*3.6. Jet Structure and the E*peak − *E*iso *Correlation*

A fraction of the GRBs (around one-third) that trigger the Burst Alert Telescope (BAT) onboard *Swift* end up with a measurement of their redshifts [213]. This allows, in most cases. The estimate of *E*iso and *L*iso require to measure the SED peak which is often possible thanks to the detection of the burst also by the *Fermi* satellite which provides a broad band (10keV-40MeV) energy spectral coverage., to estimate rest frame properties such as *E*peak, *E*iso and *L*iso. Figure 7 shows the GRBs from [214] on the (*E*iso, *E*peak) plane (black symbols), demonstrating the apparent correlation between these two quantities [191]. Such a correlation is naturally expected in a quasi-universal structured jet scenario, given the common dependence of *E*iso and *E*peak on the viewing angle (e.g., [215]). Assuming Gaussian profiles *ηγ*d*E*/dΩ = *e*<sup>0</sup> exp(−*θ* <sup>2</sup>/*θ* 2 c ) and Γ(*θ*) = 1 + (Γ<sup>c</sup> − 1) exp(−*θ* <sup>2</sup>/*θ* 2 c ), an angle-independent comoving peak SED photon energy *E* 0 peak = 1 keV [216], and a quasiuniversal structured jet scenario in which the structure parameters in the population are narrowly distributed around typical values <sup>h</sup>*e*0<sup>i</sup> <sup>=</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup><sup>53</sup> , hΓ0i = 800, h*θ*ci = 3 ◦ , the authors of [125] could reproduce both the LF of long GRBs and the observed *E*peak − *E*iso correlation, as shown by the model distribution represented in Figure 7. The horizontal dispersion in the figure corresponds to just considering a 0.5 dex log-normal dispersion of the core energy density *e*0. As shown by the color-coded viewing angle, within this interpretation the known long GRBs are observed within *θ*<sup>v</sup> . 3*θ*<sup>c</sup> ∼ 9 ◦ , which is consistent with the constraints derived by [198]. In the bottom left corner, corresponding to *E*iso < 10<sup>51</sup> erg and *E*peak < 30 keV, should reside jets observed at larger viewing angles. If these were detected with next-generation instruments with wide-field, highly sensitive hard X-ray monitors, they could probe the expected bending of the *E*peak − *E*iso correlation induced by

large viewing angles and in turn help constraining the quasi-universal jet structure scenario.

**Figure 6. Top**: luminosity function of Long GRBs as obtained by [210] (black symbols) extended to low luminosities by [206] (red and blue symbols). Models considering a uniform jet (only seen on-axis – green – or isotropically oriented – red) or a structured jet with a steep power law profile (cyan) are shown. The separation in low, intermediate, and high luminosity (LL, IL, HL) GRBs is indicated by the dashed vertical lines. **Bottom**: models of the (inverse cumulative) SGRB luminosity function. Models fitted to observed properties of short GRBs (detected at cosmological distances) are shown by the green [211] and purple [212] thick transparent lines and bands (medians and 90% credible regions). The luminosity function obtained by [114] by computing the jet structure from a semianalytical calculation of the jet propagation and breakout is shown by the blue lines (contributions by jets observed in different intervals of viewing angle are shown), arbitrarily normalized to a local rate density *R*<sup>0</sup> = 100 Gpc−<sup>3</sup> yr−<sup>1</sup> . The local BNS merger rate density constraint from [217], i.e., <sup>10</sup> <sup>≤</sup> *<sup>R</sup>*0,BNS/Gpc<sup>3</sup> yr <sup>≤</sup> 1700, is shown by the pink shaded region.

**Figure 7.** Rest frame peak energy *E*peak versus isotropic equivalent energy *E*iso of long GRBs. The data points (cross symbols) are a flux-limited sample of bright *Swift* bursts. The dashed (dotted) line shows the correlation regression line (and its 3*σ* scatter). The color-coded solid line shows the values of *E*peak and *E*iso assuming a structured Gaussian jet seen under progressively larger viewing angles-vertical color-code bar). The green shadows, representing the 1, 2, 3*σ* confidence levels around the color-coded line, are obtained considering a dispersion of the core energy of 0.5 dex around a nominal value of 3 <sup>×</sup> <sup>10</sup><sup>53</sup> erg. Figure reproduced from [125].

## *3.7. Jet Structure and 'Late-Prompt' Emission*

The observed X-ray emission of GRBs extending after the prompt phase up to a few hours is often characterized by a steep decay [218] of the flux transitioning to a shallow (so-called *plateau*) phase [219]. On top of this, X-ray flares are often observed [220–222]. Intriguingly, these features of the early X-ray emission can be explained within a structured jet scenario: the steep-plateau shape is what an observer nearly aligned with the jet axis would see, while an off-axis observer should see a more uniform power law decay [207,223]. In this interpretation, the X-ray light curve up to the end of the plateau phase would have an internal origin, being produced by prompt emission photons reaching the observer from increasingly high-latitude parts of the jet (hence the delayed arrival time, [224]). Flares have been explained, in the context of a structured jet, as late-time internal dissipation episodes whose brightness and spectral hardness are reduced by the debeaming effects for an observer with a viewing angle far from the jet axis [225].
