ResearchArticle Starobinsky Inflation: From Non-SUSY to SUGRA … · 2019. 7. 30. · 2....

Research ArticleStarobinsky Inflation From Non-SUSY to SUGRA Realizations

Constantinos Pallis and Nicolaos Toumbas

Department of Physics University of Cyprus PO Box 20537 1678 Nicosia Cyprus

Correspondence should be addressed to Constantinos Pallis kpallisgenauthgr

Received 29 December 2016 Revised 23 March 2017 Accepted 10 April 2017 Published 18 June 2017

Academic Editor Elias C Vagenas

Copyright copy 2017 Constantinos Pallis and Nicolaos Toumbas This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited The publication of this article was funded by SCOAP3

We review the realization of Starobinsky-type inflation within induced-gravity supersymmetric (SUSY) and non-SUSY modelsIn both cases inflation is in agreement with the current data and can be attained for sub-Planckian values of the inflationThe corresponding effective theories retain perturbative unitarity up to the Planck scale and the inflation mass is predicted tobe 3 sdot 1013 GeV The supergravity embedding of these models is achieved by employing two gauge singlet chiral superfields asuperpotential that is uniquely determined by a continuous 119877 and a discrete Z119899 symmetry and several (semi)logarithmic Kahlerpotentials that respect these symmetries Checking various functional forms for the noninflation accompanying field in the Kahlerpotentials we identify four cases which stabilize it without invoking higher order terms

1 Introduction

The idea that the universe underwent a period of exponentialexpansion called inflation [1ndash3] has proven useful not onlyfor solving the horizon and flatness problems of standardcosmology but also for providing an explanation for the scaleinvariant perturbations which are responsible for generatingthe observed anisotropies in the Cosmic Microwave Back-ground (CMB) One of the first incarnations of inflation isdue to Starobinsky To date this attractive scenario remainspredictive since it passes successfully all the observationaltests [4 5] Starobinsky considered adding anR2 term whereR is the Ricci scalar to the standard Einstein action inorder to source inflation Recall that gravity theories basedon higher powers of R are equivalent to standard gravitytheories with one additional scalar degree of freedom (seeeg [6 7]) As a result Starobinsky inflation is equivalent toinflation driven by a scalar field with a suitable potential soit admits several interesting realizations [8ndash29]

Following this route we show in this work that induced-gravity inflation (IGI) [30ndash38] is effectively Starobinsky-likereproducing the structure and the predictions of the originalmodel Within IGI the inflation exhibits a strong couplingto R and the reduced Planck scale is dynamically generatedthrough the vacuum expectation value (vev) of the inflation

at the end of inflation Therefore the inflation acquires aHiggs-like behavior as in theories of induced gravity [36ndash42]Apart from being compatible with data the resulting theoryrespects perturbative unitarity up to the Planck scale [29ndash31] Therefore no concerns about the validity of the corre-sponding effective theory arise This is to be contrasted withmodels of nonminimal inflation (nMI) [43ndash54] based on a 120601119899

potential with negligible vev for the inflation 120601 Althoughthese models yield similar observational predictions with theStarobinsky model they admit an ultraviolet (UV) scale wellbelow119898119875 for 119899 gt 2 leading to complicationswith naturalness[55ndash57]

Nonetheless IGI allows us to embed Starobinsky inflationwithin N = 1 supergravity (SUGRA) in an elegant wayThe embedding is achieved by incorporating two chiralsuperfields a modulus-like field 119879 and a matter-like field119878 appearing in the superpotential 119882 as well as variousKahler potentials 119870 consistent with an 119877 and discrete Z119899

symmetries [29 31 58] see also [20ndash22 28 32] In somecases [20 29 31 58] the employed 119870rsquos parameterize specificKahler manifolds which appear in no-scale models [59ndash61] Moreover this scheme ensures naturally a low enoughreheating temperature potentially consistent with the grav-itino constraint [29 62 63] if connected with a version of theMinimal SUSY Standard Model (MSSM)

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 6759267 16 pageshttpsdoiorg10115520176759267

2 Advances in High Energy Physics

An important issue in embedding IGI in SUGRA isthe stabilization of the matter-like field 119878 Indeed when 119870parameterizes the 119878119880(2 1)(119878119880(2) times 119880(1)) Kahler manifold[20 21] the inflationary trajectory turns out to be unstablewith respect to the fluctuations of 119878 This difficulty can beovercome by adding a sufficiently large term 119896119878|119878|4 with 119896119878 gt0 and |119896119878| sim 1 in the logarithmic function appearing in119870 as suggested in [64] for models of nonminimal (chaotic)inflation [47ndash49] and applied in [50ndash54 65ndash70] This solu-tion however deforms slightly the Kahler manifold [71]More importantly it violates the predictability of Starobinskyinflation since mixed terms 119896119878119879|119878|2|119879|2 with 119896119878119879 ge 001which cannot be ignored (without tuning) have an estimableimpact [31 72ndash74] on the dynamics and the observablesMoreover this solution becomes complicated when morethan two fields are considered since all quartic terms allowedby symmetries have to be considered and the analysis of thestabilization mechanism becomes tedious (see eg [31 72ndash74]) Alternatively it was suggested to use a nilpotent super-field 119878 [75] or a charged field under a gauged119877 symmetry [71]

In this review we revisit the issue of stabilizing 119878disallowing terms of the form |119878|2119898 119898 gt 1 without caringmuch about the structure of the Kahler manifold Namelywe investigate systematically several functions ℎ119894(|119878|2) (with119894 = 1 11) that appear in the choices for 119870 and we findfour acceptable forms that lead to the stabilization of 119878 duringand after IGI The output of this analysis is new providingresults that did not appear in the literature before Morespecifically we consider two principal classes of119870rsquos1198703119894 and1198702119894 distinguished by whether ℎ119894 and 119879 appear in the samelogarithmic function The resulting inflationary scenariosare almost indistinguishable The case considered in [58] isincluded as one of the viable choices in 1198702119894 class Contraryto [58] we impose here the same Z119899 symmetry on119882 and119870Consequently the relevant expressions for themass spectrumand the inflationary observables get simplified considerablycompared to those displayed in [58] As in the non-SUSYcase IGI may be realized using sub-Planckian values forthe initial (noncanonically normalized) inflation field Theradiative corrections remain under control and perturbativeunitarity is not violated up to119898119875 [31 58 76] consistently withthe consideration of SUGRA as an effective theory

Throughout this review we focus on the standard ΛCDMcosmological model [4] An alternative framework is pro-vided by the running vacuummodels [77ndash84] which turn outto yield a quality fit to observations significantly better thanthat of ΛCDM In this case the acceleration of the universeeither during inflation or at late times is not attributed toa scalar field but rather arises from the modification of thevacuum itself which is dynamical A SUGRA realization ofStarobinsky inflation within this setting is obtained in [18]

The plan of this paper is as follows In Section 2 weestablish the realization of Starobinsky inflation as IGI in anon-SUSY framework In Section 3 we introduce the formu-lation of IGI in SUGRA and revisit the issue of stabilizingthe matter-like field 119878 The emerging inflationary modelsare analyzed in Section 4 Our conclusions are summarizedin Section 5 Throughout charge conjugation is denoted by

a star ( lowast) the symbol 119911 as subscript denotes derivation withrespect to 119911 and we use units where the reduced Planck scale119898119875 = 243 sdot 1018 GeV is set equal to unity

2 Starobinsky Inflation from Induced Gravity

We begin our presentation demonstrating the connectionbetween R2 inflation and IGI We first review the formu-lation of nMI in Section 21 and then proceed to describethe inflationary analysis in Section 22 Armed with theseprerequisites we present R2 inflation as a type of nMI inSection 23 and exhibit its connection with IGI in Section 24

21 Coupling Nonminimally the Inflation to Gravity Weconsider an inflation 120601 that is nonminimally coupled to theRicci scalarR via a coupling function 119891R(120601) We denote theinflation potential by 119881119868(120601) and allow for a general kineticfunction 119891119870(120601)mdashin the cases of pure nMI [33ndash35 45 46]119891119870 = 1 The Jordan Frame (JF) action takes the form

S = int1198894119909radicminusg(minus12119891RR + 12119891119870119892120583]120597120583120601120597]120601 minus 119881119868 (120601)) (1)

where g is the determinant of the Friedmann-Robertson-Walker metric 119892120583] with signature (+ minus minus minus) We require⟨119891R⟩ ≃ 1 to ensure ordinary Einstein gravity at low energies

By performing a conformal transformation [45] to theEinstein frame (EF) we write the action

S = int1198894119909radicminusg(minus12R + 12119892120583]120597120583120601120597]120601 minus 119868 (120601)) (2)

where a hat denotes an EF quantityThe EFmetric is given by119892120583] = 119891R119892120583] and the canonically normalized field 120601 and itspotential 119868 are defined as follows

(a) 119889120601119889120601 = 119869 = radic 119891119870119891R + 32 (119891R120601119891R )2(b) 119868 = 1198811198681198912

R

(3)

For 119891R ≫ 119891119870 the coupling function 119891R acquires a twofoldrole On the one hand it determines the relation between120601 and 120601 On the other hand it controls the shape of 119881119868thus affecting the observational predictions see below Theanalysis of nMI can be performed in the EF using thestandard slow-roll approximation It is [33ndash35] completelyequivalent with the analysis in the JF We just have to keeptrack the relation between 120601 and 12060122 Observational and Theoretical Constraints A viablemodel of nMI must be compatible with a number of obser-vational and theoretical requirements summarized in thefollowing (cf [85ndash88])

(1) The number of e-foldings ⋆ that the scale 119896⋆ =005Mpc experiences during inflation must be large enough

Advances in High Energy Physics 3

for the resolution of the horizon and flatness problems of thestandard hot Big Bang model that is [4 45]

⋆ = int120601⋆

120601119891

119889120601 119868119868120601

= int120601⋆

120601119891

1198891206011198692 119868119868120601

≃ 617 + ln119868 (120601⋆)12119868 (120601119891)13 + 13 ln119879rh

+ 12 ln119891R (120601⋆)119891R (120601119891)13

(4)

where 120601⋆ [120601⋆] is the value of 120601 [120601] when 119896⋆ crosses theinflationary horizon In deriving the formula above (cf[65ndash67]) we take into account an equation-of-state withparameter119908rh = 0 [89] since 119868 can be well approximated bya quadratic potential for low values of 120601 see(20b) (32b) and(71b) below Also 119879rh is the reheating temperature after nMIWe take a representative value 119879rh = 41 sdot 10minus10 throughoutwhich results in ⋆ ≃ 53 The effective number of relativisticdegrees of freedom at temperature 119879rh is taken 119892rh = 10775in accordance with the standard model spectrum Lastly120601119891 [120601119891] is the value of 120601 [120601] at the end of nMI which in theslow-roll approximation can be obtained via the condition

max 120598 (120601119891) 10038161003816100381610038161003816120578 (120601119891)10038161003816100381610038161003816 = 1 where 120598 = 12 (119868120601119868

)2 = 121198692 (119868120601119868

)2 120578 = 119868120601120601119868

= 11198692 (119868120601120601119868

minus 119868120601119868

119869120601119869 ) sdot (5)

Evidently nontrivial modifications of 119891R and thus of 119869 mayhave a significant effect on the parameters above modifyingthe inflationary observables

(2) The amplitude 119860 119904 of the power spectrum of thecurvature perturbation generated by 120601 at 119896⋆ has to beconsistent with the data [90] that is

radic119860 119904 = 12radic3120587 119868 (120601⋆)3210038161003816100381610038161003816119868120601 (120601⋆)10038161003816100381610038161003816 = 1003816100381610038161003816119869 (120601⋆)10038161003816100381610038162radic3120587 119868 (120601⋆)3210038161003816100381610038161003816119868120601 (120601⋆)10038161003816100381610038161003816≃ 4627 sdot 10minus5

(6)

As shown in Section 34 the remaining scalars in the SUGRAversions of nMI may be rendered heavy enough so they donot contribute to 119860 119904

(3) The remaining inflationary observables (the spectralindex 119899119904 its running 120572119904 and the tensor-to-scalar ratio 119903) mustbe in agreementwith the fitting of thePlanckBaryonAcousticOscillations (BAO) andBicep2KeckArray data [4 5] with theΛCDM+119903model that is

(a) 119899119904 = 0968 plusmn 0009(b) 119903 le 007 (7)

at the 95 confidence level (cl) with |120572119904| ≪ 001 Althoughcompatible with (7)(b) all data taken by the Bicep2KeckArray CMB polarization experiments up to the 2014 obser-vational season (BK14) [5] seem to favor 119903rsquos of the order of001 as the reported value is 0028+0026

minus0025 at the 68 cl Theseinflationary observables are estimated through the relations

(a) 119899119904 = 1 minus 6120598⋆ + 2120578⋆(b) 120572119904 = 23 (41205782

⋆ minus (119899119904 minus 1)2) minus 2120585⋆(c) 119903 = 16120598⋆

(8)

where 120585 = 1198681206011198681206011206011206012119868 and the variables with subscript ⋆

are evaluated at 120601⋆(4) The effective theory describing nMI remains valid up

to a UV cutoff scale ΛUV which has to be large enough toensure the stability of our inflationary solutions that is

(a) 119868 (120601⋆)14 le ΛUV(b) 120601⋆ le ΛUV (9)

As we show below ΛUV ≃ 1 for the models analyzed in thiswork contrary to the cases of pure nMI with large 119891R whereΛUV ≪ 1 The determination of ΛUV is achieved expandingS in (2) about ⟨120601⟩ Although these expansions are not strictlyvalid [57] during inflation we take ΛUV extracted this wayto be the overall UV cutoff scale since the reheating phaserealized via oscillations about ⟨120601⟩ is a necessary stage of theinflationary dynamics

23 From Nonminimal to R2 Inflation R2 inflation can beviewed as a type of nMI if we employ an auxiliary field 120601withthe following input ingredients

119891119870 = 0119891R = 1 + 4119888R120601119868 = 1206012

(10)

Using the equation ofmotion for the auxiliary field120601 = 119888RRwe obtain the action of the original Starobinsky model (seeeg [71])

S = int1198894119909radicminusg(minus12R + 1198882RR2) (11)

As we can see from (10) the model has only one freeparameter (119888R) enough to render it consistent with theobservational data ensuring at the same time perturbative


cℛ = 23 middot 104

00

02

04

06

08

10

12

1 2 3 4 5 6 7 8 9 100

VI(10minus10)

휙 (10minus4)

휙f

휙⋆

(a)

cℛ = 77

M = 0114

14

12

10

08

06

04

02

0002 04 06 08 10 12 1400

VI(10minus10)

휙f

휙

휆 = 28 middot 10minus6

휙⋆

(b)

Figure 1 The inflationary potential 119868 as a function of 120601 forR2 inflation (a) and IGI with 120601⋆ = 1 (b) Values corresponding to 120601⋆ and 120601119891 arealso indicated

unitarity up to the Planck scale Using (10) and (3) we obtainthe EF quantities

(a) 119869 = 2radic6 119888R119891R (b) 119868 = 1206012

1198912R

≃ 1161198882R

(12)

For 119888R ≫ 1 the plot of 119868 versus 120601 is depicted in Figure 1(a)An inflationary era can be supported since 119868 becomes flatenough To examine further this possibility we calculate theslow-roll parameters Plugging (12) into (5) yields

120598 = 1121198882R1206012

120578 = 1 minus 4119888R120601121198882

R1206012

sdot (13)

Notice that 120578 lt 0 since 119868 is slightly concave downwardsas shown in Figure 1(a) The value of 120601 at the end of nMI isdetermined via (5) giving

120601119891 = max( 12radic3119888R 16119888R) 997904rArr120601119891 = 12radic3119888R sdot (14)

Under the assumption that 120601119891 ≪ 120601⋆ we can obtain arelation between ⋆ and 120601⋆ via (4)⋆ ≃ 3119888R120601⋆ (15)The precise value of 119888R can be determined enforcing (6)Recalling that ⋆ ≃ 53 we get

11986012119904 ≃ ⋆12radic2120587119888R = 4627 sdot 10minus5 997904rArr119888R ≃ 23 sdot 104

(16)

The resulting value of 119888R is large enough so that

120601⋆ ≃ ⋆3119888R ≃ 83 sdot 10minus4 ≪ 1 (17)

consistently with (9)(b) see Figure 1(a) Impressively theremaining observables turn out to be compatible with theobservational data of (7) Indeed inserting the above valueof 120601⋆ into (8) (⋆ = 53) we get

119899119904 ≃ (⋆ minus 3) (⋆ minus 1)2⋆

≃ 1 minus 2119873⋆

minus 31198732⋆

≃ 0961 (18a)

120572119904 ≃ minus(⋆ minus 3) (4⋆ + 3)24⋆

≃ minus 21198732⋆

minus 1523⋆

≃ minus76 sdot 10minus4(18b)

119903 ≃ 121198732⋆

≃ 42 sdot 10minus3 (18c)

Without the simplification of (15) we obtain numerically 119899119904 =0964 120572119904 = minus67 sdot 10minus4 and 119903 = 37 sdot 10minus3 We see that 119899119904 turnsout to be appreciably lower than unity thanks to the negativevalues of 120578 see (13) The mass of the inflation at the vacuumis

120575120601 = ⟨119868120601120601⟩12 = ⟨1198681206011206011198692 ⟩12 = 12radic3119888R≃ 125 sdot 10minus5 (ie 3 sdot 1013 GeV)

(19)

As we show below this value is salient future in all models ofStarobinsky inflation

Furthermore the model provides an elegant solution tothe unitarity problem [55ndash57] which plagues models of nMIwith 119891R sim 120601119899 ≫ 119891119870 119899 gt 2 and 119891119870 = 1 This stems fromthe fact that 120601 and 120601 do not coincide at the vacuum as (12)(a)


implies 120601 = ⟨119869⟩120601 = 2radic3119888R120601 In fact if we expand the secondterm in the right-hand side (rhs) of (2) about ⟨120601⟩ = 0 wefind

1198692 1206012 = (1 minus 2radic23120601 + 21206012 minus sdot sdot sdot) 1206012 (20a)

Similarly expanding 119868 in (12)(b) we obtain

119868 = 1206012

241198882R

(1 minus 2radic23120601 + 21206012 minus sdot sdot sdot) (20b)

Since the coefficients of the above series are of order unityindependent of 119888R we infer that the model does not face anyproblem with perturbative unitarity up to the Planck scale

24 Induced-Gravity Inflation It would be certainly benefi-cial to realize the structure and the predictions ofR2 inflationin a framework that deviatesminimally fromEinstein gravityat least in the present cosmological era To this extent weincorporate the idea of induced gravity according to which119898119875 is generated dynamically [41 42] via the vev of a scalarfield 120601 driving a phase transition in the early universe Thesimplest way to implement this scheme is to employ a double-well potential for120601 for scale invariant realizations of this ideasee [39 40] On the other hand an inflationary stage requiresa sufficiently flat potential as in (10) This can be achieved atlarge field values if we introduce a quadratic119891R [33ndash38]Moreexplicitly IGImay be defined as nMIwith the following inputingredients

119891119870 = 1119891R = 119888R1206012119881119868 = 120582 (1206012 minus 1198722)24

(21)

Given that ⟨120601⟩ = 119872 we recover Einstein gravity at thevacuum if

119891R (⟨120601⟩) = 1 997904rArr119872 = 1radic119888R (22)

We see that in this model there is one additional freeparameter namely 120582 appearing in the potential as comparedtoR2 model

Equations (3) and (21) imply

(a) 119869 ≃ radic6120601 (b) 119868 = 1205821198912

12060141198884R1206014

≃ 12058241198882R

with 119891120601 = 1 minus 119888R1206012(23)

For 119888R ≫ 1 the plot of 119868 versus 120601 is shown in Figure 1(b) Asin R2 model 119868 develops a plateau so an inflationary stage

can be realized To check its robustness we compute the slow-roll parameters Equations (5) and (23) give

120598 = 431198912120601

120578 = 4 (1 + 119891120601)31198912

120601

sdot(24)

IGI is terminated when 120601 = 120601119891 determined by the condition

120601119891 = max(radic1 + 2radic3119888R radic 53119888R) 997904rArr120601119891 = radic1 + 2radic3119888R sdot

(25)

Under the assumption that 120601119891 ≪ 120601⋆ (4) implies thefollowing relation between ⋆ and 120601⋆

⋆ ≃ 3119888R1206012⋆4 997904rArr

120601⋆ ≃ 2radic ⋆3119888R ≫ 120601119891(26)

Imposing (9)(b) and setting ⋆ ≃ 53 we derive a lowerbound on 119888R

120601⋆ le 1 997904rArr119888R ge 4⋆3 ≃ 71 (27)

Contrary toR2 inflation 119888R does not control exclusively thenormalization of (6) thanks to the presence of an extra factorofradic120582 This is constrained to scale with 119888R Indeed we have11986012

119904 ≃ radic120582⋆6radic2120587119888R = 4627 sdot 10minus5 997904rArr 119888R ≃ 42969radic120582for ⋆ ≃ 53

(28)

If in addition we impose the perturbative bound 120582 le 35 weend-up with following ranges

77 ≲ 119888R ≲ 85 sdot 10428 sdot 10minus6 ≲ 120582 ≲ 35 (29)

where the lower bounds on 119888119879 and 120582 correspond to120601⋆ = 1 seeFigure 1(b)Within the allowed ranges 120575120601 remains constantby virtue of (28) The mass turns out to be

120575120601 = radic120582radic3119888R ≃ 125 sdot 10minus5 (30)


essentially equal to that estimated in (19) Moreover using(26) and (8) we extract the remaining observables

119899119904 = (4⋆ minus 15) (4⋆ + 1)(3 minus 4⋆)2 ≃ 1 minus 2119873⋆

minus 922⋆

≃ 0961(31a)

120572119904 = minus128⋆ (4⋆ + 9)(3 minus 4⋆)4 ≃ minus 21198732

⋆

minus 2123⋆


119903 = 192(3 minus 4⋆)2 ≃ 121198732

⋆


Withoutmaking the approximation of (26) we obtain numer-ically (119899119904 120572119904 119903) = (0964 minus66 sdot 10minus4 37 sdot 10minus3) Theseresults practically coincide with those of R2 inflation givenin (18a)ndash(18c) and they are in excellent agreement with theobservational data presented in (7)

As in the previous section the model retains perturbativeunitarity up to 119898119875 To verify this we first expand the secondterm in the rhs of (1) about 120575120601 = 120601minus119872 ≃ 0 with 119869 given by(23)(a) We find

1198692 1206012 = (1 minus radic23120575120601 + 121205751206012 minus sdot sdot sdot) 1205751206012

with 120575120601 ≃ radic6119888R120575120601(32a)

Expanding 119868 given by (23)(b) we get

119868 = 1205822

61198882R

1205751206012 (1 minus radic32120575120601 + 25241205751206012 minus sdot sdot sdot) (32b)

Therefore ΛUV = 1 as for R2 inflation Practically identicalresults can be obtained if we replace the quadratic exponentsin (21) with 119899 ge 3 as first pointed out in [30] Thisgeneralization can be elegantly performed [31 32] withinSUGRA as we review below

3 Induced-Gravity Inflation in SUGRA

In Section 31 we present the general SUGRA setting whereIGI is embedded Then in Section 32 we examine a varietyof Kahler potentials which lead to the desired inflationarypotential see Section 33 We check the stability of theinflationary trajectory in Section 34

31TheGeneral Set-Up To realize IGI within SUGRA [29 3132 58] we must use two gauge singlet chiral superfields 119911120572with 1199111 = 119879 and 1199112 = 119878 being the inflation and a ldquostabilizerrdquosuperfield respectively Throughout this work the complexscalar fields 119911120572 are denoted by the same superfield symbolThe EF effective action is written as follows [47ndash49]

S = int1198894119909radicminusg(minus12R + 119870120572120573119892120583]120597120583119911120572120597]119911lowast120573 minus ) (33a)

where 119870120572120573 = 119870119911120572119911lowast120573

is the Kahler metric and 119870120572120573 its inverse

(119870120572120573119870120573120574 = 120575120572120574 ) is the EF F-term SUGRA potential given

in terms of the Kahler potential 119870 and the superpotential 119882by the following expression

= 119890119870 (119870120572120573119863120572119882119863lowast

120573119882lowast minus 3 |119882|2)

with 119863120572119882 = 119882119911120572 + 119870119911120572119882 (33b)

Conformally transforming to the JF with 119891R = minusΩ119873where 119873 is a dimensionless positive parameter S takes theform

S = int1198894119909radicminusg( Ω2119873R + 34119873Ω120597120583Ω120597120583Ωminus 1119873Ω119870120572120573120597120583119911120572120597120583119911lowast120573 minus 119881) with 119881 = Ω2

1198732 (34)

Note that119873 = 3 reproduces the standard set-up [47ndash49] Letus also relateΩ and119870 by

minusΩ119873 = 119890minus119870119873 997904rArr119870 = minus119873 ln(minusΩ119873) (35)

Then taking into account the definition [47ndash49] of the purelybosonic part of the auxiliary field when on shell

A120583 = 119894 (119870120572120597120583119911120572 minus 119870120572120597120583119911lowast120572)6 (36)

we arrive at the following action

S = int1198894119909radicminusg( Ω2119873R

+ (Ω120572120573 + 3 minus 119873119873 Ω120572Ω120573Ω )120597120583119911120572120597120583119911lowast120573

minus 271198733ΩA120583A

120583 minus 119881) (37a)

By virtue of (35)A120583 takes the form

A120583 = minus119894119873 (Ω120572120597120583119911120572 minus Ω120572120597120583119911lowast120572)6Ω (37b)

with Ω120572 = Ω119911120572 and Ω120572 = Ω119911lowast120572 As can be seen from(37a) minusΩ119873 introduces a nonminimal coupling of the scalarfields to gravity Ordinary Einstein gravity is recovered at thevacuum when

minus⟨Ω⟩119873 ≃ 1 (38)


Table 1 Definition of the various ℎ119894(119883)rsquos ℎ10158401015840119894 (0) = 1198892ℎ119894(0)1198891198832 and masses squared of the fluctuations of 119904 and 119904 along the inflationary

trajectory in (46) for119870 = 1198703119894 and 1198702119894

119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912

120601 3 sdot 2119899minus111989121206012 119890119883 minus 1 1 2(4minus119899)2119888119879120601119899 minus 2 + 21198991198912

120601 minus6 + 3 sdot 2119899minus111989121206013 ln(119883 + 1) minus1 minus2(1 + 21minus1198992119888119879120601119899) 6(1 + 2119899minus11198912

120601 )4 minus cos(arcsin 1 + 119883) 0 minus2(1 minus 2119899minus11198912120601 ) 3 sdot 2119899minus11198912

1206015 sin(arccos 1 + 119883) 0 minus2(1 minus 2119899minus11198912120601 ) 3 sdot 2119899minus11198912

1206016 tan (119883) 0 minus2(1 minus 2119899minus11198912120601 ) 3 sdot 2119899minus11198912

1206017 minus cot(arcsin 1 + 119883) 0 minus2(1 minus 2119899minus11198912120601 ) 3 sdot 2119899minus11198912

1206018 cosh(arcsin 1 + 119883) minus radic2 radic2 2(5minus119899)2119888119879120601119899 minus 2 + 21198991198912120601 3 sdot 2119899minus11198912

120601 minus 6radic29 sinh(119883) 0 minus2(1 minus 2119899minus11198912120601 ) 3 sdot 2119899minus11198912

12060110 tanh(119883) 0 minus2(1 minus 2119899minus11198912120601 ) 3 sdot 2119899minus11198912

12060111 minus coth(arcsinh 1 + 119883) + radic2 minus2radic2 21198991198912120601 minus 2(7minus119899)2119888119879120601119899 minus 2 3 sdot 2119899minus11198912

120601 + 12radic2Starting with the JF action in (37a) we seek to realize IGI

postulating the invariance of Ω under the action of a globalZ119899 discrete symmetryWith 119878 stabilized at the origin wewrite

minus Ω119873 = Ω119867 (119879) + Ωlowast119867 (119879lowast)with Ω119867 (119879) = 119888119879119879119899 + infinsum

119896=2

120582119896119879119896119899 (39)

where 119896 is a positive integer If119879 le 1 during IGI and assumingthat 120582119896rsquos are relatively small the contributions of the higherpowers of 119879 in the expression above are small and these canbe dropped As we verify later this can be achieved when thecoefficient 119888119879 is large enough Equivalently wemay rescale theinflation setting 119879 rarr = 1198881198791119899119879 Then the coefficients 120582119896 ofthe higher powers in the expression of Ω get suppressed byfactors of 119888minus119896119879 ThusZ119899 and the requirement 119879 le 1 determinethe form ofΩ avoiding a severe tuning of the coefficients 120582119896Under these assumptions119870 in (35) takes the form1198700 = minus119873 ln (119891 (119879) + 119891lowast (119879lowast)) with 119891 (119879) ≃ 119888119879119879119899 (40)where 119878 is assumed to be stabilized at the origin

Equations (35) and (38) require that 119879 and 119878 acquire thefollowing vevs

⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)

These vevs can be achieved if we choose the followingsuperpotential [31 32]

119882 = 120582119878(119879119899 minus 12119888119879) (42)

Indeed the corresponding F-term SUSY potential 119881SUSY isfound to be

119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)

and is minimized by the field configuration in (21)

As emphasized in [29 31 58] the forms of119882 andΩ119867 canbe uniquely determined if we limit ourselves to integer valuesfor 119899 (with 119899 gt 1) and 119879 le 1 and impose two symmetries

(i) An 119877 symmetry under which 119878 and 119879 have charges 1and 0 respectively

(ii) A discrete symmetry Z119899 under which only 119879 ischarged

For simplicity we assume here that both 119882 and Ω119867 respectthe same Z119899 contrary to the situation in [58] This assump-tion simplifies significantly the formulae in Sections 33and 34 Note finally that the selected Ω in (39) does notcontribute in the kinetic term involving Ω119879119879lowast in (37a) Weexpect that our findings are essentially unaltered even if weinclude in the rhs of (39) a term minus(119879 minus 119879lowast)22119873 [32] orminus|119879|2119873 [31] which yieldsΩ119879119879lowast = 1 ≪ 119888119879 the former choicethough violates Z119899 symmetry above

32 Proposed Kahler Potentials It is obvious from the con-siderations above that the stabilization of 119878 at zero duringand after IGI is of crucial importance for the viability of ourscenario This key issue can be addressed if we specify thedependence of the Kahler potential on 119878 We distinguish thefollowing basic cases

1198703119894 = minus1198993 ln (119891 (119879) + 119891lowast (119879lowast) + ℎ119894 (119883)) 1198702119894 = minus1198992 ln (119891 (119879) + 119891lowast (119879lowast)) + ℎ119894 (119883) (44)

where the various choices ℎ119894 119894 = 1 11 are specified inTable 1 and119883 is defined as follows

119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)

As shown in Table 1 we consider exponential logarithmictrigonometric and hyperbolic functions Note that 11987031 and


11987021 parameterize 119878119880(2 1)119878119880(2)times119880(1) and 119878119880(1 1)119880(1)times119880(1) Kahler manifolds respectively whereas11987023 parameter-izes the 119878119880(1 1)119880(1) times 119878119880(2)119880(1) Kahler manifold see[58]

To show that the proposed119870rsquos are suitable for IGI we haveto verify that they reproduce 119868 in (23)(b) when 119899 = 2 andthey ensure the stability of 119878 at zero These requirements arechecked in the following two sections

33 Derivation of the Inflationary Potential Substituting 119882of (42) and a choice for 119870 in (44) (with the ℎ119894rsquos given inTable 1) into (33b) we obtain a potential suitable for IGI Theinflationary trajectory is defined by the constraints

119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)

where we have expanded 119879 and 119878 in real and imaginary partsas follows

119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)

Along the path of (46) reads

119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)

From (42) we get119882119878 = 119891 minus 12 Also (44) yields

119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)

where we take into account that 119891(119879) = 119891lowast(119879lowast) along thepath of (46) Moreover 119870119878119878lowast = 1119870119878119878lowast can be obtained fromthe Kahler metric which is given by

(119870120572120573) = diag (119870119879119879lowast 119870119878119878lowast)

=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)

where a prime denotes a derivative with respect to 119883 Notethat119870119879119879lowast for119870 = 1198702119894 (and 119878 = 0) does not involve the field 119878in its denominator and so no geometric destabilization [91]

can be activated contrary to the 119870 = 1198703119894 case Inserting 119882119878

and the results of (49) and (50) into (33b) we obtain

119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)

Recall that 119891 sim 120601119899 see (40) Then 119868 develops a plateauwith almost constant potential energy density for 119888119879 ≫ 1 and120601 lt 1 (or 119888119879 = 1 and 120601 ≫ 1) if we impose the followingconditions

2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)

This empirical criterion is very precise since the data on 119899119904

allows only tiny (of order 0001) deviations [28] Actually therequirement 119888119879 ≫ 1 and the synergy between the exponentsin 119882 and 119870rsquos assist us to tame the well-known 120578 problemwithin SUGRA with a mild tuning If we insert (52) into (51)and compare the result for 119899 = 2 with (23)(b) (replacing also1205822 with 120582) we see that the two expressions coincide if we set

ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)

As we can easily verify the selected ℎ119894 in Table 1 satisfy theseconditions Consequently 119868 in (51) and the correspondingHubble parameter 119868 take their final form

(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)

with119891120601 = 21198992minus1minus119888119879120601119899 lt 0 reducing to that defined in (23)(b)Based on these expressions we investigate in Section 4 thedynamics and predictions of IGI

34 Stability of the Inflationary Trajectory We proceed tocheck the stability of the direction in (46) with respect tothe fluctuations of the various fields To this end we haveto examine the validity of the extremum and minimumconditions that is

(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)


Table 2 Mass-squared spectrum for 119870 = 1198703119894 and 1198702119894 along the inflationary trajectory in (46)

Fields Eigenstates Masses squared119870 = 1198703119894 119870 = 1198702119894

1 real scalar 120579 21205792

119868 4(2119899minus2 minus 119888119879120601119899119891120601)1198912120601 6(2119899minus2 minus 119888119879120601119899119891120601)1198912

120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601

scalar +22minus1198992119888119879120601119899ℎ10158401015840119894 (0) minus6ℎ10158401015840

119894 (0)2Weyl spinors plusmn 2120595plusmn2

119868 21198991198912120601 6 sdot 2119899minus31198912

120601

Here 2119911120572 are the eigenvalues of themassmatrixwith elements

2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)

and a hat denotes the EF canonically normalized field Thecanonically normalized fields can be determined if we bringthe kinetic terms of the various scalars in (33a) into thefollowing form

119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)

where a dot denotes a derivative with respect to the JF cosmictime Then the hatted fields are defined as follows

119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)

where by virtue of (52) and (53) the Kahler metric of (50)reads


= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)

Note that the spinor components 120595119879 and 120595119878 of thesuperfields 119879 and 119878 are normalized in a similar way thatis 119879 = radic119870119879119879lowast120595Φ and 119878 = radic119870119878119878lowast120595119878 In practice wehave to make sure that all the 2

119911120572 rsquos are not only positive

but also greater than 2119868 during the last 50ndash60 e-foldings of

IGIThis guarantees that the observed curvature perturbationis generated solely by 120601 as assumed in (6) Nonethelesstwo-field inflationary models which interpolate between theStarobinsky and the quadratic model have been analyzed in[92ndash95] Due to the large effective masses that the scalarsacquire during IGI they enter a phase of damped oscillations

about zero As a consequence 120601 dependence in their normal-ization (see (57b)) does not affect their dynamics

We can readily verify that (55)(a) is satisfied for all thethree 119911120572rsquos Regarding (55)(b) we diagonalize 2

120572120573 (56) andwe obtain the scalar mass spectrum along the trajectory of(46) Our results are listed in Table 1 together with the massessquared 2

120595plusmn of the chiral fermion eigenstates plusmn = (119879 plusmn119878)radic2 From these results we deduce the following

(i) For both classes of 119870rsquos in (44) (55)(b) is satisfied forthe fluctuations of 120579 that is 2

120579 gt 0 since 119891120601 lt 0Moreover 2

120579 ≫ 2119868 because 119888119879 ≫ 1

(ii) When119870 = 1198703119894 and ℎ10158401015840119894 (0) = 0 we obtain 2

119904 lt 0 Thisoccurs for 119894 = 1 4 7 9 and 10 as shown in Table 1For 119894 = 1 our result reproduces those of similarmodels [31 47ndash52 68ndash70] The stability problem canbe cured if we include in 1198703119894 a higher order term ofthe form 119896119878|119878|4 with 119896119878 sim 1 or assuming that 1198782 = 0[75] However a probably simpler solution arises if wetake into account the results accumulated in Table 2It is clear that the condition 2

119904 gt 2119868 can be satisfied

when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840

119894 (0)| ge 1 From Table 1 wesee that this is the case for 119894 = 2 and 8

(iii) When 119870 = 1198702119894 and ℎ10158401015840119894 (0) = 0 we obtain 2

119904 gt 0 but2119904 lt 2

119868 Therefore 119878 may seed inflationary pertur-bations leading possibly to large non-Gaussianities inthe CMB contrary to observations From the resultslisted in Table 2 we see that the condition 2

119904 ≫ 2119868

requires ℎ10158401015840119894 (0) lt 0 with |ℎ10158401015840

119894 (0)| ge 1 This occurs for119894 = 3 and 11 The former case was examined in [58]

To highlight further the stabilization of 119878 during and afterIGI we present in Figure 2 2

119904 2119868 as a function of 120601 for the

various acceptable 119870rsquos identified above In particular we fix119899 = 2 and 120601⋆ = 1 setting 119870 = 11987032 or 119870 = 11987038 in Figure 2(a)and 119870 = 11987023 or 119870 = 119870211 in Figure 2(b) The parametersof the models (120582 and 119888119879) corresponding to these choices arelisted in third and fifth rows of Table 3 Evidently 2

119904 2119868

remain larger than unity for 120601119891 le 120601 le 120601⋆ where 120601⋆ and 120601119891

are also depicted However in Figure 2(b) 2119904 2

119868 exhibits aconstant behavior and increases sharply as 120601 decreases below02 On the contrary 2

119904 2119868 in Figure 2(a) is an increasing

function of 120601 for 120601 ≳ 02 with a clear minimum at 120601 ≃ 02For 120601 ≲ 02 2

119904 2119868 increases drastically as in Figure 2(b) too

Employing the well-known Coleman-Weinberg formula[96] we find from the derived mass spectrum (see Table 1)


K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)

Figure 2 The ratio 2119904 2

119868 as a function of 120601 for 119899 = 2 and 120601⋆ = 1 We set (a) 119870 = 11987032 or 119870 = 11987038 and (b) 119870 = 11987023 or 119870 = 119870211 The valuescorresponding to 120601⋆ and 120601119891 are also depicted

the one-loop radiative corrections Δ119868 to 119868 depending onrenormalization groupmass scaleΛ It can be verified that ourresults are insensitive to Δ119868 provided that Λ is determinedby requiring Δ119868(120601⋆) = 0 or Δ119868(120601119891) = 0 A possibledependence of the results on the choice ofΛ is totally avoided[31] thanks to the smallness of Δ119868 for Λ ≃ (1 minus 18) sdot 10minus5see Section 42 too These conclusions hold even for 120601 gt 1Therefore our results can be accurately reproduced by usingexclusively 119868 in (54)(a)

4 Analysis of SUGRA Inflation

Keeping in mind that for 119870 = 1198703119894 [119870 = 1198702119894] the values 119894 = 2and 8 [119894 = 3 and 11] lead to the stabilization of 119878 during andafter IGI we proceedwith the computation of the inflationaryobservables for the SUGRA models considered above Sincethe precise choice of the index 119894 does not influence ouroutputs here we do not specify henceforth the allowed 119894values We first present in Section 41 analytic results whichare in good agreement with our numerical results displayedin Section 42 Finally we investigate the UV behavior of themodels in Section 43

41 Analytical Results The duration of the IGI is controlledby the slow-roll parameters which are calculated to be

(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)

The end of inflation is triggered by the violation of 120598 conditionwhen 120601 = 120601119891 given by

120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)

The violation of 120578 condition occurs when 120601 = 120601119891 lt 120601119891

120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)

Given 120601119891 we can compute ⋆ via (4)

⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)

Ignoring the logarithmic term and taking into account that120601119891 ≪ 120601⋆ we obtain a relation between 120601⋆ and ⋆

120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)


Table 3 Input and output parameters of the models which are compatible with (4) for ⋆ = 532 (6) and (7)

Kahler potential119870 ParametersInput Output119899 119888119879 120601⋆ 120582 (10minus3) 120601119891 119899119904 120572119904 (10minus4) 119903 (10minus3)1198703119894 1 1 545 0022 15 0964 minus63 361198702119894 1 1 80 0028 17 0964 minus66 251198703119894 2 77 1 17 017 0964 minus67 371198703119894 3 109 1 24 03 0964 minus65 371198702119894 2 113 1 2 015 0964 minus67 251198702119894 3 159 1 3 03 0964 minus67 26

Obviously IGI consistent with (9)(b) can be achieved if120601⋆ le 1 997904rArr119888119879 ge 21198992⋆120581 (61b)

Therefore we need relatively large 119888119879rsquos which increase with 119899On the other hand 120601⋆ remains trans-Planckian since solvingthe first relation in (57b) with respect to 120601 and inserting (61a)we find

120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)

where the integration constant 120601119888 = 0 and as in the previouscases we set ⋆ ≃ 53 Despite this fact our constructionremains stable under possible corrections from higher orderterms in119891119870 since when these are expressed in terms of initialfield 119879 they can be seen to be harmless for |119879| le 1

Upon substitution of (54) and (61a) into (6) we find

11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)

Enforcing (6) we obtain a relation between 120582 and 119888119879 whichturns out to be independent of 119899 Indeed we have

120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)

Finally substituting the value of 120601⋆ given in (61a) into (8) weestimate the inflationary observables For119870 = 1198703119894 the resultsare given in (31a)ndash(31c) For119870 = 1198702119894 we obtain the relations

119899119904 = 4⋆ (⋆ minus 3) minus 3(1 minus 2⋆)2 ≃ 1 minus 2119873⋆

minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)

These outputs are consistent with our results in [58] for119898 = 119899and 11989911 = 1198992 = 2 (in the notation of that reference)

42 Numerical Results The analytical results presentedabove can be verified numerically The inflationary scenariodepends on the following parameters (see (42) and (44))

119899 119888119879 and 120582 (66)

Note that the stabilization of 119878 with one of119870321198703411987023 and119870211 does not require any additional parameter Recall thatwe use 119879rh = 41 sdot 10minus9 throughout and ⋆ is computedself-consistently for any 119899 via (4) Our result is ⋆ ≃ 532For given 119899 the parameters above together with 120601⋆ can bedetermined by imposing the observational constraints in (4)and (6) In our code we find 120601⋆ numerically without thesimplifying assumptions used for deriving (61a) Inserting itinto (8) we extract the predictions of the models

The variation of 119868 as a function of 120601 for two differentvalues of 119899 can be easily inferred from Figure 3 In particularwe plot 119868 versus 120601 for 120601⋆ = 1 119899 = 2 or 119899 = 6 setting119870 = 1198703119894

in Figure 3(a) and 119870 = 1198702119894 in Figure 3(b) Imposing 120601⋆ = 1for 119899 = 2 amounts to (120582 119888119879) = (00017 77) for 119870 = 1198703119894 and(120582 119888119879) = (00017 113) for 119870 = 1198702119894 Also 120601⋆ = 1 for 119899 = 6 isobtained for (120582 119888119879) = (00068 310) for 119870 = 1198703119894 and (120582 119888119879) =(00082 459) for 119870 = 1198702119894 In accordance with our findingsin (61b) we conclude that increasing 119899 (i) requires larger 119888119879rsquosand therefore lower 119868rsquos to obtain 120601 le 1 (ii) larger 120601119891 and⟨120601⟩ are obtained see Section 43 Combining (59a) and (64)with (54)(a) we can conclude that 119868(120601119891) is independent of119888119879 and to a considerable degree of 119899

Our numerical findings for 119899 = 1 2 and 3 and 119870 = 1198703119894

or 119870 = 1198702119894 are presented in Table 2 In the two first rowswe present results associated with Ceccoti-like models [97]which are defined by 119888119879 = 119899 = 1 and cannot be madeconsistent with the imposed Z119899 symmetry or with (9) Wesee that selecting 120601⋆ ≫ 1 we attain solutions that satisfy allthe remaining constraints in Section 22 For the other caseswe choose a 119888119879 value so that 120601⋆ = 1 Therefore the presented119888119879 is the minimal one in agreement with (61b)

In all cases shown in Table 2 the modelrsquos predictions for119899119904 120572119904 and 119903 are independent of the input parameters Thisis due to the attractor behavior [30ndash32] that these modelsexhibit provided that 119888119879 is large enough Moreover these


00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)

Figure 3 The inflationary potential 119868 as a function of 120601 for 120601⋆ = 1 and 119899 = 2 or 119899 = 6 We set (a) 119870 = 1198703119894 and (b) 119870 = 1198702119894 The valuescorresponding to 120601⋆ and 120601119891 are also depicted

outputs are in good agreement with the analytical findingsof (31a)ndash(31c) for 119870 = 1198703119894 or (65a)ndash(65c) for 119870 = 1198702119894On the other hand the presented 119888119879 120582 120601⋆ and 120601119891 valuesdepend on 119899 for every selected 119870 The resulting 119899119904 ≃ 0964is close to its observationally central value 119903 is of the orderof 0001 and |120572119904| is negligible Although the values of 119903lie one order of magnitude below the central value of thepresent combined Bicep2Keck Array and Planck results [5]these are perfectly consistent with the 95 cl margin in(7) The values of 119903 for 119870 = 1198703119894 or 119870 = 1198702119894 distinguishthe two cases The difference is small at the level of 10minus3However it is possibly reachable by the next-generationexperiment (eg the CMBPol experiment [98]) is expectedto achieve a precision for 119903 of the order of 10minus3 or even05sdot10minus3 Finally the renormalization scaleΛ of theColeman-Weinberg formula found by imposingΔ119868(120601⋆) = 0 takes thevalues 78 sdot 10minus5 93 sdot 10minus5 13 sdot 10minus5 and 21 sdot 10minus5 for 119870321198703811987023 and119870211 respectively As a consequenceΛdependsexplicitly on the specific choice of 119894 used for1198703119894 or1198702119894

The overall allowed parameter space of the model for 119899 =2 3 and 6 is correspondingly77 105 310 ≲ 119888119879 ≲ 16 sdot 105

(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)

where the parameters are bounded from above as in (29)Letting 120582 or 119888119879 vary within its allowed region above we obtain

the values of 119899119904 120572119904 and 119903 listed in Table 3 for 119870 = 1198703119894

and 1198702119894 independently of 119899 Therefore the inclusion of thevariant exponent 119899 gt 2 compared to the non-SUSY modelin Section 24 does not affect the successful predictions ofmodel

43 UV Behavior Following the approach described inSection 22 we can verify that the SUGRA realizations ofIGI retain perturbative unitarity up to 119898119875 To this end weanalyze the small-field behavior of the theory expanding S in(1) about

⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)

which is confined in the ranges (00026ndash01) (0021ndash024)and (017ndash048) for the margins of the parameters in (67a)and (67b)

The expansion of S is performed in terms of 120575120601 which isfound to be

120575120601 = ⟨119869⟩ 120575120601 with ⟨119869⟩ ≃ radic120581119899⟨120601⟩ = 2(2minus119899)2119899radic1205811198991198881119899119879 (69)

where 120581 is defined in (60) Note in passing that the mass of120575120601 at the SUSY vacuum in (41) is given by

120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)

precisely equal to that found in (19) and (30) We observethat 120575120601 is essentially independent of 119899 and 120581 thanks to therelation between 120582 and 119888119879 in (64)


Expanding the second term in the rhs of (33a) about ⟨120601⟩with 119869 given by the first relation in (57b) we obtain

1198692 1206012

= (1 minus 2119899radic120581120575120601 + 311989921205811205751206012 minus 41198993120581minus321205751206013 + sdot sdot sdot)

sdot 1205751206012(71a)

On the other hand 119868 in (54)(a) can be expanded about ⟨120601⟩as follows

119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012

minus sdot sdot sdot) (71b)

Since the expansions above are 119888119879 independent we infer thatΛUV = 1 as in the other versions of Starobinsky-like inflationThe expansions above for 119870 = 1198703119894 and 119899 = 2 reduce to thosein (32a) and (32b) Moreover these are compatible with theones presented in [31] for 119870 = 1198703119894 and those in [58] for 119870 =1198702119894 and 11989911 = 2 Our overall conclusion is that our models donot face any problem with perturbative unitarity up to119898119875

5 Conclusions and Perspectives

In this review we revisited the realization of induced-gravityinflation (IGI) in both a nonsupersymmetric and super-gravity (SUGRA) framework In both cases the inflationarypredictions exhibit an attractor behavior towards those ofStarobinsky model Namely we obtained a spectral index119899119904 ≃ (0960ndash0965) with negligible running 120572119904 and a tensor-to-scalar ratio 0001 ≲ 119903 ≲ 0005 The mass of the inflationturns out be close to 3sdot1013 GeV It is gratifying that IGI can beachieved for sub-Planckian values of the initial (noncanon-ically normalized) inflation and the corresponding effectivetheories are trustable up to Planck scale although a parameterhas to take relatively high values Moreover the one-loopradiative corrections can be kept under control

In the SUGRA context this type of inflation can be incar-nated using two chiral superfields119879 and 119878 the superpotentialin (42) which realizes easily the idea of induced gravity andseveral (semi)logarithmic Kahler potentials 1198703119894 or 1198702119894 see(44)Themodels are prettymuch constrained upon imposingtwo global symmetries a continuous 119877 and a discrete Z119899

symmetry in conjunction with the requirement that theoriginal inflation 119879 takes sub-Planckian values We paidspecial attention to the issue of 119878 stabilization during IGIand worked out its dependence on the functional form of theselected 119870rsquos with respect to |119878|2 More specifically we testedthe functions ℎ119894(|119878|2) which appear in1198703119894 or1198702119894 see Table 1We singled out ℎ2(|119878|2) and ℎ8(|119878|2) for 119870 = 1198703119894 or ℎ3(|119878|2)and ℎ11(|119878|2) for 119870 = 1198702119894 which ensure that 119878 is heavyenough and so well stabilized during and after inflationThis analysis provides us with new results that do not appearelsewhere in the literature Therefore Starobinsky inflation

realized within this SUGRA set-up preserves its originalpredictive power since no mixing between |119879|2 and |119878|2 isneeded for consistency in the considered119870rsquos (cf [31 72 73])

It is worth emphasizing that the 119878-stabilization mecha-nisms proposed in this paper can be also employed in othermodels of ordinary [47ndash49] or kinetically modified [65ndash67]nonminimal chaotic (and Higgs) inflation driven by a gaugesinglet [47ndash49 53 54 65ndash67] or nonsinglet [50ndash52 68ndash70]inflation without causing any essential alteration to theirpredictions The necessary modifications involve replacingthe |119878|2 part of119870 with ℎ2(|119878|2) or ℎ8(|119878|2) if we have a purelylogarithmic Kahler potential Otherwise the |119878|2 part can bereplaced by ℎ3(|119878|2) or ℎ11(|119878|2) Obviously the last case canbe employed for logarithmic or polynomial119870rsquos with regard tothe inflation terms

Let us finally remark that a complete inflationary sce-nario should specify a transition to the radiation dominatedera This transition could be facilitated in our setting [29 6263] via the process of perturbative reheating according towhich the inflation after inflation experiences an oscillatoryphase about the vacuum given by (22) for the non-SUSYcase or (41) for the SUGRA case During this phase theinflation can safely decay provided that it couples to lightdegrees of freedom in the Lagrangian of the full theoryThis process is independent of the inflationary observablesand the stabilization mechanism of the noninflation field Itdepends only on the inflation mass and the strength of therelevant couplings This scheme may also explain the originof the observed baryon asymmetry through nonthermalleptogenesis consistently with the data from the neutrinooscillations [29] It would be nice to obtain a complete andpredictable transition to the radiation dominated era Analternative graceful exit can be achieved in the runningvacuum models as described in the fourth paper of [77ndash84]

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

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[2] A D Linde ldquoA new inflationary universe scenario a possiblesolution of the horizon flatness homogeneity isotropy andprimordial monopole problemsrdquo Physics Letters B vol 108 no6 pp 389ndash393 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 article 1220 1982

[4] P A R Ade N AghanimM Arnaud et al ldquoPlanck 2015 resultsXX Constraints on inflationrdquo Astronomy and Astrophysics vol594 article A20 2016

[5] P A R Ade Z Ahmed RWAikin et al ldquoImproved constraintson cosmology and foregrounds from bicep2 and keck arraycosmic microwave background data with inclusion of 95 GHzbandrdquo Physical Review Letters vol 116 Article ID 031302 2016


[6] K S Stelle ldquoRenormalization of higher-derivative quantumgravityrdquo Physical Review D vol 16 no 4 pp 953ndash969 1977

[7] K S Stelle ldquoClassical gravity with higher derivativesrdquo GeneralRelativity and Gravitation vol 9 no 4 pp 353ndash371 1978

[8] S V Ketov and A A Starobinsky ldquoEmbedding R + R2 inflationin supergravityrdquo Physical Review D vol 83 Article ID 0635122011

[9] S V Ketov and N Watanabe ldquoCosmological properties ofa generic Script R2-supergravityrdquo Journal of Cosmology andAstroparticle Physics vol 2011 no 3 article 11 2011

[10] S V Ketov and A A Starobinsky ldquoInflation and nonminimalscalar-curvature coupling in gravity and supergravityrdquo Journalof Cosmology and Astroparticle Physics vol 2012 no 8 article22 2012

[11] S V Ketov and S Tsujikawa ldquoConsistency of inflation andpreheating in F(R) supergravityrdquo Physical Review D vol 86Article ID 023529 2012

[12] W Buchmuller V Domcke and K Kamada ldquoThe Starobinskymodel from superconformal D-term inflationrdquo Physics LettersB vol 726 no 1ndash3 pp 467ndash470 2013

[13] F Farakos A Kehagias and A Riotto ldquoOn the Starobinskymodel of inflation from supergravityrdquo Nuclear Physics B vol876 no 1 pp 187ndash200 2013

[14] J Alexandre N Houston and N E Mavromatos ldquoStarobinsky-type inflation in dynamical supergravity breaking scenariosrdquoPhysical Review D vol 89 Article ID 027703 2014

[15] K Kamada and J Yokoyama ldquoTopological inflation from theStarobinsky model in supergravityrdquo Physical Review D vol 90Article ID 103520 2014

[16] R Blumenhagen A Font M Fuchs D Herschmann andE Plauschinn ldquoTowards axionic Starobinsky-like inflation instring theoryrdquo Physics Letters B vol 746 pp 217ndash222 2015

[17] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationvia non-geometric flux compactifications from natural tostarobinsky-like inflationrdquo Journal of High Energy Physics vol2015 no 10 article 138 2015

[18] S Basilakos N E Mavromatos and J Sola ldquoStarobinsky-likeinflation and running vacuum in the context of supergravityrdquoUniverse vol 2 no 3 article 14 2016

[19] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11ndash13 Article ID 111301 2013Erratum-ibid vol 111 no 12 Article ID 129902 2013

[20] J Ellis D Nanopoulos and K Olive ldquoStarobinsky-like infla-tionary models as avatars of no-scale supergravityrdquo Journal ofCosmology and Astroparticle Physics vol 2013 no 10 article 92013

[21] R Kallosh and A Linde ldquoSuperconformal generalizations ofthe Starobinsky modelrdquo Journal of Cosmology and AstroparticlePhysics vol 2013 no 6 article 28 2013

[22] D Roest M Scalisi and I Zavala ldquoKahler potentials for Planckinflationrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 11 article 7 2013

[23] J Ellis H-J He and Z-Z Xianyu ldquoNew Higgs inflation in ano-scale supersymmetric SU(5) GUTrdquo Physical Review D vol91 no 2 Article ID 021302 2015

[24] J Ellis A G G Marcos N Natsumi V N Dimitri and AO Keith ldquoStarobinsky-like inflation and neutrino masses in ano-scale SO(10) modelrdquo Journal of Cosmology and AstroparticlePhysics vol 2016 no 11 article 18 2016

[25] I Garg and S Mohanty ldquoNo scale SUGRA SO(10) derivedStarobinsky model of inflationrdquo Physics Letters B vol 751 pp7ndash11 2015

[26] J Ellis A G G Marcos V N Dimitri and A O Keith ldquoNo-scale inflationrdquo Classical and Quantum Gravity vol 33 no 9Article ID 094001 2016

[27] G Chakravarty G Lambiase and SMohanty ldquoTesting theoriesof Gravity and Supergravity with inflation and observationsof the cosmic microwave backgroundrdquo httpsarxivorgabs160706325

[28] A B Lahanas and K Tamvakis ldquoInflation in no-scale super-gravityrdquoPhysical ReviewD vol 91 no 8 Article ID085001 2015

[29] C Pallis ldquoLinking Starobinsky-type inflation in no-scale super-gravity to MSSMrdquo Journal of Cosmology and AstroparticlePhysics vol 2014 no 4 article 24 2014

[30] G F Giudice and H M Lee ldquoStarobinsky-like inflation frominduced gravityrdquo Physics Letters B vol 733 pp 58ndash62 2014

[31] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 8 article 57 2014

[32] R Kallosh ldquoMore on universal superconformal attractorsrdquoPhysical Review D vol 89 Article ID 087703 2014

[33] F S Accetta D J Zoller and M S Turner ldquoInduced-gravityinflationrdquo Physical ReviewD vol 31 no 12 pp 3046ndash3051 1985

[34] D S Salopek J R Bond and J M Bardeen ldquoDesigning densityfluctuation spectra in inflationrdquo Physical Review D vol 40 no6 pp 1753ndash1788 1989

[35] R Fakir and W G Unruh ldquoInduced-gravity inflationrdquo PhysicalReview D vol 41 no 6 pp 1792ndash1795 1990

[36] J L Cervantes-Cota and H Dehnen ldquoInduced gravity inflationin the SU(5) GUTrdquo Physical Review D vol 51 no 2 article 3951995

[37] N Kaloper L Sorbo and J Yokoyama ldquoInflation at the GUTscale in a Higgsless universerdquo Physical Review D vol 78 no 4Article ID 043527 2008

[38] ACerioni F Finelli A Tronconi andGVenturi ldquoInflation andreheating in spontaneously generated gravityrdquo Physical ReviewD vol 81 no 12 Article ID 123505 2010

[39] K Kannike G Hutsi L Pizza et al ldquoDynamically inducedPlanck scale and inflatiordquo Journal of High Energy Physics vol2015 no 5 article 65 2015

[40] M B Einhorn and D R T Jones ldquoInduced gravity I real scalarfieldrdquo Journal of High Energy Physics vol 2016 no 1 article 192016

[41] A Zee ldquoBroken-symmetric theory of gravityrdquo Physical ReviewLetters vol 42 article 417 1979

[42] H Terazawa ldquoCosmological origin of mass scalesrdquo PhysicsLetters B vol 101 no 1-2 pp 43ndash47 1981

[43] J L Cervantes-Cota and H Dehnen ldquoInduced gravity inflationin the standardmodel of particle physicsrdquoNuclear Physics B vol442 no 1-2 pp 391ndash409 1995

[44] F L Bezrukov and M Shaposhnikov ldquoThe Standard ModelHiggs boson as the inflatonrdquo Physics Letters B vol 659 no 3pp 703ndash706 2008

[45] C Pallis ldquoNon-minimally gravity-coupled inflationarymodelsrdquoPhysics Letters B vol 692 no 5 pp 287ndash296 2010

[46] R Kallosh A Linde and D Roest ldquoUniversal attractor forinflation at strong couplingrdquo Physical Review Letters vol 112Article ID 011303 2014


[47] M B Einhorn and D R T Jones ldquoInflation with non-minimalgravitational couplings in supergravityrdquo Journal of High EnergyPhysics vol 2010 no 3 article 26 2010

[48] S Ferrara R Kallosh A Linde A Marrani and A VanProeyen ldquoSuperconformal symmetry NMSSM and inflationrdquoPhysical Review D vol 83 no 2 Article ID 025008 2011

[49] C Pallis and N Toumbas ldquoNon-minimal sneutrino inflationPeccei-Quinn phase transition and non-thermal leptogenesisrdquoJournal of Cosmology and Astroparticle Physics vol 2011 no 2article 19 2011

[50] M Arai S Kawai and N Okada ldquoHiggs inflation in minimalsupersymmetric SU(5) grand unified theoryrdquo Physical ReviewD vol 84 no 12 Article ID 123515 2011

[51] C Pallis and N Toumbas ldquoNon-minimal Higgs inflationand non-thermal leptogenesis in a supersymmetric Pati-Salammodelrdquo Journal of Cosmology andAstroparticle Physics vol 2011no 12 article 2 2011

[52] M B Einhorn and D R T Jones ldquoGUT scalar potentials forHiggs inflationrdquo Journal of Cosmology and Astroparticle Physicsvol 2012 no 11 article 49 2012

[53] C Pallis and Q Shafi ldquoNonminimal chaotic inflation Peccei-Quinn phase transition and nonthermal leptogenesisrdquo PhysicalReview D vol 86 no 2 Article ID 023523 2012

[54] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 23 2015

[55] J L F Barbon and J R Espinosa ldquoOn the naturalness of Higgsinflationrdquo Physical Review D vol 79 Article ID 081302 2009

[56] C P Burgess H M Lee and M Trott ldquoOn Higgs inflation andnaturalnessrdquo Journal of High Energy Physics vol 2010 no 7article 7 2010

[57] A Kehagias A M Dizgah and A Riotto ldquoRemarks on theStarobinsky model of inflation and its descendantsrdquo PhysicalReview D vol 89 no 4 Article ID 043527 2014

[58] C Pallis and N Toumbas ldquoStarobinsky-type inflation withproducts of Kahler manifoldsrdquo Journal of Cosmology andAstroparticle Physics vol 2016 no 5 article 15 2016

[59] E Cremmer S Ferrara C Kounnas and D V NanopoulosldquoNaturally vanishing cosmological constant in 119873 = 1 super-gravityrdquo Physics Letters B vol 133 no 1-2 pp 61ndash66 1983

[60] J R Ellis A B Lahanas D V Nanopoulos and K TamvakisldquoNo-scale supersymmetric standard modelrdquo Physics Letters Bvol 134 no 6 pp 429ndash435 1984

[61] A B Lahanas and D V Nanopoulos ldquoThe road to no scalesupergravityrdquo Physics Reports vol 145 no 1-2 pp 1ndash139 1987

[62] T Terada YWatanabe Y Yamada and J Yokoyama ldquoReheatingprocesses after Starobinsky inflation in old-minimal supergrav-ityrdquo Journal of High Energy Physics vol 2015 no 2 article 1052015

[63] J Ellis M Garcia D Nanopoulos and K Olive ldquoPhenomeno-logical aspects of no-scale inflation modelsrdquo Journal of Cosmol-ogy and Astroparticle Physics vol 2015 no 10 article 3 2015

[64] H M Lee ldquoChaotic inflation in Jordan frame supergravityrdquoJournal of Cosmology and Astroparticle Physics vol 2010 no 8article 3 2010

[65] C Pallis ldquoKinetically modified nonminimal chaotic inflationrdquoPhysical Review D vol 91 no 12 Article ID 123508 2015

[66] C Pallis ldquoObservable gravitational waves fromkineticallymod-ified non-minimal inflationrdquo PoS PLANCK vol 2015 article 952015

[67] C Pallis ldquoKinetically modified non-minimal inflation withexponential frame functionrdquo httpsarxivorgabs161107010

[68] G Lazarides and C Pallis ldquoShift symmetry and Higgs inflationin supergravity with observable gravitational wavesrdquo Journal ofHigh Energy Physics vol 2015 no 11 article 114 2015

[69] C Pallis ldquoKinetically modified nonminimal Higgs inflation insupergravityrdquo Physical Review D vol 92 no 12 Article ID121305 2015

[70] C Pallis ldquoVariants of kinetically modified non-minimal Higgsinflation in supergravityrdquo Journal of Cosmology and Astroparti-cle Physics vol 2016 no 10 article 37 2016

[71] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[72] C Pallis ldquoReconciling induced-gravity inflation in supergravitywith the Planck 2013 amp BICEP2 resultsrdquo Journal of Cosmologyand Astroparticle Physics vol 2014 no 10 article 58 2014

[73] C Pallis ldquoInduced-gravity inflation in supergravity confrontedwith planck 2015 amp BICEP2keck arrayrdquo PoS CORFU vol 2014article 156 2015

[74] C Pallis ldquoModels of non-minimal chaotic inflation in super-gravityrdquo PoS CORFU vol 2013 article 61 2013

[75] I Antoniadis E Dudas S Ferrara and A Sagnotti ldquoTheVolkovndashAkulovndashStarobinsky supergravityrdquo Physics Letters Bvol 733 pp 32ndash35 2014

[76] A A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[77] J A S Lima S Basilakos and J Sola ldquoExpansion history withdecaying vacuum a complete cosmological scenariordquo MonthlyNotices of the Royal Astronomical Society vol 431 no 1 pp 923ndash929 2013

[78] J A Lima S Basilakos and J Sola ldquoNonsingular decayingvacuum cosmology and entropy productionrdquoGeneral Relativityand Gravitation vol 47 no 4 article 40 2015

[79] J Sola and A Gomez-Valent ldquoThe Λ cosmology from inflationto dark energy through running Λrdquo International Journal ofModern Physics D Gravitation Astrophysics Cosmology vol24 no 4 Article ID 1541003 37 pages 2015

[80] J A S Lima S Basilakos and J Sola ldquoThermodynamicalaspects of running vacuum modelsrdquo The European PhysicalJournal C vol 76 no 4 article 228 2016

[81] J Sola A Gomez-Valent and J de Cruz Perez ldquoFirst evidence ofrunning cosmic vacuum challenging the concordance modelrdquoThe Astrophysical Journal vol 836 no 1 article 43 2017

[82] J Sola A Gomez-Valent and J de Cruz Perez ldquoHints ofdynamical vacuum energy in the expanding universerdquo TheAstrophysical Journal Letters vol 811 p L14 2015

[83] J Sola J de Cruz Perez A Gomez-Valent and R C NunesldquoDynamical vacuum against a rigid cosmological constantrdquohttpsarxivorgabs160600450

[84] J Sola ldquoDark energy a quantum fossil from the inflationaryUniverserdquo Journal of Physics A Mathematical and Theoreticalvol 41 no 16 Article ID 164066 2008

[85] D H Lyth and A Riotto ldquoParticle physics models of inflationand the cosmological density perturbationrdquo Physics Reports AReview Section of Physics Letters vol 314 no 1-2 pp 1ndash146 1999

[86] G Lazarides ldquoBasics of inflationary cosmologyrdquo Journal ofPhysics Conference Series vol 53 p 528 2006


[87] A Mazumdar and J Rocher ldquoParticle physics models ofinflation and curvaton scenariosrdquo Physics Reports A ReviewSection of Physics Letters vol 497 no 4-5 pp 85ndash215 2011

[88] J Martin C Ringeval and V Vennin ldquoEncyclopaeligdia inflation-arisrdquo Physics of the Dark Universe vol 5-6 pp 75ndash235 2014

[89] M S Turner ldquoCoherent scalar-field oscillations in an expandinguniverserdquo Physical Review D Particles and Fields Third Seriesvol 28 no 6 pp 1243ndash1247 1983

[90] P A R Ade N AghanimM Arnaud et al ldquoPlanck 2015 resultsXIII Cosmological parametersrdquo Astronomy and Astrophysicsvol 594 article A13 p 63 2016

[91] S Renaux-Petel and K Turzynski ldquoGeometrical destabilizationof inflationrdquo Physical Review Letters vol 117 no 14 Article ID141301 2016

[92] J Ellis M A G Garcıa D V Nanopoulos and K A OliveldquoTwo-field analysis of no-scale supergravity inflationrdquo Journalof Cosmology and Astroparticle Physics vol 2015 no 1 article10 2015

[93] C van de Bruck and L E Paduraru ldquoSimplest extension ofStarobinsky inflationrdquo Physical Review D vol 92 no 8 ArticleID 083513 2015

[94] S Kaneda and S V Ketov ldquoStarobinsky-like two-field inflationrdquoThe European Physical Journal C vol 76 no 1 p 26 2016

[95] G Chakravarty S Das G Lambiase and S Mohanty ldquoDilatonassisted two-field inflation from no-scale supergravityrdquo PhysicalReview D vol 94 no 2 Article ID 023521 2016

[96] S R Coleman and E J Weinberg ldquoRadiative corrections as theorigin of spontaneous symmetry breakingrdquo Physical Review Dvol 7 no 6 pp 1888ndash1910 1973

[97] S Cecotti ldquoHigher derivative supergravity is equivalent tostandard supergravity coupled to matterrdquo Physics Letters BParticle Physics Nuclear Physics and Cosmology vol 190 no 1-2pp 86ndash92 1987

[98] D Baumann M G Jackson P Adshead A Amblard and AAshoorioon ldquoProbing inflation with CMB polarizationrdquo AIPConference Proceedings vol 1141 article 10 2009

Submit your manuscripts athttpswwwhindawicom

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An important issue in embedding IGI in SUGRA isthe stabilization of the matter-like field 119878 Indeed when 119870parameterizes the 119878119880(2 1)(119878119880(2) times 119880(1)) Kahler manifold[20 21] the inflationary trajectory turns out to be unstablewith respect to the fluctuations of 119878 This difficulty can beovercome by adding a sufficiently large term 119896119878|119878|4 with 119896119878 gt0 and |119896119878| sim 1 in the logarithmic function appearing in119870 as suggested in [64] for models of nonminimal (chaotic)inflation [47ndash49] and applied in [50ndash54 65ndash70] This solu-tion however deforms slightly the Kahler manifold [71]More importantly it violates the predictability of Starobinskyinflation since mixed terms 119896119878119879|119878|2|119879|2 with 119896119878119879 ge 001which cannot be ignored (without tuning) have an estimableimpact [31 72ndash74] on the dynamics and the observablesMoreover this solution becomes complicated when morethan two fields are considered since all quartic terms allowedby symmetries have to be considered and the analysis of thestabilization mechanism becomes tedious (see eg [31 72ndash74]) Alternatively it was suggested to use a nilpotent super-field 119878 [75] or a charged field under a gauged119877 symmetry [71]

In this review we revisit the issue of stabilizing 119878disallowing terms of the form |119878|2119898 119898 gt 1 without caringmuch about the structure of the Kahler manifold Namelywe investigate systematically several functions ℎ119894(|119878|2) (with119894 = 1 11) that appear in the choices for 119870 and we findfour acceptable forms that lead to the stabilization of 119878 duringand after IGI The output of this analysis is new providingresults that did not appear in the literature before Morespecifically we consider two principal classes of119870rsquos1198703119894 and1198702119894 distinguished by whether ℎ119894 and 119879 appear in the samelogarithmic function The resulting inflationary scenariosare almost indistinguishable The case considered in [58] isincluded as one of the viable choices in 1198702119894 class Contraryto [58] we impose here the same Z119899 symmetry on119882 and119870Consequently the relevant expressions for themass spectrumand the inflationary observables get simplified considerablycompared to those displayed in [58] As in the non-SUSYcase IGI may be realized using sub-Planckian values forthe initial (noncanonically normalized) inflation field Theradiative corrections remain under control and perturbativeunitarity is not violated up to119898119875 [31 58 76] consistently withthe consideration of SUGRA as an effective theory

Throughout this review we focus on the standard ΛCDMcosmological model [4] An alternative framework is pro-vided by the running vacuummodels [77ndash84] which turn outto yield a quality fit to observations significantly better thanthat of ΛCDM In this case the acceleration of the universeeither during inflation or at late times is not attributed toa scalar field but rather arises from the modification of thevacuum itself which is dynamical A SUGRA realization ofStarobinsky inflation within this setting is obtained in [18]

The plan of this paper is as follows In Section 2 weestablish the realization of Starobinsky inflation as IGI in anon-SUSY framework In Section 3 we introduce the formu-lation of IGI in SUGRA and revisit the issue of stabilizingthe matter-like field 119878 The emerging inflationary modelsare analyzed in Section 4 Our conclusions are summarizedin Section 5 Throughout charge conjugation is denoted by

a star ( lowast) the symbol 119911 as subscript denotes derivation withrespect to 119911 and we use units where the reduced Planck scale119898119875 = 243 sdot 1018 GeV is set equal to unity

2 Starobinsky Inflation from Induced Gravity

We begin our presentation demonstrating the connectionbetween R2 inflation and IGI We first review the formu-lation of nMI in Section 21 and then proceed to describethe inflationary analysis in Section 22 Armed with theseprerequisites we present R2 inflation as a type of nMI inSection 23 and exhibit its connection with IGI in Section 24

21 Coupling Nonminimally the Inflation to Gravity Weconsider an inflation 120601 that is nonminimally coupled to theRicci scalarR via a coupling function 119891R(120601) We denote theinflation potential by 119881119868(120601) and allow for a general kineticfunction 119891119870(120601)mdashin the cases of pure nMI [33ndash35 45 46]119891119870 = 1 The Jordan Frame (JF) action takes the form

S = int1198894119909radicminusg(minus12119891RR + 12119891119870119892120583]120597120583120601120597]120601 minus 119881119868 (120601)) (1)

where g is the determinant of the Friedmann-Robertson-Walker metric 119892120583] with signature (+ minus minus minus) We require⟨119891R⟩ ≃ 1 to ensure ordinary Einstein gravity at low energies

By performing a conformal transformation [45] to theEinstein frame (EF) we write the action

S = int1198894119909radicminusg(minus12R + 12119892120583]120597120583120601120597]120601 minus 119868 (120601)) (2)

where a hat denotes an EF quantityThe EFmetric is given by119892120583] = 119891R119892120583] and the canonically normalized field 120601 and itspotential 119868 are defined as follows

(a) 119889120601119889120601 = 119869 = radic 119891119870119891R + 32 (119891R120601119891R )2(b) 119868 = 1198811198681198912

R

(3)

For 119891R ≫ 119891119870 the coupling function 119891R acquires a twofoldrole On the one hand it determines the relation between120601 and 120601 On the other hand it controls the shape of 119881119868thus affecting the observational predictions see below Theanalysis of nMI can be performed in the EF using thestandard slow-roll approximation It is [33ndash35] completelyequivalent with the analysis in the JF We just have to keeptrack the relation between 120601 and 12060122 Observational and Theoretical Constraints A viablemodel of nMI must be compatible with a number of obser-vational and theoretical requirements summarized in thefollowing (cf [85ndash88])

(1) The number of e-foldings ⋆ that the scale 119896⋆ =005Mpc experiences during inflation must be large enough



⋆ = int120601⋆

120601119891

119889120601 119868119868120601

= int120601⋆

120601119891

1198891206011198692 119868119868120601

≃ 617 + ln119868 (120601⋆)12119868 (120601119891)13 + 13 ln119879rh

+ 12 ln119891R (120601⋆)119891R (120601119891)13

(4)


max 120598 (120601119891) 10038161003816100381610038161003816120578 (120601119891)10038161003816100381610038161003816 = 1 where 120598 = 12 (119868120601119868

)2 = 121198692 (119868120601119868

)2 120578 = 119868120601120601119868

= 11198692 (119868120601120601119868

minus 119868120601119868

119869120601119869 ) sdot (5)




(6)



(a) 119899119904 = 0968 plusmn 0009(b) 119903 le 007 (7)



(a) 119899119904 = 1 minus 6120598⋆ + 2120578⋆(b) 120572119904 = 23 (41205782


(8)




(a) 119868 (120601⋆)14 le ΛUV(b) 120601⋆ le ΛUV (9)



119891119870 = 0119891R = 1 + 4119888R120601119868 = 1206012

(10)






00

02

04

06

08

10

12

1 2 3 4 5 6 7 8 9 100

VI(10minus10)

휙 (10minus4)

휙f

휙⋆

(a)

cℛ = 77

M = 0114

14

12

10

08

06

04

02

0002 04 06 08 10 12 1400

VI(10minus10)

휙f

휙


휙⋆

(b)



(a) 119869 = 2radic6 119888R119891R (b) 119868 = 1206012

1198912R

≃ 1161198882R

(12)


120598 = 1121198882R1206012

120578 = 1 minus 4119888R120601121198882

R1206012

sdot (13)





(16)


120601⋆ ≃ ⋆3119888R ≃ 83 sdot 10minus4 ≪ 1 (17)


119899119904 ≃ (⋆ minus 3) (⋆ minus 1)2⋆

≃ 1 minus 2119873⋆

minus 31198732⋆

≃ 0961 (18a)

120572119904 ≃ minus(⋆ minus 3) (4⋆ + 3)24⋆

≃ minus 21198732⋆

minus 1523⋆


119903 ≃ 121198732⋆




(19)







119868 = 1206012

241198882R




119891119870 = 1119891R = 119888R1206012119881119868 = 120582 (1206012 minus 1198722)24

(21)


119891R (⟨120601⟩) = 1 997904rArr119872 = 1radic119888R (22)



(a) 119869 ≃ radic6120601 (b) 119868 = 1205821198912

12060141198884R1206014

≃ 12058241198882R

with 119891120601 = 1 minus 119888R1206012(23)



120598 = 431198912120601

120578 = 4 (1 + 119891120601)31198912

120601

sdot(24)



(25)


⋆ ≃ 3119888R1206012⋆4 997904rArr

120601⋆ ≃ 2radic ⋆3119888R ≫ 120601119891(26)


120601⋆ le 1 997904rArr119888R ge 4⋆3 ≃ 71 (27)



(28)


77 ≲ 119888R ≲ 85 sdot 10428 sdot 10minus6 ≲ 120582 ≲ 35 (29)






minus 922⋆

≃ 0961(31a)


⋆

minus 2123⋆


119903 = 192(3 minus 4⋆)2 ≃ 121198732

⋆





with 120575120601 ≃ radic6119888R120575120601(32a)


119868 = 1205822

61198882R







where 119870120572120573 = 119870119911120572119911lowast120573




= 119890119870 (119870120572120573119863120572119882119863lowast

120573119882lowast minus 3 |119882|2)

with 119863120572119882 = 119882119911120572 + 119870119911120572119882 (33b)



1198732 (34)




A120583 = 119894 (119870120572120597120583119911120572 minus 119870120572120597120583119911lowast120572)6 (36)




minus 271198733ΩA120583A

120583 minus 119881) (37a)




minus⟨Ω⟩119873 ≃ 1 (38)




119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912














119896=2

120582119896119879119896119899 (39)



⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)


119882 = 120582119878(119879119899 minus 12119888119879) (42)


119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)









119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)






119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





⋆ = int120601⋆

120601119891

119889120601 119868119868120601

= int120601⋆

120601119891

1198891206011198692 119868119868120601

≃ 617 + ln119868 (120601⋆)12119868 (120601119891)13 + 13 ln119879rh

+ 12 ln119891R (120601⋆)119891R (120601119891)13

(4)


max 120598 (120601119891) 10038161003816100381610038161003816120578 (120601119891)10038161003816100381610038161003816 = 1 where 120598 = 12 (119868120601119868

)2 = 121198692 (119868120601119868

)2 120578 = 119868120601120601119868

= 11198692 (119868120601120601119868

minus 119868120601119868

119869120601119869 ) sdot (5)




(6)



(a) 119899119904 = 0968 plusmn 0009(b) 119903 le 007 (7)



(a) 119899119904 = 1 minus 6120598⋆ + 2120578⋆(b) 120572119904 = 23 (41205782


(8)




(a) 119868 (120601⋆)14 le ΛUV(b) 120601⋆ le ΛUV (9)



119891119870 = 0119891R = 1 + 4119888R120601119868 = 1206012

(10)






00

02

04

06

08

10

12

1 2 3 4 5 6 7 8 9 100

VI(10minus10)

휙 (10minus4)

휙f

휙⋆

(a)

cℛ = 77

M = 0114

14

12

10

08

06

04

02

0002 04 06 08 10 12 1400

VI(10minus10)

휙f

휙


휙⋆

(b)



(a) 119869 = 2radic6 119888R119891R (b) 119868 = 1206012

1198912R

≃ 1161198882R

(12)


120598 = 1121198882R1206012

120578 = 1 minus 4119888R120601121198882

R1206012

sdot (13)





(16)


120601⋆ ≃ ⋆3119888R ≃ 83 sdot 10minus4 ≪ 1 (17)


119899119904 ≃ (⋆ minus 3) (⋆ minus 1)2⋆

≃ 1 minus 2119873⋆

minus 31198732⋆

≃ 0961 (18a)

120572119904 ≃ minus(⋆ minus 3) (4⋆ + 3)24⋆

≃ minus 21198732⋆

minus 1523⋆


119903 ≃ 121198732⋆




(19)







119868 = 1206012

241198882R




119891119870 = 1119891R = 119888R1206012119881119868 = 120582 (1206012 minus 1198722)24

(21)


119891R (⟨120601⟩) = 1 997904rArr119872 = 1radic119888R (22)



(a) 119869 ≃ radic6120601 (b) 119868 = 1205821198912

12060141198884R1206014

≃ 12058241198882R

with 119891120601 = 1 minus 119888R1206012(23)



120598 = 431198912120601

120578 = 4 (1 + 119891120601)31198912

120601

sdot(24)



(25)


⋆ ≃ 3119888R1206012⋆4 997904rArr

120601⋆ ≃ 2radic ⋆3119888R ≫ 120601119891(26)


120601⋆ le 1 997904rArr119888R ge 4⋆3 ≃ 71 (27)



(28)


77 ≲ 119888R ≲ 85 sdot 10428 sdot 10minus6 ≲ 120582 ≲ 35 (29)






minus 922⋆

≃ 0961(31a)


⋆

minus 2123⋆


119903 = 192(3 minus 4⋆)2 ≃ 121198732

⋆





with 120575120601 ≃ radic6119888R120575120601(32a)


119868 = 1205822

61198882R







where 119870120572120573 = 119870119911120572119911lowast120573




= 119890119870 (119870120572120573119863120572119882119863lowast

120573119882lowast minus 3 |119882|2)

with 119863120572119882 = 119882119911120572 + 119870119911120572119882 (33b)



1198732 (34)




A120583 = 119894 (119870120572120597120583119911120572 minus 119870120572120597120583119911lowast120572)6 (36)




minus 271198733ΩA120583A

120583 minus 119881) (37a)




minus⟨Ω⟩119873 ≃ 1 (38)




119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912














119896=2

120582119896119879119896119899 (39)



⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)


119882 = 120582119878(119879119899 minus 12119888119879) (42)


119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)









119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)






119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





00

02

04

06

08

10

12

1 2 3 4 5 6 7 8 9 100

VI(10minus10)

휙 (10minus4)

휙f

휙⋆

(a)

cℛ = 77

M = 0114

14

12

10

08

06

04

02

0002 04 06 08 10 12 1400

VI(10minus10)

휙f

휙


휙⋆

(b)



(a) 119869 = 2radic6 119888R119891R (b) 119868 = 1206012

1198912R

≃ 1161198882R

(12)


120598 = 1121198882R1206012

120578 = 1 minus 4119888R120601121198882

R1206012

sdot (13)





(16)


120601⋆ ≃ ⋆3119888R ≃ 83 sdot 10minus4 ≪ 1 (17)


119899119904 ≃ (⋆ minus 3) (⋆ minus 1)2⋆

≃ 1 minus 2119873⋆

minus 31198732⋆

≃ 0961 (18a)

120572119904 ≃ minus(⋆ minus 3) (4⋆ + 3)24⋆

≃ minus 21198732⋆

minus 1523⋆


119903 ≃ 121198732⋆




(19)







119868 = 1206012

241198882R




119891119870 = 1119891R = 119888R1206012119881119868 = 120582 (1206012 minus 1198722)24

(21)


119891R (⟨120601⟩) = 1 997904rArr119872 = 1radic119888R (22)



(a) 119869 ≃ radic6120601 (b) 119868 = 1205821198912

12060141198884R1206014

≃ 12058241198882R

with 119891120601 = 1 minus 119888R1206012(23)



120598 = 431198912120601

120578 = 4 (1 + 119891120601)31198912

120601

sdot(24)



(25)


⋆ ≃ 3119888R1206012⋆4 997904rArr

120601⋆ ≃ 2radic ⋆3119888R ≫ 120601119891(26)


120601⋆ le 1 997904rArr119888R ge 4⋆3 ≃ 71 (27)



(28)


77 ≲ 119888R ≲ 85 sdot 10428 sdot 10minus6 ≲ 120582 ≲ 35 (29)






minus 922⋆

≃ 0961(31a)


⋆

minus 2123⋆


119903 = 192(3 minus 4⋆)2 ≃ 121198732

⋆





with 120575120601 ≃ radic6119888R120575120601(32a)


119868 = 1205822

61198882R







where 119870120572120573 = 119870119911120572119911lowast120573




= 119890119870 (119870120572120573119863120572119882119863lowast

120573119882lowast minus 3 |119882|2)

with 119863120572119882 = 119882119911120572 + 119870119911120572119882 (33b)



1198732 (34)




A120583 = 119894 (119870120572120597120583119911120572 minus 119870120572120597120583119911lowast120572)6 (36)




minus 271198733ΩA120583A

120583 minus 119881) (37a)




minus⟨Ω⟩119873 ≃ 1 (38)




119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912














119896=2

120582119896119879119896119899 (39)



⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)


119882 = 120582119878(119879119899 minus 12119888119879) (42)


119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)









119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)






119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119868 = 1206012

241198882R




119891119870 = 1119891R = 119888R1206012119881119868 = 120582 (1206012 minus 1198722)24

(21)


119891R (⟨120601⟩) = 1 997904rArr119872 = 1radic119888R (22)



(a) 119869 ≃ radic6120601 (b) 119868 = 1205821198912

12060141198884R1206014

≃ 12058241198882R

with 119891120601 = 1 minus 119888R1206012(23)



120598 = 431198912120601

120578 = 4 (1 + 119891120601)31198912

120601

sdot(24)



(25)


⋆ ≃ 3119888R1206012⋆4 997904rArr

120601⋆ ≃ 2radic ⋆3119888R ≫ 120601119891(26)


120601⋆ le 1 997904rArr119888R ge 4⋆3 ≃ 71 (27)



(28)


77 ≲ 119888R ≲ 85 sdot 10428 sdot 10minus6 ≲ 120582 ≲ 35 (29)






minus 922⋆

≃ 0961(31a)


⋆

minus 2123⋆


119903 = 192(3 minus 4⋆)2 ≃ 121198732

⋆





with 120575120601 ≃ radic6119888R120575120601(32a)


119868 = 1205822

61198882R







where 119870120572120573 = 119870119911120572119911lowast120573




= 119890119870 (119870120572120573119863120572119882119863lowast

120573119882lowast minus 3 |119882|2)

with 119863120572119882 = 119882119911120572 + 119870119911120572119882 (33b)



1198732 (34)




A120583 = 119894 (119870120572120597120583119911120572 minus 119870120572120597120583119911lowast120572)6 (36)




minus 271198733ΩA120583A

120583 minus 119881) (37a)




minus⟨Ω⟩119873 ≃ 1 (38)




119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912














119896=2

120582119896119879119896119899 (39)



⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)


119882 = 120582119878(119879119899 minus 12119888119879) (42)


119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)









119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)






119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






minus 922⋆

≃ 0961(31a)


⋆

minus 2123⋆


119903 = 192(3 minus 4⋆)2 ≃ 121198732

⋆





with 120575120601 ≃ radic6119888R120575120601(32a)


119868 = 1205822

61198882R







where 119870120572120573 = 119870119911120572119911lowast120573




= 119890119870 (119870120572120573119863120572119882119863lowast

120573119882lowast minus 3 |119882|2)

with 119863120572119882 = 119882119911120572 + 119870119911120572119882 (33b)



1198732 (34)




A120583 = 119894 (119870120572120597120583119911120572 minus 119870120572120597120583119911lowast120572)6 (36)




minus 271198733ΩA120583A

120583 minus 119881) (37a)




minus⟨Ω⟩119873 ≃ 1 (38)




119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912














119896=2

120582119896119879119896119899 (39)



⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)


119882 = 120582119878(119879119899 minus 12119888119879) (42)


119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)









119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)






119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119894 ℎ119894(119883) ℎ10158401015840119894 (0) 2

119904 2119868119870 = 1198703119894 119870 = 11987021198941 119883 0 minus2 + 21198991198912














119896=2

120582119896119879119896119899 (39)



⟨119879⟩ ≃ 1(2119888119879)1119899 ⟨119878⟩ = 0 (41)


119882 = 120582119878(119879119899 minus 12119888119879) (42)


119881SUSY = 120582210038161003816100381610038161003816100381610038161003816119879119899 minus 12119888119879

100381610038161003816100381610038161003816100381610038162 + 12058221198992 10038161003816100381610038161003816119878119879119899minus1100381610038161003816100381610038162 (43)









119883 = minus|119878|21198993

for 119870 = 1198703119894

|119878|2 for 119870 = 1198702119894(45)






119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119878 = 119879 minus 119879lowast = 0or 119904 = 119904 = 120579 = 0 (46)


119879 = 120601radic2119890119894120579119878 = 119904 + 119894119904radic2 sdot (47)


119868 = (120579 = 119904 = 119904 = 0) = 119890119870119870119878119878lowast 100381610038161003816100381611988211987810038161003816100381610038162 (48)


119890119870 = (2119891 + ℎ119894 (0))minus1198993 for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 for 119870 = 1198702119894 (49)



=

diag(11989931198992

21206012 ℎ1015840

119894 (0)(2119891 + ℎ119894 (0))) for 119870 = 1198703119894

diag(11989921198992

21206012 ℎ1015840

119894 (0)) for 119870 = 1198702119894(50)




119868 = 1205822 (1 minus 2119891)21198882119879

sdot

(2119891 + ℎ119894 (0))1minus1198993ℎ1015840119894 (0) for 119870 = 1198703119894

119890ℎ119894(0)(2119891)1198992 ℎ1015840119894 (0) for 119870 = 1198702119894

(51)


2119899 = 119899 (1198993 minus 1) for 119870 = 11987031198941198991198992 for 119870 = 1198702119894

997904rArr1198993 = 3 for 119870 = 11987031198941198992 = 2 for 119870 = 1198702119894

(52)



ℎ119894 (0) = 0ℎ1015840119894 (0) = 1 (53)


(a) 119868 = 12058221198912120601411988841198791206012119899

(b) 119868 = 12

119868radic3 = 1205821198911206012radic31198882119879120601119899

(54)



(a) 120597119868120597120572

100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 = 0(b) 2

119911120572 gt 0 with 119911120572 = 120579 119904 119904

(55)




1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1 real scalar 120579 21205792


120601

1 complex 119904 2119904 2

119868 21198991198912120601 minus 2 3 sdot 2119899minus11198912

120601



119868 21198991198912120601 6 sdot 2119899minus31198912

120601


2120572120573 = 1205972119868120597120572120597120573

1003816100381610038161003816100381610038161003816100381610038161003816119904=119904=120579=0 with 119911120572 = 120579 119904 119904 (56)


119870120572120573120572lowast120573 = 12 ( 1206012 + 1205792) + 12 ( 1199042 + 1199042) (57a)


119889120601119889120601 = 119869 = radic119870119879119879lowast 120579 = 119869120579120601

(119904 ) = radic119870119878119878lowast (119904 119904) (57b)



= (31198992

21206012 21198992minus1

119888119879120601119899) for 119870 = 1198703119894

(1198992

1206012 1) for 119870 = 1198702119894

(57c)











120579 ≫ 2119868 because 119888119879 ≫ 1




when ℎ10158401015840119894 (0) gt 0 with |ℎ10158401015840



119904 gt 0 but2119904 lt 2


119904 ≫ 2119868






119904 2119868









K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




K = K38

K = K32

n = 2

04 06 08 10 1202

휙

1

10

m2 sH

2 I(10)

cT = 77

휙f

휙⋆

(a)

n = 2

K = K211

K = K23

04 06 08 10 1202

휙

6

8

10

16

18

20

22

24

m2 sH

2 I

휙f

휙⋆

cT = 113

(b)







(120598 120578)

=

( 2119899

31198912120601

21+1198992 (21198992 minus 119888119879120601119899)31198912120601

) for 119870 = 1198703119894

(2119899minus2

1198912120601

21198992 (21198992 minus 119888119879120601119899)1198912120601

) for 119870 = 1198702119894(58)


120598 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot

((1 + 2radic3)2119888119879 )1119899

for 119870 = 1198703119894

((1 + radic2)2119888119879 )1119899

for 119870 = 1198702119894(59a)


120578 (120601119891) = 1 997904rArr

120601119891 ≃ radic2 sdot( 56119888119879)1119899

for 119870 = 1198703119894

(radic32119888119879)1119899

for 119870 = 1198702119894(59b)


⋆ = 1205812 (21minus1198992119888119879 (120601119899⋆ minus 120601119899

119891) minus 119899 ln 120601⋆120601119891

)

with 120581 = 32 for 119870 = 1198703119894

1 for 119870 = 1198702119894(60)


120601⋆ ≃ 119899radic21198992⋆120581119888119879 (61a)






120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








120601⋆ ≃ 120601119888 + radic120581 ln(2⋆120581 ) ≃ 52 for 119870 = 119870311989446 for 119870 = 1198702119894 (62)



11986012119904 ≃

120582(3 minus 4⋆)296radic2120587119888119879⋆

for 119870 = 1198703119894

120582 (1 minus 2⋆)216radic3120587119888119879⋆

for 119870 = 1198702119894(63)


120582 ≃

6120587radic2119860 119904119888119879⋆

997904rArr 119888119879 ≃ 42969120582 for 119870 = 1198703119894

4120587radic3119860 119904119888119879⋆

997904rArr 119888119879 ≃ 52627120582 for 119870 = 1198702119894(64)



minus 31198732⋆

≃ 0961 (65a)

120572119904 ≃ 16⋆ (3 + 2⋆)(2⋆ minus 1)4 ≃ minus 21198732⋆

minus 71198733⋆

≃ minus000075 (65b)

119903 ≃ 32(1 minus 2⋆)2 ≃ 81198732

⋆

+ 81198733⋆

≃ 00028 (65c)



119899 119888119879 and 120582 (66)








00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00

02

04

06

08

10

12

14VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 77

n = 6 cT = 310

(a)

00

02

04

06

08

10

VI(10minus10)

00 04 06 08 10 1202

휙

휙⋆

휙f

n = 2 cT = 113

n = 6 cT = 453

(b)




(17 24 68) sdot 10minus3 ≲ 120582 ≲ 354for 119870 = 1198703119894

(67a)

113 159 453 ≲ 119888119879 ≲ 193 sdot 105(2 29 82) sdot 10minus3 ≲ 120582 ≲ 354

for 119870 = 1198702119894(67b)





⟨120601⟩ = 2(119899minus2)2119899119888minus1119899119879 (68)





120575120601 = ⟨119868120601120601⟩12 ≃ 120582radic2120581119888119879 ≃ 2radic6119860 119904120587⋆

≃ 125 sdot 10minus5(70)




1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198692 1206012


sdot 1205751206012(71a)


119868 ≃ 12058221206012

41205811198881198792(1 minus 119899 + 1radic120581119899 120575120601 + (1 + 119899) 11 + 7119899121205811198992

1205751206012












References











































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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ResearchArticle Starobinsky Inflation: From Non-SUSY to SUGRA … · 2019. 7. 30. · 2....

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