M easuring the cosm ologicalparam eters w ith the Ep;i { Eiso correlation ofG am m a{R ay B ursts

A B ST R A C T W ehaveused the Ep;i { Eiso correlation ofGRBsto m easurethecosm ologicalparam - eter M.By adopting a m axim um likelihood approach which allowsus to correctly quantify the extrinsic (i.e.non{Poissonian) scatter ofthe correlation,we constrain (fora atuniverse) M to 0.04{0.40 (68% condence level),with a besttvalue of M � 0:15,and exclude M = 1 at > 99.9% condence level.Ifwe release the as- sum ption ofa at universe,we stillnd evidence for a low value of M (0.04{0.50 at68% condence level)and a weak dependence ofthe dispersion ofthe Ep;i { Eiso correlation on � (with an upper lim it of � � 1:15 at90% condence level).Our approach m akes no assum ptions on the Ep;i { Eiso correlation and it does not use othercalibratorsto setthe\zero'pointoftherelation,thereforeourtreatm entofthe data is not aected by circularity and the results are independent ofthose derived via type Ia SNe(orothercosm ologicalprobes).Unlikeotherm ulti-param eterscorre- lations,ouranalysisgroundson only two param eters,then including a largernum ber (a factor � 3)ofGRBsand being lessaected by system atics.Sim ulationsbased on realisticextrapolationsofongoing (and future)GRB experim ents(e.g., Swift,Konus{ W ind,GLAST) show that:i) the uncertainties on cosm ologicalparam eters can be signicantly decreased;ii)futuredata willallow usto getclueson the\dark energy" evolution. K ey w ords: gam m a{rays:observations{ gam m a{rays:bursts{ cosm ology:cosm o- logicalparam eters.