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Appendix A: Detailed redshift discussions

GRB 020531.

HETE detected this short GRB with a duration of 0.2–1 s [GCN1399]⁴ and follow-up observations revealed various sources in the IPN error circle [GCN1408, GCN1415], including two new asteroids [GCN1400]. One of these sources was found to weaken in brightness [GCN1426] very close to an extended object, i.e. the probable host galaxy [GCN1427], which has a spectroscopic redshift of z = 1.0 [GCN1428]. A different fading source was found just outside the error box [GCN1434], making this a probable redshift estimate only.

GRB 040924.

This is a short GRB with a duration of  ~1.5 s [GCN2754] for which an optical afterglow (OA) has been identified [GCN2734]. A spectroscopic analysis of the very likely host galaxy [GCN2750] revealed a redshift of z = 0.859 [GCN2800].

GRB 050416.

This is a SWIFT/XRT GRB [GCN3264, GCN3268] with a duration of 2.4  ±  0.2 s [GCN3273], and because an optical transient was observed [GCN3265, GCN3266] the measured redshift of z = 0.6535 ± 0.0002 [GCN3542] is very likely.

GRB 050509B.

GRB 050509B was found only within the XRT error circle [GCN3395] with a radius of 8′′. Further investigations showed that there are at least four more sources in the XRT error circle, one of them a probable high-redshift galaxy [GCN3401]. The chance association of a low redshift galaxy is reported to be very small [GCN3418], hence this redshift estimation is implausible.

GRB 050709.

Although a long duration is reported for this GRB of  ~130 s [GCN3653], the lightcurve and other spectral features classify it as a type-I GRB [GCN3570, GCN3653]. The association of the afterglow with the probable host galaxy [GCN3605, GCN3612] makes the redshift estimation of z = 0.16 reliable.

GRB 050724.

GRB 050724 has a T₉₀ duration of strictly 3 ± 1 s, but because it could belong to the type-I GRB class [GCN3667] it is considered in the sample. Four objects were found in the XRT error circle [GCN3672], of which two are identified as Galactic stars [GCN3675, GCN3679]. Additional observations hale us improve our confidence, that the object labelled as “D” is the host galaxy with a redshift of 0.258 [GCN3690, GCN3700].

GRB 050813.

This SWIFT GRB with a duration of 0.6  ±  0.1 s [GCN3793] has been found at a position with several faint extended objects, probably forming a galaxy cluster at high redshift, which makes this cluster the most likely source of that GRB [GCN3798]. Measurements of the redshift of the galaxies suggest a value of z = 0.722 [GCN3801], although a redshift of z = 0.65 also seems plausible [GCN3808]. The more conservative value of 0.722 will be used in this work as a probable value.

GRB 050906.

SWIFT detected this very short GRB (T₉₀ = 0.128 ± 0.016 s [GCN3935]) without finding the source with XRT, leaving only the BAT error circle to search for an afterglow [GCN3927, GCN3935]. Since this circle contains the massive star-forming galaxy IC 328 at z = 0.031, a galaxy cluster at z = 0.43, and field galaxies of unknown redshift, any value is unlikely.

GRB 051016B.

For this SWIFT GRB with a duration of 4  ±  0.1 s [GCN4104], a probable optical afterglow was found [GCN4111], in a galaxy with redshift z = 0.9364 [GCN4186]; this makes the redshift value reliable.

GRB 051221A.

This GRB had a duration of 1.4 ± 0.2 s [GCN4365], and an optical afterglow was detected [GCN4375]. The measured redshift of z = 0.5465 [GCN4384] is therefore reliable.

GRB 060502B.

GRB 060502B was a very short GRB with a duration of 0.09  ±  0.02 s [GCN5064], for which two sources were found in the XRT error circle [GCN5066, GCN5071]. One is assumed to be a star, while the other appears to be an extended object, whose reliable redshift is measured to be z = 0.287 [GCN5238].

GRB 060505.

This GRB has a nominal T₉₀ duration of 4 ± 1 s [GCN5142], and therefore not clearly assigned to either type-I or type-II. The position of the optical afterglow was found to be 4′′.3 from the galaxy 2dFGRS S173Z112, with a redshift of z = 0.089 [GCN5123]. The distance in projection of this late-type galaxy was found to be 7 kpc [GCN5123]. No supernova was found to be associated with this GRB [GCN5161], suggesting that this might be either a merger-driven GRB or a GRB at a much larger distance. The redshift value is classified as probable.

GRB 060801.

This short GRB (duration of 0.5  ±  0.1 s [GCN5381]) was found in the SWIFT/XRT instrument [GCN5378], which revealed four objects in its field [GCN5384, GCN6386]. In the revised XRT error circle [GCN5389], two objects remained, of which one is extended. The redshift of that extended object is z = 1.131 [GCN5470], making it a reliable estimation.

GRB 061006.

An optical afterglow was found for this GRB [GCN5718], revealing the reliable redshift to be z =  0.4377  ±  0.0002 (Berger et al. 2007). The formal duration is 130  ±  10 s [GCN5704], but initial short spikes lasting  ~ 0.5 s, on which SWIFT did not trigger [GCN5702, GCN5710], places this GRB in the type-I category.

GRB 061201.

This short GRB (duration of 0.8 ± 0.1 s [GCN5882]) had an optical afterglow [GCN5896], but no galaxy was found at its position. Close-by objects include a galaxy at redshift 0.111 [GCN5952], as well as the galaxy cluster Abell 995, for which a mean redshift of z ~ 0.0835 was determined [GCN5995]. The offset of the GRB from the galaxy would be 34 kpc in the first case, and 800 kpc in the second case (from the center of the cluster). The value of z = 0.111 is used in the further analysis⁵.

GRB 061210.

This GRB has a nominal duration of T₉₀ = 85 ± 5s [GCN5905], but an initial short spike of duration  ~ 60 ms places it into the type-I regime. This GRB has been located in the XRT error circle containing three galaxies [GCN5922]. Since no optical transient was found, the association with a given galaxy as host is doubtful, which makes the redshift implausible.

GRB 061217.

This short GRB, with a T₉₀ of 0.30 ± 0.05 s [GCN5930], exhibited no optical afterglow, but the XRT position was found to be within 11 arcsec of a galaxy [GCN5948]. Two objects have been found in a more deeper observation, with the brighter object proposed to be the host galaxy of this GRB [GCN5949, GCN5953]. A subsequent observation of the proposed host galaxy yields a redshift of z = 0.827 [GCN5965], so the source would have a isotropic-equivalent energy release of about 8 × 10⁴⁹ erg [GCN5965], which is rather high for a type-I GRB. This makes the redshift probable only.

GRB 070209.

The localization of this GRB within the BAT error circle contained no single source within the BAT error circle, but three X-ray sources in its proximity [GCN6095]. None of these sources were found to be the afterglow of this GRB [GCN6119], making the measured redshift of the source closest to the GRB position of z = 0.314 [GCN6101] very unlikely.

GRB 070406.

A short GRB with a T₉₀ of 0.7 ± 0.2 s [GCN6261], for which a bright source was detected in the XRT error circle, whose redshift is 0.703 [GCN6262]. Spectral features indicate this to be a quasar, and unrelated to the burst. Further investigations showed no hint of a fading afterglow, and the large number of faint sources found in the XRT error circle make this redshift estimate of z = 0.11 [GCN6249] uncertain.

GRB 070429B.

This BAT GRB has a duration of 0.5 ± 0.1 s [GCN6365], and two faint objects were found in the XRT error circle [GCN6372]. For the brighter objects, a redshift of z =  0.9023  ±  0.0002 was determined [GCN7104] (Cenko et al. 2008), as well as evidence that this object contains a fading source [GCN7145]. Therefore, the measured redshift is reliable.

GRB 070714B.

This long GRB with a standard T₉₀ time of 64  ±  5 s shows spectral features of a type-I GRB, especially the zero spectral lag [GCN6623]. An optical transient was found in the XRT [GCN6630], and the host’s galaxy redshift is found to be z =    0.9225  ±  0.0001 [GCN6836] (Cenko et al. 2008), which makes this a reliable redshift.

GRB 070724.

This SWIFT GRB [GCN6654] had a duration of 0.4 ± 0.04 s [GCN6656], and two sources were found in the XRT error circle, none of which showed variations [GCN6673]. The redshift of one of the source was found to be z = 0.457 [GCN6665], making this a probable estimate.

GRB 070810B.

This is a short GRB with a duration of only 0.08 ± 0.01 s [GCN6753]. Several possible sources has been identified in the XRT error circle, among them a nearby bright galaxy at z = 0.0385 (source S1) and a cluster of galaxies at a redshift of z = 0.49 [GCN6756] with an X-ray source (source S2 in [GCN6754]). In a latter observation, the second source no longer was detected, making this the probable position of the afterglow with a redshift of 0.49.

GRB 071227.

For this GRB, an optical afterglow was found [GCN7157] to coincide with the single source in the XRT error circle [GCN7151]. The redshift of z = 0.384 [GCN7152, GCN7154] is therefore reliable.

GRB 080121.

The only redshift reported for this GRB is z = 0.046 for two galaxies in the BAT error circle [GCN7210]. In this field, many other galaxies are present that might belong to a group of galaxies [GCN7210]. Since no XRT position could be determined [GCN7209], the redshift value is probable.

GRB 080520.

For this short GRB with a duration of 2.8 ± 0.7 s [GCN7761], an optical afterglow was found [GCN7753] for which a redshift of z = 1.545 was determined [GCN7757]. This is a reliable redshift estimation.

GRB 090417A.

This GRB with a duration of 0.072  ±  0.018 s [GCN9138], was found to be close to a low-redshift galaxy [GCN9134] with a redshift of 0.088 [GCN9136]. Since no optical afterglow has been found for this GRB, the redshift values are implausible.

GRB 090510.

For this short GRB with a duration of 0.3 ± 0.1 s [GCN9337] an optical afterglow has been found [GCN9338], which gives a reliable redshift of z = 0.903 [GCN9353].

Appendix B: Rate models

The rate functions used to fit the data according to Eq. (1), except for the trivial case of the “constant” rate, are given by:

• 1.

  The “sfr” rate that follows the star formation rate (model “SF2”) described in Porciani & Madau (2001) and Guetta & Piran (2005): (B.1)

• The “merger” rate of two compact objects was derived in Guetta & Piran (2005) by studying six observed double neutron stars (Champion et al. 2004). This rate follows a time-delay distribution ( ∝ 1 / τ) (B.2)

• The “delay” rate similar to the “merger” rate, but with a constant time-delay distribution (B.3)

Appendix C: Luminosity models

This section describes the luminosity functions used to fit the data according to Eq. (1).

• A single power-law distribution, which is often used to describethe pdf of luminosity in astrophysics (with two parameters:Φ₀ and α) (C.1)

• A broken power-law distribution, describing e.g. two underlying populations in the luminosity (Guetta & Piran 2005) (with four parameters: Φ₀, L₀, α and β)

• 3.

  The Schechter distribution, used for example in Andreon et al. (2006) (with three parameters: Φ₀, L₀ and α) (C.4)

• The log-normal distribution, describing a standard candle, e.g. a population with about the same luminosity (following (Chapman et al. 2008a), with three parameters: Φ₀, L₀, and σ) (C.5)

Appendix D: Fit results

© ESO, 2011