The Big Constant Out, The Small Constant In

Some time ago, Tseytlin has made an original and unusual proposal for an action that eliminates an arbitrary cosmological constant. The form of the proposed action, however, is strongly modified by gravity loop effects, ruining its benefit. Here I discuss an embedding of Tseytlin's action into a broader context, that enables to control the loop 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 universe is that of unbroken AdS supergravity. A vacuum energy in our universe does not produce any curvature for us, but instead increases or decreases the AdS curvature in the other universe. I comment on how to introduce the accelerated expansion in this framework in a technically natural way, and consider the case where this is done by the self-accelerated solutions of massive gravity and its extensions.

1. Introduction: Nearly a quarter-century ago, Tseytlin [1] has proposed an approach to the old cosmological constant problem, using an original idea by Linde [2], and certain string-theory developments of that time. The proposal is technically well-framed, while a highly unconventional nature of this approach is commensurate with the magnitude and longevity 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 constant, the action itself was argued to be unstable w.r.t. 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 to incorporate the dark energy component into the theory in a technically natural way.

Tseytlin's proposal:
To set the conventions, consider the action: where ψ n n = 0, 1, 2, 3..., denote all fields of the theory beyond the metric field g µν . Ten Einstein equations can be decomposed as nine, plus one trace equation: (Unless M Pl is displayed explicitly, we use the M Pl = 1 units).
Instead of this, Tseytlin introduced a system where the trace equation is modified: where · · · denotes a certain space-time average defined as follows: 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 past and future included. These averages, when nonzero, have a pre-notion of future. In that sense, this is an acausal modification. Somewhat similar, but essentially different proposal was made in [3]; the subtle issues of defining the average, where there are more than one vacua, was also raised there. To begin with, we envision a simple universe evolving in one vacuum state, and comment on possible complications later. If V g → ∞, as in Tseytlin's approach, then for most of the stuff in the universe the r.h.s. of the trace equation in (3) is zero: For any observable, O, that is localized either in space or in time, the average < O > 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 the stuff in the universe. On the other hand, for a constant Lagrangian, L = c , the r.h.s. of the trace equation (3) is proportional to the constant c itself, and the latter subtracts the equivalent part on the l.h.s., hence leaving the equation independent of c! Therefore, the main consequence of the acausality might be that we don't observe the big cosmological constant in our universe [3].
For a scalar field the Lagrangian can be decomposed into the part that depends on the metric (derivative terms, e.g., the kinetic term) and the one that is independent of g µν : Simplest examples of V are a vacuum energy term E 4 vac , scalar mass term m 2 φ φ 2 , scalar potential λφ 4 , or a linear combination of the above. The vacuum energy, or a constant part of a potential V , would give rise to a nonzero average, V = [Const.]/V g = Const.
Thus, this quantity would be subtracted from the trace T in (3). This is equivalent to the elimination of the cosmological constant! The fact that a constant term in L is irrelevant, can also be seen by looking at the action that Tseytlin introduced as an object which has to be varied w.r.t. g µν to get the equations (3). Any constant shift, L → L + c, gives rise to a shift of the new action by the same constant,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 is being subtracted is the value in the "true" minimum, assuming that a transition from "false" to "true" is possible in the standard General Relativity context in finite time. Generically, what is being subtracted is what would have been the asymptotic future value of the vacuum energy 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 decays on solutions for which the field settles in its minimum, therefore its average · · · is zero. Thus, inflation would generically proceed in a conventional way, except the phenomenon of self-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 cosmological constant problem, at least in the limited context specified above, there are two important issues that it fails to address: First, as mentioned already in [1], the loop corrections should be problematic, and they are indeed: They strongly renormalize the form of the action (6), and thus ruin the solution of the cosmological constant problem. Can the issue of the loops be resolved, by perhaps extending the proposal?
Second, Tseytlin's mechanism eliminates entirely the cosmological constant. However, dark energy has to be included. The latter appears to have an equation of state wellparametrized by w = −1. We'll discuss how this can be accommodated in a technically natural way; one option is to invoke massive gravity for this purpose.
We proceed by making the problem of quantum corrections more explicit. In the absence of gravity, i.e., in the M Pl → ∞ limit,S differs from the standard action by an overall 1 ∞ factor; the latter factor is field independent, and thus can be rescaled away. Therefore, in the absence of gravity, one would quantize the theory (6) in a conventional way. We will thus regard the Lagrangian L as the one in which all the quantum corrections due to nongravitational interactions have already been taken into account, e.g., as an effective quantum action for all the fields but g µν .
Operationally, we define the partition function Z(g, J n ) ∼ dµexp(i d 4 xL(g, J n ,ψ n )), where dµ is an appropriate measure for all the fieldsψ n . The metric field is an external field, and so are J n 's. Then, L(g, ψ n ) is defined via the Legendre transform of ilnZ(g, J n ), done in a standard way (i.e., using ψ n ∼ δlnZ(g, J n )/δJ n . The obtained quantum effective action (the 1PI action), L, is then inserted in (6) to account for dynamical gravity.
In the end, however, g µν should also be quantized in (6) 1 . The trouble with gravity loops, even in the effective field theory approach, can be understood by observing that the 1/V g factor in (6) is an effective rescaling of the Planck's constant, → V g , which would now appear in the diagrams involving gravity loops [1]. Because of this, one would get: where L 1 , L 2 , .. contain all possible terms consistent with diffeomorphism and internal symmetries. The gravity loop corrections are huge, since V g is huge. The new terms ruin the above-presented solution of the cosmological constant problem. It should be noted, that there is yet another class of loop corrections if one quantizes graviton fluctuations in the theory (6) on a given background solution 2 . To consider the effects 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 volume factor, V −1 g , multiplying the action S in (6), can also be expanded as follows:  (6). This suggests that the loop effects discussed in the present paragraph could be assumed to be small and be neglected. Similar to these considerations apply to the proposal discussed in the next section.

4.
Dealing with the problems: To avoid the above difficulty with the quantum loops discussed in the previous section, let us use the following action instead of (6): where a second metric f AB (y), A, B = 0, 1, 2, 3, ..D − 1, has been introduced, and V f = d D y f (y), in the M Pl = 1 units. Note that while the action S is four dimensional, the f metric could live in D ≥ 4 dimensions in general.
The action of the f-universe has a certain vacuum energy scale M ∼ < M Pl , and the scale that determines the strength of its gravitational coupling is M f . Depending on details of the theory -encoded in the dots in (8) -there may or may not be a stable hierarchy between the scales M f and M (see below).
The main idea is that in (8) any shift of L by a constant, L → L + c, converts c into a cosmological constant of the f-universe, removing it from the g-universe, where we presumably reside. Thus, while the curvature in our universe is (nearly) zero, the other universe could be highly curved.
However, in order for the loops not to ruin the above classical property, one should make sure that V f >> V g : indeed, then, → (V g /V f ) and the action including the potentially dangerous gravity loops would take the form As long as V f >> V g , all the corrections proportional to V g /V f can be neglected. There are also terms similar to the ones discussed in the last paragraph on the previous section, but they are harmless for the same reasons as before 3 . This is not all however, the gravity loop diagrams in the f-universe generate two groups of new terms -first, the terms containing higher powers and derivatives of curvatures R(f ) ′ s, and second, terms containing powers of S (and their products with powers of R's and derivatives); some of these terms are displayed in (9). All these terms, however, introduce small corrections, as it will be clear from the discussion given below on the hierarchy between the scales in the g-and f-universes.
In general, both V f and V g are divergent. It is sufficient for our purposes that the condition V f /V g >> 1 is satisfied, even though V f and V g individually tend to infinity. For considerations of this ratio it is convenient to invoke the Euclidean space to get a sense of the ratio of the Euclidean four-volumes, V f /V g , as will be discussed below.
Then, how do we achieve the condition V f /V g >> 1? To fulfill this we're going to explore technically natural hierarchies between parameters of the theory. First of all, we assume that the g-universe has supersymmetry broken at some high scale, and therefore, there is a natural value of its vacuum energy density proportional to E 4 vac . The scale E vac can be anywhere between a T eV and the GUT scale, µ GU T ∼ 10 16 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 be somewhat higher than E vac . The latter condition should be natural, since without special arrangements one would expect M ∼ M f ∼ M Pl , and since E vac << M Pl , one would also get E vac < M. If so then, the vacuum energy of the g-universe, E 4 vac , would make a small contribution to the pre-existing vacuum energy of the f-universe. In short, the vacuum energy density of the f-universe, c 0 M D , would dominate over the vacuum energy density that gets delegated from the g-universe to the f-universe.
While one could try to explore a case when the f-universe has a positive vacuum energy density, it seems more straightforward to make a mild assumption that the curvature due to the term c 0 M D in the f-universe is negative (AdS like). 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 supergravity with the "Planck scale" equal to M f , and the quantity acting as its vacuum energy density. The action (8) completed to the one of the N = 1 AdS supergravity [6] would then be written as: whereẽ is the determinant of the vierbein of the f-metric,ω is its spin connection, D = ∂ − 1 2ω σ is the covariant derivative, and ψ µ is the Rarita-Schwinger field describing a fgravitino. The quantityS enters into λ, while the latter defines the cosmological constant (with AdS sign) as well as a quadratic term for the gravitino. Thus, the entire g-universe enters this action via the parameter λ defined in (10). The gravitino bilinear term in (11) 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 classical solutions. Thus this term will not change our conclusions on the cosmological constant 4 .
There is no reason for the parameter λ to be much smaller than M 2 f ; quantum corrections would renormalize the former up to the scale of the latter even if we started with a large hierarchy between them. On the other hand, we do need some small hierarchy between M f and λ 1/2 , essentially to be sure that AdS curvature of the f-universe can reliably be described in the supergravity approximation. For this, an order of magnitude hierarchy, M f ∼ 10λ 1/2 , would be more than enough. While this hierarchy could perhaps be attributed, without too much of anxiety, to the 4π 2 loop factor's here and there, we note that it could be generated dynamically if we were to introduce more general supersymmetric theory with some matter fields in the Lagrangian: the N m matter fields with characteristic scale M m would renormalize additively the Planck scale M f via the Adler-Zee mechanism producing, [8], while a renormalisation of the cosmological constant λ due to complete SUSY multiplets of matter would have been zero. Thus, we could adopt, M f ∼ 10λ 1/2 , as a technically natural choice. If so, then the hierarchies M Pl ∼ M f ∼ 10M, M ∼ > 10E vac , ensure that all the corrections in (9) are negligible in comparison with the terms in (8).
Having the scales clarified, let us see how this plays out for the cosmological constant for a general D-dimensional f-universe. First we consider the case when f is not among the fields ψ n , n = 0, 1, 2, 3, ... Then, the new terms in (8) or (11) do not affect the equations (3), except that they introduce a overall multiplier V f . Thus, the cosmological constant is eliminated from the g-universe. There is, however, a new equation due to variation w.r.t. f : The right hand side contains a vacuum energy generated in our universe,S = [E 4 vac ] Vg = E 4 vac , as well as that of the f-universe. According to our construction, the net energy density is negative, so that the f-universe has an AdS curvature. If so, then V f = ∞. Then, to reach our goal it is sufficient to have V g finite, so that V f >> V g . A de Sitter universe with Euclidean V g = H −4 0 would fit the data and satisfy the above criterium 5 . 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 −4 0 . We'll discuss below how this could nevertheless be achieved. 5. Getting the accelerated universe: One needs to get a dS metric in the g-universe without using a vacuum energy or a scalar potential. More precisely, one would need to get the small dS curvature due to the terms in the Lagrangian (5) that explicitly depend on g.
There might be a few ways of achieving this: e.g., by invoking Lorentz invariant condensates of some vector fields with a coherence length comparable with H −1 0 , or by using field theories with higher derivatives but no Ostrogradsky instabilities. Such proposals could produce dark energy due to terms that aren't potentials, but depend on the metric g, so that the last term on the r.h.s. of the trace equation (3), would define the cosmic speed-up.
We briefly comment here on a possibility to obtain this feature due to massive gravity. Nonlinear massive gravity [9,10], or some of its extensions [11,12,13,14], introduce graviton mass m as a small parameter, m ∼ H 0 , in a technically natural way [15]; these theories also produce self-accelerated solutions with a dS background [16]; moreover, the fluctuations on these backgrounds are healthy when the pure massive graviton is amended with a dilaton-like field [13,14] (for theory reviews of massive gravity see, [17,18]).
Let us briefly outline how massive gravity would produce R ∼ m 2 in the trace equation (3). For this we put aside the matter Lagrangian and assume that L equals to the diffeomorphism invariant potential of massive gravity [10]: 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 coordinate system parametrized by φ a , a = 0, 1, 2, 3. The φ a (x) fields also represent the Stückelberg fields that guarantee diffeomorphism invariance of (13). The square root of a matrix and its traces are defined via its eigenvalues, and α 3 , α 4 , are some free parameters. Note that all possible values of the three parameters of the theory, m, α 2 , α 3 , are technically natural [15]. 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 to adding into (13) the kinetic term for the quasidilaton σ (and possibly some other derivative terms [13]), and replacing γ → e 2σ/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 [ On the self-accelerated solutions this trace equals to a constant, T g ∼ M Pl 2 m 2 . Therefore, T g on the l.h.s. in (3) will cancel with T g on the r.h.s.; the remaining trace equation will take the form On the selfaccelerated solutions, however, g µν ∂U(K)/∂g µν | SA = −C(α 2 , α 3 ), is also a constant, that depends on the parameters α 2 and α 3 . Therefore, its average yields the same 5 That V AdS f /V dS g → ∞ can also be seen in Lorentzian signature, by calculating the ratio, e.g., in the global coordinate systems for the universal covering of AdS, and the dS space. constant, and we get R = 2m 2 C(α 2 , α 3 ). For a certain reasonable magnitudes, and certain signs of the parameters, one gets the dS curvature of the order, m 2 ∼ H 2 0 , in a technically natural way. Quasidilaton does not change this conclusion, it only affects (improves) dynamics of small perturbations above the solution [14]. Thus, to summarize, the above approach enables to remove the big cosmological constant, and to get a small space-time curvature determined by the graviton mass.
In the approach adopted above γ was taken to be independent of the f metric, that was used to remove the big cosmological constant. We've discussed the case when f was an AdS metric, while γ was flat. However, neither of these choices are ordained -we only require that space-time described by f to have an infinite Euclidean volume. It is intriguing, therefore, to consider 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 (12); the latter will now be modified due to the terms in (13), but the modification is proportional to m 2 ∼ H 2 0 << M 2 f , and should be negligible. If such a framework can be made to work in detail, this would provide an 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 theories at energies way below M Pl ; while this may not be in conflict with observations in our universe due to the Vainshtein mechanism [19], in its intricate cosmological and astrophysical form [20], [21], [22], 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 backgrounds probing this strong scale, are indeed artifacts to be removed in a complete treatment.

Conclusions and outlook:
The proposed approach eliminates the cosmological constant, at least in a simple setup where there are a few (non-proliferating) vacua with wellseparated hierarchy between their energy densities, and allowed transitions between them. What is eliminated is what would have been an asymptotic future value of the cosmological constant for such a potential in GR; for instance, for two vacua, "false" and "true", with allowed transitions from "false" to "true", the "true" vacuum energy is eliminated. This is similar 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 condensates of vector fields, or via derivatively interacting scalar fields. We briefly discussed how the accelerated universe could be due to massive gravity in this approach.
As an outlook, I'd like to make three comments on the literature: Ref. [3] has made arguments for a connection of a "high-pass filter" modification of gravity with a specific theory containing averages · · · . It might be interesting to see if the present proposal could also be connected to some "high-pass filter" modified gravities discussed in [3]. Conversely, one could then hope to find an action principle for the equations of [3], and address the issue of the gravity loops for them.
The original motivation of Tseytlin was to obtain the unconventional action (6) by including the winding modes of string theory. It would be interesting to see if any proposal along this idea can give the action (8), or a version of it.
Refs. [23] have recently discussed gravity equations involving the averages · · · , with the goal to sequester the Standard Model vacuum energy. The equations and physical picture obtained in [23] are different from the ones discussed in the present work. It is argued that the particle physics loops are under control in [23], while the gravity loops were not considered. It could perhaps be interesting to apply the proposal of the present work to deal with the gravity loops in [23].