Testing Mass Varying Neutrino With Short Gamma Ray Burst(2)

In this paper we study the possibility of probing for the absolute neutrino mass and its variation with short Gamma Ray Burst (GRB). We have calculated the flight time difference between a massive neutrino and a photon in two different approaches to mass v

thetdonewouldbeabletodeterminetheabsolutevalueoftheneutrinomass,howeveringeneralsincethepho-tonsaretrappedinsidethe reballandarereleasedmuchlater,theneutrinoandthephotonwillnotbeemittedatthesametime.Thiswilladdansystematicerrorinthemeasurementofthetd,consequentlyanuncertaintyinthedeterminationoftheneutrinomasses.ToreducethistypeofsystematicerrorwefocusontheshortGRBwhosedurationisgenerallylessthan2secondsorevenmuchshorter,ForexampleBurstAndTransientSourceExperiment(BATSE)

hasdiscoveredaGRBwithdura-tiononly5milliseconds[22].IthasbeenwidelyarguedthatshortGRBsareproducedbythemergeroftwocom-pactobjects(e.g.,neutronstarsorblackholes)[23],andthattheirenvironmentsarelikelytobelow-densitymediabecausethemergingplaceisfarawayfromthebirthsiteofoneneutronstar.Evenso,thepredictedafterglowsfromshortGRBsappeartobedetectablewithcurrentinstrumentsintheSwiftera[24].Oncesuchafterglowsaredetected,theredshiftsofshortGRBsmaybemea-sured.

FromFig.1onecanseethatingeneralthetimedelaytdislongerthanthedurationoftheshortGRB.Forexampleform=0.6eV,z=2,p=10Mev,tdisaround400seconds;especiallyfortheGRB030329withredshiftz=0.17,tdis112seconds.Thetimedelayforthesecasesisexpectedtobedetectableinprinciple.Ifnot,thiswillputalimitontheabsolutevalueoftheneutrinomassbetterthanthecosmologicallimits.

t

p eV

FIG.1:td(unitinsecond)asafunctionofpfordi erentred-shiftsz=0.17(solid),0.5(dashed),1(dotted),2(dashdotted).

Inthefollowingwewilldiscussthepossibilityoftest-ingonthevariationoftheneutrinomasseswiththeshortGRB.Firstofallweparametrizethevariationoftheneu-trinomassinamodelindependentway,thenwewilltaketwoconcretemodelsofQuintessenceforthecalculationofmassvariationandthetimedelayoftheneutrinos.Considerageneralcasewheremνisanarbitraryfunc-tionoftheredshiftZ,i.e.m(Z),onecanexpanditintermsofredshiftz.Forsmallz,weget:

m(Z)=m0+m′Z+

1

2

mZ+...).(3)

De ningc≡

m′(0)

m202

(1+z′)2H0

p

.Notethathere

theneutrinomassin(5)variesasafunctionofredshiftz.FromFig.2onecanseethatthevariationofthe

t

z=0.3

z=0.05

m/p

FIG.2:td(unitinsecond)asafunctionof

m0

f

ννφφ+h.c.,(6)

andmν～

v2

2Mpl

1+β

Q0

M),(8)

pl