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