code (${\rm{Foreman-Mackey}}$ et al., 2013). In the fitting, we focus on the early observational data (e.g. $t<1500$ s), which essentially reflects whether the lensing effect exists. In order to reduce the number of free parameters in our fitting we fix several of them at their typical values. For instance, we set $E_{0}=10^{53}\ \rm{ergs}$, ${\theta_{0}}=0.3$, ${R_{0}}={10^{15}}$cm, ${\epsilon_{e}}=0.1$, and ${\epsilon_{B}}={10^{-4}}$. We take the initial bulk Lorentz factor ${\Gamma_{0}}$ of the outflow, the number density $n$ of the circumburst medium, the magnifications of the two images ($\rm Mag_{1}$, $\rm Mag_{2}$), the time delay between the two images $t_{\rm delay}$ as the free parameters. Note that by fitting the observed light curve, parameters characterizing the overall lensing effect (magnifications and time delay) can be obtained directly. The lens system parameters, such as $M_{l}$, $z_{l}$, $\beta$, could hardly be resolved due to strong degeneracy, unless some of them ($z_{l}$ or $\beta$) could be directly determined with observations. Figure 3 shows our best fitting results for the lightcurve and the corresponding corner plot of the posterior probability distribution for the fitting. We can see that the early lightcurve of GRB 130831A could be well fitted with our proposed model. The best fit parameter values at the $1\sigma$ confidence level are ${\Gamma_{0}}=206_{-11}^{+13}$, $n=6.3_{-2.2}^{+2.6}\;{\rm{c}}{{\rm{m}}^{{\rm{-3}}}}$, ${\rm{Ma}}{{\rm{g}}_{\rm{1}}}{\rm{=4}}.1_{-0.6}^{+0.8}$, ${\rm{Ma}}{{\rm{g}}_{\rm{2}}}{\rm{=7}}{\rm{.8}}_{-1.2}^{+1.5}$, ${t_{{\rm{delay}}}}=492_{-3.6}^{+3.5}$ s. Constraints on all parameters associated with the external shock are consistent with typical values. Note that due to the lack of information about the baseline (unlensed) flux, the values of $\rm Mag_{1}$ and $\rm Mag_{2}$ should be highly degenerated with the afterglow modeling parameters, so that the ratio between $\rm Mag_{1}$ and $\rm Mag_{2}$ is more meaningful, compared with their absolute values. According to the constraint result, the time delay between the two lensed images is $\sim 500$ s and the flux ratio is $\sim 2$. Through external interpolation, we show that the best fitting lightcurve is generally consistent with the late time observations⁴⁴4Between 3000 s and 17000 s, the observational data show a plateau feature, which makes this part of the data slightly higher than the extrapolation extension of the best fitting lightcurve. This is acceptable since the plateau feature is likely related to the late central engine activity (see De Pasquale et al. (2016) for detailed physical interpretations for late time observations of GRB 130831A). . For completeness, we also fit the early data of GRB 130831A by using the standard afterglow model without invoking any lensing effect, and the best fit result is shown in Figure 3 as well. We then use the Bayesian method to test how significant the observation supports the lensing model (marked as model $M_{1}$) against the model without strong lensing (marked as model $M_{2}$). It turns out the lensing model is more preferable by the observations (see Appendix for details of our adopted Bayesian method and relevant results).

However, it is worth noting that although the optical data of GRB 130831A can be well fitted by the lens effect, it is highly doubtful whether the source is actually lensed for the following reasons:

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  The second arriving image has a larger magnification. Usually, for typical lens models, if there are only two lensed images, the second image should be fainter as the echo. In other words, even if GRB 130831A is actually lensed, it may have experienced very special gravitational lensing events. One possible lens model is a point mass+shear which might explain the brighter trailing images (Chen et al., 2021). Another plausible explanation for such a particular combination of time delay and the flux ratio could be the existence of dark matter subhalo of mass $\sim 10^{6}M_{\odot}$ in the galaxy resulting in the obtained value of time delay. In this case the magnification of the second image (and estimated value of the flux ratio) could be caused by microlensing effect, i.e. by the stars in the lens galaxy. Let us note that microlensing effect on GRB afterglows has been discussed in some previous works (e.g. Koopmans & Wambsganss, 2001; Gaudi & Loeb, 2001; Ioka & Nakamura, 2001).

• •

  Since GRB 130831A was triggered by Swift, we checked the historical data of BAT and XRT. As shown in Figure 3, when the second optical peak emerging, there is no signal showing up in the BAT data. Before the data recording stopped, the BAT data did show an increasing trend, but it is difficult to judge whether it is a real signal or it is caused by satellite attitude adjustment. There is an X-ray flare starting at about 500 s after the trigger, which is roughly consistent with the optical rising feature, but the flux at $\sim$ 800 s is about an order of magnitude lower than the flux at $\sim$ 100 s, inconsistent with the expected magnification ratio. We also checked the historical data of Fermi/GBM and found that the position of GRB 130831A was blocked by the Earth at the trigger time, but it entered the monitoring region of GBM after hundreds of seconds. As shown in Figure 3, when the second optical peak was emerging, there were no signal showing up in the GBM data. Interestingly, the GBM data also showed an obvious rise around the time when the second optical peak emerged. However, this may be caused by the rise of the background, because data in all the detectors of GBM have risen, and the satellite has entered the high latitude region in that period.

It has long been proposed that GRBs have potential to be gravitationally lensed into multiple images, due to their high redshift nature. The detection of lensed GRBs could be used to improve constraints on cosmological parameters such as the Hubble constant ${{\rm{H}}_{\rm{0}}}$, to test fundamental physics from the propagation speed, and to make constraints on the abundance of compact dark matter, and so on (Oguri, 2019, for a review).

The traditional searches for lensed GRBs are focused on the gamma-ray band. Here we propose that one can search for lensed GRBs by combining gamma-ray and the multi-band afterglow data. We use the standard afterglow model and two standard lens models to calculate the characteristics of the lensed afterglow lightcurves in different scenarios concerning lens mass and time delays. Based on our results, we suggest that the future search for lensed GRBs can be focused on several following cases. 1) When a lens is a compact object, the time delay is in order of $\sim 100$s or less. In this case, multiple gamma-ray images could be either separated or overlapped, depending on the comparison between $T_{90}$ and $t_{\rm delay}$. It is also possible that only one image signal can be seen, once the later arriving signal is under the detector threshold. In this case, we can use the observation of X-ray and optical afterglow data to judge whether these GRBs are lensed or they just happen to have several similar emission episodes. If the GRBs are indeed lensed, the X-ray afterglows are likely to contain several X-ray flares with similar width in linear scale (the later, the narrower in logarithmic scale) and similar spectrum. Moreover, the optical afterglow lightcurve will show rebrightening signatures. 2) When the compact object is a low mass galaxy, the time delay would be $\sim 10^{3}-10^{4}$s. In this case, there will be two independent GRB triggers in the gamma ray band, and there will be a sharp rise flare feature in both X-ray and optical afterglow lightcurve, which can help to easily justify the lensing effect. 3) When the compact object are massive galaxies, the time delay would be in order of days or even longer. In this case, there will be two independent GRB triggers in the gamma-ray band, and the late X-ray signal will rise sharply. Nevertheless, optical telescopes now usually have the ability to resolve different lensed images, so as to better help confirm the lensing effect. Note that for all three situations, radio telescopes usually have the ability to resolve multiple images. Therefore, for suspected samples selected from gamma-ray, X-ray, and optical bands, concerted radio observations could finally help to confirm whether they are real lensing events.

Through archive data searching in optical band, we find one potential candidate of lensed GRB, 130831A, with a time delay $\sim$500 s, which should belong to the first situation we discuss above. However, the gamma-ray and X-ray band observations of GRB 130831A seem not to support the lensing hypothesis. Therefore, whether this source is a lensed event remains to be further discussed. With the successful operation of many sky survey projects in multiple bands, especially the establishment of all-sky gamma-ray monitors, more lensed GRBs would be detected and accurately certified in the future.