GRAVITY WAVES FROM NON-MINIMAL QUADRATIC INFLATION

# Gravity Waves From Non-Minimal Quadratic Inflation

Constantinos Pallis and Qaisar Shafi
Departament de Física Teòrica and IFIC
Universitat de València-CSIC
E-46100 Burjassot
Bartol Research Institute
Department of Physics and Astronomy    University of Delaware
Newark
DE 19716    USA; e-mail address: shafi@bartol.udel.edu
###### Abstract

We discuss non-minimal quadratic inflation in supersymmetric (SUSY) and non-SUSY models which entails a linear coupling of the inflaton to gravity. Imposing a lower bound on the parameter , involved in the coupling between the inflaton and the Ricci scalar curvature, inflation can be attained even for subplanckian values of the inflaton while the corresponding effective theory respects the perturbative unitarity up to the Planck scale. Working in the non-SUSY context we also consider radiative corrections to the inflationary potential due to a possible coupling of the inflaton to bosons or fermions. We find ranges of the parameters, depending mildly on the renormalization scale, with adjustable values of the spectral index , tensor-to-scalar ratio , and an inflaton mass close to . In the SUSY framework we employ two gauge singlet chiral superfields, a logarithmic Kähler potential including all the allowed terms up to fourth order in powers of the various fields, and determine uniquely the superpotential by applying a continuous and a global symmetry. When the Kähler manifold exhibits a no-scale-type symmetry, the model predicts and . Beyond no-scale SUGRA, and depend crucially on the coefficient involved in the fourth order term, which mixes the inflaton with the accompanying non-inflaton field in the Kähler potential, and the prefactor encountered in it. Increasing slightly the latter above , an efficient enhancement of the resulting can be achieved putting it in the observable range. The inflaton mass in the last case is confined in the range .

Keywords: Cosmology, Supersymmetric models, Supergravity;
PACS codes: 98.80.Cq, 11.30.Qc, 12.60.Jv, 04.65.+e

Published in J. Cosmol. Astropart. Phys. 03, 023 (2015)

## 1 Introduction

The simplest model [1] of chaotic inflation (CI) based on a quadratic potential predicts a (scalar) spectral index (in good agreement with WMAP [2] and Planck [3] measurements) and a tensor-to-scalar ratio , a canonical measure of primordial gravity waves, close to or so. The Bicep2 results [4] announced earlier this year, purporting to have found gravity waves from inflation () provided a huge boost for this class of models [5, 6, 7, 8, 9]. However, serious doubts regarding the Bicep2 results have appeared in the literature [10, 11] that are largely related to the inadequate treatment of the impact on their analysis of the dust background. Furthermore, very recently, the Planck HFI 353 GHz dust polarization data [12] has been released and the first attempts to make a joint analysis of Planck and Bicep2 data have been presented [11, 13] concluding that the quadratic CI is disfavored at more than 95 confidence level (c.l.). Indeed, it is conceivable that most, if not the whole, Bicep2 polarization signal may be caused by the dust.

Be that as it may, it was shown several years ago [14] that a quadratic (or quartic) potential can, at best, function as an approximation within a more realistic inflationary cosmology. The end of CI is followed by a reheating phase which is implemented through couplings involving the inflaton and some additional suitably selected fields. The presence of these additional couplings can significantly modify, through radiative corrections (RCs), the tree level inflationary potential. For instance, for a quadratic potential supplemented by a coupling of the inflation field to, say, right-handed neutrinos, can be reduced to values close to [5] or so, at the cost of a (less efficient) reduction of , though. In this paper we briefly review this idea taking into account the recent refinements of Ref. [15], according to which an unavoidable dependence of the results on the renormalization scale arises.

Another mechanism for reducing at an acceptable level within models of quadratic CI is the introduction of a strong, linear non-minimal coupling of the inflaton to gravity [17, 16]. The aforementioned mechanism, that we mainly pursue here, can be applied either within a supersymmetric (SUSY) [16] or a non-SUSY [17] framework. The resulting inflationary scenario, named non-minimal CI (nMI), belongs to a class of universal “attractor” models [18], in which an appropriate choice of the non-minimal coupling to gravity suitably flattens the inflationary potential, such that is heavily reduced but stays close to the currently preferred value of . However, in generic Supergravity (SUGRA) settings, a mild tuning is needed [19] respecting the coefficient involved in the fourth order term that mixes the inflaton with the accompanying non-inflaton field in the Kähler potential.

In this work we reexamine the realization of nMI based on the quadratic potential implementing the following improvements:

• As regards the non-SUSY case, we also consider RCs to the tree-level potential which arise due to Yukawa interactions of the inflaton – cf. Ref. [20, 21]. We show that the presence of RCs can affect the values of nMI – in contrast to minimal CI, where RCs influence both and . For subplanckian values of the inflaton field, though, remains well suppressed and may be observable only in the next generation of experiments such as COrE+ [22], PIXIE [23] and LiteBIRD [24] which may bring the sensitivity down to .

• As regards the SUSY case, following Ref. [25], we generalize the embedding of the model in SUGRA allowing for a variation of the numerical prefactor encountered in the adopted Kähler potential. We show that (i) the tuning of can be totally avoided in the case of no-scale SUGRA which uniquely predicts and ; (ii) beyond no-scale SUGRA, increasing slightly the prefactor encountered in the adopted Kähler potential and adjusting appropriately , an efficient enhancement of the resulting , for any , can be achieved which will be tasted in the near future [26, 27].

We finally show that, in both of the above cases, the ultaviolet (UV) cut-off scale [28, 29] of the theory can be identified with the Planck scale and, thus, concerns regarding the naturalness of this kind of nMI can be safely evaded. It is worth emphasizing that this nice feature of these models was recently noticed in Ref. [30] and was not recognized in the original papers [17, 16].

The paper is organized as follows: In Sec. 2, we describe the generic formulation of CI with a quadratic potential and a non-minimal coupling to gravity. The emergent non-SUSY and SUSY inflationary models are analyzed in Secs. 3 and 4 respectively. The UV behavior of these models is analyzed in Sec. 5 and our conclusions are summarized in Sec. 6. In Appendix A we outline the implementation of nMI by the imaginary part of the inflaton superfield adopting a shift-symmetric logarithmic Kähler potential. Throughout the text, the symbol as subscript denotes derivation with respect to (w.r.t) the field (e.g., ); charge conjugation is denoted by a star, and we use units where the reduced Planck scale is set equal to unity.

## 2 Inflaton non-Minimally Coupled to Gravity

We consider below an inflationay sector coupled non-minimally to gravity within a non-SUSY (Sec. 2.1) or a SUSY (Sec. 2.2) framework. Based on this formulation, we then derive the inflationary observables and impose the relevant observational constraints in Sec. 2.3.

### 2.1 non-SUSY Framework

In the Jordan frame (JF) the action of an inflaton with potential non-minimally coupled to the Ricci scalar through a coupling function has the form:

 S=∫d4x√−g(−12fRR+fK2gμν∂μϕ∂νϕ−VCI0+Lint), (1)

where is the determinant of the background Friedmann-Robertson-Walker metric, . We allow also for a kinetic mixing through the function and a part of the langrangian which is responsible for the interaction of with a boson and a fermion , i.e.,

 Lint=12gχϕ2χ2+gψϕ¯ψψ. (2)

By performing a conformal transformation [17] according to which we define the Einstein frame (EF) metric

 ˆgμν=fRgμν  ⇒  {√−ˆg=f2R√−gandˆgμν=gμν/fRˆR=(R+3□lnfR+3gμν∂μfR∂νfR/2f2R)/fR, (3)

where and hat is used to denote quantities defined in the EF, we can write in the EF as follows:

 S=∫d4x√−ˆg(−12ˆR+12ˆgμν∂μˆϕ∂νˆϕ−ˆVCI0+ˆLint). (4)

The EF canonically normalized field, , the EF potential, , and the interaction Langrangian, turn out to be:

 \footnotesize\sf(a)(dˆϕdϕ)2=J2=fKfR+32(fR,ϕfR)2,\footnotesize\sf(b)ˆVCI0=VCI0f2R  and  \footnotesize\sf(c)ˆLint=Lintf2R⋅ (5)

Taking into account that , [31], and that the masses of these particles during CI are heavy enough such that the dependence of on does not influence their dynamics, can be written as

 ˆLint=gχϕ22fRˆχ2+gψϕ√fR¯¯¯¯ˆψˆψ. (6)

From Eq. (5) we infer that convenient choices of and assist us to obtain suitable for observationally consistent CI. Focusing on quadratic CI and following Ref. [17, 18], we select

 \footnotesize\sf(a)VCI0=12m2ϕ2%\footnotesize(b)fR(ϕ)=1+cRϕand\footnotesize\sf(c)fK=1 (7)

where is the renormalized mass of the inflaton. For we observe that a sufficiently flat through Eq. (5b) can be obtained which may decrease from its value in (minimal) quadratic CI. On the other hand, the vacuum expectation value (v.e.v) of is , and the validity of ordinary Einstein gravity is guaranteed since .

### 2.2 SUGRA Framework

A convenient implementation of nMI in SUGRA is achieved by employing two singlet superfields, i.e., , with () and ( being the inflaton and a “stabilized” field respectively. The EF action for ’s within SUGRA [32] can be written as

 S=∫d4x√−ˆg(−12ˆR+Kα¯βˆgμν∂μzα∂νz∗¯β−ˆV), (2.\theparentequationa)

where the summation is taken over the scalar fields , with , is the determinant of the EF metric , is the EF Ricci scalar curvature, is the EF F–term SUGRA scalar potential which can be extracted once the superpotential and the Kähler potential  have been selected, by applying the standard formula

 ˆV=eK(Kα¯βFαF∗¯β−3|W|2),whereFα=W,zα+K,zαW. (2.\theparentequationb)

Note that D-term contributions to do not exist since we consider gauge singlet ’s.

A quadratic potential for in this setting can be realized if we adopt the following superpotential

 W=mSΦ. (1)

To protect the form of from higher order terms we impose two symmetries: Firstly, an symmetry under which and have charges and respectively, which ensures the linearity of w.r.t ; secondly, a global symmetry with assigned charges and for and respectively. To verify that leads to the desired quadratic potential we present the SUSY limit, of , which is

 VSUSY=m2(|Φ|2+|S|2). (2.\theparentequationa)

Note that the complex scalar components of and superfields are denoted by the same symbol. From Eq. (2.\theparentequationa), we can easily conclude that for stabilized to zero, becomes quadratic w.r.t to the real (or imaginary) part of . The SUSY vacuum lies at

 ⟨S⟩=⟨Φ⟩=0. (2.\theparentequationb)

The construction of Eq. (1) can be obtained within SUGRA if we perform the inverse of the conformal transformation described in Eq. (3) with

 fR=−Ω/3(1+n), (1)

and specify the following relation between and ,

 −Ω/3(1+n)=e−K/3(1+n)⇒K=−3(1+n)ln(−Ω/3(1+n)). (2)

Here is a dimensionless (small in our approach) parameter which quantifies the deviation from the standard set-up [32]. Following Ref. [25] we arrive at the following action

 S=∫d4x√−g(ΩR6(1+n)+(Ωα¯β−nΩαΩ¯β(1+n)Ω)∂μzα∂μz∗¯β−ΩAμAμ(1+n)3−V), (3)

where is the JF potential and is [32] the purely bosonic part of the on-shell value of the auxiliary field

 Aμ=−i(1+n)(Ωα∂μzα−Ω¯α∂μz∗¯α)/2Ω. (4)

It is clear from Eq. (3) that exhibits non-minimal couplings of the ’s to . However, also enters the kinetic terms of the ’s. To separate the two contributions we split into two parts

 −Ω/3(1+n)=ΩH(Φ)+ΩH∗(Φ∗)−ΩK(|Φ|2,|S|2)/3(1+n), (2.\theparentequationa)

where is a dimensionless real function including the kinetic terms for the ’s and takes the form

 ΩK(|Φ|2,|S|2)=kNS|Φ|2+|S|2−2(kS|S|4+kΦ|Φ|4+kSΦ|S|2|Φ|2), (2.\theparentequationb)

with coefficients and of order unity. The fourth order term for is included to cure the problem of a tachyonic instability occurring along this direction [32], and the remaining terms of the same order are considered for consistency – the factors of are added just for convenience. Alternative solutions to the aforementioned problem of the tachyonic instability are recently identified in Ref. [33, 34]. On the other hand, in Eq. (2.\theparentequationa) is a dimensionless holomorphic function which, for , represents the non-minimal coupling to gravity – note that is independent of since . To obtain a situation similar to Eq. (7), we adopt

 ΩH=12+cR√2Φ, (2.\theparentequationc)

which respects the imposed symmetry but explicitly breaks during nMI. Furthermore, assuming that the phase of , , is stabilized to zero, the selected at the SUSY vacuum, Eq. (2.\theparentequationb), reads

 −⟨Ω⟩/3(1+n)=1, (1)

which ensures a recovery of conventional Einstein gravity at the end of nMI.

When the dynamics of the ’s is dominated only by the real moduli , or if for [32], we can obtain in Eq. (3). The choice , although not standard, is perfectly consistent with the idea of nMI. Indeed, the only difference occurring for – compared to the case – is that the ’s do not have canonical kinetic terms in the JF due to the term proportional to . This fact does not cause any problem since the canonical normalization of keeps its strong dependence on included in , whereas becomes heavy enough during nMI and so it does not affect the dynamics – see Sec. 4.1.

In conclusion, through Eq. (2) the resulting Kähler potential is

 K=−3(1+n)ln(1+cR√2(Φ+Φ∗)−|S|2+kNS|Φ|23(1+n)+2kS|S|4+kΦ|Φ|4+kSΦ|S|2|Φ|23(1+n)). (2)

We set throughout, except for the case of no-scale SUGRA which is defined as follows:

 n=0,kNS=0andkSΦ=kΦ=0. (3)

This arrangement, inspired by the early models of soft SUSY breaking [35, 36], corresponds to the Kähler manifold  with constant curvature equal to . In practice, these choices highly simplify the realization of nMI, thus rendering it more predictive thanks to a lower number of the remaining free parameters.

### 2.3 Inflationary Observables – Constraints

The analysis of nMI can be carried out exclusively in the EF using the standard slow-roll approximation keeping in mind the dependence of on – given by Eq. (5) in both the SUSY and non-SUSY set-up. Working this way, in the following we outline a number of observational requirements with which any successful inflationary scenario must be compatible – see, e.g., Ref. [37].

##### 2.3.1.

The number of e-folds, , that the scale experiences during CI,

 ˆN⋆=∫ˆϕ⋆ˆϕfdˆϕˆVCIˆVCI,ˆϕ=∫ϕ⋆ϕfJ2ˆVCIˆVCI,ϕdϕ, (4)

must be enough to resolve the horizon and flatness problems of standard big bang, i.e., [39, 3]

 ˆN⋆≃61.7+lnˆVCI(ϕ⋆)1/2ˆVCI(ϕf)1/3+13lnTrh+12lnfR(ϕ⋆)fR(ϕf)1/3, (5)

where is the radiatively corrected EF potential presented in Sec. 3.1 [Sec. 4.1] for the non-SUSY [SUSY] scenario. Also, we assume here that nMI is followed in turn by a decaying-inflaton, radiation and matter domination, is the reheat temperature after nMI, is the value of when crosses outside the inflationary horizon, and is the value of at the end of nMI. The latter can be found, in the slow-roll approximation for the models considered in this paper, from the condition

 max{ˆϵ(ϕf),|ˆη(ϕf)|}=1, (2.\theparentequationa)

where the slow-roll parameters can be calculated as follows:

 ˆϵ=12⎛⎜⎝ˆVCI,ˆϕˆVCI⎞⎟⎠2=12J2(ˆVCI,ϕˆVCI)2andˆη=ˆVCI,ˆϕˆϕˆVCI=1J2(ˆVCI,ϕϕˆVCI−ˆVCI,ϕˆVCIJ,ϕJ)⋅ (2.\theparentequationb)

It is worth mentioning that in our approach we calculate self-consistently with and , and do not let it vary within the interval as often done in the literature – see e.g. Ref. [3, 5]. Our estimation for in Eq. (5) takes into account the transition from the JF to EF – see Ref. [17] – and the assumption that nMI is followed in turn by a decaying-particle, radiation and matter domination – for details see Ref. [38]. During the first period, we adopt the so-called [39] canonical reheating scenario with an effective equation-of-state parameter . This value corresponds precisely to the equation-of-state parameter, , for a quadratic potential. In the nMI case we expect that will deviate slightly from this value. However, this effect is quite negligible since for low values the inflationary potential can be well approximated by a quadratic potential – see Sec. 5 below.

##### 2.3.2.

The amplitude of the power spectrum of the curvature perturbation generated by at the pivot scale must be consistent with data [3]:

 √As=12√3πˆVCI(ˆϕ⋆)3/2|ˆVCI,ˆϕ(ˆϕ⋆)|=|J(ϕ⋆)|2√3πˆVCI(ϕ⋆)3/2|ˆVCI,ϕ(ϕ⋆)|≃4.685⋅10−5, (1)

where we assume that no other contributions to the observed curvature perturbation exists.

##### 2.3.3.

The (scalar) spectral index, , its running, , and the scalar-to-tensor ratio must be in agreement with the fitting of the data [3] with CDM model, i.e.,

 \footnotesize\sf(a)ns=0.9603±0.0146,% \footnotesize\sf(b)−0.0314≤as≤0.0046and\footnotesize\sf(c)r<0.1at 95%. (2)

In Eq. (2c) we conservatively take into account the recent analyses [11, 13] which combine the Bicep2 results [4] with the polarized foreground maps released by Planck [12]. These observables are estimated through the relations:

 \footnotesize\sf(a)ns=1−6ˆϵ⋆ + 2ˆη⋆,\footnotesize\sf(b)as=2(4ˆη2⋆−(ns−1)2)/3−2ˆξ⋆and\footnotesize\sf(c)r=16ˆϵ⋆, (3)

where and the variables with subscript are evaluated at .

##### 2.3.4.

To avoid corrections from quantum gravity and any destabilization of our inflationary scenario due to higher order terms – e.g. in Eq. (7) or Eq. (2.\theparentequationc) –, we impose two additional theoretical constraints on our models – keeping in mind that :

 \footnotesize\sf(a)ˆVCI(ϕ⋆)1/4≤1and\footnotesize\sf(b)ϕ⋆≤1. (4)

As we show in Sec. 5, the UV cutoff of our model is , and so concerns regarding the validity of the effective theory are entirely eliminated.

## 3 non-SUSY Inflation

Focusing first on the non-SUSY case, we extract the inflationary potential in Sec. 3.1. Then, to better appreciate the importance of the non-minimal coupling to gravity for our scenario, we start the presentation of our results with a brief revision of the case where the inflaton is minimally coupled to gravity in Sec. 3.2. We extend our analysis to the more relevant case of nMI in Sec. 3.3.

### 3.1 Inflationary Potential

The tree-level EF inflationary potential of our model, found by plugging Eq. (7) into Eq. (5b), can be supplemented by the one-loop RCs computed in EF with the use of the standard formula of Ref. [40] – cf. Ref. [20]. To this end, we determine the particle masses as functions of the background field – see Eq. (6). Our result is

 ΔˆVCI=164π2⎛⎝ˆm4χlnˆm2χΛ2−4ˆm4ψlnˆm2ψΛ2⎞⎠,withˆm2χ=gχϕ2fRandˆm2ψ=g2ψϕ2fR⋅ (5)

Here is the renormalization scale and we assume that the on-shell masses of and are much ligther than the effective ones. Note that the only difference from the flat space case [14, 15] is the presence of the conformal factor in the denominators of the masses. We verify that these masses are heavier than the Hubble parameter during CI. On the other hand, the mass of is much lower than and thus, its contribution to Eq. (5) can be safely neglected. For numerical manipulations we find it convenient to write the one-loop corrected inflationary potential as

 ˆVCI=ˆVCI0+ΔˆVCI=m2ϕ22f2R(1+κlnϕ√fRΛ),whereκ=g2χ−4g4ψ16π2m2 (6)

expresses [15] the inflaton interaction strength. Following Ref. [15] we assume that for [], we have [], and thus or can be absorbed by redefining . Since there is no information, from particle physics about physical quantities – such as masses and coupling constants – which would assist us to determine uniquely, we consider it as a free parameter and discuss below the unavoidable dependence of the inflationary predictions on it.

At the end of CI, settles in its v.e.v and the EF (canonically normalized) inflaton,

 ˆδϕ=⟨J⟩δϕwith⟨J⟩≃√1+3c2R/2, (7)

acquires mass which is given by

 ˆmδϕ=⟨ˆVCI0,ˆϕˆϕ⟩1/2=m/⟨J⟩. (8)

The decay of is processed not only through the decay channel originating from the term in Eq. (6) which is proportional to , but also through the spontaneously arisen interactions which are proportional to [41]. The relevant lagrangian which describes these decay channels reads

 ˆLdc=ˆgψˆmδϕˆδϕ¯¯¯¯ˆψˆψ+ˆgχˆmδϕˆδϕˆχ2,whereˆgψ=gψ⟨J⟩+cRmψ2⟨J⟩andˆgχ=cRˆmδϕ4⟨J⟩ (9)

are dimensionless couplings and , the mass of , is set equal to for numerical applications. As it turns out, dominates the computation of for all relevant cases. These interactions give rise to the following decay rates of

 ˆΓψ=ˆg2ψ8πˆmδϕandˆΓχ=ˆg2χ16πˆmδϕ, (10)

which can ensure the reheating of the universe with temperature calculated by the formula [42]:

 Trh≃(725π2g∗)1/4√ˆΓδϕ,whereˆΓδϕ=ˆΓψ+ˆΓχ (11)

and we set for the relativistic degrees of freedom assuming the particle spectrum of Standard Model. Summarizing, the proposed inflationary scenario depends on the parameters:

 m,cR,κandΛ.

Following common practice [15], we consider below two optimal values which makes vanish for or .

### 3.2 Minimal Coupling to Gravity

This case can be studied if we set and , resulting in , in the formulae of Secs. 2.1, 3.1, and 2.3 – hatted and unhatted quantities are identical in this regime. In our investigation we first extract some analytic expressions – see Sec. 3.2.1 – which assist us to interpret the exact numerical results presented in Sec. 3.2.2.

#### 3.2.1 Analytic Results.

The slow-roll parameters can be calculated by applying Eq. (2.\theparentequationb) with results

 ϵ=12⎛⎜⎝2+κϕ2+4κϕ2lnϕΛϕ+κϕ3lnϕΛ⎞⎟⎠2   and   η=2+7κϕ2+12κϕ2lnϕΛϕ2+κϕ4lnϕΛ⋅ (12)

Numerically we verify that does not decline by much from its value for , i.e., . Hiding the dependence, which turns out to be not so significant, Eq. (4) yields for the number of -foldings experienced from during CI

 N⋆≃12κln1+κϕ2⋆/21+κ  ⇒   ϕ⋆=(2κ(e2κN⋆(1+κ)−1))1/2⋅ (13)

Note that the above formulae are valid for both signs of although we concentrate below on negative values which assist us in the reduction of . The normalization of Eq. (1) imposes the condition

 √As≃mϕ2⋆2√6π(2+κϕ2⋆)⇒m≃2π√6Ase2κN⋆κ(1+κ)e2κN⋆(1+κ)−1⋅ (14)

In the limit , the expressions in Eqs. (13) and (14) reduce to the corresponding ones – see Eqs. (2) and (3) with – that we obtain within the simplest quadratic CI. Upon substitution of Eqs. (12) and (13) into Eq. (3) we may compute the inflationary observables. Namely, Eq. (3a) yields

 ns≃1−2N⋆+{4κ−38κ2N⋆/3+(1/6−12N2⋆)κ3for  Λ=ϕ⋆2(2−l⋆)κ−4(11+3l⋆(7+2l⋆))κ2N⋆/3for  Λ=ϕf, (3.\theparentequationa)

where and an expansion for has been performed. Needless to say, the optimal scale or yields or respectively for – see Eq. (6). Similarly, from Eq. (3b) we get

 as≃−2N2⋆+{2κ/N⋆+128κ2/3+98κ3N⋆/3for  Λ=ϕ⋆2(1−2l⋆)κ/N⋆−4(29+3l⋆(15+4l⋆))κ2/3for  Λ=ϕf, (3.\theparentequationb)

while Eq. (3c) implies

 r≃8N⋆+{24κ+104κ2N⋆/3+32κ3N2⋆for  Λ=ϕ⋆8(3+4l⋆)κ+8(25−12l2⋆)κ2N⋆/3for  Λ=ϕf. (3.\theparentequationc)

From the expressions above we infer that a negative can reduce and, less efficiently, and below their values for .

#### 3.2.2 Numerical Results.

These conclusions are verified numerically in Table 1 where we present results compatible with Eqs. (5), (1), (2a, b) and (4a), taking and (cases A and B), or (cases A’ and B’) – note that Eq. (4b) cannot be satisfied. We observe that by adjusting we can succeed to diminish below its value in quadratic CI without RCs but not a lot lower than its maximal allowed value in Eq. (2c). Indeed, the lowest obtained is . Moreover, this reduction causes a reduction of which acquires its lowest allowed value in cases A and A’ – see Eq. (2a). The dependence of the results on can be inferred by comparing the sets of parameters in the primed and unprimed columns. Note that the reference value of is fixed in every couple of columns – i.e., in cases A and A’ and in cases B and B’. The -dependence of the results is imprinted mainly on the values of which are considerably lower for . From the definition of in Eq. (6), though, we infer that this -dependence becomes milder as regards values. Since , we also notice that and roughly equal to its value, , for .

In conclusion, the consideration of RCs arising from the coupling of the inflaton to fermions can reconcile somehow CI with data. However, the violation of Eq. (4b) and the -dependence are two severe shortcomings of this mechanism.

### 3.3 Non-Minimal Coupling to Gravity

If we employ the linear non-minimal coupling to gravity suggested in Eq. (7b) with , we can follow the same steps as in Sec. 3.2 – see Secs. 3.3.1 and 3.3.2 below.

#### 3.3.1 Analytic Results.

From Eqs. (5) and (2.\theparentequationb), we find

 J≃√321ϕ,  ˆϵ=⎛⎜ ⎜ ⎜ ⎜⎝4+κϕ2(2+cRϕ)+2κϕ2(2+cRϕ)lnϕ2Λ2fR√3fR(2+κϕ2lnϕ2Λ2fR)⎞⎟ ⎟ ⎟ ⎟⎠2, (3.\theparentequationa)

and

 ˆη=28+ϕ(16κϕ+cR(κϕ2(15+4cRϕ)−4))−2κϕ2(8+cRϕ(7+2cRϕ))lnϕ2Λ2fR3f2R(2+κϕ2lnϕ2Λ2fR)⋅ (3.\theparentequationb)

The expressions above reduce to the well known ones [17, 19] for . We can, also, verify that the formulas for , and found there [17, 19] give rather accurate results even with , i.e.,

 ˆN⋆≃3cRϕ⋆/4  ⇒  ϕ⋆≃4ˆN⋆/3cR≪ϕf≃2/√3cR, (1)

and can be subplanckian – see Eq. (4b) – if we confine ourselves to the regime

 cR≳4ˆN⋆/3≃77forˆN⋆≃54. (2)

However, may be transplanckian since integrating Eq. (5a) in view of Eq. (3.\theparentequationa) and employing then Eq. (1) we extract

 (3)

whose the absolute value is greater than unity for . Nonetheless, Eq. (4b) is enough to protect our scheme from higher order terms. Eq. (4a) does not restrict the parameters.

The relation between and implied by Eq. (1), neglecting the dependence, becomes

 √As≃mϕ⋆2π(4+κϕ2⋆(2+cRϕ⋆))⇒m≃2π√As(27c2R+16κˆN3⋆)9cRˆN⋆⋅ (4)

Plugging Eqs. (3.\theparentequationa), (3.\theparentequationb) and (1) into Eq. (3) and expanding for , we arrive at

 ns≃1−2ˆN⋆+12827κˆN2⋆c2Rδns,as≃−2ˆN2⋆−41627κˆN⋆c2Rδas,andr≃12ˆN2