Scale invariant extension of the Standard Model: a nightmare scenario in cosmology

Inflationary observables of a classically scale invariant model, in which the origin of the Planck mass and the electroweak scale including the right-handed neutrino mass is chiral symmetry breaking in a QCD-like hidden sector, are studied. Despite a three-field inflation the initial-value-dependence is strongly suppressed thanks to a river-valley like potential. The model predicts the tensor-to-scalar ratio r of cosmological perturbations smaller than that of the R 2 inflation, i.e., 0.0044 ≳ r ≳ 0.0017 for e-foldings between 50 and 60: the model will be consistent even with a null detection at LiteBird/CMB-S4. We find that the non-Gaussianity parameter f NL is O(10-2), the same size as that of single-field inflation. The dark matter particles are the lightest Nambu-Goldstone bosons associated with chiral symmetry breaking, which are decay products of one of the inflatons and are heavier than 109 GeV with a strongly suppressed coupling with the standard model, implying that the dark matter will be unobservable in direct as well as indirect measurements.


Introduction
The idea of cosmic inflation [1][2][3][4] is fully consistent with the Planck and BICEP/Keck data of CMB measurements [5][6][7].Their results show that the scalar spectral index n s of the gravitational fluctuations is close to one, and that the ratio r of tensor to scalar power spectra of fluctuations is very small; r < ∼ 0.035 at 95% confidence level [7].Accordingly, the R 2 inflation-the Starobinsky inflation- [8][9][10] and also the Higgs inflation [11] seem to be the most promising candidate models [6].In fact, future experiments [12][13][14] are planned to arrive at an accuracy of O(10 −3 ) for r, where the aforementioned models predict r to be of the same order.
The main reason for the success of these models is that their inflationary scalar potential is super-flat, when expressed in the Einstein frame after a local Weyl scaling from the Jordan frame [11,15] .The super flatness in the scalar potential in both models is caused by the relative suppression of the non-scale-invariant Einstein-Hilbert term, R, compared, respectively, to the scale invariant term of the Starobinsky inflation, γR 2 with γ ∼ O(10 9 ) [15,16] and to that of the Higgs inflation, β|H| 2 R with β ∼ O(10 4 ) [11], where R is the Ricci curvature scalar and H is the standard model (SM) Higgs doublet.(We note however that a super-flat potential can be realized for the Higgs inflation even for β ∼ O (10) if the RG improved Higgs potential is used [17,18].)Thus, we may interpret the results of the Planck and BECEP/Keck experiments as suggesting a scale invariant extension of Einstein's theory of gravity.Along this line of thought a number of models have been recently proposed .These models involve a multi-field system for cosmic inflation.Therefore, the inflation trajectory that describes the background Universe is in a multi-dimensional space of fields, which means that the initial value of the trajectory is also multi-dimensional.Note that there is no warranty a priori that the predictions of inflationary parameters do not largely depend on the multidimensional initial values.However, to our knowledge, detailed investigations of the initialvalue-dependence in the models mentioned above have not been made.
If the inflationary potential of a multi-field system has a river-valley like structure, there are a slow-roll direction (the direction along the river) and also the direction(s) perpendicular to it (the heavy-mode direction(s)) [25,43], where slow-roll conditions are likely violated in the heavy-mode direction(s).Furthermore, although the background Universe is described by the multi-field system, we have to take care of the fact that the filed fluctuations of the heavy mode(s) do not produce perturbations on cosmologically relevant scales [44,45].
In this paper, we revisit the classical scale-invariant model of Ref. [41], in which a strongly interacting QCD-like hidden sector is introduced and the mass scale is generated in a non-perturbative way via condensation that breaks chiral symmetry dynamically [46][47][48] in the hidden sector [49][50][51][52][53][54].
In Section 2 we start by briefly describing the model.In particular, we review how we use the Nambu-Jona-Lasinio (NJL) model [46][47][48] as an effective low-energy theory in a mean field approximation.In Section 3 we discuss inflation.First we derive the effective action for inflation in the Jordan frame.We next go to the Einstein frame and discuss the rivervalley like structure of the inflationary potential to reduce a three-field system to a two-filed system for inflation.Then we consider the two-field system by solving coupled equations of motion numerically and analyze the initial-value-dependence.We also identify the light mode (corresponding to the slow-roll direction) and the heavy mode to see how the slowroll conditions are violated.Before we proceed with this issue, we briefly outline the basic ingredients of the δN formalism [55][56][57][58][59][60].Then, adjusting the δN formalism to the aforementioned circumstance, we develop an algorithm to evaluate numerically the derivatives of e-foldings N with respect to background fields to compute inflationary parameters including the non-Gaussianity parameter f N L [61,62] of the primordial curvature perturbations.
In Section 4, we discuss a particular dark matter (DM) candidate in the hidden sector, the Nambu-Goldstone (NG) bosons associated with the chiral symmetry breaking, which are produced through the decay of one of the inflatons [63][64][65].The DM relic abundance depends crucially on the reheating temperature, which we estimate using the method of Refs.[6,[66][67][68].Section 5 is devoted to Conclusion.

Four elements
The model of Ref. [41] that we consider consists of four elements, which describe, respectively, (i) the generation of a robust energy scale through the formation of chiral condensate, (ii) gravity together with cosmic inflation, and (iii) the SM interactions which are slightly extended because of the Higgs portal coupling, (iv) including the right-and left-handed neutrinos: where we have imposed scale invariance, meaning that no element contains any dimensionful parameter at the classical level.In the Lagrangian (2.1), F is the matrix-valued field-strength tensor of the SU (N c ) H gauge theory, coupled with the vector-like hidden fermions ψ i (i = 1, . . ., n f ) belonging to the fundamental representation of SU (N c ) H , 1 and S is a real SM singlet scalar, where we assume N c = n f = 3.For the part (ii) we have suppressed the Ricci curvature tensor squared, R µναβ R µναβ , because it (and also R µν R µν ) can be written as a linear combination of R 2 , the Weyl tensor squared W µναβ W µναβ and the Gauß-Bonnet term which is a surface term.The chiral condensate ⟨ ψψ⟩ generates a linear term in S [49][50][51][52], leading to a nonzero VEV of S denoted by v S , which is responsible for the origin of the Planck mass as well as the right-handed neutrino mass m N = y M v S . 3We assume that m N ∼ 10 7 GeV to obtain a desired size of the radiative correction to the Higgs mass term [71][72][73][74] for triggering the electroweak symmetry breaking and at the same time to make the type-I seesaw mechanism [75][76][77][78] viable -a scenario dubbed the "Neutrino option" [79] (see also [24,[80][81][82][83][84][85][86]).Further, the Yukawa coupling in (2.1) violates explicitly chiral symmetry.Consequently, the (quasi) NG bosons, associated with the dynamical chiral symmetry breaking, acquire their masses and can become a DM candidate due to a remnant unbroken vector-like flavor group that can stabilise them [49][50][51][52].
In brief summary, the real scalar field S, involved in all the sectors, first transfers the robust energy scale, created by the chiral condensate in the hidden QCD-like sector, to the gravity sector and at the same time plays a role of inflaton and produces DM trough its decay.Another channel of the energy scale transfer is the right-handed neutrino, which becomes massive through the Yukawa coupling S-N -N , giving rise to the electroweak symmetry breaking as well as to the light active neutrino mass.
The one-loop effective potential can be obtained from L MFA by integrating out the hidden fermions: where the function I 0 is given by with a four-dimensional momentum cutoff Λ. 6 Note that the cutoff parameter Λ is an additional free physical parameter in the NJL theory.We obtain non-zero VEVs, i.e., v σ = ⟨σ⟩ ̸ = 0 and v S = ⟨S⟩ ̸ = 0, for a certain interval of the dimensionless parameters G 1/2 Λ and (−G D ) 1/5 Λ [51,53,54]: The non-zero v σ means the chiral symmetry breaking in the NJL theory.
The actual value of Λ can be fixed, once the hidden sector is connected with a sector whose energy scale is known.In our case, the hidden sector is coupled via the mediator S with the gravity sector (ii) described by the Lagrangian (2.2) as well as with the right-handed neutrino sector (iv) described by the Lagrangian (2.4), while the coupling with the SM sector (iii) is assumed to be extremely suppressed, because we assume that the portal coupling λ HS is negligibly small.The cutoff in the hidden sector Λ H can be fixed in the following way.
The NJL parameters for the SM hadrons satisfy the dimensionless relations [51,53,54] G 1/2 Λ Hadron = 1.82 and (−G D ) 1/5 Λ Hadron = 2.29 , where Λ Hadron ≃ 1 GeV.We assume that the above dimensionless relations remain satisfied while scaling-up the values of G, G D and the cutoff Λ from QCD hadron physics.That is, we assume that (2.9) are valid even if Λ Hidden = Λ H is much larger than Λ Hadron .Therefore, the free parameters G, G D are fixed when the hidden sector scale Λ H is fixed.Consequently, the only free parameter in the hidden sector is the Yukawa coupling y once Λ H is fixed.Since the (reduced) Planck mass becomes M Pl ≃ √ βv S and we can write v S = c S (y) Λ H , while the dimensionless quantity c S is calculable for a given y, we can always relate Λ H to M Pl .
We recall that the cutoff scale Λ H is the energy scale around which the hidden QCD-like sector starts to be strongly interacting so that we may use the NJL theory as a low-energy effective theory below that scale.We note however that by this energy scale a dynamical energy is meant, because static energy like the zero point energy of a potential does not influence QCD-like interactions.Since dynamical energies are created after the end of inflation, we should require that the scale of these energies be less than Λ H . Further, the spontaneously generated Planck scale M Pl is not a dynamical scale, but the VEVs, v σ and v S , should be less than Λ H .We also note that it is not only the energy scale, which plays the role for the applicability of the NJL theory, but also the scale of the explicit chiral symmetry breaking, because the NJL theory is an effective theory for chiral symmetry breaking.In our case it is the Yukawa coupling y, while in QCD the current quark masses are responsible.Therefore, since m s (strange quark mass)/Λ Hadron ≃ 0.09 GeV/1 GeV ≃ 0.1, the self-consistency requires yS/Λ H < ∼ 0.1 during inflation, which we shall check when discussing inflation.
We emphasize that the mean fields σ and ϕ a are non-propagating classical fields at the tree level.Their kinetic terms are generated by integrating out the hidden fermions at the one-loop level.Furthermore, one can see that the potential V NJL (S, σ) is asymmetric in σ, which is the reason that the chiral phase transition can become of first-order in the NJL theory [91][92][93].

Planck mass
In this subsection we discuss how the Planck mass is generated.We may start by integrating out the quantum fluctuations δS at one-loop to obtain the effective potential where m2 s = 3λ S S 2 + βR, and we have employed the MS scheme, while the constant −3/2 is absorbed into the renormalization scale µ. 7 So, the total effective potential in the Jordan frame is given by U eff (S, σ, R) = V NJL (S, σ) + U S (S, R) − U 0 , where V NJL is given in Eq. (2.7), and U 0 is the zero-point energy density.We have subtracted it, such that U eff (S = v S , σ = v σ , R = 0) = 0 is satisfied.Note that the zero-point energy density U 0 is the cosmological constant at the tree-level (which breaks super-softly the scale invariance).In other words, we put the cosmological constant problem [95] aside and ignore the existence of the problem.
To compute v S = ⟨S⟩ and v σ = ⟨σ⟩, we assume that βR < 3λ S S 2 (during inflation), such that U S (S, R) in Eq. (2.10) can be expanded in powers of βR: where We have chosen µ 2 = v 2 S /2, because for this choice the logarithmic term in (2.12) does not enter into the determination of v S and v σ .Finally, the identification of M Pl follows from the first term in Eq. (2.2) along with Eq. (2.11): Since β is larger than O( 102 ) for a successful inflation, v S is at most ∼ M Pl /10.

Effective action for inflation
If the inequality βR < 3λ S S 2 is satisfied during inflation, the higher order terms in Eq. (2.11) can be consistently neglected for inflation.We will proceed with this simplification.Similarly, if κ, the coefficient of the W µναβ W µναβ term in the Lagrangian (2.2), is small, this term has only a small effect on the inflationary parameters (see for instance Refs.[20,33,39,42,[96][97][98] and also [99]), so that we will ignore it in the following discussion.Hence the effective Lagrangian for inflation in the Jordan frame can be written as where the subscript J means quantities in the Jordan-frame, and To go the Einstein frame we first rewrite the R 2 J term in the Lagrangian (3.1) by introducing an auxiliary field χ with mass dimension two and by replacing G(S)R 2 J by 2G(S)R J χ − G(S)χ 2 .Then we perform a Weyl rescaling of the metric, Pl .Using the scalaron field φ [100, 101], which is canonically normalized and defined as φ = 3  2 M Pl ln Ω 2 , we finally obtain the Einstein-frame Lagrangian for the coupled S-σ-scalaron system: where8 with Φ (φ) = 2/3 φ/M Pl .The form of the potential (3.5) is quite general: One obtains it, after going to the Einstein frame if a R 2 term is included in the Lagrangian and is converted into a term linear in R by introducing an auxiliary field in the Jordan frame.
The auxiliary field becomes a dynamical degrees of freedom, the scalaron [8], in the Einstein frame.(See e.g. the scale invariant models of [19-24, 28, 29, 31, 33, 34, 38-42].)Note however that the definition of the scalaron is not always the same, where we adapt the definition of Refs.[100,101].The main difference of our potential U (S, σ) from those of the other scale invariant models is that U (S, σ) contains a linear term in S, which originates from the NJL part V NJL (S, σ).This will be the main reason that our model can predict a tensor-to-scalar ratio r which is smaller than that of the R 2 infaltion, as we will see in 3.3.6.

River-valley like structure of the potential
As we see from the effective Lagrangian (3.4) we have a multi-field system at hand [102].It has been found in Ref. [41] that the potential (3.5) has a river-valley like structure, such that the valley approximation [21,24,34] can be successfully applied.In this approximation the multi-field system can be reduced to a single-field system, which we have analyzed in Ref. [41].
To see the river-valley like structure, we solve the stationary point condition ∂V (S, σ, φ)/∂φ | φ=φv = 0 with respect to φ to obtain We then consider the two-field system potential Ṽ (S, σ) = V (S, σ, φ v (S, σ)).In The red line in Fig. 1 is the bottom line of Ṽ (S, σ), from which we see that σ does not change very much during inflation.We will use indeed this fact in the following sections to reduce the three-field system to a two-field system: We will assume that σ stays at v σ during inflation.
We next consider the potential V (S, σ v (S), φ), where σ v (S) is the bottom line of Ṽ (S, σ), i.e., the red line in Fig. 1 .Its contour plot is shown in Fig. 2 (left) for the same set of the parameters (3.7).The green line is φ v (S, σ v (S)), i.e., (3.6) with σ replaced by σ v (S), while the dotted red line presents the true bottom line of V (S, σ v (S), φ).In the valley approximation in Ref. [41] it is assumed that inflaton rolls down along the green line (see also Ref. [24]).
To figure out the difference between V (S, σ v (S), φ) and V (S, σ = v σ , φ), we consider the bottom line of these potentials.We denote the bottom line of V (S, σ v (S), φ) (the dotted red line in Fig. 2

Background system
The starting Lagrangian, that describes our flat and homogeneous background Universe in the Einstein frame, is (3.4), and we use it to compute inflationary observables including the non-Gaussianity of the curvature perturbation [45] (see also [103]).The original system contains three fields, S , σ and φ.As we see from Fig. 1, σ stays approximately constant during inflation.Therefore, we set σ equal to v σ to simplify the numerical integration of the equations of motion, where v σ = ⟨σ⟩ can be computed from the potential (3.3) together with v S = ⟨S⟩.Then we rewrite the Lagrangian (3.4) (with σ replaced by v σ ) in a geometric form using the metric G IJ in the space of the fields: where the "coordinates" are ϕ I = (φ , S), and the non-vanishing elements of G IJ are The equations of motion, that follow from (3.8), are: with the non-vanishing components of Γ I JK : In terms of the component fields the equations of motion (3.10) become where H is the Hubble parameter.In the subsequent discussions we assume that the fields are homogenous and depend only on time t.Under this assumption the Hubble parameter H satisfies in the Friedmann-Lemaître-Robertson-Walker metric, i.e., ds 2 = dt 2 − a 2 (t) dx 2 .Similar models have been considered for instance in Refs.[104,105] to calculate the non-Gaussianity of the curvature perturbation.The crucial difference compared with Refs.[104,105] is that our potential V (ϕ I ) in (3.8) boasts a river-valley like structure, which causes complications in calculating the inflationary parameters without using the valley approximation, as we will see.
We solve Eqs.(3.12) and (3.13) numerically with a set of initial values of φ and S at t = t 0 .The end of inflation is defined such that ϵ(t end ) = 1, where the slow-roll parameter ϵ(t) is defined as and the number of e-folds N can be calculated from During the numerical integration of Eqs.(3.12) and (3.13) we check whether the slow-roll conditions ϵ(t) ≪ 1 and β sr (t) ≪ 1 for t < t end are satisfied (except for t close to t end ), where [56] (The another slow-roll parameter η can be obtained from η = 2ϵ + 2β sr .)

Exact solution vs valley approximation
By an exact solution we mean the solution obtained by the numerical integration of Eqs.(3.12) and (3.13).In Figs. 3 and 4 we show a representative exact solution, where the input parameter values are: which give Λ H = 5.89 × 10 −2 M Pl and The initial filed values at t = t 0 and at the end of inflation for the trajectory (blue) plotted in Figs. 3 and 4   and Ṡ0 = φ0 = 0.The left panel of Fig. 3 shows the trajectory at the beginning (i.e., t ≃ t 0 ), from which we see that it is strongly oscillating, especially in the S direction.In the right panel we compare two trajectories with different initial values; the blue one is for (3.20) and the red one is for S 0 /v S = 6.06 , φ 0 /M Pl = 5.36.We see that the trajectories converge very fast to a "fixed-point" trajectory.This fixed-point solution (trajectory) is in fact very close to the approximate valley solution (3.6).Fig. 4 presents the exact solution (blue) and the valley solution (red) in later time, i.e., from just before the horizon exit to the end of inflation.This is a desired feature on one hand, because, if it converges to a fixed-point solution before the horizon exit, the dependence of the initial field values at t = t 0 do not influence the observable inflationary parameters. 9On the other hand, this means that we have effectively a single-field system, so that we expect that the non-Gassianity of the primordial curvature perturbations will be very small [62,106]. 10This is a prediction of our model.In the subsequent sections we will verify explicitly this expectation without using the full valley approximation (a single-field approximation).However, there is a complication in computing the inflationary parameters using the δN formalism [55][56][57][58][59][60], which we will point out below, and we will come back to this problem when computing the non-Gaussianity of the primordial curvature perturbations explicitly in 3.3.4.The valley structure can be quantitated by which is a mass matrix (Hessian) normalized by V /M 2 Pl .Therefore, for a prototype of the river-valley like potential, there is a small as well as a large eigenvalue of η IJ ; the small one is the second order change of the potential along the river at the bottom of the valley, and the large one is that of the direction normal to the river. 11Note that η IJ is also a slow-roll parameter.This means that the slow-roll condition on the river-valley like potential is necessarily violated in the direction normal to the river.As a result, the second time derivative of ϕ in that direction, φ⊥ , can be large.That is, can not be used to express φ⊥ in terms of ϕ I .The fact that φ⊥ as well as a mass eigenvalue is large means that the background trajectory is oscillating while slowly rolling down along the river.Note that the violation of (3.23) and (3.24) does not contradict with ϵ , β sr ≪ 1, where ϵ and β sr are give in Eqs.(3.15) and (3.17).The main reason for this is that φ⊥ can be much smaller than RHS of Eq. (3.24), such that φ⊥ φ⊥ in β sr can be small even if φ⊥ is large [105].
In the following figures, using the representative example (3.18) of the set of parameters, we demonstrate the situation described above.If we assume that N = 55 for the representative example (3.18), then the filed values and their derivatives at t = t * should be In the left panel of Fig. 5 we show the zoomed trajectory near S = S * (i.e., t = t * ).We see that the trajectory is still slightly oscillating.In the right panel, the slow roll parameter β sr (3.17) is plotted, which clearly shows that β sr ≪ 1 is satisfied.We have checked that this is satisfied not only for this period, but also for all the time during inflation.The second derivative of ϕ I with respect time in the period near t = t * is plotted in Fig. 6, where the left panel is for S and the right one is for φ.We see from the left panel that the condition (3.23) is clearly violated in the S direction, while it is satisfied in the φ direction.
A consequence of the river-valley like potential is that we can not eliminate φ⊥ using Eq.(3.24), which has an implication in calculating the derivatives of N with respect ϕ ⊥ in  the δN formalism.How we overcome this problem will be explained in the subsections 3.3.4and 3.3.5.
Before we close this subsection we make a following remark about the applicability of the NJL theory as an effective theory.This subject has been briefly touched in 2.2.Using concrete numbers given in (3.19)-(3.21)and also (3.25) for the benchmark point (3.18), we find that v σ /Λ H < 1, v S /Λ H < 1, (yS 0 , yS * )/Λ H < 0.1 are all satisfied.We have made this consistency check for other sets of parameters and found that the above requirement is satisfied for all the cases.

δN formalism
Here we briefly summarize the basic ingredients of the δN formalism [55][56][57][58][59], and we follow Ref. [60] below.We assume that the unperturbed Universe can be described by the Friedmann-Lemaître-Robertson-Walker metric which will be perturbed.The metric of the perturbed Universe in the Arnowitt-Deser-Misner (ADM) form is given by where γ ij can be written as and ψ is the spatial curvature perturbation.We assume that the scalar and vector perturbations in the tensor perturbations are eliminated (gauged away) so that h ij is traceless and transverse.A physical quantity, relevant to our purpose, is the curvature perturbation on constant density slice, ζ.In the language of Ref. [60], ζ is ψ in the uniform-density gauge, that is, ψ| UD = ζ.The perturbed system can be analyzed by expanding the non-linear field equations in the number of spatial derivatives, which corresponds to the expansion in ξ (≪ 1), where ξ is the ratio of the comoving wavenumber k and the comoving Hubble scale aH.It has been shown in Ref. [60] that, up to and including the next-to-leading order in the gradient expansions in the flat gauge, the perturbed scalar fields ϕ I satisfy the same equations of motion as the unperturbed ones φI , where ψ in the flat gauge vanishes, ψ| F = 0.This is regarded as a justification of the assumption of the separate Universe approach (see Ref. [109] and references cited therein) that each area in the perturbed Universe, flatted in a superhorizon scale, independently develops as an unperturbed Universe.
Since ψ| F = 0, one performs a gauge transformation from the flat gauge to the uniformdensity gauge to obtain the curvature perturbation on constant energy density slice, i.e., ψ| F = 0 → ψ| UD = ζ.The gauge transformation is a change of the time coordinate t, t → t = t + δ t, and it can be shown [60] that the spatial metric γ ij (3.28) transforms as a scalar up to and including the next-to-leading order in the gradient expansions.Therefore, from which one finds where Interestingly, ζ = δN can be related to scalar fiels perturbations [55][56][57][58][59], as we will see below.Once again, using the fact that the perturbed scalar fields ϕ I satisfy the same equations of motion as the unperturbed ones φI , we can express the energy density in a perturbed system from that in the unperturbed system.So, we perturb the homogeneous background solution in the unperturbed system by changing the initial field values φI * = φI (t * ) to a x dependent ones, i.e, φI where we have identified Q I with the tangent vector d φI * (λ)/dλ at λ = 0, the geodesic equation D λ (d φI * /dλ) = 0 is used to arrive at the second equation, and ϵ is absorbed into Q I to obtain the last line.Using Q I the curvature perturbation (3.33) can now be written as [111,112] (3.36)

Inflationary observables
As we have seen in the previous subsection, the spatial inhomogeneity (x dependence) of ζ in the δN formalism originates from the inhomogeneity of the initial field values at horizon exit, i.e., t = t * .Inflationary observables can be calculated from correlation functions of ζ [45].Our interest here is focused on the power spectrum and bispectrum [61,62] in is the Fourier component of ζ(x), and the expectation values are calculated with respect to the Bunch-Davies Vacuum.As we see from Eq. (3.35), RHS of Eq. (3.36) is an expansion in Q. Accordingly, we restrict ourselves to the power spectrum and bispectrum in the lowest nontrivial order in Q [111,112]: To proceed, we recall that the non-Gaussianity parameter f N L is defined as [61,62] f N L = 5 6 where c.p. stands for cyclic permutation of k 1 , k 2 and k 3 .Further, the Mukhanov-Sasaki equation [55,113] in the case of multi-component systems contains a matrix-valued parameter [56] (see also Ref. [111]) which is slowly varying in time during inflation.We assume that ϵ IJ * = ϵ IJ at the horizon exit t = t * remain constant during inflation (as it is assumed in Ref. [56]) and also that the first term on RHS of Eq. (3.38), the three-point correlation function, is very small [114,115].Under these assumptions, we obtain [45,55,56,59,111,112,114,115] where 1/a * H * is the comoving Hubble radius at the horizon exit.
We may now proceed with the application of the general formulae Eqs.(3.41)-(3.43) to our concrete system, where the evolution of the background Universe is described by a heavy as well as a light mode.To obtain reliable results, we therefore have to incorporate the fact that the fluctuations of the heavy scalar mode do not produce perturbations on cosmologically relevant scales [44] (see also Ref. [45]).To this end, we first define the light and heavy modes, φL and φH , by diagonalizing ϵ IJ * : which means that the original background fields φ and S are written as φ = cos θ φH + sin θ φL , S = − sin θ φH + cos θ φL .Then we perturb the background only in the light mode direction.That is, we consider only the change of the initial values of the form δϕ L * (x) ̸ = 0 but δϕ H * (x) = 0, which can be achieved in our two-component system by imposing δφ * (x)/δS * (x) = tan θ.In this way, the fluctuations from the heavy mode are suppressed at the horizon exit, and we may assume that they remain suppressed, because ϵ IJ is only slowly changing during inflation.The discussions above then lead us to where With ϕ L * and ϕ H * , the "mass term" is now diagonal, but not their kinetic term in general.However, in the present case, where the heavy mode δϕ H remains suppressed, the kinetic term for the light mode can be simply read off from G IJ φI φJ : where we have used: /3φ/M Pl and φ ′ = sin θ.

Numerical calculation of the derivatives of N
To use δN formalism, we have to compute the derivatives of N ( φI . The time axis.The background solution φI (t) starts to run at t 0 with a vanishing velocity, i.e., φI (t 0 ) = 0.A perturbed system is described by ϕ I T (t), which is an exact solution with the initial values at t = T ; ϕ I T (T ) = φI (T ) + ∆ϕ I T and φI T (T ) = φI (T ).The variation from the background solution at the horizon exit (t = t * ) can be numerically computed First we describe how we eliminate φI .The background solution is the one with the initial values set at t = t 0 , which we denote by φI (t).In addition to φI (t), we introduce ϕ I T (t) which is also an exact solution with the initial value ϕ I T (T ) = φI (T ) + ∆ϕ I T and φI T (T ) = φI (T ), where t 0 ≪ T < t * as it is illustrated in Fig. 7. Then at t = t * we calculate ∆ϕ I * and ∆ φI * from Since ϕ I T (t) is an exact solution, there exist unique ∆ϕ I * and ∆ φI * at t = t * for a given ∆ϕ T .If we regard ∆ϕ I * as independent, ∆ φI * is no longer an independent variation.Equivalently, we use φI * = φI T (t * ) to substitute Eq. (3.24).Note that we can not choose ∆ϕ I * freely, because the initial values of ϕ I T (t) are set at t = T .Therefore, we introduce a set of ∆ i ϕ I T with i = 1, . . ., M and then compute numerically for each i, where ∆ i ϕ I * is calculated from Eq. (3.51).At the same time, we can expand RHS of Eq. (3.52) in ∆ i ϕ I T : where ∆ i N and ∆ i ϕ I * are explicitly known, and the unknowns are derivatives, N I , N IJ , • • • .This means that, if we truncate the series at a certain order, RHS of Eq. (3.53) can define a system of linear equations for the unknowns.For instance, to obtain the first and second derivatives, we need five independent equations, because there are 3 + 2 = 5 unknowns, N S , N φ , N SS , N Sφ and N φφ .If we include the third derivatives, we need 5+4 = 9 equations.
Using the method described above we have computed the inflationary parameters, which are shown in Fig. 8, where we have included fourth derivatives N ′′′′ to improve the accuracy.
Figure 8.The spectral index n s (left), the tensor-to-scalar ratio r (right) and the non-Gaussianity parameter f N L (bottom) as a function of (T − t * )/t * .

Result
We recall that the result depends on y and λ S only slightly in the valley approximation [41] and may assume that this will be also the case in the present treatment.In the following discussions we therefore fix them at y = 0.0046 , λ S = 0.0114, as in (3.18), which gives and we are left with the free parameters β and γ.Furthermore, since the inflationay parameters, n s , r and β 2 A s , depend only on γ = γ/β 2 in the valley approximation [41], we may assume that this is also the case here, although the potential V (ϕ I ) for our two-field system (3.8)depends explicitly on γ and β.The reason is that the trajectories fast converge to a fixed-point trajectory, as it is shown in Fig. 4 (red), which is nearly the lowest part of the river-valley like potential (3.8) and that it depends only on γ.As we see from TABLE 1, which shows n s , r , f N L and β 2 A s at N = 50 , 55 and 60 for β = 4000 , 4500 and 5000, respectively, with γ = 20 fixed, this assumption is justified for these examples.Therefore, to simplify the calculation, we calculate the inflationary parameters first at fixed γ and β and assume that the experimental constraint [5,6] ln(10 10 A s ) = 3.044 ± 0.014 (3.57) can satisfied if we change β appropriately without changing γ.We now proceed with the presentation of our results.Fig. 9 shows the prediction (redyellow points) in the n s -r plane for γ = 3500 , 1500, 500, 100, 8 (from top to bottom), where the color represents N . 13The black-grey points are for the Starobinsky inflation.It is interesting to observe that the tensor-to scalar ratio r of the present model is smaller than that of the Starobinsky inflation.The green backgrounds are the LiteBird/Planck constraints [12] at 95% (light green) and 68% (green) confidence level; the upper one shows the constraint assuming the Starobinsky inflation with N = 51 as the fiducial model, while the lower one shows the constraint in the case of null detection of r.As we see from Fig. 9, there exists a parameter space of our model which will not be excluded by LiteBird/Planck in the case of null detection.We will arrive at the same conclusion even if we will include the CMB-S4 constraint [13,14].
Of the scale invariant models cited in [19]- [42], the models of Refs.[19-24, 28, 29, 31, 33, 34, 38-42] can be regarded as an extension of the Starobinsky model and in fact predict small values of r.However, only those of Refs.[20,29,33,[39][40][41] predict r which is smaller than that of the Starobinsky model.Therefore, the null observation at LiteBird and CMB-S4 can exclude many models based on an scale invariant extension of the Starobinsky model.Fig. 10 shows the prediction in the r-f N L plane for γ = 3500 , 1500, 500, 100, 8 (from right to left), where the black-grey points are for the Starobinsky inflation.The green (dark green) vertical line is the upper bound of r at 95 (68)% confidence level in the case of null detection [12].As we see from Fig. 10, the non-Gaussianity parameter f N L is O(10 −2 ), so that it will be too small to be observed [14,61].
Fig. 11 shows f N L as a function of n s for γ = 500, where the gray points are for the Starobinsky inflation.In the case of single-field inflation we have f N L = −(5/12)(n s − 1) [62,106] which is presented by the blue dotted line.As we have expected and we see from Fig. 10, our multi-field inflation model effectively behaves as a single-filed model.

Dark matter
Because of the vector-like flavor symmetry (i.e., SU (3) V or its subgroup), the CP-odd scalars ϕ a (the quasi-NG bosons) are stable and hence are good a DM candidate.Dark mater can be produced during or after the reheating phase (see e.g., Refs.[63][64][65]).It has been found in Ref. [41] that m ϕ > m S is satisfied in the most of the parameter space, so that the inflaton S can not decay into these scalars.However, if we break SU (3) V down to SU (2) V × U (1) and assume a hierarchy in the Yukawa couplings [54], i.e., y = diag.(y 1 , y 1 , y 3 ) with y 1 = y 2 < y 3 , ( where y is the Yukawa matrix in the hidden sector described by the Lagrangian (2.1), there is a sufficiently large parameter space in which S can decay into the lightest meson which we will identify as DM.To see this, we note that under (4.1) the quasi-NG bosons fall into three categories, π = π± , π0 , K = K± , K0 , K0 and η, where and η8 will mix with η0 = ϕ 0 to form the mass eigenstates η and η′ .The states in the same category have the same mass, m π0 = m π± (≡ m π) and m K± = m K0 = m K0 (≡ m K ), with m π < m K < m η.In the following discussion we will work in the parameter space, in which is realized and use the approximation for m DM (= m π) [117], m 2 DM /m 2 ϕ ≃ m u /m q ≃ y 1 /y, where m u is the current quark mass in the SU (2) V × U (1) case, and m q , m ϕ and y are the current quark mass, the dark meson mass and the Yukawa coupling, respectively, in the SU (3) V limit.
In the parameter space, in which the mass hierarchy (4.1) is satisfied, the inflaton S can decay only into a pair of DM particles -but not into the other mesons, and therefore the heavier dark mesons are not produced during the reheating stage and later [64].The decay width for S → π + π is given by where the effective coupling G ππS is calculated in Ref. [54] and is found to be G ππS /Λ H ≃ −0.012 y 1 for y 3 = 0.0046 and y 1 ≪ y 3 .The DM can also be produced by the co-annihilation of N , i.e., N N ↔ π π.However, its cross section is small due to the small y M (see the subsection 2.1).We have found that the DM production rate by the co-annihilation process is negligibly small compared with that through the decay of S.
Under the situation specified above we arrive at a system, which consists of only the inflaton S and the dark matter π and in which the evolution of their number densities, n S and n DM , can be described by the coupled Boltzmann equations [63] dn ) with Γ S being the total decay width of S. Eq. (4.5) can be simply solved [118]: where a is the scale factor at t > t end , a end is a at the end of inflation t end and ρ end = ρ S (a end ) = m S n S (a end ) is the inflaton energy density at t end .Then we follow Refs.[6,67,119] to proceed with the calculation of the DM relic abundance Ω DM h 2 and define the reheating temperature T RH as where g RH is the relativistic degrees of freedom at the end of reheating.Finally, we arrive at [41,64]: where the branching ratio γ DM /Γ S can be obtained from γ DM given in Eq. (4.4) together with the assumption that 1/Γ S is the time scale at the end of the reheating phase [63,118], which means 1/H(a Although ρ RH and hence T RH are unknown quantities, it is possible to constrain T RH for a given inflation model without specifying reheating mechanism 14 [6,[66][67][68]: where a * = k * /H * is the scale factor at the time of CMB horizon exit (t = t * ),V * = V (t = t * ), k * is the pivot scale set by the Planck mission [5,6], and H * is the Hubble parameter at a = a * .Further, we use where .10). 15 In Fig. 12 we show the points in the T RH − m DM plane, for which Ω DM h 2 = 0.1198 ± 0.0024 (2σ) is obtained.The dark matter mass m DM is calculated from m 2 DM ≃ (y 1 /y 3 )m 2 ϕ , where m ϕ is the meson mass in the SU (3) V limit (with y = y 3 ) and obtained in the NJL formalism [41].
Since Ω DM h 2 (4.9) depends on m DM and γ DM and hence on y 1 and is strongly constrained, they are closely correlated.We have varied y 1 for γ = γ/β 2 = 100 and 1500 with λ S and y 3 fixed at 0.0114 and 0.0046, respectively.(The corresponding r and n s can be found in Fig. 9.) As we see from Fig. 12, our DM is heavier than ∼ 10 9 GeV.Furthermore, there exist only indirect couplings with the SM sector: In addition to the gravitational interaction, the scalar S mediates an interaction, as we can see from the Lagrangians (2.3) and (2.4).But the coupling of S with the SM is extremely suppressed; the Higgs portal coupling λ HS ∼ O(10 −28 ) and the Yukawa coupling y M ∼ O(10 −9 ), implying that ourDM will be invisible, except if it can decay into SM particles [121], such that the decay is consistent with, e.g., the resent observation of an extremely energetic cosmic ray of O(10 11 ) GeV at Telescope Array [122]. 14The inflaton S can reheat the SM sector through the Yukawa coupling S-N -N , where the size of the coupling is very small in the present model, i.e., yM ∼ 10 −9 , because we implement the neutrino option mechanism to break the SM gauge symmetry.Another interesting way to reheat the SM sector may follow from the observation that the kinetic terms become non-canonical when going from the Jordan frame to the Einstein, such that the the inflaton, especially the scalaron φ in our model, interacts with the SM in this frame [21,120]. 15The energy scale (temperature) just after the end of inflation is approximately Emax ≃ ρ 1/8 end ( √ 8πM Pl ΓS/gRH) 1/4 (gRH ≃ 10 2 ) [63].We find Emax/ΛH ≃ 4.1 × 10 −5 < 1 for the benchmark point (3.18), which means that the applicability of the NJL theory as an effective theory is not violated, according to the discussion in 2.2.We have made this consistency check for other sets of parameters and found that the above requirement is satisfied for all the cases.

Conclusion
In this paper we have assumed that the origin of the Planck mass and the electroweak scale including the right-handed neutrino mass is the chiral symmetry breaking in a QCD-like hidden sector which couples with the SM sector only via a real scalar S, the mediator.The scalar S is not only a mediator, but also an inflaton, which makes a Higgs-inflation-like scenario possible.
The inflationary system that we have considered contains three scalar fields; σ (the chiral condensate in the NJL formalism), S (the mediator) and φ (the scalaron).The scalar potential (3.5) that depends on these fields has a river-valley like structure, and we have found that σ changes only slightly along the river, so that we have assumed that it stays during inflation at the position of the absolute minimum of the potential.In this way we have reduced the system to a two-field system of S and φ.Thanks to the river-valley like structure of the potential, the trajectories in the two-field system converge very fast to a fixed-point trajectory.If they converge to a fixed-point trajectory before the horizon exit, the dependence of the initial field values has only negligible effects on the observable inflationary parameters -a nice feature of river-valley like potential.
We have analyzed the river-valley like structure of the two-field system in detail to separate the heavy and light modes, because one of the slow-roll conditions is violated in the heavy mode direction of the potential and only the light mode contributes to the inflationary parameters.This implies that the system is effectively a single-field system, as we have expected from Ref. [41].Nevertheless we have applied the δN formalism [55][56][57][58][59][60] to compute the inflationary parameters, including the non-Gassianity parameter f N L of the primordial curvature perturbations.To compute the inflationary parameters of our model concretely we have adjusted the δN formalism to the river-valley like structure of the potential and developed an algorithm to compute numerically the derivatives of N with respect to background fields.
Though the scalar spectral index n s and the tensor-to-scalar ratio r of the present model are similar to those of the Starobinsky inflation [8][9][10], we have found that r is smaller than that of the Starobinsky inflation and found that the present model will be consistent with a null detection of r at LiteBird/CMB-S4, while the Starobinsky inflation will be excluded in this case [12].
Since we have effectively a single-field system at hand, the non-Gaussianity parameter f N L is expected to be O(10 −2 ) [62,106], which we have explicitly confirmed by using the δN formalism.This means that the Universe described by the multi-field cosmological system of the model in this paper is so isotropic, that the non-Gaussianity will not be measured in future experiments [14,61].It should be, however, noted that a considerable improvement of the experimental accuracy of f N L as well as of r can be achieved by using the fluctuations in the 21-cm signal from atomic hydrogen in the dark ages [123,124].
NG bosons are produced due to the chiral symmetry breaking in the hidden sector in a very much similar way as in QCD.Since the Yukawa coupling of S with the hidden fermions explicitly breaks the chiral symmetry, they are quasi-NG bosons and massive.They are stable because of the unbroken vector-like flavor symmetry and therefore can be a DM candidate.We have however realized that the full SU (3) V flavor group has to be explicitly broken in order to make the only viable scenario for a realistic DM in our model, the decay of the inflaton S into them, possible.The dark matter particles have turned out be heavier than ∼ 10 9 GeV, and there exist only indirect couplings with the SM sector in addition to the gravitational interaction.Unfortunately, the indirect coupling with the SM, which is mediated by S, is so suppressed, that the dark matter particles will be unobservable in direct as well as indirect measurements.

Figure 3 .
Figure 3. Left: Trajectory in the S-φ plane just after the start at t = t 0 .The initial values are given in (3.20).The trajectory is strongly oscillating.Right: Trajectories with two different initial values, where the blue dashed line is for (3.20) and the red one is for S 0 /v S = 6.06 , φ 0 /M Pl = 5.36.They converge very fast to a "fixed-point" trajectory.

Figure 9 .
Figure 9.The prediction (red-yellow points) in the n s -r plane, where we have used γ = 3500, 1500, 500, 100 and 8 from top to bottom,, and the color represents N ∈ [50, 60].The black-grey points are for the Starobinsky inflation.The green backgrounds are the LiteBird/Planck constraints [12] at 95% (light green) and 68% (green) confidence level; the upper one shows the constraint assuming the Starobinsky inflation with N = 51 as the fiducial model, while the lower one shows the constraint in the case of null detection of r.

Figure 10 .
Figure 10.The prediction in the r-f N L plane for, from right to left, γ = 3500 , 1500, 500, 100, 8, where the black-grey points are for the Starobinsky inflation.The green (dark green) vertical line is the upper bound of r at 95 (68)% confidence level in the case of null detection[12].

Figure 11 .
Figure 11.f N L as a function n s for γ = 500 (red), where the grey points are for the Starobinsky inflation.The blue dotted line stands for single-field inflation [62, 106].

Figure 12 .
Figure 12.Dark matter mass m DM against the reheating temperature T RH for γ = 1500 and 100 with y 3 = 0.0046 and λ S = 0.0114.The colored points satisfy Ω DM h 2 = 0.1198 ± 0.0024 (2σ), where the color represents the e-foldings N .The corresponding r and n s can be found in Fig. 9.