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

2P.Bagotetal.:γ-bursts

Table1.Adoptedparametersforgalaxiesofdi erentmor-phologicaltypes:timescaleofstarformationτi;contributiontotheB-bandluminosityoftheUniverseci(Phinney1991);masstoblue-lightratioM/LB(afterLipunovetal.1995andGuiderdoni&Rocca-Volmerange1987).

best ttotheexpectedbirthrateandorbitalparametersoftheGalacticpopulationofhigh-massbinarypulsars.2.Model

Theannual(ns,ns)birthrateisproportionaltotheSFRinthegalaxyΨ(t,τ),whereτisthetimescaleforstarformation.TheSFRadoptedisadecreasingexponential,whichappearswhentheSFRissupposedtobepropor-tionaltothegascontentwithouttakingintoaccountthegasejectedbystars(e.g.Bruzual&Charlot1993):E-SOSa-SbSc-Sd1410161520%40%40%1052

Ψ(tgal,τ)=τ 1exp( tgal/τ),

(1)

wheretheageofthegalaxyisgivenbytgal=t tFandtFthetimeatwhichthegalaxywasformed.Atanytimetthe(ns,ns)mergerrateµ(t,τ)hasacontributionfromsystemsthatareformedatdi erentepochst tcinthehistoryofthatgalaxy.ItcanthusbeexpressedasaconvolutionintegralofthebirthrateofstarsΨ(t,τ)andthedistributionof(ns,ns)mergingtimesf(tc):t tµ(t,τ)=µF

f(tc)Ψ(t tc,τ)dtc,(2)

Theresultsofthepopulationsynthesiscomputationsgivef(tc)andthenormalizationcoe cientµ0.Thefunc-tionf(tc)fromPZY98isapproximatedbyaGaussianwithx=log(tc/tgal(0))asaparameter,themaximumatx = 2andwithσ=1(tgal(0)istheageofthegalaxyatz=0).Weassumethatthedistributionfunctionf(tc),normalizedtounity,istime-independentandthesameforallgalaxies.Forthenormalizationofµ(t,τ)were-quirethatinareferencegalaxysimilartotheMilkyWay(typeSb,Mref=2×1011M⊙,withacurrentastrationrateof4M⊙yr 1)thecurrentrateof(ns,ns)mergersis2×10 5yr 1,whichisthemergerrateobtainedfortheGalaxybyPZY98.

Forsimplicity,wesplittheHubblesequenceintothreetypesofgalaxies:EtoSO,SatoSbandSctoSd.ForourselectedmixtureofHubbletypes,twosetsofcharacteris-ticstarformationtimescalesareused(cf.Table1).Thisparameterizationisrathersimplistic;thefractionofE/S0galaxiesmaydi erconsiderablyfrom20%(Dressler1980)andthestarformationhistoryinspiralsofthesamemor-phologicaltypemaybeafunctionoftheirmass(Gallagheretal.1984).IfindeedmoststarsintheUniverseformedindwarfstar-burstgalaxies(e.g.Babul&Ferguson1996)atz～1,thismaya ectourresultsconsiderably.

Starformationisassumedtooccurcontinuouslyac-cordingtoEq.1inallgalaxies.Wealsoinvestigatethecaseofaninitialburstofstarformationduringthe rstGyrinE-SOgalaxiesandnostarformationthereafter.Thelattermodelsaredenotedassets1band2b.

Thecomovingratedensitycanberelatedtoµ(z,τ)viatheB-bandluminositydensityfromtheUniverseLB=

Mcref

i

i

M