*3.5. The GRB Luminosity Function and the Jet Structure*

A key property of the GRB population is the luminosity function (LF). The LF can be defined as the probability density of the isotropic-equivalent luminosity at a particular redshift *P*(*L*iso | *z*) or, equivalently, as the comoving rate density in a differential luminosity bin, d*Rz*/d*L*iso = *RzP*(*L*iso | *z*), where *R<sup>z</sup>* is the event rate density (GRBs per comoving Gpc<sup>3</sup> yr) at redshift *z*. If the population does not feature a luminosity evolution with redshift, then the local LF d*R*0/d*L*iso = *R*0*P*(*L*iso) is sufficient to describe the luminosity distribution in the population. In the context of a structured jet, the luminosity of each event depends both on the intrinsic properties of the underlying jet, and on its viewing angle. Hence, the luminosity function is shaped at least in part by viewing angle effects [24]. If the jet structure is universal (i.e., all jets share the same properties) then the LF is entirely determined by viewing angle effects: for a population with isotropic orientations and a monotonic viewing-angle-dependent luminosity *L*iso(*θ*v) (common to all events), the LF can be obtained [24,206] by application of the chain rule, namely

$$\frac{\mathrm{d}R\_0}{\mathrm{d}L\_{\mathrm{iso}}} = \frac{\mathrm{d}R\_0}{\mathrm{d}\theta\_\mathrm{v}} \frac{\mathrm{d}\theta\_\mathrm{v}}{\mathrm{d}L\_{\mathrm{iso}}} = R\_0 \left(\frac{\partial}{\partial\theta\_\mathrm{v}} L\_{\mathrm{iso}}(\theta\_\mathrm{v})\right)^{-1} \sin\theta\_\mathrm{v} \Bigg|\_{\theta\_\mathrm{v} = f^{-1}(L\_{\mathrm{iso}})} \tag{8}$$

where *f* −1 is the inverse of *L*iso(*θ*v), i.e., *f* −1 (*L*iso(*θ*v)) = *θ*v. In the simple case of a power law dependence of the luminosity on the viewing angle, *L*iso(*θ*v) ∝ *θ* −*a* v , and using the small-angle approximation sin *θ*<sup>v</sup> ∼ *θ*v, one obtains [206] d*R*0/d*L*iso ∝ *L* −1−2/*a* . The fact that this asymptotes to *L* <sup>−</sup><sup>1</sup> when *<sup>a</sup>* <sup>→</sup> <sup>∞</sup> shows that, due to viewing angle effects, the LF cannot in principle be shallower than *L* <sup>−</sup><sup>1</sup> due to the contribution of off-axis jets, no matter how suppressed their emission is at large viewing angles. In practice, the spectrum of a far off-axis jet would be much softer than a typical GRB, the duration much longer, and the light curve smooth [207]. These features would lead to a different classification than a GRB. As an additional note, the *L* <sup>−</sup>1−2/*<sup>a</sup>* behavior breaks down when the small angle approximation sin *θ*<sup>v</sup> ∼ *θ*<sup>v</sup> becomes invalid. The assessment of the contribution of far off-axis jets to the GRB LF thus requires additional care.

In practice, even in the case in which the progenitor parameter space for which a relativistic jet can be launched is very narrow (as suggested, for example, by the relatively small spread in the peak luminosities of supernovae associated with long GRBs, e.g., [208,209]), some spread in the jet properties within the population is unavoidable. For that reason, even in the wildest unification fantasy the best that can be expected is a *quasi-universal* jet structure with parameters that are spread around a "typical" value. The LF in a quasi-universal structured jet scenario can then be seen as a convolution of probability distributions,

$$P(L\_{\rm iso}) = \int\_0^{\pi/2} P(L\_{\rm iso} \mid \theta\_\mathbf{v}) P(\theta\_\mathbf{v}) \, \mathrm{d}\theta\_\mathbf{v} \tag{9}$$

where *P*(*θ*v) = sin *θ*<sup>v</sup> and *P*(*L*iso | *θ*v) is the probability distribution of the isotropicequivalent luminosity at a given viewing angle, which is in turn induced by the probability distributions of the jet structure parameters (Salafia et al. in preparation).

The black symbols with error bars in the top panel of Figure 6 show the LF of long GRBs as obtained by combining samples of high luminosity (HL) GRBs with measured *z* and estimated isotropic equivalent luminosities *<sup>L</sup>*iso <sup>≥</sup> <sup>10</sup><sup>50</sup> erg/s [210]. The plot also shows the extension of the LF to intermediate luminosities (IL) <sup>10</sup><sup>48</sup> <sup>≤</sup> *<sup>L</sup>*iso/(erg s−<sup>1</sup> ) < 10<sup>50</sup> where, due to selection effects, only lower limits on the intrinsic rate density can be placed [206], and to the low luminosity range (LL – *L*iso . 10<sup>47</sup> erg s−<sup>1</sup> ) dominated by few events detected in the very local Universe [206]. In the bottom panel of Figure 6, the green and purple lines and bands show the inverse cumulative LFs of short GRBs obtained by two different studies [211,212] based on the observed properties of the *Swift* and *Fermi* samples.

The LF of both long and short GRBs extends over eight orders of magnitudes in luminosity and presents a steep decay *L* <sup>−</sup>*<sup>α</sup>* with *<sup>α</sup>* & <sup>3</sup> at high luminosities (*<sup>L</sup>* <sup>&</sup>gt; *<sup>L</sup>*break <sup>≈</sup> <sup>10</sup><sup>52</sup> erg/s). In the structured jet framework, the most luminous events are likely those observed within the core opening angle. As discussed above, the intermediate and faint end of the LF (for *L* < *L*break) is at least in part shaped by the jet structure [206]. In long GRBs, the binned LF can be reproduced by a quasi-universal structured jet model [206] with a power law dependence of the luminosity on the viewing angle *L*iso(*θ*v) = *L*<sup>c</sup> min(1,(*θ*v/*θ*c) −*a* ) with *a* & 6 and *θ*<sup>c</sup> and *L*<sup>c</sup> narrowly distributed around 5 ◦ and <sup>3</sup> <sup>×</sup> <sup>10</sup><sup>52</sup> erg/s, respectively, (cyan solid line in the top panel of Figure 6). In short GRBs, due to the scarcer data and the absence of an agreement on the general features of the LF, the situation is more unclear. Yet, an attempt at deriving a quasi-universal jet structure model from modeling the jet propagation through, and breakout from, binary neutron star merger ejecta [114] does produce an LF (blue solid line in Figure 6) with a steep decay above a break at *<sup>L</sup>*break <sup>∼</sup> <sup>3</sup> <sup>×</sup> <sup>10</sup><sup>52</sup> erg/s and a relatively less steep distribution at lower luminosities, which gets shallower as it extends to the LL range where the observation of GRB170817A [27] places some constraints [67]. A shallower (e.g., a power law *L*<sup>v</sup> ∝ *θ* −2 v ) structure would result in a steeper LF that would overproduce the LL long GRBs rate density (blue symbol in Figure 6, top panel) or, in short GRBs, the binary neutron star merger rate (pink shaded region in Figure 6, bottom panel).
