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ManuscriptpreparedforJournalname withversion3.1oftheLATEXclasscopernicus2.cls. Date:19January2017 On the cosmological vacuum and dark energy 7 R.A.Treumann1,2 1 0 1InternationalSpaceScienceInstitute,Bern,Switzerland 2 2DepartmentofGeophysicsandEnvironmentalSciences,MunichUniversity,Munich,Germany n a J Abstract.–Adegeneratefermionicvacuumpopulationis charge,spin)must be subjectto the Fermidistribution.The 7 suggested.Based on the abundanceof the darkenergyden- only bosons propagatingon it at our temperatures are pho- 1 sity in the Universe the vacuum particle mass and number tons and gravitons, the former constituting the electrody- density are estimated. The obtained mass is in reasonable namic fields. Moreover,since the vacuum is our zero-point ] h agreementwithobservationsandpredictionsoftheneutrino reference, any particle component it contains cannot have p mass.Numberdensitiesarehigherthanthoseoftherelativis- positive energy.This had already been proposed by P.A.M. - ticrelicneutrinocomponent. Diracwhendiscoveringhisrelativisticequationoftheelec- n tron.However,asitturnedoutandwasinterpreteddifferently e g in QED, neitherelectronsnor positrons,their anti-particles, . don’tpopulatethevacuum. s c 1 Introduction At negative energy levels ǫ = p2c2+m2c4= ǫ <0 i p − v − vp ys Bydefinition,thevacuumisthezeropointofeverythingwe (morci2t,saDssiuramcieadnteoqbueivlaelsesntt)haanndthleowhpytpemotpheertiactaulrepaTrvti≡cleβ−vr1es<t h measure:energy,temperatureandso on.Recentastrophysi- mvassenergym c2 of thevacuumparticles,the Fermidistri- p v [ calobservations(Riessetal.,1998;Perlmutteretal.,1999a) bution(intheabsenceofnegativetemperatures1)impliesthat tellconvincinglythatitisnotempty.Itcontainsasubstantial theaverageoccupationnumberofstatesisn¯p=1.Allnega- 1 energy density that is attributed to some unknown ingredi- tiveenergystatesareoccupied.Summedoverallstatesfrom v ent (Zeldovich, 1967, 1968; Turner, 1998; Perlmutteretal., to 0 would yield an infinite vacuum particle number, 8 1 1999b) andits quantumfluctuations.Thisenergycontentis −w∞hich cannot be true. The number of particles respectively 1 negative while playing the role of a cosmological constant theirnumberdensityN shouldbefinitealsointhevacuum. v 5 Λ (Einstein, 1917; Dirac, 1937; Wilczek, 1984; Weinberg, Thusnotallnegativeenergylevelsareoccupied.Insteadthe 0 1989; Carrolleta., 1992) in a quasi-Einstein-de Sitter Uni- negativefullypopulatedseaissomehowmirrorsymmetricto . 1 verse. thepositivedomainofparticleenergiesatzerotemperature. 0 Let us assume the vacuumhad temperatureT =0 (in en- Inspectionof theFermidistributionfornegativeenergiesat 7 ergyunits),atemperaturefarawayfromanytemperatureat T =0 1 v which anyof the high energyquantumfield theories would : iv come to work,exceptof quantumelectrodynamics.Atzero n¯p= e−βv(ǫvp+µv)+1 −1 (1) X temperature (or close to it, which in cosmological context (cid:20) (cid:21) meansattimeswellafterelementgeneration,atleast, how- r withµ thechemicalpotential,suggeststhatthechemicalpo- a ever,afterdecoupling)therealvacuuminthecosmosissub- v tential in the vacuummust be negative,µ <0, if the Fermi v jecttoelectrodynamicsonly.It,forinstance,impliesthatwe distributionshouldmakesensealsothere.Occupationofneg- canuseelectromagneticwavesascarrierofinformationand ativeenergylevelsisinthiscaselimitedbythenegativevac- can measure everything against the stable electrodynamic uumFermienergy vacuum.Anotherconsequenceisburiedinthe constancyof the electrodynamicpropertiesof the vacuum,the constancy ~2 6π2 2 of its electric andmagnetic susceptibilitiesµ0 and ǫ0 which µv≡−ǫvF=−2mv(cid:18) g Nv(cid:19)3 (2) includestheconstancyoftheresistanceofthevacuum. Thiscontemporaryelectrodynamicvacuum,ifcontaining 1For the non-existence of negative temperatures T <0 see areal(virtual)particlecomponent(virtualparticlesofmass, (DunkelandHilbert,2014).Infact,temperaturesaredefinedfrom kinetic theory through mean square velocity differences (∆v2) Correspondenceto:R.A.Treumann which, for positive occupation numbers of states, are always pos- ([email protected]) itivedefinite. 2 R.A.Treumann:Darkenergy whereN istheunknownand,inprinciple,notdirectlymea- Definearelativevacuumparticlemass̟ =m /m,where v v v sureablevacuumparticledensity.Levelsǫ <ǫ areempty, m is theelectronmass. Then,fromthe last equalityfollows p vF whilelevelsǫ ǫ .0areoccupiedwithn¯ =1.Moreover forthevacuumnumberdensity vF p p ≤ byDirac’soriginalrelativisticargumentsenergylevelsabove 3 2 tthoepnreevgeantitvleeareksatgmeaosfspeanretircglyes−fmrovmc2tahreevaalscouuemmpsteyaitnoorredael-r Nv= 2m~2ρΛ!5 6gπ2!5̟v35 (5) ityunlessinjectionofradiativeenergywouldleadtosponta- InsertingthisintotheconditiononthevacuumFermienergy neousparticlepaircreationoutofthenothing,i.e.outofthe (4) yields an estimate for the relative mass of the vacuum vacuum.Thisrequiresthatthevacuumparticlescouldinter- Fermions actwithradiationwhich,topresentknowledge,seemsnotto bethecase,exceptforthecosmologicaleffectofthenegative ̟ .ǫvF(̟v) ρΛ (6) vacuumenergydensity. v mc2 ∼mc2N v whichisanimplicitequation.Resolvingitgives 2 Estimates Little is known so far about the vacuum population. The ̟v. 43πg2!41 ρmΛcλ2C3 !14 ≈1.34 ρcmrictλ2C3!14 (7) above Fermi energy, a completely unknown quantity, con- tainstwoundeterminedquantities,thevacuumparticlemass, wheretheComptonwavelengthofanelectronλ =~/mchas C m ,whichforrealparticlesispositive,andthevacuumparti- beenintroduced.Thecentralpartofthisexpressiondepends v clenumberdensityN . only on the one free parameter Ω which is subject to de- v Λ We can, however,impose conditions.The first is that the termination making use of astronomicaldata. Ω 0.7 and Λ ≈ vacuumFermienergy,forasufficientlylargenumberofneg- g=1.75forFermionsgivesthenumericalfactor.Thisisin- ative energy states to exist in the vacuum, should at least dependenton the numberdensity N . Hence, it canbe used v equal the rest-mass energy ǫ m c2 or equivalently to obtain a limit on ̟ . Inserting into the above expression vF v v − ∼− ǫ m c2whereǫ istakenpositive.Thisgives forthevacuumdensityitalsopermitsobtaininganimportant vF v vF ∼ independentvacuumdensityestimate ~2 6π2 32 mvc2.2mv g Nv! (3) N .1.1 g 14 ρΛ 34 0.85 ρcrit 34 (8) v 6π2! mc2λ ! ≈ mc2λ ! Astronomical observations of the accelerated expansion of C C theuniverse(Riessetal.,1998;Perlmutteretal.,1999a)tell which may be considered as an upper bound. The critical thatourepochischaracterizedbytheincreasingdominance mass density in the Universe is ρ =9.21 10 27 kg/m3 m − × ofthevacuumpressurewhichdrivesthecosmologicalexpan- which,intermsoftheelectronrestmassenergymc2,yieldsa sion.Ourcurrentcosmologicalepochisspecialinthesense criticalenergydensityofρ /mc2 1.01 104.Thisnumber crit ≈ × that vacuum and baryonic energy densities are of same or- givesarestmass der of magnitude. The vacuum energy density, however, is constantwhilethemassdensitydecreaseswithcosmological mv=̟vme 5.05 10−9me (9) ∼ × expansion.Hence,itincreasinglyovercomesboththevisible ofthehypotheticalparticlesthatmakeupthevacuumpopula- as well as dark matter contributions plus the cosmic back- tion.Usingtherestmassenergyofanelectron,thisbecomes groundradiationpressureincludinganyotherradiativecom- ponents.Fromtheseobservationsanegativevacuumenergy m .2.6 10 3 eV/c2 (10) v − density ρ hasbeeninferredwhichseemsto be either ex- × Λ − actly or at least very close to the value expected for a sta- a small number which indicates that the massive vacuum tionary, i.e. time-independentvacuum. The index Λ relates Fermionsshouldbe light. With thissmall mass the vacuum the cosmological vacuum energy density to the cosmologi- numberdensityisfoundtobeoftheorder calconstantΛ(Wilczek,1984;Weinberg,1989;Carrolleta., N .1.75 1012 m 3 (11) 1992).WritingforthevacuumenergydensityNvǫvF andset- v × − tingitequaltotheobservedcosmologicalvaluewehave Suchafermionicvacuumnumberdensityissurprisinglyhigh whencomparedwiththedensitiesofinterstellar,galacticor N ǫ ρ (4) v vF Λ ∼ evenintergalacticspace. It is also fourordersof magnitude Since ρ =Ω ρ is known, with ρ the critical cosmo- larger than the relativistic relic neutrino background den- Λ Λ crit crit logical energy density, the two above equations provide a sity Nrel 3.3 108 m 3 in the Universe. Such a vacuumis ν ≈ × − meansofestimatingboththenumberdensityN ofparticles quitedensethoughbeingverydilutewhencomparedwitha v inthecosmologicalvacuumandthemassofthehypothetical vacuumcomposedofPlanckianparticlesofmassm 1019 Pl ∼ Fermionicpopulationofthevacuum. GeV/c2. Journalname www.jn.net R.A.Treumann:Darkenergy 3 Lowerlimitsonboththesenumbersarecurrentlynotknow would suggest that the degenerate vacuum consist of pairs theoretically.ConsideringtheexperimentalinferenceonΩ of virtual neutrinos which are their own antiparticles. Such Λ we note that the ever more precise measurements have de- neutrinossatisfy Majorana statistics instead of Dirac statis- creased its valueslowly fromΩ &0.7to its presently best ticsandareknownasMajorananeutrinos.Currentdegener- Λ valueΩ 0.695.Inordertobeontheveryconservativeside ateexperimentalupperboundsforMajorananeutrinomasses ≈ we may thus boldly assume that, at least to current intelli- arem .0.1eV/c2,whichisagainnotindisagreementwith ν gence,anabsolutelowerobservationallimitwillnotbeless the aboveindependentestimate. The currentdiscrepancyof thanΩ =0.6.Thisgivesa factorof1.29insteadof1.34in lessthantwoordersofmagnitudeissurprisinglysmall. Λ (7),restrictingthevacuumparticlemasstothenarrowrange Giventheindependentassumptionsmadeinourestimates, itishardtobelievethatthisclosenessrespectivelyapproxi- 2.5 10 3<m .2.6 10 3 eV/c2 (12) × − v × − matesimilarityofthenumberswouldjustbeacoincidence. Thevacuumnumberdensityofthehypotheticalparticlesis, Onemaythusbecomeinclinedtoidentifythevacuummass bythesamereasoning,subjecttothebounds withthelightest(sayelectron)neutrinomass 1.56 1012<N .1.75 1012 m 3 (13) v − m m (14) × × v≈ νe These densities are the famous disturbing 110 orders of ∼ of degenerate neutrinos populating the vacuum being in magnitudes below a hypothetical vacuum number density negative energy states. For a continuum of vacuum energy based on the assumption of a vacuum built of Planckions. levels the neutrinos form a degenerate Landau-Fermi fluid IntermsofmassratiosbetweenPlanckionsofmassm and Pl (Landauetal.,1980).Suchafluidissubjecttoquantumfluc- the vacuum particles this reduces to a “somewhat fairer” tuations, which at the low vacuum temperatures propagate m /m 30ordersofmagnitudediscrepancy.Ifthecurrent thPeloryvh∼asanythingincommonwithrealitythenPlanckions aslowintensityzerosoundatsoundspeedc≥u0&pvF/mv. Since p m c,oneconcludesthat areruledoutbyitatleastforthecontemporaryvacuumasit vF∼ v isexperiencedoveralltheastronomicalobservationaltimes u =c (15) 0 inourpast-inflationepoch. Wavenumbersk <m c/~ imply wavelengths λ &18 Mpc, 0 v 0 whichislargerthan>0.1%oftheradiusoftheUniverseand 3 Thequestionfortheparticles exceedsthediameterofallgalaxyclusters.Onshorterscales Not many Fermions this light are known to us. The most thisvacuumcanbeconsideredascompletelyhomogeneous. promising particles are neutrinos of which we know that Thusvacuumfluctuationswouldbeinterpretedasaspectrum theyinteractonlyweaklywithmatter.Thecombinedcurrent oflarge-scalezerosoundfluctuationspropagatingattheve- limits fromcosmologicalprobes(Cuestaetal., 2016) so far locityoflight.Longest“visiblewavelengths"wouldbeofthe available are upper bounds m = m <0.12 eV/c2 which orderofthediameterofthevisibleUniverse. do not rule out that the real vνacuPuimνnieutrino mass would Thevacuumenergydensityhasnotchangedovertheen- be somewhat lower. Neutrino oscillations of real (i.e. non- tiretimeofexistenceofthevacuumasweknowitafterthe vacuum) relativistic non-degenerateneutrinosindicate non- end of inflation. For the same reason the zero-sound speed vanishingmassdifferences ∆m 2betweenthethreedifferent hasbeenconstantoverthisperiod.Itsconstancyimpliesthe ν generations of neutrinos. I|n con|trast to the free relativistic constancyofthevelocityoflightasapropertyoftheinvari- neutrinobackgroundandneutrinosproducedintheUniverse ablevacuum. inβdecayandotherelementaryinteractions,vacuumneutri- Invertingtheargumentonemayconjecturethattheveloc- nosaredegenerateatT 0.Theorysuggeststhatthemasses ity of light is a constant of nature because it is a constant v ofthethreedegeneraten≈eutrinogenerationsareofsameor- propertyofthedegeneratevacuumwhichhasbeeninvariant der of magnitudem ∆m 5 10 2 eV/c2. Observations formostofthelifetimeoftheUniverse. ν ν − ∼| |∼ × (onthe2σconfidencelevel)indicate∆m2&8 10 5(eV/c2)2 ν × − yieldingmν 10−2eV/c2whichissomewhatthoughnotvery 4 Cosmologicalconsequences ∼ muchlargerthanourestimatedvacuumparticlemassrange. Whenmeasurementswillbepusheddownto<0.1eV/c2,for The visible Universe contains a non-degenerate relativistic instance by KATRIN (Mertens, 2016), this will soon be in dilute neutrino population of cosmological very-early uni- thereachtobecomechecked. versal origin and current temperature T /k 1.9 K. These ν B ≈ Thetheoreticalnormalhierarchicalschemecurrentlypre- neutrinos have decoupled at roughly t 1 s after the Big ∼ dicts m .5 10 3 eV/c2 for the electron neutrino, which Bang. Though being of finite mass they by current intelli- ν − × is very close to the above completely independentestimate genceprobablydonotcontributemuchtotheunknowncom- (12). positionofDarkMatterbecauseoftheirlowdensityandrel- The usual notion that the vacuum is not empty but hosts ativisticspeedclosetothevelocityoflight.Itiscertainthat quantum fluctuations, i.e. virtual particle-antiparticle pairs, theydonotparticipateintheDarkEnergy. www.jn.net Journalname 4 R.A.Treumann:Darkenergy Thehypotheticaldegeneratenon-relativisticvacuumneu- lapse.MasseslargerthanthePlanckmasshaveshorterinter- trinos must also have been producedin the very early Uni- nalcollapsetimes.Hawkingradiationcannotstopthemfrom verse,presumablybeforeoratleastduringtheinflationphase collapsing. as a component of the then existing “hot vacuum” which ThelifetimeofPlanckionswillinadditiondependonthe is assumed to have been unstable with respect to the phase primordialvacuumconditions,the vacuumtemperature,re- transitionthatledtoonsetofinflation.Duringinflationthey spectively the inflaton field Φ. In thermal equilibrium with cooled down to the new low, and from our positive energy their environment, Planckions will be stable because the point of view negative, degenerate energy state filling the same amount of radiation they lose by Hawking radiation vastdomainoftheUniverseuptoitsinflation-extendedsize theyabsorbinreturnfromtheenvironment.Intheothercase before settling at a time around some t 10−32 s or so into allPlanckionsshoulddecay,butpossiblyonlyslowly,ifthe ∼ its current quasi-stationary or even time-independent state. thermalequilibriumisonlyweaklyviolated.Thismayhave They form an about homogeneous background in the Uni- beenthecasewhentheinflatonwasinitsoriginalmeta-stable verse,thevacuumasweexperienceit,extendingfaroutover state.Onemayexpectthat,whenthepre-inflationaryvacuum the horizons of the visible Universe. It is then no surprise temperatureapproachedtheorderoftheelectroweaktransi- that the vacuum energy density does not vary, at least over tionatabout 10 36s,neutrinosandotherparticles(massive − ∼ thosetimesanddistanceswhichhavebeenaccessibletoob- Bosons)mighthavebecomeradiatedawayintothesurround- servations.ThisisinaccordwithDirac’soriginalhypothesis ingvacuumasvirtualparticles. (Dirac,1937)thatΛisaboutconstantinourpresentcosmo- The time difference between Planck (or Planck-Hawking logicalepoch.Thisepochisjustpost-inflationarylong. evaporation) time and onset of inflation around 10 34 − ∼ − The interesting question concerns the origin of the vac- 10 33 s, believed to be of the order of few orders of mag- − uum neutrino component. Speculatively they might have nitude in time, should provide information about the phys- been produced prior to or during inflation as the lightest ical state (temperature, density, composition) of the pre- Fermionic decay products of the Planckions, the smallest inflationary vacuum and prolongation of the decay time of known (or imaginable) Black Holes, which constituted the PlanckionsintheveryearlyUniverse. pre-inflationary vacuum in case these evaporated when the Whatenabledthevacuuminthecourseofinflationtotrap Universe started inflating.It is believedthat their interioris themajorityofneutrinos,whichintheabovescenariomight subjecttoquantumgravity. havebeenproducedpriortoinflation,lettingthemdisappear Black Holes have finite temperature, behave like black our direct inference, is not known. From their comparably bodies,andemitHawkingradiationofvirtualparticlepairs. large density one may suspect that they represent the bulk ThehorizonofPlanckionsisthePlanckradiusℓPl,withun- componentofthe pre-inflationaryneutrinopopulation,with certainty ∆ℓPl ℓPl being ofthe same orderas the radiusit- the free cosmological relic neutrino background just form- ∼ self.ThusPlanckionsappeartotheoutsideasdiffusespecks ingtheinflationaryrun-awaytailofthepre-inflationneutrino ofPlanckdensitywhichsuggeststhattheirinteriorisinthe distribution that escaped trapping by the inflating vacuum. chromo-plasmastateoffreequarksandgluons.Ifnotinther- Constitutingthevacuumcomponentthedegeneratenegative- malequilibriumwiththeirvacuumenvironment,in particu- energy vacuum neutrinos (or virtual neutrino pairs), being larwhenthevacuumisempty,theyirradiatewithinPlanck- trapped by the post-inflationary vacuum, were not affected Hawking times τH.104τPl 10−40 s, i.e. rather soon after byandthusdidnotparticipateinthesubsequentreheatingof ∼ generationintheBigBang. matter. Hawking radiation of Black Holes represent local anti- One might wonder whether today the vacuum neutrinos, gravity,awell-knownfact.ForPlanckionsitovercomesgrav- inadditiontoprovidingtheDarkEnergydensity,couldcon- ityandletsthemevaporate.Onemayaskwhichmasswould tribute to the unknowncompositionof Dark Matter. Thisis just balance the collapse. This happens when the Hawking notthecase,however,ascanbeshownbyaskinguptowhat timeτH τcollequalsthecollapsetime.ForaSchwarzschild- radial distance R from a large local mass M=αM , with Black H≈ole 2τcoll/π= 3/8πGρ(Rs) has been given for M thesolarmass,thelocalgravitationalenergydens⊙ityand a dust cloud (OppenhepimerandSnyder, 1939; Weinberg, thu⊙sgravitationalattractioncouldovercometheDarkEnergy 1972) consisting of particles that interact only gravitation- pressureofthelightdegeneratevacuumneutrinos.Thisleads ally, withcollapse startingatSchwarzschild-radiusRs, with to density ρ(R ) and zero free-fall velocity. Equating both ex- s p√rπesmsPiol,nws2hicyhieilsdssligfohrtlythlaergceorlltahpasnet-hbealPalnacnicnkgmmasass.sSmMaclble≈r R<GΩMΛ⊙ρmmcN2vα̟v≈10−9αm (16) masses, including the Planck mass, if confined to their Schwarzschild radii, will evaporate completely before col- Even the largest clusters of galaxies hosting α 1014 solar ∼ masses, if concentrated within one point, would dominate 2Thisisallowedbecausebothtimesrefertotheircommonzero theDarkEnergymerelywithinasphereof 100kmradius ∼ pointoftimeattheSchwarzschildhorizon. fromtheir centers.A lightneutrinovacuumcanthusbarely Journalname www.jn.net R.A.Treumann:Darkenergy 5 bemaderesponsiblefortheDarkMattercontentoftheUni- PerlmutterS. et al.: Astrophys. J. 517, 565, doi: 10.1086/307221, verse. 1999. PerlmutterS.,TurnerM.S.andWhiteM.:Phys.Rev.Lett.83,670, doi:10.1103/PhysRevLett.83.670,1999. References RiessA.G.etal.:Astron.J.116,1009,doi:10.1086/300499,1998. Turner M.S.: Dark matter and dark energy in the universe, CarrollS.M.,PressW.H.andTurnerE.L.:Ann.Rev.Astron.Astro- arXiv:astro-ph/9811454,1998. phys.30,499,doi:10.1146/annurev.aa.30.090192.002435,1992. WeinbergS.:GravitationandCosmology(JohnWiley&Sons,Inc., CuestaA.J.,NiroV.andVerdeL.:Phys.DarkUniverse13,77,doi: NewYork,1998). 10.1016/j.dark.2016.04.005,2016. Weinberg S.: Rev. Mod. Phys. 61, 1, doi: 10.1103/RevMod- DiracP.A.M.:Nature139,323,doi:10.1038/139323a0,1937. Phys.61.1,1989. Dunkel J. and Hilbert S.: Nature Phys. 10, 67, doi: WilczekF.:Phys.Reports104,143,doi:10.1063/1.2780139,1984. 10.1038/nphys2815,2014. ZeldovichY.B.:JETPLett.6,316,1967. EinsteinA.:Sitzungsber.Königl.Preuss.Akad.Wiss.,Phys.-Math. ZeldovichY.B.:Sov.Phys.Usp.11,381,1968. Klasse,142,8.Februar1917. LandauL.D.,LifschitzE.M.andPitaevskiiL.P.:StatisticalPhysics, Part2(Butterworth-Heinemann,OxfordUK,1998). MertensS.:J.Phys.:Conf.Series718,022013,doi:10.1088/1742- 6596/718/2/022013,2016. Oppenheimer J.R. and Snyder H.: Phys. Rev. 56, 455, doi: 10.1103/PhysRev.56.455,1939. www.jn.net Journalname