Abstract. We propose a new empirical redshift indicator for gamma-ray bursts. This indicator is easily computed from the gamma-ray burst spectral parameters and its duration, and it provides "pseudo-redshifts" accurate to a factor two. Possible applications of this redshift indicator are briefly discussed.

In this letter we discuss the flux and the behavior of the bright optical flare emission detected by the 25 cm TAROT robotic telescope during the prompt high-energy emission and the early afterglow. We combine our data with simultaneous observations performed in X-rays and we analyze the broad-band spectrum. These observations lead us to emphasize the similarity of GRB 050904 with GRB 990123, a remarkable gamma-ray burst whose optical emission reached 9th magnitude. While GRB 990123 was, until now, considered as a unique event, this observation suggests the existence of a population of GRBs which have very large isotropic equivalent energies and extremely bright optical counterparts. The luminosity of these GRBs is such that they are easily detectable through the entire universe. Since we can detect them to very high redshift even with small aperture telescopes like TAROT, they will constitute powerful tools for the exploration of the high-redshift Universe and might be used to probe the first generation of stars.

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 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 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 Vmax , 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 intr peak , T intr 90 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 H2 0 > ∼ 11 Gpc −3 yr −1 , which is intermediate between the local rate obtained by considering only the "high-luminosity" bursts (∼1 Gpc −3 yr −1 ) and that obtained by including the "low-luminosity" bursts ( > ∼ 200 Gpc −3 yr −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 characteristics in order to produce a gamma-ray burst or an X-ray flash.