# Finding high-redshift gamma-ray bursts in tandem near-infrared and optical surveys

S. Campana, G. Ghirlanda, R. Salvaterra, O.A. Gonzalez, M. Landoni, G.

Pariani, A. Riva, M. Riva, S.J. Smartt, N.R. Tanvir, et al.

► To cite this version:

S. Campana, G. Ghirlanda, R. Salvaterra, O.A. Gonzalez, M. Landoni, et al. Finding high-redshift gamma-ray bursts in tandem near-infrared and optical surveys. Nature Astronomy, 2022, 6 (10), pp.1101-1104. 10.1038/s41550-022-01804-x. hal-03830238

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# Finding high-redshift gamma-ray bursts in tandem near-infrared and optical surveys

S. Campana, G. Ghirlanda, R. Salvaterra, O. A. Gonzalez, M. Landoni, G. Pariani, A. Riva, M. Riva, S. J. Smartt, N. R. Tanvir and S. D. Vergani

<figure><img src="image_2.png" /></figure>

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Gamma-ray bursts are linked to the most distant objects in the Universe, but detecting them is a rare event. With a dedicated near-infrared telescope to observe in tandem with the optical Vera Rubin Observatory, ten or so high-redshift ($z \ge 6$) gamma-ray bursts could potentially be detected every year.

The race for the most distant object in the Universe has involved long-duration gamma-ray bursts (GRBs), star-forming galaxies and quasars. Gamma-ray bursts took a temporary lead with the discovery¹,² of GRB 090423 at a redshift $z = 8.2$, but the current record-holder³ is the galaxy GN-z11 at $z = 11.0$, although the recently released first images from the James Webb Space Telescope (JWST) have revealed a number of distant candidates in the range $z \sim 9-15$ (ref. ⁴, for example). However, galaxies and quasars are very faint (GN-z11 has a magnitude $H = 26$), hampering the study of the physical properties of the primordial Universe. On the other hand, GRB afterglows are brighter by a factor of $\ge 100$, but with the drawback of being transients, and lasting only for 1–2 days. When bright (for example, $H = 18.6$ mag at 7 hr from the explosion as observed for GRB 090423¹,²), GRB spectroscopic afterglow studies could provide independent constraints on the reionization history, on the ionizing photon escape fraction and on the cosmic history of metal enrichment. Gamma-ray bursts could be used to select high-z galaxies independently from their brightness, thus sampling the galaxy luminosity function below the sensitivity limit of deep surveys. They also are expected to provide important information about the nature of the stellar population in the first galaxies by, for example, identifying explosions from the first population of metal-free stars (Population III stars), showing the presence of their distinctive metal enrichment signature and measuring the stellar initial mass function evolution at early epochs.

The major limitation is that, with current instrumentation, high-redshift ($z \ge 6$) GRBs are extremely rare, with just nine events recognized in 17.5 years of Swift observations⁵. This is mainly due to the limited sensitivity and field of view of current (the Neil Gehrels Swift observatory) and soon to be launched (Space-based multi-band astronomical Variable Objects Monitor, SVOM) space missions, and to the relatively faint nature of high-redshift GRBs. It is also likely that some of these high-redshift GRBs are lost during the sometimes patchy follow-up process; patchy because sometimes the afterglow is followed up only at optical wavelengths and not in the near-infrared (nIR), or weather conditions do not allow for rapid observations.

Here we describe an approach to the potential discovery of high-redshift ($z \ge 6$) GRBs, exploiting their nIR emission properties.

<table><caption>**Table 1 | Free parameters for the afterglow model adopted for simulating the GRB population**</caption><thead><tr><th></th><th>μ</th><th>σ</th></tr></thead><tbody><tr><td>G(logε<sub>0</sub>)</td><td>-1.0</td><td>0.1</td></tr><tr><td>G(logε<sub>B</sub>)</td><td>-3.0</td><td>0.1</td></tr><tr><td>G(logn)</td><td>0.0</td><td>0.5</td></tr><tr><td>G(p)</td><td>2.1</td><td>0.2</td></tr></tbody></table>

Among these, the shock micro-physical parameters $f = (\epsilon_0, \epsilon_\infty)$, corresponding to the fraction of energy shared by the shock between particles ($\epsilon_\infty$) and the magnetic field ($\epsilon_0$). The circum-burst number density is parameterized with $n$. These parameters were assigned by sampling log-normal distributions $G(\log|\mu,\sigma)$ (with average $\mu$ and standard deviation $\sigma$). The index of the energy distribution of shock-accelerated electrons was sampled from a Gaussian distribution $G(p|\mu,\sigma)$. These are typical afterglow parameter values and it has been verified that they produce nIR fluxes consistent with the ranges observed in the few high-redshift GRBs known. To account for the possibility that the very early afterglow emission is characterized by a rising phase, the deceleration time has been computed by assigning to each GRB a bulk Lorentz factor $\Gamma_0$, sampled from the empirical correlation between $\Gamma_0$ and the prompt emission isotropic energy $E_{iso}$ (ref. ¹⁰). We assume negligible dust absorption, as indicated by high-z GRB afterglow observations²⁰, that in all cases show $A_\gamma < 0.15$. This is in line with the expected proprieties of high-z GRB host galaxies²¹ and the general trend of decreasing dust absorption in GRB afterglows with redshift²².

The afterglows of high-redshift GRBs are naturally absorbed, like any other source, at optical wavelengths by hydrogen along the line of sight in the intergalactic medium (Lyman-α absorption). We propose to take advantage of the deep monitoring of the sky by the Vera Rubin Observatory⁶ to simultaneously observe exactly the same fields with a dedicated nIR facility. By comparing the two streams of transients, one can pinpoint transients detected in the nIR band and not in the optical band. Fast transients detected only in the nIR and with an AB colour index $r-H \ge 3.5$ are high-redshift GRBs, with a low contamination rate (Box 1).

To estimate the number of nIR high-redshift GRBs, we carried out simulations using a population synthesis code⁷. A population of long GRBs is simulated based on their luminosity function (that is, the number of sources as a function of their luminosity or energy) and their formation rate as a function of redshift (that is, the cosmic rate density). The free parameters of these functions have been derived by reproducing the observed properties of well-selected samples of GRBs collected in the past 15 years. In particular, the population is calibrated by reproducing the fluence, peak flux, observer frame peak energy and observer-frame duration distribution of GRBs detected by Fermi and Swift satellites. To minimize the impact of observational biases, which always play a role in GRB studies, a flux-limited, 97% complete in redshift, sub-sample of bright GRBs detected by Swift⁸ was used. This sample provides constraints on the GRB rate density. Long-duration GRBs are associated with the core-collapse of massive stars, indicating they might be good tracers of the star formation. However, theoretical arguments indicate that GRBs require a low metal content in the
<table><caption>**Table 2 | Exposure time needed to reach a signal-to-noise ratio (SNR) of 5 in nIR exposures of different telescopes and instruments**</caption><thead><tr><td>Telescope (diameter)</td><td>Instrument</td><td>Total exposure time to SNR = 5</td></tr></thead><tbody><tr><td>TNG (3.6 m)</td><td>NICS</td><td>190 s</td></tr><tr><td>UKIRT (3.8 m)</td><td>WFCAM</td><td>132 s</td></tr><tr><td>CFHT (3.6 m)</td><td>WIRCAM</td><td>114 s</td></tr><tr><td>NTT (3.6 m)</td><td>SOFI</td><td>84 s</td></tr><tr><td>Gemini N (8.1 m)</td><td>NIRI</td><td>75 s</td></tr><tr><td>VISTA (4.1 m)</td><td>WIRCAM</td><td>70 s</td></tr><tr><td>Gemini S (8.1 m)</td><td>FLAMINGO2</td><td>60 s</td></tr><tr><td>LBT (8.4 m)</td><td>LUCI</td><td>47 s</td></tr><tr><td>Keck (10 m)</td><td>OSIRIS</td><td>40 s</td></tr><tr><td>Subaru (8.3 m)</td><td>MOIRCS</td><td>18 s</td></tr><tr><td>GTC (10.4 m)</td><td>CIRCE</td><td>8 s</td></tr><tr><td>VLT (8.2 m)</td><td>HAWK-I</td><td>8 s</td></tr></tbody></table>

Values have been obtained using the relevant exposure time calculator with a 0.5 full moon illumination, 1.2 airmass, 10 mm precipitable water vapour, 1 seeing and a flat power-law spectrum with index 0. We used different detector integration time combinations, with two exposures lasting 15 s being the preferred choice.

progenitor⁹, resulting in an increase of GRBs with redshift over the nominal star formation rate. This has been addressed, including a GRB density evolution with redshift⁷,⁸.

In addition to the GRB prompt properties, the population code simulates the afterglow emission based on the model for a fireball decelerating in a constant density external medium¹⁰ (Table 1 shows the adopted parameter values). The afterglow luminosity depends on the kinetic energy of the jet and on the shock efficiencies in amplifying the magnetic field and accelerating the emitting particles, and it has been demonstrated that there is a close relationship between the afterglow luminosity and the prompt-emission high-energy luminosity¹¹. This code has been adopted for rate estimates for the Transient High-Energy Sky and Early Universe Surveyor (THESEUS)¹² and the Gamow Explorer¹³.

Having set the parameters for a Monte Carlo simulation, we run the code asking what the rate of GRBs is as a function of redshift for a given limiting magnitude and at a given time after the GRB event. This is shown in Fig. 1a for $H < 21$ (AB magnitude system). The decay of the GRB rate as a function of redshift for $z > 5$ follows a power-law evolution $\propto z^{-3.4}$ (solid line in Fig. 1a). Owing to the power-law temporal decay of the afterglow light curve, the data points sampled at different epochs follow a similar $z$ decay. The dependence of the rate of high-redshift ($z \ge 6$) GRBs on the limiting $H$ magnitude is shown in Fig. 1b. We estimate that about 1/3 of high-redshift Swift GRBs are missed during the follow-up process. We also note that the population sampled by nIR observations is different from the one sampled by current high-energy instruments. Fig. 2 shows the contour levels in the plane of the prompt emission isotropic GRB energy $E_{iso}$ versus redshift for the simulated long GRB population detectable with $H < 21$. For comparison, the nine long GRBs known with redshift $z > 6$ are shown (star symbols). The nIR detection samples relatively less energetic events with respect to those currently detected by Swift, for example, and in the near future, SVOM.

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 1 | Rates of high-redshift GRBs depending on the nIR limiting magnitude and redshift.** *a*, Predicted rate per unit solid angle of $H < 21$ high-redshift ($z \ge 6$) GRB afterglows as a function of redshift and at different observing times. For reference a decay $\propto z^{-3.4}$ is shown by the solid orange curve. Horizontal bars (shown only for the circle symbol) represent the redshift bin width. *b*, Rates of high-redshift GRBs ($z \ge 6$) as a function of the limiting $H$ magnitude in the AB system at fiducial times after the prompt.</figcaption></figure>

The high-$z$ nIR detection rates are indeed very promising. The only problem is identifying these transients as GRB afterglows. Lyman-$\alpha$ absorption comes to the rescue, heavily absorbing radiation at optical wavelengths for high-redshift objects. If we can simultaneously observe in the optical and in nIR bands, transients detected only at longer wavelengths are natural candidates to be high-$z$ GRBs. We can make these candidates stronger by increasing the colour index, that is, the magnitude difference between optical and nIR, ensuring the optical observations are sufficiently deep.

In the next few years, we will have one of the most formidable optical survey machines: the Vera Rubin Observatory⁶. The Rubin Observatory will carry out the Legacy Survey of Space and Time (LSST), reaching a limiting magnitude $r \sim 24.5$ in 30-s exposures (AB magnitude system, single visit, signal-to-noise ratio of 5). Here we propose the
Other red transients that might contaminate the GRB sample

'Pure' nIR transients are rare and occur in the most heavily absorbed regions of our Galaxy. These can be easily avoided. In addition, heavily absorbed transients in nearby galaxies (for example, magnetar flares) can be avoided too, being part of a known object catalogue.

* Some classes of young stellar objects show powerful nIR outbursts. These systems are low-mass pre-main sequence objects. FU Orionis-type stars experience 4–5 mag outbursts lasting several decades; EX Lupi-type stars show dimmer outbursts (2–3 mag) that last for a few months to years²³. Their quiescent counterpart is usually brighter than H ~ 21 and they are embedded in Galactic star-formation regions. Despite the heavy absorption, they are often observed at optical wavelengths, too. We should be able to discard them, based on pre-existing optical and nIR counterparts, as well as on their location in Galactic star-formation regions.

* Late M–L dwarfs usually have very red colours, becoming redder as one progresses along the spectral sequence. Many of these M–L dwarfs show prominent flares, because they retain their magnetic field and remain active for gigayears. Fortunately, flares are significantly hotter than the star’s temperature. They are detectable in white light, often with a prominent Hα line, as testified to, for example, by the optical ASAS-SN survey²⁴. The flare spectra of ultra-cool M-dwarfs have been modelled²⁵ as black-body emission of ~10,000 K. The investigation of hundreds of M-dwarf flares from the ATLAS survey shows them to have a characteristic exponential decay on a timescale of 30–60 minutes, with a red stellar counterpart visible in Pan-STARRS, SDSS, UKIDDS, VISTA or WISE archives. Hence, these Galactic flares are easily distinguished from GRB afterglows. A combination of high Galactic extinction and large flares could potentially result in very red flare colours, but this is easily circumvented, again, by avoiding high-extinction regions.

* 'Dark' GRBs are probably the most severe contaminants for high-redshift GRBs. Naively, dark GRBs are long-duration GRBs

with no detectable optical afterglow. A more formal definition involves also the overall strength of the afterglow: a dark GRB is one for which the X-ray-to-optical spectral slope $\beta_{OX}$ is less than 0.5 (ref. ²⁶). An alternative definition involves also the X-ray spectral slope $\beta_X$, requiring $\beta_{OX} < \beta_X - 0.5$ (ref. ²⁷), resulting in a more restrictive selection (which we adopt here). Dark GRBs are likely to be caused by intrinsic absorption within the host galaxy²⁸, with a contribution from high-redshift GRBs. These two effects are, however, not easy to disentangle. Using a flux-limited GRB sample complete in redshift, it has been estimated that dark GRBs comprise ~30% of the GRB population²⁹. With a different sample, it was estimated³⁰ that only ~20% of the dark GRBs are at redshift $z > 4$. In a recent work, a sample of 21 dark GRBs were studied with Chandra and HST data³¹. For ~50% of these dark GRBs, a host galaxy with *r* magnitude brighter than 25 has been detected. These host galaxies will appear as steady sources in Rubin LSST stacked images, leading to an easy identification.

* Finally, assuming a typical afterglow spectrum with a power-law photon index $\Gamma = 2$, extending into the optical regime (without a cooling break, resulting in a more restrictive limit), one would expect a ~1 mag difference from the *r* and *H* emission, without absorption. A Galactic extinction curve, which again is a severe approximation since the extinction curve at high-z is flatter, provides an absorption of $A_R \sim 0.75$ and $A_H \sim 0.18$, for $A_V = 1$. To gain ~2.5 from extinction (and satisfy our $r-H \ge 3.5$ colour selection constraint), one would need $A_V \ge 4.5$. In the Swift-BAT6 flux-limited redshift-complete sample⁸, we have only three GRBs with such a high absorption²⁰,²⁹, indicating that the contribution to extremely dark GRBs able to pass our colour selection criteria (and contaminate high-redshift GRBs), should be at the ~6% level, at the most. Based on one GRB per day in the whole sky, the number of dark GRBs able to contaminate the search for high redshift GRBs should be ~4 GRBs yr⁻¹, in the worst case, that is, a factor of two less than the annual rate of high-z GRBs. In passing, we note that dark GRBs are a valuable scientific topic, allowing us to probe star-formation in heavily absorbed environments, too.

construction of a new nIR facility to observe in tandem with the Rubin LSST to unveil the high-redshift Universe through the identification of GRBs. To be conservative, we consider a telescope and nIR camera able to reach magnitude *H* = 21 in 30 s, with the same field of view as the Rubin telescope (9.6 deg²). This will allow us to look for transients with *r-H* ≥ 3.5, enough to assure that the transients are at high redshift¹,²,¹⁴,¹⁵. Then, we assume 10-hr observations per night, 80% of good weather nights, 30-s exposures, and 73% open shutter time as a fraction of the observing time¹⁶. Based on these metrics, we estimate that we would detect ~11.0 GRBs per year at $z \ge 6$. The corresponding rate at $z \ge 10$ is ~2.7 GRBs per year.

A nIR telescope with the required sensitivity and field of view does not currently exist. This new telescope should have the same field of view of the Rubin telescope (9.6 deg²) and should reach magnitude *H* = 21 in 30 s (in two sub-exposures of ~15 s). The system should be

optimized to observe in the nIR to improve performance and minimize the telescope diameter. In Table 2, we provide a summary of the main existing telescopes with nIR imaging instruments. It is readily apparent that, if the system is not nIR optimized, an 8 m-class telescope is needed. Using the Advanced Exposure Time Calculator¹⁷ tool (http://aetc.oapd.inaf.it/), we simulated astronomical images with customizable telescope and instrument combinations. Adopting state-of-the-art detectors (efficiency >0.8, low readout noise), we end up with a ~5 m telescope. We can reduce the need for a large mirror by slightly widening the observing band: extending the short-wavelength edge of the H band (1,320–1,940 nm) to 1,220 nm (the central wavelength of the J band), would reduce the aperture requirement to a ~3 m telescope. A dedicated trade-off study would be needed to optimize the wavelength range. Assuming a 4 m telescope with an optical design similar to Rubin's telescope (as a baseline), we can roughly estimate a cost of
<figure><img src="image_5.png" /><figcaption>**Fig. 2** | Prompt emission isotropic equivalent energy versus redshift of the simulated population of events with *H* <21. The shaded regions show the 68, 90, 99% confidence contours. The nine GRBs at *z* ≳ 6 are shown with star symbols.</figcaption></figure>

35–40 MEuro for the (robotic) telescope and 5–7 MEuro for the dome and the infrastructure on the current market. No particular requirements on the pointing speed are needed, as this telescope will slavishly mirror Rubin’s activity. To minimize the camera costs, we can foresee the use of InGaAs detectors with a ~0.8–1 arcsec per pixel scale. This camera would be the largest nIR camera ever built, even larger than the VISTA IR Camera (VIRCAM)¹⁸. Based on the VIRCAM experience, the estimated cost is 10–13 MEuro (depending on the pixel scale). No adaptive optics corrections are foreseen. The camera would comprise 128–200 Mpixels, resulting in 2–3 times the VIRCAM data rate. This nIR telescope must be located in the Southern Hemisphere, in order to work in tandem with the Rubin telescope. Favourable sites could be found in Chile: Cerros La Silla, Paranal, Tololo and Pachon.

High-redshift GRB afterglows can be directly extracted by comparing the stream of nIR transients with the (public) stream of Rubin LSST transients. Rapid forced photometry on the LSST data will ensure the objects selected are optically faint or undetected. We stress that this method is independent of high-energy facilities, such red and fast objects are unique in astronomical transient streams, and it will allow us to sample a less-extreme population of high-z GRBs (Fig. 2). Transients detected in the nIR and missing from the Rubin LSST stream are selected for follow-up observations. To take advantage of these detections, dedicated nIR spectroscopic facilities (such as X-shooter or the Son of X-shooter (SOXS) at La Silla or FLAMINGOS-2 at the Gemini South observatory, in the Southern Hemisphere) should confirm the candidates, then allowing the largest next-generation facilities, such as the JWST, the extremely large telescopes (ELT, TMT) at nIR wavelengths, the Advanced Telescope for High Energy Astrophysics (Athena) at X-ray

energies, or the Square Kilometre Array (SKA) in the radio, to point at them disclosing the line of sight and the host galaxy properties.

S. Campana Ⓓ¹, G. Ghirlanda Ⓓ¹,², R. Salvaterra³, O. A. Gonzalez⁴, M. Landoni¹, G. Pariani¹, A. Riva⁵, M. Riva¹, S. J. Smartt Ⓓ⁶, N. R. Tanvir⁷ and S. D. Vergani Ⓓ⁸

¹INAF, Osservatorio astronomico di Brera, Merate, Italy. ²INFN, Sezione di Milano-Bicocca, Milano, Italy. ³INAF, Istituto di Astrofisica Spaziale e Fisica cosmica, Milano, Italy. ⁴STFC UK Astronomy Technology Centre, The Royal Observatory, Edinburgh, UK. ⁵INAF, Osservatorio astrofisico di Torino, Pino Torinese, Italy. ⁶Astrophysics Research Centre, School of Mathematics and Physics, Queen's University Belfast, Belfast, UK. ⁷School of Physics and Astronomy, University of Leicester, Leicester, UK. ⁸GEPI, Observatoire de Paris, PSL University, CNRS, Meudon, France.

e-mail: sergio.campana@inaf.it

Published online: 13 October 2022

## References

1. Salvaterra, R. et al. *Nature* **461**, 1258–1260 (2009).

2. Tanvir, N. R. et al. *Nature* **461**, 1254–1257 (2009).

3. Jiang, L. et al. *Nat. Astron.* **5**, 256–261 (2021).

4. Castellano, M. et al. Preprint at https://arxiv.org/abs/2207.09436 (2022).

5. Melandri, A. et al. *Astron. Astrophys.* **581**, A86 (2015).

6. Izvezić, Ž. et al. *Astrophys. J.* **873**, 111 (2019).

7. Ghirlanda, G. et al. *Mon. Not. R. Astron. Soc.* **448**, 2514–2524 (2015).

8. Salvaterra, R. et al. *Astrophys. J.* **749**, 68 (2012).

9. Yoon, S.-C., Langer, N. & Norman, C. *Astron. Astrophys.* **460**, 199–208 (2006).

10. Ryan, G., van Eerten, H., Piro, L. & Troja, E. *Astrophys. J.* **896**, 166 (2020).

11. Gehrels, N. et al. *Astrophys. J.* **689**, 1161–1172 (2008).

12. Amati, L. et al. *Exper. Astron.* **52**, 183–218 (2021).

13. White, N. E. et al. *Proc. SPIE* **11821**, 1182109 (2021).

14. Cucchiara, N. et al. *Astrophys. J.* **736**, 7 (2011).

15. Tanvir, N. R. et al. *Astrophys. J.* **865**, 107 (2018).

16. Marshall, P. et al. Preprint at https://arxiv.org/abs/1708.04058 (2017).

17. Uslenghi, M., Falomo, R. & Fantinel, D. *Proc. SPIE* **9911**, 99112U (2016).

18. Sutherland, W. et al. *Astron. Astrophys.* **575**, A25 (2015).

19. Ghirlanda, G. et al. *Astron. Astrophys.* **609**, A112 (2018).

20. Covino, S. et al. *Mon. Not. R. Astron. Soc.* **432**, 1231–1244 (2013).

21. Salvaterra, R. *J. High Energy Astrophys.* **7**, 35–43 (2015).

22. Zafar, T. et al. *Astron. Astrophys.* **532**, A143 (2011).

23. Audard, M. et al. in *Protostars and Planets VI* (Beuther, H. et al. eds) 387–410 (Univ. Arizona Press, 2014).

24. Schmidt, S. J. et al. *Astrophys. J.* **876**, 115 (2019).

25. Kanodia, S. et al. *Astrophys. J.* **925**, 155 (2022).

26. Jakobsson, P. et al. *Astrophys. J.* **617**, L21–L24 (2004).

27. van der Horst, A. J. et al. *Astrophys. J.* **699**, 1087–1091 (2009).

28. Campana, S. et al. *Mon. Not. R. Astron. Soc.* **421**, 1697–1702 (2012).

29. Melandri, A. et al. *Mon. Not. R. Astron. Soc.* **421**, 1265–1272 (2012).

30. Greiner, J. et al. *Astron. Astrophys.* **526**, A30 (2011).

31. Chrimes, A. A., Levan, A. J., Groot, P. J., Lyman, J. D. & Nelemans, G. *Mon. Not. R. Astron. Soc.* **508**, 1929–1946 (2021).

## Acknowledgements

S. C. thanks M. G. Bernardini, P. D'Avanzo, A. Melandri, P. Schipani, G. Tagliaferri and F. Vitali for useful conversations.

## Competing interests

The authors declare no competing interests.

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