Vacuum energy sequestering: The framework and its cosmological consequences

Vacuum energy sequestering: The framework and itscosmological consequences

Nemanja Kaloper1,* and Antonio Padilla2,†1Department of Physics, University of California, Davis, California 95616, USA

2School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom(Received 30 July 2014; published 14 October 2014; corrected 30 October 2014)

Recently we suggested a reformulation of general relativity which completely sequesters from gravity allof the vacuum energy from a protected matter sector, assumed to contain the standard model. Here weelaborate further on the mechanism, presenting additional details of how it cancels all loop corrections andrenders all contributions from phase transitions automatically small. We also consider cosmologicalconsequences in more detail and show that the mechanism is consistent with a variety of inflationarymodels that make a universe big and old. We discuss in detail the underlying assumptions behind thedynamics of our proposal, and elaborate on the relationship of the physical interpretation of divergentoperators in quantum field theory and the apparent “acausality” which our mechanism seems to entail,which we argue is completely harmless. It is merely a reflection of the fact that any UV sensitive quantity inquantum field theory cannot be calculated from first principles, but is an input whose numerical value mustbe measured. We also note that since the universe should be compact in spacetime, and so will collapse inthe future, the current phase of acceleration with wDE ≈ −1 is just a transient. This could be tested by futurecosmological observations.

DOI: 10.1103/PhysRevD.90.084023 PACS numbers: 04.50.Kd

I. INTRODUCTION

Cosmological observations suggest that the cosmologi-cal constant is different from zero. Yet we have no clear andcompelling argument which would explain the observedscale of the cosmological constant in quantum field theory(QFT). This fact is recognized under the term “the cosmo-logical constant problem” [1–3]. In a nutshell, the problemarises because of the universality of gravity. The equivalenceprinciple of general relativity (GR), which controls howenergy distorts geometry, posits that all forms of energycurve spacetime. Since in QFT even the vacuum possessesenergy density, given by the resummation of the QFTbubble diagrams, this means that the vacuum geometrygenerically must be curved. The scale of the curvature isLvac ∼ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

GNρvacp Þ−1. Cosmological observations then con-

strain it to be Lvac ≳ 1024 cm, implying that the energydensity of the vacuum must satisfy ρvac ≲ ð10−3 eVÞ4.However, attempts to estimate the contributions to ρvac inQFT, using the available scales in nature, exceed this valuemanyfold. An early example of this contradiction led Paulito famously quip “that the radius of the world would noteven reach to the Moon” (cf. 3.8 × 107 cm) after he hadestimated the vacuum energy contributions down to thescale set by the classical radius of the electron [4].Estimates involving higher energy cutoffs yield energydensities of the vacuum far in excess of this value, at least

as high as ðTeVÞ4, and possibly as high as Planckianscales, M4

Pl.The structure of GR, namely the underlying diffeo-

morphism invariance, allows one to freely add a classicalcontribution to the cosmological constant and tune it withtremendous precision to cancel off the vacuum energy.This means one needs to pick the classical piece to bethe opposite of whatever the (regulated) result of a fieldtheory calculation of the quantum vacuum energy yields,plus an extra ð10−3 eVÞ4, once one chooses to satisfy thecosmological observations with the leftover remainingafter the cancellation. However, this choice is unstable inany perturbative scheme for the computation of vacuumenergy: any change of the matter sector parameters orinclusion of higher order loop corrections to the vacuumenergy shifts its value significantly, often by Oð1Þ in theunits of the UV cutoff.1 One must then retune theclassical term by hand order by order in perturbationtheory. This is discomforting. The classical contributionto the cosmological constant is a separately conservedquantity, coded in the initial conditions of the universe.So the required retuning of its value needed to cancel thesubsequent corrections the vacuum energy in quantum

*[email protected]†[email protected]

1If exact, supersymmetry (SUSY) and/or conformal symmetrycan enforce the vanishing of vacuum energy. There are somecases where the corrections might be smaller, of quadratic orderin the cutoff [5–7]. However the corrections are quartic in thecutoff in generic cases, with interactions; without supersymmetryand conformal symmetry, the vacuum energy is given by thefourth power of the breaking scale of these symmetries [3,8–11].

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theory means that one—in reality—must choose a com-pletely different approximate description of the cosmichistory to attain the required smallness of the vacuumcurvature at late times.One could try to argue how this might be a red herring,

by proposing to sum all the quantum contributions to thetotal vacuum energy right away. After that one couldsimply pick the required value of the classical contribution,however tuned, and be done with it. One might evenwonder if the large corrections from known field theorydegrees of freedom are merely a problem of perturbationtheory rather than a real physical issue.2 However, this isclearly impossible if one does not know the full QFT up towhatever the fundamental cutoff may be. One can merelyguess the quantum contents of the universe beyond the TeVscale, and so cannot be sure just what it is that needs to becanceled. Further, quantum corrections to the vacuumenergy coming from phase transitions at late times wouldnot be properly accounted for either.In fact, the subtleties are brought into focus when one

recounts how one really deals with UV-divergent quantities.In QFT they must be renormalized: after a suitableregularization, which at a technical level introduces a cutoff(that renders the divergence formally finite, but verysensitive to whatever lies beyond the cutoff) the diver-gence—i.e. the contribution which diverges as the cutoff issent to infinity—must be subtracted by the bare counter-term. This is the real role of the classical contribution tothe cosmological constant: it is the bare counterterm whichone must keep in the theory for the purpose of renorm-alization. This is true even in effective field theories with ahard cutoff [12], where one still renormalizes away thecutoff-dependent terms which signify the dependence onthe unknown short distance physics. The remainder isfinite. However since it depends on an arbitrary subtrac-tion scale it cannot be computed from first principles.Thus since the vacuum energy is divergent its final valuecannot be predicted within QFT. It must be measured, justas, for example, a quadratically divergent mass of a scalarfield. Once the measurement is performed, and theterminal numerical value obtained, predictions can bemade for other observables, which depend on the renor-malized vacuum energy, but are not divergent themselves.Now, the measurement of the cosmological constant may

seem to be a simple exercise, since, after all, it is just onenumber. However, this depends on the nature of themechanism of subtracting the UV-sensitive part from it.If the subtraction scheme involves local fields, then theleftover value could in principle vary in spacetime andneeds to be measured more precisely. In other words, oncemust be able to distinguish between the cosmological

constant, and the contributions which vary in the far IRvery slowly, having weak dependence on very lowmomenta. The problem with measuring the vacuum energyis that it is a quantity which characterizes an object ofcodimension zero. To measure it, one therefore ultimatelyneeds a detector of the same codimension: namely, thewhole universe. Only in this way can one assure that theremainder is really a constant, being the same everywhereand at all times. The measurement with an arbitraryprecision thus must be nonlocal, in space and in time,from the viewpoint of an observer of a smaller codimensioncohabiting the universe. Remarks about the necessity of anonlocal measurement of cosmological constant were alsonoted in [13].Thus asking why the cosmological constant is small,

large, or anything in between ultimately does not—really—make sense in the context of QFT coupled of gravity. To seewhat it is, one must regulate it, renormalize it and measureit. The one aspect of the problem which remains, however,is why is the evaluation of the leftover cosmologicalconstant radiatively unstable? Or in other words, why dothe measurements of the cosmological constant afterregularization require large modifications of the finitesubtractions between consecutive orders in perturbationtheory? This issue is fully analogous to the gauge hierarchyproblem in QFT, and is well defined in QFT coupled togravity. Yet there are precious few clues as to how toaddress it, in contrast to various extensions of the standardmodel beyond the electroweak breaking scale designedprecisely to address the gauge hierarchy problem.3 Note,that this applies to any attempt to address the cosmologicalconstant problem within the realm of local QFT.At first sight, not only does this perspective invoke

nonlocality, but it also resembles a “circular argument.”There is a precedent for this logic, however. Recall how anotion of force is defined in Newtonian mechanics. Toformulate the second law, m~a ¼ ~F, one needs to firstintroduce a standard of mass whose accelerations inresponse to various externally applied agents calibrate theirforces. Only after such a calibration has been completed,can one begin to make predictions for all other masses in aNewtonian universe. The mass standard must be taken outof the clockwork, since no predictions can be made for it:its motion defines the forces, rather than the other wayaround. Of course, since the mass characterizes objectswhose codimension is 3, the sector of the universe whoseevolution cannot be predicted is a mere worldline, a set ofmeasure zero. This is far easier to ignore. That the massscales in QFTof the standard model are all set by the Higgs,whose own mass is quadratically divergent, and therefore

2Of course, this might sound implausible given that noconvincing alternatives to perturbation theory calculations ofcosmological constant have been offered to date.

3The absence of direct signatures from the LHC of any suchnew beyond Standard Model (BSM) physics to date may raiseconcerns whether naturalness is realized in nature. In whatfollows, however, we shall ignore this.

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can only be determined a posteriori, by measurement, onlyreaffirms this view further. The challenge, therefore, is notto ask why the cosmological constant is small, but why it isradiatively stable.Motivated by this “vacuum” of natural protection mech-

anisms of the vacuum energy, in a recent paper [14] weproposed a mechanism that ensures that all vacuum energycontributions from a protected matter sector are in factsequestered from (semi)classical gravity. Further themechanism automatically renders all the contributions tovacuum energy coming from the phase transitions small,and hence observationally harmless. Crucial for the mecha-nism is that we only consider the theory in the decouplinglimit of gravity, prohibiting gravitons as internal lines of thevacuum energy corrections.4 In this paper we will expandon that proposal, providing further details of the mecha-nism and its consequences. The sequestering works at eachand every order in perturbation theory, so there is no need toretune the classical cosmological constant when higherloop corrections are included. Instead, as we noted above,one is left with a residual cosmological constant, which isautomatically radiatively stable, being completely indepen-dent of the vacuum energy contributions from the protectedsector. In a universe that grows old and big, the residualcosmological constant is naturally small, along with anycontributions from phase transitions in the early universe.To be clear, our mechanism takes care of all vacuum energycontributions from a protected matter sector, which we taketo include the standard model of particle physics, but hasnothing to say about virtual graviton loops.5

Our mechanism is based on the introduction of globalconstraints in the formulation of the action describing thematter coupled to gravity. We postulate two such con-straints, by promoting the classical contribution to thecosmological constant, Λ, and the dimensionless parameterwhich controls the protected sector scales relative to a fixedPlanck scale, λ ∝ mphys=MPl, to global variables in thevariational principle, and extend the action by adding to it aterm σðΛ; λÞ outside of the integral. Then diffeomorphisminvariance guarantees that the vacuum energy always scaleswith λ in the same way, regardless of the order in the loopexpansion, and the constraints imposed by the variationwith respect to the global variables amount to dynamically

determining the value of Λ which always precisely cancelsthe vacuum energy. Since the theory remains completelydiffeormorphism invariant and locally Poincaré invariant,no new local degrees of freedom appear. So locally thetheory looks just like a usual QFT coupled to gravity, butwith an a posteriori cosmological constant determinedby a nonlocal measurement intrinsic to the universe’sdynamics—as it should be. An alternative way of describ-ing the mechanism is to integrate out the auxiliary variablesΛ and λ. Then our mechanism can be understood simply assetting all scales in the protected matter sector to befunctionals of the 4-volume element of the universe.6

This ensures that the vacuum energy quantum correctionsconsistently drop out order by order in perturbation theory,yet the local dynamics remains the same as in the standardapproach. Furthermore, the mechanism is consistent withphenomenological requirements, specifically with largehierarchies between the Planck scale, electroweak scaleand vacuum curvature scale, and with early universecosmology including inflation. In the paper [14] we haveillustrated the last point by showing that the mechanism iscompletely harmonious with the Starobinsky inflation.Here, in contrast we will show that it can also coexistwith a model where the last 60 e-folds of inflation aredriven by a quadratic potential from the protected sector.Nevertheless, the global cosmology must differ. For theprotected matter scales to be nonzero, the universe shouldhave a finite spacetime volume, being spatially compactand crunching in the future. This represents the mainprediction of our theory to date. An immediate corollaryis that the current phase of acceleration cannot last forever,which means that wDE ≈ −1 is merely a transient phase.Ergo, the dark energy cannot be an ever present cosmo-logical constant.7 In addition, since the cosmological con-tributions of local events are weighed down by the largespacetime volume of the universe, the contributions tovacuum energy from phase transitions are automaticallysmall. Hence in our framework the residual net cosmologi-cal constant, which sources the curvature of the vacuum, is

(i) purely classical, set by the complete evolution of thegeometry;

(ii) a “cosmic average” of the values of nonconstantsources;

(iii) automatically small in universes which grow oldand big.

This paper is organized as follows. In Sec. II, we reviewthe cosmological constant problem, outlining the keyfeatures, and ending with a discussion of Weinberg’svenerable no-go theorem prohibiting a local field theoryadjustment mechanism [3]. In Sec. III we present our main

4This limit of the problem is in fact precisely how Zeldovitchformulated it in [1].

5The graviton contributions to the vacuum energy are contextdependent (see e.g. [15] for an exploration of loop contributionsto vacuum energy in string theory). The standard model con-tributions are not, once we assume QFT to describe them. Theresult of our protection mechanism might be desensitized fromgravity, for example, if we can supersymmetrize gravitationalsector down to a millimeter scale, in which case gravitationalcontributions to vacuum energy in field theory would neverexceed the observational bounds. While this looks interesting,further work is needed to determine if such an extension couldwork.

6An equivalent interpretation is to fix the protected field theoryscales, and make the Planck scale a functional of the worldvolume of the universe.

7Although it can approximate a cosmological constant veryaccurately for a long period.

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proposal in detail, explaining how the cancellation ofvacuum energy works, focusing on a pair of symmetriesthat are ultimately responsible for this cancellation. InSec. IV we study the kinematics of our theory in somedetail. In particular we estimate the relevant historic cosmicaverages, demonstrating that the residual cosmologicalconstant will not exceed the critical density today, andshow that the contributions from (many) astrophysicalblack holes in Sec. IV B are small. In Sec. IV C we outlinethe procedure for finding general cosmological solutionswhich satisfy the global constraints by their selection fromthe families of known solutions of general relativity. Thevery important question of contributions to vacuum energyfrom early universe phase transitions are the topic of Sec. V,where we show that after the transition they are automati-cally small in large old universes. In Sec. VI, we show howour mechanism is compatible with the observed mass scalesin particle physics, and how this can be achieved withoutintroducing any new hierarchies. Generalizations of ourproposal are presented in Sec. VII, including the effect ofradiative corrections to the Planck scale. Section VIII isdevoted to inflation. Commenting on consistency withStarobinsky inflation, we also show that single monomialinflationary models, and specifically quadratic inflation,driven by fields from the protected sector, are alsocompatible with our mechanism. We further comment onthe possible conflict with eternal inflation, which generi-cally yields universes with infinite volume. Finally, inSec. IX, we briefly summarize some key features andobservational signatures of our proposal.

II. THE COSMOLOGICAL CONSTANT PROBLEM

We begin with a brief review of the vacuum energyproblem from the viewpoint of QFT coupled to gravity.Further details can be found in many reviews of the subject,for example [3,16–19]. So consider a QFT of matter,which—to be complete in the flat space limit—requiresa specification of a UV regulator. By a UV regulator, wemean a procedure for rendering the divergences in renor-malizable QFTs formally finite, so that they can besystematically subtracted by the addition of bare counter-terms. This could just be a hard cutoff in nonrenormalizabletheories, which is really an avatar of the full renormaliza-tion procedure in the UV completion of the theory(assuming that one exists). Couple this theory to a covariantsemiclassical theory of gravity universally (i.e. minimally),by defining the source of gravity to be stress energy of thematter QFT. Then, by the equivalence principle, thevacuum energy of the QFT, which corresponds to theresummation of the bubble diagrams in the loop expansionand drops out from the expressions for the scatteringamplitudes in flat space8 couples to gravity via the

covariant measure, −Vvac

Rd4x

ffiffiffig

p, where g ¼ j det gμνj

is the determinant of the metric gμν. Hence the vacuumenergy sources gravity, yielding an energy-momentumtensor Tμν ¼ −Vvacgμν. Since Vvac is divergent, we shouldalso include a bare cosmological constant—i.e. the“classical contribution”—as a counterterm in the action,−Λbare

Rd4x

ffiffiffig

p, so that it is actually the combination

Λtotal ¼ Λbare þ Vvac that gravitates. Cosmology thenrequires that this combination should not exceed theobserved dark energy scale, i.e., Λtotal ¼ Λbare þ Vvac≲ð10−3 eVÞ4.In practice, one computes Vvac in perturbation theory,

using the loop expansion in flat space.9 This means, onetruncates the infinite series of bubble diagram contributionsto a desired precision in the loop expansion, evaluates thesediagrams and sums them up. For example, at, say, one loop,one simply evaluates the vacuum energy to one loop,Vvac ¼ V tree

vac þ V1-loopvac , and then subtracts the infinity in

Vvac by(i) regulating Vvac with, e.g. a hard cutoff;(ii) adding the bare term Λbare to subtract away the UV

sensitive contribution from Vvac;(iii) tuning the remainder in Λbare so that the observa-

tional constraint is satisfied.Although the definition of the calculated Λtotal ¼ Λbare þVvac ensures that it is a constant, for it to actuallybe measured one must continue measuring it across thewhole world volume of the universe, as we explained inthe Introduction. This fact is obscured by the description ofthe flat space subtraction procedure utilized here—since itis commonplace—because the subtraction procedure weemploy is local (being imposed by an external observerwho is computing Λtotal), and so the residual leftover Λtotalis a constant by consequence.Note that this implies that Λtotal can be very different

from either Λbare or Vvac. It is not predicted—it is to bemeasured, as we stressed above, and so it can be anything.This is standard in renormalization in QFT. So, saying thatΛtotal ≪ Vvac is not a problem by itself. The problem is thatthe renormalization procedure sketched above is notperturbatively stable. For example, generically, the regu-lated two-loop correction is V2-loop

vac ∼ V1-loopvac , which is much

greater than Λtotal that remains after the subtraction of theUV-sensitive contribution at the one-loop level. This meansthat the bare (“classical”) cosmological constant Λbare,which was selected with a high degree of precision tocancel the UV-sensitive contribution to the one-loopvacuum energy needs to be retuned by Oð1Þ in the unitsof the UV cutoff, in order to yield a new two-loop Λtotal thatremains compatible with the observational bound. Oncegravity is turned on, so that we can view Λbare as a classical

8Which is guaranteed by the fact that in flat space field theory onecan freely recalibrate the zero point energies of any matter fields.

9Note that the results are consistent with the UV-sensitiveterms computed in curved space using covariant regularizationtechniques [20,21].


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conserved quantity fixed by some cosmological initialcondition, this implies that the initial condition must bealtered dramatically in order to readjust the two-loop valueof Λtotal to satisfy the bound. Thus the observable value ofthe total cosmological constant is not a reliable, stableprediction of the theory. It is very sensitive to both thedetails of the UV physics, which are unknown, and to thecosmological initial conditions, which are notoriouslydifficult to reconstruct. This issue repeats at each successiveorder in the loop expansion, indicating the radiativeinstability of the small value of the observable Λtotal.A simple illustrative example is provided by a massive

scalar field theory with quartic self-coupling and minimalcoupling to gravity. At one loop, the relevant Feynmandiagrams correspond to a single scalar loop with externalgraviton legs carrying zero momenta. The sum of all suchdiagrams generates a term −Vϕ;1-loop

vacRd4x

ffiffiffig

p, so it suffices

to calculate just one. Perhaps the simplest contribution isthe tadpole diagram, given by [18]

tadpole ¼ 1

2

Zd4kð2πÞ4

�kμkν − 1

2ημνðk2 þm2Þ

k2 þm2

�

¼ −i2ημνVϕ;1-loop

vac : ð1Þ

Using dimensional regularization, one finds [18]

Vϕ;1-loopvac ¼ −

m4

ð8πÞ2�2

ϵþ finiteþ ln

�M2

UV

m2

��; ð2Þ

whereMUV is the UV regulator scale. The bare countertermone would add to cancel the divergence would therefore be

Λbare ¼m4

ð8πÞ2�2

ϵþ ln

�M2

UV

M2

��; ð3Þ

where M is the subtraction point. The one-loop renormal-ized cosmological constant would then be

Λren ¼ Vϕ;1-loopvac þ Λbare ¼

m4

ð8πÞ2�ln

�m2

M2

�− finite

�: ð4Þ

Note that the remainder depends on lnM, illustrating thedependence of the renormalized cosmological constant onthe arbitrary subtraction scale M. This is why the value ofΛren can only be fixed by a measurement. At two loops, weconsider the so-called scalar “figure of eight” with externalgraviton legs. Its contribution to vacuum energy is given byVϕ;2-loopvac ∼ λm4. For perturbative theories without finely

tuned couplings, where λ ∼Oð0.1Þ (as for example thestandard model Higgs), the higher loop corrections remaincompetitive with the leading order contributions.One could try to improve the situation by designing

dynamical mechanisms which cancel the vacuum energycontributions to the total cosmological constant order by

order in the loop expansion. An example is provided byeither supersymmetric theories or conformal field theories.In both cases, it is the underlying unbroken symmetrywhich automatically sets the vacuum energy at any order inthe loop expansion to zero. In the case of supersymmetry,this follows from the cancellations of loop diagramsbetween bosons and fermions with degenerate masses.For conformal theories, it is the unbroken conformalsymmetry which simply scales away all the dimensionalcharacteristics of the vacuum. Both examples representtechnically natural solutions of the problem: the vacuumenergy, and the total cosmological constant vanish as aconsequence of the underlying symmetry. However, in thereal world, both of these symmetries, if they exist, arebroken, at least below the electroweak breaking scale. Theprediction which follows from the breaking, and which is infact technically natural, is that the resulting vacuum energyshould be at leastM4

EW. Restoring either supersymmetry orconformal symmetry (or both) would render the vacuumenergy to be zero, and so these symmetries could beprotecting the cosmological constant from the correctionslarger thanM4

EW. However, in our world the observed valueof the cosmological constant must be much smaller, by atleast 60 orders of magnitude.An alternative approach could be to look for some

dynamical extension of the minimal framework of QFTcoupled to gravity, where a symmetry protecting a smalltotal cosmological constant could be hidden. This was anidea behind many past attempts to address the cosmologicalconstant problem via the adjustment mechanism. Thisapproach is obstructed by the venerable Weinberg’s no-go theorem [3], which precludes a dynamical adjustmentmechanism of the cosmological constant in the frameworkof an effective QFT coupled to gravity. We review it here, asit provides a very clear and important guide for theformulation of our mechanism of vacuum energy seques-tration, to which we will turn in the next section.Consider a local and locally Poincaré-invariant 4D QFT

describing finitely many degrees of freedom below acertain UV cutoffMUV. Imagine, for starters, that it couplesminimally to gravity described by the 4D metric gμν. Nextlook for a Poincaré-invariant vacuum; if it exists, the theoryadmits a vacuum with a zero cosmological constant.Clearly, the aim is to evaluate under which circumstancesthis can happen. Now, in the Poincaré state, all the fields,regardless of their spin are annihilated by the translationgenerators. Choosing them as the coordinate basis, thisimplies that the fields must be Φm ¼ const and gμν ¼ ημνmodulo a residual rigid GL(4) symmetry, inherited fromdiffeomorphisms. Hence the field equations for matter andgravity, respectively, reduce to

∂L∂Φm

¼ 0;∂L∂gμν ¼ 0: ð5Þ


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The residual rigid GL(4) symmetry of the system isrealized linearly. The metric and matter fields and theLagrangian density transform as gμν → JμαJνβgαβ, Φm →J ðJÞΦm and L → detðJÞL, respectively, where J ðJÞ issome appropriate representation of GL(4). Now, since thefield equations (5) are linearly independent, and thePoincaré-invariant ground state is unique, by virtue ofthe matter field equations in (5) and the residual GL(4)symmetry, the Lagrangian in this state is

L ¼ ffiffiffiη

pΛ0ðΦmÞ; ð6Þ

where Φm are field configurations which extremize Λ0.Here, clearly, one can think of Λ0 and Φm as the renormal-ized variables, computed to some fixed order in the loopexpansion, and using some type of a covariant regulator andsubtraction scheme to cancel the divergent contribution andobtain a finite value. The bottom line is that the finalequation of (5) then implies Λ0ημν ¼ 0, and so consistencyrequires settingΛ0 ¼ 0—by hand, and resetting it to zero—also by hand—if further loop corrections are included.Technically, the problem is that the last equation of (5) iscompletely independent of the other equations.What if that were not the case? Clearly, an adjustment

mechanism, if it exists, would set the value of Λ0 auto-matically to zero (or sufficiently close to it) once fieldvariables attain their extrema. This could be enforced byrequiring that the trace of the last equation of (5) is replacedby an equation of the form

2gμν∂L∂gμν ¼

Xm

fmðΦnÞ∂L∂Φm

: ð7Þ

If a theory exists such that one of the resulting gravitationalequations yields (7), it would open the road to constructinga successful adjustment mechanism within the domain ofeffective QFT coupled to gravity. Furthermore, to ensurethe absence of fine-tunings, one can require that fmðΦnÞ area set of smooth functions in field space, which only dependon the fieldsΦm in order to maintain Poincaré invariance. Acloser look [3] reveals that the first order (functional) partialdifferential equation (7) is in fact a requirement that thetotal action describing the theory is invariant under atransformation generated by

δgμν ¼ 2ϵgμν; δΦm ¼ −ϵfmðΦnÞ: ð8Þ

This is clearly a scaling symmetry in disguise. Now, since(i) the number of field theory degrees of freedom is finite,and (ii) the functions fmðΦnÞ are smooth, one can perform afield redefinition Φm → ~Φm ¼ ~ΦmðΦnÞ, so that in terms ofthe new field theory degrees of freedom the transformations(8) simplify to

δgμν ¼ 2ϵgμν; δ ~Φ0 ¼ −ϵ; δ ~Φm≠0 ¼ 0: ð9Þ

That such a transformation exists is guaranteed bya theorem of differential geometry on embeddings ofhypersurfaces [22] and Poincaré symmetry. The fieldspace generator of transformations (8) is a smooth finite-dimensional vector field X ¼ P

mfmðΦnÞ ∂∂Φm

, so that thetransformation (8) represents a motion along the flow linesgenerated by X. Then one defines the new field spacecoordinates by picking the parameter measuring the flowalong X (which being a single degree of freedom must be ascalar by unbroken local Poincaré invariance) and thecoordinates orthogonal to it. They are the n − 1 integrationconstants ~Φm≠1 of the solution of the system of differentialequations δΦm

δ ~Φ0

¼ fmðΦnÞ which remain after using one to

pick the origin of ~Φ0. See Fig. 1.In the new coordinates in field space, the field theory

equations, the Poincaré invariance of ground state and theresidual GL(4) symmetry imply that the Lagrangian in thisstate is

L ¼ ffiffiffiη

pΛ0ð ~Φm≠0Þe−4 ~Φ0 : ð10Þ

Again, this is the regulated and renormalized vacuumenergy coming from a calculation involving contributionsfrom all diagrams in the loop expansion involvingsome fixed, but arbitrarily chosen, finite number ofloops. Now, the modified gravity equation (7) yieldsΛ0ð ~Φm≠0Þe−4 ~Φ0 ¼ 0. This could be solved while avoidingfine-tuning Λ0 ¼ 0 if one allows ~Φ0 to run off to infinity.However, because Eq. (10) is the renormalized vacuumenergy at an arbitrary finite order in the loop expansion, andsince the mechanism suppressing it must operate inde-pendently of the order of the loop expansion to guaranteeradiative stability, the form of (10) must be preserved orderby order in perturbation theory. But this means that in orderto satisfy this, all the scales in the theory must depend on

the power of expð− ~Φ0Þ given by their engineering

m 0

~

~

m

0

FIG. 1. Field redefinition Φm → ~Φm.


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dimension. This includes the scales of the regulator and thesubtraction point, and is necessary because the corrections to(10) come as the powers of these scales and logarithms oftheir ratios. Only then will all the corrections, includingthe log-dependent terms, scale the same way as in (10).So (10) will accomplish the task if all the matter fields andthe regulator couple to the “rescaled” metric gμν ¼expð−2 ~Φ0Þgμν. However, after canonically normalizingthe fields in the matter sector on the background given bysolving the vacuum equations, one finds that all dimensionalparameters in QFT must scale as md ∝ e−d ~Φ0 . This meansthat in the limit ~Φ0 → ∞ not only does the cosmologicalconstant vanish, but so do all the other scales in the theory. Inother words, this restores conformal symmetry in the fieldtheory sector. As noted this is not our world. Hence theproblem.

III. OUR PROPOSAL

In the attempt to evade the Weinberg’s no-go theorem, in[14] we proposed a very minimal modification of generalrelativity with minimally coupled matter. The idea was topromote the classical cosmological constant Λ to that of aglobal dynamical variable, and introduce a second globalvariable λ corresponding to scales in the matter sector. Thevariation with respect to these new variables however isused to impose a global constraint on the dynamics of thetheory, similar in spirit to the constraint imposed in theclassic isoperimetric problem of variational calculus [23].To do this, we supplemented the local action with anadditive function σðΛ; λÞ which is not integrated over thespacetime. Then the variations with respect to Λ, λ selectthe values of these parameters. In particular, this proceduresets the boundary condition for the variable λ such that atany order of the loop expansion it takes precisely the rightvalue to completely absorb away the whole of vacuumenergy contribution from the matter sector at that looporder. Our approach is a simplified hybrid of thinking aboutGR as unimodular gravity, with a variable Λ specified byarbitrary cosmological initial conditions, and the proposalof Linde [24], further considered by Tseytlin [25], of usingmodified variational procedures to fix values of globalvariables such as Λ. Our variational prescription howeverdiffers from those previously considered in that it is moreminimal, and that it uses a global scaling symmetry as anorganizing principle for accounting for all quantum vacuumenergy contributions.The idea is to start with the action

S ¼Z

d4xffiffiffig

p �M2

Pl

2R − Λ − λ4Lðλ−2gμν;ΦÞ

�þ σ

�Λ

λ4μ4

�;

ð11Þ

where the matter sector described by L is minimallycoupled to the metric ~gμν ¼ λ2gμν. We imagine that the

standard model is included in it. For simplicity, we considerthe matter to only belong to this sector, henceforth referredto as the “protected sector.” One could also include othermatter sectors to the theory, which could couple to adifferent combination of λ and gμν. However, vacuumenergy contributions from such sectors would not cancelautomatically. In what follows we will focus on the matterdynamics only from the protected sector, for simplicity’ssake. The function σðzÞ is an (odd) differentiable functionwhich imposes the global constraints. The parameter μ is amass scale introduced on dimensional grounds. The preciseform of σ is determined in order to fix the particle masses inQFT in accordance with a specific phenomenologicalmodel of physics beyond the standard model, as we willdiscuss in Sec. VI. For example, an asymptotically expo-nential form of σ allows us to choose μ to be up near thePlanck scale, etc.The global variable λ sets the hierarchy between the

matter scales and the Planck scale, since

mphys

MPl¼ λ

mMPl

; ð12Þ

where mphys is a physical mass scale of a canonicallynormalized matter theory, and m is the bare mass in theLagrangian. As an illustration, consider a scalar field withbare mass m,

ffiffiffi~g

pLϕ ¼ 1

2

ffiffiffi~g

p½~gμν∂μϕ∂νϕþm2ϕ2�

¼ 1

2

ffiffiffig

pλ4½λ−2gμν∂μϕ∂νϕþm2ϕ2�

¼ 1

2

ffiffiffig

p ½gμν∂μφ∂νφþm2physφ

2�; ð13Þ

where φ ¼ λϕ is the canonical scalar and the physical massis mphys ¼ λm.It is absolutely essential for our cancellation mechanism

to enforce the UV regulator of this sector to also couple toexactly the same metric as the fields from L. This isnecessary in order to ensure the correct operational form ofthe vacuum energy sequestration from L, and makes theeffective UV cutoffMUV and the subtraction scaleM scalewith λ in exactly the sameway as mass scales from L, givenby Eq. (12). This can be accomplished, for example, byregulating the theory with a system of Pauli-Villarsregulators, which couple to the metric ~gμν.With this in place, the form of (11) guarantees that all

vacuum energy contributions coming from the protectedLagrangian

ffiffiffig

pλ4Lðλ−2gμν;ΦÞ must depend on λ only

through an overall scaling by λ4, even after the logarithmiccorrections are included. This follows since the regulator ofthe QFT introduces contributions where the scales dependon λ in the same way as those from the physical degrees offreedom from L. That ensures the cancellation of λ in loop


084023-7

logarithms, and so only the powers of λ remain. This factfollows from diffeomorphism invariance of the theory,which since λ is a global variable, is unbroken both forthe metrics gμν and ~gμν. Diffeomorphism invariance ensuresthat the full effective Lagrangian computed fromffiffiffig

pλ4Lðλ−2gμν;ΦÞ ¼ ffiffiffi

~gp

Lð~gμν;ΦÞ, including all quantumcorrections—and the divergent terms, as well—still cou-ples to the exact same ~gμν [26]. Of course, this is true onlywhen we restrict our attention to the vacuum energycontributions from loop diagrams involving only protectedsector degrees of freedom in the internal lines which areintegrated over. This is, however, all of the vacuum energyin the decoupling limit of gravity, which we calculate usingstandard flat space field theory techniques in locally freelyfalling frames. As we stressed previously, we recouplethese terms to gravity by the minimal coupling procedure,taking gravity as a purely (semi)classical field whichmerely serves the purpose of detecting vacuum energy.At present, this is the sharply formulated part of thecosmological constant problem, and as we discussed inthe Introduction we choose to focus on it alone.The field equations that follow from varying the action

(11) with respect to global auxiliary fields Λ, λ are

σ0

λ4μ4¼

Zd4x

ffiffiffig

p; 4Λ

σ0

λ4μ4¼

Zd4x

ffiffiffig

pλ4 ~Tμ

μ; ð14Þ

where ~Tμν ¼ − 2 ffiffi~g

p δSmδ~gμν is the energy-momentum tensor

defined in the “Jordan frame.” To go to the “physical”frame, in which matter sector is canonically normalized,note that

Tμν ¼ gμα

�−

2ffiffiffig

p δSmδgαν

�¼ λ2 ~gμα

�−2λ4ffiffiffi~g

p δSmλ2δ~gαν

�¼ λ4 ~Tμ

ν;

where σ0 ¼ dσðzÞdz . As long as it is nonzero,10 eliminating it

from the two equations (14) one finds

Λ ¼ 1

4hTμ

μi;

where we defined the 4-volume average of a quantity byhQi ¼ R

d4xffiffiffig

pQ=

Rd4x

ffiffiffig

p. Note that Λ is the bare cos-

mological constant, which is now however completelyfixed by this condition. One has to define these averagesmeaningfully, since after regulating the divergences theycan still be indeterminate ratios when the spacetime volumeis infinite [9]. We will address this in short order, since thisissue has very important physical implications and ties intohow our proposal evades Weinberg’s no-go theorem.

The variation of (11) with respect to gμν yields

M2PlG

μν ¼ −Λδμν þ λ4 ~Tμ

ν; ð15ÞwhereGμ

ν is the standard Einstein tensor. After eliminatingΛ and canonically normalizing the matter sector, thisbecomes

M2PlG

μν ¼ Tμ

ν −1

4δμνhTα

αi: ð16Þ

This equation is one of the two key ingredients of ourproposal. Note that this is the full system of ten fieldequations, with the trace equation included. It differs fromunimodular gravity [27–35], where although the restrictedvariation removes the trace equation that involves thevacuum energy, this equation comes back along with anarbitrary integration constant, after using the Bianchiidentity. Here there are no hidden equations nor integrationconstants, and all the sources are automatically accountedfor in (16). Crucially, however,

1

4hTα

αi

is subtracted from the right-hand side of (16). This meansthat the hard cosmological constant, be it a classicalcontribution to L in (11), or a quantum vacuum correctioncalculated to any order in the loop expansion, divergent(but regulated) or finite, never contributes to thefield equations (16). To see this explicitly, we take theeffective matter Lagrangian, Leff at any given order inloops, and split it into the renormalized quantum vacuumenergy contributions (classical and quantum) ~Vvac ¼h0jLeffð~gμν;ΦÞj0i, and local excitations ΔLeff ,

λ4ffiffiffig

pLeffðλ−2gμν;ΦÞ ¼ λ4

ffiffiffig

p ½ ~Vvac þ ΔLeffðλ−2gμν;ΦÞ�:ð17Þ

It follows that Tμν ¼ Vvacδ

μν þ τμν, where Vvac ¼ λ4 ~Vvac

is the total regularized vacuum energy and τμν ¼2ffiffig

p δδgμν

Rd4x

ffiffiffig

pλ4ΔLeffðλ−2gμν;ΦÞ describes the physical

excitations. By our definition of the historic 4-volumeaverage, hVvaci≡ Vvac and so the field equations (16)become

M2PlG

μν ¼ τμν −

1

4δμνhτααi: ð18Þ

The regularized vacuum energy Vvac has completelydropped out from the source in (16). There is a residualeffective cosmological constant coming from the historicaverage of the trace of matter excitations:

Λeff ¼1

4hτααi: ð19Þ

10And nondegenerate: it cannot be the pure logarithm. Inthat case Eq. (14) turns into two independent constraints,1Λ ¼ R

d4xffiffiffig

p, 4 ¼ R

d4xffiffiffig

pTμ

μ, the latter placing an artificialconstraint on the matter sector.


084023-8

We emphasize that this residual cosmological constant hasabsolutely nothing to do with the vacuum energy contri-butions from the matter sector, including the standardmodel contributions. Instead, after the cancellation ofthe vacuum energy contributions enforced by the globalvariables Λ, λ, the residual value of hτααi is picked as a“boundary condition,” resulting from the measurement ofthe finite part of the cosmological constant after it wasrenormalized. As we noted above, this measurementrequires the whole history of the universe for this onevariable, in effect setting the boundary condition for it atfuture infinity. Obviously, it is crucial that the numericalvalue of Λeff is automatically small—one might hope for it,since it is the contribution coming predominantly from theIR modes in the cosmological evolution, but a quantitativeconfirmation is necessary since otherwise the proposalwould have failed. It turns out that indeed Λeff is auto-matically small enough in large old universes, as we willshow in the next section. This follows from the fact that ouruniverse is large and old, which in the very least is aconsequence of extremely weak anthropic considerations.In actual fact, a large universe like ours can be formed by atleast 60 e-folds of inflation, which we will show isconsistent with the vacuum sequestration proposal. Thesmallness of Λeff is completely safe from radiative insta-bilities, and the required dynamics is essentially insensitiveof the order of perturbation theory. The apparent acausalityin the determination of Λeff is therefore of no consequence;a better terminology is a posteriority, that follows from thenature of the measuring process needed to set the numericalvalue of the renormalized cosmological constant. This hasno impact on local physics and so cannot lead to anypathologies normally associated with any local acausality.One might only worry if there do not appear someunexpected restrictions on a possible range of numericalvalues which the renormalized cosmological constantmight have, that follow from the global constraints whichwe introduced. We will turn to this later, when we addresshow the solutions of the theory are constructed.The second key ingredient of our mechanism is that

the field theory spectrum has a nonzero gap, which canbe arbitrarily large compared to jhτααij1=4. Otherwise, themechanism would fail to provide a way around Weinberg’sno-go theorem—reducing yet again to a framework with atleast scaling symmetry. This means that on the solutionsthe parameter λ must be nonzero, since λ ∝ mphys=MPl.But by virtue of the first equation of (14), since σðzÞ is adifferentiable function, if λ is nonzero,

Rd4x

ffiffiffig

pmust be

finite. Fortunately this can be accomplished in a universewith spatially compact sections, which is also temporallyfinite: it starts with a bang and ends with a crunch. In otherwords, the spacelike singularities regulate the worldvolume of the universe without destroying diffeomorphisminvariance and local Poincaré symmetry. Therefore, in ourframework the universes which support nonscale invariant

particle physics must be spatiotemporally finite. Infiniteuniverses are solutions too, however their phenomenologyis not a good approximation to our world, since all scalesin the protected sector vanish. Note that the quantity whichcontrols the value of λ, and therefore mphys=MPl, is theworld volume of the universe, and not hτααi. This isessential for the phenomenological viability of our pro-posal, since it separates the scales of the observedcosmological constant and masses of particle physics.We will address these points in more detail in Sec. IV,

focusing on quantitative statements, demonstrating, inparticular, that the residual cosmological constant is nat-urally small, never exceeding the critical density of theuniverse today. This is of course the final touché of ourmodel, guaranteeing that the cancellations of the vacuumenergy did not—in turn—necessitate large classical valuesto appear. Before we address these, and other phenomeno-logically important issues, let us look in more depth at justhow the vacuum energy contributions get canceled.A holy grail of the past attempts to protect the

cosmological constant from radiative corrections, whichis partially realized with supersymmetry and conformalsymmetry, was to find a symmetry which will insulate thefinite term left after renormalization from higher orderloop corrections. As it turns out, a system of twosymmetries is the reason why our cancellation works.Our action (11) has approximate scale invariance, brokenonly by the Einstein-Hilbert term,

λ → Ωλ; gμν → Ω−2gμν; Λ → Ω4Λ; ð20Þ

such that the action changes by

δS ¼ M2Pl

2Ω−2

Zd4x

ffiffiffig

pR ¼ M2

Pl

2Ω−2hRi

Zd4x

ffiffiffig

p: ð21Þ

The other symmetry appears as an approximate shiftsymmetry

L → Lþ ϵm4; Λ → Λ − ϵλ4m4; ð22Þ

under which the action changes by

δS ¼ σ

�Λ

λ4μ4− ϵ

m4

μ4

�− σ

�Λ

λ4μ4

�≃ −ϵσ0

m4

μ4: ð23Þ

The scaling symmetry ensures that the vacuum energy atan arbitrary order in the loop expansion couples to thegravitational sector exactly the same way as the classicalpiece. The “shift symmetry” of the bulk action thencancels the matter vacuum energy and its quantumcorrections. The scaling symmetry breaking by the gravi-tational sector is mediated to the matter only by thecosmological evolution, through the scale dependence onRd4x

ffiffiffig

p, and so is weak. This is why the residual


084023-9

cosmological constant is small: substituting the first equa-tion of (14) and using λ ¼ mphys=m, we see thatδS≃−ϵm4λ4

Rd4x

ffiffiffig

p ¼−ϵðmphys

MPlÞ4½M4

Pl

Rd4x

ffiffiffig

p �. Holdingthe volume fixed, this is small when mphys=MPl ≪ 1,vanishing in the conformal limit11 λ ∝ mphys → 0. Thisrestoration of symmetry renders a small residual curvaturetechnically natural.Let us see how this occurs at the level of the field

equations (14) and (16). First off, note that the historicaverage of the trace of the right-hand side of (16) is zero,hRi ¼ 0. Hence, by (21), the action (11) is in fact invariantunder scaling (20) on shell. Next, the variation of the matterLagrangian under the shift L → Lþ ϵm4 is equivalent toletting ~Tμ

ν → ~Tμν − ϵm4δμν. Then if gμν, Λ and λ solve

Eqs. (14) and (15) for a source ~Tμν, by manipulating the

constraint equations (14) one finds that

gμν ¼ gμν; Λ ¼ Λzσ0ðzÞzσ0ðzÞ ; λ4 ¼ λ4

σ0ðzÞσ0ðzÞ ; ð24Þ

are solutions with a source ~Tμν − ϵm4δμν, where z ¼

Λ=λ4μ4, and z ¼ zð1 − 4ϵm4=h ~TiÞ. Crucially, gμν remainsunchanged, which means that a shift of the vacuumenergy—by adding higher order corrections in the loopexpansion—is absorbed by an automatic readjustment ofthe global variables, and so has no impact whatsoever onthe geometry.Finally, since the role of the two global auxiliary fields, λ

and Λ, is to enforce global constraints, they only appearalgebraically in the theory and can be integrated outexplicitly. The resulting formulation of the theory providesfurther insight into the protection mechanism. So using thevariable z defined above, the constraints (14) are

σ0ðzÞλ4μ4

¼Z

d4xffiffiffig

p; z ¼ h ~Tα

αi4μ4

: ð25Þ

Solving for λ, Λ yields

λ ¼�σ0ðh ~Tα

αi=4μ4Þμ4

Rd4x

ffiffiffig

p�1=4

;

Λ ¼ σ0ðh ~Tααi=4μ4Þh ~Tα

αi=4μ4Rd4x

ffiffiffig

p : ð26Þ

Substituting these back into the action, we obtain

S ¼Z

d4xffiffiffig

p �M2

Pl

2R − λ4Lðλ−2gμν;ΦÞ

�þ Fðh ~Tα

αi=4μ4Þ;

ð27Þ

where λ is given by Eq. (26), and ~Tαα ¼

~gαβð 2 ffiffi~g

p δδ~gαβ

R ffiffiffi~g

pLð~gμν;ΦÞÞ. An important point here is that

the function FðzÞ ¼ σðzÞ − zσ0ðzÞ is the Legendre trans-form of σ. This explicitly shows that the independentvariable z (or Λ) has been traded for the new independentvariable σ0ðzÞ—which, by the first equation of (26), isλ4μ4

Rd4x

ffiffiffig

p. Thus, the independent variable of the theory

is really not the cosmological counterterm Λ, (like in GR orin unimodular extension of GR) but the world volume ofthe universe,

Rd4x

ffiffiffig

p. So this in fact reveals how our

mechanism operates. The cosmological system respondsinstead to changes in the spacetime volume, rather thanchanges in Λ. Furthermore, in GR, when higher ordercorrections to the vacuum energy are included, Λ staysfixed forcing the spacetime volume

Rd4x

ffiffiffig

pto absorb the

corrections (inflating a lot due to a large vacuum energy).For our scenario it is exactly the opposite: it is thespacetime volume that remains fixed, forcing Λ to adjust.As a result, the cosmological system is stable againstradiative corrections to the vacuum energy.Indeed, let us consider the gradient expansion of the

effective matter Lagrangian, including any number of loopcorrections. To the lowest order, we retain only the fulleffective potential of the theory, and truncate it to the zeromomentum limit, which represents the vacuum energy ~Vvac.We find that Leff ¼ ~Vvac, h ~Tα

αi ¼ −4 ~Vvac, and the action is

S ¼Z

d4xffiffiffig

p �M2

Pl

2R −

�σ0ð− ~Vvac=μ4Þ

μ4R ffiffiffi

gp

�~Vvac

�

þ Fð− ~Vvac=μ4Þ

¼Z

d4xffiffiffig

p �M2

Pl

2R

�þ σð− ~Vvac=μ4Þ: ð28Þ

We see how the choice of coupling of gμν to the standardmodel given by Eq. (27) guarantees that all gμν dependenceis canceled in the vacuum energy contribution to the action.This is how the standard model vacuum energy is seques-tered. Diffeomorphism invariance guarantees that the fulleffective Lagrangian computed from

ffiffiffi~g

pLð~gμν;ΦÞ, includ-

ing all quantum corrections, still couples to the same

~gμν ¼ ½σ0ðh ~Tααi=4μ4Þ

μ4R ffiffi

gp �1=2gμν. The loop corrections are accom-

modated by (small) changes of the scaling factor

½σ0ðh ~Tααi=4μ4Þ

μ4R ffiffi

gp �1=2, while the metric gμν remains completely

unaffected. This works to any order in the loop expansion,as is the core element of the adjustment mechanism.So to summarize, we see that the key point of our

mechanism is a dramatically altered role ofRd4x

ffiffiffig

p, which

provides the way around Weinberg’s no-go theorem [3].Instead of Λ in GR or in its unimodular formulation, nowRd4x

ffiffiffig

pis the independent variable. Taking all the

physical scales in the protected matter sector to depend11Defined by fixing

Rd4x

ffiffiffig

pand taking μ → ∞ in the first

equation of (14).


084023-10

on it, automatically removes all the vacuum energy con-tributions to drop out. For example, if we were to take thelinear function σðzÞ ¼ z in (11) and declare L to be literallythe standard model, integrating Λ and λ out and rewriting(11) as just Einstein-Hilbert action coupled to the standardmodel, the only modification is that the Higgs vev v isreplaced by v=ðμ4 R d4x

ffiffiffig

p Þ1=4. Further, in a collapsingspacetime, since

Rd4x

ffiffiffig

pis finite, the protected sector

QFT has a nonzero mass gap, by virtue of (14), while theresidual cosmological constant hτααi=4 ≠ 0, but it iscompletely independent of all the vacuum energy correc-tions, and as we will show below is automatically small in alarge old universe. The particle sector scales are affected bythe historic value of the world volume.12 However, thisdependence of field theory scales on

Rd4x

ffiffiffig

pis completely

invisible to any nongravitational local experiment, bydiffeomorphism invariance and local Poincaré symmetry.Locally the theory looks just like standard GR, in the (semi)classical limit—but without a large cosmological constant,and without its radiative instability, at least in the limit of(semi)classical gravity.

IV. HISTORIC INTEGRALS AND QUANTITATIVECONSIDERATIONS

We have already stated that in a universe which iscompact, starting in a bang and ending in a crunch, theworld volume

Rd4x

ffiffiffig

pis finite. Also the residual cosmo-

logical constant given by the historic average of the trace ofstress-energy tensor −hτααi=4 is automatically small in bigand large universes, as long as τμν satisfies the dominantand null energy conditions (DEC and NEC, respectively).In this section we will give a proof of this statement at theclassical level. Semiclassically, it has been shown that theintegrals hτααi are finite if NEC is valid in [36]. Then forbounded ταα our argument automatically extends to thesemiclassical case, too. Further, we will consider thetechniques for finding solutions of field equations in oursetup. From a practical point of view, solving the equa-tion (18) requires a bootstrap method: allow hτααi ¼ C tobe arbitrary to start with, find the family of solutionsparametrized by this integration constant, and finallysubstitute this family of solutions back into hτααi in orderto show that a subset of them is compatible with the initialchoice. This is akin to the gap equation in superconduc-tivity. Here we will focus on the proof of existence ofsolutions to this procedure in the case of backgroundFriedmann-Roberston-Walker (FRW) cosmologies. In theforthcoming paper [37] we will show that consistentsolutions of this type including the current epoch of

(transient) acceleration exist. Specific dynamics will resortto the earliest quintessence with linear potential dating backto, at least, 1987 [38], as the really relevant model oftransient cosmic acceleration. We will show in [37] that ithas a natural embedding in our proposal. Other scenariosand aspects of transient acceleration have also beenconsidered (see e.g. [39]).

A. Historic integrals and the cosmological background

The integrals which appear in the definition of ourhistoric averages are

Zd4x

ffiffiffig

p⊃ Vol3

Ztcrunch

tbang

dta3; ð29Þ

Zd4x

ffiffiffig

pταα ⊃ Vol3

Ztcrunch

tbang

dta3ð−ρþ 3pÞ; ð30Þ

where the cosmology takes place over a finite proper timeinterval tbang < t < tcrunch, with a scale factor a, and finitespatial comoving volume Vol3. We will approximate theintegrals for the most part by the FRW geometry with fluids,whose energy density and pressure are ρ and p respectively.We will eventually impose DEC and NEC, which combinedtogether require jp=ρj ≤ 1. The integrals are regulated—i.e.,finite—because we assume that spatial sections are compact,and that the universe starts at a bang and ends in crunch. Thisguarantees that (29) is finite, and that the QFT spectrum hasa nonzero mass gap. Further, wewill see that validity of NECthen also guarantees that (30) is also finite, and bounded bythe contributions from near the turning point, being esti-mated by the product of the minimal energy density duringcosmological evolution and the age of the universe. At thetechnical level, we will take the evolution to be symmetric intime for simplicity. This does not impair the generality of ouranalysis because the integrals are dominated by the con-tributions near the turning point. Approximating the spatialvolumes by homogeneous 3-geometries, we will first con-sider time integrals. We will briefly return to contributionsfrom inhomogeneities later, when we consider the effectsfrom black holes.For starters, note that the DEC and NEC together

—jp=ρj ≤ 1—put a very strong bound on the contributionsto the integral (30) from the bang and crunch singularities.Namely while at the singularities the energy density andpressure diverge, the spacelike volume which they occupyshrinks, and jp=ρj ≤ 1 ensures that the rate of shrinking isfaster than the divergence of ταα. Indeed, near the singu-larities ρ scales as ρ ∼ 1=ðt − tendÞ2, by virtue of theFriedman equation, where tend is either of the singularinstants. Since in this limit a3 ∼ ðt − tendÞ2=ð1þwÞ, theintegrand ∝ a3ρ ∼ ðt − tendÞ−2w=ð1þwÞ will not diverge pro-vided jwj < 1. For w ¼ þ1 the divergence is at mostlogarithmic, with coefficients≃Oð1Þ so that when properly

12In the case where σ is a linear function, they would be toosensitive to the initial conditions in the early universe. This is whythe forms of σ which are asymptotically exponential are pre-ferred, since they reduce the sensitivity to merely logarithmiccorrections.


084023-11

cut off at the physical Planckian density surfaces thesecontributions are still much smaller than the cutoff. Henceour historic averages will always be finite in a bang/crunchuniverse for all realistic matter sources. Having shown that(30) is bounded, we can proceed with a more carefulcomparison of contributions to (30) from different cosmo-logical epochs.Now, to get a more accurate estimate of various con-

tributions to (30) we can split the history of the universeinto epochs governed by different matter sources, which wecan approximate as perfect fluids. So let us consider onesuch epoch, during which a fluid with equation of state wicontrols the evolution of a in an interval ai < a < aiþ1.The contribution of this epoch to the temporal integralin (29) is

Ii ¼Z

tiþ1

ti

dta3 ¼Z

aiþ1

ai

daa2

H

¼ a3iþ1

Hiþ1

Zaiþ1

ai

daaiþ1

�a

aiþ1

�2 Hiþ1

H; ð31Þ

whereH ¼ _a=a andHi,Hiþ1 are the values ofH at a ¼ ai,aiþ1 respectively. Using the Friedmann equation,H2 ≃H2

iþ1ðaiþ1

a Þ3ð1þwiÞ, and so

Ii ¼2

3ð3þ wiÞ�a3iþ1

Hiþ1

��1 −

�aiaiþ1

�3ð3þwiÞ=2�

: ð32Þ

For jp=ρj ≤ 1, this integral is always finite, even if ai → 0.Similarly, the contribution of this epoch to the temporalintegral in (30) is

Ji ¼Z

tiþ1

ti

dta3ð−ρþ 3pÞ ¼ ð3wi − 1Þ ρiþ1a3iþ1

Hiþ1

×Z

aiþ1

ai

daaiþ1

�a

aiþ1

�2�Hiþ1

H

�ρ

ρiþ1

¼ ð3wi − 1Þ ρiþ1a3iþ1

Hiþ1

Zaiþ1

ai

daaiþ1

�a

aiþ1

�2�

HHiþ1

�;

ð33Þ

where in the last step we have used H2 ≃H2iþ1

ρρiþ1

. Since

H2 ≃H2iþ1ðaiþ1

a Þ3ð1þwiÞ,

Ji ¼2ð3wi − 1Þ3ð1 − wiÞ

�ρiþ1a3iþ1

Hiþ1

��1 −

�aiaiþ1

�3ð1−wiÞ=2�

:

ð34Þ

Clearly these integrals describe contributions from bothexpanding and contracting regimes.Note that (33) is logarithmically divergent near the

singularity for a stiff fluid wi ¼ 1, as we noted above.

But this divergence is a red herring. We cut the evolutionoff at a time when the density reaches Planck scale,lpl ≲ a≲ a⋆, and so

Jsingularitystiff ¼ 2

�ρ⋆a3⋆H⋆

�log

�a⋆lpl

�≃

�H⋆MPl

�log

�a⋆lpl

�;

ð35Þ

where H⋆ and ρ⋆ are the Hubble scale and energy densitywhen a ¼ a⋆. Hence Jsingularitystiff ≤ 1. For all other cases with−1 ≤ w < 1, the integral Ji is automatically finite—and small.Next we consider the contributions from the turning point

at a time T. During this time, the universe is approximatelystatic, with the scale factor roughly a constant, a≃ amax,over a time interval Δt. The turning point happens roughlyat the time given by the total age of the universe,T ≃ Δt≃ 1=Hage. The effective Hubble parameter at thattime is approximately zero, by virtue of a cancellationbetween different contributions to the energy density, wheresomemust be negative to trigger the collapse (e.g., a positivespatial curvature or a negative potential). But since 1=a2maxmeasures the characteristic curvature of the universe at thattime, jRj ∼H2

age ∼ 1=a2max, we obtain

Iturn ¼Z

TþΔt=2

T−Δt=2dta3 ≃ a3maxΔt ∼

a3max

Hage∼

1

H4age

: ð36Þ

The contribution from the turning point to the temporalintegral in (30) is

Jturn ¼Z

TþΔt=2

T−Δt=2dta3ð−ρþ 3pÞ≃Oð1Þa3maxρageΔt

∼ρageH4

age; ð37Þ

where ρage ∼M2PlH

2age corresponds to the characteristic

energy density of the universe at that time.Clearly, (36) and (37) are dominant contributions to (29)

and (30), respectively. To see this, consider first twoconsecutive epochs away from the turning point. Aftersimple algebra one finds Ii=Ii−1 ¼ Oð1Þðaiþ1

aiÞ3ð3þwiÞ=2,

Ji=Ji−1 ¼ Oð1Þðaiþ1

aiÞ3ð1−wiÞ=2 using the Friedmann equa-

tion. Clearly, the epoch with a larger value of the scalefactor at any of its end points gives a dominant contribution.This means that the larger contributions to (29) and (30)come from the regimes of evolution nearer the turningpoint. Indeed, at the turnaround, we find that the contri-butions from the quasistatic interval at during the turning

and the adjacent epochs are Iturn=Ii ¼ Oð1ÞðHiþ1

HageÞða3max

a3iþ1

Þ andJturn=Ji ¼ Oð1ÞðHage

Hiþ1Þða3max

a3iþ1

Þ. Now Hage < Hiþ1, and

amax > aiþ1. Next, as long as jwij ≤ 1, we also have


084023-12

H2age ≳H2

iþ1ðaiþ1

amaxÞ6, which follows from the fact that the

stiff fluid yields the fastest allowed decrease of H withexpansion. From these inequalities we conclude that thecontributions to (29) and (30) are indeed dominated by theevolution near the turning point. Thus we obtain

�Zd4x

ffiffiffig

p �FRW

¼ Oð1ÞVol3H4

age; ð38Þ

�Zd4x

ffiffiffig

pταα

�FRW

¼ Oð1ÞVol3ρageH4

age: ð39Þ

Therefore the residual cosmological constant is

Λeff ¼1

4hτααi≃Oð1Þρage ≃Oð1ÞM2

PlH2age ≲M2

PlH20: ð40Þ

This is guaranteed to be bounded by the current criticaldensity of the universe as long as the universe lives at leastH−1

age ≳H−10 ∼ 1010 years. This proves the previously stated

claim that Λeff is automatically small enough in large olduniverses, which is a crucial check of our proposal. On theother hand, this cannot—by itself—be taken as a predictionof the late epoch of cosmic acceleration. For one, the signof Λeff is determined by equation of state of the dominantfluid close to the turning point; if w > 1=3, Λeff is positive,and if w < 1=3 it is negative. Second, if the residualcosmological constant were dominant today, future col-lapse would have been impossible as long as all othermatter satisfies standard energy conditions. Thus Λeffcannot be identified with dark energy, since the universemust collapse in order to be phenomenologically accept-able within the context of our proposal. The currentcosmological acceleration must be a transient phenomenon,with the net potential turning negative some time in thefuture, and/or our universe were spatially closed, with asmall but nonzero positive spatial curvature. We will returnto this issue in the future [37].

B. Historic integrals and black holes

One may worry13 that astrophysical black holes mayprovide a significant contribution to the spacetime volumeof the universe. The point is that the familiar black holesolutions in GR have infinite volumes in their interiors.However, black holes in compact spacetimes are different.For simplicity let us model a small black hole in acollapsing universe as a Schwarzschild black hole on aspacetime where time is an interval limited by the total ageof the universe, Δt≲ 1=Hage. The analytic extension acrossthe horizon then trades the radial and time coordinates,making t spacelike inside. But because t is compactified,

that automatically means the internal volume is finite.Further reduction of black hole contributions may comefrom the fact that they evaporate, although this seems to bemuch less important for astrophysical black holes whichmainly gain weight in the course of their lifetime.Properties of black holes in compactified spacetimes havebeen studied in [40].Now we estimate the black hole internal volume. Wewill

stick with the model of a small Schwarzschild black holewith a compactified time direction. The geometry isapproximately ds2¼−ð1−rH=rÞdt2þð1−rH=rÞ−1dr2þr2dΩ2, where dΩ2 is the metric on the unit 2-sphere,and rH is the Schwarszchild radius. So the interiorspacetime volume is

Zinterior

d4xffiffiffig

p ¼ 4π

ZΔt

0

dtZ

rH

0

drr2

¼ 4π

3r3Ht≲ 4π

3r3H=Hage: ð41Þ

The total contribution from all black holes to the spacetimevolume of the universe cannot significantly exceed thecontribution from the largest black holes known to exist,with a mass ∼109M⊙ [41]. Assuming there is one suchblack hole in all 1011 galaxies in the Hubble volume today,and assuming that our Hubble volume is typical, weextrapolate the galaxy population in the whole collapsinguniverse to be about Ngal ∼ 1011H3

0a30, where H0 ∼

10−33 eV is the current Hubble scale and a0 > 10=H0

the current scale factor. This yields

�Zd4x

ffiffiffig

p �galBH

≲ 1011H3

0a30

Hage

�109

M⊙M2

Pl

�3

¼ 10−19�

a0amax

�3 1

H4age

; ð42Þ

as the contribution of all black hole interiors to the totalspacetime volume of the universe. We have used that thescale factor near the turning point is amax ∼ 1=Hage. Sincea0 < amax, we see that the galactic black hole interiors’contribution to the spacetime volume is completely neg-ligible in comparison to the background cosmology.Estimating the contribution of their mass to hτααi is now

straightforward. Since the total volume integral is unaf-fected, and since

Rinterior d

4xffiffiffig

pταα ≲MBHΔt≲MBH=Hage,

where MBH is the total mass in all black holes, the blackhole contribution to hτααi is bounded by MBHH3

age. Againusing the extrapolated number of black holes in the wholeuniverse to be Ngal ∼ 1011H3

0a30, the total mass in black

holes is MBH ∼ 1020H30a

30M⊙. So after a straightforward

calculation, using M⊙ ≃ 1039MPl, we find that the con-tribution to hτααi is bounded by

13We thank Paul Saffin and Alex Vikman for raising thisquestion.


084023-13

hτααigalBH ≲ ρnow10

�a0amax

�3

; ð43Þ

where ρnow ¼ M2PlH

20 is the critical density today. As

a0 ≲ amax, this is clearly subleading even with our over-estimate of the black hole population, ρBH ∼ MBH

a30

∼ 0.1ρnow.

C. Historic integrals and FRW cosmology

So far we have been considering the implications of thehistoric integrals and averages on the phenomenology ofsolutions, assuming they exist. Do they? Now we addressthis question.14 In principle, one should expect that suchsolutions should exist, on the grounds that one couldalways start with a family of solutions in GR withan arbitrary value of cosmological constant, constructthe geometry, and then pick the specific value of the—initially arbitrary—bare cosmological constant to satisfyΛeff ¼ hτααi=4. Equivalently, this merely means taking thespecial solution from each one parameter family of sol-utions, parametrized by Λbare, for which hRi ¼ 0. In a way,the freedom of picking Λ and the arbitrariness of initialconditions appear to guarantee the existence of specificsolutions. Nevertheless, an explicit proof is still requiredgiven that the determination of Λeff ¼ hτααi=4 involvesspecific future boundary conditions which must be pickedto find the self-consistent solution. This “bootstrapping”logic is very similar to what one encounters in BCS theoryof superconductivity, where one also has to solve a nonlocalequation—the gap equation—to decide if the solutionsexist in the first place. Here, we will review the conditionsunder which such solutions do exist in the family of FRWcosmologies. In a forthcoming publication [37] we willconsider the requirements to build a fully realistic model,consistent with cosmic phenomenology, that includes atransient phase of acceleration like the one we see today.For simplicity we will focus on solutions which are

dominated by a single fluid and the residual effectivecosmological constant. Adding more ingredients infact makes the existence of solutions easier to prove, byadding additional parameters. So, FRW cosmologies withspatial curvature k, a cosmological constant Λeff and asingle perfect fluid with equation of state parameterw ¼ p=ρ ¼ const, which obeys DEC and NEC, jwj ≤ 1,are described by the Friedmann equation,

3M2Pl

�H2 þ k

a2

�¼ ρ0

�a0a

�3ð1þwÞ

þ Λeff : ð44Þ

The special solution of (44) which also satisfies Eq. (19),which in this case reduces to

Λeff ¼ −ð1 − 3wÞ

4

Rdta3ρRdta3

; ð45Þ

is the solution of our theory. We will assume that thespatial volume is compactified, so that the integral over thespatial coordinates stays finite. We will further assume thatρ0 > 0. It is instructive to rewrite (44) in the formresembling the energy conservation equation for a particlein one dimension,

_a2 þ VeffðaÞ ¼ −k; Veff ¼ −κ2

a1þ3w −Ω2a2; ð46Þ

where κ2 ¼ ρ0a3ð1þwÞ0

3M2Pl

> 0, but Ω2 ¼ Λeff3M2

Plcan take

either sign.Let us first consider the (simpler) case of universes with

k ≤ 0. When w ≥ 1=3, by (45) Λeff and Ω2 are non-negative. So the “potential” Veff in (46) is negative definite.For the total conserved “energy” −k ≥ 0, the solutions existfor all _a ≠ 0, since the lines Veff and _a2 ¼ const > 0 neverintersect. An expanding solution expands forever.Therefore the temporal “volume”

Rdta3 is infinite, and

so Λeff vanishes. However, this also means thatRd4x

ffiffiffig

pdiverges. Hence these solutions are all cosmologies inwhich field theory has no mass gap, and are not goodcandidates to accommodate our universe.When −1 < w < 1=3, Λeff and Ω2 are negative. The

potential Veff is a sum of two powers, a−ð1þ3wÞ and a2, withopposite coefficients. However, since w > −1, the quad-ratic always wins at large a. This means that a “particle”moving from the origin (an expanding universe startingwith a bang) with a total energy which is non-negative(−k ≥ 0) encounters a potential barrier at some finite afrom the origin and turns around. So such configurationsalways admit a collapsing solution, and for the given “freeparameters” describing the solutions of (44) in GR (Λeffand the value of a at the turning point, specified by theintegration constant), one needs to pick the combinationwhich solves (45), which as we see exists. Note that thisdoes not mean tuning the initial conditions for a to find thesolution. It means, for the given initial conditions speci-fying the solution, one needs to pick the right value ofthe a posteriori parameter Λeff. The same logic applies toall cases.The case k > 0 is slightly more subtle. In the language of

the one-dimensional dynamics (46) we are now lookingfor states with negative conserved energies. Now, when−1 < w < −1=3, since Λeff and Ω2 are negative, and so is1þ 3w, the potential Veff is a sum of two positive powersof a, with opposite coefficients. This sum is negativebetween the origin and some (large) value of a, beyondwhich it turns positive, going again as a2 at large a. Butsince we are looking for trajectories with −k < 0 now, itmeans that the total energy −k can be greater than theeffective potential Veff only in a finite interval of a’s. So

14We thank Guido D’Amico and Matt Kleban for discussionson the subject of this section.


084023-14

classical motion is only possible between these two turningpoints, and it continues forever. This case corresponds tooscillating cosmologies of [42], which have finite spatialvolume, and where the negative effective cosmologicalconstant, needed for oscillating motion, is generated by ourconstraint. Here, even though

Rd4x

ffiffiffig

pis infinite, and so

the QFT is gapless, Λeff remains finite as can be seenreadily by splitting the integration into the sum of integralsover full periods. Yet, such universes are not phenomeno-logically viable since QFT is scale invariant.When −1=3 < w < 1=3, Λeff and Ω2 are still negative,

but 1þ 3w > 0. Thus Veff is a combination of an infinitepotential well at the origin and a quadratic barrier far away.So solutions with −k < 0 always exist, again representinguniverses which start with a bang, expand to the maximumradius and subsequently crunch. The cases w ¼ −1=3 andw ¼ 1=3 are special limits. In the former, the barrier hasfinite depth, again admitting collapsing solutions, while inthe later case the barrier is pushed to infinity since Λeff ¼ 0by scale invariance. This latter case corresponds to aradiation dominated universe with vanishing cosmologicalconstant.Finally, when w > 1=3, Λeff and Ω2 are positive. Hence

the potential Veff is negative definite, diverging at the originand infinity, with a maximum in between. Since −k < 0,solutions exist if the “conserved energy” −k is smaller thanthe value of Veff at the maximum, representing againcosmologies that start with a bang and end with a crunch.The solutions would not have existed if −k were larger thanthe maximum. But this is not the case; the limiting case,where −k is exactly equal to VeffðmaxÞ, would have been astatic Einstein universe, that would have been eternal. Thiswould require a ¼ _a ¼ 0. Taking the derivative of (46) onecan easily check that this requires w ¼ −1, contradictingw > 1=3. So therefore the closed cosmologies describingbang/crunch always exist for w > 1=3.Before closing this section, we should clarify the role

and the implications of the constraint hRi ¼ 0 whichfollows from tracing and integrating Eq. (18) and usingEq. (19). On FRW geometries with spatially compactsmooth sections it reduces to

Rdta3ð _H þ 2H2 þ k

a2Þ ¼ 0after factoring out the finite spatial integral. Integrating theterm ∝ _H by parts yields

a3Hjtcrunchtbang ¼Z

dta3�H2 −

ka2

�: ð47Þ

If we were to take the limits of integration to be the timeintervals where the scale factor literally vanishes, and thedominant stress-energy sources determining the behaviorofH obey NEC, the left-hand side would have vanished forprecisely the same reasons as we have already discussed inSec. IVA. This would appear to have ruled out spatiallyopen and flat FRWuniverses, as is clear from Eq. (47) since

in this case the right-hand side could only have vanished ifH and k were exactly zero at all times.This argument is a little hasty, however. The point is that

the integration must be ended at times when the curvature Rreaches the Planck scale, and not when the scale factorexactly vanishes. Thus the left-hand side does not vanish,but simply represents a boundary condition which theterminal geometry must satisfy to ensure that (1) it is asolution of Einstein’s equations and (2) that the vacuumenergy is sequestered. Indeed, the left-hand side is merely anonzero number which represents the difference betweena3H at the beginning and the end of cosmology, encodingthe posterior determination of the effective cosmologicalconstantΛeff ¼ hτi=4, communicated to the on-shell geom-etry by way of Einstein’s equations. For small universes,the contributions to this term from beyond the cutoff aresufficiently important that we can treat it as an arbitraryboundary condition allowing any value of k. However, forlarge and old universes, the integrals on the right-hand sideare dominated by the contributions from large volumes nearthe turning point, and thus essentially insensitive to theboundary conditions. Since the left-hand side is muchsmaller, the integrals on the right-hand side must includenegative contributions that yield cancellations to match theleft-hand side. Thus having a large old approximately FRWuniverse does require k ¼ 1 (and so Ωk < 1). Smallinhomogenous universes may evade this. A similar obser-vation was made in a different context in [34].

V. VACUUM ENERGY SEQUESTERING ANDPHASE TRANSITIONS

In the course of the cosmological evolution the fieldtheory dynamics may go through many phase transitions:QCD phase transition, electroweak symmetry breaking, etc.This means that the local vacuum of the theory changes. Asa result, since after renormalization the effective potentialgoverning the dynamics is fixed, this implies that the netrenormalized cosmological constant may actually changein time and space. The variation is typically of the order ofΔV ∼M4

PT, whereMPT is the energy scale of the transition.So the question arises, if the universe has undergone manyphase transitions in its past, how come the initial finitevalue of cosmological constant was just right to exactlycancel the phase transitions and yield a very small valuewhich obeys current observational bounds [8–10]?In our framework, such contributions do not drop out

from (16) and (18), but they become automatically smallat times after the transition in a large and old universe. Letus assume, for simplicity, a single phase transition thattook place in the past. If the phase transition is first order,it proceeds through bubble formation, which will percolatesome time after the transition. If it is second order, then itis characterized by some order parameter smoothly tran-sitioning from one value to another in the universe, wherethe onset of this rolling in different regimes is local. In any


084023-15

case the process takes some time to complete, and there-fore involves gradients of fields or background geometry.Either way, the energy density in the gradients will notexceed the total potential energy density change. So forsimplicity we will model the phase transitions as a suddenjump in the potential across the whole universe. Thismeans that we ignore the gradients of fields, as they justgive an Oð1Þ correction, and model the transition with asimple step function potential V ¼ Vbeforeð1 − Θðt − t�ÞÞþVafterΘðt − t�Þ, where Θðt − t�Þ is the step function, and t�the transition time. Substituting into (18), we find

τμν −1

4δμνhτααi ¼

�−hVbefore − Viδμν t < t�;−hVafter − Viδμν t > t�:

ð48Þ

Hence, after the transition, the historic average in (48) is

hVafter − Vi ¼ −ΔV

R t�tbang dta

3R tcrunchtbang dta3

; ð49Þ

where ΔV ¼ Vbefore − Vafter. The denominator is just thespacetime volume, computed in Sec. IVA, and is given by∼1=H4

age. To estimate the numerator, we use the result ofSec. IVA, where we showed that the largest contributionsto the temporal integrals during a particular epoch comefrom the region where the scale factor is largest. Since weare considering the phase transitions in the past, as dictatesby the structure of the standard model, the dominantcontributions come from just before the transition itself.Hence,

Zt�

tbang

dta3 ∼Oð1Þ a3�

H�;

where a� is the scale factor and H� the curvature scaleduring the transition. Therefore

hVafter − Vi ¼ Oð1ÞρageΔV

M2PlH

2�

�H�Hage

��a�amax

�3

; ð50Þ

where we have used the fact that ρage ∼M2PlH

2age, and

amax ∼ 1=Hage. Assuming that the cosmological evolutionafter the transition to the turning point is dominated by asingle perfect fluid with an equation of state w, so thatH2

age ¼ H2�ð a�amax

Þ3ð1þwÞ we then find that

hVafter − Vi ¼ Oð1ÞρageΔV

M2PlH

2�

�Hage

H�

�1−w1þw

: ð51Þ

If the phase transition is followed by several differentepochs dominated by different matter distributions, thecorrection is merely a factor of Oð1Þ. Now assumingDEC and NEC, jwj ≤ 1, along with the fact thatH� ∼

ffiffiffiffiffiffiffiffiffiffiffiffiffiVbefore

p=MPl ≳

ffiffiffiffiffiffiffiΔV

p=MPl, we conclude that

jhVafter − Vij ≲ ρage < ρnow; ð52Þ

which means that the contributions of phase transitions tothe net effective vacuum energy are automatically sup-pressed to below the critical energy density of the universetoday. As we also see, the earlier they are, the moresuppressed: the reason is the historic weighting of thecontributions, since the “imbalance” is weighted only bythe spacetime volume before the transition. Dividing bythe total spacetime volume of the universe then suppressesthe contributions from the transition more efficiently ifthey occur earlier. For the standard model all the phasetransitions occur early, with H� ≫ H0 > Hage, and so thesuppression is considerable.What about the period before the transition? Using

similar arguments, we find that hVbefore − Vi ∼Oð1ÞΔV.Since ΔV ∼M2

plH2�, this contribution could be large, but it

affects the universe only at early times before the transition,when H ≳H�. This means that this a posteriori contribu-tion is subdominant to the contributions from other energysources, except at most during the period immediatelybefore the transition. Indeed, it is consistent with thestandard model and a reheating scale ≫ TeV to assumethat all transitions occur during the radiation era. So thiscould at most yield several short bursts of inflation (of atmost a few e-folds) in the run up to a transition, as long as itis sharp. Such phases are only a small perturbation of theradiation driven cosmology, and occurring at very shortscales in the present universe. If, however, the transition isvery slow, say a second order phase transition where theorder parameter takes a long time to move from its initialposition to the new vacuum, this in fact can drive a longinflation responsible for making the universe subsequentlybig and old. This, in fact, is beneficial, and we will show inSec. VIII that it is the key reason why our proposal isconsistent with inflation, even in the protected sector.

VI. PARTICLE MASSES AND PHENOMENOLOGY

So far we have shown that vacuum energy can besequestered away from curvature yielding a small residualnet cosmological constant, while particle physics still has amass gap as long as the universe is compact in spacetime,with a finite spacetime volume. Here we expand on thelatter point, and discuss in greater detail the properties thatthe function σ must have in order to ensure that the particlemass gap is phenomenologically reasonable. Since thenumerical value of the parameter λ sets the physical scalesin L, setting mphys ¼ λm, where m is the bare mass, thefunction σ must be carefully engineered to generate theright hierarchies between the various scales in a givenmodel of (beyond) the standard model physics. This cannotprotect the hierarchy between mphys and MPl, nor thehierarchies between different physical masses in L, butit can help set the scale hierarchy, and it can coexist withspecific mechanisms designed by model builders to protect


084023-16

particle physics hierarchies, by removing the vacuumenergy contributions.Since by virtue of the Eq. (14)

mphys

m¼ λ ¼

�σ0ðh ~Tα

αi=4μ4Þμ4

R ffiffiffig

p�1=4

; ð53Þ

to compute it we need to specify the cutoff for the regulatedvacuum energy contributions. We will assume that thecutoff is large enough that the vacuum energy gives thedominant contribution to ~Tα

α. We approximate ~Tαα ≈

−4 ~Vvac, where ~Vvac ¼ h0jLeffð~gμν;Φj0i is the regulatedvacuum energy of the effective matter Lagrangian in theprotected sector L at any given order in the loop expansion.It is related to the physical regulated vacuum energy via therelation Vvac ¼ λ4 ~Vvac. The regulator scale, i.e., the cutoff,of the protected sector isMphys

UV ¼ λMUV, and in principle itcan be as high as MPl. Depending on the model, the “bare”regulator,MUV, could also be as high asMPl, in which casewe must ensure that the solutions saturate λ ∼Oð1Þ. Evenin this most extreme scenario, the regulated vacuum energyVvac ∼ ~Vvac ∼M4

Pl is still completely sequestered fromgravity by our mechanism. The scale μ needs to be chosenappropriately in order to be consistent with λ ∼Oð1Þ, viathe condition

Oð1Þ ¼�σ0ð�Oð1ÞM4

Pl=μ4Þ

μ4R ffiffiffi

gp

�1=4

∼ ½σ0ð�Oð1ÞM4Pl=μ

4Þ�1=4Hage

μ; ð54Þ

where the � reflects the fact that the vacuum energy maytake either sign. Since we are introducing μ by hand, we cantake it to have as natural a value as possible, and pick itclose to the cutoff in the protected sector. Note that thisparameter is entirely external—a mere normalizationrequired for dimensional purposes—since neither it northe function σ ever gets explicitly renormalized. The valueof the function σ can only change because the ratio Λ=λ4

which appears in the argument of σ may change betweendifferent orders in the loop expansion. So settingμ ∼MPl=

ffiffiffiffiffi10

p, we now require

Oð1Þ ∼ ½σ0ð�Oð100ÞÞ�1=4Hage

MPl: ð55Þ

Provided the lifetime of the universe, 1=Hage, does notexceed its lower bound, 10=H0, by too much, getting λ ∼Oð1Þ will follow if σðzÞ≃ expðzÞ for z > 1. To allow forboth positive and negative vacuum energy, we need an oddfunction, so we can simply take σðzÞ ∼ sinh z.We should note that the exponential form of σ, which

easily sets up the required hierarchy between the electro-weak scale and the Planck scale, is also highly beneficialby protecting particle physics masses from a significant

sensitivity to the cosmological initial conditions. This canbe easily seen as follows. Consider Eq. (14) and imagine,for example, that σðzÞ ¼ z. In this case, the first equation of(14) translates to λ ¼ 1=½μðR d4x

ffiffiffig

p Þ1=4�, which byEq. (38) implies λ≃Hage=μ. In this case, if we take thecutoff to be MUV ∼MPl, to get mphys ∼ TeV, we needλ ∼ 10−15. If Hage ∼H0 ∼ 10−33 eV, this requiresμ ∼ 10−18 eV. Besides having such a tiny μ, this setup iseven more problematic from extreme sensitivity to thecosmological conditions that yield the terminal value ofHage. For example, if they are changed so that one has a fewmore or less e-folds of inflation,Hage will change by ordersof magnitude. If this happens, one must redial all theparticle scales by those amounts by hand. Having anexponential form of σ reduces this sensitivity to logarithms,which are far tamer.Of course, the standard model may be embedded in L via

some of its BSM extensions, such as some version ofsupersymmetry, in which case we could get a much lowervalue of the vacuum energy. For example, the SUSYbreaking scale might still be close to TeV, in which caseΛvacuum ≳ ðTeVÞ4. In this case, to retain λ ∼Oð1Þ as thesolution, and keep the exponential form for σ, whichprotects particle scales from sensitivity to cosmologicalinitial conditions, we ought to rescale μ so that it is closer toTeV. Either way, in principle we can adjust the externalparameters μ and/or σðzÞ to fit with a dynamical mechanismwhich protects the hierarchy within L. Then our mecha-nism can complement the QFT hierarchy protection bysequestering those vacuum energy contributions which theBSM extension cannot remove.

VII. PLANCK MASS RENORMALIZATON ANDGENERALIZED ACTIONS

So far we have conspicuously ignored discussing therenormalization of the Planck mass. In the context of QFTcoupled to (semi)classical gravity, however,we cannot avoidfacing this issue. The reason is precisely the logic we haveadopted in addressing the problem of vacuum energycorrections to the cosmological constant. These are calcu-lated in the loop expansion in QFT, and involve loops withgravitons only in external lines. However, similar loopdiagrams also give rise to graviton vacuum polarization—i.e., wave function renormalization. These are precisely thecorrections to the Planck mass. For example, if one includesone-loop corrections to the graviton wave function, onefinds that each field theory species contributes to theregulated Planck mass an expression of the form ΔM2

Pl ≃Oð1Þ× ðMphys

UV Þ2 þOð1Þ×m2phys lnðMphys

UV =mphysÞþOð1Þ×m2

phys þ � � �, where MphysUV ¼ λMUV is the matter UV regu-

latormass andmphys ¼ λmbare themass of the virtual particlein the loop [20]. It is important to note that here, as well as inthe corrections to the vacuum energy, the dependence on λ is


084023-17

multiplicative, since λ’s in the logs cancel by virtue of thecareful choice of the regulator. Renormalization ofMPl thencorresponds to picking a suitable subtraction scale M and

canceling the MUV dependent pieces, which yields M2Pl →

M2Pl þ Oð1Þ × M2

phys þ Oð1Þ × m2phys lnðMphys

mphysÞ þ Oð1Þ×

m2phys þ � � �. But, since Mphys ¼ λM, this implies that the

counterterm in the renormalization of the Planck scale mustalso go as λ2, implying that the consistent semiclassicaltheory really is given by the action

S ¼Z

d4xffiffiffig

p �M2

Pl þ λ2M2


�

þ σ

�Λ

λ4μ4

�: ð56Þ

Here, λM is the total finite renormalization of MPl whichremains after subtracting the infinities. As long as Mphys

UV ¼λMUV < MPl, a condition automatically satisfied by pickinga cutoff of the QFT belowMPl and maintained by evolutionin a large old universe,15 the value of the Planck scale isradiatively stable in our original model. The change of theform of (56) relative to (11) then does not affect thedynamics of vacuum energy sequestering at all in the limitof (semi)classical gravity.To see this, note that the field equations that follow from

varying the action (56) with respect to Λ, λ now take aslightly different form,

σ0

λ4μ4¼

Zd4x

ffiffiffig

p;

4Λσ0

λ4μ4¼

Zd4x

ffiffiffig

p ðλ2M2Rþ λ4 ~TμμÞ; ð57Þ

yielding

Λ ¼ 1

4hλ2M2Rþ λ4 ~Tμ

μi: ð58Þ

The variation of (56) with respect to gμν givesðM2

Pl þ λ2M2ÞGμν ¼ −Λδμν þ λ4 ~Tμ

ν. After eliminating Λfrom this equation with the help of Eq. (58) and canonicallynormalizing the matter sector, this becomes

ðM2Pl þ λ2M2ÞGμ

ν ¼ Tμν −

1

4δμνhλ2M2Rþ Tα

αi: ð59Þ

Rewriting this as M2PlG

μν ¼ Tμ

ν − λ2M2Gμν − δμνhTα

α−λ2M2Gα

αi=4, we see that the right-hand side vanishesidentically upon taking the trace and the spacetime average.Thus, as long as M2

Pl ≠ 0, we again obtain hRi ¼ 0, and so

the gravitational equations reduce to

ðM2Pl þ λ2M2ÞGμ

ν ¼ Tμν −

1

4δμνhTα

αi: ð60Þ

The theory is identical to the one described previously butwith a renormalized Planck mass ðMren

Pl Þ2 ¼ M2Pl þ λ2M2

and everything we have said so far remains unchanged.A curious limiting case occurs if we take the bare Planck

scale to zero, M2Pl → 0, in effect thinking of the whole

theory as “induced gravity” [43,44]. The action becomes

S ¼Z

d4xffiffiffig

p �λ2M2


�

þ σ

�Λ

λ4μ4

�: ð61Þ

Now the resulting gravitational field equations have avanishing historic trace:

0 ¼ Tμν − λ2M2Gμ

ν −1

4δμνhTα

α − λ2M2Gααi: ð62Þ

This degeneracy of the averaged trace is an artifact of theglobal scale invariance (20) being unbroken, so that thefield equations no longer force the constraint hRi ¼ 0.Nevertheless, the vacuum energy from the protected mattersector is again sequestered from gravity in this model—allloop corrections still automatically cancel. Indeed, asabove, at any given order in loops, we can separate offvacuum energy and local excitations by writingTμ

ν ¼ Vvacδμν þ τμν. Then the vacuum energy is seen to

drop out, and the field equations become

λ2M2Gμν ¼ τμν −

1

4δμνhταα þ λ2M2Ri: ð63Þ

With the vacuum energy eliminated, the theory nowbehaves like GR with an effective Planck massMphys

Pl ¼ λM, and a residual cosmological constantΛeff ¼ hταα þ λ2M2Ri=4. Unlike the generic class of the-ories, in this case the choice of Λeff does not impose anyboundary conditions on the matter sources. This of courseis a consequence of the fact that the whole theory now hasunbroken scale invariance. Choosing Λeff breaks it sponta-neously, but it does not fix hτααi. Thus this limit of ourmechanism has more in common with unimodular gravitythan the main case, in that it leaves an unspecifiedintegration constant as the effective residual cosmologicalconstant. However, the vacuum energy corrections from theQFT are still completely canceled, unlike in the standardformulation of unimodular gravity. Thus the residualcosmological constant is completely classical.The unbroken scale invariance of the theory (61) makes

it tempting to argue that the sequestering of the vacuum15This is because the physical masses scale as λ, and λ

decreases with increasing spacetime volume.


084023-18

energy corrections from gravity may now be extended toinclude also the corrections involving virtual graviton lines.This, however, does not happen. While some of the loopsinvolving internal graviton lines might indeed cancel, thepresence of R in the historic average in (63) is indicative ofthe problem. It will be renormalized by quantum gravityeffects, both quantitatively and by the fact that other higherderivative operators will appear in the action, which impliesthat the radiative stability of the residual cosmologicalconstant will be lost. In general however, allowing grav-itons in the loops takes us on a road to the—as yetunexplored—realms of quantum gravity, so it is difficultto say anything concrete. To sequester contributions fromgraviton loops at each and every order we would presum-ably need stronger symmetry requirements than just globalscale invariance which appears here. Therefore we willrefrain from delving into this complex issue here, hoping toreturn to this in the future.Note that the phenomenology of this limiting case of our

mechanism also differs. In this case λ does not set thehierarchy between the physical matter scales and the Planckmass, since mphys

MphysPl

¼ λmλM ¼ m

M. This implies that now the

spacetime volume is allowed to be infinite, and in particularit admits Minkowski spacetime as a solution. This immedi-ately raises the specter of Weinberg’s no-go theorem, whichwe have evaded previously by requiring a collapsingspacetime. In the Minkowski limit, our scale invarianttheory corresponds to the runaway behavior as ~Φ0 → −∞discussed at the end of our review of Weinberg’s theorem inthe previous section. There we argued that such a runawayalso set all QFT scales to zero, in obvious conflict with theuniverse we see. The caveat here is that in the case ofWeinberg’s no-go, the Planck mass were held fixed. That isnot the case here. The effective Planck mass experiencesexactly the same runaway behavior, dependent on thevolume, so that ratios between masses are maintained.So it appears that at least in the limit of classical gravity wecan consistently take all bare masses to infinity such thatthe physical masses remain finite. However, it is not clearthis will survive when quantum effects are in fact includedon the gravitational side, even in infrared. Again this is aquestion which we leave to future considerations.Finally, let us comment here on the main difference

between our proposal and the suggestion by Tseytlin [25]which seems to be similar (for follow-ups to Tseytlin’swork,exploring his proposal at the classical level, see [45]).This proposal posits that the standard action of a QFTminimally coupled to gravity should be divided by the

spacetime volume of the universe, ST ¼ Rd4x

ffiffiffig

p ½M2Pl2R−

Lðgμν;ΦÞ�=½μ4 R d4xffiffiffig

p �. Clearly, this trick immediatelyremoves the classical and zero-point (tree-level) correctionsto the cosmological constant. However it does not removethe loop corrections to vacuum energy. To see that, rewritethe theory as

ST ¼Z

d4xffiffiffig

p �λ4M2

Pl

2R − Λ − λ4Lðgμν;ΦÞ

�þ Λλ4μ4

;

ð64Þ

with the introduction of the global variables Λ, λ. Ignoringany phenomenological problems associated with the linearfunction σ, we see that the main difference between (64) andour proposal (11) is the dependence of the bulk terms on λ.Here, the Einstein-Hilbert term has a λ4 prefactor, and thematter sector does not have the kinetic energy scaling 1=λ2.Now, it is convenient to normalize the Einstein-Hilbert termcanonically, by taking gμν → λ−4gμν. The action (64) then

becomes ST ¼ Rd4x

ffiffiffig

p ½M2Pl2R − Λ

λ8− 1

λ4Lðλ4gμν;ΦÞ� þ Λ

λ4μ4.

Now it is clear that the tree-level vacuum energy scaleslike 1=λ4 and so will be automatically eliminated from thedynamics once λ is integrated out. However, after canoni-cally normalizing the QFT Lagrangian, it is straightforwardto see that the physicalmasses scale asmphys ¼ m=λ2, and sothe radiative corrections to the vacuum energy scale as∼1=λ8. Thus they will not automatically cancel, and willrestore the vacuum energy radiative instability in much thesame way as in GR (or unimodular gravity). In effect, the λdependence of (64) does not correctly count the engineeringdimension of the vacuum energy loop corrections. Further,as already alluded in [25], the Planck mass is not radiativelystable. The reason is that the corrections to it come asΔM2

Pl ≃Oð1Þ ×m2phys ≃Oð1Þm2=λ4, and so they are

extremely large in old and large universes. Therefore in alarge and old universe like ours, the Planck scale wouldreceive very large radiative corrections (unless of course thebare particle masses are incredibly small to start with). Thisis in complete contrast to what happens in our model.

VIII. INFLATION

So far, we have shown that in our framework the residualeffective cosmological constant is automatically small inlarge old universes. How does a universe become large andold? A common approach to answering this question is toresort to the inflationary paradigm, and postulate that atsome early epoch the universe was dominated by a transientlarge vacuum energy which made it very big and smoothquickly. But, we want to get rid of vacuum energy here. Sohow do we reconcile these two requirements? Is ourframework compatible with inflation? If so, why doesour mechanism not sequester away the vacuum energyduring inflation?In [14] we noted that in principle one could add an extra

sector to (11) which contains an inflaton, outside of theprotected sector L. However, since inflation must end andthe universe must reheat, this means that the inflaton mustcouple to the protected sector fields in L. Then one mustworry about quantum cross-contamination between thetwo sectors and how this may spoil the vacuum energy


084023-19

sequestration. In particular, loops involving inflaton inter-nal lines could spoil the scaling of the vacuum energycorrections with λ, and so they could end up yielding largecorrections to the residual cosmological constant. We notedhowever that there is an easy way out: since we are alreadytreating gravity (semi)classically, prohibiting internal grav-iton lines in vacuum energy loops, we could just embed theinflaton in the gravitational sector, and imagine that some—yet unknown—mechanism protects the residual vacuumcorrections involving virtual inflaton lines in the sameway it protects them from corrections involving virtualgravitons.Such a model is readily provided by the original inflation

model of Starobinsky [46]. This model has—until veryrecently—been favored by the data [47], and even postBICEP2 [48], there may still be variants in play [49]. Themodel can be simply embedded in our framework byextending the action (11) by

S ¼Z

d4xffiffiffig

p �M2

Pl

2Rþ βR2 − Λ − λ4Lðλ−2gμν;ΦÞ

�

þ σ

�Λ

λ4μ4

�; ð65Þ

where β ∼Oð106Þ is a dimensionless parameter. Such alarge parameter is radiatively stable to the corrections fromthe loops of the protected sector fields, and retains its formas long as we pick the UV regulators that couple to ~gμν, asbefore. We can now go to the axial gauge, by first extractingthe Starobinsky scalaron χ by the field redefinition

gμν ¼ ð1þ 4βM2

PlRÞgμν, χ ¼

ffiffi32

qMPl ln ð1þ 4β

M2PlRÞ [50]. The

scalaron has the potential V ¼ M4Pl

16β ½1 − exp ð−ffiffi23

qχ=MPlÞ�

2

and the protected matter sector couples to both gμν and toχ, via

S ¼Z

d4xffiffiffig

p �M2

Pl

2R −

1

2ð∂χÞ2 − V − Λe−2

ffiffi23

pχ

MPl

− λ4e−2ffiffi23

pχ

MPlL�e

ffiffi23

pχ

MPl

λ2gμν;Φ

��þ σ

�Λ

λ4μ4

�: ð66Þ

Clearly, the sequestering of vacuum energy in the protectedsector goes through unaffected, which follows from thescaling of the protected sector Lagrangian with λ.The Starobinsky inflation χ does not spoil it, since comingfrom the gravitational sector it is also treated purelyclassically, like the graviton. By a direct variation of(66) one could directly verify this, noting that a substitutionTμ

ν ¼ −Vvacδμν þ τμν, where Vvac is the physical vacuum

energy computed up to any given order in loops, and τμν isthe energy momentum tensor describing local on-shellmodes, still leads to a complete cancellation of Vvac.

Furthermore, we find that the deviations fromthe original Starobinsky scenario go like hτααiχ=V,where the χ-modified average is defined as hQiχ ¼R

d4xffiffig

pe−2

ffiffi23

pχ

MPlQRd4x

ffiffig

pe−2

ffiffi23

pχ

MPl

. Since χ ≠ 0 only during inflation, the

dominant contribution to the χ-modified averages comesfrom the full history of the universe, which means thathτααiχ ≈ hτααi ∼ ρage. This is very small compared to theinflaton potential V ∼M4

pl=β during inflation. We see, then,that the dynamics of Starobinsky inflation with sequester-ing is identical to the standard case to an accuracyof ∼ρage=ðM4

Pl=βÞ≲ 10−110.However, if the BICEP2 claim survives the ongoing

scrutiny, the inflationary mechanism which shaped ouruniverse may be different from Starobinsky. In fact,BICEP2 appears to favor large field chaotic inflation[51], which can be UV completed consistently [52,53].It turns out that such inflationary models are also consistentwith vacuum energy sequestering. This means, the inflatonitself resides in the protected sector—so that the exit frominflation, reheating and all the usual inflationary phenom-enology may proceed without spoiling the cancellation ofvacuum energy loop corrections. A clue that this is the casecomes from the consideration of phase transition contri-butions to vacuum energy in Sec. V. There we have notedthat while the phase transition contributions are small afterthe transition, they could end up dominating before. Slow-roll large field inflation can in fact be viewed as one suchslow, second-order phase transition—very slow, in fact, toensure that at least Oð60Þ e-folds of accelerated expansionmay occur.To check this, the simplest way to proceed is to consider

the field equations that follow from the action

S ¼Z

d4xffiffiffig

p �M2

Pl

2R − Λ −

λ4

2

�ð∂φÞ2λ2

þm2φ2

�

− λ4Lðλ−2gμν;Φ;φÞ�þ σ

�Λ

λ4μ4

�; ð67Þ

where φ is the inflaton avatar and Φ are the fields that itdecays into after the end of inflation, including the standardmodel. This guarantees that the vacuum energy correctionsremain completely sequestered even if they involve inflatonloops. Further, the inflaton sector needs to have an internalmechanism which protects the flatness of its potential [inthis case, the smallness of m in (67)], and allows forsufficient reheating, as explained, for example, in [53]. Wewill ignore these—important—details here, and merelyfocus on showing that (67) supports inflation driven by thequadratic potential of φ, which is sufficient to prove theconsistency between vacuum sequestration and large fieldinflation. This means, we will focus only on the quadratic φpotential in the matter sector.


084023-20

For our purposes, it suffices to show that there exists aninflating FRW flat (toroidal) cosmology driven by the fieldφ in the slow roll, and including the historic averagecosmological term. With this, after canonically normalizingφ → φ=λ, m → m=λ, the variational equations (14) and(16) reduce to

3M2PlH

2 ¼ 1

2_φ2 þ 1

2m2φ2 þ Λeff ;

Λeff ¼ −Rdta3ð1

2m2φ2 − 1

4_φ2ÞR

dta3;

φþ 3H _φþm2φ ¼ 0: ð68Þ

In slow-roll approximation, the field kinetic energyand acceleration terms are negligible, so this systemsimplifies to

3M2PlH

2 ¼ 1

2m2φ2 þ Λeff ; Λeff ¼ −

m2

2

Rdta3φ2Rdta3

;

3H _φþm2φ ¼ 0: ð69Þ

Now, it is clear that the integral for Λeff picks up significantcontributions only from the time interval during whichφ ≠ 0. In other words,

Rdta3φ2 ≃ R tend

0 dta3φ2 where tendis the end of inflation. We can bound this by taking φ ≤few ×MPl during the last 60 e-folds, which implies thatRdta3φ2 ≲ 100M2

Pl

R tend0 dta3 ≃ 100M2

Ple3N=H4, where H

is the Hubble scale during inflation, and N the number of

e-folds. So we find that jΛeff j≲ 50m2M2

Pl

H4R

dta3e3N . But the

integral in the denominator involves the full cosmic history,and so by Eq. (36) it is ≃1=H4

age. This means

jΛeff j≲ 50m2M2

PlH4agee3N

H4: ð70Þ

So relative to the scale of inflation, this givesjΛeff jM2

PlH2 ≲ 50

m2H4agee3N

H6 . Now, to have slow-roll inflation we

must enforce m < H. More importantly, for inflation tosolve the horizon problem, we must require thatH0eN=H ≤ 1. Since Hage ≤ H0, this immediately showsthat

jΛeff jM2

PlH2≪ 1: ð71Þ

This means that the historic average contribution to theresidual cosmological constant from the inflationary dynam-ics is completely negligible during inflation. Similar con-siderations also apply to other power law potentials. Hencelarge field inflation proceeds as in GR.The one remaining point of potential conflict between

our proposal and inflation is the regime of eternal inflation[54]. For our proposal to work we need the spacetime

volume of the universe to be finite. On the other hand, ineternal inflation large quantum fluctuations of the inflatonfield restart inflation in various regions of space. In effectthis allows the inflaton to utilize an entire ensemble ofinitial conditions, many of which yield never-endingexponential expansion creating infinite spacetime volumes.Although the dynamics of eternal inflation remains a topicof debate, it is nevertheless an important ingredient of theinflationary paradigm and hence we cannot ignore it.The success of our mechanism in its present form

precludes eternal inflation. We can imagine two ways toaccomplish this. The simplest possibility is to imagine thatthe true minima of the QFT have negative potential, inwhich case the vacuum would have been an anti–de Sitter(AdS). This is even consistent with the fact that our historicterm hτααi generically yields a very small negative residualcosmological constant. If this happens, then a randomobserver will eventually find herself in the global AdSminimum and experience an apocalyptic crunch. Of course,this does not rule out an infinite universe, as the birth rate ofinflating bubbles could exceed the death rate due tocrunches. Whether or not this is the case depends on theshape of the potential barrier separating inflationary vacuafrom the AdS minimum. A scenario in which the rate ofterminal collapse of a universe exceeds their birth ratecorresponds to the case where the barrier is neither too widenor too steep [55,56]. One could reasonably expect toachieve this and still allow the inflaton field to explore anensemble of inflationary initial conditions. The convincing,quantitative estimates of the likelihood of such a scenario atthis point are obstructed by the ambiguities in the definitionof the cosmological measures, and dynamics of eternalinflation. Another possibility is to imagine that inflationarypotential at high energies—or large displacements of theinflaton from the vacuum—is dramatically modified pre-venting inflationary slow roll altogether. For this, one needsto have very steep potentials away from the minimum. Thisdoes appear to be a designer model, but it is neverthelesspossible to construct such theories [55–58]. In such setups,one can avoid eternal inflation but then one needs adifferent approach for addressing initial conditions forinflation.

IX. SUMMARY

We have proposed a mechanism for stabilizing thecosmological constant from vacuum energy corrections.The main idea behind it is to modify the dynamics byintroducing two global variables which yields a theory withtwo new approximate symmetries: global scale invarianceof the matter sector, which is broken only by the Einstein-Hilbert term, and a shift symmetry that allows us to shift thematter Lagrangian by a constant without affecting thegeometry. These symmetries provide the key insight intohow the vacuum energy is sequestered. At a fixed scalebelow the cutoff, the shift symmetry is responsible for the


084023-21

cancellation of the vacuum energy after it is renormalized.The scaling symmetry then guarantees that the shiftsymmetry remains operational at all scales below thecutoff. Thus, all the vacuum energy corrections comingfrom a sector whose dynamics is constrained by the globalvariables is completely removed from the gravitational fieldequations. Since the two symmetries are approximate,being broken by the gravitational sector, the net residualcosmological constant is not zero, but it is automaticallysmall in old large universes, being given by the historicaverage hτααi ¼

Rd4x

ffiffiffig

pταα=

Rd4x

ffiffiffig

p, where the trace

involves only the contributions from the fluctuating on-shell sources that affect cosmic evolution.This term is nonlocal. However this nonlocality is not

pathological, but a mere consequence that the startingcosmological constant is a UV-divergent variable in thetheory. It must be renormalized, which means that the finiteleftover part is a quantity that cannot be predicted, but mustbe measured. Since the cosmological constant is a globalvariable, a parameter of a system of codimension zero, andits measurement requires carefully separating it from all theother long-wavelength modes in the universe, it takes adetector of the size of the universe to measure it. Oncedetermined, however, it is independent of any vacuumenergy corrections, and can be used to predict other, UV-insensitive, observables. Note that the cancellation of theUV-sensitive contributions is perfectly local in spacetime.This a posteriority of the measurement of Λeff represents

a self-consistent determination of a single number, and doesnot give rise to any pathologies one typically associateswith local violations of causality. It is interesting to noteColeman has argued that the universe should possess adegree of what he calls precognition—it knows from thebeginning that it intends to grow old and big [59].All this follows from the modification of the gravita-

tional sector, accomplished with the introduction of twoglobal variables and two global constraints. This approach,equivalent to the isoperimetric problem of variationalcalculus, does not affect local particle physics at all. Thetheory still has diffeomorphism invariance and localPoincaré symmetry, and so local QFT behaves exactlythe same as in the conventional formulation. Further,because of these symmetries the number of local degreesof freedom in gravity is still just two, the usual spin-2helicities of GR. The global variables disappear from localdynamics, one canceling the quantum vacuum energy and

the other being absorbed into the definition of physicalscales of the local QFT, without affecting particle physics.A key ingredient of the proposal, however, which allows forthis disappearance of the variable λ, is that the universeshould be compact in space and time. This is necessary inthe present formulation in order to have nonzero mass gapin the QFT sector. So far we have not specified the detailedmechanism which produces collapse, but have shown thatthis is consistent with the dynamics of the proposal. In aforthcoming paper [37] we will return to this issue.The main phenomenological differences of our proposal

relative to the standard formulation of QFT coupled togravity, and so predictions, follow from this requirement.This completely changes cosmic eschatology since we nowrequire the universe to be spatially closed and to exist for afinite proper time, starting out with a big bang and endingwith a big crunch. A spatially closed universe can haveobservational signatures, yielding doubling of images dueto nontrivial spatial topology or affecting CMB at thelargest scales. Currently there is no direct evidence of suchtopology [60], but the search continues. Further, it may bepossible to forecast the impending collapse observationally,as noted in [61], particularly if the impending doom istriggered by whatever is driving the current acceleration.The key consequence, however, is the prediction that thecurrent epoch of cosmic acceleration with wDE ≈ −1 is atransient. So finding deviations away from this equation ofstate of dark energy may indeed be a harbinger of a futurecosmological collapse.

ACKNOWLEDGMENTS

We would like to thank Guido D’Amico, Ido Ben-Dayan, Ed Copeland, Savas Dimopoulos, Kiel Howe,Justin Khoury, Matt Kleban, Albion Lawrence, AndreiLinde, Adam Moss, Fernando Quevedo, Paul Saffin,Martin Sloth, Lorenzo Sorbo, David Stefanyszyn,Natalia Toro, Alex Vikman, Andrew Waldron andAlexander G. Westphal for useful discussions. N. K. thanksthe School of Physics and Astronomy, University ofNottingham, for hospitality in the course of this work.N. K. is supported by the DOE Grant No. DE-FG03-91ER40674, and received support from a Leverhulmevisiting professorship. A. P. was funded by a RoyalSociety URF.

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Vacuum energy sequestering: The framework and its cosmological consequences

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