*10.2. High-Redshift GRB Rate Excess*

Although appealing, the above developments do not show why GRBs do not follow the star formation history being enhanced by hidden high-redshift mechanisms [242–247]. In particular, the star formation rate at high-redshift, namely *z* > 6, appears too large if confronted with the star formation rate obtained from high-redshift galaxy surveys [248].

A natural origin of the high-redshift GRB rate excess can be found in the metallicity evolution, as LGRBs seem to prefer a low-metallicity environment, as supported by recent studies that favor such a requirement31. Typical mass bounds on stars suggest <sup>&</sup>gt; <sup>30</sup>*<sup>M</sup>*, being responsible for BH remnants.

## *10.3. Gravitationally-Lensed GRBs*

Gravitationally lensed GRBs (GLGRBs) have been proposed in Ref. [49], where it was speculated that such a phenomenon would result in multiple light curves detected at different times, as due to the different light paths of the produced multiple GRB images.

Quests for GLGRBs were mostly based on strong lensing<sup>32</sup> and similarities among GRB light curves with identical spectra and close locations in the sky, as primary search criteria [251–255]. However, such searches led to null results, possibly due to Poisson noise uncertainties, affecting GRB light curves especially at low signal-to-noise ratios, which may have introduced large differences between the lensed GRB images [254]. On the other hand, some GLGRB could exhibit time delays shorter than (or comparable to) the burst duration, hence leading to unresolved (or locally separated) peaks separated by the time delay [256–258].

Several searches have been performed in the literature, resulting in a few or null candidates, based on different techniques and lens models, such as globular cluster with a mass of <sup>≈</sup> <sup>10</sup>5–10<sup>7</sup> *<sup>M</sup>* [256], Population III stars with a mass range of <sup>10</sup>2–10<sup>3</sup> *<sup>M</sup>* [257], diverse objects with a mass range of <sup>10</sup>2–10<sup>7</sup> *<sup>M</sup>* [259], or a supermassive BH with a mass in the range of <sup>≈</sup> <sup>10</sup>5–10<sup>7</sup> *<sup>M</sup>* [260].

Considering models where the lens is a supermassive BH [259,260], GLGRB candidates can provide an opportunity to estimate the number density of massive compact objects at cosmological distances by calculating the rate of GRB lensing. Assuming such supermassive BH lenses with mass <sup>≈</sup> <sup>10</sup><sup>6</sup> *<sup>M</sup>* as dark matter compact objects [261], the density parameter of BHs is ΩBH = 0.007 ± 0.004, or equivalently Ω*BH*/Ω*<sup>M</sup>* = 0.027 ± 0.016 [260]. In this respect, finding more GLGRBs candidates from supermassive BH may enhance our understanding of the matter content of the Universe.
