3.3. Interpretation of the Peak Flux Distribution
The counts distribution or the Peak flux distribution of the bursts observed by BATSE show a paucity of weak burst. A homogeneous count distribution, in an Eucleadian space should behave like: N(C) C-3/2, where N(C) is the number of bursts with more than C counts (or counts per second). The observed distribution is much flatter (see Fig. 10). This fact is reflected by the low <V / Vmax> value of the BATSE data: there are fewer distant sources than expected.
The observed distribution is compatible with a cosmological distribution of sources. A homogeneous cosmological distribution displays the observed trend - a paucity of weak bursts relative to the number expected in a Eucleadian distribution. In a cosmological population four factors combine to make distant bursts weaker and by this to reduce the rate of weak bursts: (i) K correction - the observed photons are red-shifted. As the photon number decreases with energy this reduces the count rate of distant bursts for a detector at a fixed energy range. (ii) The cosmological time dilation causes a decrease (by a factor 1 + z) in the rate of arrival of photons. For a detector, like BATSE, that measures the count rate within a given time window this reduces the detectability of distant bursts. (iii) The rate of distant bursts also decreases by a factor 1 + z and there are fewer distant bursts per unit of time (even if the rate at the comoving frames does not change). (iv) Finally, the distant volume element in a cosmological model is different than the corresponding volume element in a Eucleadian space. As could be expected, all these effects are significant only if the typical red-shift to the sources is of order unity or larger.
The statistics <V / Vmax> > is a weighted average of the distribution N( > f ). Already in 1992 Piran  compared the theoretical estimate of this statistics to the observed one and concluded that the typical redshift of the bursts observed by BATSE is zmax ~ 1. Later Fenimore et al.  compared the sensitivity of PVO (that observes N( > f ) f-3/2) with the sensitivity of BATSE and concluded that zmax(BATSE) ~ 1 (the maximal z from which bursts are detected by BATSE). This corresponds to a peak luminosity of ~ 1050 ergs/sec. Other calculations based on different statistical methods were performed by Horack & Emslie , Loredo & Wasserman [181, 182], Rutledge et al.  Cohen & Piran  and Mészáros and collaborators [187, 188, 189, 190] and others. In particular Loredo & Wassermann [181, 182] give an extensive discussion of the statistical methodology involved.
Consider a homogeneous cosmological distribution of sources with a peak luminosity L, that may vary from one source to another. It should be noted that only the luminosity per unit solid angle is accessible by these arguments. If there is significant beaming, as inferred , the distribution of total luminosity may be quite different. The sources are emitting bursts with a count spectrum: N() d = (L / h ) () d, where h, is the average energy. The observed peak (energy) flux in a fixed energy range, [Emin, Emax] from a source at a red-shift z is:
where dl(z) is the luminosity distance .
To estimate the number of bursts with a peak flux larger than f, N( > f ), we need the luminosity function, (L, z): the number of bursts per unit proper (comoving) volume per unit proper time with a given luminosity at a given red-shift. Using this function we can write:
where the red-shift, z(f, L), is obtained by inverting Eq. 3 and rp(z) is the proper distance to a red-shift z. For a given theoretical model and a given luminosity function we can calculate the theoretical distribution N(f ) and compare it with the observed one.
A common simple model assumes that (L, z) = (L) (z) - the luminosity does not change with time, but the rate of events per unit volume per unit proper time may change. In this case we have:
The emitted spectrum, N(), can be estimated from the observed data. The simplest shape is a single power law (Eq. 2). with = 1.5 or = 1.8 . More elaborate studies have used the Band et al.  spectrum or even a distribution of such spectra .
The cosmic evolution function (z) and the luminosity function (L) are unknown. To proceed one has to choose a functional shape for these functions and characterize it by a few parameters. Then using maximum likelihood, or some other technique, estimate these parameters from the data.
A simple characterization of (z) is:
Similarly the simplest characterization of the luminosity is as standard candles:
with a single parameter, L0, or equivalently zmax, the maximal z from which the source is detected (obtained by inverting Eq. 3 for f = fmin and L = L0).
There are two unknown cosmological parameters: the closure parameter, , and the cosmological constant . With the luminosity function given by Eqs. (5) and (6) we have three unknown parameters that determine the bursts' distribution: L0, 0, . We calculate the likelihood function over this five dimensional parameter space and find the range of acceptable models (those whose likelihood function is not less than 1% of the maximal likelihood). We then proceed to perform a KS (Kolmogorov-Smirnov) test to check whether the model with the maximal likelihood is an acceptable fit to the data.
The likelihood function is practically independent of in the range: 0.1 < < 1. It is also insensitive to the cosmological constant (in the range 0 < < 0.9, in units of the critical density). This simplifies the analysis as we are left only with the intrinsic parameters of the bursts' luminosity function.
There is an interplay between evolution (change in the bursts' rate) and luminosity. Fig. 9 depicts the likelihood function in the (zmax, ) plane for sources with a varying intrinsic rate. The banana shaped contour lines show that a population whose rate is independent of z ( = 0) is equivalent to a population with an increasing number of bursts with cosmological time ( > 0) with a lower L0 (lower zmax). This tendency saturates at high intrinsic evolution (large ), for which the limiting zmax does not go below .5 and at very high L0, for which the limiting does not decrease below -1.5. This interplay makes it difficult to constraint the red shift distribution of GRB using the peak flux distribution alone. For completeness we quote here "typical" results based on standard candles, no evolution and an Einstein-DeSitter cosmology .
Figure 9. The likelihood function (levels 33%, 10%, 3.3% 1% etc..) in the (, L) plane for standard candles = 1.5, O = 1, and evolution given by (z) = (1 + z)-. Superimposed on this map is the luminosity of GRB970508, (solid curve), GRB971214 (dashed curve) and GRB980703 (dahsed-dotted curve). We have used h75 = 1. From .
Recall that <V / Vmax> of the short bursts distribution is rather close to the homogeneous Eucleadian value of 0.5. This means that when analyzing the peak flux distribution one should analyze separately the long and the short bursts . For long bursts (bursts with t90 > 2 sec) the likelihood function peaks at zmax = 2.1 (see Fig. 10) . The allowed range at a 1% confidence level is: 1.4 < zmax < 3.1 (zmax(=2) = 1.5(+.7)(-.4) for = 2). The maximal red-shift, zmax = 2.1(+1.1)(-0.7), corresponds, with an estimated BATSE detection efficiency of 0.3, to 2.3(+1.1)(-0.7) . 10-6 events per galaxy per year (for a galaxy density of 10-2 h3 Mpc-3; ). The rate per galaxy is independent of H0 and is only weakly dependent on . For = 1 and = 0 the typical energy of a burst with an observed fluence, F, is 7(+11)(-4) . 1050(F / 10-7 ergs / cm2) ergs. The distance to the sources decreases and correspondingly the rate increases and the energy decreases if the spectral index is 2 and not 1.5. These numbers vary slightly if the bursts have a wide luminosity function.
Figure 10. The observed long burst peak flux distribution and three theoretical cosmological distributions with = 1, = 0, = - 1.5, standard candles and no source evolution: L = 3.4 . 1050 ergs/sec (solid line: best fit), L = 7.2 . 1050 ergs/sec (dashed line: lower 1% bound), L = 1.4 . 1050 ergs/sec (dashed-dotted line: upper 1% bound).
Short bursts are detected only up to a much nearer distances: zmax(short) = 0.4+1.1, again assuming standard candles and no source evolution. There is no significant lower limit on zmax for short bursts and their distribution is compatible with a homogeneous non-cosmological one. The estimate of zmax(short) corresponds to a comparable rate of 6.3(-5.6) . 10-6 events per year per galaxy and a typical energy of 3(+39) . 1049 F-7 ergs (there are no lower limits on the energy or and no upper limit on the rate since there is no lower limit on zmax(short)). The fact that short bursts are detected only at nearer distances is also reflected by the higher <V / Vmax> of the population of these bursts .
Relatively wide luminosity distributions are allowed by the data . For example, the KS test gives a probability of 80% for a double peaked luminosity distribution with luminosity ratio of 14. These results demonstrate that the BATSE data alone allow a variability of one order of magnitude in the luminosity.
The above considerations should be modified if the rate of GRBs trace the SFR - the star formation rate [193, 194, 195]. The SFR has been determined recently by two independent studies [196, 197, 198]. The SFR peaks at z ~ 1.25. This is a strongly evolving non monotonic distribution. which is drastically different from the power laws considered so far. Sahu et al.  find that (z) SFR(z) yields N( > f ) distribution that is compatible with the observed one (for q0 = 0.2, H0 = 50 km/sec-1 Mpc-1) for a narrow luminosity distribution with L = 1051 ergs/sec. Wijers et al.  find that the implied peak luminosity is higher L = 8.3 . 1051 ergs/sec and it corresponds to a situation in which the dimmest bursts observed by BATSE originate from z 6!
The direct red-shift measure of GRB970508  agrees well with estimates made previously using peak-flux count statistics ([200, 182, 183]). The red-shift of GRB971214, z = 3.418, and of GRB980703, z = 0.966, and the implied luminosities disagree with these estimates. A future detection of additional red-shifts for other bursts will enable us to estimate directly the luminosity function of GRBs. It will also enable us to determine the evolution of GRBs. Krumholz et al.  and Hogg & Fruchter  find that with a wide luminosity function both models of a constant GRB rate and a GRB rate following the star formation rate are consistent with the peak flux distribution and with the observed redshift of the three GRBs.