Planck mass and inflation as consequences of dynamically broken scale invariance

Classical scale invariance represents a promising framework for model building beyond the Standard Model. However, once coupled to gravity, any scale-invariant microscopic model requires an explanation for the origin of the Planck mass. In this paper, we provide a minimal example for such a mechanism and show how the Planck mass can be dynamically generated in a strongly coupled gauge sector. We consider the case of hidden SU(N_c) gauge interactions that link the Planck mass to the condensation of a scalar bilinear operator that is nonminimally coupled to curvature. The effective theory at energies below the Planck mass contains two scalar fields: the pseudo-Nambu-Goldstone boson of spontaneously broken scale invariance (the dilaton) and a gravitational scalar degree of freedom that originates from the R^2 term in the effective action (the scalaron). We compute the effective potential for the coupled dilaton-scalaron system at one-loop order and demonstrate that it can be used to successfully realize a stage of slow-roll inflation in the early Universe. Remarkably enough, our predictions for the primordial scalar and tensor power spectra interpolate between those of standard R^2 inflation and linear chaotic inflation. For comparatively small gravitational couplings, we thus obtain a spectral index n_s ~= 0.97 and a tensor-to-scalar ratio as large as r ~= 0.08.

If conformal symmetry is imposed in the presence of a scalar field and if it is an anomaly-free local symmetry, the scalar field can be eliminated by a gauge fixing [4][5][6][7][8][9][10]. Alternatively, a certain boundary condition may be responsible for the generation of Einstein gravity [7,20]. In contrast to this, if global conformal symmetry is spontaneously broken, the Nambu-Goldstone (NG) boson associated with the breaking appears as a physical degree of freedom (DOF). This bosonic DOF may be present from the start and become the NG boson after symmetry breaking.
Bosonic DOFs are welcome because they may play the role of the inflaton field, which is a crucial element [21][22][23] to overcome the problems of old inflation [24] (see, e.g., [25] for a historical account). Since the inflaton field should roll down very slowly at first, the scalar potential for the inflaton field needs to satisfy the so-called slow-roll conditions (see, e.g., [26,27] for reviews). Moreover, the shape of the potential is these days very restricted in order to agree with the PLANCK observations [28]. Symmetry principles such as the concept of conformal symmetry can help to explain the origin of the inflaton field in combination with its specific potential. It can also help if the inflaton field is absent at the beginning as in the case of R 2 inflation [29][30][31] (see also [32] and references therein), or if the scalar field plays a dual role as in the case of Higgs inflation [33] (see also [34] and references therein).
Motivated by these observations and guided by the hints above, we consider in this paper a model with global conformal symmetry, which is spontaneously broken by the strong dynamics in a hidden SU (N c ) gauge theory. The model contains a complex scalar field S in the fundamental representation of SU (N c ) with the curvature portal coupling S † S R, where R represents the Ricci curvature scalar. Due to the strong dynamics, a nonzero scalar bilinear condensate forms, S † S = 0, which breaks conformal symmetry spontaneously [35,36]. The Planck mass is generated dynamically in this way, and the lowest excitation around the symmetry-breaking condensate can be identified with the NG boson, i.e., the dilaton. Note that this mechanism eliminates the generic asymmetry between gravity with a built-in scale and quantum field theory, where overall scales have no meaning. The breaking of conformal symmetry in our scenario sets the scale in both sectors in a symmetric way. The potential for the dilaton field and its coupling to the gravitational field is generated at the same time in this scenario. Through an appropriate Weyl transformation, the resulting action can be brought into the Einstein frame. In this way, we are able to identify the origin of the Planck mass and the inflaton; it is a mixture of the dilaton and the scalaron, which appears due to the R 2 term.
Our basic idea to generate the Planck mass and at the same time to induce inflation is similar to that of [37] in that our construction is based on global scale invariance in the starting classical theory. 1 However, our concept to arrive at the Einstein-Hilbert kinetic term for the gravitational field that couples to the inflaton field is different: We rely on the strong dynamics in a non-Abelian gauge theory that break scale invariance spontaneously, while the 1 Refs. [38][39][40][41][42][43] is a partial list of references that discuss inflation in classically scale-invariant models. None of them uses strong dynamics in a non-Abelian gauge theory to break scale invariance. Nonperturbative chiral symmetry breaking to produce a robust energy scale in a classically scale-invariant hidden sector [44][45][46] has been applied to an inflation model [47], but scale invariance is explicitly broken in their starting classical theory.
model of [37] has no strongly interacting gauge sector, so that the hard breaking of conformal symmetry by the conformal anomaly [48][49][50] (i.e., the running of coupling constants) plays the crucial role. Finally, we mention that the supersymmetric models discussed in [51][52][53][54][55][56][57] also make use of strongly coupled gauge dynamics in a hidden sector to generate the energy scale of inflation. However, these models simply assume the presence of the Einstein-Hilbert term from the very beginning and hence offer no dynamical explanation for the origin of the Planck scale.
Our starting tree-level action in Eq. (1) below has the the most general form consistent with general diffeomorphism invariance, SU (N c ) local gauge invariance and global scale invariance. Then, along the lines of the Nambu-Jona-Lasinio (NJL) model [58,59], we proceed to an effective-theory description of the scalar bilinear condensate. In the mean-field approximation, we can not only identify the dilaton, but also derive its potential by integrating out the fluctuations of the scalar field S around its background. In doing so, we arrive at the one-loop effective action, which is the starting action in the Jordan frame in order to subsequently discuss inflation in section III. In this part of our analysis, we assume that the Weyl tensor squared is negligible for the purposes of inflation, which is why we suppress it in the effective action. The final effective Lagrangian in the Einstein frame involves the scalaron field as well. We find that the scalar potential is such that the coupled dilaton-scalaron system gives rise to an effective single-field model of inflation. In the last part of section III, we perform a numerical analysis to compute the inflationary observables encoded in the cosmic microwave background (CMB) and to compare our predictions with the latest data from the PLANCK satellite mission [28]. Our model contains four independent parameters, of which two are used to fix the values of the reduced Planck mass M Pl and the amplitude A s of the primordial scalar power spectrum. We are hence left with two free parameters to describe the inflationary Universe. The last section is devoted to conclusion.

II. DYNAMICAL ORIGIN OF THE PLANCK MASS
A. Spontaneous breaking of scale invariance in a hidden strongly coupled sector The starting point of our analysis is the most general action that complies with the symmetry principles of general diffeomorphism invariance, local gauge invariance, and global scale invariance at the classical level, Here, R denotes the Ricci curvature scalar, R µν is the Ricci tensor, R µναβ is the Riemann tensor, F is the matrix-valued field-strength tensor of the SU (N c ) gauge theory, and S is the complex scalar field in the fundamental representation of SU (N c ), with D µ being the covariant derivative. Note that the action S C also features an accidental global U (1) symmetry corresponding to S number conservation. The last two terms in Eq. (1) can be written as where W µναβ = R µναβ + 1 2 (−g µα R νβ + g µβ R να + g να R µβ − g νβ R µα )+ 1 6 (g µα g νβ − g να g µβ ) R denotes the Weyl tensor, and E = R µναβ R µναβ − 4 R µν R µν + R 2 is the Gauß-Bonnet term, which we will suppress because the corresponding action is a surface term. The R 2 term in Eq. (2) can be absorbed into a redefinition of the coefficient γ, so that the last two terms in Eq. (1) can be replaced by a term of the form κ W µναβ W µναβ , which, multiplied with √ −g, is invariant under local conformal transformations, g µν (x) → g µν (x) = Ω 2 (x) g µν (x).
In our scenario [60][61][62], the condensation of the gauge-invariant scalar bilinear S † S in the confining phase breaks scale symmetry spontaneously, in a similar way as the chiral fermion condensate breaks chiral symmetry in QCD.
However, a consistent formulation of the quantum theory for the classical action in Eq. (1) is not yet available. One possibility of such a formulation utilizes the notion of asymptotic safety [63] (see, e.g., [64] and references therein), which assumes the existence of a nontrivial ultraviolet fixed point. At the ultraviolet fixed point, conformal symmetry is restored as an unbroken symmetry of the full quantum theory. In the following discussion, we are implicitly assuming that the hard breaking of scale invariance by the scale anomaly is weak enough, so that we may ignore it for the purposes of describing the spontaneous breaking of scale invariance. Related to that, we recall Ref. [65], where it has been pointed out that spontaneous scale symmetry breaking can be associated with the chiral condensate in QCD if the running of the gauge coupling is slow (i.e., if the hard breaking by scale anomaly is weak). To realize a slow running of the gauge coupling as well as of the quartic couplingλ in our model, a certain set of fermion fields should be introduced [66]. But we will not go into such details, which are beyond the scope of the present work.
Even if gravity is switched off, it is nontrivial to investigate whether or not a scalar condensate really forms [35,36].
In the following, instead of investigating this problem from first principles, we will rely on an effective theory for the order parameter S † S (strictly speaking, it is an approximate order parameter [35,36]). Let us outline the basics of the effective theory that has been proposed in [61,62] by applying the concepts of the NJL model [58,59], which is an effective theory for the chiral condensate, i.e., the order parameter of chiral symmetry breaking. As in the case of the NJL model, the effective Lagrangian does not contain the gauge fields, because they are integrated out, while it contains the "constituent" scalar field S, similar to the constituent quarks in the NJL model, This is again the most general action that is consistent with general diffeomorphism invariance, global SU (N c ) × U (1) symmetry, and classical global scale invariance. 2 Note that the couplingsβ,γ, andλ in the fundamental action S C are not the same as β, γ, and λ in the effective action S C,eff , because the latter are effective couplings that are dressed by hidden-gauge-field contributions. Introducing the auxiliary field (mean field) f , we can rewrite Eq. (3) as The action in Eq. (3) can be obtained from Eq. (4) by inserting the equation of motion for the auxiliary field f , To obtain the effective potential for f , we integrate out the fluctuations of the field S around its background valueS, where Λ = e 3/4 µ MS . We used dimensional regularization together with renormalization according to the modified minimal subtraction scheme, 3 where the divergences are absorbed in the coupling constants as follows, λ → λ − 4 λ 2 L, If we assume R = 0, the absolute minimum of V eff is located at This is an important result of our analysis: In view of the action in Eq. (3), together with Eq. (5), we now see that the strong dynamics in the non-Abelian gauge theory can break conformal symmetry spontaneously and consequently 2 An attempt at taking into account the effect of confinement was made in [67]. 3 The Gaussian path integral has been performed assuming a flat spacetime metric. If the fluctuations around the flat metric are taken into account, a term ∝ √ −g Rµν R µν − R µναβ R µναβ as well as an additional term ∝ √ −g R 2 will be generated at one-loop order [68,69]. They can be absorbed into redefinitions of κ and γ in Eq. (4), respectively, if the topological E term is ignored. generate the Planck mass dynamically, where M Pl is the reduced Planck mass. Note that the potential value at the minimum in Eq. (7) is given by Therefore, the spontaneous breaking of scale symmetry generates a negative vacuum energy, which is −8 × 10 −9 M 4 Pl for λ = 1, N c = 5, β = 10 3 . This implies, however, that the assumption made above, R = 0, is not consistent, unless the vacuum energy density vanishes, because what we are looking for is the absolute minimum of . The zero-point energy density U 0 is finite in dimensional regularization because of the scale invariance of the action in Eq. (4). If we use another regularization scheme, it can be different or even divergent. This reflects the fact that the zero-point energy density can not be uniquely determined within the framework of quantum field theory in flat spacetime: The cosmological constant problem remains still unsolved, although we established a link between particle physics and gravity via spontaneous breaking of global conformal symmetry. Here we are not attempting to solve this problem and proceed with our discussion in the hope that there will be a mechanism in, e.g., quantum gravity that solves this problem. In the following discussion, we thus simply subtract this vacuum energy density from the potential and continue with the potential V eff → V eff + U 0 , so that the potential has zero cosmological constant. Consequently, the effective potential at the minimum with nonvanishing R can be written as from which we see that the Planck mass is correctly identified in Eq. (8) and that the parameter γ is shifted tô where Eq. (7) has been used.

B. Effective action for the dilaton field at low energies
The mean field f is a scalar field with canonical mass dimension two, and it can describe the excitations above the vacuum expectation value f 0 . At the one-loop level, the mean field f obtains a kinetic term, and the excitations can be described by a propagating real scalar field. Suppressing the complex scalar S, we proceed with the effective action where The wave function renormalization constant Z −1 is given by An explicit derivation of this result for the wave function renormalization constant can be found in [70]. Next, we proceed from f with canonical dimension two to the canonically normalized field χ with canonical dimension one, i.e., where is defined in Eq. (11). We find that the effective action in Eq. (12) can be written as Here, the ellipsis stands for higher-order terms such as g µν ∂ µ χ ∂ ν χ β R/χ 2 0 , β 3 R 3 /χ 2 0 , etc. The dilaton-dependent functions B (χ), G (χ), and U (χ) are given by U 0 andγ are defined in Eqs. (9) and (11), respectively. To arrive at L eff in Eq. (16), we used Eqs. (7) and (15) to find V eff in Eq. (13) is obtained from Eq. (6) by expanding in powers of β R. This expansion makes sense only if 2 λ f β R, Otherwise, it is impossible to generate Einstein gravity. Furthermore, (16) still vanishes as soon as B (χ) = 0.

III. DILATON-SCALARON INFLATION
A. Full scalar potential of the dilaton-scalaron system In the previous section, we discussed how the condensation of a scalar bilinear operator nonminimally coupled to the Ricci scalar leads to the dynamical generation of the Planck mass. In this section, we shall now turn to the effective theory at energies below the Planck scale and demonstrate how the scalar DOFs at low energies can give rise to a stage of inflationary expansion in the early Universe. The starting point of our analysis is the effective Jordan-frame Lagrangian in Eq. (16), truncated after the second order in the Jordan-frame Ricci scalar R J , Here, g µν J and g J denote the inverse and the determinant of the Jordan-frame spacetime metric g J µν , respectively. By construction, the kinetic term of the dilaton field χ is canonically normalized in the Jordan frame. In this work, we shall assume that the term involving the Weyl tensor squared is negligible for the purposes of inflation. The equation of motion for the dilaton field is, in any case, independent of this term. Moreover, we recall that the dimensionless function B(χ) given in Eq. (17) parametrizes the nonminimal coupling of the dilaton field χ to the Ricci scalar R J , while the dimensionless function G(χ) given in Eq. (18) denotes the field-dependent coefficient of the R 2 J term. The dimensionful function U (χ) given in Eq. (19) stands for the Jordan-frame scalar potential.
The Lagrangian in Eq. (21) admits solutions of the equations of motion that describe an inflationary stage of expansion. To see this more clearly, we need to transform Eq. (21) from the Jordan frame to the Einstein frame. In a first step, let us introduce an auxiliary field ψ with mass dimension two that allows us to remove the R 2 J term, Indeed, varying this Lagrangian w.r.t. the auxiliary field, δ ψ L J eff = 0, results in ψ = R J , which restores the Lagrangian in Eq. (21). In a second step, we perform a Weyl rescaling of the metric, g µν = Ω 2 g J µν , where we choose the conformal factor Ω such that the kinetic term for the gravitational field obtains its standard Einstein-Hilbert form, At this point, it is helpful to remember that the Ricci scalar R J behaves as follows under a Weyl transformation, Neglecting the total derivative in this expression, we thus arrive at the following Einstein-frame Lagrangian, where V denotes the scalar potential in the Einstein frame, The Lagrangian in Eq. (25) indicates that the function ln Ω 2 , which we extracted from the Jordan-frame Ricci scalar R J in Eq. (24), can be regarded as a propagating scalar field that possesses its own kinetic term. In the following, we will refer to this new scalar field as the scalaron φ; the canonically normalized scalaron is defined as follows, Making use of this definition, the conformal factor Ω 2 can be written as an exponential function of the scalaron field, Similarly, thanks to Eqs. (23) and (27), the auxiliary field ψ can be traded for a function of the fields χ and φ, The Einstein-frame Lagrangian for the coupled dilaton-scalaron system thus obtains the following form, where the Einstein-frame scalar potential V as a function of χ and φ now reads This scalar potential can be used as the starting point for the analysis of slow-roll inflation.
B. Effective single-field description along the inflationary trajectory The scalar potential in Eq. (31) is a complicated function in the two-dimensional field space spanned by χ and φ.
In principle, one could therefore imagine that Eq. (31) allows one to realize inflation in various parameter regimes and different parts of field space. In particular, this might include scenarios in which both scalar fields act as slowly rolling inflatons, such that inflation needs to be described by a full-fledged two-field analysis. In the following, however, we will refrain from attempting to identify such intricate scenarios. Instead, we will focus on a particularly simple inflationary solution that can be described by an effective single-field model. Our key observation is that the scalar potential in Eq. (31) always possesses exactly one local extremum in the scalaron direction at φ = φ , where we introduced the dilaton-dependent function A, which will allows us to simplify our notation in the following, The presence of this extremum motivates us to seek inflationary solutions along the following contour in field space, where σ can be regarded as the physical inflaton field that parametrizes the one-dimensional contour C in the twodimensional field space. Along this trajectory, the dilaton-scalaron system can be effectively described by a single-field model in terms of the real scalar σ. Evaluating Eqs. (30) and (31) along C, we obtain the following Lagrangian, Note that the field σ is not canonically normalized; its kinetic term is multiplied by the field-dependent function N σ , Meanwhile, the scalar potential for the noncanonically normalized inflaton field σ obtains the following form, The canonically normalized inflaton fieldσ follows from integrating the function N σ over an appropriate field range, The relation between the three scalar fields χ, φ, andσ is illustrated in Fig. 1  contour C will describe an unstable ridge in the potential landscape. Therefore, a necessary condition for successful inflation is that, along the trajectory C, the scalaron direction is always stabilized by a positive mass squared, Making use of the function A introduced in Eq. (33), the scalaron mass during inflation can be written as Successful inflation requires a positive vacuum energy density, V inf > 0. The expression for the scalar potential in Eq. (37) thus implies that A must always be larger than −1/4. For this reason, the condition of a positive scalaron mass squared in Eq. (39) translates into the statement that the function G must always be positive during inflation, 4 We remark that the radiative corrections to the Lagrangian parameter γ that are encoded in the function G always come with a negative sign. As a consequence, it is, in fact, impossible to satisfy the condition G > 0 for all values of the scalar field σ. As soon as σ exceeds a certain critical value σ crit , the function G turns negative, However, for all practical purposes, it is sufficient if this critical field value is large enough compared to the typical field values during inflation. In addition, it may well be that the scalar potential receives further gravitational corrections at field values far above the Planck scale that regularize the singularity at G = 0. For these reasons, we content ourselves with the requirement that σ crit should be at least as large as the Planck scale, This condition ensures that the available field range in the σ direction will always be large enough to support inflation.
(ii) Merely requiring m 2 φ to be positive is not enough to sufficiently stabilize the scalaron at φ = φ . In fact, from the perspective of slow-roll inflation, we can only ignore the motion in the scalaron direction away from the trajectory C if the scalaron mass m φ is large compared to the inflationary Hubble rate H inf for all times during inflation, Here, we used the standard expression for H inf in the slow-roll approximation, In view of Eq. (44), we therefore conclude that the function A must always remain small during inflation, A 1.
Once the condition A 1 becomes violated, we can no longer fully trust our ansatz in Eqs. (32) and (34) and be sure that inflation will indeed proceed along the contour C. In this case, we may alternatively look for local minima in the dilaton direction χ and attempt to construct an inflationary trajectory C along these minima, However, this approach is complicated by the fact that the form of the scalar potential in Eq. (31) does not allow us to compute the local minima in the dilaton direction analytically. The best we can do in order to determine the function χ is to adopt an iterative procedure. First, let us ignore for a moment the field dependence of all logarithmic terms in Eq. (31). In this approximation, we are able to solve the extremal problem in Eq. (46) exactly. Then, in the next step, we can use the result of the first step to find a better approximation for the logarithmic terms, which provides in turn us with a slightly improved solution for χ . After n iterations, this leads to the following expression for χ , with the logarithmic terms after n iterations, L (n) and G (n) , being defined as follows, The initial x value is arbitrary and must be chosen by hand. For definiteness, we may, e.g., choose x (0) = √ e such that L (0) = 1. In an explicit numerical analysis, we are able to confirm that this iterative procedure performs very well. Over a broad range of parameter and field values, Eq. (47) manages to approximate the exact numerical result for x to high precision. Moreover, we are able to show that, for A 1, the contours C and C always lie very close together in field space. In this case, both approaches lead to the same inflationary dynamics and hence to identical predictions for the primordial power spectra. But on top of that, we are also able to confirm numerically that, even for A 1, inflation along the contours C and C leads to very similar predictions. For this reason, we will from now on only focus on inflation along the trajectory C. In the regions of parameter space where A 1, this is still a sufficiently good approximation. A dedicated numerical analysis of inflation along the contour C is left for future work. (iii) The fact that the dilaton-scalaron system can be effectively described by a simple single-field model is also apparent from the scalar mass spectrum. After diagonalizing the dilaton-scalaron mass matrix and identifying the two mass eigenstates, one typically finds that one mass eigenvalue is extremely large. It is therefore well justified to integrate out the heavy DOF and describe inflation in terms of a single-field model. To see this explicitly, one requires knowledge of the scalar mass matrix along the contour C. The mass squared of the scalaron field is given in Eq. (40).
The mass squared of the canonically normalized dilaton fieldχ as well as the dilaton-scalaron mass mixing parameter can be calculated as follows, Here, we accounted for the noncanonical normalization of the dilaton field in Eq. (30) by means of the factor N χ , For simplicity, we shall neglect the field dependence of N χ for the time being and simply work with a symmetric mass matrix, m 2 φχ = m 2 χφ , in the following. An explicit computation of the partial derivatives in Eq. (49) results in Together, the mass parameters m 2 φ , m 2 χ , and m 2 χφ can be used to determine the scalar mass eigenvalues m 2 ± during inflation. Typically, we find one heavy mass eigenstate, 0 < H 2 inf m 2 + , as well as one light tachyonic mass eigenstate, −H 2 inf m 2 − < 0. The tachyonic mass eigenstate appears as a consequence of the negative curvature of the scalar potential in the direction of the inflationary trajectory. Similarly, the heavy mass eigenstate reflects the fact that the scalar potential is steeply rising in the direction orthogonal to the contour C. In fact, an important condition for the validity of our analysis is that the mass eigenvalue m + never exceeds the Planck scale, Otherwise, we would no longer be able to trust our standard quantum field theory analysis. In our numerical study (iv) The fourth and last condition for successful inflation is concerned with the perturbativity of our model. Recall that our construction is based on a strongly coupled gauge theory at high energies. This implies the risk that our effective single-field model at low energies may inherit some strong-coupling effects that would render a standard analysis in terms of perturbation theory invalid. Therefore, it is essential to study the self-interactions of the canonically normalized inflaton fieldσ and examine whether or not they remain in the perturbative regime. The brute-force approach to this problem would be to explicitly compute the scalar potential V inf as a function ofσ and to study its properties in the most general terms. However, to do so, one would have to invert the relation between σ andσ in Eq. (38), which, in practice, can only be done numerically. For this reason, we will adopt a different approach in the following that by-passes this technical complication. The crucial point is that we are mostly interested in the properties of the scalar potential in the vicinity of the origin in field space, where we expect the inflaton self-interactions to be the strongest [see Fig. 1]. For small field values, we are then able to expand V inf in powers of the inflaton field, where we made use of the fact that the origin corresponds to a global Minkowski vacuum, V inf (0) = V inf (0) = 0, by construction. Thanks to the chain rule, the coefficients in this Taylor expansion can be computed solely in terms of the known expressions for V inf and N σ as functions of the noncanonically normalized inflaton field σ, Based on these results, one can show that the self-interactions of the fieldσ remain perturbative for all parameter values of interest. On the analytical side, it is easy to see that the overall magnitude of the coefficients is universally controlled by U 0 /M 4 Pl , i.e., the normalization of the Jordan-frame scalar potential in units of the Planck mass, As long as this parameter is small, the coefficientsâ 2,3,4 automatically obtain small values in Planck units. With regard to the quartic self-couplingâ 4 , this means in particular that a sub-Planckian potential energy density in the Jordan frame, U 0 M 4 Pl , readily results in perturbatively small values,â 4 1. In our numerical slow-roll analysis, we are able to confirm these results by explicitly evaluating the coefficientsâ 2,3,4 at each point in parameter space.

C. Numerical analysis of the slow-roll dynamics
The effective single-field model specified in Eqs. (35), (36), and (37) allows us to perform a standard single-field slow-roll analysis of inflation. The goal of this analysis is to compute the usual observables related to the primordial power spectra encoded in the temperature fluctuations of the CMB: the amplitude A s and the spectral tilt n s of the scalar power spectrum as well as the tensor-to-scalar ratio r. These three observables are measured or constrained by the latest data from the PLANCK satellite mission [28]. At 68 % C. L., the scalar amplitude A s is measured to be A s = e 3.044±0.014 × 10 −10 (TT + TE + EE + lowE + lensing) .
In the following, we will work with the best-fit value, A obs s = e 3.044 × 10 −10 2.1 × 10 −9 , for definiteness. The combined constraints on n s and r for various combinations of data sets are shown in the right panel of Fig. 2.
Our predictions for A s , n s , and r can be conveniently computed in terms of the slow-roll parameters ε and η, where all field-dependent quantities (V inf , ε, and η) are supposed to be evaluated at the time of CMB horizon exit, i.e., N e 50 · · · 60 e-folds before the end of inflation. The parameters ε and η are defined in terms of partial derivatives of the scalar potential w.r.t. the canonically normalized inflaton fieldσ. However, thanks to the chain rule, these derivatives can again be readily expressed as derivatives w.r.t. the noncanonically normalized field σ, The inflaton field value at the time of CMB horizon exit can be computed by solving the slow-roll equation of motion, with the field value at the end of inflation, σ 0 = σ (N e = 0), defined by the requirement that max {ε (σ 0 ) , |η (σ 0 )|} = 1.
Together, the relations in Eqs. (57), (58), and (59) provide us with the necessary tools to determine our theoretical predictions for A s , n s , and r. Because of the complicated form of the functions N σ and V inf in Eqs. (36) and (37), this can only be done numerically -at least as long as one intends not to make any simplifying assumptions.
Consequently, we will content ourselves with a purely numerical analysis in this paper. Any analytical investigation based on further assumptions is left for future work. The parameter space of our inflationary model is spanned by three dimensionless coupling constants: (i) the quartic self-coupling λ of the scalar quark field S, (ii) the strength β of the nonminimal coupling between the field S and the Ricci scalar R, and (iii) the coefficient γ of the bare R 2 term before accounting for any radiative corrections. We also recall that all three parameters are understood to correspond to those couplings that appear in the effective action for the constituent scalar field S after integrating out the strong SU (N c ) gauge dynamics. In the following, one of these parameters can be eliminated by requiring that the amplitude of the scalar power spectrum obtains the correct value, A s = A obs s . For definiteness, we will take this parameter to be the nonminimal coupling constant β. This leaves us with a two-dimensional parameter space spanned by λ and γ, in which we can compute our predictions for n s and r. The outcome of this analysis is shown in Fig. 2.
A remarkable feature of our results is that they include and extend the well-known results of R 2 inflation [29].
This connection can be illustrated in the following intuitive way: First of all, let us consider the orientation of the inflationary trajectory C in the dilaton-scalaron field space at the time of CMB horizon exit. At each point in parameter space, we would like to know: Does inflation rather proceed in the scalaron or dilaton direction? Or in other words: In view of the scalar potential in Eq. (31), are the inflationary dynamics rather determined by the function e Φ or by the functions B, G, and U ? A convenient way to answer these questions is to introduce the inclination angle α, which measures the angle between the dilaton axis in field space and the inflationary trajectory. 5 An inclination angle close to 0 indicates that the dynamics of inflation are mostly governed by the dilaton-dependent functions in the scalar potential. Conversely, an inclination angle close to π points to inflation being dominated by the scalaron-dependent part of the scalar potential. In Fig. 2, we refer to these two scenarios as dilaton and scalaron inflation, respectively.
In particular, we indicate the dependence of α as a function of λ and γ by the color shading in the left panel of Fig. 2.
Interestingly enough, α converges towards α π as soon as γ approaches a characteristic value of O 10 9 . This is nothing but the R 2 -inflation limit of our model. In this limit, the dynamics of the dilaton field become increasingly irrelevant from the perspective of inflation. In fact, for large values of γ, the value of β that is required to obtain the correct scalar amplitude decreases. According to Eq. (17), this results in a large value of χ 0 , such that Quartic scalar-quark self-coupling at high energies, λ Coefficient of the R 2 term at tree level, γ  are taken from [28]. For more details on the various reference models of inflation, see the discussion and analysis in [28]. The black-and-white circle in both plots corresponds to the benchmark parameter point chosen in Fig. 1.
As a consequence, the scalar potential in Eq. (31) approaches the well-known potential of R 2 inflation [29], In this potential, we recognize the radiatively corrected parameterγ as the coefficient of the R 2 term in R 2 inflation.
The requirement of a correctly normalized scalar power spectrum then uniquely fixesγ at a value of O 10 9 , A s = A obs s , 50 ≤ N e ≤ 60 ⇒ 0.5 × 10 9 γ 0.7 × 10 9 .
Up to terms suppressed by higher powers of N −1 e , R 2 inflation results in the following predictions for n s and r, For N e 50 · · · 60, this means that r is expected to remain rather small, 3 r/10 −3 5, in the R 2 -inflation limit.
However, this situation changes as soon as we make use of the parametric freedom of our model. For γ values smaller than those required by R 2 inflation, the dilaton dynamics begin to become important. This helps us increase the tensor-to-scalar ratio [see Fig. 2]. In fact, for small values of γ and λ, the predictions of our model approach those of linear chaotic inflation [71], which is described by a scalar potential that is linear in the inflaton field, V inf ∝ σ.
This can be understood as follows: For small values of γ and λ, we may neglect the function A in Eqs. (36) and (37).
In this approximation, it is possible to simplify the scalar potential V inf and the normalization factor N σ as follows, where x is defined similarly as in Eq. (47), x (σ) = 1 + σ/χ 0 . Given this form of the normalization factor N σ and making use of the relation in Eq. (38), we are able to determine the canonically normalized inflaton fieldσ, Solving this relation for σ and inserting the result into V inf in Eq. (66) provides us with the scalar potential forσ, At large field values,σ σ 0 , this is a linear potential just like in the case of linear chaotic inflation! At the same time, the exponential correction regularizes the derivative of the scalar potential around the origin in field space. In fact, thanks to this exponential term, the scalar potential exhibits a stable global minimum atσ = 0. These observations explain why small values of γ and λ lead us to the same predictions as in the case of linear chaotic inflation, For N e = 50, the tensor-to-scalar ratio can hence become as large as r 0.08, while for N e = 60, one can achieve values as large as r 0.07. These are remarkable results, as they promise that our model can be probed in future CMB experiments that are sensitive to r values of the order of r few × 10 −2 . Moreover, it is important to note that our model populates regions in the n s -r plane that are inaccessible in other popular inflation models [see Fig. 2].

IV. CONCLUSIONS
In this paper, we studied the dynamical generation of the Planck mass in consequence of spontaneous conformalsymmetry breaking. To this end, we constructed an explicit field-theoretical model based on the idea that classical scale invariance becomes spontaneously broken because of the strongly coupled gauge dynamics in a hidden sector.
Our construction led us to a remarkable realization: The stage of cosmic inflation in the early Universe may be a consequence of the microscopic physics that is responsible for the dynamical generation of the Planck mass! If correct, this observation would point to a deep connection between particle physics and cosmology. We emphasize that the pivotal element of our construction is the concept of classical scale invariance -a new approximate global symmetry that forbids any dimensionful coupling constant in the tree-level Lagrangian. At the quantum level, scale invariance is explicitly broken by the conformal anomaly, which reflects the logarithmic running of the renormalized coupling constants. However, as a working assumption, one may speculate that this explicit breaking is eventually going to vanish at very high energies because of the theory's renormalization group flow reaching an ultraviolet fixed point.
Within the paradigm of classical scale invariance, explicit mass scales such as the Planck mass M Pl can only be generated radiatively by means of spontaneous symmetry breaking. In our model, we follow this philosophy and show how classical scale invariance may be spontaneously broken by means of dimensional transmutation in a hidden strongly coupled SU (N c ) gauge sector. In particular, we consider a scalar bilinear operator S † S that is nonminimally coupled to the Ricci scalar and that condenses in the confining phase of the SU (N c ) gauge theory at low energies.
Of course, a similar construction making use of a chiral condensate is also conceivable. In this work, we focused on a scalar condensate for simplicity. A study of the fermionic case is left for future work. Up to a dimensionless coupling constant β, the vacuum expectation value of the scalar condensate then directly determines the Planck mass, The vacuum expectation value of the scalar condensate can be computed according to the following recipe: 1. Write down the most general action compatible with all global and local symmetries of our model.
2. Construct an effective model for the constituent scalar field S along the lines of the NJL model in QCD.
3. Apply the mean-field approximation to describe the dynamics of the order parameter f = S † S.
4. Compute the quantum effective action for f by integrating out the heavy constituent scalar S.

5.
Determine the symmetry-breaking minimum of the effective scalar potential for the mean field f .
This procedure results in a Coleman-Weinberg-like correction to the Lagrangian of the gravitational sector, For small values of R, the logarithmic term can be expanded in powers of R, which generates among other terms the Einstein-Hilbert term in the gravitational action. The coefficient of this term is then identified as the Planck mass. An important aspect of this construction is that the gravitational scale M Pl originates from an ordinary energy scale in quantum field theory, i.e., the vacuum expectation value of the scalar condensate [see Eq. (70)]. Likewise, the absolute energy scale of the scalar potential after spontaneous conformal-symmetry breaking, U 1/4 0 , is initially computed in field theory [see Eq. (9)]. Therefore, before going the next step and making contact with gravity, the potential energy scale U 1/4 0 is expected to be subjected to appropriate renormalization conditions on the field-theory side. This observation may be useful in more realistic attempts at solving the problem of the cosmological constant.
A remarkable consequence of our mechanism to generate the Planck mass is that it automatically results in the presence of an additional scalar DOF: the dilaton field χ, which parametrizes the fluctuations around the symmetrybreaking vacuum and which plays the role of the pseudo-NG boson of spontaneously broken scale invariance. As we were able to show in the second part of this paper, this dilaton field can be used to construct a successful model of inflation. In fact, the dilaton couples to a second scalar DOF, the scalaron field φ, which originates from the R 2 term in the effective action. A main result of our analysis is that the coupled dilaton-scalaron system gives rise to an effective single-field model of inflation in the Einstein frame. The predictions of this model for the inflationary CMB observables interpolate between those of standard R 2 inflation and those of linear chaotic inflation. This important result represents a characteristic phenomenological feature of our model. As a consequence, we typically find values of the scalar spectral index in the range 0.970 n s 0.975 and values for the tensor-to-scalar ratio as large as r 0.08 [see Fig. 2]. These are exciting predictions that will be tested in future CMB experiments.
Our study presented in this paper should be understood as a first step towards a better understanding of inflation in classically scale-invariant models. There are certainly several directions into which our discussion may be extended: (i) It would be worthwhile to study the full two-field dynamics of our model (see, e.g., [72,73]) and to investigate which other inflationary solutions one might be able to identify. In this context, one should also compute the degree of non-Gaussianities generated by our model and compare them to the current observational bounds. (ii) One may refrain from expanding the effective scalar potential in powers of R and keep working with the full expression in Eq. (71).
This would amount to the study of a typical f (R) model, where f (R) would schematically be given by [74,75] f (R) = γ R 2 + V CW (R) .