The Effective Theory of Gravity and Dynamical Vacuum Energy

Gravity and general relativity are considered as an Effective Field Theory at low energies and macroscopic distances. The effective action of the conformal anomaly of light or massless quantum fields has significant effects on macroscopic scales, due to associated light cone singularities that are not captured by an expansion in local curvature invariants. A compact local form for the WZ effective action of the conformal anomaly and stress tensor is given, requiring the introduction of a new light scalar field, which it is argued should be included in the low energy effective action for gravity. The scalar conformalon couples to the conformal part of the spacetime metric and allows the effective value of the vacuum energy, described as a condensate of a 4-form abelian gauge field strength F=dA, to change in space and time. This is achieved by the identification of the torsion dependent part of the Chern-Simons 3-form of the Euler class with the gauge potential A, which enters the effective action of the conformal anomaly as a J.A interaction analogous to electromagnetism. The conserved 3-current J describes the worldtube of 2-surfaces that separate regions of differing vacuum energy. The resulting EFT thus replaces the fixed constant Lambda of classical gravity, and its apparently unnaturally large sensitivity to UV physics, with a dynamical condensate whose ground state value in empty flat space is Lambda = 0 identically. By allowing Lambda to vary rapidly near the 2-surface of a black hole horizon, the proposed EFT of dynamical vacuum energy provides an effective Lagrangian for gravitational condensate stars, as the final state of complete gravitational collapse consistent with quantum theory. The possible consequences of dynamical vacuum dark energy for cosmology, the cosmic coincidence problem, and the role of conformal invariance for other fine tuning issues in the SM are discussed.


I. The Quantum Vacuum and Vacuum Energy in Black Holes and Cosmology
Observations of type Ia supernovae (SN) at moderately large redshifts indicate that the expansion of the universe is accelerating [1,2]. This is possible in classical general relativity (GR) only if the dominant energy of the universe has an effective mean eq. of state satisfying ρ + 3p < 0, i.e. assuming positive energy density ρ > 0, it must have negative pressure. As the cosmological term Λ enters Einstein's eqs. as a constant with p Λ = −ρ Λ = −Λ/8πG N pervading all space, the SN observations taken at face value imply a Λ value of [3] [4][5][6][7]. If Λ is such a UV sensitive quantity, and the short distance cutoff is of order of L Pl , then the value of Λ SN in Planck units represents the most severe scale hierarchy problem in all of physics, clashing with expectations of 'naturalness' developed over several decades of successful application of Effective Field Theory (EFT) methods [8][9][10].
On the other hand, if one adopts the EFT hypothesis that macroscopic gravity and the value of Λ at cosmological scales should be decoupled from and not require detailed knowledge of extreme UV physics, then (1.1) suggests instead that the EFT of gravity is incomplete at low energies, and one or more additional EFT degrees of freedom are needed to account for a vacuum energy naturally of order of the Hubble scale.
If the EFT of gravity relates Λ eff to the cosmological Hubble scale 3H 2 0 rather than the microscopic Planck scale L −2 Pl , the further implication is that Λ eff would have to become a dynamical quantity, i.e. dependent upon the content and evolution of the universe, as the Hubble 'constant' itself is [6].
Related to both possibilities of additional low energy gravitational degrees of freedom other than the metric of classical GR and of Λ eff becoming dynamical as a result, it is worth noting that current cosmological models already require at least one additional scalar (inflaton) field of unknown origin, in order to generate the present small (∼ 10 −5 ) CMB anisotropies during a very early epoch of cosmic inflation [11]. This epoch is assumed to have been dominated by a much larger effective Λ eff vacuum energy, that is supposed to have dynamically 'relaxed' to its present much smaller value.
Indications that the EFT of gravity may require some additional degree(s) of freedom relevant at macroscopic scales come also from the quite different domain of black hole (BH) physics. EFT methods in gravity for BHs have been put into question by both the extreme blueshifting of energy scales in the presence of horizons, invalidating the EFT assumption of decoupling of short distance from long distance physics [12], and by the BH 'information paradox' [13][14][15][16][17][18][19][20], and apparent conflict with unitary evolution it entails [21]. The various forms of this paradox arise from ascribing an enormous entropy to a BH, equal to 1/4 its horizon area A H in Planck units, despite the assumed classical nature of the BH horizon as a mathematical causal boundary only, with no independent degrees of freedom of its own. Significant quantum effects on the macroscopic scale of the BH horizon are also in apparent conflict with the usual EFT approach to gravity, which relies on an expansion in local curvature invariants [22][23][24], since these can yield only negligibly small corrections for large BHs with small local curvatures at their horizons. In fact, the enormous entropy (1.2) and Hawking effect upon which it is predicated rely crucially upon the specification of the quantum vacuum state, which (as always in quantum theory), requires non-local boundary conditions, that are not determined solely by the local curvature [25][26][27].
In previous work it has been noted that large quantum effects on the horizon follow quite generally from the stress tensor of the conformal anomaly in both BH and cosmological spacetimes [28][29][30].
The importance of the conformal anomaly in the near horizon behavior of the stress tensor is a consequence of the conformal scaling behavior of the metric near the horizon and the extreme blueshifting of local frequencies, which renders all finite mass scales irrelevant there [30]. It is just this extreme blueshifting of frequencies, hence energies, that can lead to effects not accounted for in local EFT expansions based on the assumption of decoupling and strict separation of scales. State dependent quantum vacuum entanglement and polarization effects are contained in the effective action of the conformal anomaly, which by its nature spans multiple scales.
That the physics of BHs and vacuum energy are related is inherent also in the proposed resolution of the BH information paradox by the formation of a gravitational vacuum condensate star with inte-rior Λ eff eq. of state p = −ρ [31,32]. This Λ eff eq. of state in the interior of a gravastar, collapsed to its gravitational radius r M = 2G N M/c 2 prevents further collapse to a BH singularity for the same reason that it causes the Hubble expansion of the universe to accelerate, namely by defocusing (rather than the usual focusing) of worldline geodesics, avoiding the classical singularity theorems [33][34][35]. The localized formation of such a p = −ρ gravitational vacuum condensate within an ultra compact star can occur only if there is at least one additional degree of freedom in the low energy EFT of gravity, whose variation allows Λ eff to change abruptly at or near r = r M . This turns the BH horizon from a mathematical surface to a physical phase boundary layer with a positive surface tension [36][37][38].
The purpose of this paper is to propose and develop the EFT of low energy gravity, deduced from general principles of QFT in curved space and the conformal anomaly, in which finite dynamical vacuum energy is consistently described as a scalar vacuum condensate. The description of vacuum energy by the scalar dual to an exact 4-form abelian field strength F = dA requires Λ eff ≥ 0 with Λ eff = 0 the unique value of lowest energy in flat space, independently of UV physics. When the 3-form potential A is identified with the Chern-Simons 3-form of the Euler class, a J · A interaction is induced by the conformal anomaly effective action, in analogy with electromagnetism. The 3-current J source for F describes the worldtube of 2-surfaces that separate regions of differing vacuum energy Λ eff , which therefore becomes spacetime dependent. The observational implications of this EFT extension of classical GR and identification of the relevant low energy degrees of freedom describing Λ eff as a dynamical gravitational vacuum condensate for both BH physics and cosmological vacuum dark energy can then be studied in detail.
The metric and curvature conventions of the paper are those of MTW [39], while the definitions and conventions for tetrads and differential forms used are reviewed in appendix A. A second appendix B is devoted to the topological aspects of the Euler density, associated Chern-Simons 3-form and physical interpretation of the new constant κ introduced in the EFT, as a torsional topological susceptibility of the gravitational vacuum.

II. Relevance of the Conformal Anomaly to Macroscopic Gravity
If one takes as the basic building block of a gravitational theory the spacetime metric g µν (x), with the requirements that the field eqs. must transform as tensor eqs. under general coordinate transformations, and be no higher than second order in derivatives of the metric, one arrives at the classical theory of general relativity (GR). This is described by the classical action namely the Einstein-Hilbert (EH) action involving the Ricci curvature scalar R, second order in derivatives of the metric, or first order in derivatives of the symmetric Christoffel connection As É. Cartan pointed out soon after the appearance of GR [40,41], the most general setting of an affine geometry and differential manifold allows also for a non-zero torsion, which is described by an anti-symmetric part of the connection Γ λ [µν] 0 that Einstein had assumed to be vanishing, as the simplest realization of the Equivalence Principle. An anti-symmetric part of Γ λ µν drops out of the geodesic eq. for the worldlines of freely falling point particles in any case. Einstein-Cartan theory allows the connection Γ λ µν and functions of it to be treated as dynamical variables in their own right, a priori independent of the spacetime metric [41], a property that will be exploited in Sec. V.
Let us note that the problems of reconciling classical GR with QFT first appear with the stressenergy tensor T µν , which is treated as a completely classical source in Einstein's eqs., whereasT µν is a UV divergent operator in QFT. Since matter and radiation in the Standard Model (SM) are certainly quantum in nature, replacing the quantum operatorT µν by its renormalized expectation value T µν in a semi-classical approximation amounts to two different assumptions, which should be recognized and distinguished at the outset.
The first assumption is that UV divergences of QFT are to be removed by counterterms involving up to dimension-four curvature invariants, such as R αβµν R αβµν , R αβ R αβ , R 2 , by adding them to the effective action with finite renormalized coefficients, leaving the low energy EFT unchanged. These terms ∼ (∂ 2 g µν ) 2 , involve up to four derivatives of the metric, in contrast to the dimension-two EH action.
Since quantum theory introduces a new scale L Pl , this standard renormalization procedure amounts to the assumption that such higher derivative terms may be important only on the corresponding Planck energy scale of M Pl c 2 = 1.221 × 10 19 GeV. Since this scale is so much higher than those generally encountered either in terrestrial accelerators or astrophysics, the reasonable assumption of an EFT approach is that Planck scale physics decouples from the low energy EFT, so that knowledge of the UV completion or full quantum theory is not needed to describe macroscopic gravitation.
Less often noted is a second critical assumption in the replacement of the stress tensor source in Einstein's eqs. by a finite renormalized T µν , treated classically, namely that the operatorT µν can be well approximated by its sharply peaked mean value. It is easily verified in QFT that even after renormalization there are quantum fluctuations from the mean, and e.g. T αβ (x)T µν (y) − T αβ (x) T µν (y) 0 at one-loop order [42]. Connected higher point correlation functions of this kind probe the polarization and entanglement properties of the quantum vacuum even at macroscopic scales. These quantum correlators exhibit operator product singularities as x → y. All is well with these UV singularities since dimensional analysis, as well as explicit calculations show that these quantum correlations grow large as L 2 Pl /ℓ 2 relative to T µν only for metric variations on the length scale ℓ L Pl . Neglecting these short distance correlations at scales ℓ ≫ L Pl reduces then to the first decoupling assumption of EFT.
There is however a second kinematic regime where the Lorentz invariant distance (x − y) 2 → 0, even for x y, and where two and higher point quantum correlation functions ofT µν can become large, namely on the light cone. Since light cones extend over arbitrarily large distances, lightlike correlations are not limited to the ultrashort L Pl , but can lead to macroscopic quantum effects, in particular on null horizons [28], which can be relevant in both BH and cosmological spacetimes with positive Λ, such as de Sitter space. The two very different sorts of quantum effects, short distance UV vs. macroscopic lightlike correlations are distinct and require two quite different EFT treatments.
The short distance UV quantum corrections to GR are taken into account by adding to the action (2.1) of classical GR the expansion in ascending powers of higher derivatives of local invariants, divided by appropriate powers of the UV cutoff scale, expected to be the Planck scale M Pl for gravity. This is the most common EFT approach [22][23][24]. As already mentioned, it amounts to the decoupling assumption common to all EFT approaches, based on the decoupling theorem of massive states in the UV from the low energy degrees of freedom [43].
On the other hand it has also been known for some time that QFT anomalies are not captured by such an expansion in higher order local invariants, nor are they suppressed by any UV scale. Anomalies are associated instead with the fluctuations of massless fields which do not decouple, and which lead to 1/k 2 poles in momentum space correlation functions, that grow large on the light cone k 2 → 0 rather than the extreme UV regime k 2 ∼ M 2 Pl . The prototype of this light cone pole is the Schwinger model of 1 + 1 dimensional massless electrodynamics and its chiral anomaly [44], which extends to the two-dimensional conformal anomaly in curved space [45,46]. In 3+1 dimensional flat space, light cone poles are found in explicit calculations in the triangle anomaly diagrams of Ĵ λ 5Ĵ αĴβ , T µνĴαĴβ in massless QED 4 [29,47], and in the stress tensor three-point correlator T αβT γλT µν of a general conformal field theory (CFT), by solution of the conformal Ward Identities in momentum space [48,49].
Quite contrary to the decoupling hypothesis, quantum anomalies lead instead to the principle of anomaly matching from UV to low energy EFT [50]. In the strong interactions, the chiral anomaly of the UV theory, QCD, survives to low energies, requiring a specific Wess-Zumino (WZ) addition to the low energy meson EFT [8,51], which is not suppressed by any high energy scale, and is technically a marginally relevant operator in the infrared (IR). Indeed, as befits being associated with light cone singularities, the chiral anomaly has both UV and IR features. The successful prediction of the low energy π 0 → 2γ decay rate provides a window into the UV and evidence for the SU(3) color group and fractional charge assignments of quarks that helped establish QCD as the UV theory of the strong interactions [52][53][54].
For gravitational theory it is the conformal anomaly in the trace of the stress-energy tensorT µν that is associated with 1/k 2 anomaly poles in higher point correlation functions, such as T αβT γλT µν .
Such light cone singularities imply the existence of at least one additional light (a priori massless) degree of freedom in the low-energy EFT of macroscopic gravity, that is not accounted for in the classical action of GR (2.1), nor by an expansion in higher order local curvature invariants.
A representative of the light cone singularities and massless pole associated with conformal anomalies is afforded by the effective action of two-dimensional gravity coupled to conformal matter [45,55] S NL, 2D where c m = N s + N f is the central charge, given by the sum of the numbers of massless scalar and fermion fields, and by the introduction of the scalar field ϕ describing a collective spin-0 degree of freedom, which is linearly coupled to R, and whose massless propagator gives rise to the light cone singularities of the underlying massless CFT [46]. Variation of (2.4) with respect to ϕ gives its eq. of motion − ϕ = R, which when solved for ϕ = − −1 R and substituted back into (2.4) returns the non-local form of the effective action (2.3). At the same time variation of (2.4) with respect to the metric g µν yields the stress tensor T µν A [g; ϕ] whose trace is −(c m /24π) ϕ = c m R/24π which is the 2D conformal anomaly. Taking two variations with respect to g αβ (x) and g µν (y) yields the connected two-point CFT correlator T αβ (x)T µν (y) which exhibits a massless 1/k 2 light cone pole in flat space [46]. Clearly such a massless scalar degree of freedom affects the macroscopic behavior of 2D gravity [56], but cannot be described by a local action in curvature invariants alone, as is the very nature of an anomaly.

III. The Effective Action of the Conformal Anomaly and Conformalon Scalar
The anomalous Ward identities for all higher point quantum correlation functions of the stress tensor containing anomalous light cone singularities can be derived by functional variation of the basic one-point expectation value of T µν in a general curved space background [49]. In D = 4 this mean value, defined and renormalized by any method that preserves its covariant conservation, results in it acquiring an anomalous trace in background gravitational and gauge fields, the general form of which is [57][58][59] T even if the underlying QFT is conformally invariant at tree level, and one might have expected this trace to vanish. In (3.1) are the Euler-Gauss-Bonnet invariant and the square of the Weyl conformal tensor respectively. The where (N s , N f , N v ) are the number of free conformal scalar, Dirac fermion, and gauge vector fields respectively. Interactions of the massless or light QFT degrees of freedom are taken into account by the L i terms in (3.1), which denote invariant Lagrangians to which these fields are coupled, such as L F = F αβ F αβ for light charged particles coupled to electromagnetism, or L G = tr G αβ G αβ for light quarks coupled to the SU(3) color gluonic gauge fields of QCD. The β i are proportional to the β-functions of these couplings.
All terms in (3.1) are dependent only upon the low energy QFT particle content, independently of a UV cutoff or the Planck scale. They are therefore independent of UV physics and with fixed matter content cannot be removed in any metric theory of gravity with a covariantly conserved stress tensor.
Successive variations of (3.1) with respect to the arbitrary metric background yield the anomalous trace Ward identities that the stress tensor correlators must satisfy, even in flat space. Note that these variations are independent of the purely local counterterms needed to define a renormalized T µ µ .
To construct the effective action corresponding to (3.1) one notes that upon multiplying (3.1) by √ −g, the various terms transform as under the conformal variation of the metric g µν → e 2σ g µν . Here is the (unique) fourth order scalar differential operator that is conformally covariant [60,61] for arbitrary σ(x). It is thus the four-dimensional analog of the second order wave operator which has the conformal property Using these properties under conformal variations and the fact that the anomaly density is the conformal variation of an effective action of the anomaly, it is straightforward to find an action satisfying (3.7). The non-local form of this anomaly effective action is [61][62][63][64] S NL anom [g] = 1 4 where ∆ −1 4 xy denotes the Green's function inverse of the fourth order differential operator (3.5) between the spacetime points x and y, indicated by subscripts in (3.8), so that d 4 x √ −g∆ 4 x ∆ −1 4 xy = 1, for all y. The non-local effective action (3.8) of the conformal anomaly in D = 4 is the analog of non-local effective action (2.3) in D = 2 spacetime dimensions. Note that due to the appearance of the fourth order curvature invariants in (3.1) and (3.8) of D = 4, a single light cone 1/k 2 pole of the anomaly first appears for three variations of (3.8) with respect to the metric, in the 3-point correlation function T αβT γλT µν in flat space. This has been checked explicitly for a general CFT in [49].
The anomaly effective action (3.8) is one term in the full one-particle irreducible (1PI) effective action obtained by integrating out all the matter/radiation fields in a fixed but arbitrary metric and background gauge fields. It is possible to classify all the terms in the full 1PI effective action of QFT into three general classes, so that the full 1PI action may be expressed as the sum [65] of (i) purely local, (ii) non-local anomalous, and (iii) non-local invariant under the action of the local Weyl transformation g µν → e 2σ g µν . The classification of terms (3.9) for the possible quantum corrections to the effective action for gravity and the conformal anomaly effective action (3.8) applies in a general curved space background, and hence is considerably more general than expansions around flat space [66][67][68]. The local terms S local are the ones usually considered in EFT approaches including the clearly IR relevant cosmological constant and EH term of (2.1) which scale as e 4σ 0 and e 2σ 0 under global Weyl rescalings respectively, together with terms higher order in local curvature invariants, divided by some high energy UV energy scale, which scale as e −2nσ 0 for n ≥ 0. For n > 0 these local terms are irrelevant in the IR. The n = 0 terms are marginal and require special care.
The local R 2 and C 2 actions are neutral under global Weyl rescalings, while S NL anom of (3.8) is unique (up to the b, b ′ , β i coefficients and S inv ) in scaling linearly with σ 0 , i.e. logarithmically under the global rescalings of the metric and distance scales. Hence the anomaly terms (3.8) can grow to importance in the IR and are classified as marginally relevant. All other n = 0 terms satisfying S inv [e 2σ g] = S inv [g] are neutral under all Weyl rescalings, and hence do not grow in the IR. While contributions to low energy gravity of the Weyl invariant terms S inv [g] cannot be excluded, the minimal assumption is to take only the relevant n < 0 local terms of (2.1) and the logarithmic non-local (3.8) as the basis for an EFT treatment of gravity. One could add to (3.1) a R term with an arbitrary coefficient, but since this term is the trace of the metric variation of a local R 2 action, it is classified with the local terms S local and not as part of the true anomaly, which is not the trace of a variation of any local action [65].
Like the chiral anomaly in QCD, the conformal anomaly is intrinsically a non-local quantum effect and like (2.3), S NL anom [g] is non-local in terms of the original metric and curvature variables. The nonlocal 1PI effective action of the conformal anomaly (3.1) in 4D can nevertheless be rendered in a compact local form [30,61,64,69] by introducing an additional local scalar field ϕ and making the replacement under Weyl conformal transformations. The scalar degree of freedom ϕ is therefore closely related (by the shift ϕ → ϕ + 2σ) to the conformal factor of the spacetime metric to which it couples.
The general classification (3.9) does not preclude the possibility that S inv might contain additional light cone singular terms relevant in the IR, implying additional low energy degrees of freedom besides ϕ. However only a single scalar ϕ is necessary to account for the known anomaly. Although it is possible to introduce additional low energy degrees of freedom, as was done in [28,69], there is no general symmetry-based reason to do so, and the simplest possibility is the minimal one of S A [g; ϕ], with ϕ the single scalar carrying all the conformal transformation properties of the WZ effective action and identity (3.11). Thus (3.10) is the minimal addition to the classical action (2.1) needed to take into account the light cone singularities of correlation functions of the stress tensor of quantum matter. Also although ϕ resembles the dilaton of string theory in some respects, it should be distinguished from it, because of the specific WZ identity (3.11) required by the Weyl cohomology of the anomaly [65], which reflects its quite different physical origin. The scalar ϕ is a collective mode, composed of a quantum correlated pair of massless SM fields contributing to T µν [29,30], similar to the Schwinger boson in QED 2 , or a U(1) ch flavor singlet (η or η ′ ) meson in low energy QCD. For this reason the distinct term of conformalon is reserved for ϕ.
As an explicit solution of the WZ consistency condition (3.11), the effective action (3.10) is a generating functional that reproduces all the anomalous conformal Ward identities of a CFT. Since ϕ has zero scaling dimension, and its propagator ∆ −1 4 xy ∼ log(x − y) 2 in position space, (3.10) scales logarithmically under global Weyl rescalings, and is a marginally IR relevant operator under finite size renormalization group scaling [70]. In S A [g; ϕ] the b, b ′ coefficients are to be fixed by experiment, as they may receive contributions from the gravitational field(s) themselves [63]. These coefficients may become dependent upon the energy scale as well, since the number and spin of fields that can be considered light enough to be effectively massless, and contributing to the conformal anomaly, is scale dependent.
As in 2D the WZ effective action of the anomaly S A is not purely a local functional of higher order curvature invariants, unlike the higher dimensional quantum corrections to GR usually considered [22,24,71]. Being derived from non-local quantum fluctuations of massless fields, the effects of (3.10) need not be negligible on macroscopic scales far greater than L Pl , and generally are relevant on null horizons, even when local curvatures are small there [28,30,72]. The massless excitations described by ϕ do not generally decouple, so they can have physically relevant effects at low energies, which must be studied on a case by case basis, and particularly in cases where naive EFT decoupling arguments would seem to fail. In fact the covariantly conserved stress tensor (3.12) derived from (3.10), whose trace is (3.1), generally grows without bound like (r−r M ) −2 , resp. (r−r H ) −2 , as either the Schwarzschild or de Sitter horizons at r M or r H are approached [28], cf. section VIII.
The effective action S A thus amounts to a specific addition to Einstein's GR, consistent with, and in fact required by first principles of QFT, general covariance, and the general form of the conformal anomaly (3.1). It is a relevant addition in both the mathematical and physical sense [65], capturing the macroscopic light cone singularities of anomalous correlation functions. It is therefore a necessary part of the low energy EFT of gravity, and should be added to (2.1) of classical GR, much as the WZ term must be added to the low energy meson theory to account for the chiral anomaly of QCD.

IV. The Cosmological Term as a 4-Form Gauge Field
The action (3.10) of the conformal anomaly is the first essential element in the EFT of gravity taking macroscopic quantum effects into account. Quite apart from and independent of the anomaly, there is a second element that plays an essential role in the characterization of vacuum energy and resolution of the naturalness problem of Λ. This is the observation that the constant Λ term in (2.1) can be reformulated in terms of an abelian gauge theory as follows [73][74][75][76][77]. Let be a 4-form field strength which is the curl of a totally anti-symmetric 3-form gauge potential i.e. F is an exact 4-form. As a natural generalization of ordinary electromagnetism where F = dA is an exact 2-form exterior derivative of the 1-form vector gauge potential A = A µ dx µ , let F be provided with the 'Maxwell' action is the scalar Hodge star ⋆ dual to F, cf. (A6), and κ is a free parameter whose significance as the topological susceptibility of the gravitational vacuum is discussed in appendix B.
Now the point is that when the rank of the D-form F is matched to the number of D = 4 spacetime dimensions, the free 'Maxwell' theory (4.4) has two very special properties, namely: (i) F is constrained to be a constant, with no propagating degrees of freedom, and (ii) its stress tensor T µν F is proportional to the metric g µν , hence equivalent to a cosmological term. The simplest example of this is usual 2-form electrodynamics in D = 2 spacetime dimensions, where the classical Maxwell action is with the dual F = 1 2 ε αβ F αβ = F 01 the electric field in one spatial dimension. The stress tensor corresponding to (4.6) is −g µν F 2 /2e 2 , provided F αβ = ∂ α A β − ∂ β A α is taken to be independent of the metric, with spacetime indices raised by g αβ . Since the electric field F is constrained by Maxwell's eqs. ∂ ν F µν = j µ in D = 2, to be a spacetime constant of integration, i.e. F = const. in the absence of sources j µ = 0, the stress tensor of electric field energy, proportional to g µν is equivalent to a cosmological vacuum energy. Classical Maxwell theory contains no propagating degrees of freedom at all in one space plus one time dimension, in the absence of sources, and the constant F simply parametrizes the energy of the vacuum. In D = 2 the electric charge e has mass dimension one, while in D = 4, the constant κ also carries mass dimension one if the field strength tensor F αβγλ is mass dimension four.
The exactly analogous situation obtains in D = 4 for (4.1)-(4.5), and in fact may be generalized to any even D spacetime dimension [76]. The equivalence to Λ in D = 4 follows once again the fact that the energy-momentum-stress tensor corresponding to (4.4) is proportional to the metric tensor, if the convention that F αβγλ with all lower indices is independent of the metric is again adopted. Analogous to the D = 2 case that F is a constant follows from the sourcefree 'Maxwell' eq. obtained by variation of (4.4) with respect to A αβγ , viz.
and ∂ λ F = 0, so that F = F 0 is a spacetime constant -in the complete absence of any sources J = 0. Hence the effective (necessarily non-negative) cosmological constant term for κ and F = F 0 real constants. In In either case this is simply the classical state of lowest energy, as well as the unique state that is even under the discrete symmetry of space parity inversion.
Moreover this setting of the value of the free constant F = F 0 , which is a priori independent of geometry, to zero in empty flat space is required by the sourcefree Einstein's eqs.
viewed as a low energy EFT. This shows already that flat space QFT estimates of vacuum energy in any way dependent upon UV cutoffs or heavy mass scales are inconsistent with Einstein's eqs. It is well-known that QFT in flat space is sensitive only to differences in energy. Hence the absolute value of quantum zero point energy in flat space, and its dependence upon cutoffs or UV regularization schemes is arbitrary and of no physical significance, a point made before, e.g. [78], but well worth emphasizing. The value of Λ eff is significant only through its gravitational effects, and hence cannot be evaluated in isolation, but only within the context of a gravitational EFT, 'on shell' as in (4.10), and only if each side of (4.10) can be evaluated independently. This becomes possible only if Λ eff is a free constant of integration, not a fixed parameter of the Lagrangian.
What has been gained then is that whereas in the usual treatment of Λ as a fixed parameter of the classical theory (2.1), which receives quantum corrections, dependent upon zero point energies and is apparently sensitive to UV physics, treating Λ eff instead as an integration constant of the classical 'Maxwell' eq. ∂ λ F = 0 of a 4-form gauge field, the free constant F = F 0 is independent of UV physics.
Initially independent also of spacetime curvature, its value is uniquely determined by evaluating both sides of (4.10) in flat space. If Λ eff is given by (4.9), F and Λ eff necessarily vanish in the flat space limit of GR, if classical (or semi-classical) GR is to be a consistent approximation to the low energy EFT of gravity.
Since the condition and a boundary condition through (4.9) is a significant step conceptually. For in addition to removing the fine tuning or naturalness problem of Λ, introducing an independent 4-form field F in place of constant Λ also allows for the introduction of sources in (4.8) that will enable F (and hence Λ eff ) to change, departing from its zero value in infinite sourcefree flat space in finite calculable ways, and eventually to become a full-fledged dynamical variable of the low energy EFT in its own right.

V. The Chern-Simons 3-Form and Anomaly Current Source for F
In section IV the 4-form field strength F = dA was postulated as an independent degree of freedom, with the observation that it contributes to the EFT of low energy gravity in the same way as an effective vacuum energy and cosmological term according to (4.9). This has the advantage of reformulating the naturalness problem of a vanishing cosmological term as simply the solution of the sourcefree 'Maxwell' eq. (4.8) that minimizes the 'electric' energy in flat space, required by consistency of GR in its flat space limit (4.10). In this section a fundamental geometric origin of F is proposed by iden- Of the several terms in the trace anomaly, the Euler-Gauss-Bonnet invariant E in (3.2) is distinguished by its topological character. Its integral is a topological invariant insensitive to local variations, and therefore can be related to global macroscopic effects, analogous to the index theorems associated with the ε αβµν F αβ F µν topological density of the axial anomaly [54]. Just as the axial anomaly density can be expressed as the total divergence of a gauge dependent Chern-Simons current, the topological character of E implies that it is also is a total divergence of a topological current, E = ∇ µ Ω µ , with Ω µ dependent upon the choice of local Lorentz frame, through the SO(3, 1) gauge connection.
The explicit form of the topological 3-form gauge field and Ω µ associated with E follows from its relation to the 4-form field strength that gives the Riemann curvature components R ab µν in terms of the affine connection 1-form which specifies the law of parallel transport of orthonormal frames in tangent space. Thus ω ab may be regarded as a local gauge potential for the Lie algebra of the G = SO(3, 1) Lorentz group, in close analogy to Yang-Mills gauge potentials for any internal group G, for which R ab would be the 2-form field strength tensor.
which has the spacetime coordinate components completely anti-symmetrized in its three indices α, β, γ. These relations imply, cf. (A13) demonstrating that the integrand √ −g E is in fact a total derivative of a coordinate frame dependent abelian current dual to A, with Ω µ = ε αβγµ A αβγ /3!, analogous to the topological density ε αβµν F αβ F µν of the axial anomaly which is the total derivative of the gauge dependent Chern-Simons current. Eq. (5.7) holds in the absence of torsion, since the it has been so far implicitly assumed that the connection ω ab is the usual Riemannian or Levi-Civita connection of the metric g µν .
To show that A is indeed an abelian gauge field, consider the response of the gauge connection to an infinitesimal local SO(3, 1) tangent frame rotation where θ ab (x) = −θ ba (x) are the 6 functions of the local Lorentz frame transformation. A short exercise then shows that the gauge potential 3-form A of (5.5) transforms under (5.8) as the exact differential δ θ A = ǫ abcd dθ ab ∧ dω cd = d ǫ abcd θ ab dω cd (5.9) and hence δ θ F = 0, so that the corresponding abelian field strength tensor is invariant under the particular form of the local 2-form gauge parameter Θ = ǫ abcd θ ab dω cd . This gauge transformation establishes the Chern-Simons 3-form A as an abelian gauge potential. Since in order for the transformation (5.9) to be non-null, Θ itself must be co-exact, i.e. not itself an exact 2-form, the gauge transformation (5.9) has just 3 independent components. This means that of the 4 independent components of A, three of them are pure gauge, and only one is gauge invariant under (5.8). Hence the single scalar ⋆F carries the full gauge invariant field content of the 3-form A.
Now the essential point is that although E is given in terms of the metric and its derivatives in Riemannian spacetime by (3.2), the 3-form potential A given by (5.5) is defined in terms of the SO(3, 1) spin connection ω ab in an orthonormal basis, and a priori independently of the spacetime metric g µν .
This distinction becomes clear when one considers Cartan's second equation of structure which defines the torsion 2-form T a [41,80]. This definition may be solved algebraically for the spin connection, viz. [41] ω ab µ = −ω ba µ = υ ν a η bc ∇ µ e c ν − K abc e c µ (5.11) whose first term is a purely Riemannian part in terms of the torsionless covariant derivative with respect to the symmetric Levi-Civita connection Γ λ µν of (2.2), which depends upon the metric and its derivatives. The second part of (5.11) is dependent upon the torsion through the contorsion tensor K abc = 1 2 T abc + T bca − T cab (5.13) which is defined by (5.10) independently of the metric. The definitions and properties of the vierbein field e a µ and its inverse υ µ a used here are given by eqs. (A1)-(A3) of appendix A. It follows from (5.11) that if all components of torsion vanish, T a bc = 0, then the affine connection ω ab µ reduces to the Levi-Civita connection, which in holonomic coordinates is just the usual symmetric Riemann-Christoffel symbol Γ λ µν of (2.2), that is fully specified by the metric and its derivatives. In that case of vanishing torsion the Chern-Simons 3-form (5.5) is purely Riemannian A = A R and metric dependent in the standard way. Conversely, if torsion (5.10) is non-vanishing, both ω ab µ and A will contain a torsion dependent part A T , which may be treated as dynamical variables that can be varied independently of the metric g µν . In the general case (5.5) will contain both purely Riemannian and torsion dependent terms, so that since the the spin connection (5.11) itself by which A is defined contains purely Riemannian and torsional terms. In (5.14) E is understood to be the purely Riemannian, torsionless Euler-Gauss-Bonnet integrand of (3.2), derived from A R alone, so that E = −(⋆dA R ) replaces (5.7) when torsion is present and A T 0. The independent variation of the affine connection is the basis of the first order or Palatini formalism of GR [39,81]. In the case of the EH action this first order formalism leads to T a bc = 0 in the absence of spin currents, and hence turns out to be equivalent to the more common approach to GR where the connection is fixed to be the torsionless Christoffel connection (2.2) from the start. For more general actions, including that of the conformal anomaly, independent variation of the affine connection and the metric generally leads to different Euler-Lagrange eqs., so that the resulting Einstein-Cartan theory differs in general from the torsionless theory.
In the EFT based on the conformal anomaly it is only the particular dependence on the spin connection through the 3-form gauge field A of (5.5) that enters, which has only 4 (not the 24 of ω ab µ ) independent components. Hence rather than adopting the full first order formalism, the proposal for the EFT is that the torsional parts of the Chern- up to a surface term which does not affect local variations and may be taken to vanish for suitable boundary conditions. Then defining the 3-current the last term in (5.16) can be expressed in the form analogous to a J · A interaction of ordinary electromagnetism. The analogy with electromagnetism is apt because the current (5.17) is covariantly conserved, i.e.
since √ −g ε αβγµ is independent of the spacetime metric cf. (A13). Thus the J · A interaction (5.18) is invariant under the abelian gauge transformation where Θ is an arbitrary 2-form, by another integration by parts. This means that (5.17) is a candidate source term for the 'Maxwell' eq. deduced from the anomaly effective action have the consequence that the vacuum energy defined by Λ eff of (4.9) will change if and when ϕ does and J αβγ 0, provided torsion (5.10) and A T 0.

VI. The Effective Theory of Gravity in the Absence of Torsion
Assembling the elements of the previous sections, the effective action for low energy gravity in the absence of any torsional contribution to the Chern-Simons 3-form (5.5) is is the conformal anomaly effective action (3.10) and S F is the 'Maxwell' action The classical Euler-Lagrange eqs. following from variation of (6.1) are for the conformalon scalar ϕ, together with the semi-classical Einstein eq.
with T µν F and Λ eff given by (4.7) and (4.9), and with the stress tensor resulting from the metric variation of S A [g; ϕ]. The first contribution here is which is the metric variation of all the b ′ terms in (3.10), both quadratic and linear in ϕ [28,30,64], are the metric variations of the last two b and β i terms in (3.10), both of which are only linear in ϕ.
In (6.4) a classical matter/radiation stress tensor T cl µν independent of ϕ has been allowed as well. If any β i L i or T cl µν terms are non-zero, the Euler-Lagrange eqs. of the fields (or fluid constitutive relations) upon which these additional degrees of freedom depend must be appended to (6.3) and (6.4), to close the system (6.3)-(6.7).
Note that the stress tensor T µν A [g; ϕ] derived from S A [g; ϕ] with ϕ treated as a classical field, is the finite renormalized stress tensor of the underlying quantum conformal QFT, after all short distance UV divergences have been removed, with T µν flat in infinite flat space defined to be zero, by the consistency condition (4.10), where ϕ may be taken to vanish as well. If boundary conditions different from infinite empty flat space are considered, T µν F and Λ eff of (4.7) parametrize any finite vacuum energy in place of Λ by (4.7), and ϕ may be different from zero. For example the stress tensor (6.6) with ϕ ∝ z 2 /a 2 , 2 ϕ = 0 and Λ eff = 0 can account for the Casimir energy and force between two parallel plates a distance a apart in the z direction in flat space, when boundary conditions appropriate to that situation are imposed on ϕ [28].
The scalar-tensor theory (6.1), based on first principles of QFT and the conformal anomaly SM fields, is quite distinct from other modifications of GR, such as Brans-Dicke theory, Hordenski theory or massive gravity. As the possible anomaly sources for ϕ in (6.3) are negligibly small in our local neighborhood, this effective theory easily passes the most constraining solar system tests, as well as the laboratory bounds on ad hoc modified gravity or other scalar-tensor theories [64,82]. Being generally covariant, the theory described by (6.1)-(6.7) is consistent with the Weak Equivalence Principle (WEP) and local Lorentz invariance. The solutions of (6.3) propagate at c, the speed of both light and of gravitational waves, consistent with present gravitational wave observations.
There are nevertheless two situations where the effects of (6.5) are significant at macroscopic scales and lead to effects qualitatively different from the purely classical theory: (i) In the vicinity of horizons, where there are large local blueshifts and the light cone singularities of anomalies come to the fore, relevant for BHs and cosmology, cf. section VIII; (ii) When the source for ϕ in (6.3) is non-gravitational in origin and sufficiently strong, such as from the QCD trace anomaly L G in dense nuclear matter, where scalar gravitational waves may be generated in compact binary mergers and in the hot, dense early universe [64].
Since the eq. of motion (6.3) for ϕ involves the fourth order conformal Panietz-Riegert operator ∆ 4 of (3.5) [60,61], and typically, differential eqs. higher than second order possess negative energy and/or unstable solutions growing in time, a few additional comments are in order here. Note first that S A and ∆ 4 do not occur in isolation, but as part of the EFT of gravity, subject to the first class constraints of diffeomorphism invariance. These constraints restrict the class of physically allowed solutions both classically and quantum mechanically. Since S A is quadratic in ϕ, a simple case in which the specific effects of the higher derivative terms in the anomaly effective action can be studied is the exactly solvable limit of G −1 N → 0, where the EH term is neglected, and on the product space R× S 3 , where E, C 2 and R all vanish. The result of this analysis is that only a small subset of solutions survive the constraints of diffeomorphism invariance, and correspondingly, only a small subspace of physical states with positive norm, and no propagating modes whatsoever, survive quantization [83].
When G −1 N 0 and the EH term of classical GR is added, the resulting EFT (6.1) becomes nonlinear, but then can be studied in linearized perturbation theory around flat space. The mixing of the scalar ϕ with the conformal factor of the metric turns this constrained mode of classical GR into a propagating one. The result is that (6.1)-(6.6) predicts the existence of scalar gravitational waves of positive energy [64], i.e. a 'breather' mode polarization in addition to the two transverse, traceless gravitational wave modes of classical GR. The active linearized solutions of (6.3) are those with 2 ϕ = 0, but Υ = ϕ 0. In other words, the modes with ϕ = 0 decouple entirely, and the remaining solutions satisfy the second order eq. Υ = 0, with positive energy. Thus in linearization around flat space, half of the solutionsof the fourth order ∆ 4 operator are eliminated by the constraints, and do not appear in the physical asymptotic states and S -matrix of the EFT. There is no instability in this case either, consistent with general results on the stability of flat space to quantum corrections [42,84].
Since S A is derived from the conformal anomaly of well-behaved SM fields, each with a unitary S -matrix in flat space, the EFT incorporating the anomaly is not expected to lead to unphysical instabilities in weakly curved or asymptotically flat space at low energies, within its range of validity.
While this interesting issue certainly deserves further investigation, and should be revisited now with the introduction of the 4-form gauge field, the detailed studies of the EFT obtained from the conformal anomaly to date are consistent with this physical expectation.

VII. The Effective Theory of Gravity in the Presence of Torsion
In the presence of torsion the Chern-Simons 3-form potential (5.5) acquires a torsional dependent term A T , which may be identified with the 3-form potential of Sec. IV and makes the additional contribution (5.18) to the effective action, which in this case becomes in which S int is given by (5.18). Since A = A T is independent of the metric, the independent variables of (7.1) are (g µν , ϕ, A αβγ ).
A possible additional torsion dependent term arising from the conformal anomaly of N f massless fermions is [68,85,86] in terms of the transverse part of the axial vector field W µ = 1 4 ǫ abcd K abc e d µ = 1 8 ǫ abcd T abc e d µ (7.3) dependent upon torsion. This term has been omitted from (7.1), assuming that it can be varied independently of (g µν , ϕ, A αβγ ), and the resulting Euler-Lagrange eq.
admits the solution W ⊥ µ = 0. Since √ −g ε αβγµ appearing in (5.18) is in fact independent of the metric, cf. (A13), the additional term S int makes no contribution to the stress tensor, and variation of (7.1) with respect to the metric gives the Einstein eqs., identical in form to (6.4).
Variation of (7.1) with respect to ϕ yields instead of (6.3). Since the 'Maxwell' action (4.4) and interaction term (5.18) may be varied independently with respect to A αβγ , with g µν and ϕ held fixed, the novel feature of (7.1) is the 'Maxwell' eq.
with the source current (5.17). Upon taking its dual, with (4.5), this becomes which is an eq. of constraint that is immediately solved by in which F 0 is a spacetime constant. Thus F can be eliminated in favor of ϕ and (7.5) becomes For asymptotically flat boundary conditions F 0 may be set to zero, but is retained here for more general cases.
For the second form of the effective action (7.1), the result is that F given by (7.8) is no longer a constant, and will change, as will the effective cosmological term Λ eff , when ϕ changes according to (7.9). Eq. (7.9) for ϕ and the Einstein eq. (6.4) together with (4.7) and (6.5)-(6.7) is the form of the proposed EFT of low energy gravity and dynamical vacuum energy in the presence of torsion.
In deciding which which form of the EFT (6.1) or (7.1) applies, the critical question is whether spacetime acquires a non-vanishing torsion. This question remains open at present. Since the vanishing of torsion is equivalent to the vanishing of the covariant derivative of thevierbein, according to the definition (5.10), it has been suggested that a natural place for torsion to appear is where the vierbein vanishes, and the locking together of the SO(3, 1) tangent space gauge group and GL(4, R) group of coordinates transformations is broken [87]. This hypothesis will be adopted in the application of the following section.

VIII. Gravitational Vacuum Condensate Stars in the EFT of Gravity
In [31,32,36] it was proposed that the solution of the multiple BH paradoxes is that the final state of complete gravitational collapse is a gravitational condensate star rather than a BH. The proposed gravastar is a compact object with a physical surface of positive surface tension replacing the BH horizon, and a static region of de Sitter space with the eq. of state p = −ρ replacing the singular interior of a BH. Because such an object is both horizonless and non-singular, with low entropy, it suffers from no information paradox, and is consistent with quantum unitary evolution. The EFT of Sec. VII provides a fundamental first principles Lagrangian basis for this proposal.
The requirements for a gravitational condensate star to be realized in gravitational collapse are first, that quantum vacuum polarization effects can grow large in the vicinity of a BH horizon and second, they can induce a phase transition to a non-vanishing interior gravitational Bose-Einstein condensate (GBEC) with p = −ρ, equivalent to a non-zero Λ eff .
That the anomaly effective action S A [g; ϕ] and stress tensor (6.6)-(6.7) of section VI can have substantial effects in spacetimes with horizons, satisfying this first part of the gravastar hypothesis may be seen in the case of the exterior static Schwarzschild spacetime tuting the solution (8.3) into the anomaly stress tensor (6.5)-(6.6), one finds diverging on the Schwarzshild BH horizon for any c S 0.
This leading order (r − r M ) −2 divergence of the stress tensor on the horizon may be understood from the kinematic blueshifting of local frequencies in (8.2) near the horizon according to (8.6) relative to that at r = ∞. The corresponding energy ω loc (r) diverges as r → r M and therefore becomes much greater than any finite mass scale. This is reflected in the fact that the wave eq. for a quantum field of arbitrary finite mass and spin becomes indistinguishable from that of a massless conformal field in the horizon limit r → r M [30], and hence the conformal anomaly effects come to the fore.
Since the stress tensor is a dimension four, conformal weight four operator, it behaves generically as the fourth power of ω loc in (8.5), i.e. ∝ f −2 . Noting that ϕ is a scalar, as is the norm of the static Killing field K = ∂ t of (8.1), which is −K µ K µ = √ −g tt = f (r), the divergence of (8.5) depending on the inverse fourth power of this norm is also a coordinate invariant scalar, and observer independent.
The diverging behavior of the local stress tensor (8.5) shows that the anomaly stress tensor can there is no solution of (6.3) in Schwarzschild spacetime with ϕ = ϕ(r) only, corresponding to a fully Killing time t invariant and spherically symmetric quantum state, with a finite stress tensor at both regular singular points r = r M and r = ∞ of the differential eq. (6.3). This result, following simply and directly from the conformal anomaly effective action, confirms results of previous studies of the stress tensor expectation value in specific states in Schwarzschild spacetime [27].
The divergences on either the future or past BH horizon (but not both) can be cancelled by allowing linear time dependent solutions of (6.3), which give rise to a Hawking flux T t r , such as in the Unruh states; or by relaxing the regularity condition at infinity which gives rise to a non-zero stress tensor there, as in the Hartle-Hawking state. This thermal state is both incompatible with asymptotically flat boundary conditions and unstable. Usually the assumption of regularity of the semi-classical T µ ν on the horizon is used to argue for the necessity of Hawking radiation flux T t r > 0 [88]. However the converse is also true: if a truly static and stable asymptotically flat solution of the final state of gravitational collapse is sought with T t r = 0, then quantum effects at the horizon, specifically due to the conformal anomaly T µν A cannot be neglected, and imply instead the breakdown of regularity there. A similar behavior is observed in de Sitter spacetime, and indeed in any spherically symmetric static spacetime (8.1) with a horizon at which −K µ K µ = f (r) → 0. With Λ positive, the static patch of de Sitter space is of this form with f (r) ∝ h(r) = 1 − H 2 r 2 = 1 − Λr 2 /3. The general spherically symmetric static solution of (6.3) for ϕ = ϕ(r) which is regular at the origin in this case is [28,30] ϕ dS (r) = ln 1 − H 2 r 2 + c 0 + q 2 ln 1 − Hr where the constant c ′ = c 0 + (2 − c H ) ln 2. Substituting this into the anomaly stress tensor (6.5) gives which also diverges as f −2 for any c H 0 as the de Sitter static horizon is approached.
As in the Schwarzschild case this divergence can be removed if a thermal state is considered, but then only if the temperature is precisely matched to the Hawking temperature T H = H/2π associated with the horizon, which in the de Sitter case leads to the maximally O(4, 1) symmetric state [89,90].
However, this state is not a vacuum state of QFT and is unstable to particle pair creation, much as a uniform, constant electric field is [91][92][93][94][95], and for essentially the same reason. Due to the nonexistence of a global static time by which positive and negative frequency (particle and anti-particle) Killing time and no horizon, as also required by the low energy EFT consistency condition (4.10).
With Λ eff replaced by a dynamical condensate according to (4.9), vacuum energy can be non-zero only if localized in space, within the static patch r < r H = H −1 of de Sitter space. If one seeks a stable spherically symmetric static solution of the EFT, the light cone enhanced effects of the anomaly stress tensor (3.12) at both the Schwarzschild and de Sitter horizons, (8.5) and (8.8) respectively, should be taken into account, and the assumption that either horizon is a mathematical boundary only should be re-examined. In the gravastar proposal [31,32], the BH horizon is the location of a physical surface phase boundary layer between two different phases characterized by different values of the vacuum energy Λ eff , regarded as a gravitational condensate. On general thermodynamic grounds the Gibbs relation ρ + p = sT + µn = 0 implies that the eq. of state of a zero temperature condensate with no conserved particle number should be p = −ρ. An argument based on non-relativistic condensed matter analogs given in [96] reached a similar conclusion.
In [36] the gravitational condensate star was shown to follow directly from Schwarzschild's con- group of coordinates transformations, where torsion may be expected to arise [87]. Thus it is natural to describe this region where f (r), h(r) → 0 by the EFT (II) of section VII. This EFT (II) then provides a mechanism and Lagrangian description for the second part of the gravitational condensate star proposal of [31,32,36], by allowing the value of Λ eff to change from exterior to interior of the gravastar, through (4.9), (7.8) and (7.9).
The scalar F, dual to the 4-form field strength of sections IV-V, is a classical coherent field that provides an explicit realization of a gravitational condensate Λ eff interior. When coupled to the conformalon scalar through the 3-form abelian current J αβγ , concentrated on a three-dimensional extended world tube of topology R × S 2 where ∂ µ ϕ grows large, ϕ, F and the condensate Λ eff all change rapidly in the radial direction. This is precisely the appropriate description of a thin shell phase boundary layer of a gravastar with S 2 spatial topology sweeping out a tube in spacetime.
One may now search for static, rotationally invariant solutions of the EFT eqs. of Secs. VI-VII which are asymptotically Schwarzshild-like with Λ eff vanishing in the exterior region, changing rapidly but continuously near the Schwarzschild r M = 2G N M/c 2 or de Sitter r H = H −1 classical hori-zons by (7.8) and (7.9) in the phase boundary region, and then remaining nearly constant 3H 2 in the interior region. The blueshifting of local frequencies ∼ f − 1 2 as in (8.6) leads to the ϕ field having an increasingly large radial derivative in the vicinity of r M ≃ r H , so that (5.17) and (6.5) and the torsional effects become significant there. The physical thickness of the phase boundary surface layer where the anomaly stress tensor (8.5) or (8.8) becomes significant and large enough to compete with the classical terms and where the EFT (II) of VII must be used is of the order of r M L Pl . The effects of this surface layer and regular de Sitter interior on binary BH mergers, gravitational waves, ringdown and 'echoes' can then be investigated in the EFT.

IX. Discussion and Outlook: Vacuum Energy as a Dynamical Condensate
In this paper an EFT of gravity has been proposed taking account of the most significant macroscopic light cone effects of the conformal anomaly of massless or light SM fields. In this EFT Λ is no longer a fundamental constant, whose value appears to be sensitive to ultra high energy physics, but rather a dynamical condensate described by a classical 4-form field strength and 3-form abelian gauge potential of sections IV-V.
The EFT based on the conformal anomaly (3.1) introduces two relevant scalar degrees of freedom to low energy gravity beyond classical GR. This follows from the local form of the effective action of the anomaly S A [g; ϕ] of (3.10) in terms of the scalar ϕ conformalon field, which satisfies a fourth order eq. of motion (6.3)- (7.5). Before the addition of S A [g; ϕ], the conformal factor of the metric in GR is constrained to be non-propagating by the classical diffeomorphism constraints of Einstein's eqs. Once S A [g; ϕ] is added to the effective action, the conformal part of the metric mixes with one of the two ϕ modes, and gives rise to scalar gravitational waves [64].
Likewise before the identification (5.15) of the potential A of the 4-form gauge field (4.1) with the torsional part of the Chern-Simon 3-form, F is constrained to be a constant and simply equivalent to a cosmological constant term in Einstein's eqs. by (4.9). Once the interaction S int [ϕ, A] is added to the effective action, the 3-form current J of (5.17) provides a source for F through (7.6), and F, hence Λ eff becomes dynamical through ϕ. It is this second scalar degree of freedom of ϕ in S A [g; ϕ] that allows (and requires) the vacuum energy to become a dynamical variable through (4.9) and (7.8).
Thus in the EFT each of the two scalar conformalon degrees of freedom present in the anomaly action for ϕ mix with, and source the previously classically constrained conformal factor of g µν and F, turning each into full-fledged dynamical degrees of freedom. Assuming the SM, the only new free parameter of the EFT is the constant κ, a kind of gravitational vacuum susceptibility cf. appendix B, determining the coupling of the current J to the gauge field A, field strength F, and Λ eff of (4.9).
For the longstanding problem of the cosmological term when QFT is coupled to gravity, in either form of the EFT the classical state of minimum energy is that of vanishing 4-form condensate: F = F 0 = 0. By the identification of Λ eff in (4.9), this condition automatically sets the value of the cosmological term to zero in infinite flat Minkowski space. By simply allowing a flat Minkowski solution, this removes one oft-stated obstacle and 'no-go theorem' to solution of the 'cosmological constant problem' [4,97]. Moreover Λ eff = 0 is required by consistency with the classical or semiclassical Einstein eqs. (4.10) or (6.4) in infinite flat Minkowski space, in the complete absence of matter and radiation. The condition (4.10) is independent of UV physics or cutoffs, as simply the unique classical state of minimum energy with zero curvature and zero torsion, and the stable ground state of the EFT of low energy gravity.
Empty flat space with zero external fields is also the unique classical state where the total anomaly A = 0, so that the WZ anomaly action (3.10)-(3.11) has an additional global shift symmetry under ϕ → ϕ + ϕ 0 in that state, and F 0123 = 0 is also required by spatial parity invariance of the ground state.
These enhanced global or discrete symmetries may be regarded as replacing the 't Hooft naturalness criterion [50] for Λ eff = 0.
The consistency condition of Λ eff = 0 on the classical condensate in the low energy EFT of gravity in flat space indicates that estimates of vacuum energy as sensitive to UV physics are not applicable if GR, with or without the conformal anomaly addition, is to be a well-behaved EFT at macroscopic scales. Arguments or estimates of the cosmological term in flat space as proportional to the fourth power of a UV cutoff (which break Lorentz invariance) or the fourth power of all masses i m 4 i ln m i in QFT (as in dimensional regularization) have no physical meaning in the absence of gravitation, as well as being in conflict with observations even on non-cosmological, solar system scales [7].
Instead a way around the 'naturalness' problem of the Λ term is to extend the EFT of gravity beyond classical GR to contain the additional scalar conformalon degree(s) of freedom inherent in the conformal anomaly effective action (3.10), and to replace the fixed parameter Λ of the classical theory by Λ eff of (4.9) in terms of a 4-form gauge field F. As a result of the 'Maxwell' eq. F satisfies, this abelian gauge field becomes a fully dynamical degree of freedom of low energy gravity by (7.6) and (7.8). The setting of an integration constant to zero at the minimum of energy in flat space is a solution of the naturalness problem of the cosmological term that involves no fine tuning of any fixed parameters of the EFT Lagrangian (6.1) of macroscopic gravity.
At lowest order all fields (g µν , ϕ, A) in (6.1)-(7.1) are treated as classical. Quantum loop corrections are then to be computed by the usual EFT method of appending local terms as needed to absorb UV divergences in an expansion in powers of 1/M 2 pl , maintaining the physical meaning of the constants of the lowest order EFT. Thus at one-loop order the energy of the vacuum will continue to be defined to be identically zero in flat space by the consistency condition (4.10) on the constant F 0 = 0, hence Λ eff = 0 in the absence of any sources, and with ϕ = 0. All formal divergences in T µν flat in purely flat space, quartic or otherwise, are treated as without physical significance and removed by this consistency condition on F 0 . There is no sensitivity of this free integration constant, or Λ eff defined in terms of it by (4.9), on UV divergences or UV mass scales.
Following the usual logic of EFT, divergences of quantum loops require the introduction of additional local terms which are higher order in powers of the Riemann curvature tensor and its derivatives, divided by higher powers of a UV scale, presumably the Planck mass scale M Pl [22][23][24], which ultimately limits the range of applicability of the low energy EFT. These higher order effects and their renormalization should be defined so as not to disturb the meaning of the low energy parameters at lowest order, such as G N , or ground state boundary condition on the condensate, Λ eff = 0 in flat space.
Logarithmic divergences in curved space (regulated by any covariant method) require the introduction of counterterms proportional to the local R 2 and C 2 curvature invariants, together with the finite logarithmic running of their dimensionless couplings. These terms and local terms involving still higher numbers of derivatives are not treated as fundamental, but rather as suppressed at energy scales far below the Planck energy M Pl c 2 , remaining negligibly small at macroscopic distance scales much greater than L Pl , as consistent with existing EFT results [22][23][24].
One may also turn the EFT logic around, to conjecture that the important role of the conformal anomaly and anomalies in general as windows into the UV, and exceptions to the usual decoupling hypothesis of EFT, may indicate that in the fundamental theory all masses vanish and conformal invariance is restored -broken perhaps only spontaneously, at asymptotically high energies. Speculations of this kind for resolution of the naturalness problem of the cosmological term and possible relation with that of the Higgs mass hierarchy have been advanced by a number of authors, e.g. [98][99][100][101][102]. Although no clearly successful complete theory has emerged from these speculations, the idea of fundamental conformal invariance and relation between these large hierarchy problems and the properties of the quantum vacuum rather than UV physics remains intriguing.
That both Λ and Higgs hierarchies may be resolvable only by the consistent inclusion of gravity receives some support from the EFT approach to vacuum energy and the cosmological 'constant' as a dynamical condensate proposed in this paper. To the extent that ∂ λ F 0, this dynamical condensate necessarily requires non-vanishing spacetime torsion. The role of relaxing the torsionless condition of classical GR in describing the condensate by (5.15) as relevant to resolving BH singularities was anticipated in [87], where specific models generating torsion were proposed. The possible extensions of the EFT proposed in this paper to generate torsion dynamically and self-consistently remain to be explored. The coupling of fermions to a condensate with torsion through the spin connection and possible relation to neutrino mass generation is another intriguing direction for future research. The microscopic constituents of the gravitational Bose-Einstein condensate (GBEC) described by F from which its superfluid nature is emergent remain to be elucidated [103,104].
As a more immediate matter, the description of Λ eff as a dynamical condensate of a 4-form gauge field in the EFT of sections VI-VII makes possible calculations of vacuum energy in numerous applications, first and foremost for a gravitational condensate star interior and surface. For cosmology, the universe as the interior of a gravastar realizes the hypothesis made in section I of automatically relating the effective value of the vacuum energy Λ eff to 3H 2 , and hence to the horizon Hubble scale H −1 , with no fine tuning. More realistic cosmological models, with Ω Λ < 1, require dynamical EFT solutions including matter and radiation, rather than a purely static de Sitter vacuum condensate.
The tying of the value of Λ eff to the Hubble scale is clearly relevant to the 'cosmic coincidence problem' of the ΛCDM model, and immediately suggests a rather different set of possibilities for cosmological models, in which spatial inhomogeneities and/or boundary conditions at the Hubble scale H −1 play an important role. The EFT of the conformal anomaly coupled to the 3-form potential and 4-form abelian field strength term presented in sections VI and VII thus provides a distinctly new framework for dynamical dark energy in cosmology based on fundamental theory.
From the form of the anomaly A in (3.1), any deviation from exact homegeneity and isotropy will lead in general to F µν F µν 0 for the photon radiation field, and tr {G µν G µν } 0 for the electroweak and QCD color gauge fields in the unconfined phase of the early universe. This will induce changes in the conformalon field ϕ through its eq. of motion (6.3), which will then cause the field strength F and hence the vacuum energy Λ eff to change. After the transition to the confining phase of QCD, baryonic matter will still contain non-vanishing gluonic condensates and thus still act as a source for ϕ, thereby coupling non-relativistic baryonic matter to dynamical vacuum energy as well.
Thus although ϕ is not an inflaton, it is a dynamical scalar that is well-grounded in QFT of the SM and can produce backreaction effects on the vacuum energy when fluctuations away from exact homogeneity and isotropy are admitted. It permits interaction between both radiation and matter with dynamical dark energy, in which adiabaticity of the matter and radiation components will no longer be satisfied in general, in effect introducing a bulk viscosity into the cosmological fluid. If Λ eff ∝ F 2 does not remain constant in the de Sitter phase, deviations from the ΛCDM cosmological model are to be expected, and evolution away from a pure de Sitter phase due to cosmological horizon modes [105] becomes calculable, and testable by the cosmological data of large scale structure.
In addition to removing the singularity and paradoxes of BHs, developing detailed predictions from the EFT proposed will allow study of gravastar stability, normal modes of oscillation, surface modes and 'echoes' that can be tested with gravitational wave and multi-messenger signals from binary merger events, in the increasing data samples expected in the future. The prediction of scalar gravitational waves can also be tested by the coming global array of gravitational wave antennae [106].
The effects of rotational angular momentum have been neglected in the simplest gravastar solution, although the first steps in including those effects have been taken in [37,38]. The EFT solutions of static or stationary gravastars also leaves unexamined the process of their formation, and in particular the behavior of the stress tensor near the would-be horizon of collapsing matter, which would have to activate the dynamical condensate terms of (7.1)-(7.5). These and many other interesting questions remain to be addressed in the context of the EFT of dynamical vacuum energy proposed in this paper.

A. Metrics, Tetrads, Differential Forms, and Hodge Star Dual
In this first Appendix conventions and mathematical details used in the text are collected and catalogued. The metric and curvature conventions used in this paper are those of Misner, Thorne & Wheeler [39]. Greek indices are four-dimensional coordinate (holonomic) indices, while Latin indices refer to local othonormal tangent space.
In the tetrad or vierbein formalism the metric line element is written with η ab = diag (−1, 1, 1, 1) the flat spacetime tangent space Minkowski metric tensor, and e a (x) are the 1-forms e a = e a µ dx µ satisfying e a µ e b ν η ab = g µν , g µν e a µ e b ν = η ab .
The dual basis of vectors υ a satisfy e a µ υ µ b = δ a b (A3a) e a ν υ µ a = δ µ ν (A3b) υ µ a υ ν b g µν = η ab (A3c) υ µ a υ ν b η ab = g µν (A3d) defining the orthonormal basis in tangent space. Coordinate indices are lowered (resp. raised) by the metric tensor g µν (resp. its inverse g µν ). Tangent space indices are lowered or raised by the flat Minkowski tensor η ab or η ab . The covariant derivative of the vierbein field in (5.12) is defined with respect to the torsionless Levi-Civita connection, that in holonomic coordinates is the familiar Christoffel symbol (2.2), which is specified entirely by the metric tensor and its first derivatives.
The exterior derivative operator d maps the general p-form into the p + 1-form where the square brackets denote anti-symmetrization with respect to the enclosed indices, and both p and p + 1 must be ≤ D in D dimensions, by the anti-symmetry of the wedge product.
That S int of (5.18) with (5.17) is independent of the spacetime metric g µν (x), and hence makes no contribution to the Einstein eq. (6.4), also follows from (A13).

B. Topological and Torsional Susceptibility of the Gravitational Vacuum
In this appendix we consider the physical interpretation of the parameter κ in (4.4), as a kind of torsional topological susceptibility of the gravitational vacuum. This depends upon the identification of the 3-form gauge field and associated field strength F, and follows by close analogy with the cases of the topological susceptibility of the QED 2 vacuum, and chiral susceptibility of QCD in D = 4.
If one were to define the topological susceptibility of the Riemannian E χ E (k 2 ) = d 4 x e ik·x E(x)E(0) (B3) directly in terms of the curvature invariants one would encounter the correlator of two dimension-four operators, with the expected short distance singularity of 1/x 8 as x → 0. Thus the integral in (B3) is undefined in perturbative quantum gravity and badly (in fact, quartically) UV divergent. To define it requires promoting the correlator to a distribution with a 2 2 δ 4 (x), a 1 δ 4 (x) and a 0 δ 4 (x) local contact terms added with arbitrary finite coefficients a 2 , a 1 , a 0 , as in [109]. This corresponds to adding three local counterterms to the action in order to make the three subtractions necessary to remove the quartic, quadratic and logarithmic divergences from (B3), and thence to obtain a finite renormalized result in terms of these three finite but unknown parameters. In the quenched approximation where the remaining finite terms vanish, only the local δ-functions and derivatives thereof remain, and for k 2 = 0, χ E, 0 (0) = a 0 = κ 4 .
When F is identified with the torsional part of the topological density of the Euler class, as in (5.15), κ 4 in the action (4.4) parametrizes the logarithmic short distance κ 4 δ 4 (x) renormalized singularity of this torsional density. Because of (B1b)-(B2), this is the leading order effect of quantum gravitational vacuum fluctuations at short distances that is physically relevant to the k 2 → 0 low energy (light cone) correlations of the EFT.
Since F involves just one derivative of the gauge potential A in the low energy EFT, F/κ 2 is a quantum operator of mass dimension 2 in terms of A/κ 2 , in contrast to E which is fourth order in metric derivatives. By this accounting (4.4) is a dimension 4 (rather than dimension 8) operator which is marginally IR relevant in the Wilsonian EFT sense in D = 4, just as (4.6) is in D = 2. In the QED 2 Schwinger model case there are no UV divergences whatsoever and e 2 is a UV finite coupling, despite having dimensions of (mass) 2 . This may provide an interesting prototype of how parameters with positive mass dimensions can nevertheless remain finite and insensitive to UV corrections. If the matter contributions to the vacuum polarization self-energy d 4 x e ik·(x−y) J αβγ (x)J λµν (y) are also UV finite, as suggested by its bosonized form (5.17), which converts this self-energy to a classical tree graph in terms of ϕ, just as occurs in D = 2 [46], then the torsional topological susceptibility κ will also be UV finite in D = 4. This interesting possibility also merits an independent investigation.
Although one might expect the distance scale 1/κ of non-trivial vacuum topology change to be of order L Pl , and the value of κ to be of order of M Pl , there is no a priori relation between the two scales.
They are initially distinct, just as Λ QCD and f π are in QCD, to become possibly related only in a UV complete theory of quantum gravity. Otherwise κ and M Pl are treated as independent and unrelated dimensionful constants in the low energy EFT of gravity proposed in this paper.