Abstract
Gamma-ray bursts (GRBs) are brief, intense, gamma-ray flashes in the universe, lasting from a few milliseconds to a few thousand seconds. For short gamma-ray bursts (sGRBs) with duration less than 2 seconds, the isotropic energy (E
iso) function may be more scientifically meaningful and accurately measured than the luminosity (L
p) function. In this work we construct, for the first time, the isotropic energy function of sGRBs and estimate their formation rate. First, we derive the L
p – E
p correlation using 22 sGRBs with known redshifts and well-measured spectra and estimate the pseduo redshifts of 334 Fermi sGRBs. Then, we adopt the Lynden-Bell c
− method to study isotropic energy functions and formation rate of sGRBs without any assumption. A strong evolution of isotropic energy E
iso ∝ (1+z)5.79 is found, which is comparable to that between L
p and z. After removing effect of the cosmic evolution, the isotropic energy function can be reasonably fitted by a broken power law, which is
ϕ
(
E
iso
,
0
)
∝
E
iso
,
0
−
0.45
for dim sGRBs and
ϕ
(
E
iso
,
0
)
∝
E
iso
,
0
−
1.11
for bright sGRBs, with the break energy 4.92 × 1049 erg. We obtain the local formation rate of sGRBs is about 17.43 events Gpc−3 yr−1. If assuming a beaming angle is 6° to 26°, the local formation rate including off-axis sGRBs is estimated as ρ
0,all = 155.79 – 3202.35 events Gpc−3 yr−1.
Gamma-ray bursts: the isotropic-equivalent-energy function and the cosmic formation rate
