Predicting and analyzing topological correlations in the CMB

The Cosmic Microwave Background (CMB) power spectrum derived from the first year WMAP data demonstrates an intriguing lack of power at large scales that cannot be accounted for within the framework of the standard cosmological model. We explore the possibi

PredictingandanalyzingtopologicalcorrelationsintheCMB

A.Niarchou andA.H.Jaffe

arXiv:astro-ph/0602214v1 9 Feb 2006AstrophysicsGroup,BlackettLaboratory,ImperialCollegeLondon,PrinceConsortRoad,LondonSW72AZ,UKAbstract.TheCosmicMicrowaveBackground(CMB)powerspectrumderivedfromthe rstyearWMAPdatademonstratesanintriguinglackofpoweratlargescalesthatcannotbeaccountedforwithintheframeworkofthestandardcosmologicalmodel.Weexplorethepossibilitythatthisanomalycouldbeexplainedbywaivingtheimplicitassumptionofasimplyconnectedtopology,ratherthanmodifyingthephysicsofthestandardmodel.Inparticular,weassumethattheUniverseisslightlyclosedanditsspatialsectioncanbedescribedbyoneofthesimplestsphericalmulti-connectedmanifolds(thequaternionic,theoctahedral,thetruncatedcubeandthePoincarespace).WediscusstheimplicationsfortheCMBineachcase.INTRODUCTIONTheCMBhasprovedtobeaninvaluabletoolinourquesttounveilthemysteriesofthecosmos.Itdatesbacktotheepochofmatter-radiationdecouplingandtheexpansionoftheUniversehascauseditsmeantemperaturetodroptoamere2.726Ktoday.Itexhibitsminutetemperature uctuationswhosepropertiesdependonfundamentalcosmologicalparametersandinitialconditions.Itiscustomarytoexpandthetemperature uctuationinagivendirectionn ontheskyinaseriesofsphericalharmonics: T

T(m ) T

4π∑ C P (cosθ)(3)

whereP aretheLegendrepolynomials.Sincetheaforementionedanisotropiesareverysmall(oftheorder10 5),extremelyaccuratemeasurementsoftheCMBtemperatureare

The Cosmic Microwave Background (CMB) power spectrum derived from the first year WMAP data demonstrates an intriguing lack of power at large scales that cannot be accounted for within the framework of the standard cosmological model. We explore the possibi

FIGURE1.Left:TheCMBpowerspectrumfortheWMAPbest- tmodel.Thereisade citofpoweratlargescales(speci cally =2,3).Right:ThetemperatureanisotropycorrelationfunctionfromtheWMAPdata,aspredictedfromtheWMAPbest- tΛCDM,forthebest- tΛCDMwithC2andC3changedtotheobservedvaluesandfromtheWMAPdatawithC2andC3matchingthepredictionsofthebest- tΛCDM.Itisclearthatthedisagreementbetweentheoryandobservationisduetothelackofpowerat =2,3.

oftheessence.TheWMAPsatellite,launchedin2001,hasbeenthemostaccurateCMBexperimenttodateintherangeoflargetomediumscales.Its rstyeardataseemtobeoverallconsistentwiththestandardcosmologicalmodel(ΛCDM),a atcolddarkmatteruniverse,dominatedbydarkenergyandwhoseprimordialperturbationsareadiabaticandnearlyscale-invariant[1].However,thereisastrikingdisagreementbetweenthepredictionsofthatmodelandtheactualdataatlargescales,whichisprominentinthe ≤3powerspectrumandthecorrelationfunction(Fig.1).Therehavebeensomeattemptstoremovethisdiscrepancybyinvokingnewphysics,butmostofthemresultincontrivedmodelswithlowstatisticalsigni cance[2].However,thisphenomenonarisesnaturallyinthecontextofnon-trivialtopologies,apossibilitywewillconsidernext.

SPHERICALMULTI-CONNECTEDMANIFOLDS

AslightlycloseduniverseismarginallyconsistentwiththeWMAPdata( 0=1.02±0.02).Ifithasanon-trivialtopology,itsspatialsectioncouldbedescribedbyoneofthesphericalmulti-connectedmanifolds.Closedmulti-connectedmanifoldsarethequotientspacesS3/Γ,whereS3isthe3-sphereandΓisthesymmetrygroupparticulartoagivenmanifold.AccordingtothewayitactsonS3,weobtaindifferentcategoriesofmanifolds.Weareonlyinterestedinsomeofthesimplestmanifolds(singleactionmanifolds)[3]:

thequaternionicspace,whereΓ=D 2,thebinarydihedralgroupoforder4.Thefundamentaldomainisa4-sidedprism.

theoctahedralspace,whereΓ=T ,thebinarytetrahedralgroupoforder24.Thefundamentaldomainisaregularoctahedron.

The Cosmic Microwave Background (CMB) power spectrum derived from the first year WMAP data demonstrates an intriguing lack of power at large scales that cannot be accounted for within the framework of the standard cosmological model. We explore the possibi

FIGURE2.Fundamentaldomainsofsomesphericalmulti-connectedmanifolds.Fromlefttorightdepictedaretheregularoctahedron,thetruncatedcubeandtheregulardodecahedron.

thetruncatedcubespace,whereΓ=O ,thebinaryoctahedralgroupoforder48.Thefundamentaldomainisatruncatedcube.

thePoincaréspace,whereΓ=I ,thebinaryicosahedralgroupoforder120.Thefundamentaldomainisaregulardodecahedron.

Ineachcase,thenumberoffundamentaldomainsthattilethe3-sphereisequaltotheorderofthegroupΓ.Figure2showsfundamentaldomainsofsomesphericalmulti-connectedmanifolds.

IMPRINTSOFANON-TRIVIALTOPOLOGY

Themostobviouseffectofanon-trivialtopologywouldbeasuppressionofpoweratlargescales,duetothe nitenessofthefundamentaldomain:evidently,nowave-lengthslargerthanthefundamentaldomaincanexist,thereforetheprimordialpowerspectrumshouldbetruncatedatscalesclosetothesizeofthefundamentaldomain.Thelatterdependsonthecurvatureradius,sincethenumberoffundamentaldomainsthattilethe3-sphereisconstrained.Anothercrucialeffectwouldbethebreakdownofglobalisotropy,asallthemanifoldsweconsiderareorientable.Thiswouldmanifestastheemergenceofnon-diagonaltermsinthetemperatureanisotropycorrelationmatrix,appearingasnon-Gaussianity.Table1summarizesthedifferencesbetweensimplycon-nectedandmulti-connectedspaces.Weobservethatthecausalmicro-physicsgoverningtheevolutionofCMBanisotropiesremainunchanged;whatchangesaretheeigenmodesoftheLaplacian.ThisaffectsCMBobservablesintwoways: rstly,inmulti-connectedspacesnotallwavenumbersβ≥3correspondtoLaplacianeigenmodes(incontrasttosimplyconnectedspaces)sothattheβ-contributiontoeachmultipoleisnotthesame;secondly,sincetheeigenmodesaffecttheevolutionof uctuationsalongthelineofsight,weexpectthelattertoevolvedifferentlyinspacesofdifferenttopology.TheLaplacianeigenmodesofamulti-connectedspacearetheΓ-invariantei …… 此处隐藏：8272字，全部文档内容请下载后查看。喜欢就下载吧 ……