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since gamma - ray bursts ( grbs ) were first discovered [ 1 ] , thousands of bursts have been detected by a wide variety of instruments , most notably , the burst and transient source experiment ( batse ) on the _ compton gamma - ray observatory ( cgro ) _ , which detected 2704 bursts by the end of _ cgro _ s more than 9 year mission in 2000 june ( see , e.g. , [ 2 ] ) .
however , the distance scale of the bursts remained uncertain until 1997 , when bepposax began localizing long - duration bursts to a few arcminutes on the sky , and distributing the locations to observers within hours of the bursts .
this led to the discovery of x - ray [ 3 ] , optical [ 4 ] , and radio [ 5 ] afterglows , as well as host galaxies [ 6 ] .
subsequent observations led to the spectroscopic determination of burst redshifts , using absorption lines in the spectra of the afterglows ( see , e.g. , [ 7 ] ) , and emission lines in the spectra of the host galaxies ( see , e.g. , [ 8 ] ) . to date , redshifts have been measured for 13 bursts .
recently , [ 9 ] ( see also [ 10 ] ) , [ 11 ] ( see also [ 12 ] ) , and [ 13 ] ( see also [ 14 ] ) have proposed trends between burst luminosity and quantities that can be measured directly from burst light curves , for the long - duration bursts
. using 1310 batse bursts for which peak fluxes and high resolution light curves were available , [ 9 ] have suggested that simple bursts ( bursts dominated by a single , smooth pulse ) are less luminous than complex bursts ( bursts consisting of overlapping pulses ) ; however , see [ 15 ] . using a sample of 7 batse bursts for which spectroscopic redshifts , peak fluxes , and high resolution light curves were available , [ 11 ]
have suggested that more luminous bursts have shorter spectral lags ( the interval of time between the peak of the light curve in different energy bands ) .
using the same 7 bursts , [ 13 ] have suggested that more luminous bursts have more variable light curves .
these trends between luminosity and quantities that can be measured directly from light curves raise the exciting possibility that luminosities , and hence luminosity distances , might be inferred for the long - duration bursts from their light curves alone .    in this paper
( see also [ 15,16 ] ) , we present a possible luminosity estimator for the long - duration bursts , the construction of which was motivated by the work of [ 14 ] and [ 13 ] .
we term the luminosity estimator `` cepheid - like '' in that it can be used to infer luminosities and luminosity distances for the long - duration bursts from the variabilities of their light curves alone .
we also present a preliminary application of this luminosity estimator to 907 long - duration bursts from the batse catalog .
we discuss the construction of our measure @xmath0 of the variability of a burst light curve  2 . in  3 , we discuss our expansion of the original [ 14 ] sample of 7 bursts to include a total of 20 bursts , including 13 batse bursts , 5 _ wind_/konus bursts , 1 _ ulysses_/grb burst , and 1 near / xgrs burst .
also in  3 , we discuss the construction of our luminosity estimator .
we present our preliminary application of this luminosity estimator in  4 .
qualitatively , @xmath0 is computed by taking the difference of the light curve and a smoothed version of the light curve , squaring this difference , summing the squared difference over time intervals , and appropriately normalizing the result .
we rigorously construct @xmath0 in [ 15 ] .
we require it to have the following properties : ( 1 ) we define it in terms of physical , source - frame quantities , as opposed to measured , observer - frame quantities ; ( 2 ) when converted to observer - frame quantities , all strong dependences on redshift and other difficult or impossible to measure quantities cancel out ; ( 3 ) it is not biased by instrumental binning of the light curve , despite cosmological time dilation and the narrowing of the light curve s temporal substructure at higher energies [ 17 ] ; ( 4 ) it is not biased by poisson noise , and consequently can be applied to faint bursts ; and ( 5 ) it is robust ; i.e. , similar light curves always yield similar variabilities . also in [ 15 ]
, we derive an expression for the statistical uncertainty in a light curve s measured variability , and we describe how we combine variability measurements of light curves acquired in different energy bands into a single measurement of a burst s variability .
we plot the @xmath1 kev light curves of the most and least variable cosmological batse bursts in our sample in figure 1 .
we list our sample of 20 bursts in table 1 of [ 15 ] ; it consists of every burst for which redshift information is currently available .
spectroscopic redshifts , peak fluxes , and high resolution light curves are available for 11 of these bursts ; partial information is available for the remaining 9 bursts .
we rigorously construct the luminosity estimator in [ 15 ] , applying the bayesian inference formalism developed by [ 18 ] .
we plot the data and best - fit model of the distribution of these data in the @xmath2-@xmath3 plane in figure 2 .
we apply the best - fit luminosity estimator ( figure 2 ) to 1667 batse bursts , which are all of the batse bursts for which the necessary information was available as of the summer of 2000 . to remove the short - duration bursts
, we conservatively cut the bursts with durations of @xmath4 sec from the sample , reducing the number to 907 .
we plot the distribution of variability redshifts in figure 3 .
a rigorous analysis of how statistical and systematic errors affect this distribution will be presented elsewhere , but the largest effect is the statistical uncertainty in the fitted slope of the luminosity estimator .
we plot how reasonable variations of this slope affect the distribution also in figure 3 .
although the original distribution has too many low-@xmath5 and high-@xmath5 bursts for comfort , reasonable variations appear to yield at least qualitatively acceptable distributions ( see figure 5 of [ 19 ] ) .
we plot the joint redshift and luminosity distribution of the qualitatively acceptable redshift distribution of figure 3 ( dashed histogram ) in figure 4 .
if systematic effects can be ruled out near batse s detection threshold , this distribution suggests that the luminosity distribution of the bursts is evolving , in which case no more than about 15% of bursts have redshifts greater than @xmath6 , in contrast to the results of [ 13 ] and [ 20 ] .
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we present a possible cepheid - like luminosity estimator for the long - duration gamma - ray bursts based on the variability of their light curves . we also present a preliminary application of this luminosity estimator to 907 long - duration bursts from the batse catalog .