Scholarly article on topic 'Cosmological dark energy: prospects for a dynamical theory'

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New Journal of Physics

The open-access journal for physics

Cosmological dark energy: prospects for a dynamical theory

Ignatios Antoniadis1'4, Pawel O Mazur2 and Emil Mottola35

1 Department of Physics, CERN, Theory Division, CH-1211 Geneva 23, Switzerland

2 Department of Physics and Astronomy, University of South Carolina, Columbia SC 29208, USA

3 T-8, Theoretical Division, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

E-mail:, and

New Journal of Physics 9 (2007) 11 Received 20 October 2006 Published 22 January 2007 Online at


Abstract. We present an approach to the problem of vacuum energy in cosmology, based on dynamical screening of A on the horizon scale. We review first the physical basis of vacuum energy as a phenomenon connected with macroscopic boundary conditions, and the origin of the idea of its screening by particle creation and vacuum polarization effects. We next discuss the relevance of the quantum trace anomaly to this issue. The trace anomaly implies additional terms in the low-energy effective theory of gravity, which amounts to a non-trivial modification of the classical Einstein theory, fully consistent with the Equivalence Principle. We show that the new dynamical degrees of freedom the anomaly contains provide a natural mechanism for relaxing A to zero on cosmological scales. We consider possible signatures of the restoration of conformal invariance predicted by the fluctuations of these new scalar degrees of freedom on the spectrum and statistics of the cosmic microwave background radiation (CMB), in light of the latest bounds from WMAP. Finally we assess the prospects for a new cosmological model in which the dark energy adjusts itself dynamically to the cosmological horizon boundary, and therefore remains naturally of order H2 at all times without fine tuning.

4 On leave from CPHT (UMR CNRS 7644) Ecole Polytechnique, 91128 Palaiseau Cedex, France.

5 Author to whom any correspondence should be addressed.

New Journal of Physics 9 (2007) 11 1367-2630/07/010011+47$30.00

PII: S1367-2630(07)35484-0 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft


1. Vacuum fluctuations and the cosmological term 2

2. Quantum effects in de Sitter spacetime 7

2.1. Classical de Sitter spacetime......................................................7

2.2. Particle creation in de Sitter space................................................9

2.3. Thermodynamic instability of de Sitter spacetime................................14

2.4. Graviton fluctuations in de Sitter spacetime........................................15

3. Quantum gravity in 2D 18

4. Quantum conformai factor in 4D 21

5. Finite volume scaling and infrared screening of X 24

6. Conformai invariance and the CMB 26

7. The low energy EFT of gravity 32

8. The horizon boundary 36

9. A new cosmological model of dark energy 40 Acknowledgment 44 References 44

1. Vacuum fluctuations and the cosmological term

Vacuum fluctuations are an essential feature of quantum theory. The attractive force between uncharged metallic conductors in close proximity, discovered and discussed by Casimir more than half a century ago, is due to the vacuum fluctuations of the electromagnetic field in the region between the conductors [1]. At first viewed perhaps as a theoretical curiosity, the Casimir effect is now being measured with increasing accuracy and sophistication in the laboratory ([2] and references therein). The Casimir force directly confirms the existence of quantum fluctuations, and the field theory methods for handling the ultraviolet (UV) divergences they generate, to obtain finite answers at macroscopic distance scales. When combined with the Equivalence Principle, also well established experimentally, this success of relativistic quantum field theory should permit the treatment of the effects of vacuum fluctuations at macroscopic distances in the context of general relativity as well.

In classical general relativity, the requirement that the field equations involve no more than two derivatives of the metric tensor allows for the possible addition of a constant term, the cosmological term A, to Einstein's equations,

r - R A^ = 8^7?. (i)

If transposed to the right-hand side of this relation, the A term corresponds to a constant energy density pA = c4A/8nG and isotropic pressure pA = -c4A/8nG permeating all of space uniformly, and independently of any localized matter sources. Hence, even if the matter 7ab = 0, a cosmological term causes spacetime to become curved with a radius of curvature of order |A|-1/2.

In purely classical physics, there is no natural scale for A. Indeed if h = 0 and A = 0, there is no fixed length scale at all in the vacuum Einstein equations, G/c4 being simply a conversion factor between the units of energy and those of length. Hence A may take on any value whatsoever with no difficulty (and with no explanation) in classical general relativity.

As soon as we allow h = 0, there is a quantity with the dimensions of length that can be formed from h, G, and c, namely the Planck length,

(h G\ 1/2

— J = 1.616 x 10-33 cm. (2)

Hence when quantum theory is considered in a general relativistic setting, the quantity

A - A^pl = —, (3)

becomes a dimensionless pure number, whose value one might expect a theory of gravity incorporating quantum effects to address.

Some eighty years ago W Pauli was apparently the first to consider the question of the effects of quantum vacuum fluctuations on the curvature of space [3].6 Pauli recognized that the sum of zero point energies of the two transverse electromagnetic field modes in vacuo,

d3k hwk 1 hc (2n)3 2 = 8n2 L4

Pa = 2 ^ x3 ^ =——— = -pA, (4)

contribute to the stress-energy tensor of Einstein's theory as would an effective cosmological term A > 0. Since the integral (4) is quartically divergent, an UV cutoff L-n of ( ) at large k is needed. Taking this short distance cutoff Lmin to be of the order of the classical electron radius e2/mc2, Pauli concluded that if his estimate were correct, Einstein's theory with this large a A would lead to a universe so curved that its total size 'could not even reach to the moon.' If instead of the classical electron radius, the apparently natural but much shorter length scale of Lmin ~ Lpl is used to cutoff the frequency sum in (4), then the estimate for the cosmological term in Einstein's equations becomes vastly larger, and the entire universe would be limited in size to the microscopic scale of Lpl (2) itself, in even more striking disagreement with observation.

Clearly Pauli's estimate of the contribution of short distance modes of the electromagnetic field to the curvature of space, by using (4) as a source for Einstein's equations (1) is wrong. The question is: why? Here the Casimir effect may have something to teach us. The vacuum zero point fluctuations being considered in (4) are the same ones that contribute to the Casimir effect, but this estimate of the scale of vacuum zero point energy, quartically dependent on a short distance cutoff Lmin, is certainly not relevant for the effect observed in the laboratory. In calculations of the Casimir force between conductors, one subtracts the zero point energy of the electromagnetic field in an infinitely extended vacuum (with the conductors absent) from the modified zero point energies in the presence of the conductors. It is this subtracted zero point energy of the electromagnetic vacuum, depending upon the boundary conditions imposed by the conducting surfaces, which leads to experimentally well-verified results for the force between the conductors. In this renormalization procedure, the UV cutoff L^ drops out, and

6 W Nernst had given an estimate of zero point energy similar to (4) in connection with aether theories even earlier [4].

the distance scale of quantum fluctuations that determine the magnitude of the Casimir effect is not the microscopic classical electron radius, as in Pauli's original estimate, nor much less the even more microscopic Planck length Lpl, but rather the relatively macroscopic distance d between the conducting boundary surfaces. The resulting subtracted energy density of the vacuum between the conductors is

Pv = -720 d4. (5)

This energy density is of the opposite sign to (4), leading to an attractive force per unit area between the plates of 0.013 dyne cm-2 (jxm/d)4, a value which is both independent of the UV cutoff L^n, and the microscopic details of the atomic constituents of the conductors. This is a clear indication, confirmed by experiment, that the measurable effects associated with vacuum fluctuations are infrared phenomena, dependent upon macroscopic boundary conditions, which have little or nothing to do with the extreme UV modes or cutoff of the integral in (4).

By the Equivalence Principle, local short distance behaviour in a mildly curved spacetime is essentially equivalent to that in flat spacetime. Hence on physical grounds we should not expect the UV cutoff dependence of (4) to affect the universe in the large any more than it affects the force between metallic conductors in the laboratory.

In the case of the Casimir effect a constant zero point energy of the vacuum, no matter how large, does not affect the force between the plates. In the case of cosmology it is usually taken for granted that any effects of boundary conditions can be neglected. It is not obvious then what should play the role of the conducting plates in determining the magnitude of pv in the universe, and the magnitude of any effect of quantum zero point energy on the curvature of space has remained unclear from Pauli's original estimate down to the present. In recent years this has evolved from a question of fundamental importance in theoretical physics to a central one of observational cosmology as well. Observations of type Ia supernovae at moderately large redshifts (z ~ 0.5 to 1) have led to the conclusion that the Hubble expansion of the universe is accelerating [5]. According to Einstein's equations this acceleration is possible if and only if the energy density and pressure of the dominant component of the universe satisfies the inequality,

p + 3p = p(1+3w)< 0. (6)

A vacuum energy with p > 0 and w = pv/pv = — 1 leads to an accelerated expansion, a kind of 'repulsive' gravity in which the relativistic effects of a negative pressure can overcome a positive energy density in (6). Taken at face value, the observations imply that some 74% of the energy in the universe is of this hitherto undetected w = — 1 dark variety. This leads to a nonzero inferred cosmological term in Einstein's equations of

3H 2 c3

— ___ hG

Ameas - (0.74) —0 - 1.4 x 10-56cm-2 - 3.6 x 10-122 —. (7)

Here H0 is the present value of the Hubble parameter, approximately 73 km s 1 Mpc 1 ~ 2.4 x 10-18s-1. The last number in (7) expresses the value of the cosmological dark energy

—2 3

inferred from the SN Ia data in terms of Planck units, Lpl = j£, i.e. the dimensionless number in (3) has the value

A ~ 3.6 x 10-122. (8)

Explaining the value of this smallest number in all of physics is the basic form of the 'cosmological constant problem.'

We would like to emphasize that the naturalness problem posed by the very small value of the cosmological vacuum energy X of (7) arises only when quantum fluctuations (h = 0) and gravitational effects (G = 0) are considered together. As we have already noted, if the universe were purely classical, Lpl would vanish and A, like the overall size or total age of the universe, could take on any value whatsoever without any technical problem of naturalness. Likewise as the Casimir effect makes clear, if G = 0 and there are also no boundary effects to be concerned with, then the cutoff-dependent zero point energy of flat space (4) can simply be subtracted, with no observable consequences. A naturalness problem arises only when quantum vacuum fluctuations are weighed by gravity, and the effects of vacuum zero point energy on the large scale curvature of spacetime are investigated. This is a problem of the quantum aspects of gravity at macroscopic distance scales, very much greater than Lpl. Indeed, what is measured in the supernova data (7) is not directly the energy density of the vacuum (4) at all, but rather the geometry of the universe at very large distance scales. The dark energy content of the universe and the equation of state pv/pv ~ — 1 are inferred from the observations, by assuming the validity of (6) and therefore of Einstein's equations (1) as the effective theory of gravity applicable at cosmological distances.

The treatment of quantum effects at distances much larger than any UV cutoff is precisely the context in which effective field theory (EFT) techniques should be applicable. The EFT point of view is the one we shall adopt for this paper. This means that we assume that we do not need to know every detail of physics at extremely short distance scales of 10—33 cm or even 10—13 cm in order to discuss cosmology at 1028 cm scales. What is important instead is the Equivalence Principle, i.e. invariance under general coordinate transformations, which greatly restricts the form of any EFT of gravity. In his search for field equations for a metric theory with universal matter couplings, which incorporate the Equivalence Principle automatically but which is no higher than second order in derivatives of the metric, Einstein was using what we would now recognize as EFT reasoning. In an EFT treatment quantum effects and any UV divergences they generate at very short distance scales are absorbed into a few, finite low-energy effective parameters, such as G and A.

General coordinate invariance of the low-energy EFT does requires a more careful renormalization procedure than a simple normal ordering subtraction, which suffices for the original Casimir calculations in flat space. The UV divergent terms from the stress tensor must be isolated and carefully removed in a way consistent with the Equivalence Principle to extract physical effects correctly. These more general renormalization procedures, involving e.g. proper time, covariant point splitting or dimensional regularization have been developed in the context of quantum field theory in curved spacetime ([6] and references therein). The non-renormalizability of the classical Einstein theory poses no particular obstacle for an EFT approach. It requires only that certain additional terms be added to the effective action to take account of UV divergences which are not of the form of a renormalization of G or A. The result of the renormalization program for quantum fields and their vacuum energy in curved space is that general relativity can be viewed as a low-energy quantum EFT of gravity, provided that the classical Einstein-Hilbert classical action is augmented by these additional terms, a necessary by product of which is the quantum trace anomaly of massless fields [7]. We do not review the technology of renormalization of the stress tensor here, referring the interested reader to the literature for details [6]. We shall make extensive use of one important result of the renormalization of the stress tensor however,

namely the trace anomaly and its effects at large distance scales. Hence it is the renormalization of the quantum stress tensor in curved space which provides the rigorous basis for an EFT approach to gravity at distance scales much larger than LPl.

The essential physical assumption in an EFT approach is the hypothesis of decoupling, namely that low-energy physics is independent of very short distance degrees of freedom and the details of their interactions. All of the effects of these short distance degrees of freedom is presumed to be encapsulated in a few phenomenological coefficients of the infrared relevant terms of the EFT. Notice that this will not be the case in gravity if the low-energy A relevant for dark energy and cosmology depends upon the quantum zero point energies of all fields up to some UV cutoff, as in (4). If (4) is to be believed, then the introduction of every new field above even some very large mass scale would each contribute its own zero point energy of the order of L—4 and generate additional terms relevant to the large scale curvature of spacetime. Clearly this contradicts any intuitive notion of decoupling of very massive states from low-energy physics. Despite the severe violation of decoupling this represents, the usual presumption is that the 'natural' scale for A is of order unity in Planck units, i.e. X ~ 1.

In order to make the naturalness problem of small numbers and large hierarchies more precise, 't Hooft gave the intuitive notion of fine tuning a technical definition [8]. According to his formulation, a parameter of an EFT can be small naturally, only if setting it equal to zero results in a larger symmetry of the theory. Then quantum corrections will not upset the hierarchy of scales, once imposed. An example of such a naturally small parameter is the ratio of the pion mass to the p meson or nucleon mass in quantum chromodynamics (QCD). In the limit of vanishing u and d quark masses, QCD possesses an enhanced SUch(2) chiral symmetry, and mn ^ 0 in the chiral limit. Even if the quark masses are finite, this is a 'soft' breaking of SUch (2), and Goldstone's theorem protects mn from receiving large loop corrections at the otherwise natural scale of the strong interactions, mp ~ 770 MeV. Of course, this approximate symmetry does not enable one to predict the magnitude of chiral symmetry breaking in the strong interactions, and the actual small values of mn ~ 140 MeV or the u and d quark masses in QCD remain unexplained. However, the enhanced symmetry as these masses go to zero does permit the soft breaking scale of chiral symmetry to be quite different in principle from the UV cutoff scale of the pion EFT, since at least any large hierarchy of scales and a small value of mn/mp is not automatically upset by quantum loop corrections.

In the case of the cosmological term A, the problem is that Einstein's theory does not possess any apparent enhanced symmetry if A is set equal to zero. This is hardly surprising since as we have already pointed out, the classical theory contains no natural scale with which to compare A, any value being equally allowed a priori. Supersymmetry does not help here since in order to account for the absence of supersymmetric partners to the standard model particles at low energies, supersymmetry must be spontaneously broken at an energy scale no lower than approximately 1 TeV. Then the natural scale of A is still some 57 orders of magnitude larger than that measured by the acceleration of the universe in (7). Similar considerations apply to any new symmetry invoked at very short distances, and the problem persists even in apparently more microscopic description of quantum effects in gravity at the Planck scale. This impasse emphasizes once again that the problem of vacuum energy arises on macroscopic distance scales, and suggests that there is some basic ingredient missing in our EFT estimate of supposedly very weak quantum effects in gravity at large distances.

A fine tuning problem potentially related to the naturalness problem for the dimensionless X posed by (7) is that of the dimensionless ratio of pA to the closure density,

8nGpA c2A

Qa = "A = ~ "m2eas ~ 0.74. (9)

A 3c2H02 3H02 V ;

According to the conventional adiabatic expansion history of the universe, in which little or no entropy is generated during large epochs of time, QA is strongly time-dependent, being very much smaller than unity at early times when matter or radiation dominates, but approaching unity exponentially rapidly at late times, as these other components are diluted by the expansion. Thus the value inferred from observations (9) would seem to imply that we are living in a very special epoch in the history of the universe, when the vacuum energy has grown to be an appreciable fraction (74%) of the total, but just before the matter and radiation have been redshifted away completely, as conventional adiabatic theory indicates they soon will be, exponentially rapidly. This 'cosmic coincidence problem,' independently of the naturalness problem of the very small value of A also suggests that something basic may be missing from current cosmological models. The first indication as to what that element may be emerges from consideration of quantum effects in curved spacetimes such as de Sitter spacetime.

2. Quantum effects in de Sitter spacetime

2.1. Classical de Sitter spacetime

The simplest example of accelerated expansion is a universe composed purely of vacuum energy, i.e. QA = 1. This is de Sitter spacetime with H2 = Ac2/3 a constant. In classical general relativity de Sitter spacetime is the stable maximally symmetric solution to Einstein's equations (1) with positive A and rab = 0 otherwise. In spatially flat, homogeneous and isotropic Robertson-Walker (RW) form de Sitter spacetime has the line element,

ds2 = -c2dT2 + a2(x) dx2. (10)

Here t is the proper time of a freely falling observer and a(x) is the RW scale factor, determined from the Friedman equation,

0 /à\2 8nG 2

H s W = -3? >. (11)

with the overdot denoting differentiation with respect to t. The cosmological constant is included in the right-hand side of (11) as a vacuum energy contribution with equation of state, pA = — pA = c4A/8nG. With Qa = 1, and H2 = c2A/3 a constant, the RW scale factor is

ades(T) = eH (12)

in de Sitter spacetime.

At the beginning of modern cosmology the de Sitter model was proposed to account for the Hubble expansion [9]. This model was soon abandoned in favour of Friedman-LeMaitre-Robertson-Walker (FLRW) models, with apparently more physical ordinary matter and radiation

replacing the unknown pA as the dominant energy density in the universe. Unlike the eternal expansion of (12), these FLRW models possess an origin of time at which the universe originates at a spacelike singularity of infinite density, i.e. a big bang. Evidence for the radiation relic of this primordial explosion was discovered in the cosmic microwave background radiation (CMB). The CMB is remarkably uniform (to a few parts in 105) over the whole sky, and hence its discovery immediately raised the question of how this uniformity could have been established. In the classical FLRW models with their spacelike initial singularity, there is no possibility of causal contact between different regions of the present microwave sky, and hence these models possess a causality or horizon problem. In the 1960s Sakharov and Gliner [10] observed that a de Sitter epoch in the early universe could remove this causality problem of the initial singularity in the standard matter or radiation-dominated cosmologies. In the 1980s such a de Sitter phase in the early history of the universe was proposed under the name of inflation, with a vacuum energy scale h H of the order of 1015 GeV, associated with the unification scale of the strong and electroweak interactions [11]. A key success of inflationary models is that the small deviations from exact homogeneity and isotropy of the CMB, now measured in beautiful detail by the COBE (Cosmic Background Explorer) and WMAP (Wilkinson Microwave Anisotropy Probe) satellites [12], can originate from quantum zero point fluctuations in the de Sitter phase, with a magnitude determined by the ratio h H/MPl c2 [13]. Thus if inflationary models are correct, microscopic quantum physics at the tiny unification scale of 10—29 cm is responsible for the large scale classical inhomogeneities of the matter distribution in the universe at 1025 cm and above. The supposition of inflationary models that microscopic quantum physics might be responsible for the classical distribution of galaxies at cosmological distance scales is no less remarkable now than when it was first proposed a quarter of a century ago.

Despite the central importance of quantum fluctuations in the de Sitter phase of inflationary models, and the cosmological vacuum energy responsible for the inflationary epoch itself rooted in quantum theory, current inflationary models remain quite classical in character. For example one large class of models involves the 'slow roll' of a postulated (and so far unobserved) scalar inflaton field in a classical potential V [13]. In models of this kind the classical evolution of the inflaton must be slow enough to allow for a sufficiently long-lived de Sitter phase, to agree with the presently observed size and approximate flatness of the universe, a condition that requires fine tuning of parameters in the model. The value of the cosmological A term responsible for the present acceleration of the universe (generally assumed to be identically zero prior to the supernova data) is unexplained by scalar inflaton models and requires additional fine tuning. Indeed fine tuning issues are characteristic of all phenomenological treatments of the cosmological term, with a large hierarchy of scales and no fundamental quantum theory of vacuum energy.

Beginning in the 1980s, quantum fluctuations and their backreaction effects in de Sitter spacetime were considered in more detail by a number of authors, including the present ones [14]-[17]. Several of these studies indicated that fluctuations at the horizon scale c/H could be responsible for important effects on the classical de Sitter expansion itself. Based on these studies, a mechanism for relaxing the effective value of the vacuum energy to zero over time dynamically was proposed [14, 18]. Although a fully satisfactory cosmological model based on these ideas does not yet exist, a dynamical theory of vacuum energy still appears to be the most viable alternative to fine tuning or purely anthropic considerations for the very small but nonzero value of X. For a recent overview of approaches we refer the reader to [19].

Given the incomplete and scattered nature of the results in the technical literature, the need for a review of the subject accessible to a wider audience has been apparent for some time. With the discovery of dark energy in the universe, and the recent twenty-fifth anniversary of inflationary models, we have thought it worthwhile to review at this time the status and prospects for a dynamical resolution to the problem of vacuum energy. In this section, we begin by reviewing the infrared quantum effects in de Sitter space, which first suggested a dynamical relaxation mechanism, in roughly the historical order that they were first discussed.

2.2. Particle creation in de Sitter space

A space or time-dependent electric field creates particles. J Schwinger first studied this effect in quantum electrodynamics (QED) in a series of classic papers [20]. Later, Parker, Fulling, Zel'dovich and many others realized that a time-dependent gravitational metric should also produce particles [6]. The study of these effects formed the beginning of the subject of quantum fields in curved spacetime. From this point of view the exponential de Sitter expansion (12) provides a time-dependent background field which can create particle pairs from the 'vacuum,' converting the energy of the classical gravitational background into that of particle modes.

Both the concept of 'vacuum' and 'particle' in a background gravitational (or electromagnetic) field merit some comment. Since particle number generally does not commute with the Hamiltonian in spacetime-dependent backgrounds, a unique definition of a 'vacuum' state devoid of particles does not exist in this situation. In flat space with no background fields, relativistic wave equations have both positive energy (particle) solutions and negative energy (anti-particle) solutions, which are clearly distinguishable by time reversal symmetry. In time-dependent backgrounds, this is no longer the case and the two solutions mix on the scale of the time variation. Physically this is because in quantum theory a particle cannot be localized to a region smaller than its de Broglie wavelength. When this wavelength becomes large enough to be of the same order of the scale of spacetime variation of the background field, the particle concept begins to lose its meaning and it is better to think of matter as waves rather than as particles. Of course, this is precisely the wavelike effect of quantum matter which we characterize as 'particle' creation. A fair amount of the technical literature on particle creation is concerned with defining what a particle is and what sort of detector in a particular state of motion can detect it. Much of this is irrelevant to the backreaction problem, and will not concern us here. Provided one sticks to questions of evaluating conserved currents in well-defined initial states, one can bypass any technical discussion of definitions of 'particles' per se. The solutions of the wave equation of the quantum field(s) undergoing the particle creation, and the evolution of its electric current or energy-momentum tensor once renormalized are well-defined, and independent of arbitrary definitions of particle number. For a consistent physical definition of adiabatic particle number see [21].

Let us consider the electromagnetic case in more detail. To motivate the discussion, we note that there is an analogy to the vacuum energy problem in electromagnetism as well, the 'cosmological electric field problem' [18]. It consists of the elementary observation that Maxwell's equations in vacuo, i.e. in the absence of all sources admit a solution with constant, uniform electric field Ecosm of arbitrary magnitude and direction. Why then are all electric fields we observe in nature always associated with localized electrically charged matter sources? Why do we not observe some macroscopic uniform electric field pointing in an arbitrary direction in space?

A nonzero Ecosm selects a preferred direction in space. Nevertheless, and perhaps somewhat surprisingly, relativistic particle motion in a uniform constant electric field has precisely the same number of symmetry generators (ten) as those of the usual zero field vacuum. These ten generators define a set of modified spacetime symmetry transformations, leaving Ecosm fixed, whose algebra is isomorphic to that of the Poincare group [22]. In this respect field theory in a spacetime with a constant, non-dynamical Ecosm = 0 is similar to field theory in de Sitter spacetime with a constant, non-dynamical A = 0. The isometry group of de Sitter spacetime is O(4, 1), which also has ten generators, exactly the same number of flat Minkowski spacetime. In the absence of a unique choice of vacuum in a spacetime-dependent background, the point of view often adopted is to choose the state with the largest possible symmetry group permitted by the background. In de Sitter spacetime this maximally symmetric state is called the Bunch-Davies (BD) state [23].

Ordinarily one would say that the ground state should also be the one of lowest energy, and as a nonzero electric field has a nonzero energy density, the ground state should be that with Ecosm = 0. However, once we appeal to an energy argument we must admit that we do not know the absolute energy of the vacuum, and because of (4) must allow for an arbitrary shift of that zero point, perhaps compensating for the electromagnetic field energy. Again A = 0 is similar to Ecosm = 0 in having zero energy in flat space. As with A classically we are free to set Ecosm = 0 by an appropriate boundary condition at very large distances, but this leaves unanswered the question of why this is the appropriate boundary condition for our universe. The suggestion that A may be regarded also as a constant of integration has been made by a number of authors [24]. In quantum theory the boundary condition which determines a constant of integration in the low-energy description requires information about the macroscopic quantum state of the system at large distance scales. Is there any evidence that quantum effects are relevant to the question of the cosmological electric field or of A?

The answer is almost certainly yes, because of quantum vacuum polarization and particle creation effects. Suppose that Ecosm were not zero. Then the quantum vacuum of any charged matter fields interacting with it is polarized. This is described by a polarization tensor,

nab(x, x'; E) = i<Tja (x)jV)>E, (13)

where ja is the charge current operator and the expectation value is evaluated in a state with classical background field E. This could be taken to be the state of maximal symmetry allowed by the background, in particular a state with discrete time reversal symmetry. The time ordering symbol T enforces the Feynman boundary conditions on the polarization operator. These boundary conditions are not time symmetric. Time-ordering (with an k prescription in the propagator) defines a different polarization tensor or Feynman Green's function from anti-time-ordering (with a -k prescription). This time asymmetry is built into quantum theory by the demands of causality which distinguishes retarded from advanced effects in the polarization function (13). Correspondingly, the polarization operator nab contains two pieces, an even and an odd piece under time reversal. The even piece describes the polarizability of the vacuum, since the vacuum fluctuations of virtual charge pairs may be thought of as giving rise to an effective polarizability of the vacuum (dielectric constant). The time reversal odd piece describes the creation of real particle anti-particle pairs from the vacuum by the Schwinger mechanism. The creation of real charged pairs means that a real current flows j ^ 0 in the direction of the electric field, even if none existed initially. This implies that the electric field does work at a rate, j ■ E. To the extent that this power cannot be recovered because the created particles interact and

lose the coherence they may have had in their initial state, this is the rate of energy dissipation. The vacuum can behave then very much like a normal conductor with a finite conductivity and resistivity, due to the random, uncorrelated motions of its fluctuating charge carriers.

At the same time, the electric field is diminished by the Maxwell equation,

f = -<»• "4>

in the case of exact spatial homogeneity of the average current. The important question is that of the timescale of the effective dissipation. Does the degradation of the coherent electric field take place rapidly enough to effectively explain why there is no observed Ecosm today? The original Schwinger calculation of the decay rate of the vacuum into charged pairs involves a tunnelling factor, exp (—nm2c3/eEh) for the creation of the first pair from the vacuum under the assumption of particular initial conditions. An exponential tunnelling factor like this would greatly suppress the effect. However, if any charged matter is present initially, i.e. if we are not in precisely the maximally symmetric 'vacuum' state, the charges are accelerated, radiate, and pair produce without any tunnelling suppression factor. Thus the timescale for an induced cascade of particle pairs to develop and degrade the electric field energy in any state save a carefully tuned initial vacuum state will be very much faster than the Schwinger spontaneous vacuum rate. Hence on physical grounds there is an instability of the vacuum in a background electric field and extreme sensitivity to boundary conditions. These are the necessary conditions for the spontaneous breaking of time reversal symmetry [18].

The particle creation effects in homogeneous electric fields, including the backreaction of the mean current on the field through (14) have been studied in some detail in the large N or mean field approximation [25]. In this approximation the direct scattering between the created particles is neglected. Inclusion of scattering processes opens additional channels and faster timescales for dissipation, as in classical plasmas or Boltzmann gases. Allowing for these dissipative processes should permit any long range coherent Ecosm present initially to relax to very small values on timescales much shorter than a Hubble expansion time, H0—1.

The electromagnetic analogy is that of a giant capacitor discharging. Any cosmological electric field initially present in the universe eventually shorts itself out, and degrades to zero, when the vacuum polarization effects described by nab and realistic particle interactions are taken into account. The actual value of the electric field at late times may then be very much less than its 'natural' value of m2c3/he or any other scale related to short distance physics. The real and imaginary parts of nab are related by a dispersion relation which is one form of a fluctuation-dissipation theorem for the electrically polarized quantum vacuum. It is simply a consequence of causality, and the existence of a positive Hilbert space of quantum states that quantum vacuum fluctuations and quantum vacuum dissipation are inseparably related. One necessarily implies the other. This is the modern, relativistically covariant formulation for quantum fields of the relation between an equilibrium quantity (mean square displacement) and a time asymmetric dissipative quantity (diffusion coefficient or viscosity) first discussed by Einstein in his theory of Brownian movement [26].

These observations are of a very general nature, and apply equally well to vacuum fluctuations in a gravitational background field. Since the geometry couples to the energy-momentum stress tensor, it is the fluctuations in this quantity which govern the dynamics of the gravitational field, and we are led to consider the corresponding polarization tensor,

nabcd(x, x!; g) = i{TTab (x)Tcd(xr)),, (15)

where gab (x) represents some classical background metric, for example that of de Sitter spacetime (10)-(12). This polarization tensor may be handled by exactly the same techniques as (13), and the analogous fluctuation-dissipation theorem relating its real and imaginary parts may be proven [16]. If the background gab (x) possesses a timelike Killing field, and therefore a Euclidean continuation with periodicity f = h/kBT, it is natural to introduce the Fourier transform with respect to the corresponding static coordinate time difference t — t'. Defining the Fourier transforms of the symmetric and anti-symmetric parts of (15) by

dt<{Tab(x), Tcd(x0}+> e1M(t-t° = Sabcd(r, r'; m),

d t([Tab(x), Tcd(x')]—) eia(t—t,) = Dabcd(r, r'; m),

we find that the two pieces are related via

^abcdz „/. x _ —. ( }m\ Cabcd,

Dabcd (r, r; m) = tanh J Sabcd(r, r; m). (17)

One can show that the condition for particle creation effects to occur in the background electric or gravitational field is just the condition that the time asymmetric piece of the exact polarization function diverges for large t — t', in particular that

Dabcd(r, r'; m) a m—1, m ^ 0. (18)

In fact it is precisely the residue of this m—1 pole which determines the particle creation rate in the adiabatic limit of slowly varying backgrounds [16]. This singular behaviour at low frequencies means that the background is unstable to small perturbations, and is the signal for spontaneous breaking of time reversal symmetry. This sensitivity to infrared fluctuations in nabcd is why the inclusion of quantum fluctuations and correlation functions higher than the average (Tab) can change the behaviour of a macroscopic quantum system over long times.

In cosmology the Friedman equation (11) together with the equation of covariant energy conservation,

p + 3H(p + p) = 0, (19)

imply that

H =--r(p + p). (20)

This relation from Einstein's equations is to be compared to the Maxwell equation (14). In both cases there is a classical static background that solves the equation trivially, namely H or E a constant with zero source terms on the right-hand side. In the case of (20) this is de Sitter spacetime with pA + pA = 0. In order to exhibit explicitly the static nature of de Sitter spacetime, we make the coordinate transformation,

2Ht = 2Ht + ln(1 — H2r2/c2), (21a)

|x| = re-HT = re-Ht(1 — H2r2/c2)-1/2, (21ft)

to bring the de Sitter line element (10) with (12) into the form

= -(c2 — H2r2) dt2 + --_ 2, 2 + r2 (d92 + sin2 9 d02), (22)

deS 1 — H r ¡C

in which the metric is independent of the static time variable t. The translation dt defines a isometry of de Sitter spacetime or Killing field, which is timelike for r < c/H. This is the time translation invariance of de Sitter space which permits the introduction of a time stationary state with an equilibrium thermal distribution at the inverse temperature f of (17). In the representation (22) the static nature of de Sitter space and the existence of the observer horizon at rH = c/H are manifest, but an arbitrary particular point in space r = 0 is chosen as the origin, so that the spatial homogeneity of the RW coordinates (12) is no longer manifest. Spacetime events for r > rH are causally disconnected and unobservable to a freely falling observer at the origin of static coordinates.

In the static coordinates (22) the de Sitter metric becomes space dependent rather than time-dependent, just as a constant electric field can be expressed either in a time-dependent (A = —Et) or static (A0 = — E ■ r) gauge. The effects of the de Sitter background on the polarization of the vacuum and particle creation should be independent of the coordinates or gauge. In each case the static H or E background field is stable to classical matter perturbations. Classical matter (if charged in the E case) is simply accelerated, and swept out by the constant electric or gravitational field. In the de Sitter case classical matter (obeying p + p > 0) is redshifted away by (19). However if quantum matter fluctuations possess a spectrum with a singular m—1 dependence for small frequencies (18), then the classically stable background with an event horizon is unstable to quantum fluctuations. In that case the system will be driven away from the quasi-static initial state towards a final state in which the classical field energy has been dissipated into matter or radiation field modes. In [16] it was argued that the polarization function corresponding to scalar (i.e. metric trace) perturbations of the de Sitter background has precisely the required singular m—1 behaviour. This behaviour describes the response of the system to perturbations on length scales of order of the horizon size rH or larger. Then de Sitter spacetime satisfies the condition for dissipation of curvature stress-energy into matter and radiation modes, much as the electric field background considered previously.

Actually computing the evolution away from the initial state by this effect requires that we go beyond the simple replacement of the quantum stress tensor Tab by its expectation value {Tab) and consider fluctuations about the mean as in (15), as well as possibly higher correlation functions of the stress tensor as well. This is not an easy task. Nevertheless it is worth noting that whereas the Schwinger suppression factor for vacuum tunnelling occurs in electrodynamics because there are no massless charged particles, massless particles do couple gravitationally, and would be expected to dominate the dissipative process. In both cases the creation of matter particle pairs with j ■ E > 0 and p + p > 0 causes the background field parameter, Ecosm or H to decrease. In fact, unlike the electric current which is a vector and may change sign, p + p for realistic matter or radiation is always positive, so we should expect H to decrease monotonically to zero, without the plasma oscillations that can occur in the electrodynamic case [25].

The problems with implementing these promising ideas in a realistic model are mainly technical. As we have already noted, fluctuations about the mean stress tensor and their backreaction on the mean geometry must be taken into account in a consistent calculation. Infrared divergences are encountered in any direct attempt to evaluate the diagrams contributing to the dissipation process perturbatively. Infrared problems of this kind due to long range forces

are well known in the low frequency hydrodynamic limit of plasmas and gauge field theories at finite temperatures [27]. In many analogous situations of this kind, the m ^ 0 limit is outside the range of validity of perturbative expansions, due to collective effects. In hydrodynamics the long time behaviour of correlation functions (17) are non-perturbative, requiring at the least a resummation of perturbative processes. The problem of dissipating vacuum energy by microscopic particle creation effects may be compared in order of difficulty with the problem of evaluating the viscosity of water and the damping of eddy currents in a stream from the electronic structure and interactions of the H2O molecule.

Although a full calculation including self-interactions and backreaction along these lines has not been done, even in electromagnetism, the formal similarity between charge carrier fluctuations and Maxwell's equation for the displacement current on the one hand, and fluctuations in stress-energy and Einstein's equations (1) on the other suggests that a dissipative relaxation of the vacuum energy into ordinary matter and radiation is possible via this mechanism. The bulk viscosity of the cosmological 'fluid' of vacuum fluctuations is the quantity controlling this dissipation. Quantum fluctuations on or near the horizon scale are the relevant ones which need to be handled in a consistent, reliable, non-perturbative framework, in order to convert the coherent vacuum energy of de Sitter space to matter/radiation modes on the timescales relevant for cosmology.

2.3. Thermodynamic instability ofde Sitter spacetime

A second set of considerations points to the role of dynamical quantum effects on the horizon scale in de Sitter space. The existence of the observer horizon at r = rH leads in conventional treatments to a Hawking temperature for freely falling observers [28]. The Hawking temperature is closely related to the particle creation effect since both depend upon a mixing between positive and negative frequency components of quantum fields in de Sitter space at the horizon scale. It is worth emphasizing that like the Casimir effect, this is a global effect of the spacetime, where the horizon now sets the scale, playing the role of the boundary conditions on the conducting plates in (5). In the BD state, particle creation and annihilation effects are exactly balanced in a precisely time symmetric manner and a configuration formally similar to that of thermodynamic equilibrium is possible. The Hawking temperature in de Sitter space,

is the temperature a freely falling detector would detect in the BD state. This temperature becomes the one which enters the fluctuation-dissipation formulae (16), (17) of the previous subsection in de Sitter space. Although this temperature is very small for X ^ 1, a thermodynamic argument similar to Hawking's original one for black holes implies that the BD equilibrium state in de Sitter space is thermodynamically unstable [15].

Consider as the 'vacuum' state in de Sitter spacetime that state defined by the analytic continuation of all of its Green's functions by t ^ it. Since the resulting geometry in (22) is a space of uniform positive curvature, i.e a sphere S4 with O(5) isometry group and radius cH—1, the Euclidean Green's functions are periodic in imaginary time with period 2n/H. Continuing back to Lorentzian time, the BD state defined by this analytic continuation is a thermal state with temperature TH [29], invariant under the full O(4, 1) de Sitter symmetry group [6]. Because of that symmetry, the expectation value of the energy-momentum tensor of any matter fields in this

state must itself be of the form p = — p = constant. This is the BD state usually assumed, tacitly or explicitly, by inflationary model builders.

The existence of such a maximally symmetric state does not guarantee its stability against small fluctuations, any more than the existence of a state invariant under the 10 isometries of space with a constant electric field guarantee the stability of the vacuum with an electric field. In fact, both the energy within one horizon volume, and the entropy of the de Sitter horizon SH are decreasing functions of the temperature, i.e.,

c5 h c5

Eh = PaVh = 2gh = 4GTh ; (24a)

Ah nc5 kB h c5 3n, nAU,

SH = kB—t- =-- =-t = —kB. (24b)

H B4Lj2l h GH2 4nGkBTH X V ;

Hence, by considering a small fluctuation in the Hawking temperature of the horizon which (like the black hole case) causes a small net heat exchange between the region interior to the horizon and its surroundings, we find that this interior region behaves like a system with negative heat capacity [15],

E = —— = —< 0. (25)

dTH Th X V 7

However, negative heat capacity is impossible for a stable system in thermodynamic equilibrium. It corresponds to a runaway process in which any infinitesimal heat exchange between the regions interior and exterior to the horizon will drive the system further away from its equilibrium configuration. Since the choice of origin of static coordinates in de Sitter space is arbitrary, the entire space is unstable to quantum/thermal fluctuations in its Hawking temperature nucleating a kind of vacuum bubble at an arbitrary point, breaking the global O(4, 1) de Sitter invariance down to O(3) rotational invariance.

This thermodynamic consideration is consistent with the previous one based on particle creation and the fluctuation-dissipation theorem, and again suggests that collective quantum effects on the horizon scale are the relevant ones. The enormously negative heat capacity for de Sitter space in (25) for X ^ 1, if taken literally suggests that the timescale for the instability to develop may not be exponentially large given any initial perturbation. Despite this signal of thermodynamic instability, an evaluation of nabcd and full dynamical analysis of the fluctuations about the BD state in de Sitter space in real time has not been carried out, again mostly for technical reasons. The framework for such a linear response analysis has been laid down only recently in [30]. A linearized analysis exhibiting an unstable mode would still be only the first step in a more comprehensive non-perturbative treatment of its effects.

2.4. Graviton fluctuations in de Sitter spacetime

A third route for investigating quantum effects in de Sitter spacetime leading to the same qualitative conclusions is through studies of the fluctuations of the metric degrees of freedom themselves. The propagator function describing these metric fluctuations is a function of two spacetime points xa and x'a. In flat spacetime, assuming a Poincare invariant vacuum state,

the propagator becomes a function only of the invariant distance squared between the points, i.e. (x — x')2. Likewise full invariance of the graviton propagator under the global O(4, 1) de Sitter isometry group is a necessary condition for the gravitational vacuum in de Sitter spacetime to exist and to be stable to perturbations. Using such covariant methods to evaluate the propagator encounters a problem however. If one requires de Sitter invariance by computing the Euclidean propagator on S4 with O(5) invariance group, and then analytically continuing to de Sitter spacetime, then one obtains a graviton propagator with rather pathological properties. Both the transverse-tracefree (spin-2) and trace (spin-0) projections of the Feynman propagator grow without bound both at large spacelike and large timelike separations [31]-[33]. Since this propagator leads to infrared divergences in physical scattering processes [32], this large distance behaviour of the propagator function cannot be removed by a gauge transformation. This infrared behaviour is a striking violation of cluster decomposition properties of the de Sitter invariant vacuum state.

A similar situation had been encountered before in de Sitter spacetime, in the quantization of a massless, minimally coupled free scalar field, obeying the wave equation, DO = 0. A covariant construction of the propagator function for O meets the obstacle that □ has a normalizable zero mode (namely a constant mode) on the Euclidean continuation of de Sitter space, S4. Hence a de Sitter invariant propagator inverse for the wave operator □ does not exist [34]. Formally projecting out the problematic mode leads to a propagator function which grows logarithmically for large spacelike or timelike separations of the points x and x'. Since the wave equation for spin-2 graviton fluctuations is identical to that of two massless, minimally coupled scalar fields (one for each of the two polarization states of the graviton) in a certain gauge [31, 35], it is not surprising that it shares many of the same features as the massless scalar case.

If one abandons the method of Euclidean continuation from S4 and quantizes either the massless scalar or the graviton by canonical methods, starting on a fixed spacelike surface rather than imposing global de Sitter invariance, one finds a Feynman propagator function of x and x' that is not de Sitter invariant [36]. In other words, canonical quantization of either the massless scalar or graviton field necessarily breaks de Sitter invariance, and no de Sitter invariant vacuum exists in either case.

The closest analog of this behaviour in flat spacetime is that of massless scalar field in two dimensions (2D). The Feynman propagator G(x,xr) for a massless field in (1 + 1) dimensions satisfies

— DG(x,x') = h 82 (x — x'). (26)

If we require Lorentz invariance, the propagator must be a function of the invariant s = (x — x')2. In that case, the wave operator □ becomes an ordinary differential operator in s and the only solutions to the homogeneous wave equation (for x = x') are ln s and a constant. The coefficient of the ln s solution is fixed by the delta function in (26), and we find

G(x,xr) = —— ln [¡2 (x — x')2], (27)

with ¡i an arbitrary constant which acts as an infrared cutoff. The logarithmic growth at large distances and infrared cutoff dependence of the propagator implies that free asymptotic particle states do not exist for a massless field in 2D. In a canonical treatment the origin of the infrared problem is the constant k = 0 Fourier mode of the field, which grows linearly with time. A Lorentz

invariant normalizable ground state does not exist in the Fock space. In the generic case either the massless field must develop a mass or otherwise become modified by its self-interactions or Lorentz invariance is necessarily broken, and secular terms develop in the evolution from generic initial conditions. From the EFT point of view, a mass and other interaction terms for a scalar field in 2D are generically allowed, and would be expected to control the behaviour of the theory at large distances and late times. These additional interactions are relevant operators in the infrared, and cannot be treated as small perturbations to the massless theory. Conversely, if the masslessness of the scalar boson is protected by a global symmetry, then that symmetry is restored by quantum fluctuations and there are again no Lorentz invariant massless Goldstone scalar states in the physical spectrum [37, 38].

Infrared divergences in even classical scattering amplitudes show that there is no analog of a scattering matrix for gravitational waves in global de Sitter spacetime [39], much as in the 2D massless scalar theory in flat spacetime. The similar logarithmic behaviour of the graviton propagator indicates that infrared quantum fluctuations of the gravitational field are important in de Sitter space, and self-interactions or additional relevant terms in the effective action will control the late time behaviour. This is the same conclusion we reached from particle creation, fluctuation-dissipation, and thermodynamic considerations. de Sitter invariance is spontaneously broken by quantum effects, and the ground state of the gravitational field with a cosmological term is not global de Sitter spacetime [17].

The authors of [40] have performed a perturbative analysis of long wavelength gravitational fluctuations in non de Sitter invariant initial states up to two-loop order. They focus on the self-interactions of gravitons generated by nonlinearities in the classical Einstein theory itself. This study indicates the presence of secular terms in the quantum stress tensor of fluctuations about de Sitter space, tending to decrease the effective vacuum energy density, consistent with our earlier considerations. The authors of [41] have studied the stress tensor for long wavelength cosmological perturbations in inflationary models as well, and also found a backreaction effect of the right sign to slow inflation. Some serious technical questions concerning the gauge invariance of these results have been raised [42], addressed [43] and raised again recently in [44].

Even if free of technical problems the perturbative backreaction of [40], [41] is suppressed by an effective coupling constant of order X2. This is already very small (of order 10—16) for an initial cosmological term at the unification scale of 1015 GeV. For the present value of X in (8) any secular backreaction effect of order X2 on vacuum energy would be completely negligible. A much larger, essentially non-perturbative effect is needed to be relevant to the naturalness question of dark energy in cosmology.

Actually, as should be clear from the previous discussion, none of the considerations based on particle creation, thermodynamic fluctuations or the infrared behaviour of the gravitational fluctuations in de Sitter space suggest that backreaction effects can be treated in a uniform perturbative expansion in a small parameter like X. On the contrary, perturbation theory about a state which is itself infrared singular would be expected to break down and require a non-perturbative resummation to capture the dominant effects. In statistical systems perturbation theory certainly breaks down when there are additional infrared relevant terms in the low-energy EFT, and their effects do dominate the purely perturbative contributions. For this reason we initiated the study of new infrared relevant operator(s) in gravity in the conformal or trace sector of gravity, generated by the quantum trace anomaly. An essentially non-perturbative treatment of this sector indicates the existence of a infrared renormalization fixed point in

which the cosmological term is driven to zero. We review and update the present status of this proposal next.

3. Quantum gravity in 2D

The sensitivity to horizon scale quantum fluctuations in de Sitter spacetime reviewed in the last section strongly suggests that there is an infrared relevant operator in the low-energy effective theory of gravity, not contained in the classical Einstein-Hilbert action. This is also the implication of the naturalness considerations in EFT with which we began our discussion. Since there is no preferred value of A in the purely classical theory, a dynamical mechanism for the relaxation of A ^ 0 must be sought in the larger quantum framework.

There are several hints for the source of new infrared relevant terms in the quantum framework which are not contained in the classical Einstein theory. First it is the fluctuations in the scalar or conformal sector of the metric field which are the most infrared divergent in de Sitter spacetime [45]. These are associated with the trace of the polarization tensor, and are parameterized by the conformal part of the metric tensor,

gab (x) = e2a(x)gab(x), (28)

where ea is called the conformal factor and gab (x) is a fixed fiducial metric. The RW scale factor a(x) in (10) is an example of a conformal factor, fixed classically by the Friedman equation (11). The second clue that this is the important sector to look for non-perturbative infrared effects is that a couples to the trace of the energy-momentum tensor Taa, an operator known to have an anomaly for massless quantum fields in curved spacetime [7]. An anomaly implies that the effects of quantum fluctuations can remain relevant at the longest length and timescales, and therefore modify the purely classical theory. This is certainly the lesson of 2D quantum gravity, which we review next. Most importantly of all, the scalar a field is constrained in the Einstein theory in 4D, but acquires dynamics through the trace anomaly. It is the effective action and dynamics of this field which we have proposed as the essential new ingredient to gravity at cosmological distance scales which can provide a natural mechanism for screening the cosmological term [45].

Classical fields satisfying wave equations with zero mass, which are invariant under conformal transformations of the spacetime metric, gab ^ e2agab have stress tensors with zero classical trace, Taa = 0. In quantum theory, the stress tensor Tab becomes an operator with fluctuations about its mean value. The mean value itself (Tab) is formally UV divergent, due to its zero point fluctuations, as in (4), and requires a careful renormalization procedure. The result of this renormalization consistent with covariant conservation in curved spacetime is that classical conformal invariance cannot be maintained at the quantum level. The trace of the stress tensor is generally nonzero when h = 0, and any UV regulator which preserves the covariant conservation of Tab (a necessary requirement of any theory respecting general coordinate invariance and consistent with the Equivalence Principle) yields an expectation value of the quantum stress tensor with a nonzero trace [6, 7].

In 2D, the trace anomaly takes the simple form,

(Taa) = nr (d = 2) (29)

where N = NS + NF is the total number of massless fields, either scalar (NS) or (complex) fermionic (NF). The fact that the anomalous trace is independent of the quantum state of the

matter field(s), and dependent only on the geometry through the local Ricci scalar R suggests that it should be regarded as a geometric effect. However, no local coordinate invariant action exists whose metric variation leads to (29). This is important because it shows immediately that understanding the anomalous contributions to the stress tensor will bring in some non-local physics or boundary conditions on the quantum state at large distance scales.

A non-local action corresponding to (29) can be found by introducing the conformal parameterization of the metric (28) and noticing that the scalar curvature densities of the two metrics gab and gab are related by

Ry—g = Rv— — 27—g (d = 2) (30)

a linear relation in a in two (and only two) dimensions. Multiplying (29) by y—g, using (30) and noting that y^—g<Taa} defines the conformal variation, 8r(2) /8a of an effective action r(2), we conclude that the a dependence of r(2) can be at most quadratic in a. Hence the Wess-Zumino effective action [46] in 2D, rWZ is

N r _ _

rWZ[g; a] = — J dV—g(—aa Da + Ra). (31)

Mathematically the fact that this action functional of the base metric gab and the Weyl shift parameter a cannot be reduced to a single local functional of the full metric (28) means that the local Weyl group of conformal transformations has a non-trivial cohomology, and rWZ is a one-form representative of this cohomology [47,48]. This is just a formal mathematical statement of the fact that a effective action that incorporates the trace anomaly in a covariant EFT consistent with the Equivalence Principle must exist but that this Sanom[g] is necessarily non-local. It is straightforward in fact to find a non-local scalar functional Sanom[g] such that

rWZ[g; a] = S^Jg = e2ag] — Sinojg]. (32)

By solving (30) formally for a, and using the fact that y—gD = y—g □ is conformally invariant

in 2D, we find that rWZ can be written as a Weyl shift (32) with

S^Ug] = Y62/ d2xy—gj d2x'V—g'R(x)D—1(x,x')R(x'), (33)

and D—1 (x, x') denoting the Green's function inverse of the scalar differential operator □. The parameter Q2 is —N/6 if only matter fields in a fixed spacetime metric are considered. It becomes (25 — N)/6 if account is taken of the contributions of the metric fluctuations themselves in addition to those of the N matter fields, thus effectively replacing N by N — 25 [49]. In the general case, the coefficient Q2 is arbitrary, related to the matter central charge, and can be treated as simply an additional free parameter of the low-energy effective action, to be determined.

The anomalous effective action (33) is a scalar under coordinate transformations and therefore fully covariant and geometric in character, as required by the Equivalence Principle. However since it involves the Green's function 1(x, x'), which requires boundary conditions for its unique specification, it is quite non-local, and dependent upon more than just the local curvature invariants of spacetime. In this important respect it is quite different from the classical terms in the action, and describes rather different physics. In order to expose that physics it is

most convenient to recast the non-local and non-single valued functional of the metric, S^Om into a local form by introducing auxiliary fields. In the case of (33) a single scalar auxiliary field, y satisfying

- Dy = R (34)

is sufficient. Then varying

W-g(SabVa& - 2R&) (35)

with respect to y gives the equation of motion (34) for the auxiliary field, which when solved formally by y = —□^R and substituted back into San,m[g; y] returns the non-local form of the anomalous action (33), up to a surface term. The non-local information in addition to the local geometry which was previously contained in the specification of the Green's function □—1 (x, x') now resides in the local auxiliary field y(x), and the freedom to add to it homogeneous solutions of (34).

The variation of (35) with respect to the metric yields a stress-energy tensor,

r(2)[g y] = _ ^L ^Ug; y]

Tab [g; y] " y—g 8gab Q2

= Q [ — Va^ + gab^y — 2(Vay)(yby) + 4gab(Vcy)(Vcy)], (36)

which is covariantly conserved, by use of (34) and the vanishing of the Einstein tensor, Gab = Rab — Rgab/2 = 0 in two (and only two) dimensions. The classical trace of the stress tensor,

b (2) Q2 Q2

gabTi2) [g; = Q = -QR (37)

reproduces the quantum trace anomaly in a general classical background (with Q2 proportional to h). Hence (35) is exactly the local auxiliary field form of the effective action which should be added to the action for 2D gravity to take the trace anomaly of massless quantum fields into account.

Since the integral of R is a topological invariant in 2D, the classical Einstein-Hilbert action contains no propagating degrees of freedom whatsoever in d = 2, and it is Sanom which contains the only kinetic terms of the low-energy EFT. In the local auxiliary field form (35), it is clear that

Sanom describes an additional scalar degree of freedom y, not contained in the classical action

Sc). This is reflected also in the shift of the central charge from N — 26, which would be expected from the contribution of conformal matter plus ghosts by one unit to N — 25. Quantum gravity in 2D acquires new dynamics in its conformal sector, not present in the classical theory. Once the anomalous term is treated in the effective action on a par with the classical terms, its effects become non-perturbative and do not rely on fluctuations from a given classical background to remain small.

Extensive study of the stress tensor (37) and its correlators, arising from this effective action established that the 2D trace anomaly gives rise to a modification or gravitational 'dressing' of critical exponents in conformal field theories at second order critical points [49]. Since critical

exponents in a second order phase transition depend only upon fluctuations at the largest allowed infrared scale, this dressing is clearly an infrared effect, independent of any UV cutoff. These dressed exponents are evidence of the infrared fluctuations of the additional scalar degree of freedom y which are quite absent in the classical action. The scaling dimensions of correlation functions so obtained are clearly non-perturbative in the sense that they are not obtained by considering perturbatively small fluctuations around flat space, or controlled by a uniform expansion in X ^ 1. The appearance of the gravitational dressing exponents and the anomalous effective action (33) itself have been confirmed in the large volume scaling limit of 2D simplicial lattice simulations in the dynamical triangulation approach [50, 51]. Hence there can be little doubt that the anomalous effective action (35) correctly accounts for the infrared fluctuations of 2D geometries.

The importance of this 2D example is the lessons it allows us to draw about the role of the quantum trace anomaly in the low-energy EFT of gravity, and in particular the new dynamics it contains in the conformal factor of the metric. The effective action generated by the anomaly in 2D contains a new scalar degree of freedom, relevant for infrared physics, beyond the purely local classical action. It is noteworthy that the new scalar degree of freedom in (34) is massless, and hence fluctuates at all scales, including the very largest allowed. In 2D its propagator D-1 (x, x') is logarithmic, as in (27), and hence is completely unsuppressed at large distances. Physically this means that the quantum correlations at large distances require additional long wavelength information such as macroscopic boundary conditions on the quantum state.

The action (35) due to the anomaly is exactly the missing relevant term in the low-energy EFT of 2D gravity responsible for non-perturbative fluctuations at the largest distance scales. This modification of the classical theory is required by general covariance and quantum theory, and essentially unique within the EFT framework.

4. Quantum conformal factor in 4D

The line of reasoning in d = 2, just sketched to find the conformal anomaly and construct the effective action may be followed also in 4D. In d = 4, the trace anomaly takes the somewhat more complicated form,

(Taa) = bF + b'(E - §DR) + b"DR + J] frHu (38)

in a general 4D curved spacetime. This is the 4D analog of (29) in 2D. In equation (38) we employ the notation,

E = RabcdRabcd = RabcdRabcd - 4RabRab + R2, (39a)

F = CabcdCabcd = RabcdRabcd - 2RaR + y. (39b)

with Rabcd the Riemann curvature tensor, *Rabcd = £abefRed/2 its dual, and Cabcd the Weyl conformal tensor. Note that E is the 4D Gauss-Bonnet combination whose integral gives the Euler number of the manifold, analogous to the Ricci scalar R in d = 2. The coefficients b, b' and b" are dimensionless parameters multiplied by h. Additional terms denoted by the sum J]i jiiHi

in (38) may also appear in the general form of the trace anomaly, if the massless conformal field in question couples to additional long range gauge fields. Thus in the case of massless fermions coupled to a background gauge field, the invariant H = tr(FabFab) appears in (38) with a coefficient f determined by the anomalous dimension of the relevant gauge coupling.

As in d = 2 the form of (38) and the coefficients b and b' are independent of the state in which the expectation value of the stress tensor is computed, nor do they depend on any UV short distance cutoff. Instead their values are determined only by the number of massless fields [6, 7],

b = 120(4^2 Ns + 6Nf + 12Nv), (40a)

1/11 \

b' =--- NS + — NF + 62NV , (40b)

360(4^)2 V S 2 F ) ' V ;

with (NS, Nf' Nv) the number of fields of spin (0, 1/2, 1) respectively and we have taken h = 1. Notice also that b > 0 while b' < 0 for all fields of lower spin for which they have been computed. Hence the trace anomaly can lead to stress tensors of either sign, and in particular, of the sign needed to compensate for a positive bare cosmological term. The anomaly terms can also be utilized to generate an effective positive cosmological term if none is present initially. Such anomaly driven inflation models [52] require curvatures comparable to the Planck scale, unless the numbers of fields in (4) is extremely large. It is clear that conformally flat cosmological models of this kind, in which the effects of the anomaly can be reduced to a purely local higher derivative stress tensor, are of no relevance to the very small cosmological term (7) we observe in the acceleration of the Hubble expansion today. Instead it is the essentially non-local boundary effects of the anomaly on the horizon scale, much larger than LPl which should come into play.

Three local fourth order curvature invariants E, F and OR appear in the trace of the stress tensor (38), but only the first two (the b and b') terms of (38) cannot be derived from a local effective action of the metric alone. If these terms could be derived from a local gravitational action we would simply make the necessary finite redefinition of the corresponding local counterterms to remove them from the trace, in which case the trace would no longer be nonzero or anomalous. This redefinition of a local counterterm (namely, the R2 term in the effective action) is possible only with respect to the third b" coefficient in (38), which is therefore regularization dependent and not part of the true anomaly. Only the non-local effective action corresponding to the b and b' terms in (38) are independent of the UV regulator and lead to effects that can extend over arbitrarily large, macroscopic distances. The distinction of the two kinds of terms in the effective action is emphasized in the cohomological approach to the trace anomaly [48].

The number of massless fields of each spin (NS, NF, NV) is a property of the low-energy effective description of matter, having no direct connection with physics at the ultrashort Planck scale. Indeed massless fields fluctuate at all distance scales and do not decouple in the far infrared, relevant for cosmology. As in the case of the chiral anomaly with massless quarks, the b and b' terms in the trace anomaly were calculated originally by techniques usually associated with UV regularization [6]. However just as in the case of the chiral anomaly in QCD, or 2D gravity, the trace anomaly can have significant new infrared effects, not captured by a purely local metric description.

To find the WZ effective action corresponding to the b and b' terms in (38), introduce as in 2D the conformal parameterization (28), and compute

y-gF = J-gF (41a)

y-g (E - 3□R) =J—g(~E - §□R + 4y-gÄ4a > (41b)

whose a dependence is no more than linear. The fourth order differential operator appearing in this expression is [45, 48, 53]

A4 = □ + 2RabVa Vb — 3 R^ + 1 (Va R)Va, (42)

which is the unique fourth order scalar operator that is conformally covariant, namely

y—gA4 = V—A (43)

for arbitrary smooth a(x) in four (and only four) dimensions. Thus multiplying (38) by y—g and recognizing that the result is the a variation of an effective action rWZ, we find immediately that this quadratic effective action is

rwz[g; a] = b J d4^^—gFa + b' j d'x^—g{(E — 2OR)a + 2aA,a}, (44)

up to terms independent of a. This Wess-Zumino action is a one-form representative of the non-trivial cohomology of the local Weyl group in 4D which now contains two distinct cocycles, corresponding to the two independent terms multiplying b and b'. By solving (41b) formally for a and substituting the result in (44), we obtain

Twz[g; a] = Sanom[g = e^g] — Sanom[g], (45)

with the non-local anomalous action

E_ □R\ 2 3

Sanom[g] = 2 / d4xyg j d4xry~g' ^E — □R) A—1 (x, x?) bF + b^E —

and A—1(x, x') denoting the Green's function inverse of the fourth order differential operator defined by (42). From the foregoing construction it is clear that if there are additional Weyl invariant terms in the anomaly (38) they should be included in the Sanom by making the replacement bF ^ bF + ; fyHi in the last square bracket of (46).

Notice from the derivation of Sanom that although the a independent piece of the gravitational action cannot be determined from the trace anomaly alone, the a dependence is uniquely determined by the general form of the trace anomaly for massless fields. Thus, whatever else may be involved in the full quantum theory of gravity in 4D at short distance scales, the anomalous effective action (46) should be included in the gravitational action at large distance scales, i.e. in the far infrared. Again the physics is that the quantum fluctuations of massless fields do not decouple and contribute to gravitational effects at arbitrarily large distances. Graviton (i.e. spin-two) fluctuations of the metric should give rise to an effective action of precisely the

same form as Sanom with new coefficients b and b', which can be checked at one-loop order [46]. The effective action (46) or (47) is the starting point for studying the new physics of quantum fluctuations of the conformai factor and infrared renormalization in gravity which allows the bare cosmological term of the classical Einstein theory to be screened. We postpone until section 7 the discussion of the auxiliary field description of (46) and the full low-energy EFT of 4D gravity.

5. Finite volume scaling and infrared screening of X

Let us consider the dynamical effects of the anomalous terms in the simplest case that the fiducial metric is conformally flat, i.e. gab = e2ffnab. Then the Wess-Zumino effective action simplifies to

Twz[n; ff = - jQ^ J d4x(Da)2, (47)

)2 — '

Q2 = -32nb. (48)

This action quadratic in a is the action of a free scalar field, albeit with a kinetic term that is fourth order in derivatives. The propagator for this kinetic term is (p2)-2 in momentum space, which is a logarithm in position space,

Ga(x, x') = -Q ^(x - x')2], (49)

the same as (27) in 2D. Of course this is no accident but rather a direct consequence of the association with the anomaly of a conformally invariant differential operator, □ in 2D and A4 in 4D, a pattern which continues in all higher even dimensions. Because of this logarithmic propagator we must expect the similar sort of infrared fluctuations, conformal fixed point and dressing exponents as those obtained in 2D gravity.

The classical Einstein-Hilbert action for a conformally flat metric gab = e2anab is

8nG , d4x[3e2o(daa)2 - Ae4a], (50)

which has derivative and exponential self-interactions in a .It is remarkable that these complicated interactions can be treated systematically using the the fourth order kinetic term of (47). In fact, these interaction terms are renormalizable and their anomalous scaling dimensions due to the fluctuations of a can be computed in closed form [45, 54]. Direct calculation of the conformal weight of the Einstein curvature term shows that it acquires an anomalous dimension f 2 given by the quadratic relation,

f = 2 + 2Q * (51)

In the limit Q2 ^^ the fluctuations of a are suppressed and we recover the classical scale dimension of the coupling G-1 with mass dimension two. Likewise the cosmological term in

(50) corresponding to the four volume acquires an anomalous dimension given by

ßo = 4 +

A 2ß2

Again as Q2 ^ro the effect of the fluctuations of the conformal factor are suppressed and we recover the classical scale dimension of A/G, namely four. The solution of the quadratic relations (51) and (52) determine the scaling dimensions of these couplings at the conformal fixed point at other values of Q2. This can be extended to local operators of any non-negative integer mass dimension p, with associated couplings of mass dimension 4 — p, by

ßP = 4 - p +

ip_ 2ß2

In order to obtain the classical scale dimension 4 — p in the limit Q2 root is determined so that

ro the sign of the square

ßp = Q2

(8 - 2p)

valid for Q2 ^ 8 — 2p for all p ^ 0, and thus Q2 ^ 8. These scaling dimensions were computed both by covariant and canonical operator methods. In the canonical method, we also showed that the anomalous action for the conformal factor does not have unphysical ghost or tachyon modes in its spectrum of physical states [55].

In the language of statistical mechanics and critical phenomena the quadratic action (47) describes a Gaussian conformal fixed point, where there are no scales and conformal invariance is exact. The positive corrections of order 1/Q2 (for Q2 > 0) in (51) and (52) show that this fixed point is stable in the infrared, that is, both couplings G—1 and A/G flow to zero at very large distances. Because both of these couplings are separately dimensionful, at a conformal fixed point one should properly speak only of the dimensionless combination h GA/c3 = X. By normalizing to a fixed four volume V = f d4x one can show that the finite volume renormalization of X is controlled by the anomalous dimension,

28 - 1 = 2 — - 1 =

84 1---1--

Q2 V Q2

1 + -/1 - A-

This is the anomalous dimension that enters the infrared renormalization group volume scaling relation [47],

V—X = 4(28 - 1)X. dV v ;

The anomalous scaling dimension (55) is negative for all Q2 ^ 8, starting at 1 — V^ = —0.414 at Q2 = 8 and approaching zero as —1/Q2 as Q2 ^ ro. This implies that the dimensionless cosmological term X has an infrared fixed point at zero as V ^ ro. Thus the cosmological term

is dynamically driven to zero as V ^^ by infrared fluctuations of the conformal part of the metric described by (47).

We emphasize that no fine tuning is involved here and no free parameters enter except Q2, which is determined by the trace anomaly coefficient b' by (48). Once Q2 is assumed to be positive, then 28 — 1 is negative, and A is driven to zero at large distances by the conformal fluctuations of the metric, with no additional assumptions.

The result (56) does rely on the use of (47) or its curved space generalization (46) as the free kinetic term in the effective action for gravity, treating the usual Einstein-Hilbert terms as interactions or 'marginal deformations' of the conformal fixed point. This conformal fixed point represents a new phase of gravity, non-perturbative in any expansion about flat space. In this phase conformal invariance is restored and the mechanism of screening A due to quantum effects proposed in [45] is realized.

Identifying the fluctuations responsible for driving A to zero within a framework based on quantum field theory and the Equivalence Principle, free of ad hoc assumptions or fine tuning is an important first step towards a full solution of the cosmological constant problem. However, the application of this screening mechanism to cosmology, in which we presume a classical or semiclassical line element of the form (10), is unclear. Near the conformal fixed point the inverse Newtonian constant G—1 is also driven to zero when compared to some fixed mass scale m [54]. This is clearly different from the situation we observe in our local neighbourhood. Under what conditions and where exactly (46) can dominate the classical Einstein terms, and moreover how the screening mechanism could be used to relax the vacuum energy to zero in a realistic cosmological model are questions not answered by our considerations to this point. Absent such a complete theory of cosmological vacuum energy we proposed that the conformally invariant phase might be relevant on horizon scales in cosmology. In that case the signatures of conformal invariance should be imprinted on and observable in the spectrum and statistics of the CMB.

6. Conformal invariance and the CMB

Our earlier studies of fluctuations in de Sitter space suggest that the fluctuations responsible for the screening of X take place at the horizon scale. In that case then the microwave photons in the CMB reaching us from their surface of last scattering should retain some imprint of the effects of these fluctuations. It then becomes natural to extend the classical notion of scale invariant cosmological perturbations, pioneered by Harrison and Zel'dovich [56] to full conformal invariance. In that case the classical spectral index of the perturbations should receive corrections due to the anomalous scaling dimensions at the conformal phase [57].7 In addition to the spectrum, the statistics of the CMB should reflect the non-Gaussian correlations characteristic of conformal invariance. This generic dynamical prediction of non- Gaussian correlations in the CMB due to conformal invariance was made for the first time to our knowledge in [57].

Scale invariance was introduced into physics in early attempts to describe the apparently universal behaviour observed in turbulence and second order phase transitions, which are independent of the particular short distance dynamical details of the system. The gradual refinement and development of this simple idea of universality led to the modern theory of

7 This paper contains a misprint in equation (16) which is corrected by equation (73) of the present paper. We thank G Shiu for bringing this to our attention.

critical phenomena, one of whose hallmarks is well-defined logarithmic deviations from naive scaling relations [58]. A second general feature of the theory is the specification of higher point correlation functions of fluctuations according to the requirements of conformal invariance at the critical point [59].

In the language of critical phenomena, the observation of Harrison and Zel'dovich that the primordial density fluctuations should be characterized by a spectral index n = 1 is equivalent to the statement that the observable giving rise to these fluctuations has engineering or naive scaling dimension p = 2. This is because the density fluctuations Sp are related to the metric fluctuations by Einstein's equations, SR ~ GSp, which is second order in derivatives of the metric. Hence, the two-point spatial correlations (Sp(x)Sp(y)} ~ (SR(x)SR(y)} should behave like |x — y|—4, or jkj1 in Fourier space, according to simple dimensional analysis.

One of the principal lessons of the modern theory of critical phenomena is that the transformation properties of observables under conformal transformations at the fixed point is not given by naive dimentional analysis. Rather one should expect to find well-defined logarithmic deviations from naive scaling, corresponding to a (generally non-integer) dimension A = p. The deviation from naive scaling A — p is the 'anomalous' dimension of the observable due to critical fluctuations, which may be quantum or statistical in origin. Once A is fixed for a given observable, the requirement of conformal invariance determines the form of its two- and three-point correlation functions up to an arbitrary amplitude, without reliance on any particular dynamical model.

Consider first the two-point function of any observable OA with dimension A. Conformal invariance requires [58, 59]

(Oa(x1)OA(x2)} ~ jx1 — X2|—2A (57)

at equal times in 3D flat spatial coordinates. In Fourier space this gives

G2(k) = (OA(k)dA(-k)) - |k|2A-3. (58)

Thus, we define the spectral index of this observable by

n = 2A - 3. (59)

In the case that the observable is the primordial density fluctuation Sp, and in the classical limit where its anomalous dimension vanishes, A ^ p = 2, we recover the Harrison-Zel'dovich spectral index of n = 1.

In order to convert the power spectrum of primordial density fluctuations to the spectrum of fluctuations in the CMB at large angular separations we follow the standard treatment [60] relating the temperature deviation to the Newtonian gravitational potential y at the last scattering surface, ST/ T — Sy, which is related to the density perturbation in turn by

V2 Sy = 4nGSp. (60)

Hence, in Fourier space,

8T 1 8p

~ ~ _ > (61) T k2 p

and the two-point function of CMB temperature fluctuations is determined by the conformal dimension A to be

18T 8T \ C ( 1 \2 C2 (9) = ( — (ri) — (h)) - j d3k i -) Gi(k) eik'r12 - r(2 - A)^)2-A, (62)

where r12 = (r1 — r2)r is the vector difference between the two positions from which the CMB photons originate. They are at equal distance r from the observer by the assumption that the photons were emitted at the last scattering surface at equal cosmic time. Since r22 = 2(1 — cos 9)r2, we find then

C2(9) - r(2 — A)(1 — cos9)2-A (63)

for arbitrary scaling dimension A.

Expanding the function C2(9) in multipole moments,

C2(9) = 4- J](2i + 1)cf(A)Pi(cos9), (64)

cf](A) - r(2 - A) sin [n(2 - A)]

- 2 + A) T(i + 4 - A)'

shows that the pole singularity at A = 2 appears only in the i = 0 monopole moment. This singularity is just the reflection of the fact that the Laplacian in (60) cannot be inverted on constant functions, which should be excluded. Since the CMB anisotropy is defined by removing the isotropic monopole moment (as well as the dipole moment), the i = 0 term does not appear in the sum, and the higher moments of the anisotropic two-point correlation function are well defined for A near 2. Normalizing to the quadrupole moment c22)(A), we find

:?\A) = cf](A)

r(6 - A)F(l - 2 + A) F(A) Y(l + 4 - A)'

which is a standard result [60]. Indeed, if A is replaced by p = 2 we obtain 1(1 + 1)c()

(p) = 6c2 )(p), which is the well-known predicted behaviour of the lower moments (i ^ 30) of the CMB anisotropy where the Sachs-Wolfe effect should dominate.

If the conformal fixed point behaviour described in the previous section dominates at these scales then the scaling dimension of an observable with classical dimension p is given by [61]

Ap = 4

(8-2p)_ / ß2 V Q2

1 - Q2

Hence consideration of the trace anomaly generated by the zero-point fluctuations of massless fields leads necessarily to well-defined quantum corrections to the naive scaling dimensions of

observables in cosmology. In the limit Q2 ^œ, the effects of fluctuations in the metric are suppressed and one recovers the classical scaling dimension p,

ap = P + - P) + •••• (68)

The quantity Q—2 is determined in principle by the trace anomaly coefficient b' through (38) and (48), but we may regard it as simply a free parameter characterizing the universality class of the conformal metric fluctuations, which should be determined from the observations. From this slightly more general perspective, the conformal invariance considerations that lead to (67) are quite independent of any particular model of their origin.

In the analysis of physical observables in the conformal sector of gravity, the operator with the lowest non-trivial scaling dimension corresponds, in the classical limit, to the scalar curvature R with p = 2 [51]. Since the fluctuations which dominate at large distances correspond to observables with lowest scaling dimensions, the conformal factor theory in this limit selects precisely Harrison's original choice. With p = 2, we find a definite prediction for deviations from a strict Harrison-Zel'dovich spectrum according to equations (59) and (67) in terms of the parameter Q2. The resulting spectral index is always greater than unity for all finite Q2 ^ 8, approaching one as

n = 1 + m + • • • • (69)

as Q2 ^ ro.

The latest WMAP three year CMB results now favour a spectral index for scalar perturbations of about 0.95, some three standard deviations lower than unity. From (38) and (48), the value of Q2 for free conformally invariant fields is

Q2 = 180 + TNf + 62Nv — 28^ + Q2rav, (70)

where Q^rav is the contribution of spin-2 gravitons. The —28 contribution is that of the conformal or spin-0 part of the metric itself. The main theoretical uncertainty in determining Q2grav is that the Einstein theory is neither conformally invariant nor free, so that a method for evaluating the infrared effects of spin-2 gravitons is required which is insensitive to Uv physics. A purely one-loop computation gives Qgrav — 7.9 for the graviton contribution [46]. Taking this estimate at face value and including all known fields of the Standard Model of particle physics (for which Nf = 45 and NV = 12) we find

QSM - 13.2 and n - 1.45, (71)

which is now firmly excluded by the WMAP data. If we require that n be within 0.05 of unity, then Q2 > 80 is needed.

It is possible that the effective number of massless degrees of freedom was much higher at the surface of last scattering from which the CMB was emitted, leading to a much higher value of Q2. It is also to be noted that the same conformal fixed point fluctuations that led to (59) and (67) also drive the cosmological term A and the inverse Newtonian constant to zero. These

non-classical effects on the geometry have not been taken into account in the essentially classical calculation of (60)-(63). The scaling relations at the conformal fixed point were also derived in a 4D Euclidean theory. Yet in (60)-(63) we assumed spatially flat FLRW 3D sections. The actual geometry through which the CMB photons propagate from their last scattering surface to us may be different. Finally, the cosmological parameters are also extracted from the data through the use of models such as Л CDM or variants thereof. Some of the assumptions in these models may have to be re-examined if gravity itself is modified by the anomalous terms. Accounting for these possible effects on the spectral index requires a more complete cosmological model.

Turning now from the two-point function of CMB fluctuations to higher point correlators, we find a second characteristic prediction of conformal invariance, namely non-Gaussian statistics for the CMB. The first correlator sensitive to this departure from Gaussian statistics is the three-point function of the observable OA, which takes the form [59]

(OA(xl)OA(x2)OA(x3)) - |X1 - X2|-A|X2 - Хз|-А|хз - хх|-А, (72)

or in Fourier space [57],

G3(*i,*2) - / d3p\p\A-3\p + кх\А-ъ\р - k2\

A 3 - A

.1 pi л л

x / du dv[u(1 - u)v] 2 (1 - v)2 1[u(1 - u)(1 - v)k1

+ v(1 - u)k2 + uv(k1 + k2)2]-(3-A\ (73)

This three-point function of primordial density fluctuations gives rise to three-point correlations in the CMB by reasoning precisely analogous as that leading from equations (58) to (62). That is,

C3(012, 023, 031) - (^1)^2)^3)) ~f /2/d23kld3k2 G3(ki, k2) eiklr13 eik2r23 (74) \T T T I J kfk|(ki + &2)2

where rj — (rt - rj)r and = 2(1 - cos dij)r2.

From (73) and (74), it is easy to extract the global scaling of the three-point function,

G3(sk1, sk2) — s3(A-2)G3(k1, k2), C3 — r3(2-A). (75)

In the general case of three different angles, the expression for the three-point correlation function (74) is quite complicated, although it can be rewritten in parametric form analogous to (73) to facilitate numerical evaluation, if desired. An estimate of its angular dependence in the limit A ^ 2 can be obtained by replacing the slowly varying G3(k1, k2) by a constant. Then (74) can be expanded in terms of spherical harmonics,

C3^ — £ (2/1 + 1^(2m/12i2+213)m(32/3 + 1) (/T+lb/3 + /1+12 + Z3+3)Yl1m1 (r1)Yl2m2(r2)Yl3m3^

where £/^1^2/3m3 — /drYi1m1 (r)Y/2m2(r)Yim3(r).

In the special case of equal angles 9j = 9, it follows from (75) that the three-point correlator is

C3 (9) - (1 — cos 9) 2(2—A). (77)

Expanding the function C3(9) in multiple moments as in equation (64) with coefficients c(3), and normalizing to the quadrupole moment, we find

ci»(A) = c»)(A)r(4 + |(2 — A)) F(i — 2(2 — A)) . (78)

* 2 T(2 — 2 (2 — A))F(i + 2 + 3 (2 — A))

In the limit A ^ 2, we obtain i(i + 1)c(3) = 6c23), which is the same result as for the moments cf} of the two-point correlator but with a different quadrupole amplitude.

The value of this quadrupole normalization c23) (A) cannot be determined by conformal symmetry considerations alone. A naive comparison with the two-point function which has a small amplitude of the order of 10—5 leads to a rough estimate of c23) — 0(10—7 5), which would make it very difficult to detect. However, if the conformal invariance hypothesis is correct, then these non-Gaussian correlations should exist at some level, in distinction to the simplest inflationary scenarios. Their amplitude is model dependent and possibly much larger than the above naive estimate. The present WMAP data does not show any evidence of these non-Gaussian statistics [62]. Again we are in need of a calculation of the amplitude of the non-Gaussianity based on a more complete model. In the meantime any detection of non-Gaussian statistics of the CMB would be an important clue to their origin and possibly an important test for the hypothesis of conformal invariance.

For higher point correlations, conformal invariance does not determine the total angular dependence. Already the four-point function takes the form,

Oa(X1)0 A (X2) O A (X3 ) O A (X4 )} - --^, (79)


where the amplitude A4 is an arbitrary function of the two cross-ratios, r^r^/r^^ and Analogous expressions hold for higher p-point functions. However, in the equilateral case 9j = 9, the coefficient amplitudes Ap become constants and the angular dependence is again completely determined. The result is

Cp(9) - (1 — cos9)p(2—A), (80)

and the expansion in multiple moments yields coefficients c^ of the same form as in equation (78) with 3/2 replaced by p/2. In the limit A = 2, we obtain the universal i-dependence

i(i + 1)cip) = 6c2p).

Thus the conformal invariance hypothesis applied to the primordial density fluctuations predicts deviations both from the classical Harrison-Zel'dovich spectrum and Gaussian statistics, which should be imprinted on the CMB anisotropy. A particular realization of this hypothesis is provided by the metric fluctuations induced by the known trace anomaly of massless matter fields which gives rise to fixed point with a spectral index n > 1. Although this is disfavoured by the WMAP data, lacking a complete cosmological model which takes dark energy, dark matter, the

CMB and possibly other effects into account in a consistent way, it is premature to draw a final conclusion on the conformal invariance hypothesis. The possibility of explaining the small value of X in a natural way is a strong reason to pursue a more complete cosmological model within this framework. In order to establish the firm theoretical basis for a consistent cosmological model, we return to consideration of the full EFT of gravity, reporting on recent progress in the auxiliary field description of the non-local terms (46) generated by the trace anomaly.

7. The low energy EFT of gravity

The quantization of the conformal factor in certain specialized cases carried out in the 1990s and reviewed in sections 4 and 5 shows that there are new degrees of freedom and infrared quantum effects in gravity which are not contained in classical general relativity. Moreover these new scalar degrees of freedom and their fluctuations appear to be the relevant ones for cosmological scales and the screening of the cosmological term, which was our initial motivation for studying them. In order to place these new degrees of freedom on a solid footing together with Einstein's theory, we give in this section a self-contained systematic EFT treatment of 4D gravity.

The EFT of gravity is determined by the same general principles as in other contexts ([63] and references therein), namely by an expansion in powers of derivatives of local terms consistent with symmetry. Short distance effects are parameterized by the coefficients of local operators in the effective action, with higher order terms suppressed by inverse powers of an UV cutoff scale M. The effective theory need not be renormalizable, as indeed Einstein's theory is not, but is expected nonetheless to be quite insensitive to the details of the underlying microscopic degrees of freedom, because of decoupling [63]. It is the decoupling of short distance degrees of freedom from the macroscopic physics that makes EFT techniques so widely applicable, and which we assume applies also to gravity.

As a covariant metric theory with a symmetry dictated by the Equivalence Principle, general relativity may be regarded as just such a local EFT, truncated at second order in derivatives of the metric field gab(x) [64]. When quantum matter is considered, the stress tensor Tab becomes an operator. Because the stress tensor has mass dimension four, containing up to quartic divergences, the proper covariant renormalization of this operator requires fourth order terms in derivatives of the metric. However, the effects of such higher derivative local terms in the gravitational effective action are suppressed at distance scales L ^ LPl in the low-energy EFT limit. Hence surveying only local curvature terms, it is often tacitly assumed that Einstein's theory contains all the low-energy macroscopic degrees of freedom of gravity, and that general relativity cannot be modified at macroscopic distance scales, much greater than LPl, without violating general coordinate invariance and/or EFT principles. As we have argued previously in 2D, this presumption should be re-examined in the presence of quantum anomalies.

When a classical symmetry is broken by a quantum anomaly, the naive decoupling of short and long distance physics assumed by an expansion in local operators with ascending inverse powers of M fails. In this situation, even the low-energy symmetries of the effective theory are changed by the presence of the anomaly, and some remnant of the UV physics survives in the low-energy description. An anomaly can have significant effects in the low-energy EFT because it is not suppressed by any large energy cutoff scale, surviving even in the limit M ^ro. Any explicit breaking of the symmetry in the classical Lagrangian serves only to mask the effects of the anomaly, but in the right circumstances the effects of the non-local anomaly may still

dominate the local terms. We have mentioned in section 1 the chiral anomaly in QCD with massless quarks, whose effects are unsuppressed by any inverse power of the EFT UV cutoff scale, in this case M ~ Aqcd. Although the quark masses are nonzero, and chiral symmetry is only approximate in nature, the chiral anomaly gives the dominant contribution to the low-energy decay amplitude of n0 ^ 2y in the standard model [65,66], a contribution that is missed entirely by a local EFT expansion in pion fields. Instead the existence of the chiral anomaly requires the explicit addition to the local effective action of a non-local term in four physical dimensions to account for its effects [46, 63].

Although when an anomaly is present, naive decoupling between the short and long distance degrees of freedom fails, it does so in a well-defined way, with a coefficient that depends only on the quantum numbers of the underlying microscopic theory. In fact, since the chiral anomaly depends on the colour charge assignments of the short distance quark degrees of freedom, the measured low-energy decay width of n0 ^ 2y affords a clean, non-trivial test of the underlying microscopic quantum theory of QCD with three colours of fractionally charged quarks [63, 66, 67]. The bridge between short and long distance physics which anomalies provide is the basis for the anomaly matching conditions [8].

The low-energy effective action for gravity in 4D contains first of all, the local terms constructed from the Riemann curvature tensor and its derivatives and contractions up to and including dimension four. This includes the usual Einstein-Hilbert action of general relativity,

SEeM = / d4xV-g(R - 2A) (81)

as well as the spacetime integrals of the fourth order curvature invariants,

Cd[g] = 2 / V-g(aCabcdCabcd + fiR2) d4x, (82)

with arbitrary dimensionless coefficients a and fi. There are two additional fourth order invariants, namely E =Rbcd Rabcd and DR, which could be added to ( ) as well, but as they are total derivatives yielding only a surface term and no local variation, we omit them. All the possible local terms in the effective action may be written as the sum,

Slocal[g] = — I d4xy-g(R - 2A) + Sol + £ S™. (83)

with the terms in the sum with n ^ 3 composed of integrals of local curvature invariants with dimension 2n ^ 6, and suppressed by M-2n+4 at energies E ^ M. Here M is the UV cutoff scale of the low-energy effective theory which we may take to be of order Mpl. The higher derivative terms with n ^ 3 are irrelevant operators in the infrared, scaling with negative powers under global rescalings of the metric, and may be neglected at macroscopic distance scales. On the other hand the two terms in the Einstein-Hilbert action n = 0, 1 scale positively, and are clearly relevant in the infrared. The fourth order terms in (82) are neutral under such global rescalings.

The exact quantum effective action also contains non-local terms in general. All possible terms in the effective action (local or not) can be classified according to how they respond to global Weyl rescalings of the metric. If the non-local terms are non-invariant under global rescalings, then they scale either positively or negatively under (28). If m-1 is some fixed length

scale associated with the non-locality, arising for example by the integrating out of fluctuations of fields with mass m, then at much larger macroscopic distances (mL ^ 1) the non-local terms in the effective action become approximately local. The terms which scale with positive powers of ea° are constrained by general covariance to be of the same form as the n = 0, 1 Einstein-Hilbert terms in Slocal, (81). Terms which scale negatively with eff0 become negligibly small as mL ^ 1 and are infrared irrelevant at macroscopic distances. This is the expected decoupling of short distance degrees of freedom in an EFT description, which are verified in detailed calculations of loops in massive field theories in curved space. The only possibility for contributions to the EFT of gravity at macroscopic distances, which are not contained in the local expansion of (83) arise from fluctuations not associated with any finite length scale, i.e. m = 0. These are the non-local contributions to the low-energy EFT which include those associated with the anomaly.

The non-local form of the anomalous effective action was given in (46). To cast this into local form and exhibit the new scalar degrees of freedom the Sanom contains, it is convenient as in the 2D case to introduce auxiliary fields. Two scalar auxiliary fields satisfying

A4$ = 2 (E — 3OR), Af = 1F, (84)

respectively may be introduced, corresponding to the two non-trivial cocycles of the b and b' terms in the anomaly [48]. It is then easy to see that

Sanom[g; y, f ] = 2 d4xy—g{—yA4y + (E — 2OR)y| b f

+ 2 d4xy—g{—yA4 f — fA4y + Fy + (E — | OR)f |, (85)

is the desired local form of the anomalous action (46) [68, 69]. Indeed the variation of (85) with respect to the auxiliary fields y and f yields their equations of motion (84), which may be solved for y and f by introducing the Green's function A—1 (x, x'). Substituting this formal solution for the auxiliary fields into (85) returns (46). The local auxiliary field form (85) is the most useful and explicitly contains two new scalar fields satisfying the massless fourth order wave equations (84) with fourth order curvature invariants as sources. The freedom to add homogeneous solutions to y and f corresponds to the freedom to define different Green's functions inverses A—1(x, x') in (46). The auxiliary scalar fields are new local massless degrees of freedom of 4D gravity, not contained in the Einstein-Hilbert action.

The terms in the classical Einstein-Hilbert action scale with positive powers (L4 and L2) under rescaling of distance, and are clearly relevant operators of the low-energy description. The non-local anomalous terms, rendered local by the introduction of the auxiliary fields y and f scale logarithmically (-logL) with distance under Weyl rescalings. Unlike local higher derivative terms in the effective action, which are either neutral or scale with negative powers of L, the anomalous terms should not be discarded in the low-energy, large distance limit. The auxiliary fields are new local scalar degrees of freedom of low-energy gravity, not contained in classical general relativity. The addition of the anomaly term(s) to the low-energy effective action of gravity amounts to a non-trivial infrared modification of general relativity, fully consistent with both quantum theory and the Equivalence Principle [48].

The fluctuations generated by Sanom define a non-perturbative Gaussian infrared fixed point, with conformal field theory anomalous dimensions analogous to the 2D case [45, 70]. This is possible only because new low-energy degrees of freedom are contained in Sanom which can fluctuate independently of the local metric degrees of freedom in SEH. Thus the effective action of the anomaly Sanom should be retained in the EFT of low-energy gravity, which is specified

then by the first two strictly relevant local terms of the classical Einstein-Hilbert action (81), and the logarithmic Sanom, i.e.

Seff[g] = Seh [g] + Sanom[g]

contains all the infrared relevant terms in low-energy gravity for E ^ Mpl.

The low-energy (Wilson) effective action (86), in which infrared irrelevant terms are systematically neglected in the renormalization group program of critical phenomena is to be contrasted with the exact (field theoretic) effective action, in which the effects of all scales are included in principle, at least in the approximation in which spacetime can be treated as a continuous manifold. Ordinarily, i.e. absent anomalies, the Wilson effective action should contain only local infrared relevant terms consistent with symmetry [58]. However, like the anomalous effective action generated by the chiral anomaly in QCD, the non-local Sanom must be included in the low-energy EFT to account for the anomalous Ward identities, even in the zero momentum limit, and indeed logarithmic scaling with distance indicates that Sanom is an infrared relevant term. Also even if no massless matter fields are assumed, the quantum fluctuations of the metric itself will generate a term of the same form as Sanom [47].

By using the definition of A4 and integrating by parts, we may express the anomalous action also in the form,

= b's(EL + bsF

S(E) = Sanom

^ i ti mn

2 j d4*v-g

/ d4 *V-

- (□ y)L + 2 1 Rab - ^ gae I (Vay)(VbV) + I E - ^ DR I «

- (□«) (Df) + 2 Rab - 3gab ) (Va«)(Vbf)

11 / 2 \

+ - F« + - iE - - DR j f

It is this final local auxiliary field form of the effective action which is to be added to classical Einstein-Hilbert action to obtain the effective action of low-energy gravity in (86). We note also that in this form the simple shift of the auxiliary field y by a spacetime constant,

« ^ « + 2a0,

yields the entire dependence of Sanom on the global Weyl rescalings (28), namely

Sanom[g; V, f] ^ Sanom[e2ff°g; V + 2a°, f ] = Sanom [g; V, f]

+ a0J d4xy-g[bF + b'(E - 3DR)],

owing to the strict invariance of the terms quadratic in the auxiliary fields under (28) and equations (4). Thus the auxiliary field form of the anomalous action (88) contains the same information about the global Weyl anomaly and large distance scaling as rWZ.

One may ask if there are any other modifications of classical general relativity at low energies that are consistent with general covariance and EFT principles. The complete classification of the terms in the exact effective action [48, 69] into just three categories means that all possible

infrared relevant terms in the low-energy EFT, which are not contained in Slocal of (83) must fall into Sanom, i.e. they must correspond to non-trivial co-cycles of the local Weyl group. The Weyl invariant terms in the exact effective action are by definition insensitive to rescaling of the metric at large distances. Hence these (generally quite non-local) terms do not give rise to infrared relevant terms in the Wilson effective action for low-energy gravity.

Furthermore, the form of the non-trivial co-cycles in Sanom is severely restricted by the locality and general covariance of quantum field theory. The UV divergences in the stress-energy tensor of quantum fields in curved spacetime are purely local. It is these divergences when renormalized consistently with covariant conservation of the local operator Tab(x) that give rise to the purely local form of the trace anomaly. Since all the local gauge invariant terms with mass dimension four matched to the physical dimension of spacetime are easily cataloged, the only non-trivial terms in Sanom at low energies which can arise from short distance renormalization effects are exactly those generated by the known form of the local trace anomaly (38). Decoupling fails in local quantum field theory only in the very narrow and well-defined way dictated by local anomalies, and these uniquely determine the non-local additions to the effective action, up to any contributions from Sinv, which in any case have negligibly small effect on very large distance scales. The form of the effective action Sanom at macroscopic distances L ^ Lpl is not expected to change substantially even if the condition of strict locality of the underlying quantum theory is relaxed or replaced eventually by a more fundamental, microscopic description of gravitational interactions at very large mass scales of order Mpl. If this were not the case, then the classical Einstein theory could be overwhelmed by all sorts of non-local quantum corrections from unknown microscopic physics, and would lose all predictive power for macroscopic gravitational phenomena. Instead, under the defining assumptions of general covariance and the EFT hypothesis of decoupling of physics associated with massive degrees of freedom, the infrared modification of Einstein's theory specified by (86)-(88) is tightly constrained and becomes essentially unique. Notice in particular that the EFT logic precludes any inverse powers of the Ricci scalar or other local curvature invariants appearing in the denominators of terms in the effective action. It is the EFT of gravity defined by (86) which is the basis for the new degrees of freedom and their fluctuations leading to the conformal fixed point of sections 4 and 5. It should also be the basis upon which a more comprehensive cosmological model of dark energy incorporating these effects is constructed.

8. The horizon boundary

Quantum effects in global de Sitter space indicate infrared effects on the horizon scale are important. These effects are contained in two new infrared relevant operators in the EFT of low-energy gravity described by (87) above. If we hope to construct a consistent model of dark energy in which the quantum effects of the trace anomaly can modify classical theory, then we must identify first where the effects of the new terms in the EFT can be significant. This involves again a careful investigation of horizon and near horizon effects in de Sitter spacetime.

Two familiar examples of spacetimes with horizons are the Schwarzschild metric of an uncharged non-rotating black hole, and the de Sitter metric. Both can be expressed in static, spherically symmetric coordinates in the form,

ds2 = — f(r)c2dt2 + — + r2 (d92 + sin2 9 d02). (91)

In the Schwarzchild case,

rS 2GM

fs (r) = 1 - ri, rs = —— (92)

while in the de Sitter case, from (22),

r2 / 3\1/2

fds (r) = 1 - n, rH = j) , (93)

respectively. The first metric approaches flat space at large r but becomes singular at the finite radius rs. The second is de Sitter spacetime in the static coordinates (22), describing the interior of a spherical region with a coordinate singularity at the finite radius rH. In both cases the metric singularities may be regarded as pure coordinate artifacts, in the sense that they can be removed entirely by making a singular coordinate transformation to a different set of well-behaved coordinates in the vicinity of the horizons. Indeed by undoing (21b) we can transform de Sitter spacetime back to FLRW coordinates. However, as we have observed already in section 2 certain global effects such as the temperature associated with the horizon in the standard treatments cannot be transformed away by a local coordinate transformation.

It is important to recognize that the Equivalence Principle implies invariance under regular coordinate transformations, whereas singular transformations and the analytic extensions of spacetime they involve require a physical assumption, namely that there are no stress-energy sources or discontinuities of any kind at coordinate singularity. Even in the classical theory the hyperbolic nature of Einstein's equations allows for sources and/or discontinuities transmitted at the speed of light on a null hypersurface, such as the Schwarzschild or de Sitter horizon. Analytic continuation amounts then to a physical assumption that no such discontinuities are present.

Moreover, when quantum fields are considered, matter is no longer described as pointlike particles following classical geodesics, but as matter waves, especially on the horizon scale. This is the origin of the particle creation and Hawking effects discussed in section 2. The matter wave equations such as the Dirac or Klein-Gordon equations couple to the electromagnetic and metric potentials, not the local Maxwell or Riemann curvature tensors. Hence quantum matter effects can depend on gauge invariant global functions of the potentials such as exp (i£ A^dx^), through the boundary conditions imposed on the solutions. Like the Aharonov-Bohm effect in electron scattering or the Abrikosov vortex in superconductors, a singular 'gauge' transformation may have global physical effects at the quantum level, even though the field curvature tensor is small or even vanishing nearly everywhere. This is because it is not truly an allowed symmetry transformation of the quantum theory at all. Clearly such global quantum effects of matter waves cannot be captured by a description of matter as pointlike particles of infinitesimal size following classical geodesics.

These quantum wavelike effects show up in the behaviour of the renormalized expectation value of the stress-energy tensor, Tab | as r ^ rS or r ^ rH. This expectation value depends upon the quantum state of the field theory, specified by choosing particular solutions of the wave equation DO = 0. The Schwarzschild or de Sitter horizon is a characteristic surface and regular singular point of the wave equation in static coordinates (91). Hence the general solution of the wave equation is singular there, and so is the expectation value, (^|Tab|^} in the corresponding quantum state. This generic singular behaviour is essentially kinematical in

origin, since a photon with frequency m and energy hm far from the horizon has a local energy Eloc = hMf—1/2. The stress-energy is dimension four, and its generic behaviour near the event horizon is Ej4oc ~ f—2. Calculations of <^|Tab|^} both directly from quantum field theory and through the anomaly action (85) show this diverging behaviour in the vacuum state defined by absence of quanta with respect to the static time coordinate t in (91)[ 69,72,73]. In this 'vacuum' state the stress tensor behaves like the negative of a radiation fluid at the local temperature, Tloc = hc|f '|f—1/2/4nkB. Hence the 'vacuum' near a spacetime horizon is sensitive to arbitarily high-frequency components of quantum fluctuations, whose backreaction effects through the stress tensor expectation, <Tab} may become arbitrarily large.

Although this has been known for some time [72, 73], the attitude usually adopted is that states which lead to such divergences on the horizon are to be excluded, and only states regular on the horizon are allowed. However, this particular boundary condition is only one possibility and is not required by any general principles of quantum field theory in one causally connected region of spacetime. The essence of an event horizon is that it divides spacetime into regions which are causally disconnected from each other. In both the de Sitter and Schwarzschild cases, there are globally regular states such as the BD and Hartle-Hawking states respectively [23,71], but these specify that precise quantum correlations be set-up and maintained in regions of the globally extended spacetimes which never have been in any causal contact with each other. Despite their mathematical appeal, it is by no means clear physically why one should restrict attention to quantum states in which exact phase correlations between causally disconnected regions are to be rigorously enforced. This is the quantum version of the causality or horizon problem encountered with respect to the CMB in classical FLRW cosmologies. As soon as one drops this acausal requirement, and on the contrary restricts attention to states with correlations that could have been arranged causally in the past, within the region r ^ rH in de Sitter spacetime, for example, then states with divergent <^| Tab | on the horizon become perfectly admissible (ignoring for the moment the large backreaction such a stress-energy must exert on the background geometry). This is the conclusion reached also by consideration of the solutions of the auxiliary field equations (84) in Schwarzschild and de Sitter spacetimes [69]. In fact, the y and f fields generally diverge on the horizon, even though the local Riemann curvature is classically small there. The quantum EFT which allows these states is then quite different in its basis and consequences from a presciption to exclude these states are a priori.

Again our experience with the Casimir effect suggests a physical interpretation and resolution of the divergences in these states. In the calculation of the local stress tensor <^| Tab | in flat spacetime with boundaries, one also finds divergences in the generic situation of non-conformally invariant fields and/or curved boundaries [74]. The divergences may be traced to the boundary conditions imposed on modes of the high-frequency components of a quantum field, and cannot be removed by standard renormalization counterterms in the bulk. As the theory and applications of the Casimir effect have developed, it became clear that the material properties of the real conductors involved in the experiments must be taken into account to regulate these mathematical divergences. Casimir's idealized boundary conditions on the electromagnetic field, appropriate for a perfect conductor with infinite conductivity, must give way in more realistic calculations to boundary conditions incorporating the finite conductivity response function of real metals [2]. The idealized boundary conditions which led to the divergences are not to be excluded from consideration by mathematical fiat; indeed they are the correct boundary conditions at low to moderate frequencies, and the local stress tensor would continue to grow larger as the boundary is approached, if no new physics were to intervene. The appropriate modification

of the boundary conditions at higher frequencies and the cutoff of this growth is obtained by correctly incorporating the physics of the conducting boundary. Then finite results confirmed in detail by experiment are obtained [2]. At still smaller length and timescales approaching atomic dimensions, the approximation of a continuous or average conductivity response function of the metal surface will have to be modified again, to take account the electron band structure and microscopic graininess of the conductors, which are composed finally of discrete atomic constituents. Only the physical response function of the metal, not the mathematics of analytic continuation (which involves an unchecked assumption of arbitrarily high Fourier components) can determine the correct boundary conditions to be imposed at the boundary, and the behaviour of the stress tensor as the boundary is approached.

In the case of the Schwarzschild or de Sitter horizon this boundary condition requires additional physical input. The wave equation in the spherically symmetric static coordinates (91) can be separated by writing O = e—iatYlm^fi, and the second order ordinary differential equation for the radial function, may be cast in the form,

= o>2fai. (94)

The change of radial variable from r to r* = fr dr/f(r) has been made in order to put the second derivative term into standard form, and the scattering potential for the mode with angular quantum number l is

Vi = f

1df + (+1) r dr r

Since f ^ 0 linearly as r approaches the horizon, the variable r* c logarithmically, and the potential goes to zero there, vanishing exponentially in r*. Hence the solutions of (94) define 1D scattering states on an infinite interval (in r*), and the boundary conditions on the horizon that ensure that the scattering matrix is Hermitian are free ones, allowing both incoming and outgoing plane wave modes as r* ^ — cc The vacuum state defined by zero occupation number with respect to these scattering states is the Boulware vacuum |^B}, which has a divergent (^B|Tab|^B}, behaving in fact like — Tl4oc diag (—3, 1, 1, 1) - f—2 on the horizon [72, 73]. In contrast to the Casimir effect in flat spacetime with curved boundaries, this divergence does not arise from hard Dirichlet or Neumann boundary conditions, but from an infinite redshift surface with free boundary conditions. Hence the properties of no ordinary material at the boundary can remove this divergence, and the effective cutoff of horizon divergences can arise only from new physics in the gravitational sector at ultrashort scales, i.e. in the structure of spacetime itself very near to the horizon.

The divergence of the expectation value (Tab} at the horizon is completely described by the auxiliary potentials of the anomalous action (87), (88) [69]. As expected from their derivation and the analogous situation in 2D, these auxiliary fields carry non-local information about the global quantum state and boundary conditions. Their fluctuations describe the higher point correlators of the stress-energy tensor in the given quantum state. From the particle creation, thermodynamic and fluctuation-dissipation discussion in section 2, and the conformal fixed point considerations of sections 4 and 5, it is these fluctuations and the higher point correlators of the quantum Tab that generate the backreaction on the mean geometry necessary to relax the effective cosmological term to zero. These become significant in the limit f(r) ^ 0 as the

horizon boundary is approached. Thus the locus of important quantum effects from Sanom is not on superhorizon scales in the FLRW coordinates (10), but in a very thin boundary layer very close to the Schwarzschild or de Sitter horizon in static coordinates. In other words, the physical location of the conformally invariant phase of gravity discussed somewhat abstractly in sections 4 and 5 should be just in this boundary layer very close to r = rH.

9. A new cosmological model of dark energy

The suggestion that a quantum phase transition may occur in the vicinity of the classical Schwarzschild horizon rS has been made in [75] and [76]. The fluctuations of the auxiliary fields of Sanom which we found previously describe a conformally invariant phase of gravity with vanishing cosmological term, and may be responsible for this transition near the horizon. At a first order phase transition in which the quantum ground state rearranges itself, the vacuum energy of the state can change. Hence the region interior to rS of the Schwarzschild geometry may have a different effective value of A than the exterior region. We have suggested that the cosmological term itself may be viewed as the order parameter of a kind of gravitational Bose-Einstein condensate (GBEC), and the phase transition near the horizon where this condensate disorders would then become similar to BEC phase transition observed in cold atomic systems [76]. Moreover, since due to (6) the vacuum dark energy equation of state with pv = — pv > 0 acts as an effective repulsive term in Einstein's equations, a positive value of A in the interior serves to support the system against further gravitational collapse. For this to work, the effective value of A in the interior would have to adjust itself dynamically to the total mass of the system in order to reach a non-singular state of stable equilibrium with rH ~ rS. The de Sitter interior is free of any singularities and the entropy of this state is much less than the Bekenstein-Hawking entropy of a black hole. It therefore suffers from no 'information paradox' [76].

It is interesting to remark that the non-singular configuration described in [76] may be viewed as the gravitational analog of the model of an electron, which was one of the motivations of some of the original investigations of the Casimir effect. Since the Casimir force on a conducting charged sphere is repulsive, it cannot cancel the classical repulsive Coulomb self-force [77]. However a repulsive Casimir force with interior vacuum energy pv = —pv < 0 is exactly what is needed to balance the attractive force of gravity to prevent collapse to a singularity. The Casimir proposal to model an elementary particle such as the electron as a conducting spherical shell does not work as originally proposed, but the analogous model for the non-singular final state of gravitational collapse of a macroscopic self-gravitating object [78] appears to be perfectly viable.

The model we arrive at is one with the de Sitter interior matched to a Schwarzschild exterior, sandwiching a thin shell which straddles the region near to rH ~ rS, cutting off the divergences in (rb) as rH is approached from inside and rS is approached from outside. This thin shell is the boundary layer where the new physics of a quantum phase transition takes place. In the EFT approach, this new physics is described by the fluctuations of the auxiliary scalar degrees of freedom y and ty in Sanom.

In a true quantum boundary layer, fluctuations in all the higher point correlators of Tab are to be expected. This boundary layer is therefore quite non-perturbative. In effect the coupling constant A is multiplied by inverse factors of f(r) which greatly enhance the quantum effects in the boundary layer. Thus even if A ^ 1a critical surface is reached when one approaches the

horizon boundary from the interior de Sitter phase. As a first approximation we may treat the quantum boundary layer in a mean field approximation, in which Einstein's equations continue to hold, but with an effective equation of state of the material making up the layer. This 'material' is the quantum vacuum itself, with a stress tensor described by the auxiliary scalar fields of the effective action (87), (88). Then the divergences in this stress tensor are cutoff by backreaction on the classical geometry, replacing an infinite redshift surface at the horizon with a finite one. This ultrarelativistic vacuum effect at a causal boundary suggests that the most extreme equation of state consistent with causality should play a role here, namely the Zel'dovich equation of state p = p, where the speed of sound becomes equal to the speed of light. This is the critical equation of state at the limit of stability for a phase transition to a new phase with a different value of the vacuum energy. It also arises naturally as one component of the stress-energy tensor, (Tab} = diag (—p, p, p±, p±) in a state such as the Boulware state. The conservation equation,

V, r = ^ + f f + 2 = 0, (96)

dr 2f dr r

implies three independent components in the most general static, spherically symmetric case. It is clear from (96) that the three independent components can be taken to be that with p = p/3, behaving like f—2, p = p, behaving like f—1, and p = —p, behaving like f0, reflecting the allowed dominant and subdominant classical scaling behaviours of the stress tensor near the horizon.

In the simplest model possible we make the further approximation of setting the tangential pressure p± = p and consider only two independent components of the stress tensor in non-overlapping regions of space. In that case we have three regions, namely,

I. Interior (de Sitter) 0 ^ r < r1, p = — p,

II. Thin Shell r1 < r < r2, p = +p, (97)

III. Exterior (Schwarzschild) r2 < r, p = p = 0.

Because of (96), p = — p is a constant in the interior, which becomes a patch of de Sitter space in the static coordinates (91), for 0 ^ r ^ r1 < rH. The exterior region is a patch of Schwarzschild spacetime for rS < r2 ^ r < c. The p = p/3 component of the stress tensor and the smooth transition that it would make possible from one region to another has been neglected in this simplest model. In the Boulware state, this p = p/3 traceless stress tensor has negative sign near rH or rS. Tangential stresses have been considered by the authors of [79].

The location of the interfaces at r1 and r2 can be estimated by the behaviour of the stress tensor near the Schwarzschild and de Sitter horizons. If 1 — rS/r1 is a small parameter e, then the location of the outer interface occurs at an r1 where the most divergent term in the local stress-energy a M—4e—2, becomes large enough to affect the classical curvature —M—2, i.e. for

e — M = 10—38 (M0 \ , (98)

where Mpl is the Planck mass ^/hc/G ~ 2 x 10—5 gm. Thus e ^ 1 for an object of the order of a solar mass, M = M©, with a Schwarzschild radius of order of a few kilometres.

If instead of a collapsed star one considers the interior de Sitter region to be a model of cosmological dark energy, then the radius rH is set by measured value of (7),

i.e. the size of the entire visible universe, and M ~ 5 x 1022Mo — 1056 gm becomes of the order of the total mass-energy of the visible universe. In that case e — 2 x 10—61 — VA is very small indeed.

Since the function f(r) is of order e ^ 1 in the transition region II, the proper thickness of the shell is

Although very small, the thickness of the shell is very much larger than the Planck scale (2). For rH given by (99), the physical thickness of the shell is macroscopic: t ~ .04 mm. The energy density and pressure in the shell are of order M—2 and far below Planckian for M ^ Mpl, so that the geometry can be described reliably by Einstein's equations, essentially everywhere, except within the thin shell. The details of the solution in region II, the matching at the interfaces, r1 and r2, and analysis of the thermodynamic stability of the gravitational vacuum condensate star ('gravastar') were studied in [76].

We note from (21b) that in this kind of cosmological model, the past boundary at r = rH is at the infinite past of the RW coordinates. Thus one trades a possible special origin of time and the spacelike singularity of the big bang in FLRW cosmologies for a special spatial origin and location of the boundary wall. The redshift of primordial radiation is then the gravitational redshift due to the potential change from the cosmological horizon to an observer in the interior de Sitter geometry. For most of the interior volume, the effects of the past boundary are observed only as relic CMB radiation and a residual vacuum energy, which would be difficult to distinguish from a FLRW model far from the boundary. Since the universe is apparently 74% vacuum dark energy, the cosmological model first proposed by de Sitter [9], in which QA is unity becomes again a good first approximation to the observations.

This simple model is clearly very far from complete. The boundary layer has been posited from the allowed behaviour of the stress-energy near the horizon, rather than a full solution of the EFT equations for the auxiliary fields following from (86). The value of A in the interior is constant and can take on any value, but the solution has QA = 1 with no matter or radiation whatsoever in the interior. Since the fluctuations of the auxiliary scalar fields in the trace anomaly action are necessary for the dynamical relaxation of the vacuum energy, we can obtain only a solution which is static without considering those fluctuations in detail. This static solution does not describe the evolution of the dark energy towards smaller values with time or the very small present value of A. No attempt has been made to construct a fully dynamical cosmological model which would have to pass the many successful tests of the standard FLRW models, including the magnitude, spectrum, and statistics of the CMB. On the other hand if we start with a simplified cosmological model of pure dark energy in which QA is exactly one, our challenge becomes to explain why it is actually 0.74, rather than unity, instead of misestimating A by 122 orders of magnitude.

Despite its drastic simplifications the interior de Sitter gravastar model of dark energy does illustrate the possibility of pv being a kind of order parameter which is spacetime-dependent, whose value depends on boundary conditions at macroscopic scales. Thus the dark energy becomes a boundary effect, analogous to the Casimir energy (5), with the role of the conducting plates being taken by a critical boundary layer in the spacetime vicinity of the cosmological horizon. In order for the bulk vacuum energy to scale quadratically with H, i.e. pv ~ c4/GrH, rather than hc/rH as might be expected from (5), it is necessary for fluctuations of the metric itself, i.e. the stress-energy of gravitational waves to play an important role. These fluctuations are generated at the horizon boundary near rH. Because of the anomalous trace terms in (38), the stress-energy of the gravitational waves is not tracefree, but can contain also a trace part with p = — p of either sign which can change the effective value of A in the interior. Once A becomes dynamical then its value is determined not by naturalness considerations at the UV cutoff scale as in (4), but by the physics at the boundary r = rH and its dynamical evolution.

The basic assumption required for a solution of this kind to exist is that gravity, i.e. spacetime itself, must undergo a quantum vacuum rearrangement phase transition in the vicinity of the horizon, r — rH. Clearly this cannot occur in the strictly classical Einstein theory of general relativity with A constant. It requires that the fluctuations <rab(x)Tcd(x/)} and higher stress tensor correlators about its mean value be taken into account near the horizon. These higher order correlators are generated in the EFT approach by the additional scalar degrees of freedom in Sanom given by (87) and (88). The addition of Sanom is the minimal modification of Einstein's theory required by quantum theory and stress tensor renormalization consistent with the Equivalence Principle. The new degrees of freedom allow the vacuum energy density pv to change and adjust itself dynamically in the interior spacetime. Einstein's theory otherwise continues to apply almost everywhere, sufficiently far from the quantum boundary layer, with a dynamical bulk pv coupled to and determined by the fluctuations at the horizon.

Since allowing the effective value of A to change with time would require the generation of gravitational and other radiation at the horizon boundary, this would realize the dissipative mechanism of relaxation of coherent vacuum energy into matter/radiation modes which we discussed in section 2. From (24b) a continuously decreasing pv necessary to give a cosmologically acceptable solution to the problem of dark energy requires a continuous energy inflow through the cosmological horizon. The stress tensor of such an inflow would be expected to contain dissipative terms arising from the bulk viscosity of fluctuations of Taa from the trace anomaly terms, consistent with the general fluctuation-dissipation considerations of section 2. This relaxation of A to smaller and smaller values would not require any detailed information about Planck scale physics, but instead be consistent with the general hypothesis of decoupling, with the only short distance effects essential for macroscopic cosmology coupling at the horizon boundary, described by the effective action (86). In the limit A ^ 0, the boundary layer is removed to infinity, the auxiliary fields in (84) have zero sources, and empty flat space is the only stable asymptotic solution.

The analogy with atomic Bose-Einstein condensates (BEC) may also be a fruitful one to pursue. The vacuum energy density pv is a kind of gravitational vacuum condensate [76]. This condensate is self-trapped by its own gravity. The interior vacuum energy density depends on the total number of 'atoms' in the trap. The condensate is described by EFT methods analogous to the Gross-Pitaevskii EFT of BEC's in terms of the long wavelength collective modes of the system. Near the horizon boundary the condensate disorders, due to the quantum fluctuations of the auxiliary degrees of freedom in the EFT of gravity. At a finer level of resolution, and in

particular near the boundary, the continuum mean field description must give way to a more fundamental treatment in terms of the analogous of the atomistic degrees of freedom that make up the vacuum condensate [80]. An important step towards such a description would be to include fluctuations about the mean field, and their associated dissipative effects.

Clearly much more work remains to be done before a consistent dynamical theory of dark energy based on this interconnected set of ideas can be proposed. Yet the essential physical basis and EFT elements of such a dynamical theory would seem to be in place. Only when a comprehensive cosmological model incorporating these effects is available can we determine if it passes all observational tests of standard cosmology, and make unambiguous predictions for future measurements. The prediction of the magnitude of deviations from the classical Harrison-Zel'dovich spectrum at large angles and non-Gaussian correlations in the CMB remain the most promising tests of the conformal invariance hypothesis [57]. If cosmological dark energy is a finite size effect of the universe in the large, whose value is determined by conformal fluctuations at the infrared horizon scale, its dynamical relaxation to smaller values over time provides a natural resolution to the dilemma of quantum zero-point energy, originally raised by Pauli eighty years ago, but now made urgent by its detection (7)-(9) in the cosmos.


IA was supported in part by the European Commission under the RTN contract MRTN-CT-2004-503369.


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