Constraining the intrinsic population of long gamma-ray bursts: Implications for spectral correlations, cosmic evolution, and their use as tracers of star formation

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Table 1.

Summary of the properties and calculated quantities used to describe the LGRBs in our population model.

Properties describing an LGRB:
Property	Units	Type	Distribution	Description

z	–	–	Adjusted	The redshift of the LGRB source
L	erg s⁻¹	Pk	Adjusted	The isotropic-equivalent bolometric peak luminosity
E_p	keV	Pk	Adjusted	The peak energy of the L_E/E photon spectrum in the source frame

α	–	Pk	Fixed	The low-energy slope of the photon spectrum L_E/E in the source frame
β	–	Pk	Fixed	The high-energy slope of the photon spectrum L_E/E in the source frame

T₉₀	s	Ti	Computed	The duration over which 5–95% of photons are emitted in the source frame
C_var	–	Ti	Computed	The variability coefficient defined as the mean luminosity divided by the peak luminosity

Calculated quantities:

Quantity	Units	Type	Description

E_p obs	keV	Pk	The peak energy of the E² N_{E_obs} spectrum in the observer frame, E_p obs = E_p/(1 + z).
N^pk	ph s⁻¹ cm⁻²	Pk	The peak photon flux, calculated from z, L, E_p, α, β for any [E_min, obs;E_max, obs] (see Eq. (1))
F^pk	erg s⁻¹ cm⁻²	Pk	The peak energy flux, calculated from z, L, E_p, α, β for any [E_min, obs;E_max, obs] (see Eq. (3))

T_90 obs	s	Ti	The duration over which 5–95% of photons are received in the observer frame, T_90 obs = T₉₀ (1 + z)
𝒩	ph cm⁻²	Ti	The photon fluence, calculated from z, L, E_p, α, β, T₉₀ and C_var for any [E_min, obs;E_max, obs] (see Eq. (15))
ℱ	erg cm⁻²	Ti	The energy fluence, calculated from z, L, E_p, α, β, T₉₀ and C_var for any [E_min, obs;E_max, obs] (see Eq. (16))
E_iso	erg	Ti	The isotropic-equivalent bolometric energy, E_iso = C_var T₉₀ L

Notes. In each table, the Pk type refers to quantities defined at peak brightness of the LGRB while the Ti indicates the quantity is time-integrated (i.e., it depends on the light curve of the LGRB). The first table lists the properties describing a single LGRB, always defined in the source frame. The top three properties’ distributions are adjusted by comparing the population to the observational constraints described in Sect. 3. They are completed by the next two properties, with fixed distribution (see Sect. 2.4) to entirely describe the emission at the peak brightness. The last two properties are only used for the additional crosschecks described in Sect. 3.5. Their distributions are computed (see Sect. 2.5). The second table lists the main quantities that are calculated from the physical properties in the first table, usually in the observer frame. They are used to build the mock samples as described in Sect. 2.6.

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