1. INTRODUCTION:

THE CASE FOR/AGAINST THE DISCRETE/DIFFUSE APPROACH

    All AMANDA gamma-ray burst (GRB) analyses to date (Bay 2000, Hardtke et al. 2003, Kuehn 2004, Taboada 2005 & Hughey 2005) have featured a coincident neutrino signal assumption that is based upon a diffuse ensemble of bursts, described by averaged electromagnetic parameters, and integrated over cosmological (Hubble) time (Waxman & Bahcall 1997, Waxman & Bahcall 1999). Despite the following facts:
 

1. The theoretical underpinnings [hadronic acceleration in the context of fireball phenomenology (Piran 2004c)] of the expected neutrino energy flux directly map the signal neutrino spectrum to the intrinsic electromagnetic observables associated with prompt and afterglow GRB emission,

2. over three decades of ground-based and satellite observations have demonstrated that the experimental parameter space of each GRB observable is described by a distribution, often spanning orders of magnitude, which vary for individual (discrete) GRBs both within and among different burst classes (i.e. short, long, non-triggered, etc.),

3. neutrino telescopes make observations of individual (discrete) GRBs,

a single diffuse formulation has been used in order to model the neutrino energy spectra of leptons in spatial and temporal coincidence with the prompt electromagnetic emission from GRBs detected via the Burst and Transient Source Experiment (BATSE) and the International Planetary Network (IPN3). This methodology explicitly demands that each individual GRB is described by a single neutrino energy spectrum, whose construction is implicitly a consequence of invoking average values for each observable, which in many cases is in direct contradiction with the mean of a given observed distribution. The inescapable conclusion is that the assumptions associated with diffuse formulations, when applied to discrete observations are

1. intrinsically inconsistent within the theoretical framework and

2. are clearly falsified when confronted with observational reality.

A solution to this rather straight forward problem has been the focus of my thesis and work in AMANDA. Although scientifically transparent, the challenge against the application of a diffuse formulation upon discrete GRBs has been met with much unnecessary controversy.

    It has been argued in the past (Halzen & Hooper 1999, Muniz et al. 2000), that fluctuations in burst parameters may enhance the expected neutrino event rate, by several orders of magnitude, within the context of hadronic acceleration in the fireball phenomenology. The need for a discrete GRB analysis was initially addressed during my first GRB presentation (Stamatikos 2003) at the AMANDA Collaboration meeting at Laguna Beach. Ironically, the said talk was given within a couple of hours of the triggering of the High Energy Transient Experiment (HETE-II, burst number 2652) on March 29, 2003, i.e. GRB030329, which is the burst featured in this case study. A prescription for analyzing individual BATSE GRBs, utilizing the unique set of parameters gleaned from the observation of individual GRBs, was given a solid theoretical framework in the literature (Guetta et al. 2004, Muniz et al. 2004). Due to the overwhelming evidence against using an averaged diffuse formulation for the analysis of individual bursts, a new discrete GRB analysis was formerly announced at the AMANDA Collaboration Meeting in Madison (Stamatikos 2003b). The distributions of GRB electromagnetic observables for various burst classes, the methods used to fit the prompt g-ray photon energy spectra (Band et al. 1993) and the methods used to estimate the redshift (Fenimore & Ramirez-Ruiz 2000, Band et al. 2004, Yonetoku et al. 2004), were all presented at the AMANDA/IceCube Collaboration Meeting in Uppsala, Sweden (Stamatikos 2004c). Currently, these same methods are being applied to a subset of the bursts from the BASTE catalog for the 1997-2000 data sets (Stamatikos et al. 2004). Recently, an internal report has been released which further illustrates the discrepancy between summed signal spectra based upon average/diffuse and individual/discrete formulations (Becker et al. 2005).

    This analysis demonstrates the effects of burst parameter fluctuations in the context of AMANDA-II observations for GRB030329. The goal was to provide a very straightforward example in which the predictions of a diffuse and discrete spectrum could be compared and contrasted in the context of AMANDA detector response. Regarding GRB030329, the application of the fireball description has been well-motivated, all of the required electromagnetic observables have either been measured or calculated, and for the first time, we have direct evidence for a progenitor. Hence this is the perfect specimen for a case study. A similar case study, for IceCube has been performed elsewhere (Razzaque et al. 2004).

    This work represents a more detailed follow-up to the preliminary analysis presented at the IceCube Collaboration Meeting in Berkeley (Stamatikos 2005). At LBNL, it was demonstrated that the average burst parameterization, which is the foundation of the diffuse formulation, produces an expected muon neutrino energy spectrum that is discrepant with the discrete spectrum constructed using the electromagnetic observables gleaned from the prompt gamma-ray and afterglow emissions. In many cases, the differences in flux normalization, spectral indices, and break energies exceeded an order of magnitude and can not be reconciled within the errors or propagated errors of measurement. This represents the first unequivocal demonstration that discrete spectra yield very real differences in the observables of a neutrino telescope. A fact which can be leveraged to more accurately interpret a null detection in an astrophysical context. In light of these facts, it is clear that current diffuse searches are observationally misleading since

1. they do not describe the bursts observed by AMANDA and

2. any detectable signal would most likely come from a rare exceptional burst rather than a summation of a hundreds of average construction.

This can be directly seen in figure 1.1 below, which illustrates the following two important motivations for discrete modeling:

1. the number of neutrinos expected from a given GRB spans five orders of magnitude and

2. only a small fraction of bursts are expected to produce a detectable neutrinos.

A re-evaluation of the analysis methods currently used is compulsory, with implications for future searches with IceCube.

FIGURE 1.1: Distribution of muon neutrino events from GRBs in a km-scale detector (Guetta et al. 2004). Represented are the 568 bright (peak flux  ≥ 1.5 photons/(cm² s) long duration (T₉₀ ≥ 2 seconds) GRBs from the current BASTE catalog, modeled via a broken power law fit to the prompt photon energy spectrum and redshifts estimated via the variability-luminosity method (Fenimore & Ramirez-Ruiz 2000). The solid line indicates a model where the proton efficiency, the fraction of the proton's energy imparted to the pion during photomeson reactions, is calculated on an individual basis while the dashed line represents a model where the proton efficiency has been fixed at 20%.