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 Physics , 1998, DOI: 10.1086/306978 Abstract: With the discovery that the afterglows after some bursts are coincident with faint galaxies, the search for host galaxies is no longer a test of whether bursts are cosmological, but rather a test of particular cosmological models. The methodology we developed to investigate the original "no host galaxy" problem is equally valid for testing different cosmological models, and is applicable to the galaxies coincident with optical transients. We apply this methodology to a family of models where we vary the total energy of standard candle bursts. We find that total isotropic energies of E<2e52~erg are ruled out while log(E)~53 erg is favored.
 Physics , 1997, DOI: 10.1086/304271 Abstract: Two models of the gamma ray burst population, one with a standard candle luminosity and one with a power law luminosity distribution, are chi^2-fitted to the union of two data sets: the differential number versus peak flux distribution of BATSE's long duration bursts, and the time dilation and energy shifting versus peak flux information of pulse duration time dilation factors, interpulse duration time dilation factors, and peak energy shifting factors. The differential peak flux distribution is corrected for threshold effects at low peak fluxes and at short burst durations, and the pulse duration time dilation factors are also corrected for energy stretching and similar effects. Within an Einstein-de Sitter cosmology, we place strong bounds on the evolution of the bursts, and these bounds are incompatible with a homogeneous population, assuming a power law spectrum and no luminosity evolution. Additionally, under the implied conditions of moderate evolution, the 90% width of the observed luminosity distribution is shown to be < 10^2, which is less constrained than others have demonstrated it to be assuming no evolution. Finally, redshift considerations indicate that if the redshifts of BATSE's faintest bursts are to be compatible with that which is currently known for galaxies, a standard candle luminosity is unacceptable, and in the case of the power law luminosity distribution, a mean luminosity < 10^57 ph s^-1 is favored.
 Physics , 1998, Abstract: The quoted intrinsic luminosity of objects has a dependency on the assumed cosmology, and the (less often considered) assumed gravitational lensing due to the matter content of the viewing beam; the so called filled' or empty' beam cases. We consider the implications of filled vs empty beam assumptions for the derived total luminosities of recent gamma-ray bursts (GRBs) in which counterparts at cosmological distances have been confirmed. Conversion factors between filled and empty beam bolometric luminosities are presented graphically for a range of cosmologies and redshifts. The tendency for sources to be most probably demagnified further supports the need for non-isotropic GRB emission in neutron star merger models, or the need for at least one massive star in other models. In the most extreme case the true energy of GRB971214 at z=3.418 could be as high as ~ 8 x 10^53 ergs, a factor ~ 2 more than previously estimated. Similarly, the effect may account for some of the observed bias towards fainter host magnitudes.
 Physics , 2001, DOI: 10.1086/342123 Abstract: Within the framework of the internal shock model, we study the luminosity and the variability in gamma-ray bursts from collimated fireballs. In particular we pay attention to the role of the photosphere due to $e^\pm$ pairs produced by internal shock synchrotron photons. It is shown that the observed Cepheid-like relationship between the luminosity and the variability can be interpreted as a correlation between the opening angle of the fireball jet and the mass included at the explosion with a standard energy output. We also show that such a correlation can be a natural consequence of the collapsar model. Using a multiple-shell model, we numerically calculate the temporal profiles of gamma-ray bursts. Collimated jets, in which the typical Lorentz factors are higher than in wide jets, can produce more variable temporal profiles due to smaller angular spreading time scales at the photosphere radius. Our simulations quantitatively reproduce the observed correlation.
 Physics , 2001, DOI: 10.1086/339317 Abstract: The source population of gamma-ray bursts (GRBs) declines towards the present epoch being consistent with the measured decline of the star formation rate. We show this using the brightness distribution of 3255 long BATSE GRBs found in an off-line scan of the BATSE continuous 1.024 s count rate records. The significance of this conclusion is enhanced by the detection of three GRBs with known redshifts brighter than 10^{52} erg/s during the last two years. This is an argument in favor of the generally believed idea that GRBs are strongly correlated with the star production, at least on cosmological time scales, and favors the association of long GRBs with collapses of supermassive stars. However, we still cannot rule out neutron star mergers if the typical delay time for binary system evolution is relatively short. If we assume a steep decline of the GRB population at z>1.5, then their luminosity function can be clearly outlined. The luminosity function is close to a power law, dN/dL ~ L^{-1.4}, for low luminosities over at least 1.7 orders of magnitude. Then the luminosity function breaks to a steeper slope or to an exponential decline around L = 3*10^{51} erg/s in the 50-300 keV range assuming isotropic emission.
 Physics , 2015, Abstract: We derive, adopting a direct method, the luminosity function and the formation rate of long Gamma Ray Bursts through a complete, flux-limited, sample of Swift bursts which has a high level of completeness in redshift z (~82%). We parametrise the redshift evolution of the GRB luminosity as L = L_0(1+ z)^k and we derive k = 2.5, consistently with recent estimates. The de-evolved luminosity function of GRBs can be represented by a broken power law with slopes a = -1.32 +- 0.21 and b = -1.84 +- 0.24 below and above, respectively, a characteristic break luminosity L_0,b = 10^51.45+-0.15 erg/s. Under the hypothesis of luminosity evolution we find that the GRB formation rate increases with redshift up to z~2, where it peaks, and then decreases in agreement with the shape of the cosmic star formation rate. We test the direct method through numerical simulations and we show that if it is applied to incomplete (both in redshift and/or flux) GRB samples it can misleadingly result in an excess of the GRB formation rate at low redshifts.
 Physics , 1999, DOI: 10.1086/308847 Abstract: Gamma-ray bursts are believed to be some catastrophic event in which material is ejected at a relativistic velocity, and internal collisions within this ejecta produce the observed $\gamma$-ray flash. The angular size of a causally connected region within a relativistic flow is of the order the angular width of the relativistic beaming, $\gamma^{-1}$. Thus, different observers along different lines of sights could see drastically different fluxes from the same burst. Specifically, we propose that the most energetic bursts correspond to exceptionally bright spots along the line of sight on colliding shells, and do not represent much larger energy release in the explosion. We calculate the distribution function of the observed fluence for random angular-fluctuation of ejecta. We find that the width of the distribution function for the observed fluence is about two orders of magnitude if the number of shells ejected along different lines of sight is ten or less. The distribution function becomes narrower if number of shells along typical lines of sight increases. The analysis of the $\gamma$-ray fluence and afterglow emissions for GRBs with known redshifts provides support for our model i.e. the large width of GRB luminosity function is not due to a large spread in the energy release but instead is due to large angular fluctuations in ejected material. We outline several observational tests of this model. In particular, we predict little correlation between the $\gamma$-ray fluence and the afterglow emission as in fact is observed. We predict that the early (minutes to hours) afterglow would depict large temporal fluctuations whose amplitude decreases with time. Finally we predict that there should be many weak bursts with about average afterglow luminosity in this scenario.
 Physics , 1997, DOI: 10.1086/305768 Abstract: It is often stated that gamma-ray bursts (GRBs) have typical energies of several hundreds $\keV$, where the typical energy may be characterized by the hardness H, the photon energy corresponding to the peak of $\nu F_{\nu}$. Among the 54 BATSE bursts analyzed by Band et al. (1993), and 136 analyzed by us, more then 60% have 50 keV < H < 300 keV. Is the narrow range of H a real feature of GRBs or is it due to an observational difficulty to detect harder bursts? We consider a population of standard candle bursts with a hardness distribution: rho(H) d log H \propto H^gamma d log H and no luminosity - hardness correlation. We model the detection algorithm of BATSE as a function of H, including cosmological effects, detector characteristics and triggering procedure, and we calculate the expected distribution of H in the observed sample for various values of gamma. Both samples shows a paucity of soft (X-ray) bursts, which may be real. However, we find that the observed samples are consistent with a distribution above H=120 keV with gamma \sim -0.5 (a slowly decreasing numbers of GRBs per decade of hardness). Thus, we suggest that a large population of unobserved hard gamma-ray bursts may exist.
 Physics , 2014, DOI: 10.1093/mnras/stu2482 Abstract: The structure of Gamma Ray Burst (GRB) jets impacts on their prompt and afterglow emission properties. The jet of GRBs could be uniform, with constant energy per unit solid angle within the jet aperture, or it could instead be structured, namely with energy and velocity that depend on the angular distance from the axis of the jet. We try to get some insight about the still unknown structure of GRBs by studying their luminosity function. We show that low (1e46-1e48 erg/s) and high (i.e. with L > 1e50 erg/s) luminosity GRBs can be described by a unique luminosity function, which is also consistent with current lower limits in the intermediate luminosity range (1e48-1e50} erg/s). We derive analytical expressions for the luminosity function of GRBs in uniform and structured jet models and compare them with the data. Uniform jets can reproduce the entire luminosity function with reasonable values of the free parameters. A structured jet can also fit adequately the current data, provided that the energy within the jet is relatively strongly structured, i.e. E propto theta^{-k} with k > 4. The classical E propto theta^{-2} structured jet model is excluded by the current data.
 Maarten Schmidt Physics , 2001, DOI: 10.1086/320450 Abstract: We show that the Euclidean value of for gamma-ray bursts (GRB) selected on a timescale of 1024 ms is correlated with spectral hardness. The value of ranges from 0.42 for soft bursts to 0.26 for the hardest bursts. Given that the Euclidean value of for cosmological objects in a well defined sample is a distance indicator, the hard bursts must reside at larger redshifts and therefore be more luminous than the soft bursts. The resulting luminosity-hardness correlation cannot be shown explicitly due to the small number of observed GRB redshifts at the present time. Based on the -hardness correlation, we derive the luminosity function of GRBs without using any redshifts, but we have to make an assumption how the comoving GRB rate varies with redshift. We present luminosity functions for three models of the GRB rate as a function of redshift, based on star formation rates. The peak luminosity functions are approximately broken power laws with an isotropic-equivalent break luminosity of around 10^51.5 erg s^-1 in the 50-300 keV range and total local rate densities of 0.5 Gpc^-3 y^-1. Predicted GRB counts as a function of flux and redshift are presented. Based on the GRB luminosity function, we carry out a simulation to produce the luminosity-hardness correlation, which shows that the hardest GRBs are about 20 times more luminous than the softest ones.
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