The evolution of the comoving cosmic merger-rate density of neutron star binaries n_c(z) is calculated using a distribution of their merging times provided by population-synthesis computations of binary stars. We adopt an exponential law for the star forma

P.Bagotetal.:γ-bursts

3

Fig.1.Evolutionofthecomoving(ns,ns)mergerrateden-sityasafunctionofredshift,normalizedtounityatz=0.Theshadedareacorrespondstotheallowedregionaccordingtonc(z)∝(1+z)βwithβintherange1.5to2.Theuppersolid(dotted)lineisforset1b(set1)withtgal(0)=12.16Gyr.Thedashedlinesgivetheratesforset2withtgal(0)=12.16Gyr(uppercurve)and9.92Gyr(lowercurve).Thelowersolidlinehastobecomparedwiththedottedlineandcorrespondsto

themergertimedistributionf(tc)∝t 1

cwithalowercut-o at0.02Gyr(likeinTotani

1997).

Fig.2.Relativecontributionofgalaxiesofdi erenttypestothe(ns,ns)mergerratedensity.Thecurrentgalacticageistgal(0)=12.16Gyrandset1oftheSFtimescalesisused.

TheupperandlowerdashedlinesinFig.1(set2)demonstratethee ectoftheageofgalaxiesontheco-movingmergerrate,i.e.;thein uenceofthecosmologicalparameters.

Figure2givestherelativecontributiontothe(ns,ns)merger-ratedensityforeachselectedsubclassofgalaxies.Themajorityofeventswhicharepotentiallydetectablebygravitational-waveobservatories(LIGO/VIRGO)arelocatedinearly-typespiralgalaxies.IfGRBsoriginatefrom(ns,ns)coalescencethedimmestburstsareexpectedtobehostedinellipticalgalaxies.

Followingthestandardprocedure(e.g.Horacketal.1996)wecomputethenumberofburstsN(>P)withapeak uxgreaterthanP.Weassumethatburstsarestan-dardcandlesandtheintrinsicluminositydoesn’tevolve.

Thespectralformoftheburstsimilartothatobservedisadopted:Φ(E)∝E 1exp( E/E0),whereE0=350keVisacharacteristicenergy.Forcomparison,weusetheob-servedintegralbrightnessdistributionfromtheBATSE3Bcatalogintheenergyrange50-300keVmeasuredatatimespanof1024ms(Meeganetal.1996).Figure3showstheexpectedbrightnessdistributionscomputedforsetsofparameters1and1b,superimposedontheBATSEdata.Thecurvesarenormalizedatthepeak uxthresholdP=0.4photonscm 2s 1.Figure4providestheresultsofaKolmogorov-SmirnovtestoftheBATSE3Bcatalogtotheresultsofourcomputations.Onlydataabovethepeak uxof0.4photonscm 2s 1areusedtoavoidthresholde ects.Thehighestcon dencelevel(CL)isobtainedforsets1and1bifthelimitingredshiftz0.4atthepeak uxthresholdis～2.4and3.0respectively.Forothervaluesofthestarformationtimescalesandcosmologicalparametersz0.4isfoundtorangefrom1.9to2.7(withaCL>80%)forthemodelswithaburstofstarformationinellipticalgalaxies(models1band2b).Withoutaninitialburstofstarformationz0.4rangesfrom2.9to3.3.SimilarresultsarefoundforE0intherange300-400keV.

Notethatforhigherlimitingredshift,Fig.4showstheexistenceofotherpossible tswithalowerCL(i.e.model1b,z0.4=3.8).Theprimarypeakatz0.4=2.4corre-spondstothe rstchangeoftheslopeofthecomovingratedensity(Fig.1).Forhigherredshifts,thesuddenin-creaseofthemergerratewouldrequirethesamebehavioroftheBATSEdataforconsistency.Asaconsequence,thesecondarypeakatz0.4=3.8,althoughprovidingagood tforpeak uxvaluesnearthethreshold,tendstode-partmoreandmorewiththedataforhighervaluesofP(Fig.3).Therefore,theredshiftrangerelatedtoanas-sumedinitialburstofstarformationinellipticalgalaxiesislikelytobebeyondtheactuallimitingredshiftforGRBs.

Finally,ithastobepointedoutthatthemodelscanhardlyreproducethepeak uxvaluesofBATSE’sfaintestbursts(seealsoTotani1997).AsshownbyReichard&M´esz´aros(1997),thisfeatureresultsfromtheassumptionthatGRBsarestandardcandles.

4.Conclusion

Thecomputedrelativemergerrateasafunctionofred-shiftisinagreementwiththedetectedrateofγ-rayburstsup-tothelimitingredshiftofBATSE.Forthemodelswhereellipticalgalaxiesexperienceaninitialburstofstarformationtheoccurrencerateof(ns,ns)coalescencein-creasessuddenlybymorethanafactortwoataredshiftof2.5,whichisbeyondourcurrentdetectionlimit.ThesyntheticlogN logPdistributioniscompatiblewiththeobservationsdowntothecompletenesslimitofγ-raycata-logues.Thesameistrueformodelsbasedonobservation-allyinferredcosmicstarformationhistory(seee.g.Sahuetal.1997)andformodelsbasedongalacticevolution