Gamma-ray bursts and density evolution of neutron star binary mergers

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

A&Amanuscriptno.

(willbeinsertedbyhandlater)

8

991 ebF 9 1v4902089/hp-ortsa:viXra1.Introduction

Therecentdiscoveriesofopticalcounterpartsoftheγ-rayburstsGRB970228(Grootetal.1997)andGRB970508(Bond1997)andthemeasuredemission-lineredshiftofz=0.853(Metzgeretal.1997)forthelatterprovideev-idencefortheircosmologicalorigin.Apossiblesourceforγ-raybursts(GRBs)maybethemergingoftwoneutronstarsinaclosebinary(Blinnikovetal.1984).

Forcosmologieswithnovacuumenergy,thebright-nessdistributionofburstintensitiesexpectedforauni-formsourcepopulationisconsistentwiththeBATSEdis-tributionifthelimitingredshiftisaboutunity(e.g.Der-mer1992;Mao&Paczy´nski1992).However,cosmologicaltime-dilatione ectsintheBATSEsampleindicatethatthedimmestsourcesshouldbelocatedatz≈2(Nor-risetal.1995).Ifthisistheactuallimitingredshift,asourcepopulationwithacomovingratedensityoftheformnc(z)∝(1+z)βiscompatiblewiththeBATSEdis-tributionfor1.5～<β～

<2(Horacketal.1995).Therefore,itisworthwhiletoexaminewhetherthecomovingmergerratedensityofneutron-starbinariesevolvesinasimilarfashion.

Bymeansofpopulationsynthesiscomputationsforbi-narystarsthemergerrateofneutron-starbinaries[here-after(ns,ns)]canbecomputed(seePortegiesZwart&Yungelson1998andreferencestherein).Usingthisap-proachLipunovetal.(1995)computedtheevolutionof(ns,ns)mergersasafunctionofredshiftandlogN logSdistributionsofGRBsforacosmicpopulationwhichcon-tainsgalaxieswithaconstantstar-formationrateandalsogalaxieswithinitialburstsofstarformationindi erentproportions.Totani(1997)computedtheevolutionoftheGRBsratedensityfrom(ns,ns)mergersinamodelbasedontheobservationallydeterminedhistoryofcosmicstarformationandinamodelderivedfromdetailedgalaxyevolution.Totaniassumedadistributionofmergingtimes

f(tc)∝t 1

for(ns,ns)systems1.Wegoastepfurtherbyadoptingc

forallgalaxiesexponentiallydecreasingstarformationrates(SFR)withdi erenttimescalesdepend-ingongalaxymorphology(e.g.Sandage1986).WeapplythedistributionoftcfrommodelBofPortegiesZwart&Yungelson(1998,henceforthPZY98)whichprovidesthe

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

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

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

4P.Bagotetal.:γ

-bursts

Fig.3.CumulativecountsofburstsN(>P)oftheBATSE3Bcatalog(histogram)andmodelcurvescomputedforparame-tersshowninthe gure(fortgal(0)=12.16

Gyr)

Fig.4.Redshiftz0.4necessarytoachieveagreementwiththe

BATSE3BlogN logPdistribution.Thesolidlinecorre-spondstoset1andthedottedlineisforset1b.Inbothcases,thecurrentgalacticageusedistgal(0)=12.16Gyr.

(Totani1997).They,however,predictadi erentbehaviorifstarburstsoccurathighz.

Theabsolutevalueofthe(ns,ns)mergerrateisfoundtobe～100timeslargerthantheGRBsfrequency.Es-capefromthisconundrumisobtainediftheopeningangleoftheobservedphenomenonisafewdegrees,whichiscon-sistentwith re-ballmodelswhereleptonsareconvertedintobulkbarionicmotion(M´esz´aros&Rees1992).

Ifγ-rayburstsindeedoriginatefrom(ns,ns)mergersandourmodelforstarformationiscorrect,interestingimplicationsfollow.TheprogenitorsofbrightGRBsorburstsatlowredshift(z<～1.5)mostlikelybelongedtoearly-typespiralgalaxieswhereastheprogenitorsofthedimmestburstsandthoseathighredshift(z>～2)werelocatedinellipticalgalaxies.Themajorityoftheburstsofgravitationalwavesinthismodelareexpectedtoorig-inatefromearly-typespiralgalaxies.IfmoststarsintheUniverseformedindwarfstar-burstgalaxies,asubstan-tialfractionoftheparentalpopulationmayoriginatefromthem.Suggestedbycurrentobservationaldatathedecline

inthestarformationratebeyondz≈1.2(e.g.Connollyetal.1997),ifreal,willshow-upasaturnoverinthelogN logPdistributionforGRBs.

AsnoticedbyTutukov&Yungelson(1994)andcon- rmedbyPZY98(theirFigs.6and8),spacevelocitiesof(ns,ns)binariesmaywellexceedescapevelocitiesofdwarfaswellasgiantgalaxiesandtheymaytravelupto～1Mpcbeforecoalescence.Thus,asigni cantfractionofthesitesofGRBsmaynotbedirectlyassociatedwithstarformingregions.

Acknowledgements.WethankJanvanParadijsforreadingthemanuscript.ThisworkwaspartiallysupportedthroughNWOSpinozagrant08-0toE.P.J.vandenHeuvelandRFBRGrant96-02-16351.L.R.YacknowledgeshospitalityoftheAs-tronomicalInstitute“AntonPannekoek”andMeudonObser-vatory.

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