[

# [

C. Pallis
###### Abstract

Abstract

A novel realization of the Starobinsky inflationary model within a moderate extension of the Minimal Supersymmetric Standard Model (MSSM) is presented. The proposed superpotential is uniquely determined by applying a continuous and a discrete symmetry, whereas the Kähler potential is associated with a no-scale-type Kähler manifold. The inflaton is identified with a Higgs-like modulus whose the vacuum expectation value controls the gravitational strength. Thanks to a strong enough coupling (with a parameter involved) between the inflaton and the Ricci scalar curvature, inflation can be attained even for subplanckian values of the inflaton with and the corresponding effective theory being valid up to the Planck scale. The inflationary observables turn out to be in agreement with the current data and the inflaton mass is predicted to be . At the cost of a relatively small superpotential coupling constant, the model offers also a resolution of the problem of MSSM. Supplementing MSSM by three right-handed neutrinos we show that spontaneously arising couplings between the inflaton and the particle content of MSSM not only ensure a sufficiently low reheating temperature but also support a scenario of non-thermal leptogenesis consistently with the neutrino oscillation parameters for gravitino heavier than about .

Keywords: Cosmology, Supersymmetric models, Supergravity, Modified Gravity;
PACS codes: 98.80.Cq, 11.30.Qc, 12.60.Jv, 04.65.+e, 04.50.Kd
Published in J. Cosmol. Astropart. Phys. 04, 024 (2014); J. Cosmol. Astropart. Phys. 07, 01E (2017)

in no-Scale Supergravity to MSSM

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[

\@hangfromDepartament de Física Teòrica and IFIC,
Universitat de València-CSIC,
E-46100 Burjassot, SPAIN

Department of Physics, University of Cyprus,
P.O. Box 20537, Nicosia 1678, CYPRUS

\@xsect

After the announcement of the recent PLANCK results [1, 2], inflation based on the potential of the Starobinsky model [3] has gained a lot of momentum [4, 5, 6, 8, 9, 7, 10, 11] since it predicts [3, 12] a (scalar) spectral index very close to the one favored by the fitting of the observations by the standard power-law cosmological model with cold dark matter (CDM) and a cosmological constant (CDM). In particular, it has been shown that Starobinsky-type inflation can be realized within extensions of the Standard Model (SM) [13] or Minimal SUSY SM (MSSM) [14]. However, the realization of this type of inflation within Supergravity (SUGRA) is not unique. Different super- and Kähler potentials are proposed [6, 7, 5] which result to the same scalar potential. Prominent, however, is the idea [4, 5] of implementing this type of inflation using a Kähler potential, , corresponding to a Kähler manifold inspired by the no-scale models [15, 16]. Such a symmetry fixes beautifully the form of up to an holomorphic function which exclusively depends on a modulus-like field and plays the role of a varying gravitational coupling. The stabilization of the non-inflaton accompanying field can to be conveniently arranged by higher order terms in . In this context, a variety of models are proposed in which inflaton can be identified with either a matter-like [4, 5, 14] or a modulus-like [6, 5] inflaton. The former option seems to offer a more suitable framework [14] for connecting the inflationary physics with a low-energy theory, such as the MSSM endowed with right handed neutrinos, , since the non-inflaton modulus is involved in the no-scale mechanism of soft SUSY breaking (SSB). On the other hand, the inflationary superpotential, , is arbitrarily chosen and not protected by any symmetry. Given that, the inflaton takes transplanckian values during inflation, higher order corrections – e.g., by non-renormalizable terms in – with not carefully tuned coefficients may easily invalidate or strongly affect [8, 17] the predictions of an otherwise successful inflationary scenario.

It would be interesting, therefore, to investigate if the shortcoming above can be avoided in the presence of a strong enough coupling of the inflaton to gravity [18, 19], as done [20, 21, 22, 24, 23, 25] in the models of non-minimal Inflation (nMI). This idea can be implemented keeping the no-scale structure of , since the involved can be an analytic function, selected conveniently. In view of the fact that depends only on a modulus-like field, we here focus on this kind of inflaton – contrary to Ref. [14]. As a consequence, the direct connection of the inflationary model with the mechanism of the SSB is lost. Note, in passing, that despite their attractive features, the no-scale models [14] of SSB enface difficulties – e.g., viable SUSY spectra are obtained only when the boundary conditions for the SSB terms are imposed beyond the Grand Unified Theory (GUT) scale and so the low energy observables depend on the specific GUT.

Focusing on a modulus-like inflaton, the link to MSSM can be established through the adopted . Its form in our work is fixed by imposing a continuous symmetry, which reduces to the well-known -parity of MSSM, and a discrete symmetry. As a consequence, resembles the one used in the widely employed models [26, 27] of standard F-term Hybrid Inflation (FHI) – with singlet waterfall field though. As a bonus, a dynamical generation of the reduced Planck scale arises in Jordan Frame (JF) through the vacuum expectation value (v.e.v) of the inflaton. Therefore the inflaton acquires a higgs-character as in the theories of induced gravity [28, 29]. To produce an inflationary plateau with the selected , is to be taken quadratic, in accordance with the adopted symmetries. This is to be contrasted with the so-called modified Cecotti model [30, 6, 8, 7, 5] where the inflaton appears linearly in the super- and Kähler potentials. The inclusion of two extra parameters compared to the original model – cf. [6, 8, 5] – allows us to attain inflationary solutions for subplanckian values of the inflaton with the successful inflationary predictions of the model being remained intact. As a bonus, the ultaviolet (UV) cut-off scale [31, 32, 9] of the theory can be identified with the Planck scale and so, concerns regarding the naturalness of the model can be safely eluded.

Our inflationary model – let name it for short no-scale modular inflation (nSMI) – has ramifications to other fundamental open problems of the MSSM and post-inflationary cosmological evolution. As a consequence of the adopted symmetry, the generation [27, 33] of the mixing term between the two electroweak Higgses is explained via the v.e.v of the non-inflaton accompanying field, provided that a coupling constant in is rather suppressed. Finally, the observed [34] baryon asymmetry of the universe (BAU) can be explained via spontaneous [35, 36] non-thermal leptogenesis (nTL) [37] consistently with the constraint [38, 39, 40], the data [41, 42] on the neutrino oscillation parameters as long as the masses of the gravitino () lie in the multi-TeV region – as dictated in many versions [43, 44, 45] of MSSM after the recent LHC [46, 47] results on the Higgs boson mass.

The basic ingredients – particle content and structure of the super- and Kähler potentials – of our model are presented in Sec. [. In Sec. [ we describe the inflationary potential, derive the inflationary observables and confront them with observations. Sec. [ is devoted to the resolution of the problem of MSSM. In Sec. [ we outline the scenario of nTL, exhibit the relevant imposed constraints and describe our predictions for neutrino masses. Our conclusions are summarized in Sec. [. Throughout the text, the subscript of type denotes derivation with respect to (w.r.t) the field (e.g., ) and charge conjugation is denoted by a star.

\@xsect

We focus on a moderated extension of MSSM with three ’s augmented by two superfields, a matter-like and a modulus-like , which are singlets under . Besides the local symmetry of MSSM, , the model possesses also the baryon number symmetry , a nonanomalous symmetry and a discrete . Note that global continuous symmetries can effectively arise [48] in many compactified string theories. The charge assignments under the global symmetries of the various matter and Higgs superfields are listed in Table [. We below present the structure of the superpotential (Sec. [) and the Kähler potential (Sec. [) of our model.

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The superpotential of our model naturally splits into two parts:

 W=WMSSM+WMI, (2.1a)

where is the part of which contains the usual terms – except for the term – of MSSM, supplemented by Yukawa interactions among the left-handed leptons and

 WMSSM=hijEeciLjHd+hijDdciQjHd+hijUuciQjHu+hijNNciLjHu. (2.1b)

Here the th generation doublet left-handed quark and lepton superfields are denoted by and respectively, whereas the singlet antiquark [antilepton] superfields by and [ and ] respectively. The electroweak Higgs superfields which couple to the up [down] quark superfields are denoted by [].

On the other hand, is the part of which is relevant for nSMI, the generation of the term of MSSM and the Majorana masses for ’s. It takes the form

 WMI=λS(T2−M2/2)+λμSHuHd+12M[Nci]Nc2i+λ[Ncij]T2NciNcj/2mP, (2.1c)

where is the reduced Planck mass. The imposed symmetry ensures the linearity of w.r.t . This fact allows us to isolate easily via its derivative the contribution of the inflaton into the F-term SUGRA scalar potential, placing at the origin. The imposed prohibits the existence of the term which, although does not drastically modifies our proposal, it complicates the determination of SUSY vacuum and the inflationary dynamics. On the other hand, the imposed symmetries do not forbid non-renormalizable terms of the form where is an integer. For this reason we are obliged to restrict ourselves to subplanckian values of .

The second term in the right-hand side (r.h.s) of Eq. (2.1c) provides the term of MSSM along the lines of Ref. [27, 33] – see Sec. [. The third term is the Majorana mass term for the ’s and we assume that it overshadows (for sufficiently low ’s) the last non-renormalizable term which is neglected henceforth. Here we work in the so-called right-handed neutrino basis, where is diagonal, real and positive. These masses together with the Dirac neutrino masses in Eq. (2.1b) lead to the light neutrino masses via the seesaw mechanism. The same term is important for the decay [35, 36] of the inflaton after the end of nSMI to , whose subsequent decay can activate nTL. As a result of the imposed , a term of the form is prohibited and so the decay of into is processed by suppressed SUGRA-induced interactions [35], guaranteing thereby a sufficiently low reheat temperature compatible with the constraint and successful nTL – see Sec. [.

Since the no-scale SUGRA adopted here leads to the non-renormalizable F-term (scalar) potential in Eq. (3.1), we expect that it yields an effective SUSY theory which depends not only on the superpotential, in Eq. (2.1c), but also on the Kähler potential, in Eq. (2.6a) – see Sec. [. To trace out this behavior we apply the generic formula for the SUSY F-term potential [49]:

 VSUSY=Kα¯βWMIαW∗MI¯β, (2.2)

which is obtained from the SUGRA potential in Eq. (3.1) – see Sec. [ below – if we perform an expansion in powers and take the limit . The Kähler potential, , employed here can not be expanded in powers of , since unity is not included in the argument of the logarithm – in contrast to the ’s used in Ref. [50, 51]. In Eq. (2.2) is the inverse of the Kähler metric with and where the complex scalar components of the superfields and are denoted by the same symbol whereas this of by . We find that reads

 (2.3)

where is given in Eq. (2.7c) and we neglect the fourth order terms since we expect that these are not relevant for the low energy effective theory. The inverse of the matrix above is

 (2.4)

Substituting this in Eq. (2.2), we end up with the following expression

 VSUSY = −Ω3(λ2∣∣T2+λμHuHd/λ−M2/2∣∣2+λ2μ(|Hu|2+|Hd|2)|S|2+M[Nci]2|˜Nci|2 (2.5a) +2λ23cT|S|2(T2+T∗2−M2/2)+λλμcT(HuHd+H∗uH∗d)|S|2 +λ3cTM[Nci](S∗˜Nc2i+S˜Nc∗2i)).

The three first terms in the r.h.s of the expression above come from the terms of Eq. (2.2) for . The fourth one comes from the terms

 Kα¯αWMIαW∗MI¯α+Kα¯βWMIαW∗MI¯β+Kβ¯αWMIβW∗MI¯α

for and ; the residual terms arise from terms of the form ; for and the fifth one and the last one.

From the potential in Eq. (2.5a), we find that the SUSY vacuum lies at

 ⟨Hu⟩=⟨Hd⟩=⟨˜Nci⟩=0,\nobreak \ignorespaces⟨S⟩≃0and√2|⟨T⟩|=M. (2.5b)

Contrary to the Cecotti model [30, 6, 5] our modulus can take values at the SUSY vacuum. Also, breaks spontaneously the imposed and so, it can comfortably decay via SUGRA-inspired decay channels – see Sec. [ – reheating the universe and rendering [36] spontaneous nTL possible. No domain walls are produced due to the spontaneous breaking of at the SUSY vacuum, since this is broken already during nSMI.

With the addition of SSB terms, as required in a realistic model, the position of the vacuum shifts [27, 33] to non-zero and an effective term is generated from the second term in the r.h.s of Eq. (2.1c) – see Sec. [. Let us emphasize that SSB effects explicitly break to the matter parity, under which all the matter (quark and lepton) superfields change sign. Combining with the fermion parity, under which all fermions change sign, yields the well-known -parity. Recall that this residual symmetry prevents the rapid proton decay, guarantees the stability of the lightest SUSY particle (LSP) and therefore it provides a well-motivated CDM candidate. Needless to say, finally, that such a connection of the Starobinsky-type inflation with this vital for MSSM -symmetry can not be established within the modified Cecotti model [30, 6, 7], since no symmetry can prohibit a quadratic term for the modulus-like field in conjunction with the tadpole term in .

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According to the general discussion of Ref. [15], the Kähler manifold which corresponds to a Kähler potential of the form

 K=−3m2Pln(fK(T)+f∗K(T∗)−ΦAΦ∗¯A3m2P+kSΦA|S|2|ΦA|23m4P+⋯), (2.6a)

with being an holomorphic function of , exhibits a global symmetry. Here is the number of scalar components of and the MSSM superfields which are collectively denoted as

 ΦA=~eci,\nobreak \ignorespaces~uci,\nobreak \ignorespaces~dci,\nobreak \ignorespaces˜Nci,\nobreak \ignorespaces~Li,\nobreak \ignorespaces˜Qi,\nobreak \ignorespacesHu,\nobreak \ignorespacesHdandS. (2.6b)

Note that summation over the repeated (small or capital) Greek indices is implied. The third term in the r.h.s of Eq. (2.6a) – with coefficients being taken, for simplicity, real – is included since it has an impact on the scalar mass spectrum along the inflationary track – see Sec. [. In particular, the term with coefficient assists us to avoid the tachyonic instabilities encountered in similar models [6, 5, 7, 18, 19] – see Sec. [. The ellipsis represents higher order terms which are irrelevant for the inflationary dynamics since they do not mix the inflaton with the matter fields. This is, in practice, a great simplification compared to the models of nMI – cf. Ref. [25]. Contrary to other realizations of the Starobinsky model – cf. Ref. [6, 5, 7] –, we choose to be quadratic and not linear with respect to , i.e.,

 fK(T)=cTT2/m2P (2.6c)

in accordance with the imposed symmetry which forbids a linear term – the coefficient is taken real too. As in the case of Eq. (2.1c), non-renormalizable terms of the form , with integer , are allowed but we can safely ignore them restricting ourselves to .

The interpretation of the adopted in Eq. (2.6a) can be given in the “physical” frame by writing the JF action for the scalar fields . To extract it, we start with the corresponding EF action within SUGRA [18, 21, 25] which can be written as

 S=∫d4x√−ˆg(−12m2PˆR+Kα¯β˙Φα˙Φ∗¯β−ˆVMI0+⋯), (2.7a)

where with , is the determinant of the EF metric , is the EF Ricci scalar curvature, is defined in Sec. [, the dot denotes derivation w.r.t the JF cosmic time and the ellipsis represents terms irrelevant for our analysis. Performing then a suitable conformal transformation, along the lines of Ref. [21, 25] we end up with the following action in the JF

 S=∫d4x√−g(−m2P2(−Ω3)R+m2PΩα¯β˙Φα˙Φ∗¯β−VSUSY+⋯), (2.7b)

where is the JF metric with determinant , is the JF Ricci scalar curvature, and we use the shorthand notation and . The corresponding frame function can be found from the relation

 −Ω3=e−K/3m2P=fK(T)+f∗K(T∗)−ΦAΦ∗¯A3m2P+kSΦA|S|2|ΦA|23m4P+⋯⋅ (2.7c)

The last result reveals that has no kinetic term, since . This is a crucial difference between the Starobinsky-type models and those [25] of nMI, with interest consequences [9] to the derivation of the ultraviolet cutoff scale of the theory – see Sec. [. Furthermore, given that , recovering the conventional Einstein gravity at the SUSY vacuum, Eq. (2.5b), dictates

 fK(⟨T⟩)+f∗K(⟨T∗⟩)=1\nobreak \ignorespaces⇒\nobreak \ignorespacesM=mP/√cT. (2.8)

Given that the analysis of inflation in both frames yields equivalent results [12, 52], we below – see Sec. [ and [ – carry out the derivation of the inflationary observables exclusively in the EF.

\@xsect

In this section we outline the salient features of our inflationary scenario (Sec. [) and then, we present its predictions in Sec. [, calculating a number of observable quantities introduced in Sec. [. We also provide a detailed analysis of the UV behavior of the model in Sec. [.

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The EF F–term (tree level) SUGRA scalar potential, , of our model – see Eq. (2.7a) – is obtained from in Eq. (2.1c) and in Eq. (2.6a) by applying the standard formula:

 (3.1)

where . Setting the fields and at the origin the only surviving term of is

 ˆVMI0=eK/m2PKSS∗WMI,SW∗MI,S∗=λ2|2T2−M2|24(fK+f∗K)2⋅ (3.2)

It is obvious from the result above that a form of as the one proposed in Eq. (2.6c) can flatten sufficiently so that it can drive nSMI. Employing the dimensionless variables

 xϕ=ϕ/mP,fT=1−cTx2ϕandxM=M/mPwithϕ=|T|/√2 (3.3)

and setting , and the corresponding Hubble parameter read

 ˆVMI0=λ2m4P(x2ϕ−x2M)24c2Tx4ϕ=λ2m4Pf2T4c4Tx4ϕandˆHMI=ˆV1/2MI0√3mP≃λmP2√3c2T, (3.4)

where we put – by virtue of Eq. (2.8) – in the final expressions.

Expanding and in real and imaginary parts as follows

 T=ϕ√2eiθ/mPandXα=xα+i¯xα√2withXα=S,Hu,Hd,˜Nci (3.5)

we can check the stability of the inflationary direction

 θ=xα=¯xα=0wherexα=s,hu,hd,~νci, (3.6)

w.r.t the fluctuations of the various fields. In particular, we examine the validity of the extremum and minimum conditions, i.e.,

 ∂ˆVMI0∂ˆχα∣∣ ∣∣Eq.\nobreak\ \ignorespaces(???)=0andˆm2χα>0withχα=θ,xα,¯xα. (3.7a)

Here are the eigenvalues of the mass matrix with elements

 ˆM2αβ=∂2ˆVMI0∂ˆχα∂ˆχβ∣∣ ∣∣Eq.\nobreak\ \ignorespaces(???)with\nobreak \ignorespacesχα=θ,xα,¯xα (3.7b)

and hat denotes the EF canonically normalized fields. Taking into account that along the configuration of Eq. (3.6) defined below Eq. (2.7a) takes the form

 (Kα¯β)=diag⎛⎜ ⎜ ⎜ ⎜⎝6/x2ϕ,1/cTx2ϕ,...,1/cTx2ϕ8\nobreak \ignorespaces\footnotesize elements⎞⎟ ⎟ ⎟ ⎟⎠ (3.8)

– here we take into account that and are doublets –, the kinetic terms of the various scalars in Eq. (2.7a) can be brought into the following form

 Kα¯β˙Φα˙Φ∗¯β=12(˙ˆϕ2+˙ˆθ2)+12(˙ˆxα˙ˆxα+˙ˆ¯¯¯xα˙ˆ¯¯¯xα), (3.9a)

where the hatted fields are defined as follows

 dˆϕ/dϕ=J=√6/xϕ,ˆθ=√6θ,ˆxα=xα/√cTxϕ% andˆ¯xα=¯xα/√cTxϕ. (3.9b)

Upon diagonalization of the relevant sub-matrices of , Eq. (3.7b), we construct the scalar mass spectrum of the theory along the direction in Eq. (3.6). Our results are summarized in Table [, assuming in order to avoid very lengthy formulas for the masses of and . The various unspecified there eigenvalues are defined as follows:

 ˆh±=(ˆhu±ˆhd)/√2,ˆ¯h±=(ˆ¯hu±ˆ¯hd)/√2andˆψ±=(ˆψT±ˆψS)/√2, (3.10a)

where the spinors and associated with the superfields and are related to the normalized ones in Table [ as follows:

 ˆψS=√6ψS/xϕ,ˆψT=ψT/√cTxϕandˆNci=Nci/√cTxϕ. (3.10b)

We also use the shorthand notation:

 fSH=2+3kSHcTx2ϕandfS˜Nci=2+3kS˜NcicTx2ϕ. (3.11)

Note that, due to the large effective masses that the ’s in Eq. (3.7b) acquire during nSMI, they enter a phase of oscillations about with reducing amplitude. As a consequence – see Eq. (3.9b) –, since the quantity , involved in relating to , turns out to be negligibly small compared with – cf. Ref. [24]. Moreover, we have numerically verified that the various masses remain greater than during the last e-foldings of nSMI, and so any inflationary perturbations of the fields other than the inflaton are safely eliminated – see also Sec. [.

From Table [ it is evident that assists us to achieve – in accordance with the results of Ref. [6, 5, 7]. On the other hand, given that , requires

 λfTfSH+6λμc2Tx2ϕ<0\nobreak \ignorespaces⇒\nobreak \ignorespacesλμ<−λfTfSH6c2Tx2ϕ≃λ3cT+12λkSHx2ϕ≃2⋅10−5−10−6, (3.12)

as decreases from to . Here we have made use of Eqs. (3.16a) and (3.20b) – see Sec. [. We do not consider such a condition on as unnatural, given that in Eq. (2.1b) is of the same order of magnitude too – cf. Ref. [53]. In Table [ we also present the masses squared of chiral fermions along the trajectory of Eq. (3.6), which can be served for the calculation of the one-loop radiative corrections. Employing the well-known Coleman-Weinberg formula [54], we find that the one-loop corrected inflationary potential is

 ˆVMI=ˆVMI0 + 164π2(ˆm4θlnˆm2θΛ2+2ˆm4slnˆm2sΛ2+4ˆm4h+lnˆm2h+Λ2+4ˆm4h−lnˆm2h−Λ2 (3.13) + 23∑i=1(ˆm4iνclnˆm2iνcΛ2−ˆm4iNclnˆm2iNcΛ2)−4ˆm4ψ±lnm2ˆψ±Λ2⎞⎟⎠,

where is a renormalization group (RG) mass scale. As we numerically verify the one-loop corrections have no impact on our results. The absence of gauge interactions and of a direct renormalizable coupling between and assists to that direction – cf. Ref. [25, 55]. Based on , we can proceed to the analysis of nSMI in the EF, employing the standard slow-roll approximation [57, 56]. It can be shown [28] that the results calculated this way are the same as if we had calculated them using the non-minimally coupled scalar field in the JF.

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A successful inflationary scenario has to be compatible with a number of observational requirements which are outlined in the following.

\@xsect

The number of e-foldings, , that the scale suffers during nSMI has to be adequate to resolve the horizon and flatness problems of the standard Big Bag cosmology. Assuming that nSMI is followed in turn by a decaying-particle, radiation and matter domination and employing standard methods [20], we can easily derive the required for our model, with the result:

 ˆN⋆≃19.4+2lnˆVMI(ϕ⋆)1/41\nobreak \ignorespacesGeV−43lnˆVMI(ϕf)1/41\nobreak \ignorespacesGeV+13lnTrh1\nobreak \ignorespacesGeV+12lnfK(ϕ⋆)fK(ϕf)1/3, (3.14)

where is the value of when crosses the inflationary horizon. Also is the value of at the end of nSMI determined, in the slow-roll approximation, by the condition:

 \footnotesizemax{ˆϵ(ϕf), |ˆη(ϕf)|}=1, (3.15a)

 ˆϵ=m2P2⎛⎜⎝ˆVMI,ˆϕˆVMI⎞⎟⎠2=m2P2J2(ˆVMI,ϕˆVMI)2≃43f2T (3.15b)

and

 ˆη=m2P\nobreak \ignorespacesˆVMI,ˆϕˆϕˆVMI=m2PJ2(ˆVMI,ϕϕˆVMI−ˆVMI,ϕˆVMIJ,ϕJ)≃4(1+fT)3f2T⋅ (3.15c)

The termination of nSMI is triggered by the violation of the criterion at a value of equal to , which is calculated to be

 (3.16a)

since the violation of the criterion occurs at such that

 ˆη(~ϕf)=1⇒~ϕf=mP(5/3cT)1/2<ϕf. (3.16b)

On the other hand, can be calculated via the relation

 ˆN⋆=1m2P∫ˆϕ⋆ˆϕfdˆϕˆVMIˆVMI,ˆϕ=1m2P∫ϕ∗ϕfdϕJ2ˆVMIˆVMI,ϕ⋅ (3.17)

Given that , we can find a relation between and as follows

 ˆN⋆≃3cT4m2P(ϕ2⋆−ϕ2f)⇒ϕ⋆≃2mP√ˆN⋆/3cT. (3.18a)

Obviously, nSMI with subplanckian ’s can be achieved if

 ϕ⋆≤mP\nobreak \ignorespaces⇒\nobreak \ignorespacescT≥4ˆN⋆/3≃76 (3.18b)

for . Therefore we need relatively large ’s.

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The amplitude of the power spectrum of the curvature perturbation generated by at the pivot scale is to be confronted with the data [1, 2], i.e.

 A1/2s=12√3πm3PˆVMI(ˆϕ⋆)3/2|ˆVMI,ˆϕ(ˆϕ⋆)|=12πm2P ⎷ˆVMI(ϕ⋆)6ˆϵ(ϕ⋆)≃4.685⋅10−5. (3.19)

Since the scalars listed in Table [ are massive enough during nSMI, the curvature perturbations generated by are solely responsible for . Substituting Eqs. (3.15b) and (3.18a) into the relation above, we obtain

 √As=λm2PfT(ϕ⋆)28√2πc2Tϕ2⋆\nobreak \ignorespaces⇒\nobreak \ignorespacesλ≃6π√2AscT/ˆN⋆. (3.20a)

Combining the last equality with Eq. (3.19), we find that is to be proportional to , for almost constant . Indeed, we obtain

 λ≃3.97⋅10−4πcT/ˆN⋆⇒cT≃41637λforˆN⋆≃52. (3.20b)
\@xsect

The (scalar) spectral index , its running , and the scalar-to-tensor ratio must be consistent with the fitting [1, 2] of the observational data, i.e.,

 \footnotesize(a)ns=0.96±0.014,% \footnotesize(b)−0.0314≤as≤0.0046and\footnotesize(c)r<0.11 (3.21)

at 95 confidence level (c.l.). The observable quantities above can be estimated through the relations: