4 GRB distance scale and rate

The breakthrough made possible by the discovery that GRBs have X-ray (Costa et al. 1997), optical (Galama et al. 1997) and radio (Frail et al. 1997) afterglows cannot be overstated. The discovery by Metzger et al. (1997a) of redshifted absorption lines at z = 0.83 in the optical spectrum of the GRB 970508 afterglow showed that most, perhaps all, GRB sources lie at cosmological distances. Yet we must remember that GRB 970508 remains the only GRB whose distance we have measured directly. The current situation is summarized below, in order of increasing uncertainty in the redshift determination.

In the cases of two other bursts, GRB 980703 (Bloom et al. 1999) and GRB 971214 (Kulkarni et al. 1998; Kulkarni 1999), we infer the redshifts (z = 0.96 and 3.42) of the bursts from the redshift of a galaxy coincident with the burst afterglow (and therefore likely to be the host galaxy - but recall my earlier comments).

In the case of a fourth burst, GRB 980329, a redshift $z \approx 5$ was inferred by attributing the precipitous drop in the flux of the optical afterglow between the I- and R-bands to the Ly${\alpha}$ forest (Fruchter 1999; Lamb et al. 1999). However, Djorgovski et al. (1999) recently reported that this burst must lie at a redshift z < 3.9, based on the absence of any break longward of 6000 Å in the spectrum of the host galaxy.

In the case of GRB 980703 (Piro 1999) and of a fifth burst, GRB 980828 (Yoshida 1999), there are hints of an emission-like feature in the X-ray spectrum, which, if interpreted as a redshifted Fe K-shell emission line, would provide redshift distances for these bursts. However, substantial caution is in order because the statistical significance of these features is slight. Indeed, all three "indirect'' means of establishing the redshift distances of GRBs need verification by cross-checking the redshift distances found using these methods against those measured directly using redshifted absorption lines in the optical spectra of their afterglows. One arcminute or better angular positions in near real-time, like those that HETE-2 will provide (Ricker et al. 1999; Kawai et al. 1999), will greatly facilitate this task.

The table below summarizes the current situation, in order of increasing uncertainty in the redshift determination:

$$\begin{array} {p{0.85\linewidth}r} Redshifts of Afterglows &... ...s in X-Ray Afterglows (??) & 0 \\ & -- \\ Total & 3.\end{array}$$

Even with the paucity of GRB redshift distances currently known, and the uncertainties about these distances, it is striking how our estimate of the GRB distance scale continues to increase. Not so long ago, adherents of the cosmological hypothesis for GRBs favored a redshift range , derived primarily from the brightness distribution of the bursts under the assumption that GRBs are standard candles. (Of course, adherents of the galactic hypothesis argued for much smaller redshifts!). Now we routinely talk about redshift distances in the range , and such a redshift range is supported by the three burst redshifts that have been determined so far.

Much of the motivation for considering such a redshift range for GRBs comes from the appealing hypothesis that the GRB rate is proportional to the star-formation rate (SFR) in the universe, an hypothesis that arose partly in response to the accumulating evidence, described earlier, that GRBs occur in star-forming galaxies, and possibly near or in the star-forming regions themselves. How far have we been able to go in testing this hypothesis? The answer: Not very far. First of all, as Madau (1999) discussed at this meeting, our knowledge of the SFR as a function of redshift is itself as yet poorly known. The few points derived from the relatively small Hubble Deep Field may not be characteristic of the SFR in the universe at large, not to mention concerns about star-forming galaxies at high redshift whose light might be extinguished by dust in the star-forming galaxies themselves, as well as uncertainties in the epoch and magnitude of star formation in elliptical galaxies. Second, we have redshift determinations for only three GRBs and R-band magnitudes for only eight GRBs. Much further work establishing the star formation rate as a function of redshift in the universe, as well as the redshift distances for many more GRBs, will be needed before this hypothesis can really be tested.

One thing is now clear: GRBs are a powerful probe of the high-z universe. GRB 971214 would still be detected by BATSE and would be detected by HETE-2 at a redshift distance $z \approx 10$, and it would be detected by Swift (whose sensitivity threshold is a factor of 5 below that of BATSE and HETE-2) at $z \approx 20$! If GRBs are produced by the collapse of massive stars in binaries, one expects them to occur out to redshifts of at least $z \approx 10 - 12$, the redshifts at which the first massive stars are thought to have formed, which are far larger than the redshifts expected for the most distant quasars. The occurrence of GRBs at these redshifts may give us our first information about the earliest generation of stars; the distribution of absorption-line systems in the spectra of their infrared afterglow spectra will give us information about both the growth of metallicity at early epochs and the large-scale structure of the universe, and the presence or absence of the Lyman-${\alpha}$ forest in the infrared afterglow spectra will place constraints on the Gunn-Peterson effect and may give us information about the epoch at which the universe was re-ionized (Lamb & Reichart 1999a).

The increase in the GRB distance scale also implies that the GRB phenomenon is much rarer than was previously thought. This implication has been noted at this meeting by Schmidt (1999), who finds that the GRB rate must be

$$R_{\rm GRB} \sim 10^{-11}\ {\rm GRBs}\ {\rm yr}^{-1}\ {\rm Mpc}^{-3}$$ (1)

in order both to match the brightness distribution of the bursts and to accommodate the redshift distance of z = 3.42 inferred for GRB 971214.

By comparison, the rates of neutron star-neutron star ($\rm NS-NS$) binary mergers (Totani 1999) and the rate of Type Ib-Ic supernovae (Cappellaro et al. 1997) are

$$R_{\rm NS-NS} \sim 10^{-6}\ {\rm mergers}\ {\rm yr}^{-1}\ {\rm Mpc}^{-3}$$ (2)

$$R_{\rm Type\ Ib-Ic} \sim 3 \ 10^{-5}\ {\rm SNe}\ {\rm yr}^{-1}\ {\rm Mpc}^{-3}.$$ (3)

The rate of neutron star-black hole ($\rm NS-BH$) binary mergers will be smaller. Nevertheless, it is clear that, if either of these events are the sources of GRBs, only a tiny fraction of them produce an observable GRB. Even if one posits strong beaming (i.e., $f_{\rm beam} \approx 10^{-2}$; see below), the fraction is small:

$$R_{\rm GRB}/R_{\rm NS-NS} \sim 10^{-3}\ (f_{\rm beam}/10^{-2})^{-1}$$ (4)

$$R_{\rm GRB}/R_{\rm Type\ Ib/c} \sim 3 \ 10^{-5}\ (f_{\rm beam}/10^{-2})^{-1}.$$ (5)

Therefore, if such events are the sources of GRBs, either beaming must be incredibly strong ($f_{\rm beam} \sim 10^{-5} - 10^{-3}$) or only rarely are the physical conditions necessary to produce a GRB satisfied. Can any theoretical astrophysicist be expected to explain such incredible beaming, or alternatively, such a non-robust, "flaky'' phenomenon? I have a solution - at least in the case of SNe: We theorists merely need define those supernovae that produce GRBs to be a new class of SNe (Type I$_{\rm grb}$ SNe), and then challenge the observers to go out and find the other observational criteria that define this class!