Scalar Gravitational Waves in the Effective Theory of Gravity

As a low energy effective field theory, classical General Relativity receives an infrared relevant modification from the conformal trace anomaly of the energy-momentum tensor of massless, or nearly massless, quantum fields. The local form of the effective action associated with the trace anomaly is expressed in terms of a dynamical scalar field that couples to the conformal factor of the spacetime metric, allowing it to propagate over macroscopic distances. Linearized around flat spacetime, this semi-classical EFT admits scalar gravitational wave solutions in addition to the transversely polarized tensor waves of the classical Einstein theory. The amplitude of the scalar wave modes, as well as their energy and energy flux which are positive and contain a monopole moment, are computed. Astrophysical sources for scalar gravitational waves are considered, with the excited gluonic condensates in the interiors of neutron stars in merger events with other compact objects likely to provide the strongest burst signals.


I. INTRODUCTION: EFFECTIVE FIELD THEORY AND QUANTUM ANOMALIES
Einstein's classical theory of General Relativity (GR) predicts the existence of gravitational waves, created in significant amounts by collapsing and rotating binary systems [1]. Almost a century after this prediction, the Laser Interferometer Gravitational-wave Observatory (LIGO) and LSC collaboration have reported the first direct detection of gravitational waves from the inspiral and coalescence of two intermediate mass black hole candidates of approximately 30M each [2].
As this confirmation of classical GR is added to its already impressive successes in the astrophysical domain, quantum field theory (QFT) has also achieved a parallel and even more detailed confirmation in the microscopic domain, culminating in the Standard Model unifying the weak and electromagnetic forces, and the discovery of the Higgs boson [3]. At the most microscopic scales probed by experiment, the fundamental principles of quantum mechanics and special relativistic Lorentz invariance, the twin pillars of QFT, clearly hold.
These successes of classical GR and QFT, each in their respective domains, require that they be brought together in some consistent framework, preserving the main features of each, at least in an approximate sense, and in a quantitatively controllable manner. Absent a unifying synthesis which is able to completely describe and predict phenomena across the broad spectrum of scales from microphysics to cosmology, the best techniques presently at our disposal for a partial union of these disparate theories with a minimum of additional assumptions are those of effective field theory (EFT) and semi-classical background field methods, such as QFT in curved spacetime.
EFT methods have been developed and successfully applied in a number of areas from nuclear and particle physics to condensed matter physics [4,5]. The key hypothesis in the EFT approach is the decoupling of short and long distance scales, making possible a description of physics at energies far below an ultraviolet (UV) cutoff scale M by a set of local operators consistent with low energy symmetries. The low energy effective action is constructed by truncating at some finite order the sum of such local terms in increasing numbers of derivatives divided by powers of M , and fixing their coefficients from experiment. The resulting effective theory is expected then to apply to all physical processes with energy-momenta k M , or equivalently, at distance scales M −1 , with an error of the first order of terms in k/M which have been neglected in the derivative expansion. This is a systematic approach, consistent with general QFT principles, relying on the decoupling theorem of ultra-short distance physics at very large energy scales from the low energy, longer wavelength processes of interest [6], which applies to both renormalizable and non-renormalizable effective theories. Indeed every QFT, including also the Standard Model itself may be regarded as an EFT [4,5].
The EFT method can be extended to gravity with no essential difficulty. Classical GR itself is best understood as an EFT for energies much less than the Planck energy M P l c 2 = ( c 5 /G) 1 2 , or on distance scales much greater than the Planck length L P l = ( G/c 3 ) 1 2 2 × 10 −33 cm. Since the local EFT corrections to the gravitational action involve higher powers of the Riemann curvature tensor and its contractions, these gravitational corrections to classical GR are highly suppressed in the weak field limit of nearly flat spacetime or for distances r L P l [7]. If these local higher order curvature corrections would be the only possible QFT induced corrections to classical gravity, quantum effects would be entirely negligible and could safely be ignored on macroscopic distance scales, at least if the Einstein Equivalence Principle and metric foundation of GR are rigorously adhered to.
The semi-classical approach to EFT relies on the other hand upon the development of QFT in background curved spacetimes, treating the metric classically (at least at first), while the fields propagating on this classical curved background are allowed to be fully quantum. Since the metric is otherwise arbitrary, QFT in curved spacetime is a particular application of the background field method [8], not relying upon any perturbative or local expansion around flat space, static sources or weak gravitational fields. When the quantum fields are completely 'integrated out' in an arbitrary curved background and QFT divergences are either cut off at short distances, or renormalized in the standard way consistent with general covariance, an exact effective action is produced which contains the higher curvature squared terms characteristic of the previous EFT expansion in local invariants.
Hence the semi-classical background field method and the EFT method of cataloguing local operators are closely related. However, in integrating over the quantum fields at all scales, the background field method and quantum effective action approach captures one important but somewhat more subtle consequence of QFT easily overlooked in the Wilsonian EFT expansion in local curvature invariants, namely the generally non-local and infrared effects of quantum anomalies.
The best known and most instructive example of a quantum anomaly is the axial current anomaly in either QED or QCD. At the classical level the axial current j µ 5 is a Noether current whose divergence is expected to vanish in the limit of massless fermions. As is well known, it turns out to be impossible for any regulator to enforce the conservation of j µ 5 at the one-loop quantum level without violating gauge invariance, and a well-defined finite divergence ∂ µ j µ 5 = (N/16π 2 ) FF results in a background gauge field F µν with N massless fermions [9]. This axial current divergence cannot be represented in terms of the original gauge fields by any local effective action. If it could be, there would be the possibility of shifting the coefficient of this term by a suitable constant and cancelling the anomaly.
That this is impossible without violating gauge invariance implies that the axial anomaly is a genuine quantum infrared (IR) effect, characterized by a non-local effective action in pure QED or QCD.
The IR implications of the anomaly are confirmed by the appearance of massless poles in anomalous amplitudes, signaling the existence of a low energy massless effective excitation and degree of freedom not present in the original classical Lagrangian [10].
Since the U A (1) symmetry is explicitly broken by the axial anomaly, any local EFT expansion assuming this symmetry will necessarily miss the anomaly and its associated low energy massless excitation. Thus if one adopts a standard Wilsonian EFT approach to low energy meson physics, expanding only in polynomials of local derivatives of the meson fields divided by a characteristic mass scale of the order of Λ QCD or the ρ-meson mass of M ρ 770 MeV, no anomalous U A (1) contribution will be present. This local EFT is necessarily incomplete, neither correctly predicting the η pseudoscalar meson mass nor the decay constant of π 0 → 2γ. Capturing these effects requires adding to the standard local EFT meson Lagrangian an additional term, explicitly breaking the U A (1) symmetry, in a procedure that has come to be known as 'anomaly matching' [5,11].
The necessity of modifying the standard EFT approach when quantum anomalies are present be- EFT approach to gravity including the effects of logarithmally scaling QFT anomalies should apply.
In QFT in curved space the important quantity coupling to gravity is the matter stress-energy tensor T µν . For massless fields conformally coupled to gravity, the stress tensor is classically traceless T µ µ = 0. As in the case of the axial current, the classical conformal symmetry cannot be maintained at the one-loop quantum level without violating another invariance, in this case general coordinate invariance expressed by the covariant conservation law ∇ µ T µν = 0. Enforcing the latter results in the trace being necessarily non-zero and given by in a general four dimensional curved spacetime, where is the Euler-Gauss-Bonnet integrand and the square of the Weyl conformal tensor respectively. Here R αβγδ the Riemann curvature tensor, * R αβγδ its dual, and R αβ and R is the Ricci tensor and scalar.
Additional dimension four invariants denoted by L i in (1.2) may also appear in the general form of the trace anomaly, if the massless conformal fields in question couple to additional long range fields.
For example in the case of massless fermions coupled to a gauge field, there are contributions from the scalar invariants L F = F µν F µν of electromagnetism or of the strong or electroweak non-abelian gauge fields, with coefficients determined by the β function of the corresponding gauge coupling. Note that these gauge field terms are certainly not Planck suppressed.
All the anomaly coefficients b, b , b and β i are dimensionless parameters multiplied by , and are known for any number of free conformal scalars (N s ), four-component Dirac fermions (N f ) or vectors (N v ) respectively [13,14]. The b coefficient in (1.2) is ultraviolet (UV) regularization dependent and can be changed at will or set to zero by the addition of a local R 2 term in the effective action, and hence is not a true anomaly. In contrast, the b and b terms in (1.2) are independent of the UV regulator, cannot be removed by any local counterterm, and depend on the number of massless fields of each spin, which is a property of the low energy effective description of matter coupled to gravity.
Note that b > 0 but b < 0 for all non-gravitational free fields of spin less than or equal to one.
Since the terms in the quantum effective action of the trace anomaly, cf.  [12,15,16].
It is this semi-classical EFT, consisting of the usual Einstein-Hilbert action of classical General Relativity together with the quantum effective action generated by the conformal trace anomaly, Eqs.
(2.10) and (2.15) below, which lead to an additional massless dynamical scalar in the conformal sector, and scalar gravitational waves that propagate over macroscopic distances, that is the main subject of this paper.
The paper is organized as follows. The effective action of the conformal trace anomaly of the stress-tensor is briefly reviewed in both its non-local and equivalent local form by the introduction of an additional scalar field ϕ and the Wess-Zumino consistency condition it satisfies is given in Sec. II.

II. THE CONFORMAL ANOMALY AND LOW ENERGY EFT OF GRAVITY
The derivation of the effective action encapsulating the trace anomaly (1.2) is straightforward.
Introducing the local conformal parameterization of the metric in terms of an arbitrary fixed fiducial metricḡ µν , the conformal dependences of the terms in (1.2) are where the coordinate label x subscript on √ −g x = √ −g is generally suppressed when it causes no confusion to do so. All quantities with an overbar are evaluated in the fixed fiducial metricḡ µν . The fourth order differential operator [12,[17][18][19]] is the unique fourth order scalar kinetic operator that is conformally covariant for arbitrary σ(x). R in (2.2c) plays a special role. It is this linear combination and this combination only whose conformal variation is linear in σ and defines the differential operator ∆ 4 . This is a general feature in any even d = 2n dimensional spacetime, as there exists a particular set of local invariants which when added to the Euler-Gauss-Bonnet topological density in any even dimension defines a conformally covariant (2n) th order scalar differential operator operating linearly on σ. All other terms in the general form the conformal anomaly (1.2) are either conformally invariant as (2.2a)-(2.2b) or cohomologically trivial, as is R by itself, in the sense that they can be removed by appropriate local counterterms in the effective action.

If (1.2) is multiplied by
√ −g the result is the conformal σ variation of a Wess-Zumino effective action which gives rise to the anomalous trace. Because of (2.1) the linear σ dependence in (2.2) is a logarithmic scaling with metric distances, which makes Γ WZ a marginally relevant effective action.
This Wess-Zumino effective action is easily found by inspection of the linear relations (2.2), viz.
up to conformally invariant terms terms independent of σ, which being insensitive to rescalings do not generate infrared relevant terms. The action Γ WZ satisfies the Wess-Zumino consistency condition [20,21] By its construction the WZ action Γ WZ depends separately on σ and the arbitrary background metric g ab and is therefore not fully generally coordinate invariant. However, owing to (2.7) Γ WZ is a cohomological representation of the group of local Weyl transformations of the metric, and consequently can be expressed as a difference of fully covariant but non-local effective actions evaluated at the metric g ab (x) andḡ ab (x) [12]. This non-local difference form is obtained by solving (2.2c) formally for σ, thereby inverting the differential operator ∆ 4 , substituting the result into (2.6) and using (2.4). The resulting covariant non-local effective action is given by where D(x, x ) ≡ ∆ −1 4 (x, x ) denotes the Green's function inverse of the fourth order differential operator ∆ 4 defined by (2.3). The particular Green's function depending on appropriate boundary conditions must be selected by the physical application. In Sec. VI we solve for the retarded Green's function D R (x, x ) explicitly with classical retarded boundary conditions in flat space. Similar to the retarded Green's function for the usual second order wave operator in two dimensions, is a constant for all points x within the entire past light cone of x, and the corresponding Feynman propagator grows logarithmically at large distances. This feature allows the effects of the non-local quantum correlations induced by the conformal trace anomaly (1.2), contained in the anomaly effective action to be cumulative and non-negligible at macroscopic distance scales. Adding a R term to (1.2) with an arbitrary b = 0 would add a local R 2 term to the effective action, which affects the UV behavior but which only produces Planck suppressed effects at low energies, without altering the non-local IR contribution (2.9) of the true anomaly.
It bears emphasizing that the anomaly effective action (2.10) is completely distinct from all terms local in the curvature and its derivatives in EFT, as well as any other possible non-local but conformally invariant terms that might be generated by integrating out quantum matter fields, in both its logarithmic scaling with distance, and the special role of the ∆ 4 operator in producing a particular kinetic energy term for σ. It is this kinetic energy term that leads to an additional scalar degree of freedom in the low energy EFT of gravity, unassociated with the Planck scale, quite unlike possible additional degrees of freedom introduced by local higher derivative curvature terms. The effects of such higher local higher derivative terms have been considered in [7], and are suppressed by (L pl /r) 2 where r is a typical macroscopic scale. This need not be the case for (2.9), the effects of which are nonlocal in the invariants including L i , and which are not uniformly suppressed by the Planck scale. The effective semi-classical action (2.9) in the gravitational sector, in which b, b proportional to appear as parameters, may be regarded as an efficient bookkeeping device to take into account the anomalous vacuum polarization effects of the known light fields in the Standard Model in the gravitational sector, without having to calculate loops containing these light fields in every individual process.
To make the scalar excitation associated with the anomaly and its macroscopic consequences manifest, it is convenient to recast the generally covariant non-local effective action (2.9) in local form by the introduction of (at least one) scalar field. Because it is asymmetrical in the invariants E and (C 2 , L i ), two additional scalar fields would be necessary to render the minimal non-local action (2.9) into a local form [15]. However, one scalar is sufficient if one adds to (2.10) the Weyl invariant terms necessary to symmetrize the terms in x and x , and complete the square in (2.9) [22]. Since the anomalous effective action is determined only up to such Weyl invariant terms in any case, adding these additional Weyl invariant terms does not affect the trace anomaly or otherwise affect the conclusions of this paper.
Thus, one may consider varying the local effective action with respect to ϕ, which results in its linear eq. of motion for its one scalar field. When this eq. is solved for ϕ by formally inverting ∆ 4 and substituting the result into (2.10) the non-local effective action S N L anom [g] in (2.9) is reproduced, up to surface terms and Weyl invariant terms.
Note that unlike Γ WZ [ḡ; σ], S anom [g; ϕ] depends upon the full physical metric g µν and is therefore fully general coordinate invariant, with ϕ an additional spacetime scalar field with kinetic terms not contained in classical Einstein gravity. It is clear that ϕ is nevertheless very closely related to the conformal factor σ of the metric itself in (2.1). Indeed (2.13) which shows that conformal transformations of the metric are related to linear shifts in the spacetime scalar ϕ. For this reason and because the identity (2.13) exposes its origin and relationship to variations of the conformal frame of the metric, ϕ may be termed the scalar conformalon field, which also serves to distinguish it from dilatons and dilaton-like fields that arise in other contexts. Because of the fourth order kinetic term the scalar conformalon ϕ has canonical mass dimension zero, which also distinguishes it from other dilaton-like fields.
The method of obtaining (2.10) outlined relies upon integrating the trace anomaly eq. (1.2) with the values of the coefficients b, b , β i dependent upon the underlying QFT content. In that case the scalar conformalon ϕ does not introduce any genuinely 'new' degrees of freedom, but rather simply reexpresses in a convenient form certain two-particle correlations present in the underlying QFT vacuum [23]. For example, the Casimir effect and vacuum polarization effects in fixed geometries such as the Schwarzschild and de Sitter backgrounds may be computed for a large variety of states for fields of arbitrary spin by use of S anom and its corresponding stress tensor (3.2) [15]. On the other hand, since the form of the trace anomaly (1.2) is prescribed by locality of QFT and general coordinate invariance, the effective action (2.10) associated with it may be regarded as a necessary part of the general effective action of low energy gravity, based on dimensional scaling and invariance principles alone. From that perspective the form of (2.10) and the existence of a scalar conformalon field coupling to the spacetime metric and satisfying (2.13) is independent of any specific matter field content, and the coefficients b, b , β i may be treated as free parameters of low energy gravity, to be determined by experiment.
When (2.10) is added to the usual Einstein-Hilbert term of classical General Relativity we obtain the semi-classical effective action which defines a scalar-tensor low energy effective theory of gravity. Since ϕ(x) is a spacetime scalar field, the trace anomaly action (2.10) is invariant under general coordinate transformations, just as the classical action (2.14) is. As a metric theory that leaves unaltered the direct minimal gravitational couplings of matter through covariant derivatives, the effective action (2.15) is thus consistent with the Einstein Equivalence Principle, by admitting local freely falling Lorentz frames at every point where the non-gravitational laws of physics take on their familiar Lorentz invariant form.
Because the scalar conformalon couples to gauge fields through the β i L i terms in the trace anomaly and influences the metric and gravitational interactions through its stress tensor, cf.  [24]. In particular, the scalar conformalon ϕ does not couple to the full trace of the classical stress tensor of massive matter, as the FJBD scalar does, but only linearly to those dimension four conformal invariants with no dimensionful mass parameters, such as (2.2b), dictated by the quantum trace anomaly (1.2) and Wess-Zumino consistency (2.13), which additional couplings would violate. Since the possible sources for ϕ in (2.11) are negligibly small in our local neighborhood, this allows (2.15) to easily pass the stringent solar system tests which constrain the coupling(s) in FJBD theory, as well as the laboratory constraints on other modified gravity theories [25].
The presence of four derivatives in ∆ 4 and (2.10) raises questions about unitarity and ghosts.
These were addressed in the simple context of the pure anomaly theory (2.6) quantized on R × S 3 in [26]. In that case the negative norm ghost states are eliminated from the physical spectrum by the In [27] the full effective action (2.15) with a cosmological term was considered perturbatively in linear response around de Sitter space. There again the higher derivative mode decouples if one restricts attention to excitations far below the Planck scale, for which the EFT description is valid, but a physical scalar mode of ϕ with second order kinetic term coupled to the metric, not present in the classical Einstein theory, does survive. In this paper this analysis is carried out for the somewhat simpler case of linear perturbations around flat space, with the result that again a single second order physical scalar excitation survives at low energies, and is responsible for the new phenomenon of scalar gravitational waves. Unitarity is properly defined only in this context of small perturbations from asymptotically flat space, where asymptotic states clearly exist, and it is possible to define a unitary S-matrix. In this case the quantization of the linear scalar modes may be carried out straightfowardly, c.f. Appendix A, and no problems appear to arise.

III. STRESS-ENERGY OF THE SCALAR CONFORMALON
Variation of the last form of the anomaly action (2.10) with respect to the metric yields the stressenergy tensor of the scalar conformalon field is the metric variation of the terms proportional to b in (2.10), and are the metric variations of the last two b and β i terms in (2.10), linear in ϕ. Thus for example, for L = F µν F µν , which is proportional to ϕ and the Maxwell stress tensor.
Since the effective action is a coordinate invariant scalar, the stress-energy tensor (3.1) is covariantly for any b, b upon making use of the ϕ eq. of motion (2.11), and for any β i , if supplemented by the eqs.
of motion for any additional fields coupled through L i . It is easily verified that the terms quadratic in ϕ in (3.2), as well the terms (3.3)-(3.4) are conformally invariant and traceless, so that the total trace of the anomaly stress tensor is obtained from the terms in (3.2) linear in ϕ, yielding which reproduces (1.2), by again using the classical eq. of motion (2.11). This relation may also be derived as a direct consequence of Wess-Zumino consistency (2.13).
The stress tensor (3.1) appears as an additional source to the Einstein field equations obtained by varying the full gravitational effective action (2.15) with respect to the metric, which thus become in the EFT. Here T around the vacuum solution of flat spacetime with η µν = diag(− + ++) and ϕ = 0 is where surface terms have been discarded. The last term in (4.2) shows that there is a mixing between the trace of the metric perturbation h µν and the scalar conformalon ϕ. Note also that this mixing of ϕ to the metric perturbation is only through ϕ. Thus any scalar perturbations with ϕ = 0 are completely decoupled from the metric perturbations at linear order.
One can analyze the coupling between the anomaly scalar and the linearized metric perturbation in either a covariant or a canonical manner. In a covariant treatment space and time indices are treated on an equal footing, so that Lorentz invariance is manifest, whereas in a canonical treatment spacetime is split into space + time, and true propagating modes containing kinetic terms may be more clearly separated from constrained modes.
Considering first the spacetime covariant decomposition of the metric perturbation where u ⊥ µν and v ⊥ µ are transverse and u ⊥ µν is traceless in the four-dimensional covariant sense, i.e.
then under the infinitesimal coordinate transformation where ξ µ = ξ ⊥ µ + ∇ µ ζ is also decomposed into transverse and longitudinal components in the same covariant four-dimensional sense. Thus, of the ten independent metric components of the linearized metric perturbation h µν only the subset of depending only upon the gauge invariant tensor and scalar parts of the linearized metric perturbations with T (ϕ) µν f lat = b E µν f lat and C µν f lat = 0 (4.10) in vacuo when no other sources are present. Note that (4.9) contains terms both linear and quadratic in ϕ. The terms linear in ϕ and the scalar part of (4.7) give or in component form, to linear order in h µν and ϕ, with the spin-2 tensor part δG (T ) µν dependent upon u ⊥ µν unaffected. The general solution of the constraints (4.12) in the scalar sector is where F (t) is a function only of time and K(x) is a function only of space, with zero spatial Laplacian.
Since the only solution of ∇ 2 K = 0 which is non-singular everywhere, including spatial infinity, is K(x) = const., it may be absorbed into F (t). Thus we may effectively set K = 0 in (4.13).
In the pure Einstein theory (i.e. b = 0) (4.13) implies that the gauge invariant scalar metric perturbation h − w = 8F (t) contains no local propagating degree of freedom, in the same way that the longitudinal part of the electric field is locally constrained by Gauss' Law ∇ · E = ρ and is non-propagating in pure electromagnetism (with ρ = 0). In that case as well the longitudinal electric field can contain at most a spatially homogeneous mode that is a function only of time: As is well known in finite temperature field theory, plasmas or generally whenever there is a polarizable fluctuating medium, the fluctuations of the charges in the medium induce through Gauss' law a collective plasmon mode described by a propagating longitudinal electric field [28]. Thus the longitudinal mode of the electric field, completely constrained and non-propagating with fixed sources, becomes propagating in the presence of dynamical charged sources, inheriting the collective dynamics of the sources.
An analogous phenomenon can occur due to quantum fluctuations and anomaly of a charged field in the vacuum, as evidenced for example by the Schwinger model of massless QED in one 1+1 dimensions.
Integrating out of the massless fermion field leads to a propagating massive scalar boson equivalent to the longitudinal electric field, classically constrained, which becomes propagating through Gauss' Law and the quantum axial anomaly [23]. In either case the collective mode arises from two-particle correlations of the fluctuating source field (either thermal or quantum).
Eq. with non-trivial propagating scalar gravitational wave solutions. In the next subsection a (noncovariant) space + time splitting of the metric perturbations, appropriate for identifying the unconstrained propagating local scalar degree of freedom in a canonical framework is given, in order to verify this conclusion.

B. Space + Time Decomposition and Constraints
In order to properly characterize the dynamical propagating degrees of freedom in the EFT of gravity vs. the constrained modes, we introduce next the non-covariant space + time splitting of the linearized metric perturbations in the standard Hodge decomposition adapted from that commonly employed in spatially homogeneous, isotropic cosmological models [27,29], with η ij = δ ij in Cartesian coordinates of flat R 3 . The components A, B, C, D are four scalars with respect to the flat spatial metric on R 3 , the 3-vectors B ⊥ i , E ⊥ i are transverse, satisfying with resulting two independent components each, and H ⊥ ij = H ⊥ ji is a symmetric, transverse, traceless tensor (now in the spatial three-dimensional sense) satisfying resulting in two independent polarization components.
In this space + time splitting the four independent linearized coordinate transformations can be similarly decomposed with X ⊥ i a transverse vector satisfying ∇ i X ⊥ i = 0. The linearized coordinate gauge transformation induces the linearized changes in the metric components provided the spatial gradients of B, D, T and L are all non-vanishing. Thus are invariant under linearized coordinate transformations, whenever the spatial dependences of these functions are non-vanishing. In the special case of metric perturbations independent of the spatial coordinate x, B, D and L may be set to zero. Then only Υ C = C is gauge invariant, while Υ A = A is still subject to the gauge transformation Υ A → .
T , with T (t) independent of the spatial coordinate.
The non-zero components of the Riemann tensor can be expressed in the form .
from which follow the linearized variation of the Ricci tensor and Ricci scalar ..
and the linearized Einstein tensor all expressed in terms of the six gauge invariant components (4.21). From (4.24a) it is clear that the tt component of the Einstein's eqs., G tt = 8πG T tt is for small perturbations about flat space. Since δG ij → 0 in the non-relativistic limit for slowly moving weak sources, Υ A = −Υ C from (4.24c) becomes the Newtonian potential for quasi-static weak sources.
On the other hand for relativistic sources, δG ij = 0 and Υ A = −Υ C in general.

The quadratic effective action S
(2) ef f around flat space in the decomposition (4.16) takes the form while the linearized Einstein equations become since the linearly independent components in the tensor, vector and scalar sectors must be satisfied separately. Note that although the variation of the scalar effective action together with (4.14) show the non-trivial coupling of the anomaly scalar into the scalar sector of metric perturbations in the space + time splitting.
Analogously to the covariant analysis in the previous subsection, the constraint eqs. (4.27c) and (4.27d) imply with F some function only of time. When (4.27c) is added to one half the trace of (4.27e) we obtain .. which is sourcefree in vacuo. Finally (4.29) together with (4.28) and (4.30) imply that .. . For completeness one can also verify the relations between the covariant metric decomposition (4.3) and 3 + 1 decomposition (4.16) is which shows that the scalar wave solution (4.33) with Υ A = Υ C and Ψ ⊥ i = 0 leaves the transverse, traceless gravitational wave sector of the classical Einstein theory, with its 2 gauge invariant propagating polarizations u ⊥ ij = H ⊥ ij , unaffected.

V. HAMILTONIAN AND ENERGY
Substituting the (on-shell) solution of the linearized eqs. into the quadratic effective action (4.26) gives simply allowing us to concentrate on the conformalon scalar sector, with the scalar metric potentials deter- with ω k ≡ |k|. Since the first pair of solutions obey ϕ = 0, only the second set, with ϕ ∼ e ±iω k t e ik·x = 0 couple to and determine the metric perturbations through (4.33).
In order to characterize the canonical structure of this higher derivative theory, we follow Ostragradsky's method and define a new variable ψ = .
ϕ treated as independent, together with its independent canonical momentum π ψ by ψ ≡ .
is the field momentum conjugate to ϕ itself. The canonical Hamiltonian corresponding to (5.1) is then easily found to be where E tt is the tt component of the covariantly conserved stress-energy tensor of the scalar conformalon field T It is straightforward to check that the two Hamiltonian eqs.
reproduce the equation of motion (4.30).
Normalizing the solutions (5.2) in a cubical box of volume V , the general ϕ field in flat space can be expressed as the linear superposition where the A k , B k are dimensionless complex Fourier coefficients. Note that unlike a canonically normalized scalar field with a second order kinetic term, the conformalon scalar ϕ field is dimensionless.
The reason for choosing the normalizations in (5.8) will become apparent shortly. Computing ..
and substituting these expressions into (5.5) gives for the conserved Hamiltonian.
As expected, the quadratic energy of the scalar perturbations is not positive definite, owing to the second Re (A k B * k ) or −|B k | 2 terms in (5.10), and the energy may have have either sign. Indeed H is unbounded from below for |B k | → ∞ and Re (A k B * k ) < 0. However the physical scalar metric perturbations depend only upon ϕ through (4.33), and (5.9c) shows that ϕ depends only upon the A k modes. Explicitly so that the possible negative energy harmonic B k modes of the ϕ field are decoupled from the scalar wave metric perturbations at linear order. Thus the B k modes may be treated as a fixed passive background field in flat spacetime in vacuo. There is no instability of flat space due to one-loop quantum effects in the semi-classical EFT [30]. At higher orders the non-linear corrections to Einstein's eqs. in curved space must be taken into account, while the scalar conformalon sector remains linear, and any instability is related to the quantum vacuum state of the specific curved spacetime in question.
For arbitrary but fixed B k , it is clear from the second form of (5.10) that for each k mode the energy attains a minimum value of −ω k |B k | 2 /2 when A k = −B k /2, for b < 0. Thus for fixed passive background B k in flat spacetime, the energy (5.10) is bounded from below, by the negative finite value of H min for b < 0, provided the |B k | 2 approach zero as |k| → ∞ fast enough for the sum (integral) over k in (5.12) to converge. It is noteworthy that the b < 0 condition is exactly the one satisfied by the b topological anomaly coefficient of all known free non-gravitational fields of any spin, cf. (1.4b).

VI. SCALAR GRAVITATIONAL WAVES FROM LOCALIZED SOURCES
Having found that the semi-classical EFT of gravity admits scalar gravitational wave solutions in vacuo for any b < 0, which are not present in classical GR, in this section we consider sources for these scalar waves, arising from the anomaly stress tensor, and solve the linearized equations due to these sources of scalar gravitational waves.
In flat spacetime the eq. of motion (2.11) for the anomaly scalar field ϕ is the source, which is to be treated as a weak perturbation in the nearly flat space limit. The factor of 8π is inserted in (6.1) for later convenience. This linear equation can be solved in terms of the classical retarded Green's function ( and which is represented by the Fourier integral with the infinitesimal > 0 prescription enforcing retarded boundary conditions. Since the second order poles of the integrand at ω = ±|k| are displaced into the lower half complex by this prescription, the ω contour may be closed in the upper half complex plane for t − t < 0, giving zero, while for t − t > 0 the ω contour may be closed in the lower half complex plane, yielding the result which vanishes if t ≤ t , and where eqs. 3.741 (2) and (3) of [31] and have been used.
Thus whereas the usual second order wave operator has a retarded Green's function with support only on the past light cone in 3 + 1 spacetime dimensions, the fourth order operator 2 has a retarded Green's function (6.5) with uniformly constant support everywhere within the past light cone. This is similar to the retarded Green's function for the second order wave operator but in 1 + 1 spacetime dimensions, and results in pronounced long distance infrared behavior of the conformalon scalar ϕ. Indeed the solution of (6.1) is which does not fall off with large |x−x |, and which can even become large without bound for persistent sources. Inspection of (3.2) shows that ϕ appears only under derivatives, so that these largest infrared effects are removed for localized sources.
Relabeling the point of observation x → r, and the integration variable over the spatial extent of the source x → x, the first derivatives of (6.7) are .
is the retarded time, the spatial derivatives being taken with respect to r. The second derivatives of (6.7) are ..
from which follows Thus from (4.13) the gauge invariant scalar metric perturbation propagated from a distant localized source is in the far or radiation zone where r ≡ |r| |x|.
For time harmonic sources, where the Real Part is understood, we have in the far radiation zone, and therefore from (6.12), which has the form of an outgoing spherical scalar gravitational wave.
This result is to be compared with the linearized metric perturbation of transverse, traceless gravitational waves in the radiation zone, where Π lm ij (r) = 1 2 δ l i δ m j + δ m i δ l j +r irjr lrm − η ij η lm + η ijr lrm + η lmr irj −δ l ir jr m −δ m jr ir l (6.17) is the projector onto transverse, traceless tensors. For a time harmonic source in the quadrupole approximation, where the exponential factor in the integral in (6.18) is replaced by unity, for slow moving non-relativistic sources, and where is the Fourier component of the source quadrupole moment.
Thus whereas the source for electromagnetic radiation is the time varying current transverse to the line of sight, whose first non-vanishing multipole is the dipole term, and the source for transverse, traceless gravitational radiation in classical GR is the time varying transverse stress-energy whose first non-vanishing multipole is the quadrupole term, the source for the scalar gravitational wave in (6.12)-(6.15) is the trace anomaly scalar conformalon, apparently independently of its time variation. The scalar gravitational wave receives a contribution from the monopole source term even for arbitrarily slowly varying anomaly sources independent of the directionr in the expansion of the exponent in (6.15). However, due to the decoupling of the anomaly gauge field sources when ω k < m e (QED) or ω k < m u,d (QCD) an effective threshold for space and time variation of these sources is effectively introduced for scalar monopole radiation as well.
The linearized perturbation of the Riemann tensor corresponding to the scalar gravitational wave (6.15) may be computed with the help of (4.7a) to be in the far region where the terms dropped by neglecting the effect of the derivatives upon 1/r fall off faster than 1/r as r → ∞. In this region because of the dependence of the integrand on the retarded timet = t − |r − x|, one can also make the replacement ∇ i → −r i ∂ t . From (6.20) and the methods of Appendix B the components of the Riemann tensor perturbation then become . . .
δR (A) = 0 (6.23d) and for time harmonic sources (6.13) in the radiation zone we may make the replacement .
From eqs. (4.23c) the tensor perturbations have vanishing linearized Ricci tensor and Ricci scalar, but non-vanishing linearized Riemann and Weyl tensor, befitting a spin-2 tensor gravitational wave.
Otherwise the amplitude of the scalar gravitational wave generated by the trace anomaly effective action is of similar form as that of the transverse, traceless waves generated in the classical Einstein theory, falling off as 1/r from a distant source, with the anomaly A playing the role of the transverse, traceless stress tensor T ⊥ ij .

VII. ENERGY FLUX AND POWER RADIATED IN SCALAR GRAVITATIONAL WAVES
The terms linear in ϕ in the conformalon stress-energy tensor have already been considered in the linearization around flat space. At quadratic order in ϕ the terms in the energy flux in (4.9) are with the sums over j understood. Thus in order to calculate the energy flux of the scalar graviational waves both the terms with one and two derivatives of the scalar conformalon far field given by (6.8)-(6.11) are required, as well as the terms .
containing three derivatives of the anomaly conformalon scalar. All of the terms can be evaluated in the far or radiation zone r → ∞ in terms of the integrals over the source distribution. The details of the calculation of the various terms in (7.1) in terms of the moments (7.3) are given in Appendix B, with the final result for the power radiated into scalar gravitational waves in the radial directionr per unit solid angle.
Note that for b < 0 the first (monopole) term in (7.4) is positive definite, and has no time derivatives. The further tems in (7.4) involving time derivatives are of order ωa or (ωa) 2 relative to the first term, where a is a maximum size of the localized source. For non-relativistic sources ωa 1 whereas even for ultra-relativistic sources ωa cannot exceed unity, since by causality the timescale of temporal variation of the source cannot be less than the light travel time across its extent. The additional terms involving powers of ωa in (7.4) arise even prior to any multipole expansion of the source because of the more complicated form of the conformalon stress tensor and flux (7.1).
Because of the monopole term in (7.4), with no time derivatives, there are no powers of ωa in the leading multipole term and the power radiated is of order for a time harmonic anomaly source. There is no explicit power of the gravitational constant G in (7.4), these potentially arising only through the curvature invariants in the anomaly being generated themselves by matter sources in Einstein's equations. In the next section estimates of the scalar gravitational wave amplitude and power radiated for possible astrophysical sources are considered.

VIII. ASTROPHYSICAL SOURCES OF SCALAR GRAVITATIONAL WAVES
The typical curvature invariant in the vicinity of a completey collapsed star is of order at the star's Schwarzschild radius. This corresponds to a very small energy density of In the QED case, highly magnetized neutron stars ('magnetars') are believed to have magnetic fields up to 10 15 Gauss, which exceeds the electrodynamic critical field of B c 4 × 10 13 Gauss corresponding to 2m e c 2 [32]. Taking into account the coefficient of the trace anomaly in QED, the magnetar field provides a source of scalar gravitational waves of strength Since a typical neutron star radius is 12 km, the maximum volume over which this energy density applies is of order 7×10 18 cm 3 , giving a total magnetic field energy of order of 6×10 45 ergs. The scalar gravitational wave produced at a distance r from such a highly magnetized source has magnitude where the factors of c have been reinserted. The estimate (8.5) is many orders of magnitude greater than (8.3) but still several orders of magnitude below the sensitivity of present gravitational wave detectors. When the electron mass is not neglected, because of decoupling of the triangle diagram responsible for the QED trace anomaly, A mag is suppressed by a factor of order ( ω/m e c 2 ) 2 where ω is a characteristic frequency of time variation of the electromagnetic field strength [10]. Thus (8.5), small as it is, will be suppressed further as a source of scalar gravitational radiation for more slowly varying magnetic fields, for weaker fields, or for strong fields exceeding the critical field extending over smaller volumes.
This suppression due to decoupling of the anomaly at low frequencies is also the reason why the power radiated in scalar gravitational waves is negligible compared to classical quadrupolar radiation in binary pulsar systems, such as the Hulse-Taylor binary and double pulsar. For the double pulsar the orbital period of 2.454 hours gives an angular frequency ω 7.11 × 10 −4 sec [33]. This gives a suppression factor in the anomaly source A of ( ω/m e c 2 ) 2 2.10 × 10 −49 , which enters the scalar power radiated (7.5) squared again. Taking all the factors in (7.5) into account gives a total power radiated of order where a magnetic field of 2 × 10 12 Gauss was assumed and the value of b from (1.4b) was used with N v = 1 for the electromagnetic field. In principle the energy radiated in scalar gravitational waves will cause the orbit of the double pulsar to decay and its orbital period to decrease, above the decrease predicted due to quadrupolar radiation in Einstein's classical theory. However the comparable estimate for classical transverse tensor gravitational radiation from the same double pulsar system is of order 4 × 10 32 erg/sec. Thus the effect of the additional scalar radiation is smaller than one part in 10 20 in this system, and completely negligible. The effect in the Hulse-Taylor binary pulsar is even smaller. Although a more accurate analysis would be worthwhile, these rough estimates indicate that the anomaly EFT will easily pass all existing pulsar tests from electromagnetic sources The most promising source for generating scalar gravitational waves in significant and detectable quantities is the SU (3) color trace anomaly for N c = 3 colors and N f = 2 light fermion species, where the last value is that of an effective bag 'constant' of ρ bag = −p bag 750 Mev/fm 3 in nuclear matter [34]. The chemical potential dependence of the bag 'constant' of dense nuclear matter has been estimated, with values of the QCD trace anomaly similar to or even higher than (8.7) possible at baryon chemical potentials of µ 1.6 GeV thought to exist in neutron star cores [35]. This energy density is more than 10 orders of magnitude larger than the QED value in a strong magnetar field (8.4), and thus capable in principle of producing scalar gravitational waves 10 orders of magnitude larger than (8.5), well within the detectable range.
As with the QED source, a caveat for this estimate is that the QCD gauge field anomaly pro- The most vigorous disturbance of the QCD gluonic vacuum which can excite this astrophysical source of the conformal anomaly is in the initial formation of the neutron star (NS), or its collision, coalescence and merger with another compact stellar object. In such processes density estimates such as (8.7) may be applicable. Thus the scalar gravitational wave amplitude from neutron star formation or binary coalescence is estimated to be where f < 1 is volume fraction of the neutron star in which the gluonic condensate is excited by high energy interactions above the quark mass threshold (> 10 MeV) in the formation or coalescence event.
The estimate (8.8) indicates that scalar gravitational radiation produced by neutron star formation or merger events as far as 100 Mpc away or greater could be detectable by aLIGO or the next generation of GW detectors. Clearly f is very uncertain and could be orders of magnitude less than unity. Moreover, the frequency range in which the scalar gravitational waves may be emitted may not lie within the frequency sensitivity bands of present or near future GW detectors. Clearly more careful estimates are needed, and will require detailed modeling of the time dependence of the nuclear constituents and gluonic condensate in realistic neutron star formation and merger events. Still, the signal generated by a time dependent QCD anomaly by nuclear matter in a NS formation or merger event is almost certainly the strongest astrophysical source of scalar gravitational waves from the trace anomaly, potentially detectable by present or planned gravitational wave detectors, certainly meriting such a detailed study. If NS binary coalescence events are the sources of detectable scalar gravitational radiation, this emission should follow closely upon transverse, traceless GW emission from binary inspiral, suggesting the time co-incidence study of production of both GW polarizarization states by the same event(s) may be a promising observational strategy. Conversely, non-detection of scalar gravitational waves from NS or black hole candidate coalescence events could provide potentially interesting constraints on nuclear equations of state and/or the mechanisms of gravitational wave generation by the QCD trace anomaly.

IX. SUMMARY AND CONCLUSIONS
In this paper the extension and modification of Einstein's theory of classical General Relativity by the infrared relevant effects of the conformal trace anomaly have been described. The effective action (2.15) with an additional scalar field due to the quantum effects of the anomaly defines the low energy EFT of gravity consistent with general quantum principles. The scalar field ϕ is called a conformalon because it is closely related to and mixes with the conformal mode of the usual spacetime metric, making it dynamical. Although reminiscent of the scalar-tensor theory introduced by Fierz, Jordan, Brans and Dicke [24], it is quite different in several important respects, foremost among them that it is required by known quantum effects, and its form is dictated by that of the conformal anomaly in QFT. As such the EFT presented here does not fall into the class of scalar-tensor theories or effective field theories of scalars or modified gravity theories usually considered [25]. Most notably, the scalar ϕ does not couple directly to the trace of the matter stress tensor, but only to higher order curvature invariants E, C 2 of (1.3), or the gauge field scalars such as F µν F µν , or G a µν G a µν . This feature of the scalar conformalon most clearly distinguishing it from other possible scalar dilaton-like degrees of freedom or other modified gravity theories is a result of the Wess-Zumino consistency relation (2.13) derived from the conformal anomaly (2.6)-(2.7), which strictly constrains the form of the effective action.
The eqs. of the resulting semi-classical EFT are Einstein's eqs. Linearized around flat spacetime with ϕ = 0, the EFT describes scalar gravitational waves, in addition to the transverse, traceless waves of the linearized Einstein theory. Although the anomaly EFT has a fourth order differential operator and ϕ solves 2 ϕ = 0, only the half of these solutions with ϕ = 0 couples to the gauge invariant scalar gravitational metric perturbations through (4.33).
The other half of the solutions to (2.11) satisfying ϕ = 0 are a passive fixed background which are not coupled to the metric and do not contribute to scalar gravitational radiation at linearized order. The higher derivative canonical system of eqs. for the conformalon scalar may be treated by the Ostragradsky method, and the Hamiltonian (5.10) quadratic in ϕ may be of either sign, as vacuum energy may be. Its value is bounded from below, provided the anomaly coefficient b < 0. The quantization of these modes is considered in Appendix A, although this is not necessary for the scalar gravitational wave sector, which may be treated as a c-number. The effects of the EFT at non-linear order and in other backgrounds has been previously considered in [15] and [27].
When localized sources for the scalar gravitational waves are considered, the scalar metric perturbations have a monopole form, falling off with distance from the source as 1/r according to (6.12), or Although all possible terms in the trace anomaly A can act as sources for the scalar conformalon field and therefore scalar gravitational radiation, the higher derivative curvature sources are far too small to contribute appreciably. The electromagnetic trace anomaly of QED is much larger but still very likely too weak to produce observable scalar GWs or indirect effects, such as energy loss from even the most highly magnetized objects known. Thus the EFT associated with the conformal anomaly easily passes all present observational tests from the binary and double pulsar systems.
The most promising non-negligible sources of scalar gravitational waves in the EFT are those due to the QCD trace anomaly, excited in neutron star formation, or binary coalescence with another NS or other collapsed star. In such systems the rough preliminary estimate (8.8) indicates that scalar gravitational radiation may be produced with large enough amplitude to be observed by present and future planned GW detectors as a burst event. The power radiated in scalar gravitational waves from the QCD anomaly is potentially many orders of magnitude larger than (8.6) from strong magnetic sources, but as proportional to a factor f 2 < 1 subject to considerable uncertainty. A more careful quantitative calculation of the sources of scalar GWs in NS merger events is clearly warranted, with the development of detailed models of the scalar waveforms expected in the EFT of gravity as now becoming available in standard GR [37]. Even at this preliminary stage an open search for scalar 'breather mode' gravitational waves by present and future detectors is indicated [38], with a search strategy for burst events coincident with inspirals producing transverse, traceless GWs seeming to be the most promising approach.
Energy densities as large as (8.7) from strong interactions and the QCD trace anomaly also imply that the non-linear effects on the geometry of the anomaly stress-energy tensor, neglected in the present analysis around flat space, should be taken into account in a fully consistent treatment. These nonlinear effects of the anomaly stress-energy may be comparable in importance to classical GR effects in the final stages of NS inspiral and coalescence with a second compact object, either itself another NS or a black hole/gravastar candidate [36]. The gravastar alternative to black holes may also be a source of scalar radiation through its interior scalar condensate being partly composed of and coupled to gluonic degrees of freedom in QCD.
In the early universe when temperatures and energies greater than 10 MeV were reached, the QCD anomaly was fully unsuppressed. Thus these epochs up to the QCD phase transition, or at higher energies he electroweak phase transition are possible sources of a cosmological stochastic background of scalar gravitational radiation, albeit at much lower frequency scales, potentially detectable by the next generation of space-based gravitational wave detectors.
forward to quantize the field by treating the coefficients A k , B k and their complex conjugates in the With the commutation relations (A5) in hand it is straightforward then to compute the unequal time commutator function and show that the classical retarded Green's function D R of the fourth order operator ∆ 4 in (6.5) may be written in terms of it as which is also a c-number.
The fact that the negative metric norm states in the B k sector do not appear in the physical Hilbert space satisfying the constraints of diffeomorphism invariance linearized around flat spacetime is consistent with earlier studies of the fourth order anomaly action, stress tensor and constraints of diffeomorphism invariance on R × S 3 and linearized around de Sitter space [26,27]. .
with the ellipsis representing terms that fall off as 1/r 3 or faster for large r. Collecting all the terms in (B3), we see that the 1/r 0 and 1/r 1 terms cancel identically, while the remaining 1/r 2 terms are strictly proportional to the radius unit vectorr i . Thus the energy flux is directed radially from the localized source, and this radial component is given by as r → ∞ in the far or radiation zone. This leads to (7.4) of the text for the differential power radiated in scalar gravitational waves. Note that the first term in (B4) involving I 2 and no time derivatives arises only in the second term (B3a), − . ϕ( ∇ i ϕ) of (7.1).