Intrinsic properties of a complete sample of HETE-2 gamma-ray bursts : A measure of the GRB rate in the Local Universe

Context. As a result of the numerous missions dedicated to the detection of Gamma-ray bursts (GRBs), the observed properties of these events are now well known. However, studying their parameters in the source frame is not simple since it requires having measurements of both the bursts' parameters and of their distances. Aims. Taking advantage of the forthcoming Catalog of the High Energy Transient Explorer 2 (HETE-2) mission, the aim of this paper is to evaluate the main properties of HETE-2 GRBs-the peak energy (E_{peak}), the duration (T sub(90)) and the isotropic energy (E_{iso})-in their source frames and to derive their unbiased distribution. Methods. We first construct a complete sample containing all the bursts localized by the Wide-Field X-ray Monitor (WXM) on-board HETE-2, which are selected with a uniform criterion and whose observed parameters can be constrained. We then derive the intrinsic E_{peak}, T sub(90) and E_{iso} distributions using their redshift when it is available, or their pseudo- redshift otherwise. We finally compute the "volume of detectability" V_{max} of each GRB, i.e. the volume of the universe in which the burst is bright enough to be part of our sample, and the corresponding number of GRB within their visibility volume N_{V{max}}, in order to derive a weight for each detected burst accounting both for the detection significance and the star formation history of the universe. Results. We obtain unbiased distributions of three intrinsic properties of HETE-2 GRBs: E_{peak}{intr}, T_{intr} and the isotropic energy of the burst. These distributions clearly show the predominence of X-ray flashes (XRFs) in the global GRB population. We also derive the rate of local GRBs: R_{0}{H2} ga 11 Gpc super(-3) yr super(-1), which is intermediate between the local rate obtained by considering only the "high-luminosity" bursts ( similar to 1 Gpc super(-3) yr super(-1)) and that obtained by including the "low-luminosity" bursts (\ga200 Gpc super(-3) yr super(-1)). Conclusions. This study shows that the XRFs are predominent in the GRB population and are closely linked to the "classical" GRBs. We show that HETE- 2 detected no low-luminosity GRB like GRB 980425 or XRF 060218, due to the small size of its detectors, excluding this type of burst from our statistical analysis. The comparison of the GRB rate derived in this study with the known rate of type Ib/c supernovae clearly shows that the progenitors of SNe Ib/c must have some special charac