On the cosmological constant problem

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Reference

On the cosmological constant problem

LOMBRISER, Lucas

Abstract

An additional variation of the Einstein-Hilbert action with respect to the Planck mass provides a constraint on the average Ricci scalar that prevents vacuum energy from gravitating.

Consideration of the evolution of the inhomogeneous matter distribution in the Universe with evaluation of the averaging constraint on disconnected matter cells that ultimately form isolated gravitationally bound structures yields a backreaction effect that self-consistently produces the cosmological constant of the background. A uniform prior on our location in the formation of these isolated structures implies a mean expectation for the present cosmological constant energy density parameter of ΩΛ=0.704, giving rise to a late-time acceleration of the cosmic expansion and a coincident current energy density of matter.

LOMBRISER, Lucas. On the cosmological constant problem. Physics Letters. B , 2019, vol.

797, p. 134804

DOI : 10.1016/j.physletb.2019.134804

Available at:

http://archive-ouverte.unige.ch/unige:128782

Disclaimer: layout of this document may differ from the published version.

1 / 1

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

On the cosmological constant problem

Lucas Lombriser

DépartementdePhysiqueThéorique,UniversitédeGenève,24quaiErnestAnsermet,1211Genève4,Switzerland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received17April2019

Receivedinrevisedform21May2019 Accepted22July2019

Availableonline25July2019 Editor:M.Trodden

An additional variation of the Einstein-Hilbert action with respect to the Planck mass provides a constraint ontheaverageRicci scalarthat preventsvacuum energyfromgravitating. Considerationof theevolutionoftheinhomogeneousmatterdistributionintheUniversewithevaluationoftheaveraging constraint ondisconnected mattercellsthat ultimately form isolatedgravitationallybound structures yieldsabackreactioneffectthatself-consistentlyproducesthecosmologicalconstantofthebackground.

Auniformprioronourlocationintheformationoftheseisolatedstructuresimpliesameanexpectation forthepresentcosmologicalconstantenergydensityparameterof=⁰.704,givingrisetoalate-time accelerationofthecosmicexpansionandacoincidentcurrentenergydensityofmatter.

©^{2019TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The physical nature of the cosmological constant remains a persistentenigma immanent to Einstein’s Theoryof GeneralRel- ativity.Itisgenerallythoughttorepresentthegravitationalcontri- butionofvacuumfluctuations,anticipatedofadequatemagnitude toaccountfortheobservedlate-timeacceleratedexpansionofour Universe [1,2]. Quantumtheoretical calculations, however,exceed measurementbyseveralordersofmagnitude [3,4].Thismayimply amissingprescriptionforthecorrectcomputationofstandardvac- uum contributions but alsomotivates conjectures ofan undeter- mined mechanismthat preventsvacuum energyfromgravitating infullextentandthepossibilityofattributingcosmicacceleration to a different origin. The growing wealth of cosmological obser- vations,however,alsoputs strongconstraintsonalternatives toa cosmologicalconstantasexplanationoftheacceleratedexpansion suchasdarkenergyoradeparturefromGeneralRelativityatlarge scales [5–8]. A curiosity of cosmic acceleration can furthermore be found in the comparable magnitude of its associated current energydensitywiththat of matter, provoking theWhyNow? co- nundrum [4,9].

ThisLetter re-examinesthecosmologicalconstantproblemun- deranadditionalvariationofthegravitationalandmatteractions withrespecttothePlanckmass,whichallowsforaninterpretation ofthePlanckmassasglobalLagrangemultiplierthatimposesgen- eralrelativisticdynamicsonthemetricprescribingthespace-time forthematter fields.The resultingadditionalconstraintequation

E-mailaddress:lucas.lombriser@unige.ch.

isevaluatedrespectingtheinhomogeneousnatureoftheUniverse atsmallscales,andananalysisispresentedfortheimplicationof thisnewframeworkfortheoldcosmologicalconstantproblemof the non-gravitatingvacuum aswell asits new aspects ofcosmic accelerationandthecoincidenceproblem.

2. Non-gravitatingvacuumenergy ConsidertheEinstein-Hilbertaction S

=

^MPl2

2

M d⁴x

√

−

g

(

R

−

2

) +

M d⁴x

√

−

gLm

+

b

.

t

. ,

(1)

whereM^{denotes thecosmicmanifold,} isa freeclassical cos- mologicalconstant,matterfieldsareminimallycoupled,thestan- dard Gibbons-Hawking-York boundary term is adopted, and c=

¯

h=^{1.Variationoftheactionwithrespecttothemetricg}μνyields theEinsteinfieldequations

G_μν

+

g_μν

=

M⁻_Pl²T_μν

,

(2) where Tμν ≡ −²

δ(√

−^gLm)/δgμν /√

−^{g. In addition to the} metric variation, we shall now perform a variation of the ac- tion withrespectto thequadratic Planckmass M²_Pl. Thismaybe interpretedasusingM²_PlasaglobalLagrangemultiplierforatopo- logicalconstraint onthe matter action.Boundary conditionsmay be adaptedasinRef. [10] (alsoseeRef. [11]).Thisgivesthecon- straintequation

1 2

M d⁴x

√

−

g

(

R

−

2

) =

0

.

(3)

https://doi.org/10.1016/j.physletb.2019.134804

0370-2693/©^{2019TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

2 L. Lombriser / Physics Letters B 797 (2019) 134804

UsingthetraceofEq. (2) inEq. (3),itfollowsthat

M_Pl²

=

¹ 2

Md⁴x

√

−

^{g T}

Md⁴x

√

−

^g

≡

¹

2

^T

⁽⁴⁾

suchthattheEinsteinfieldequations (2) become G_μν

+

¹

2M_Pl⁻²

^T

^gμν

=

^M−_Pl^2Tμν

,

(5) replacingthefreecosmologicalconstantwithaspace-timeaverage ofthetraceoftheenergy-momentumtensor.

This new metric field equation is reminiscent of vacuum en- ergysequestering [12,13] but withadifferentfractionof M⁻_Pl^2T contributingdynamicallyasthecosmologicalconstant.Similarlyto thesequestering mechanism,vacuumcontributions to thematter sectorcancelout inEq. (2).However, thecancellationoccursdif- ferently. Forthisobservation, consider first the one-loop vacuum cosmologicalconstantincurvedspace-time [4]

vac

=

M⁻_Pl²

i

ni

m⁴_i 64

π

^2ln

m²_i

μ

²_i

+

^EW_vac

+ . . . ,

(6)

where m_i denote particle masses of species i, n_i represent the respective number of degrees of freedom with +/− ^{sign for} bosons/fermions,and

μ

iareunknownrenormalizationmassscales.

The electroweak vacuum contribution is given by ^EW_vac =

−^M−_Pl²(√

2/16)(m²_H/GF) with Higgs boson mass mH and Fermi constantGF,andonemayalsowishtoincludefurtherphasetran- sitions.As anexampleforcancellation, one maynow rewritethe massesasfractionsofthe Planckmassmi=λiMPl andusingthe scale

μ

i=^miexp(−^M2/M²_Pl)forsomerenormalizationmassMin- dependentofM_Pl,onefindsthatM²_Plvac∝^M2_Pl^{M2.Thesamescal-} ingis,forinstance,alsofoundfortheleading-ordervacuumcontri- butionofaWheeler space-timefoamdescription [14].Separating outthe vacuumandbare componentfromthe matterLagrangian density,Lm= ¯Lm−^M_Pl²(vac+bare),onefindsaftervariationof the actionwithrespect to gμν andM²_Pl that Gμν+(+vac+ _bare)gμν=^M−_Pl²

τ

μν andM²_Pl(+vac+_bare)=

τ

/2,where

τ

isspecifiedbyL¯m,andtherefore, G_μν

+

¹

2M_Pl⁻²

τ

^gμν

=

^M−_Pl²

τ

μν

.

(7) Hence,the vacuumandbarecosmologicalconstants donot grav- itate.Ratherthana cancellationbetweentheleft- andright-hand sidesofEq. (5),asinvacuumenergysequestering,forgivenvac and_bare,thevalueofissetbythetopologicalconstraintequa- tion (3) suchthat thesumofthecosmologicalconstantsmatches

τ

/^2.

Importantly, however, there is no guarantee that the residual cosmological constant obtained by the scaling of masses in the one-loop vacuum term (6) is radiatively stable or that the vac- uum contribution should even scale as M²_Pl. To simultaneously meet both caveats we shall first consider a scaling of the total vacuum energy density withthe Planck mass as M²α

Pl such that thetotalvacuumcontributiontotheaction (1) willbe−^M_Pl^2α¯vac withtheoverbarindicatinganindependenceofM²_Pl.Inparticular, thisallowsfor

α

=^{0.Whatisnecessaryforthe}cancellationisthe additionofaclassicalcounterterm−^M2_Pl^α¯_α.Morespecifically,fol- lowingthesameprocedureasforEq. (7) andsolvingfor¯α,one obtains

G_μν

+

¹

2

− α ⁽

¹

⁻ α ) +

^M−

2 Pl

τ

2

g_μν

=

^M−_Pl²

τ

_μν

,

(8)

where remains a free classical cosmological constant that is radiatively stable and determined by measurement. As observed next fromtheevolutionofmatterinhomogeneities,simplycor- responds tothe totalmeasured cosmologicalconstant.For

α

=^1, onerecoversEq. (7),andfor

α

=^{0,thedynamicalequationsofthe} local sequestering mechanism [13] with tot=¹/4

τ

^M−_Pl²+, where ≡/2. Notethat one may alsoconsider a seriesex- pansion of vac in M²_Pl, for instance from considering graviton loops [15].Withacounterpartexpansionoftheclassicalcosmolog- icalconstant,theresultingequivalentexpressionforEq. (3) cancels thesetermsandreproducesEq. (8).Similarly,quantumcorrections withhigher-derivativetermsinEq. (1) donotcontributetoEq. (3) orthefieldequationsifindependentofPlanckmass,andifdepen- dent on MPl,are cancelledby thesame classical expansion(also seeRef. [15]).

Finally, variations with respect to the Planck mass have also been performed in Ref. [13,15] and are in nature similar to a scalar-tensor theory in Jordan-Brans-Dicke representation with constant scalar field across the observable universe. A transfor- mationintoEinsteinframeremovesthevariationinM²_Plbutleaves variations with respect to an effective and a coupling in the matter sector, sharingsimilarities withthe proposalsofRefs. [12, 16–18], however also differing from them, e.g., by not impos- ing a constant four-volume as in unimodular gravity. The scalar field becomesaspace-timeconstant,forexample,byaδ-function generated through appropriate boundary conditions on an addi- tionalvectorfield [16,19] orbyasquaredfour-formfieldstrength contribution of a three-form gauge field as arises in supergrav- ity [13,15,19–22]. Variations in M²_Pl therefore find fundamental motivationrangingfromscalar-vector-tensororhigher-dimensional scalar-tensor theories to supergravity, string theory, or a type II multiverse.

3. Backreactionfromstructureformation

The four-volumeterminEq. (4) determiningtheresidualcos- mologicalconstantinEq. (5) growslargeforanold universe,and withintegrationoverthebackgroundmatterdensity

ρ

¯m,assumed spatially perfectlyhomogeneousandisotropic,

τ

_{= ¯}

ρ

m^eventu- allyvanishes.Fortheresidualtoreproducetheobservedcosmolog- icalconstantwithPlanckparameters [5],theUniverseshouldhave undergone animmediatecollapseatthescalefactora=⁰.926,at an ageof0.88H⁻₀¹,thus,about1 Gyrinthepast,andincontrast, animmediatecollapseatthecurrentepochwouldaccountfor81%

oftheobservedvaluewithadecreasingfractionforalongerfuture (cf. [23]).Whileitisinterestingthatthisvalueisclosetomeasure- ment,itisnotanexactrecoveryandmoreoverstandardcosmology doesnotforeseeanimminentcollapseoftheCosmos.

Importantly, however, the Universe is not perfectly homoge- neous and isotropic with structures growing to be even more pronouncedinthefuture.Weshallthereforenextexaminetheim- pactoftheevolution ofinhomogeneitiesinthegeneration ofthe residualcosmologicalconstantthroughEq. (4),thusabackreaction effectofstructureformation.Todescribetheinhomogeneousmat- terdistributionintheUniverse,consideraseparationofthematter content into disconnected cells Ui, whichare of maximal extent such that the mattercontainedwill remain gravitationally bound throughouttheirevolution,eventuallyformingisolated clustersin the far future.By a halo modelinterpretation, all matter is con- tained in these cells with each particle uniquely assigned to a singleUi.ThematteractioninEq. (1) becomes

S_m

=

i

Ui

dV₄Lm,i

+

M\

iUi

dV₄L∅

,

(9)

where Lm,i denote the Lagrangian densities of the matter in Ui and L∅ represents a Lagrangian density for the empty space.

L∅ ischosen such thattheresidual cosmologicalconstant inthe empty space matches that of the matter cells. We shall adopt L∅=^M2_Pl

/n+^M2n_Pl¯_∅

. Variation of the newaction with re- specttothemetricandM_Pl² oneachUigives

=

¹

2M_Pl²

|

_Ui

^T

Ui

,

(10)

where ^T_Ui indicate averages over the manifolds Ui. Variations onM\

iUi donotimpose aconstraintonsuch thatthereis noconflict withtheconstraintobtainedfrom thematter patches inEq. (10). It iseasy to verify that forn→(

α

₋1) the vacuum andbare cosmologicalconstants remainnon-gravitatinginanyof the patches, and asobserved in the following by examining the evolution of critical mass shells existing at turnaround between expansionandcollapse,

τ

_Ui=

τ

_Uj=^2M2_Pl∀ⁱ,j.

For simplicity, consider first the evolution and formation of nonlinearstructuresemployingthespherical collapsemodelwith approximationof forminghalos byspherically symmetrictop-hat overdensitiesanda furtherrestrictiontoonlypressurelessmatter andacosmologicalconstant ina spatially-flatcosmologicalback- ground.IthasbeencheckedinRef. [23] thatradiationcomponents onlymarginallyaffectthecomputation.The tophatisdefinedby its density

ρ

m≡ ¯

ρ

m+δ

ρ

m≡ ¯

ρ

m(1+δ) andmass M.The back- grounddensitycorresponds tothe spatialcell volumeintegration of all top-hat densities with respect to the total spatial volume oftheUniverse,whichfollowsfrommassconservationandrepro- ducesthestandardFriedmannequations.Fromenergy-momentum conservation in each matter cell ∇^μTμν|Ui =^{0, one derives the} evolutionequation [23]

y

+

2

+

^H H

y

+

¹ 2

m

(

a

)

y⁻³

−

1

y

=

0 (11)

for the dimensionless physical top-hat radius y=(

ρ

m/

ρ

¯m)^−1/3, whereprimesdenotederivativeswithrespecttolnaandm(a)≡ M_Pl⁻²

ρ

¯m/(3H²) with Hubble function H. The evolution of y can be determined settinginitial conditionsin the matter-dominated regimeai^{1,where y}i≡^y(ai)=¹−δi/3 and y_i= −δi/3 foran initialtop-hatoverdensityδi.Notethatfor y≈^1,Eq. (11) reduces tothefamiliardifferentialequationdeterminingthegrowthoflin- earmatterdensityperturbationsinCDM.

EvaluatingEq. (4) onagivenpatchUi,onefinds [23]

τ =

d⁴x

√

−

^g

τ

d⁴x

√ −

g

= ¯ ρ

m0

dlna H⁻¹

dlna H⁻¹a³y³

,

(12)

wheresubscripts of zero denote presentvalues. Importantly, the evolutionofy,andthereforethevalueof

τ

,dependsonlyonthe initialoverdensityδiandisindependentofthetop-hatmass.From thecompetitionbetweentheexpansionofthecosmologicalback- groundandthe selfgravity ofthemassivecell one cantherefore findauniversalminimal,criticalδ_ibelowwhichtheexpansionrate inthe future willexceed the effectof self gravitation andabove whicha given patchwill collapse. Hence, forcritical mass shells

τ

₌

τ

_Ui=

τ

_{Uj ∀}i,j.Using the symmetryofa(t)y(t) around t_turn=^tmax/2 formassshellsthateventuallycollapse,onefurther- moreobtains

τ

M²_Pl

obs

=

m

2

(

1

−

m

)

t_max

_tturn

0 dt a³y³

,

(13)

where obs denotes the observed cosmological constant that drivesthelate-timeaccelerationofthebackground.FromEq. (11)

it followsthatd(ay)/dt=^{0 andd2}(ay)/dt²=^{0 att}turn suchthat a³y³|tturn=m/(1−m)/2,andthus,

t_turn

0

dt a³y³

<

m

2

(

1

−

m

)

^tturn

,

(14)

implyingthat

τ

M_Pl²

_obs

>

^tmax

tturn

=

2

.

(15)

Thelongertheevolutionremainsata³y³|tturn,whichisthecasefor nearly criticalmatter patchesthat onlycollapseinthe farfuture, theclosertheratioapproximatesthislimit.Hence,inEqs. (7) and (8) itfollowsthatobs=^M−_Pl²

τ

/²=suchthatwerecoverthe EinsteinfieldequationGμν+_obsgμν=^M−_Pl²

τ

μν.Thisresultisin- dependentofassumingatop-hatdescriptionforthedisconnected mattercells.Criticalmatterpatchesthatexistlongenoughintothe cosmologicalconstantdominatedregimealwaysreachtheconstant valueofayinEq. (14) [24] andhencereproduceEq. (15).Forex- ample, usingthe analytic description for theevolution of critical massshellsinRef. [24] fortheintegrationinEq. (12) recoversthe limitinEq. (15).Intheveryfarfuturethecriticalcells willeven- tually collapseor fragment intosmaller structuresdueto energy lossfromradiation.Theseisolatedhalosmaythenfurthercollapse intoultramassiveblackholesthat eventuallyevaporate.It isalso possible thatthe entireUniversewill collapsedueto aHiggs in- stability prior to that. Theseprocesses are, however, expected to occur on much longer time scales than required forapproaching thelimitinEq. (15).Butasaconsequence,theintegralsinEq. (4) arenotexpectedtodiverge.Importantly,evenforan eternalinte- gration,thelimitinEq. (4) remainswelldefinedunderl’Hôpital’s rule with constant

τ

in the de Sitter future (also see Ref. [13]).

It isalsoworth emphasizingthat becauseof theintegrationover theentireexistenceintime,incontrasttosimilarapproacheswith causal settheory [25] orbackreaction fromlong-wavelengthper- turbations [26],

τ

^andobs inEq. (15) aretimeindependent.

Finally, note that the recovery of the observed cosmological constant in the Einstein equations is not a circular argument. A differentfractionofM⁻_Pl²

τ

contributingastheresidualcosmolog- icalconstantinEq. (7) wouldnotprovideaself-consistentsolution (cf. [23]).Moreover, aswe willseenext, theevaluationof

τ

by theevolutionofisolatedcriticalmatterpatchesoffersdirectimpli- cationsforthecoincidenceproblem.

4. Late-timeaccelerationandcoincidenceproblem

Theadditionalvariation ofthe Einstein-Hilbertactionwithre- spectto thePlanckmasscombinedwiththeevolution ofdiscon- nected critical matter cells provides a self-consistent framework toalleviatingtheoldcosmologicalconstantprobleminpreventing vacuum energy from gravitating. The observational evidence for a late-timeaccelerated expansion [1,2] addsa new aspect to the problemby the smallnon-vanishing _obs that furthermorecoin- cideswiththecurrentenergydensityofmatter.ThisWhyNow?co- incidenceproblemisnotself-evidentlyaddressedbyamechanism that gives rise to a free classical, radiatively stable cosmological constant.Whilealate-timeepochofcosmicaccelerationwillgen- erallybeassumedinthepresenceofanon-vanishingpositivecos- mologicalconstant,foran estimationoftherelativemagnitudeof its current contribution to the total energy density, one needs a sense of likelihood of our particular location in the cosmic his- tory.Onemay,forinstance,inspectthestar-formationhistory,and allowing theSuntohaveformedwithin 3

σ

ofthestar-formation

4 L. Lombriser / Physics Letters B 797 (2019) 134804

peakconstrainsthecurrentenergydensitiesofmatterandthecos- mologicalconstanttodiffernotbeyondafactorof10² [23].

The relation of the residual cosmologicalconstant to isolated matter cells however provides a more direct estimate of ≡ /(3H²₀) that is intrinsically linked to the evolution in Eq. (11).

Thedimensionlessphysicaltop-hatradius y lendsitselfheretoa meaningfulmeasureoflikelihood [23],describingtherelative ex- tentofthematter cellsandevolving withina finiterangefroma smallperturbation−δi/3 fromunityatearlytimestozeroatthe timeofitscollapseinthefarfuture.Adoptingauniformprioron y yieldsameanexpectationvalueof^y=¹/2,whichshallbeas- sumedhereasa likelylocation forourpresenttimet₀,implying thatthereisasmuchoftheevolutionofy aheadofusasthereis inthepast.Tocompute att₀onecanadoptamoreconvenient normalizationforthescalefactorathateliminatesthedependence of Eq. (11) on m. As long as the cosmological constant is not strictly vanishing,thereisalways a timewhen

ρ

¯m= ¯

ρ

₌M_Pl². Hence, normalizing the scale factor at the time of this equality, one therefore finds Heq≡^H(a=^aeq≡¹) with the correspond- ing energy density parameters defined at aeq≡1 simplifying to =m=¹/2.Requiring y(t₀)=¹/2andthennormalizingresults backtoa(t0)≡^{1 givesapresentenergydensityparameterforthe} cosmologicalconstantof

=

⁰

.

704

,

(16)

which is consistent with current cosmological observations such asfromthePlancksatellite [5] andtheDarkEnergySurvey [6] at the3

σ

and1

σ

levels,respectively.Notethatthedifferencetothe prediction of=⁰.697 in Ref. [23], found for an extension of theglobalsequesteringmechanism,isduetotherequiredcollapse ofthematterpatchesinthatscenarioattmax=^4t0 whereasthese structuresareconsideredmuchlonger-livinghere.

Finally,one observesthat y=¹/2isclosetotheepochofmin- imal y, which is another manifestation of a change from mat- terdominationtocomicaccelerationandhenceofthe coincident energy densities. This also implies, for instance, a current prox- imity to the maximal ratio of the densityof critical mass shells tothecriticalbackgrounddensity [24]. Cosmologicalcoincidences havefurthermorebeenpointedoutatrecombination,reionization, and the star-formation peak [9]. It may be interesting to elabo- ratewhethertheframeworkpresentedherecould establishalink betweentheseobservations orprovidenew approachesforother cosmologicalproblems,leavinga rangeof furtherexplorations to futurework.

5. Conclusions

The cosmological constant problemencompassing the weakly or non-gravitatingvacuum energy, thelate-time accelerated cos- micexpansion,andthecoincidentcurrentenergydensitiesofthe cosmologicalconstantandmatterremainsadifficultpuzzletocos- mology. Anew framework was presentedherethat proposesthe additionalvariation ofthe Einstein-Hilbertaction withrespectto the quadratic Planck mass on top of the usual metric variation.

It offers the interpretation of identifying the Planck mass with a globalLagrange multiplier that imposes generalrelativistic dy- namics on the metric prescribing the space-time for the matter

fields.Thevariationprovidesaconstraintequationontheaverage Ricci scalar that actsto prevent vacuumenergyfrom gravitating.

The evaluationof thisconstraintunder considerationof the evo- lution of the inhomogeneous matter distribution in the Universe in form ofdisconnected matter cells representing ultimatelyiso- latedgravitationally boundstructuresyields a backreaction effect that self-consistently produces the cosmological constant of the background.Withtheapplicationofauniformprioronthedimen- sionlessphysicalsizeofthesestructuresasameasureoflikelihood determining ourlocationin thecosmichistory,one findsa mean expectation forthecurrent energydensityparameter ofthe cos- mologicalconstantof=⁰.704.Theresultisingoodagreement withcurrent cosmologicalobservations,givingrise toa late-time accelerationofthecosmicexpansionandacoincidentcurrenten- ergydensityofmatter.Futureanalysiswillrevealifthepresented frameworkallowsforareinterpretationandpossibleunravellingof some cosmologicalobscurities such ascoincidencesidentified for the epochs ofrecombination, reionization, andthestar-formation peak with thetimes ofequality betweenthe energydensities of radiation,baryons,andthecosmologicalconstant.Itmayalsomo- tivate newapproaches forother unresolvedproblems of cosmol- ogy.

Acknowledgements

Thiswork wassupported bya SwissNationalScience Founda- tion Professorship grant (No. 170547). Please contact the author foraccesstoresearchmaterials.

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