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the origin of grb is one of the most fascinating outstanding problems in astronomy .
their observed energy injection in the universe is sufficiently large to possibly resolve another long - standing puzzle : the origin of the highest energy cosmic rays@xcite .
mounting evidence suggests that grb emission is produced by a relativistically expanding fireball , energized by a process involving neutron stars or black holes@xcite . in the early stages the fireball , its radiation trapped by the very large optical depth , can not emit photons efficiently .
the fireball s kinetic energy is therefore dissipated until it becomes optically thin  a scenario that can explain the observed energy and time scales of grbs , provided the bulk lorentz factor of the expanding flow , @xmath0 , is @xmath1 .
protons accelerated in shocks in the expanding fireball interact with photons to produce charged pions , the parents of high - energy neutrinos @xcite . assuming that particles accelerated in the grb sources produce the observed cosmic rays above the  ankle " of the energy spectrum near @xmath2  ev
, one derives that the average single burst produces only @xmath3 neutrino events in a high energy neutrino telescope with @xmath4 effective area .
although the expected rate is therefore low , the neutrino signal should be relatively easy to observe provided the detector is large enough : grb neutrinos will have a hard spectrum extending well beyond the background from atmospheric neutrinos and , even more important , the high - energy grb neutrino events should coincide with observed grb photon events within a narrow time window .    in this paper
we calculate the experimental signatures of grb in a kilometer - scale neutrino detector such as the proposed icecube @xcite .
we emphasize the importance of taking into account burst - to - burst fluctuations @xcite as well as absorption of the neutrino signal in the earth for both event rates and experimental signatures .
both effects produce additional and striking signatures with discriminating sensitivity to the value of the bulk lorentz factor whose value is only indirectly inferred from other astronomical observations @xcite .
the observation of grb neutrinos over a cosmological baseline has scientific potential beyond testing the  best - buy " fireball model : the observations can test with unmatched precision special relativity and the equivalence principle , and study oscillating neutrino flavors over the ultimate baseline of @xmath5 @xcite .
in calculating the event rates and experimental signatures of grb neutrinos in a high energy neutrino telescope we follow the model of waxman and bahcall @xcite as implemented by halzen and hooper @xcite .
we have normalized the neutrino flux to the energy rate injected in the universe needed to explain the observed cosmic ray ( cr ) spectrum above @xmath6 ev , @xmath7 .
this energy rate was calculated in reference @xcite , assuming a cosmological distribution of sources and taking into account cr propagation in the cosmic microwave background radiation ( cmbr ) .
the value quoted above corresponds to the `` low redshift '' ( @xmath8 ) energy generation rate of crs .
note that , because of the absorption of highest energy cosmic rays by cmbr photons , one could further increase @xmath9 without directly affecting their observed flux .
waxman and bahcall @xcite have calculated an upper limit to the cosmic rate production rate in the whole universe , assuming the energy generation rate evolves rapidly with redshift following the luminosity density evolution of quasi - stellar sources .
they obtain an upper limit which is @xmath10 . it is interesting however to mention that @xmath9 is comparable to that produced in @xmath11-rays by cosmological grbs ( which are not expected to be absorbed by the intervening backgrounds since the typical photon energy is below 1 mev ) . assuming the efficiency with which electrons ( which ultimately produce the observed photons by synchrotron radiation ) and protons is the same inside the grb fireball
, the value of @xmath9 quoted above might well be closer to the actual value .
for this , and other reasons , it is anyway unlikely that our calculations are accurate to better than a factor 3 or so .
moreover our neutrino event rate calculation might be conservative in this respect .
the neutrino flux is given by    @xmath12    where @xmath13 is a normalization constant that is determined from energy considerations as explained above and in reference @xcite .
the observed neutrino rates are calculated by folding this generic flux over the distributions of individual bursts in distance , @xmath14 , energy , @xmath15 and boost factor , @xmath16 : @xmath17 where @xmath18 is the energy emitted by a particular grb , @xmath19 its redshift and @xmath0 the boost factor .
@xmath20 represents the efficiency of detecting a neutrino of energy @xmath21 .
the first two distributions can be modelled after observations : a cosmological distribution in distance , and an energy distribution which assumes that ten percent of grb produce more energy than average by a factor of ten , and one percent by a factor of 100@xcite .
most important however are the fluctuations in the @xmath0 factor around its average phenomenological value of @xmath22@xmath23 .
the fluctuations in @xmath0 affect the efficiency for producing pions in the @xmath24 collisions in the fireball as @xmath25 @xcite , as well as the break energy , @xmath26 , which varies as @xmath27 .
unfortunately the distribution in @xmath0 can not even be guessed at . nevertheless , it is critical in making quantitative predictions@xcite .
the physics is clear . in grbs ,
high luminosities are emitted over short times , therefore the large photon density would render grb opaque unless @xmath0 is very large .
only transparent sources with large boost factors emit photons .
they are however relatively weak neutrino sources because the actual photon target density in the fireball is diluted by the large lorentz factor .
an even moderately reduced value of @xmath0 will produce a prolific neutrino source .
we remind the reader that the results obtained from eq .
[ eq : folding ] are at variance with the neutrino rate obtained by multiplying the average rate per burst by the number of bursts . even neglecting the all important fluctuations in @xmath0 , there is no such concept as an average grb .
e.g.for @xmath28 , the correct computation of eq .
[ eq : folding ] yields a rate of @xmath29 events per km@xmath30 and year , roughly an order of magnitude larger than the prediction obtained by neglecting the observed burst - to - burst fluctuations in distance and energy .
another consequence of fluctuations is that the signal is dominated by a few very bright bursts , which greatly simplifies their detection .
as pointed out above , the average neutrino energy varies with the square of boost factor and therefore the calculated event rates , especially their dependence on @xmath0 , is strongly affected by the fact that higher energy neutrinos are preferentially absorbed in the earth before reaching the detector@xcite .
also this effect has been neglected in all previous calculations .
for instance , the 75 events just mentioned are reduced by a factor 3 by absorption .
one should on the other hand remember that by oscillations , a large fraction of @xmath31 can oscillate into @xmath32 which penetrate the earth @xcite .
it is in this context important to realize that a kilometer - scale detector such as icecube can measure the energy of the neutrinos .
therefore , signal events can be separated from the low energy atmospheric background by energy measurement , which allows the identifications of neutrinos from all directions and not just in the hemisphere where they pass through the earth .
we calculate the flux of neutrinos from grb in the fireball model following reference  @xcite .
the number of protons in the fireball is fixed by the assumption that they are the source of the ultra high energy cosmic rays above @xmath6 ev .
the results are shown in figure [ grbfluxes ] which shows the @xmath33 flux from grbs for different values of @xmath0 .
the fluxes have been multiplied by @xmath34 so that they represent the energy emitted in the form of neutrinos .
notice the variation of the break in the spectrum @xmath26 as @xmath35 .
also , for values of @xmath0 below @xmath36 100 the fireball becomes opaque to protons and at this point the total amount of energy available for neutrino production is converted .
the neutrino flux , which roughly scales as @xmath25 , saturates and no longer grows with decreasing boost factor ; see figure [ grbfluxes ] .    as usual @xcite
, we calculate the event rate per grb by folding the neutrino flux with the probability of detecting a muon produced in a @xmath37 or @xmath38 interaction .
fluctuations in distance @xmath19 and total grb energy , or fluency , @xmath18 are accounted for by a monte carlo simulation we have developed for this purpose .
we simulate a large number of grbs at different zenith angles assuming an isotropic distribution and subsequently obtain the event rate per year by assuming 1000 grbs / year .
absorption of the neutrinos in the earth prior to reaching the detector is taken into account using the density profile of reference @xcite .
it is also important to implement the fact that above the detector there is a limited column density of atmosphere and ice available for neutrino detection .    in table
[ numuevents ] we separately show , for different @xmath0 s , the event rate of upgoing , i.e. neutrinos that cross the earth before interacting near the detector , and downgoing neutrinos .
the first two columns show for comparison the number of events when neither absorption nor the limited amount of target above the detector are taken into account .
it is clear that both effects play an important role in obtaining the correct event rate .
this is not surprising : the earth becomes opaque to neutrinos of energy around 100 tev , and the muon range exceeds the @xmath36 2 km vertical depth of the icecube detector at energies around @xmath39 tev .
.@xmath33 events @xmath40
. only fluctuations in distance and energy are taken into account .
[ numuevents ] [ cols="^,^,^,^,^ " , ]
this research was supported in part by the us department of energy under grant de - fg02 - 95er40896 and in part by the university of wisconsin research committee with funds granted by the wisconsin alumni research foundation .
thanks the department of physics , university of wisconsin , madison and the fundacin caixa galicia for financial support .    99 e. waxman and j. n. bahcall , phys .
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following waxman and bahcall @xcite we calculate the event rate , energy and zenith angle dependence of neutrinos produced in the fireball model of gamma ray bursts ( grb ) . 
 we emphasize the primary importance of i ) burst - to - burst fluctuations and ii ) absorption of the neutrinos in the earth . from the astronomical point of view , we draw attention to the sensitivity of neutrino measurements to the boost lorentz factor of the fireball @xmath0 , which is central to the fireball model , and only indirectly determined by follow - up observations . 
 fluctuations result in single bursts emitting multiple neutrinos , making it possible to determine the flavor composition of a beam observed after a baseline of thousands of megaparsecs .    6.5 in    * pacs number(s ) 
 : * 96.40.tv , 98.70.rz , 98.70.sa    * keywords : * gamma ray bursts , neutrino detection