Inflationary cosmology from quantum Conformal Gravity

We analyze the functional integral for quantum Conformal Gravity and show that with the help of a Hubbard-Stratonovich transformation, the action can be broken into a local quadratic-curvature theory coupled to a scalar field. A one-loop effective action calculation reveals that strong fluctuations of the metric field are capable of spontaneously generating a dimensionally transmuted parameter which in the weak-field sector of the broken phase induces a Starobinsky-type f(R)-model with a gravi-cosmological constant. A resulting non-trivial relation between Starobinsky'sparameter and the cosmological constant is highlighted and implications for cosmic inflation are briefly discussed and compared with recent PLANCK and BICEP2 data.

Introduction -The idea that Einsten's gravity may be considered as a large-distance effective theory arising from a spontaneous or dynamical symmetry breakdown in some underlying scale invariant quantum field theory dates back to works of Minkowski [1], Smolin [2], Adler [3], Zee [4], Spokoiny [5], Kleinert and Schmidt [6], and others (see, e.g., Ref. [7] for recent review), even though the incentives can be traced back to 1960's seminal papers of Zeldovich [8] and Sakharov [9]. Ensuing mechanisms for symmetry breaking are realized typically by spontaneously breaking a scale invariance in appropriate scale-invariant quantum field theory propagating in a curved spacetime [3] or by a Conformal Gravity (CG) which is dynamically broken via additional scalar fields [10,11].
In particular, the CG has recently attracted renewed attention because local conformal invariance seems to be the key component in number of observationally supported cosmological models. This activity was substantially fueled by Mannheim et al., no-ghosts result [12][13][14][15], Smilga's benign-ghost result [16], and related works on conformal anomaly [17]. The CG has been since revisited from various points of view, e.g., as an alternative to standard Einstein gravity giving a (partial) resolution of a flatness problem [18], or as an explanatory frame for missing matter in galaxies [19] and a possibly vanishing cosmological constant [20]. The CG has also been explored recently in number of theoretical and observational frameworks including conformal supergravity [21], Twistor-String theory [22], asymptotic safety theories [23,24], black-hole complementarity issue [25], AdS/CFT correspondence [26], and the type Ia supernova (SNIa) and H(z) observational data [27].
Unfortunately, the particle-spectrum of CG does not contain (at least not on-shell) a scalar field. In fact, CG has 6 (on-shell) propagating degrees of freedom; massless spin-2 graviton, massless spin-1 vector boson and massless spin-2 ghost field [21,28]. Should the Einstein gravity be induced within CG at low energies, the absence of a fundamental scalar poses two imminent problems: a) it is difficult to break a conformal symmetry (either spontaneously or dynamically) without a fundamental spinless boson [21], b) scalar degree of freedom is of a central importance to generate correct primordial density perturbations during inflation [7]. For these reasons an external scalar field is sometimes artificially coupled to CG [10,11]. In this Letter we wish to point out a subtle fact that a non-dynamical spurion scalar field can be introduced to CG via the Hubbard-Stratonovich transformation without spoiling particle spectrum, (nonperturbative) unitarity, and renormalizability of the CG. The spurion field is actually an imprint of a scalar degree of freedom that would normally be present in the theory if the (local) conformal symmetry would not decouple it from the on-shell spectrum. The spurion field morphs into a physical scalar field (scalaron or gravi-scalar) after its kinetic term gets generated radiatively. The field then mediates a dynamical breakdown of the conformal symmetry. In the broken phase the scalaron field acquires a non-trivial vacuum expectation value (VEV) via dimensional transmutation. Resulting low-energy behavior in the broken phase can be identified with Starobinsky's f (R)-model (SM) with a gravi-cosmological constant or, in a dual picture, with a two-field hybrid inflationary model. A scalaron field helps to form (composite) inflaton, and assists in inflaton decay in the reheating phase.
Quantum Conformal Gravity -CG is a pure metric theory that possesses general coordinate invariance, which augments standard gravity with the additional Weyl symmetry, i.e., invariance under a local rescaling of the metric g µν (x) → e 2α(x) g µν (x), with α(x) being an arbitrary local function. The simplest CG action, i.e., action with both reparametrization and Weyl invariance reads [29,30] Here α c is a dimensionless coupling constant and C λµνκ is the Weyl tensor which in 4 space-time dimensions reads with R λµνκ being the Riemann curvature tensor, R λν = R λµν µ the Ricci tensor, and R ≡ R µ µ the scalar curvature. Throughout we adopt signature (+, −, −, −) and sign conventions of Landau-Lifshitz. With the help of the Gauss-Bonnet theorem one can cast A conf into equivalent form (modulo topological term) Variation of A conf with respect to the metric yields Bach's field equation [30] 2D where B µν is the Bach tensor and D α the Riemannian covariant derivative. We formally define a quantum field theory of gravity by a functional integral ( = c = 1) Here Dg µν denotes the functional-integral measure whose proper treatment involves the Faddee-Popov gauge fixing of the gauge symmetry Diff×Weyl(Σ i ) plus ensuing Fadeev-Popov determinant [31]. Potential local factors [− det g µν (x)] ω with Misner's (ω = −5/2) or De Witt's (ω = (D − 4)(D + 1)/8) are omitted in the measure because they do not contribute to the Feynman rules. Their effect is to introduce terms ωδ (4) (0) dx 4 log(−g) into the action, which by Veltman's rule are set to zero in dimensional regularization. The sum in (5) is a sum over four-topologies, that is, a sum over topologically distinct manifolds Σ i (analogue to the sum over genus in string theory or sum over homotopically inequivalent vacua in the Yang-Mill theory) which can potentially contain topological phase factors, e.g., Euler number of Σ i , cf. Refs. [32,33]. It should be remarked that despite a fourth-order nature of the Bach equation (4) descending from A conf , it has been recently shown that the would-be ghost states [34] disappear from the energy eigenspectrum and that CG is stable (i.e., unitary) [12,13]. Also, the conformal instability typical for the Euclidean quantum gravity is not presents in CG.
Uncompleting the R 2 -term -Here we wish to point out that the large number of derivatives in the free graviton propagator implied by (3) makes fluctuations so violent so that the theory might spontaneously create a new mass term. This phenomenon is indeed known to happen in number of higher-derivative systems ranging from biomembranes [36] through string theories with extrinsic curvature [37,38], to gravity-like theories [39]. For instance, in biomembranes and stiff strings the ensuing mass term can be identified with a tension. We shall now show that an analogous mechanism spontaneously generates the Starobinsky action [40] Here κ 2 = 8πG N where G N = 1/m 2 p is Newton's (gravitational) constant and m p is the Planck mass. Starobinsky's parameter ξ is related to the inflational scale and by the Planck satellite data ξ/κ ∼ 10 5 (cf. Ref. [41]). The minus sign in front of R 2 -term is a consequence of the Landau-Lifshitz convention [42].
In order to see how the spontaneous generation of (6) comes about we first observe that the R 2 -part of the action (3) is the global scale-invariant expression. This is because under infinitesimal Weyl transformation Since g → (1 + 8α(x))g, the R 2 -term part of the action will be scale invariant provide D 2 α(x) = 0. The R 2 -part of the action can be decomposed by using the Hubbard-Stratonovich (HS) transformation [43][44][45] Although an auxiliary field λ(x) in (7) does not have a bare kinetic term, the conformal symmetry allows to rescale the metric so that a kinetic term can easily be generated. For instance, when g µν → |λ| −1 g µν then A gsi goes to (and other higher-order derivatives of λ will come from the remaining R µν R µν -term). Since the λ-kinetic term depends on the conformal scaling λ-kinematics is gauge dependent, implying that λ cannot represent a physical field. On the other hand, when the conformal symmetry breaks down then the λ-field is trapped in a particular (broken) phase with specific kinetic and potential terms. This will be seen shortly.
To proceed, it is helpful to separate the λ-field into a background fieldλ corresponding to the VEV of λ and fluctuations δλ which have only nonzero momenta. Of course, the fluctuations must be included to make the theory completely equivalent to the original (5). In the following we employ the standard effective-action strategy, i.e., neglect all terms involving δλ, and take the saddle-point approximation to the remaining integral overλ.
As will be seen shortly, λ spontaneously develops a positive VEV, so that the sign of the R-term in (7) coincides with the sign of the Einstein term. With the benefit of hindsight we introduce an arbitrary mixing angle θ and write formally Applying the HS-transformation only to the (S 2 A gsi )-part we get, after a formal replace- Here the additional rescaling λ → λ/κ 2 was included because we want our theory to eventually induce Einstein's action after λ(x) acquires a VEV.
Let us now show that the fluctuations of the metric g µν can achieve this. In particular, we find a set of parameters in the model parameter space for whichλ = λ = 1. As a result, the long-range behavior of our theory will coincide with that of Starobinsky's f (R)-model.
Emergence of Starobinsky's model -We proceed by splitting the spacetime metric into the flat Minkowski background plus a fluctuation h µν defined by g µν = η µν + α c h µν (realizing that α c ∼ Cκ/ξ), and then expanding the Lagrangian in (3) (including the explicit form (9)) to the second order in α c . Omitting total derivatives, using the simple weak-field relations of Appendix A and setting λ =λ, we end up with the following outcome where A phenomenologically consistent long-range behavior of the gravitational field is ensured ifλ = 1. To see that such a solution exists at low enough energies we calculate the one-loop contribution to the Minkowski effective action. This is obtained by functionally integrating out the fields h µν in the exponential e iA conf in which λ is approximated by its VEV, i.e.,λ. The result is e −iΩ4V eff , where Ω 4 is the total four-volume of the universe, and V eff is the effective potential. Form (10) is particularly convenient for the gauge fixings [10,21]: Here ζ ν (x) and ζ(x) are arbitrary functions of x. Using 't Hooft's averaging trick [46]: (H is an arbitrary symmetric operator) and doing some straightforward computations we obtain the zero-genus (fixed topology) contribution to partition function ((M FP ) µν = − η µν − ∂ µ ∂ ν is the Faddeev-Popov operator for coordinate gauge [47]). The factor {[det(− )] −1/2 } 6 correctly indicates that that number of propagating modes in the linearized CG is 6 (cf. Ref. [28]). From (12) the one-loop V eff reads The prime indicates a trivial subtraction of the zeromode. Note that for (assumed)λ > 0 the ensuing massive pole is physical only when θ ∈ (−arcsinh(1/4), ∞).
The integral over k can be evaluated, e.g., with the help of dimensional regularization (D = 4 − 2ǫ) in which case it yields where Λ = √ 4πµe −γ/2 e 1/2ǫ , µ is an arbitrary renormalization scale and γ is the Euler-Mascheroni constant. To obtain a finite result as ǫ → 0 we utilize the MS renormalization scheme. This fixes the counterterm so that with µ 2 being the subtraction point. The saddle point inλ corresponding to the VEV is determined by the vanishing of Vλ ≡ ∂V eff /∂λ. This yields the minimal V eff for In this case V eff < 0 for S 2 > ( √ 6 − 2)/8 ≈ 0.056, irrespective of actual values of α and κ. A trivial solution of Vλ = 0, namelyλ(S) = 0 yields V eff = 0 and hence it represents a local maximum (i.e., unstable solution) for the above range of S 2 .
Although the full theory described by the action (10) is independent of the mixing angle θ, the truncation of the perturbation series after a finite loop order in the fluctuating h µν -field spoils this independence. The optimal result is obtained by utilizing the principle of minimal sensitivity [48] known from the renormalization-group calculus. The principle of minimal sensitivity is at the heart of the δ-perturbation expansion [49] and variational perturbation expansion [50,52]. There, if the perturbation theory depends on an unphysical parameter, say θ, the best result is achieved if each order has the weakest possible dependence on the parameter θ. Consequently, at the one-loop level the value of θ is determined from the vanishing of the derivative of V eff with respect to S 2 . By setting V S 2 ≡ ∂V eff /∂S 2 , we have This is equivalent to the equation which admits two branches of real solutions; either S 2 = 0.0259237 − 0.0000197α 2 + O(α 4 ) which, however, does not give a stableλ(S) (as V eff > 0) or infinite (butλ(S)stable) which withing the range of validity of our oneloop approximations means that S is at most ∼ ξ/κ ∼ In particular, for any value of the dimensionless coupling strength α c , we can choose the renormalization mass scale µ, in such a way thatλ has the value 1, that will guarantee phenomenologically correct gravitational forces at long distances. VEVλ is thus the dimensionally transmuted parameter of the massless CG. Its role here is completely analogous to the role of the dimensionally transmuted coupling constant in the Coleman-Weinberg treatment of the massless scalar electrodynamics [53]. Namely, we have traded a dimensionless parameter α c for a dimensionfull parameterλ/κ 2 (which does not exist in the symmetric phase). By assuming that in the broken phase a cosmologically relevant metric is that of Friedmann-Lamaître-Robertson-Walker (FRLW), then modulo topological term the additional condition holds due to a conformal flatness of the FRLW metric [54]. Combining (9), (19), and (20), the low-energy limit of A conf in the broken phase reads with We should stress that Λ is entirely of a geometric origin (it descends from the CG) and it enters in (21) with the opposite sign in comparison with the matter-sector induced (de Sitter) cosmological constant. Gradient term for λ -The local conformal symmetry dictates that the scalar degree of freedom must decouple from the on-shell spectrum of the CG [21,28], whereas in theories without conformal invariance (but with the same tensorial content) the scalar field does appear in spectrum [11,28,34]. When the conformal symmetry is broken the scalar field reappears through radiatively induced gradient term of the spurion field λ. The explicit form of the kinetic term (namely its overall sign!) can be decided from the momentum-dependent part of the λ self-energy Σ λ . This can be streamlined by considering in (10) slowly fluctuating λ instead of fixedλ. Since the lowest-order contribution to Σ λ comes from coupling tō h, the only relevant substitutions in (10) are;λ h → λ h (which stops to be a total-derivative) andhλ h → λh µν 3P µν,αβ = π µν π αβ /3 is the spin-0 projection, and π µν = η µν − ∂ µ −1 ∂ ν is the transverse vector projection). In the leading α c -order, one can neglect α 2 c ∂ µ λ with respect to α c ∂ µ λ and complete the square in (10) as follows The last approximation holds for 1 ≪ α 2 c −1 /(Cκ) 2 ∼ −1 /ξ 2 ∼ 10 28 −1 , and thus in the large-scale cosmology where only low-frequency modes of scalar fields (e.g., λ) are observationally relevant. The square completion procedure employed in (23) changes the (conformal) gauge fixing condition, albeit the only effect of this modification is a redefinition of the function ζ.
In passing, we note that since V eff in the broken phase is bounded from bellow and the kinetic energy is positive (i.e., vacuum decay is prevented), the broken one-loop linearized CG does not possess ghost states.
Cosmological implications -Recent polarisation data from Planck and WMAP satellites [41] support inflationary models with small tensor-to-scalar ratio: r < 0.12 at 95% CL. These include, e.g., the Starobinsky model (6), the non-minimally coupled model (∝ φ 2 R/2) with a V (φ) ∝ φ 4 /4-potential, and an inflation model based on a Higgs field [41]. In the SM the linear Einstein term determines the long-wavelength behavior while the R 2term dominates short distances and drives inflation. In phenomenological cosmology, the SM represents metric gravity with a curvature-driven inflation. In particular, it does not contain any fundamental scalar field that could be an inflaton, even though a scalar field/inflaton formally appears when transforming the SM to the Einstein frame [55].
SM emerges naturally in CG in the weak-field sector of the broken phase where the action A conf.b.,λ reads Similarly, as in the usual SM, one can set up for A conf.b.,λ a dual description in terms of a non-minimally coupled auxiliary scalar field φ with the action [7, 55] This is a HS-transformed A conf.b.,λ with φ being the HS-field. To analyze (25) we choose to switch from the Jordan frame (25) to the Einstein frame [7] where the curvature R enters without a non-minimally coupled fields λ and φ. This is obtained via rescaling: g µν → (λ + 2ξφ) −1 g µν , giving The above metric rescalling is valid only for the metricsignature-preserving transformation, i.e., only when (λ + 2ξφ) > 0. The action (26) can be brought into a diagonal form if we pass from fields {λ, φ} to {λ, ψ} where the new field ψ is obtained via the redefinition φ = [exp( 2/3|ψ|) − λ]/(2ξ). In terms of ψ the action reads where U (ψ, λ) = 1 8ξ 2 1 − 2λe − √ 2/3|ψ| , with ξ from (22). The strength of λ-field oscillations is controlled by the size of a coefficient in front of the λ-gradient term [50], i.e., e − √ 2/3|ψ| /κ 2 (more precisely, the local fluctuations square width (λ(x)−λ) 2 ∼ κ 2 e √ 2/3|ψ(x)| ). At large values of the dimensionless scalar field ψ, i.e., at values of the dimensionful fieldψ = ψ/κ that are large compared to the Planck scale, the gradient coefficient is very small and λ-field severely fluctuates. Assuming that CG was broken before the onset of inflation, then after a brief period of violent oscillations the λ-fluctuations are strongly damped [51] atψ 10m p . From then on, the λ-field settles at its potential minimum atλ = 1. Note that U (ψ,λ) ≤ 1/(8ξ 2 ) ≪ m 2 p , which is a necessary condition for a successful inflation. At values ofψ ∼ 10m p , the potential U (ψ,λ) is sufficiently flat to produce the phenomenologically acceptable slow-roll inflation, with the (collective) scalar field ψ playing the role of inflaton. Using the slow-roll parameters (∂ ψ ≡ ∂/∂ψ) one can write down the tensor-to-scalar ratio r and the spectral index n s in the slow-role approximation as [41] r = 16ǫ, n s = 1 − 6ǫ + 2η .
In terms of the number N of e-folds left to the end of inflation (ψ f represents the values of the inflaton at the end of inflation, i.e., when e − √ 2/3|ψ| ∼ 1) one gets which for N = 50 − 60 (i.e., values relevant for the CMB) is remarkably consistent with Planck data [41]. While during the inflation, the λ-field is constant (due to a large coefficient in front of the gradient term) allowing a large-valued inflaton field descend slowly from potential plateau, inflation ends gradually when λ regains its canonical kinetic term, and a small-valued inflaton field picks up kinetic energy. From (27) the dominant interaction channels at small |ψ| is (∂ µ λ) 2 |ψ|, hence the vacuum energy density stored in the inflaton field is transferred to the λ field via inflaton decay ψ → λ+λ (reheating), possibly preceded by a non-perturbative stage (preheating).
Note also, that the gravi-cosmological constant Λ that was instrumental in setting the inflaton potential in (27) has the opposite sign when compared with ordinary (matter-sector induced) cosmological constant. Since the conformal symmetry prohibits the existence of a (scalefull) cosmological constant, the gravi-cosmological constant must correspond to a scale at which the conformal symmetry breaks, which in turn determines the cut-off scale of the scalaron. The magnitude of ξ in the SM is closely linked to the scale of inflation [7]. Using the values relevant for the CMB with 50 − 60 e-foldings, the Planck data [41] require ξ ∼ 10 −13 GeV −1 or equivalently ξ/κ ∼ 10 5 . Thus from (22) the vacuum energy density is ρ Λ ≡ Λ/κ 2 ∼ 10 −10 (10 18 GeV) 4 ∼ 2 × 10 100 erg/cm 3 , which corresponds to a zero-point energy density of a scalaron with an ultraviolet cut-off at 10 15 − 10 16 GeV. This is in a range of the GUT inflationary scale. For compatibility with an inflationary-induced large structure formation the conformal symmetry should be broken before (or during) inflation [56]. This can be naturally included in a broader theoretical context of "conformal inflation" paradigm, which has been the thrust of much of the recent research [57][58][59]. Let us also notice that an existence of a single scalar field with cutoff at the GUT scale and coupled to broken CG (e.g., λ or GUT Higgs field) would contribute with a zero-point energy that could substantially reduce or eliminate Λ.
Conclusions -To conclude, we have shown that a spurion-field mediated spontaneous symmetry breakdown of CG is capable of transforming a purely metric conformal gravity to an effective scalar-tensor gravity. This offers a new paradigm to understand inflationary and large-scale cosmology. In particular, we have shown that the low-energy dynamics in the broken phase is described by a Starobinsky-type f (R)-model which can be mapped on a two-field hybrid inflationary model. A dimensional transmutation ties up together values of Starobinsky's inflation parameter ξ and cosmological constant Λ. This fixes the symmetry-breakdown scale for CG to be roughly the GUT inflationary scale. Despite its simplicity, the presented paradigm reproduces not only phenomenologically acceptable picture of the largescale Universe that is compatible with present Planck and WMAP data but it also provides a viable mechanism for reheating mechanism.
If BICEP2 data that support a large r > 0.16 (i.e, large-field) inflationary models are confirmed by other subsequent experiments, the conventional Starobinskytype inflationary potential will be excluded. In turn, this would also invalidate the outlined scenario. Clearly one could present modified scheme in which value of r would be large, e.g., by considering a tensorial HS transformation for R 2 µν or superconformal supergravity. These scenarios seem, however, at present less appealing.