aa r X i v : . [ h e p - t h ] S e p NYU-TH-10/07/14
The Big Constant Out, The Small Constant In
Gregory Gabadadze
Center for Cosmology and Particle Physics, Department of Physics,New York University, New York, NY, 10003, USA
Abstract
Some time ago, Tseytlin has made an original and unusual proposal for an actionthat eliminates an arbitrary cosmological constant. The form of the proposed action,however, is strongly modified by gravity loop effects, ruining its benefit. Here I discussan embedding of Tseytlin’s action into a broader context, that enables to control theloop effects. The broader context is another universe, with its own metric and dynamics,but only globally connected to ours. One possible Lagrangian for the other universeis that of unbroken AdS supergravity. A vacuum energy in our universe does notproduce any curvature for us, but instead increases or decreases the AdS curvature inthe other universe. I comment on how to introduce the accelerated expansion in thisframework in a technically natural way, and consider the case where this is done by theself-accelerated solutions of massive gravity and its extensions. . Introduction:
Nearly a quarter-century ago, Tseytlin [1] has proposed an approachto the old cosmological constant problem, using an original idea by Linde [2], and certainstring-theory developments of that time. The proposal is technically well-framed, while ahighly unconventional nature of this approach is commensurate with the magnitude andlongevity of the problem, hence suggesting the approach may have a chance of being viable.While the proposed action of [1] enables one to eliminate an arbitrary cosmological con-stant, the action itself was argued to be unstable with respect to quantum corrections,therefore making the proposal not workable in its original form (see the note added in [1]).The goal of this work is to extend the proposal to avoid the quantum loop problem, and toincorporate the dark energy component into the theory in a technically natural way.
2. Tseytlin’s proposal:
To set the conventions, consider the action: S = Z d x √ g (cid:18) πG N R + L ( g, ψ n ) (cid:19) , (1)where ψ n n = 0 , , , ... , denote all fields of the theory beyond the metric field g µν . TenEinstein equations can be decomposed as nine traceless and one trace equation: R µν − g µν R = T µν − g µν T , R + T = 0 . (2)(Unless G N or M Pl are displayed explicitly, we use the πG N = 1 units).Instead of these equations, Tseytlin introduced a system where the trace equation ismodified (see the corresponding action below in eq. (6)): R µν − g µν R = T µν − g µν T , R + T = h T i − h g µν ∂L∂g µν i , (3)where h· · · i denotes a certain space-time average defined as follows: h· · · i ≡ R d x √ g ( · · · ) R d x √ g ≡ [ · · · ] V g . (4)The modification of the trace equation in (3) is dramatic: local observables on the l.h.s. are affected by space-time averaged quantities, where the averaging is done over the pastand future. These averages, when nonzero, have pre-notion of future. In that sense, this isan acausal modification. Somewhat similar, but essentially different proposal was made in[3]; a subtle issues of defining the average, where there are more than one vacua, was alsoraised there. To begin with, we envision a simple universe evolving in one vacuum state, andcomment on possible generalizations later.If V g → ∞ , as in Tseytlin’s approach, then for most of the stuff in the universe ther.h.s. of the trace equation in (3) is zero: For any observable, O , that is localized eitherin space or in time, the average hOi is zero due to the volume factor suppression. Hence,the acausality of the trace equation does not manifest itself in the dynamics of most of thestuff in the universe. On the other hand, for a constant Lagrangian, L = c , the r.h.s. of Throughout the paper we use commonly accepted acronyms: "l.h.s." and "r.h.s.", for left and right handside respectively, "w.r.t." for with respect to, "UV" for ultraviolet, and "1PI" for 1-particle irreducible. c itself, and the latter subtracts theequivalent part on the l.h.s., hence leaving the equation independent of c ! Therefore, themain consequence of the acausality might be that we don’t observe the big cosmologicalconstant in our universe [3].For a scalar field the Lagrangian can be decomposed into the part that depends on themetric (derivative terms, e.g., the kinetic term) and the one that is independent of g µν : L = L g ( g, ψ n ) − V ( ψ n ) . (5)Simplest examples of V are the vacuum energy term E vac , scalar mass term m φ φ , scalarpotential λφ , or a linear combination of the above. The vacuum energy, or a constant part ofa potential V , would give rise to a nonzero average, h V i = [ Const. ] /V g = Const.
Thus, thisquantity would be subtracted from the trace T in (3). This is equivalent to the eliminationof the cosmological constant!The fact that a constant term in L is irrelevant, can also be seen by looking at the action ¯ S = SV g , (6)that Tseytlin introduced [1] as an object which has to be varied w.r.t. g µν to get the equationsof (3). Any constant shift, L → L + c , gives rise to a shift of the new action by the sameconstant, ¯ S → ¯ S + c , that does not affect the equations of motion.Furthermore, if the potential has two minima, one "false" and one "true", then what isbeing subtracted is the value in the "true" minimum, assuming that a transition from "false"to "true" is possible in finite time in the standard General Relativity context. Generically,what is being eliminated is what would have been the asymptotic future value of the vacuumenergy density in GR, as discussed in detail in [3].As to the second term on the r.h.s. of the trace equation in (3), it contains only the L g part of the Lagrangian (5); for homogeneous scalar fields this part eventually decayson solutions for which the field settles in its minimum, therefore its average h· · · i is zero.Thus, inflation would generically proceed in a conventional way, except the phenomenon ofself-replications is not straightforward to incorporate in this framework [2, 4].
3. Problems with the loops:
While the above approach appears to solve the big cosmologi-cal constant problem, at least in the limited context specified above, there are two importantissues that it fails to address:First, as mentioned already in [1], the loop corrections should be problematic, and theyare indeed. They strongly renormalize the form of the action (6), and thus ruin the solutionof the cosmological constant problem. Can the issue of the loops be resolved, by perhapsextending the proposal?Second, Tseytlin’s mechanism eliminates entirely the cosmological constant. Later on, itwas discovered that the expansion of the universe is accelerating [5]. This acceleration canbe accounted for by some form of dark energy, with an equation of state parametrized by w = − . A cosmological constant has precisely that equation of state. Then, the questionarises: if one eliminates the cosmological constant how does one get to retain dark energywith w = − ? We’ll discuss how the accelerated expansion can be accommodated in thisscheme in a technically natural way; one option is to invoke massive gravity for this purpose.3e proceed in this work by discussing the path integral formulation of the theory moreexplicitly. This requires an introduction of a special algorithm for path integral quantizationof (6). As a result, we’ll end up with two different path integrals: one for all non-gravitationalinteractions quantized with the Planck’s constant ~ , and another path integral for gravityquantized with a different, dynamically determined Planck’s constant.In the absence of gravity – in the M Pl → ∞ limit – ¯ S differs from the standard action byan overall ∞ factor; the latter is field independent, and thus can be rescaled away . Thus, inthis limit one would quantize the theory (6) in a conventional way. When dynamical gravityis included, however, one needs to specify rules of quantization. One would not immediatelyworry about a UV completion, but a low-energy effective field theory quantization of gravityshould certainly be a matter of concern: The Einstein gravity is a good low-energy effectivequantum field theory below the Planck energy scale [6], and any of its extension should striveto retain this virtue, below a certain energy scale. It will be our goal to define such a theoryin what follows.We assume that gravity is quantized at some energy scale, M QG (the Planck scale, orstring scale), that is at least an order of magnitude higher than the UV scale, M SM , ofnon-gravitational interactions, loosely referred below as Standard Model (SM) interactions. M SM could be a scale at which the SM interactions themselves become UV complete, forinstance, by grand-unifying into an asymptotically free theory. In such a setup it’s notunnatural to have two orders of magnitude hierarchy between M QG and M SM ; if so, thengravity should be well approximated by a classical field theory below the energy scale M SM .At these "low energies" the path integral can be defined with all the SM fields quantizedusing ~ , while treating gravity as an external classical field, pending specification of the rulesof quantization for gravity. The latter should give rise to further tiny corrections to alreadyquantized SM processes (see below).Thus, at low energies the path integral for quantized SM interactions reads as follows: Z ( g, J n ) = const Z dµ ( ˜ ψ n )exp (cid:18) i Z d x √ g (cid:16) L ( g, ˜ ψ n ) + J n ˜ ψ n (cid:17)(cid:19) , (7)where dµ ( ˜ ψ n ) is a measure for all the SM fields ˜ ψ n , that appropriately modes out gaugeequivalent classes. The metric field g is an external field, and so are the sources, J n ’s,introduced for every single SM field. Then, the effective Lagrangian L ( g, ψ n ) used in (6) isdefined as a Legendre transform of W ( g, J n ) = − i ln Z ( g, J n ) Z d x √ gL ( g, ψ n ) ≡ W ( g, J n ) − Z d x √ gJ n ψ n , (8)where √ gψ n ≡ − iδ ln Z ( g, J n ) /δJ n , is √ g times the vacuum expectation value of the SMfield ˜ ψ n , in the presence of a source J n . The obtained quantum effective action (8) is a 1PIaction. Thus, all the quantum corrections due to non-gravitational interactions are alreadytaken into account in the effective Lagrangian L . This Lagrangian is then inserted into(6) to account for dynamical gravity. Note that the effective quantum Lagrangian L ( g, ψ n ) depends on the classical fields, g and ψ n ’s, only. The difference between these two sets of All four-volume infinities throughout the paper are assumed to be first regularized to yield finite quan-tities, and the regulator removed only after the equations of motion are derived. ψ n ’s, while the gravity quantum loops have not been taken into considerationyet. In what follows we will find it helpful to define an effective generating functional Z SM ( g, ψ n ) ≡ exp (cid:18) i Z d x √ gL ( g, ψ n ) (cid:19) = Z ( g, J n )exp (cid:18) − i Z d x √ gJ n ψ n (cid:19) , (9)that includes all the SM loops, but does not include quantized gravity.In the end, g µν in (6) should also be quantized. The corresponding quantum effects arelikely to become of order one at scales M QG , and they should be taken care of by a putativeUV completion of the theory, presumably via new degrees of freedom that could appearat energies ∼ M QG . These considerations can be postponed for a UV complete theory ofgravity, such as string theory, perhaps along the lines proposed in [1]. However, there isan immediate issue, irrespective of the form of UV completion. It concerns the low-energyeffective theory: the quantum gravity corrections should be small at scales well below M GQ for our approximations above to be meaningful. For instance, in Einstein’s gravity, suppliedwith a diff-invariant UV cutoff for gravity loops (that requires additional counter-termsto retain diff invariance), one generates higher dimensional operators that make negligiblecontributions at energies below M QG . In the present case, however, one first needs to definethe rules of calculation of the gravity loops given that the classical action (6) has an unusualform. To define these rules, and check whether gravity loop corrections are small, is our goalin the reminder of this section.We define an extended action: ¯ S q,λ = 1 q Z d x √ g (cid:18) R + L (cid:19) + λ ( V g − q ) , (10)and write down the path integral for gravity as follows Z g = const Z dµ ( g ) dq dλ exp( i ¯ S q,λ ) , (11)where dµ ( g ) is a measure over diff-inequivalent metric fields. Note, that the fields of the 1PISM action, ψ n ’s, play a role of external fields in the path integral for gravity. Furthermore,one also integrates w.r.t. the parameters q and λ in this path integral.The expression in (11) can be rewritten in terms of the path integral for the SM fields Z SM given in (9): Z g = const Z dµ ( g ) dq dλ (cid:0) e iS EH Z SM ( g, ψ n ) (cid:1) q e iλ ( V g − q ) , (12)where S EH is the Einstein-Hilbert action for gravity. The above path integral defines analgorithm for calculating quantum corrections – both due to the SM interactions and gravity:The SM loops are done in a conventional way, assuming the metric to be an external classicalfield; this gives rise to Z SM ( g, ψ n ) . Furthermore, for calculation of gravity loops one is invitedto use an unconventional prescription specified either by (12), or equivalently, by (11).In this proposal, the parameter q may be regarded as a second Planck’s constant thatgoverns the gravity loops (recall that SM loops governed by the standard ~ are already taken5nto account in (11)). Furthermore, one integrates w.r.t. the second Planck’s constant,however, the value of the latter is also constrained by the value of the invariant four-volumedue to integration w.r.t. λ . The form of the extended action (10), unlike that in (6), is usefulfor thinking of the formulation of the path integral, or canonical momenta and Hamiltonianof the theory . Having the path integral set up in (11) , we can integrate w.r.t. q and λ thatwould give rise to Z g = const Z dµ ( g ) exp( i ¯ S ) , (13)with ¯ S defined in (6).The trouble with the gravity loops in the effective field theory approach, can be under-stood either in the language of (12) or of (13). The latter presentation is shorter, so wereiterate it here from [1] by observing that the /V g factor in (6) is rescaling what wouldhave been the Planck’s constant for the gravity loops in a conventional effective field the-ory approach to Einstein’s gravity; that is, we should take all the quantum gravity loopcorrections calculated in the conventional approach and make a replacement, ~ → ~ V g [1].Adopting this procedure for the gravity loops, one would get: ¯ S Ren = 1 V g Z d x √ g (cid:18) R + L ( g, ψ n ) + V g L ( g, ψ n ) + V g L ( g, ψ n ) + ... (cid:19) , (14)where L , L , .. contain all possible terms consistent with diffeomorphism and SM internalsymmetries. The gravity loop corrections are huge, since V g is huge. The new terms ruinthe above-presented solution of the cosmological constant problem.It should be noted, that there is yet another class of loop corrections if one quantizesgraviton fluctuations in the theory (6) on a given background solution . To consider theeffects of these fluctuations, let us decompose the metric as a background and fluctuation,schematically, g = g b + h , where h is being treated as small. Then, the inverse volumefactor, V − g , multiplying the action S in (6), can also be expanded as follows: V − g = V − b − V − b H h + ... , where V b = R d x √ g b and H h ≡ R d x √ g b tr( g − b h ) / , and so on. It is clearthat the term, − V − b H h (and all the other subsequent terms containing higher powers of h ),will produce new unconventional interaction vertices when Wick-contracted with powers of h in the expansion of the action S . While an extra effort would be required to work out allthese unconventional vertices, one should point out that all of them will be suppressed bypowers of the background volume, V b . Indeed, in the expression − V − b H h one power of theinverse volume V − b will be spent to offset the volume factor in the expression H h , while thesecond power of V − b will be suppressing the fluctuations of h . Thus, all the new verticeswill come suppressed by as many powers of V − b as the power of the fluctuation h arisingfrom the expansion of V − g in (6). This suggests that the loop effects discussed in the presentparagraph could be assumed to be small and be neglected. Similar considerations apply tothe proposal discussed in the next section. The idea of integration w.r.t. the parameters is adopted from [1], although the path integral here, andits interpretation, differ somewhat from that in [1]. I thank Arkady Tseytlin for bringing this point to my attention. . Dealing with the problems: To avoid the difficulty with the quantum loops discussedin the previous section, let us introduce the following action instead of (6): A = V f V g S + Z d D y p f M D − f R ( y ) + c M D · · · ! , (15)where a second metric f AB ( y ) , A, B = 0 , , , , ..D − , has been used, and V f = R d D y p f ( y ) ,in the M Pl = 1 units. Note that while the action S defined in (1) is four dimensional, the f -metric could live in D ≥ dimensions in general.The action of the f -universe has a certain vacuum energy scale M , and a scale thatdetermines the strength of its gravitational coupling is M f . Depending on details of thetheory – encoded in the dots in (15) – there may or may not be a stable hierarchy betweenthe scales M f and M (see below).The main idea is that in (15) any shift of L by a constant, L → L + c , converts c intoa cosmological constant of the f -universe, thus removing it from the g -universe, where wepresumably reside. Therefore, while the curvature in our universe is (nearly) zero, the otheruniverse could be highly curved.An analog of the extended action (10) now takes the form: A q,λ = 1 q Z d x √ g (cid:18) R + L (cid:19) + λ (cid:18) V g V f − q (cid:19) + Z d D y p f M D − f R ( y ) + c M D ! . (16)This can be used to define the path integral that includes integration w.r.t. q and λ , asdiscussed in detail in the previous section. Since all the essential steps of that constructioncarry through with a straightforward extension to include the dynamics of the second metric f , we will not repeat them here . Furthermore, in what follows we will use, for brevity, theform of the action (15), obtained from the extended action (16) by integrating out q and λ .In order for the gravity loops not to ruin the crucial classical property of the action,one should make sure that V f >> V g : then, the rescaling of what would have been thePlanck’s constant for gravity loops in a conventional approach is ~ → ~ ( V g /V f ) , and theaction including the gravity loop corrections would take the form V f V g (cid:20) R + L + V g V f L · · · (cid:21) + Z d D y p f M D − f R + c M D + c R + c ¯ S M Df · · · ! . (17)As long as V f >> V g , all the corrections proportional to V g /V f can be neglected. There arealso terms similar to the ones discussed in the last paragraph on the previous section, butthey are harmless for the same reasons as before . This is not all however, the gravity loopdiagrams in the f -universe generate two groups of new terms – first, the terms containinghigher powers and derivatives of curvatures R ( f ) ′ s , and second, terms containing powersof ¯ S (and their products with powers of the R ’s and derivatives); some of these terms are As it’s evident from the above, f is quantized in a conventional way with ~ . Thus, the gravity loop corrections to both gravity itself, and the standard model processes, either vanishor are very small in this prescription. However, the theory still needs UV completion to make sense of itsunusual form for g and f gravities. I thank David Pirtskhalava for very useful discussions on these points. g - and f -universes given below.In general, both V f and V g are divergent. It is sufficient for our purposes that thecondition V f /V g >> is satisfied, even though V f and V g individually tend to infinity . Forconsiderations of the ratio, V f /V g , it is convenient to invoke the Euclidean space to get asense of the ratio of the Euclidean four-volumes, V f /V g , as will be done below.Then, how do we achieve the condition V f /V g >> ? To fulfill this we’re going to exploretechnically natural hierarchies between parameters of the theory. First of all, we assumethat the g -universe has supersymmetry broken at some high scale, and therefore, there isa natural value of its vacuum energy density proportional to E vac . The scale E vac can beanywhere between a few T eV and the GUT scale, µ GUT ∼ GeV . As to the f -universe,it’s presumably uncontroversial to set M f ∼ M Pl , but also we’d need the scale M to besomewhat higher than E vac . The latter condition should be natural, since without specialarrangements one would expect M ∼ M f ∼ M Pl , and since E vac << M Pl , one would alsoget E vac < M . If so, then the vacuum energy of the g -universe, E vac , would make a smallcontribution to the pre-existing vacuum energy of the f -universe. In short, the vacuumenergy density of the f -universe, c M D , would dominate over the vacuum energy densitythat gets delegated to the f -universe, from the g -universe .While one could try to explore a case when the f -universe has a positive vacuum energydensity, it seems more straightforward to make a mild assumption that the curvature dueto the term c M D in the f -universe is negative (AdS like) to begin with. In that case, the f -universe can be exactly supersymmetric, described by an unbroken supergravity.For instance, if we were to consider D = 4 , the f -universe could be described by super-gravity with the "Planck scale" equal to M f , and the quantity λ ≡ ¯ S + c M , (18)acting as its vacuum energy density. The action (15) completed to the one of the N = 1 AdS supergravity [7] would then be written as: A SUGRA = Z d y ˜ e (cid:18) M f R (˜ e, ˜ ω ) − ǫ µναβ ¯ ψ µ γ γ ν D α ψ β + 3¯ λ − λM f ¯ ψ µ σ µν ψ ν (cid:19) , (19)where ˜ e is the determinant of the vierbein of the f -metric, ˜ ω is its spin connection, D = ∂ − ˜ ωσ is the covariant derivative, and ψ µ is the Rarita-Schwinger field describing a f -gravitino. While a supergravity action is not the only one that can help reach our goal, themotivation to consider it can perhaps be attributed to the fact that supergravities naturallyemerge in the low-energy limit of superstrings.The quantity ¯ S enters into ¯ λ in (19), while the latter defines the cosmological constant(with AdS sign) as well as a quadratic term for the gravitino . Thus, the entire g -universeenters this action via the parameter ¯ λ defined in (18). The gravitino bilinear term in (19)would also give a nonzero contribution into the equation of motion for the metric g , however,the respective new term will be proportional to the gravitino bilinear, which is zero on As emphasized above, we assume that these infinities are first regularized, and the regularization isremoved at the end of calculations. This is certainly not a gravitino mass term [8]. .There is no reason for the parameter ¯ λ to be much smaller than M f ; quantum correctionswould renormalize the former up to the scale of the latter, even if we started with a largehierarchy between them. On the other hand, we do need some small hierarchy between M f and ¯ λ / , essentially to be sure that AdS curvature of the f -universe can reliably be describedin the supergravity approximation. For this, an order of magnitude hierarchy, M f ∼ λ / ,would be more than sufficient. While this hierarchy could perhaps be attributed, withouttoo much of anxiety, to the π loop factor’s arising at various places, we note that it couldbe generated dynamically if we were to introduce more general supersymmetric theory withsome matter fields in the Lagrangian: the N m matter fields with a characteristic scale M m would renormalize additively the Planck scale M f via the Adler-Zee mechanism producing, M f → M f + ∆ M f , where ∆ M f ∼ N m M m [9], [10], while renormalisation of the cosmologicalconstant ¯ λ due to the complete SUSY multiplets of matter would have been zero. Thus,we could adopt, M f ∼ λ / , as a technically natural choice. If so, then the hierarchies M Pl ∼ M f ∼ M , M ∼ > E vac , ensure that all the corrections in (17) are negligible incomparison with the terms in (15).Having the scales clarified, let us see how this plays out for the cosmological constantfor a general D -dimensional f -universe. First we consider the case when f is not among thefields ψ n , n = 0 , , , , .. . Then, the new terms in (15) or (19) do not affect the equations(3), except that they introduce a overall multiplier V f . Thus, the cosmological constant iseliminated from the g -universe. There is, however, a new equation due to variation w.r.t. f : M D − f ( R AB ( y ) − f AB R ( y )) = f AB ( ¯ S + c M D ) + · · · . (20)The right hand side contains the vacuum energy generated in our universe, ¯ S = [ E vac ] V g = E vac ,as well as that of the f -universe. According to our construction, the net energy density isnegative, so that the f -universe has an AdS curvature. If so, then V f = ∞ even in Euclideanspace. Then, to reach our goal it is sufficient to have Euclidean V g finite, so that V f >> V g .A de Sitter universe with Euclidean V g = H − would fit the data and satisfy the abovecriterium . However, the entire cosmological constant has been eliminated from the g -universe, and thus it’s not easy any more to get V g = H − . We’ll discuss below how thiscould nevertheless be achieved.
5. Getting the accelerated universe:
One needs to get a dS metric in the g -universewithout using a vacuum energy or a scalar potential. More precisely, one would need to getthe small dS curvature due to the terms in the Lagrangian (5) that explicitly depend on g .There might be a few ways of achieving that: e.g., by invoking Lorentz invariant con-densates of some vector fields with a coherence length comparable with H − , or by usingfield theories with higher derivatives but no Ostrogradsky instabilities. Such proposals couldproduce dark energy due to terms that aren’t potentials, but depend on the metric g , sothat the last term on the r.h.s. of the trace equation (3), would define the cosmic speed-up. For ¯ λ to be positive the scale M should be (somewhat) higher than the scale E vac . This could bearranged without any fine tunings as discussed above. That V AdSf /V dSg → ∞ can also be seen in Lorentzian signature, by calculating the ratio, e.g., in theglobal coordinate systems, for the universal covering of AdS, and the dS space.
9e briefly comment here on a possibility to obtain this feature due to massive gravity.Nonlinear massive gravity [11, 12], or some of its extensions [13, 14, 15, 16], introduce gravitonmass m as a small parameter, m ∼ H , in a technically natural way [17]; these theories alsoproduce self-accelerated solutions with a dS background [18]; moreover, the fluctuations onthese backgrounds are healthy when the pure massive graviton is amended with a dilaton-likefield [15, 16] (for theory reviews of massive gravity see, [19, 20]).Let us briefly outline how massive gravity would produce R ∼ m in the trace equation(3). For this we put aside the matter Lagrangian and assume that L represents instead thediffeomorphism invariant potential of massive gravity [12]: L = M Pl2 m U ( K ) = M Pl2 m ( det ( K ) + α det ( K ) + α det ( K )) , K = 1 − p g − γ , (21)where, the matrix K is defined via an inverse of the metric g and a fiducial metric γ ; we chose γ to be a metric of Minkowski space, γ µν = ∂ µ φ a ∂ ν φ b η ab , written in an arbitrary coordinatesystem parametrized by φ a , a = 0 , , , . The φ a ( x ) fields also represent the Stückelbergfields that guarantee diffeomorphism invariance of (21). The square root of a matrix andits traces are defined via its eigenvalues, and α , α , are some free parameters. Note thatall possible values of the three parameters of the theory, m, α , α , are technically natural[17]. Furthermore, the quasidilaton is introduced by requiring that the rescaling of the φ a coordinates w.r.t. the x µ coordinates be promoted into a global symmetry; this amounts toadding into (21) the kinetic term for the quasidilaton σ (and possibly some other derivativeterms [15]), and replacing γ → e σ/M Pl γ .Let us now look at the trace equation in (3): the trace of the stress-tensor, call it T g ,is obtained by the standard variation of [ √ gL ] = M Pl2 m [ √ gU ] . On the self-acceleratedsolutions this trace equals to a constant, T g ∼ M Pl2 m . Therefore, T g in the l.h.s. of (3)will cancel with h T g i on the r.h.s.; the remaining trace equation will take the form R = − m h g µν ∂U ( K ) ∂g µν i . (22)On the selfaccelerated solutions, however, g µν ∂U ( K ) /∂g µν | SA = − C ( α , α ) , is also a con-stant, that depends on the parameters α and α . Therefore, its average yields the sameconstant, and we get R = 2 m C ( α , α ) . For a certain reasonable magnitudes, and certainsigns of the parameters, one gets the dS curvature of the order, m ∼ H , in a technicallynatural way. Quasidilaton does not change this conclusion, it only affects (improves) dynam-ics of small perturbations above the solution [16]. Thus, to summarize, the above approachenables to remove the big cosmological constant, and to get a small space-time curvaturedetermined by the graviton mass.In the approach adopted above γ was taken to be independent of the f -metric, that wasused to remove the big cosmological constant. We’ve discussed the case when f was an AdSmetric, while γ was flat. However, neither of these choices are ordained – we only require thatspace-time described by f to have an infinite Euclidean volume. It is intriguing, therefore, toconsider γ to be related (perhaps identified?) with f . In that case, γ cannot be fixed a priori,but will be determined by the f equation of motion (20); the latter will now be modifieddue to the terms in (21), but the modification is proportional to m ∼ H << M f , andshould be negligible. If such a framework can be made to work in detail, this would provide10n additional arguments for amending Tseytlin’s approach by the f -metric, and conversely,would introduce an out-of-our-universe dynamics for the fiducial metric of massive gravity.On a more sobering note, massive gravity and its extensions are strongly coupled theoriesat energies way below M Pl ; while this may not be in conflict with observations in our universedue to the Vainshtein mechanism [21] in its intricate cosmological and astrophysical form [22],[23], [24], nevertheless, it still remains to be understood how to go above the strong scale,and show that superluminal phase and group velocities obtained on certain backgroundsprobing this strong scale, are indeed artifacts to be removed in a complete treatment.
6. Conclusions and outlook:
The proposed approach eliminates the cosmological con-stant, at least in a simple setup where there are a few (non-proliferating) vacua with well-separated hierarchy between their energy densities, and allowed transitions between them.What is eliminated is what would have been an asymptotic future value of the cosmologicalconstant for such a potential in GR; for instance, for two vacua, "false" and "true", withallowed transitions from "false" to "true", the "true" vacuum energy is eliminated. This issimilar to the proposal of [3], but here the action functional is available and it is stable w.r.t.quantum loop corrections, including loops of gravity in an effective field theory approach.The dark energy component can be introduced via the Lorentz invariant condensatesof vector fields, or via derivatively interacting scalar fields. We briefly discussed how theaccelerated universe could be due to massive gravity in this approach.The proposed scheme is rather unusual, as it involves nonlocal terms in otherwise localEinstein’s equations, making it difficult to be satisfied with this aspect. However, the cos-mological constant problem is a long-standing enigma of a tremendous magnitude, and anyinsight into its possible dynamical solution within the well-defined rules of the low-energyfield theory approach, is extremely important, and should be welcomed.As an outlook, just three comments on the literature:Ref. [3] has made arguments for a connection of the "high-pass filter" modification ofgravity with a specific theory containing the averages h· · · i . It might be interesting to seeif the present proposal could also be connected to some "high-pass filter" modified gravitiesdiscussed in [3]. Conversely, one could then hope to find an action principle for the equationsof [3], and address the issue of the gravity loops for them.The original motivation of Tseytlin was to obtain the unconventional action (6) by in-cluding the winding modes of string theory. It would be interesting to see if any proposalalong this idea can give the action (15), or a version of it.Refs. [25] have recently discussed gravity equations involving the averages h· · · i , with thegoal to sequester the Standard Model vacuum energy. The equations and physical pictureobtained in [25] are different from the ones discussed in the present work. It is arguedthat the particle physics loops are under control in [25], while the gravity loops were notconsidered. It could perhaps be interesting to apply the proposal of the present work to dealwith the gravity loops in [25].
Acknowledgements:
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