Geometry for Stochastically Inhomogeneous Spacetimes －Lambda (2000)

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1、GeometryforStohasti ally
nhomogeneousSpa etimesC.T..Dodson,Departmentofathemati s,Universityofan hester
nstituteofS ien eandTe hnology,an hester601D,Udodsonumist.a .uk2000Abstra tanifoldsofgammaprobabilitydensityfun tions,whi h arryaRiemannianinformationmetri ,havebeenusedtomodeltheevo

2、lutionoftheintergala ti voidstatisti sand orre-spondinggala ti densitystatisti s.Thisallowedrepresentationofgalaxy lusteringusingaderivedmanifoldoflog-gammadistributionfun tionstomodelstohasti spatialvariationsandprovidedametri thatmeasureddeparturesfromtherandomorhaoti state.Weshowthatthislog-

3、gammamanifoldisanisometri dieomorphofthegammamanifoldandpro-poseamethodtoapproximatematterdensitystatisti sinspa etimethrougha ouplingofspa etimeandstatisti almanifoldgeometries.1
ntrodu tionStatisti almanifolds[1℄areRiemannianmanifoldsrepresentingtheparameterspa esof ontinuousfamiliesofprobab

4、ilitydensityfun tions.Su hspa eshaveanaturalmetri indu edbytheFisherinformationmatrixwhi hisderivedfromthegradientofthelog-likelihoodfun tion,animportantfun tioninstatisti alparameterestimation[3℄.Riemannianmanifoldsofgammaprobabilitydensityfun tionshavebeenusedtomodeltheevolu-tionofintergala t

5、i voidstatisti s[4℄andallowedrepresentationofstohasti matterdistributions.Thisgaveamodelforgalaxy lustering[5℄,andprovidedametri thatmeasureddeparturesfromtherandomorhaoti state.Forfurtherdetailsonstatisti alstudiesofmatterdistributionintheuniverseseeforexample[6,13,18℄.Herewe onsider ouplingth

6、espa etimegeometryofgeneralrelativitywiththeinformationgeom-etryofalog-gammastatisti almanifoldSthatrepresentsspatiallystohasti matterdistributions.Therearepolynomialexpansionsofmetri -dependentobje tsinspa etimelo algeodesi neigh-bourhoodsthatwouldrelatetoobservablestatisti s(egauto ovarian eg

7、radientoflo almeanmatterdensities)intermsofparametersfromthelog-gammamanifolds.Thiswouldbearststeptowardsa oupledstru turewhi hmightdevelopoverspa etimeabundleorsheafofstatisti almanifoldstorepresentavailablestohasti evolutionary