# Planck mass and inflation as consequences of dynamically broken scale invariance

###### Abstract

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 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 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 inflation and linear chaotic inflation. For comparatively small gravitational couplings, we thus obtain a spectral index and a tensor-to-scalar ratio as large as .

###### Contents

## I Introduction

What is the origin of the Planck mass? This fundamental question, which we will address in this paper, attracted much attention in the past, particularly in field theory Fujii:1974bq ; Minkowski:1977aj ; Zee:1978wi ; Englert:1975wj ; Englert:1976ep ; Chudnovsky:1976zj ; Fradkin:1978yw ; Smolin:1979uz ; Zee:1979hy ; Nieh:1982nb ; Akama:1977nw ; Akama:1977hr ; Adler:1980bx ; Adler:1980pg ; Zee:1980sj ; Adler:1982ri . If we start with a theory that contains dimensionful parameters, we can not explain its origin. Thus, within the framework of Einstein’s theory of gravity, the origin of the Planck mass can not be explained. In the references cited above, conformal symmetry is imposed, as the Planck mass and hence Einstein gravity can arise through its breaking. Conformal symmetry can be global Fujii:1974bq ; Minkowski:1977aj ; Zee:1978wi or local Englert:1975wj ; Englert:1976ep ; Chudnovsky:1976zj ; Fradkin:1978yw ; Smolin:1979uz ; Zee:1979hy ; Nieh:1982nb ; Akama:1977nw ; Akama:1977hr ; Adler:1980bx ; Adler:1980pg ; Zee:1980sj ; Adler:1982ri , and the theory can contain fundamental scalar fields from the beginning Fujii:1974bq ; Minkowski:1977aj ; Zee:1978wi ; Englert:1975wj ; Englert:1976ep ; Chudnovsky:1976zj ; Fradkin:1978yw ; Smolin:1979uz ; Zee:1979hy ; Nieh:1982nb , as in the Brans–Dicke theory Brans:1961sx , or no scalar field as in the case of induced gravity Akama:1977nw ; Akama:1977hr ; Adler:1980bx ; Adler:1980pg ; Zee:1980sj ; Adler:1982ri . Conformal gravity, i.e., gravity supplemented by a local conformal symmetry, has also been strongly motivating because of its possible renormalizability (see, e.g., Adler:1982ri ; tHooft:2011aa ; Mannheim:2011ds ). 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 Englert:1975wj ; Englert:1976ep ; Chudnovsky:1976zj ; Fradkin:1978yw ; Smolin:1979uz ; Zee:1979hy ; Nieh:1982nb . Alternatively, a certain boundary condition may be responsible for the generation of Einstein gravity Fradkin:1978yw ; Maldacena:2011mk . 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 Linde:1981mu ; Linde:1982zj ; Albrecht:1982wi to overcome the problems of old inflation Guth:1980zm (see, e.g., Linde:2007fr 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., Baumann:2009ds ; Martin:2013tda for reviews). Moreover, the shape of the potential is these days very restricted in order to agree with the PLANCK observations Akrami:2018odb . 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 inflation Starobinsky:1980te ; Mukhanov:1981xt ; Starobinsky:1983zz (see also Kehagias:2013mya and references therein), or if the scalar field plays a dual role as in the case of Higgs inflation Bezrukov:2007ep (see also Rubio:2018ogq 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 gauge theory. The model contains a complex scalar field in the fundamental representation of with the curvature portal coupling , where represents the Ricci curvature scalar. Due to the strong dynamics, a nonzero scalar bilinear condensate forms, , which breaks conformal symmetry spontaneously Osterwalder:1977pc ; Fradkin:1978dv . 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 term.

Our basic idea to generate the Planck mass and at the same time
to induce inflation is similar to that of Kannike:2015apa
in that our construction is based on global scale invariance in
the starting classical theory.^{1}^{1}1Refs. Farzinnia:2015fka ; Rinaldi:2015uvu ; Ferreira:2016wem ; Ghilencea:2018thl ; Benisty:2018fja ; Barnaveli:2018dxo
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 Hur:2011sv ; Heikinheimo:2013fta ; Holthausen:2013ota
has been applied to an inflation model Inagaki:2015eza ,
but scale invariance is explicitly broken in their starting classical theory.
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 model of Kannike:2015apa
has no strongly interacting gauge sector, so that the hard breaking of
conformal symmetry by the conformal anomaly Callan:1970yg ; Symanzik:1970rt ; Capper:1974ic
(i.e., the running of coupling constants) plays the crucial role.
Finally, we mention that the supersymmetric models discussed
in Harigaya:2012pg ; Harigaya:2014sua ; Harigaya:2014wta ; Schmitz:2016kyr ; Domcke:2017xvu ; Harigaya:2017jny ; Domcke:2017rzu 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, local gauge invariance and global scale invariance. Then, along the lines of the Nambu–Jona-Lasinio (NJL) model Nambu:1961tp ; Nambu:1961fr , 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 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 Akrami:2018odb . Our model contains four independent parameters, of which two are used to fix the values of the reduced Planck mass and the amplitude 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

### ii.1 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,

(1) |

Here, denotes the Ricci curvature scalar, is the Ricci tensor, is the Riemann tensor, is the matrix-valued field-strength tensor of the gauge theory, and is the complex scalar field in the fundamental representation of , with being the covariant derivative. Note that the action also features an accidental global symmetry corresponding to number conservation. The last two terms in Eq. (1) can be written as

(2) |

where is the Gauß-Bonnet term, which we will suppress because the corresponding action is a surface term. The term in Eq. (2) can be absorbed into a redefinition of the coefficient , so that the last two terms in Eq. (1) can be replaced by a term of the form , which, multiplied with , is invariant under local conformal transformations, . denotes the Weyl tensor, and

In our scenario Kubo:2014ova ; Kubo:2015cna ; Kubo:2015joa , the condensation of the gauge-invariant scalar bilinear 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 weinberg (see, e.g., Eichhorn:2018yfc 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. Bardeen:1985sm , 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 coupling in our model, a certain set of fermion fields should be introduced Cheng:1973nv . 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 Osterwalder:1977pc ; Fradkin:1978dv . In the following, instead of investigating this problem from first principles, we will rely on an effective theory for the order parameter (strictly speaking, it is an approximate order parameter Osterwalder:1977pc ; Fradkin:1978dv ). Let us outline the basics of the effective theory that has been proposed in Kubo:2015cna ; Kubo:2015joa by applying the concepts of the NJL model Nambu:1961tp ; Nambu:1961fr , 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 , similar to the constituent quarks in the NJL model,

(3) |

This is again the most general action that is consistent with general diffeomorphism
invariance, global symmetry, and classical global scale invariance.^{2}^{2}2An attempt at taking into account the effect of confinement
was made in Kubo:2018vdw .
Note that the couplings , , and
in the fundamental action are not the same as , ,
and in the effective action , because the latter are
effective couplings that are dressed by hidden-gauge-field contributions.
Introducing the auxiliary field (mean field) , we can rewrite
Eq. (3) as

(4) |

The action in Eq. (3) can be obtained from Eq. (4) by inserting the equation of motion for the auxiliary field ,

(5) |

To obtain the effective potential for , we integrate out the fluctuations of the field around its background value ,

(6) |

where .
We used dimensional regularization together with renormalization
according to the modified minimal subtraction scheme,^{3}^{3}3The Gaussian path integral has been performed assuming
a flat spacetime metric.
If the fluctuations around the flat metric are taken into account, a term
as well as an additional term will be generated
at one-loop order tHooft:1974toh ; Casarin:2018odz .
They can be absorbed into redefinitions of and
in Eq. (4), respectively, if the topological term is ignored.
where the divergences are absorbed in the coupling constants as follows,
,
,
,
together with the shift
.
, where

If we assume , the absolute minimum of is located at

(7) |

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 generate the Planck mass dynamically,

(8) |

where is the reduced Planck mass. Note that the potential value at the minimum in Eq. (7) is given by

(9) |

Therefore, the spontaneous breaking of scale symmetry generates a negative vacuum energy, which is for , , . This implies, however, that the assumption made above, , is not consistent, unless the vacuum energy density vanishes, because what we are looking for is the absolute minimum of , and is not satisfied for Eq. (7). The zero-point energy density 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 , so that the potential has zero cosmological constant. Consequently, the effective potential at the minimum with nonvanishing can be written as

(10) | ||||

from which we see that the Planck mass is correctly identified in Eq. (8) and that the parameter is shifted to

(11) |

where Eq. (7) has been used.

### ii.2 Effective action for the dilaton field at low energies

The mean field is a scalar field with canonical mass dimension two, and it can describe the excitations above the vacuum expectation value . At the one-loop level, the mean field obtains a kinetic term, and the excitations can be described by a propagating real scalar field. Suppressing the complex scalar , we proceed with the effective action

(12) |

where

(13) | ||||

The wave function renormalization constant is given by

(14) |

An explicit derivation of this result for the wave function renormalization constant can be found in Kubo:2016kpb . Next, we proceed from with canonical dimension two to the canonically normalized field with canonical dimension one, i.e.,

(15) |

where is defined in Eq. (11). We find that the effective action in Eq. (12) can be written as , where

(16) |

Here, the ellipsis stands for higher-order terms such as , , etc. The dilaton-dependent functions , , and are given by

(17) | ||||

(18) | ||||

(19) |

and are defined in Eqs. (9) and (11), respectively. To arrive at in Eq. (16), we used Eqs. (7) and (15) to find

(20) |

in Eq. (13) is obtained from Eq. (6) by expanding in powers of . This expansion makes sense only if , which means . Otherwise, it is impossible to generate Einstein gravity. Furthermore, even if is satisfied, the coefficient of in Eq. (16) still vanishes as soon as .

## Iii Dilaton-scalaron inflation

### iii.1 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 ,

(21) |

Here, and denote the inverse and the determinant of the Jordan-frame spacetime metric , respectively. By construction, the kinetic term of the dilaton field 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 given in Eq. (17) parametrizes the nonminimal coupling of the dilaton field to the Ricci scalar , while the dimensionless function given in Eq. (18) denotes the field-dependent coefficient of the term. The dimensionful function 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 with mass dimension two that allows us to remove the term,

(22) |

Indeed, varying this Lagrangian w.r.t. the auxiliary field, , results in , which restores the Lagrangian in Eq. (21). In a second step, we perform a Weyl rescaling of the metric, , where we choose the conformal factor such that the kinetic term for the gravitational field obtains its standard Einstein-Hilbert form,

(23) |

At this point, it is helpful to remember that the Ricci scalar behaves as follows under a Weyl transformation,

(24) |

Neglecting the total derivative in this expression, we thus arrive at the following Einstein-frame Lagrangian,

(25) |

where denotes the scalar potential in the Einstein frame,

(26) |

The Lagrangian in Eq. (25) indicates that the function , which we extracted from the Jordan-frame Ricci scalar 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 ; the canonically normalized scalaron is defined as follows,

(27) |

Making use of this definition, the conformal factor can be written as an exponential function of the scalaron field,

(28) |

Similarly, thanks to Eqs. (23) and (27), the auxiliary field can be traded for a function of the fields and ,

(29) |

The Einstein-frame Lagrangian for the coupled dilaton-scalaron system thus obtains the following form,

(30) |

where the Einstein-frame scalar potential as a function of and now reads

(31) |

This scalar potential can be used as the starting point for the analysis of slow-roll inflation.

### iii.2 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 and . 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 ,

(32) |

where we introduced the dilaton-dependent function , which will allows us to simplify our notation in the following,

(33) |

The presence of this extremum motivates us to seek inflationary solutions along the following contour in field space,

(34) |

where can be regarded as the physical inflaton field that parametrizes the one-dimensional contour in the two-dimensional field space. Along this trajectory, the dilaton-scalaron system can be effectively described by a single-field model in terms of the real scalar . Evaluating Eqs. (30) and (31) along , we obtain the following Lagrangian,

(35) |

Note that the field is not canonically normalized; its kinetic term is multiplied by the field-dependent function ,

(36) |

Meanwhile, the scalar potential for the noncanonically normalized inflaton field obtains the following form,

(37) |

The canonically normalized inflaton field follows from integrating the function over an appropriate field range,

(38) |

The relation between the three scalar fields , , and is illustrated in Fig. 1 for a particular benchmark point in parameter space. The left panel of this figure shows a contour plot of the scalar potential in Eq. (31) as a function of and , together with the inflationary trajectory in Eq. (34), while in the right panel of this figure, we plot the effective potential during inflation in Eq. (37), , as a function of .

Let us now discuss the various conditions that need to be satisfied so that the effective Lagrangian in Eq. (35) can be used as a starting point for the description of inflation. (i) A first obvious condition is that inflation can only occur along the contour if the extremum in the scalaron direction corresponds to a local minimum. In this case, inflation will proceed along a slightly sloped valley. Otherwise, i.e., if the extremum corresponds to a local maximum, the contour will describe an unstable ridge in the potential landscape. Therefore, a necessary condition for successful inflation is that, along the trajectory , the scalaron direction is always stabilized by a positive mass squared,

(39) |

Making use of the function introduced in Eq. (33), the scalaron mass during inflation can be written as

(40) |

Successful inflation requires a positive vacuum energy density, .
The expression for the scalar potential in Eq. (37) thus implies
that must always be larger than .
For this reason, the condition of a positive scalaron mass
squared in Eq. (39) translates into the statement
that the function must always be positive during inflation,^{4}^{4}4 implies in turn that cannot become negative.
As a matter of fact, thus never obtains values in the interval .

(41) |

We remark that the radiative corrections to the Lagrangian parameter that are encoded in the function always come with a negative sign. As a consequence, it is, in fact, impossible to satisfy the condition for all values of the scalar field . As soon as exceeds a certain critical value , the function turns negative,

(42) |

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 . For these reasons, we content ourselves with the requirement that should be at least as large as the Planck scale,

(43) |

This condition ensures that the available field range in the direction will always be large enough to support inflation.

(ii) Merely requiring to be positive is not enough to sufficiently stabilize the scalaron at . In fact, from the perspective of slow-roll inflation, we can only ignore the motion in the scalaron direction away from the trajectory if the scalaron mass is large compared to the inflationary Hubble rate for all times during inflation,

(44) |

Here, we used the standard expression for in the slow-roll approximation,

(45) |

In view of Eq. (44), we therefore conclude that the function must always remain small during inflation, .

Once the condition 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 . In this case, we may alternatively look for local minima in the dilaton direction and attempt to construct an inflationary trajectory along these minima,

(46) |

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 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 . After iterations, this leads to the following expression for ,

(47) |

with the logarithmic terms after iterations, and , being defined as follows,

(48) |

The initial value is arbitrary and must be chosen by hand. For definiteness, we may, e.g., choose such that . 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 to high precision. Moreover, we are able to show that, for , the contours and 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 , inflation along the contours and leads to very similar predictions. For this reason, we will from now on only focus on inflation along the trajectory . In the regions of parameter space where , this is still a sufficiently good approximation. A dedicated numerical analysis of inflation along the contour 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 . The mass squared of the scalaron field is given in Eq. (40). The mass squared of the canonically normalized dilaton field as well as the dilaton-scalaron mass mixing parameter can be calculated as follows,

(49) |

Here, we accounted for the noncanonical normalization of the dilaton field in Eq. (30) by means of the factor ,

(50) |

For simplicity, we shall neglect the field dependence of for the time being and simply work with a symmetric mass matrix, , in the following. An explicit computation of the partial derivatives in Eq. (49) results in

(51) |

Together, the mass parameters , , and can be used to determine the scalar mass eigenvalues during inflation. Typically, we find one heavy mass eigenstate, , as well as one light tachyonic mass eigenstate, . 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 . In fact, an important condition for the validity of our analysis is that the mass eigenvalue never exceeds the Planck scale,

(52) |

Otherwise, we would no longer be able to trust our standard quantum field theory analysis. In our numerical study of slow-roll inflation [see Sec. III.3], we explicitly compute along the inflationary trajectory at each point in parameter space and check whether the condition in Eq. (52) is always fulfilled. As it turns out, sometimes does become as large as , but the bound in Eq. (52) is never severely violated. We thus argue that the mass spectrum of our model is under control and that we can trust our analysis in all relevant regions of parameter space.

(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 field 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 as a function of and to study its properties in the most general terms. However, to do so, one would have to invert the relation between and 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 in powers of the inflaton field,

(53) |

where we made use of the fact that the origin corresponds to a global Minkowski vacuum, , by construction. Thanks to the chain rule, the coefficients in this Taylor expansion can be computed solely in terms of the known expressions for and as functions of the noncanonically normalized inflaton field ,

(54) | ||||

Based on these results, one can show that the self-interactions of the field 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 , i.e., the normalization of the Jordan-frame scalar potential in units of the Planck mass,

(55) |

As long as this parameter is small, the coefficients automatically obtain small values in Planck units. With regard to the quartic self-coupling , this means in particular that a sub-Planckian potential energy density in the Jordan frame, , readily results in perturbatively small values, . In our numerical slow-roll analysis, we are able to confirm these results by explicitly evaluating the coefficients at each point in parameter space.

### iii.3 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 and the spectral tilt of the scalar power spectrum as well as the tensor-to-scalar ratio . These three observables are measured or constrained by the latest data from the PLANCK satellite mission Akrami:2018odb . At C. L., the scalar amplitude is measured to be

(56) |

In the following, we will work with the best-fit value, , for definiteness. The combined constraints on and for various combinations of data sets are shown in the right panel of Fig. 2.

Our predictions for , , and can be conveniently computed in terms of the slow-roll parameters and ,

(57) |

where all field-dependent quantities (, , and ) are supposed to be evaluated at the time of CMB horizon exit, i.e., -folds before the end of inflation. The parameters and are defined in terms of partial derivatives of the scalar potential w.r.t. the canonically normalized inflaton field . However, thanks to the chain rule, these derivatives can again be readily expressed as derivatives w.r.t. the noncanonically normalized field ,

(58) | ||||

The inflaton field value at the time of CMB horizon exit can be computed by solving the slow-roll equation of motion,

(59) |

with the field value at the end of inflation, , defined by the requirement that .

Together, the relations in Eqs. (57), (58), and (59) provide us with the necessary tools to determine our theoretical predictions for , , and . Because of the complicated form of the functions and 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 of the scalar quark field , (ii) the strength of the nonminimal coupling between the field and the Ricci scalar , and (iii) the coefficient of the bare 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 after integrating out the strong 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, . For definiteness, we will take this parameter to be the nonminimal coupling constant . This leaves us with a two-dimensional parameter space spanned by and , in which we can compute our predictions for and . 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 inflation Starobinsky:1980te . This connection can be illustrated in the following intuitive way: First of all, let us consider the orientation of the inflationary trajectory 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 or by the functions , , and ? A convenient way to answer these questions is to introduce the inclination angle ,

(60) |

which measures the angle between the dilaton axis in field space and
the inflationary trajectory.^{5}^{5}5Here, we consider the
dimensionless quantities and as the relevant
field variables in the scalar potential.
Only after this rescaling of the fields and are we able to compare
the importance of the functions , , , and in a meaningful way.
An inclination angle close to indicates that the dynamics of inflation
are mostly governed by the dilaton-dependent functions in the scalar potential.
Conversely, an inclination angle close to 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 as a function of
and by the color shading in the left panel of Fig. 2.

Interestingly enough, converges towards as soon as approaches a characteristic value of . This is nothing but the -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 , the value of that is required to obtain the correct scalar amplitude decreases. According to Eq. (17), this results in a large value of , such that

(61) |

As a consequence, the scalar potential in Eq. (31) approaches the well-known potential of inflation Starobinsky:1980te ,

(62) |

In this potential, we recognize the radiatively corrected parameter as the coefficient of the term in inflation. The requirement of a correctly normalized scalar power spectrum then uniquely fixes at a value of ,