On SUSY Restoration in Single-Superfield Inflationary Models of Supergravity
AAPCTP Pre2016-011IPMU 16-0077
On SUSY Restoration in Single-SuperfieldInflationary Models of Supergravity
Sergei V. Ketov a, b, c and Takahiro Terada d a Department of Physics, Tokyo Metropolitan UniversityMinami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan b Kavli Institute for the Physics and Mathematics of the Universe (IPMU)The University of Tokyo, Chiba 277-8568, Japan c Institute of Physics and Technology, Tomsk Polytechnic University30 Lenin Ave., Tomsk 634050, Russian Federation d Asia Pacific Center for Theoretical Physics, Pohang 37673, South Korea [email protected], [email protected]
Abstract
We study the conditions of restoring supersymmetry (SUSY) after inflation in thesupergravity-based cosmological models with a single chiral superfield and a quarticstabilization term in the K¨ahler potential. Some new, explicit, and viable inflationarymodels satisfying those conditions are found. The inflaton’s scalar superpartner isdynamically stabilized during and after inflation. We also demonstrate a possibilityof having small and adjustable SUSY breaking with a tiny cosmological constant. a r X i v : . [ h e p - t h ] A ug Introduction
Inflation is the excellent scenario to solve the fundamental difficulties of the hot big-bangcosmology such as the horizon, flatness, and monopole problems [1–6]. Moreover, it predictsgeneration of the curvature perturbations from quantum fluctuations of a scalar calledinflaton. Its adiabatic, scale-invariant, and Gaussian features have been precisely measuredby the cosmic microwave background (CMB) observations like WMAP [7,8] and Planck [9,10]. The observed small deviation from the scale invariance of CMB is measured bythe spectral index n s = 0 . ± . r < .
07 [11].Inflation should be described not only phenomenologically but also consistently withparticle physics expected beyond the Standard Model. One of the most motivated ap-proaches is supersymmetry (SUSY), or its gauged version called supergravity [12–15]. Insupergravity, it is known to be non-trivial to obtain a sufficiently flat inflaton scalar po-tential needed to trigger slow-roll inflation [16]. It is called the η problem, and its primarycause is the presence of the exponential factor e K in the scalar potential (see Appendix A).For a small-field inflation, one may attempt to tune the parameters of the model to makethe potential flat, but it becomes much more difficult for large-field inflationary models.A simple way to suppress the exponential steepness is to invoke an (approximate) shiftsymmetry of the K¨ahler potential [17, 18] , though it often leads to a potential unboundedfrom below [17, 18].Apart from tuning of the superpotential [24–26], there are two known generic solutionsto the unboundedness problem. The first one is to introduce a stabilizer superfield S whosevalue is required to vanish on-shell [17,18,27,28]. The superpotential is taken to be propor-tional to that superfield, so that the negative contribution − | W | is effectively removed.This approach can also be used by assuming the S to be a nilpotent superfield, thus effec-tively eliminating the need of its stabilization and invoking non-linear realizations of localsupersymmetry and its breaking [29–33], see e.g. Refs. [34–36] and references therein formore recent contributions. The other approach is to introduce a (shift-symmetric) quar-tic term of the inflaton in the K¨ahler potential [23, 37, 41, 42], instead. The quartic term The shift symmetry may be viewed as a non-linear realization of U(1) symmetry. The U(1)-symmetricformulation of inflation in supergravity was studied in Refs. [19–22] with a stabilizer field, and in Ref. [23]without a stabilizer field. A model without shift symmetry was proposed in Ref. [37]. It is related to the single-superfield α -attractor proposed in Refs. [25, 26, 38]. See also Refs. [39, 40] for more. Some tuned and complicated K¨ahler potentials with a similar stabilization mechanism were usedin Ref. [41]. Their (shift-symmetric) simplifications were proposed in Refs. [37, 42] where the quartic sinflaton ). It also has another important (dual) role by lifting up the inflatonpotential to make it positive, and fixing the value of the sinflaton during inflation, in or-der to make inflationary dynamics to be the single-field one (not just the single- super fieldone). The latter approach reduces the matter degrees of freedom (needed for inflation andsupersymmetry) to half of the former approach with a stabilizer (super)field. In this sense,it is a minimal approach to (large-field) inflation in supergravity. Therefore, it is worthstudying the properties of those relatively new supergravity-based inflationary models inmore detail.Generic models with a single chiral superfield (in other words, without a stabilizer)were introduced in Ref. [37], where it was found that SUSY is generically not restored afterinflation. Hence, without a hierarchically small parameter, SUSY tends to be broken invacuum at a scale comparable to the inflation scale, i.e. the gravitino mass is approximatelyof the same order as the inflaton mass. The conditions to restore SUSY after inflationwere not addressed in Ref. [37]. Besides, in Ref. [42], we introduced the special logarithmicK¨ahler potential that allows an approximate embedding of arbitrary positive semi-definitescalar potentials. In that class of models, it is also possible to fine-tune the cosmologicalconstant and SUSY breaking after inflation to zero. So, it can be a good starting point ofthe model building to obtain a small positive cosmological constant and SUSY breakingthat would be parametrically smaller than the inflation scale. It is also worth investigatingwhether this feature is maintained after taking into account corrections of the order 1 /ζ ,with ζ being the strength of the quartic stabilisation term. Whether SUSY is restoredafter inflation (in the absence of the hidden SUSY breaking sector) is quite important,being related to the grand unification of the gauge coupling constants and to the gaugehierarchy problem of the Standard Model of particle physics. Should SUSY be broken ata scale higher than the intermediate scale, the electroweak vacuum may be unstable [48].Moreover, gravitino production from inflaton decay in the early Universe is enhanced wheninflaton breaks SUSY in vacuum [49–53], which leads to a cosmological disaster.In this paper, we study SUSY breaking (and its preservation) in vacuum, by usingthe supergravity setup utilising a quartic stabilisation term in the K¨ahler potential. Morespecifically, we study the conditions to restore SUSY after inflation. We find that SUSYrestoration is intact in the presence of finite corrections in 1 /ζ . In Section 2, we discuss stabilization mechanism was pointed out to be applicable to generic inflaton potentials with appropriatesuperpotentials. As regards further developments of this approach, see Refs. [43–46]. The importance ofK¨ahler curvature (represented by a quartic term in our K¨ahler potential) was emphasized in Ref. [47]. α -attractor potential [54–56], which leads to slightlydifferent predictions compared to the original α -attractor’s. In Section 4 we outline how toget an adjustable cosmological constant with SUSY breaking. Section 5 is our Conclusion.Our setup is described in Appendix A. Various models which restore SUSY after inflationare presented in Appendix B. We adopt the natural units, c = (cid:126) = M Pl / √ π = 1. We consider the “generic” shift-symmetric K¨ahler potential of Ref. [37]. We take theconvention that the real part φ of the leading component of a chiral superfield, Φ | =( φ + iχ ) / √
2, defines the shift-symmetric direction, i.e. K = ic (cid:0) Φ − ¯Φ (cid:1) − (cid:0) Φ − ¯Φ (cid:1) − ζ (cid:0) Φ − ¯Φ (cid:1) , (1)where c is a real constant and ζ is a real positive constant. Inflaton is identified with φ that enters a superpotential. The sinflaton χ is stabilized by the quartic term duringinflation [37, 40]. We choose the origin of sinflaton χ in such a way that it coincides withits stabilized value, i.e. (cid:104) χ (cid:105) → ζ → ∞ . That is why the cubic term in Eq. (1), whichwould induce unsuppressed (cid:104) χ (cid:105) , is assumed to be negligible. With this choice, the higherorder terms in i (Φ − ¯Φ) have negligible effects because of the suppression (cid:104) χ (cid:105) (cid:39)
0. It isworth mentioning that these choices are different from those in Ref. [37], though beingequivalent via a field redefinition. The quartic term is needed only for the stabilization ofthe sinflaton and, hence, the related uplifting of the inflaton potential via the linear term.During inflation, the sinflaton χ can be integrated out, while the impact of the quarticterm results in the appearance of the terms inversely proportional to ζ in the effectivesingle-field potential for inflaton φ , due to the sinflaton value (cid:104) χ (cid:105) ∼ ζ − . This is becausethe quartic term itself is multiplied by the power of the sinflaton value, and vanishes inthe limit ζ → ∞ . Though we take into account the corrections in 1 /ζ in our calculationsof the inflationary observables, essential features of our models can be already seen in theleading (zeroth) order in 1 /ζ . For instance, the K¨ahler potential of Ref. [42] — see Eq. (6) — can be approximately expressed byEq. (1) as a Taylor series. Then the linear coefficient is c = −√
3, the quartic coefficients are relatedas ζ = ξ − /
3, and a small cubic term, i (Φ − ¯Φ) / (3 √
3) is needed, whose presence merely results insubdominant effects. V = | W Φ | + ic (cid:0) W ¯ W ¯Φ − ¯ W W Φ (cid:1) + ( c − | W | . (2)Note that the coefficient of the last term is only positive for c > √
3. In the presence ofcorrections of the order 1 /ζ , the critical value of the lower bound increases, as is shownbelow. When assuming for simplicity that all parameters in the superpotential are real,the second term in the V given above vanishes, so we get a simplified formula, V = | W Φ | + ( c − | W | . (3)The requirement of SUSY preservation in vacuum with the vanishing cosmologicalconstant is not a severe condition, since it merely requires W = W Φ = 0 (in vacuum). (4)This leads to the vanishing F -term, D Φ W = W Φ + K Φ W = 0, and V = 0. (We discusssmall SUSY breaking and a cosmological constant in Section 4.) As follows from Eqs. (1)and (4), it is self-consistent to assume χ = 0 exactly in vacuum for arbitrary c and ζ .Namely, V = V φ = V χ = 0 (in vacuum with χ = 0) , (5)with the SUSY mass squared for φ and χ given by | W ΦΦ | ( i.e. no tachyon). Both φ and χ are canonically normalized at χ = 0, including the vacuum state. Those facts simplify ouranalysis. Moreover, they prevent large field excursion of χ at the final phase of inflation, asobserved in Ref. [43], because the relevant expectation values (both in vacuum and duringinflation) are close to each other.In summary, we established the following statement: once one constructs an inflationarymodel in the ideal stabilization limit ζ → ∞ in such a way that the vacuum is at φ = φ (here φ is a constant) and χ = 0, the vacuum position is unchanged when ζ becomes finite.In fact, this is valid for general shift-symmetric K¨ahler potentials, K (Φ , ¯Φ) = K ( i (Φ − ¯Φ)).It is straightforward to show that Eq. (5) holds under the assumption (4). In the generalcase, the squared mass of φ and χ is given by e K (0) | W ΦΦ | /K (cid:48)(cid:48) (0), where the primes denotethe differentiation with respect to the given argument. Note that the value of χ = 0 ina SUSY vacuum is not guaranteed by the quartic stabilization term because the SUSY4reaking mass stabilizing χ vanishes in a SUSY vacuum.The important corollary of our statement exists for a class of the inflationary modelswith the special K¨ahler potential K = − (cid:16) i (Φ − ¯Φ) / √ ξ (Φ − ¯Φ) / (cid:17) . (6)It makes possible to incorporate an arbitrary positive semi-definite inflaton potential viathe formula V = | W Φ | + √ i (cid:0) ¯ W W Φ + W ¯ W ¯Φ (cid:1) = | W Φ | , (7)where the second equality holds when one assumes all the coefficients in the superpotentialto be real. It was shown in Ref. [42] that one can always fine-tune both a cosmologicalconstant and SUSY breaking in vacuum to zero in the infinite ξ limit. The above statementgeneralizes it to the case of arbitrary ξ . Let us study in more detail the conditions (4) for SUSY preservation and the vanishingcosmological constant. By shifting Φ with a real constant, we can set φ = 0 withoutchanging the form of the K¨ahler potential. Without loss of generality, we set it first andstudy the structure of the superpotential satisfying Eq. (4). Then the vacuum is at theorigin Φ = 0. The superpotential can be written down as a Taylor expansion without thezeroth and first order terms, W = (cid:88) n ≥ c n Φ n . (8)This satisfies Eq. (4), while any holomorphic function satisfying Eq. (4) can be expressedas above. Given an even function W ( − Φ) = W (Φ) satisfying W Φ = 0 at the origin, it isalways possible to subtract a constant to obey the remaining condition W = 0.We focus on a particular example in this Section, though there exist many modelswith SUSY restoration after inflation. Their possible classification within our approach isoutlined in Appendix B.The CMB data favours a flat potential similar to the potential of the R model [1]5r Higgs inflation model [57]. It is, therefore, reasonable to consider a potential thatasymptotes to a constant at the infinite inflaton field, φ → ∞ . An asymptotically constantpotential is generated by an asymptotically constant superpotential. Such a superpotentialcan be expanded as W = (cid:88) n ≥ a n e − b n Φ , (9)where b = 0 and b n > b m for n > m . The condition (4) then reads (cid:88) n ≥ a n =0 and (cid:88) n ≥ a n b n =0 . (10)It does not have a non-trivial solution when there are only two terms in the expansion.Hence, let us consider the simplest non-trivial case with W = a + a e − b Φ + a e − b Φ . (11)This superpotential is the same as that in the so-called racetrack model [58], though ourK¨ahler potential is different. By solving the constraint (10), we can eliminate a and a as a = − a b b − b and a = a b b − b . (12)Then the potential (3) in the leading order takes the form V = a ( b − b ) (cid:20) b b (cid:16) e − ˜ b φ − e − ˜ b φ (cid:17) + ( c − (cid:16) b − b + b e − ˜ b φ − b e − ˜ b φ (cid:17) (cid:21) , (13)where ˜ b i = b i / √ i = 1 , W Φ , isnot suitable for a plateau inflation. In fact, when c is closer to the critical value √ b and b are large (the rough criterium is b b (cid:38) ( c − b − b ) ), a bump in the potentialappears. The red solid and purple dot-dashed lines in Fig. 1a are such examples.To study the b i -dependence in our models, we consider the large c limit in which the firstterm in Eq. (13), originating from the derivative of the superpotential, is negligible. When b becomes much larger than b , the terms with the higher-oder exponentials also becomenegligible. It means that the potential is Starobinsky-like. More precisely, it coincides withthe E-model-type realization of the α -attractor potential [59]. Actually, a large b with a6 ϕ ( ϕ ) (a) The examples of the potential (13). - ( ) (b) The limit b → b in Eq. (15). Figure 1: The examples of the potential in the model (11). (1a) the parameters are chosento be ( a , b , b , c ) = (1 , , , . , (1 , , , , (1 , , , . , (1 , , , , and (1 , , ,
2) for thered solid, yellow long-dashed, green dashed, blue short-dashed, and purple dot-dashedlines, respectively. (1b) the potential (15) with b = (cid:112) /
3. The Starobinsky potential isshown by the dashed line for comparison.fixed c implies that the neglected term becomes important. When b is closer to b , thesituation is more complicated. The most nontrivial case is the limit b → b . Then, thesuperpotential and the potential become W = a (cid:2) − (1 + b Φ) e − b Φ (cid:3) , (14) V = a ( c − (cid:2) − (1 + bφ ) e − bφ (cid:3) , (15)where b ≡ b / √
2. This potential is different from the Starobinsky one, as is shown inFig. 1b. In this limit, the neglected derivative-originated term is √ a b φe − bφ , so that theabove approximation is valid for b (cid:46) c . As a rough estimate of the inflationary observablesfor these cases including the Starobinsky-like limit, we have n s (cid:39) − N , and r (cid:39) b N . (16)In our numerical calculations we took into account a shift of the sinflaton χ from theorigin up to the first order. For this purpose, we expanded the potential in terms of χ upto the second order and minimized it. After integrating out χ , we obtained the effectivesingle-field potential of the inflaton φ , including the corrections having the ζ -dependent7erms. So, we neglected the time derivatives of χ . To do those calculations efficiently, weapproximated the potential by an interpolation method of Mathematica, and solved theequation of motion to obtain the inflaton trajectory as well as the inflationary observables.For a reduction of the multi-parameter space, we set the stabilization parameter to be ζ = 1. The overall scale of the potential merely affects the time-scale of simulations, so weset a = 1. n s t en s o r - t o - sc a l a rr a t i o r Figure 2: A rapid change of predictions near the critical value of the parameter c . Thevalue of c is varied from 1.98359 to 1.98500 with a 0.00001 step each. The points circulatecounter-clockwise with increasing c . The other parameters are set to ζ = a = 1, b =2 / √ b = 2 b , and N = 60.With those corrections, the critical value of c , needed to obtain a potential boundedfrom below, increases a bit. The critical value of c is also dependent on the other param-eters, in particular b and b . One may also study the critical values of b i at a fixed c .Near their critical values, the effect of the derivative-induced term is not negligible. Whendecreasing c from a large value to the critical value, the potential around the origin is grad-ually deformed, a short flat region appears, and finally it becomes a bump to trap inflatoninto a local minimum. Some of these features can be seen in Fig. 1a. Around that smallparameter range of c , the e -folding number earned in the short flat region changes rapidly,and it is reflected in a rapid change of the corresponding predictions for ( n s , r ). This isdemonstrated in Fig. 2 where one looses predictability against flexibility of predictions in8his very special case. n s t en s o r - t o - sc a l a rr a t i o r Figure 3: The b i -dependence in the model with two exponentials (11). The set of right(left) lines with darker (lighter) colors corresponds to N = 60 (50). In each set, the redsolid (the very left), yellow long-dashed, blue dashed, and purple short-dashed (the veryright) lines show the cases of c = 2 , ,
4, and 5, respectively. The lines with c = 3 , b = 2 b and ζ = a = 1, and vary b from 0.01above in the figure to 3.0 (1.2 for c = 2) below in the figure. For comparison, the result ofthe b → b limit with large ζ and c (see Eq. (15)) is shown as the gray dot-dashed line.The green contours are 1 σ and 2 σ Planck constraints combined with other observations(Planck TT+lowP+BKP+lensing+BAO+JLA+H0) [9].For larger values of c , observational predictions vary less rapidly, as usual. The pre-dictions of our models are shown in Fig. 3 where we have fixed the relation b = 2 b fordefiniteness, and have taken c = 2 , ,
4, and 5. The value of b is varied from 0.01 to 3.0at c = 3 ,
4, and 5. When c = 2, the maximum value is taken to be 1.2 that is almost thecritical value for c = 2. For this reason, only the red solid line for c = 2 is deflected to thelower n s region. It is worthwhile to comment on the related issue, as to how to obtain small
SUSY breakingand a very small cosmological constant, by some minimalistic extension of our approach.9or simplicity, let us consider a nilpotent F -term SUSY breaking superfield S [30], whichis essentially a Polonyi superfield [60] subject to the nilpotent condition S = 0. We takethe minimal K¨ahler potential of the nilpotent superfield, and combine the two sectors asfollows: K (Φ , S, ¯Φ , ¯ S ) = K (inf) ( i (Φ − ¯Φ)) + ¯ SS, (17) W (Φ , S ) = W (inf) (Φ) + W + µ S, (18)where W and µ are constants, and (inf) denotes the quantities in the inflation sectordiscussed in this paper. In particular, W (inf) satisfies Eq. (4), and we set K (inf) (0) = 0 asour convention by using a K¨ahler transformation. In the presence of the superfield S , theVEV of the inflaton gets shifted as (cid:104) Φ (cid:105) = − W K (inf)Φ W (inf)ΦΦ , (19)where we have taken the convention (cid:104) Φ (cid:105) = 0 before introducing W and S , and the termsof the higher order in | W | or | µ | have been neglected. Accordingly, the vacuum energybecomes V = | µ | − | W | , (20)in the leading order of | W | or | µ | . Therefore, the SUSY breaking scale | D S W | = | µ | canbe chosen freely, while the cosmological constant can be chosen arbitrarily small by fine-tuning between | µ | and 3 | W | . For many purposes, we may simply set | µ | ≈ √ | W | , sothat the gravitino mass is m / = | W | = | µ | / √ S in our approach is limited to uplifting the vacuum en-ergy. The inflaton potential is solely constructed from the inflaton superfield Φ. If theSUSY breaking scale is much lower than the inflation scale, the effects of S or W on theinflationary dynamics are negligible.Our results are consistent with the argument in Ref. [43] and the “no-go” statement inRef. [61], which claim that any SUSY preserving Minkowski vacuum without flat directions cannot be uplifted to a de Sitter vacuum by a small continuous deformation of the model.However, there exists a loophole in those arguments, also noticed in Ref. [61], namely, via One may impose further constraints, ¯ SS (Φ − ¯Φ) = 0 and ¯ SS D α Φ = 0, to eliminate sinflaton andinflatino, respectively [34].
In this paper we investigated the SUSY breaking properties of the supergravity-basedinflationary models without a stabilizer superfield by using a shift-symmetric quartic sta-bilization term in the K¨ahler potential. The shift symmetry is a global symmetry imposedon the K¨ahler potential at the tree level, which is likely to be broken by quantum (gravity)corrections. Though quantum corrections from the superpotential are suppressed due tothe relatively small scale controlling the amplitude of CMB perturbations, one may expectnon-negligible quantum corrections from the inflaton-matter couplings, depending upon thereheating temperature. We assumed those terms to be suppressed in our phenomenologicalapproach. Possible origins of our superpotentials and K¨ahler potential, and, in particular,measuring quality of the shift symmetry, are beyond the scope of our investigation.In Section 2, we found that the vacuum expectation value (VEV) of the inflaton multi-plet is not sensitive to the parameters of the K¨ahler potential as far as the condition (4) issatisfied. This demonstrates robustness of the SUSY preservation property in our models.In addition, we showed that the large-field excursion of sinflaton at the end of inflation,observed in Ref. [43], can be suppressed by tuning the sinflaton VEV to be equal to itsstabilized value during inflation ( i.e. zero in our conventions).A relatively simple, racetrack-like model was studied in Section 3. It shares the essentialqualitative features with some other models in Appendix B. The observational aspects ofthe model extend those of the α -attractor, including the R model and the Higgs inflation— see e.g. , Eqs. (13) and (15), and Figs. 1b and 3 for details.In summary, our single-superfield model building with the quartic stabilization is apowerful tool to construct inflationary models in supergravity, which are consistent withobservations. Its inflationary sector has the minimal number of physical degrees of free-dom, i.e. has the inflaton supermultiplet only. Since sinflaton is stabilized, isocurvatureperturbations and non-Gaussianity are negligible in our models, see e.g. , Ref. [62] for more.The SUSY breaking by the inflaton supermultiplet driving inflation is restored after infla-tion when the condition (4) is satisfied. On top of that, we found that it is possible toobtain a tunable SUSY breaking and a tiny cosmological constant in vacuum. It fills agap in our earlier work on the single-superfield approach to inflation in supergravity, asregards its SUSY breaking structure after inflation.11 cknowledgements The authors are grateful to H. Murayama for raising the issue of the 1 /ζ -corrections tothe SUSY breaking properties, which has led us to our investigation in Section 2. SVKis supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS)under No. 26400252, a grant of the President of Tokyo Metropolitan University, the WorldPremier International Research Center Initiative (WPI Initiative), MEXT, in Japan, andthe Competitiveness Enhancement Program of the Tomsk Polytechnic University in Russia.A part of TT’s work was supported by the Grant-in-Aid for JSPS Fellows and the Grant-in-Aid for Scientific Research on Scientific Research No. 26 · A Basic facts about inflationary model building
The most important inflationary observables are (i) the amplitude of the curvature per-turbations A s , (ii) the scalar spectral index n s , and (iii) the tensor-to-scalar ratio r . Theycan be expressed in terms of the slow-roll parameters as follows: A s = V π (cid:15) , n s =1 − (cid:15) + 2 η , r =16 (cid:15) , (21)at the horizon exit, in terms of the inflaton scalar potential V . The slow-roll parametersare defined as (cid:15) = 12 (cid:18) V (cid:48) V (cid:19) , η = V (cid:48)(cid:48) V , (22)where the primes denotes the differentiation with respect to the canonical inflaton field φ .The e -foldings number N ≡ log( a end /a ∗ ) can be expressed in terms of the inflaton field as N = (cid:90) φ ∗ φ end √ (cid:15) d φ , (23)where a is the scale factor of the FLRW metric, the subscript “end” denotes the end ofinflation (at (cid:15) = 1), the subscript ( ∗ ) stands for the horizon exit of the observed scale, andwe set φ end < φ ∗ without loss of generality.In four-dimensional N = 1 supergravity, an inflationary model is specified by a K¨ahlerpotential K = K ( φ i , ¯ φ ¯ j ), a superpotential W = W ( φ i ), and a gauge kinetic function h AB = h AB ( φ i ) of chiral superfields φ i . The kinetic and potential terms of their leading12calar field components φ i and ¯ φ ¯ j (where a bar denotes complex conjugation, and we usethe same notation for chiral superfields and their leading field components) in Einsteinframe are given by √− g − L kinetic = − K i ¯ j ∂ µ φ i ∂ µ ¯ φ ¯ j , (24) V = e K (cid:16) K i ¯ j D i W ¯ D ¯ j ¯ W − | W | (cid:17) + 12 h R AB D A D B , (25)where the subscripts i , ¯ j , etc. denote the differentiation with respect to the correspondingfields φ i , ¯ φ ¯ j , etc., and D i W ≡ W i + K i W . The D -term (proportional to D A D B ) is irrelevantfor our investigation in this paper.The minimal K¨ahler potential K = ¯ φφ leads to a scalar potential having the overallexponentially steep factor e ¯ φφ in large-field inflationary models. A detailed review of the η -problem in supergravity can be found, e.g. , in Ref. [63]. B Towards a classification of inflationary models insupergravity with SUSY restoration
Type 1a: the single-term-model
The simplest option is merely a single term in Eq. (8), W = c Φ . (26)Then the leading-order scalar potential is a sum of quadratic and quartic terms, V = | c | φ (cid:2) (cid:0) c − (cid:1) φ (cid:3) . (27)However, both quadratic and quartic potentials are too steep to be consistent with obser-vations. Tensions with observations become milder when a small negative quartic term isadded to a quadratic potential [64]. Though the potential then becomes unbounded frombelow in the large-field limit, it may not be a problem if the tunnelling time is longer thanthe age of the Universe. 13 ype 1b: the two-terms-model After adding a cubic term to the previous model, we get W = c Φ + c Φ . (28)The potential in the leading order is polynomial, V = 2 c φ + 3 √ c c φ + 94 c φ + 18 ( c − φ (cid:16) c + 2 √ c c φ + c φ (cid:17) , (29)where we have taken c and c to be real for simplicity. Some examples of such potentialare shown in Fig. 4. In the limit of c = √
3, the potential is a quartic function and has thedouble-well form. The positions of the minima (with the vanishing cosmological constant)are φ = 0 and φ = − √ c / (3 c ). SUSY is preserved in the former minimum and isbroken in the latter. A hilltop inflation of the quartic order is possible between the minima,but it gives n s (cid:39) . ∼ .
95 which is smaller than the observational bound. When weincrease c , the nontrivial minimum is uplifted, but the local minimum still exists. - ( ) Figure 4: The examples of the potential (29). The parameters are chosen as ( c, c ) =(1 . , − . , (2 . , − . , and (1 . , − .
13) for the red solid, blue dashed, and green dot-dashed lines, respectively. In all cases we set c = 1.In the two examples above, we merely considered the simplest options. Next, we requirethat the potential asymptotes to a constant in the large-field limit (with some values of theparameters). The leading-order potential (3) has two parts: the derivative part and thenon-derivative part. Accordingly, there exist two possibilities where one of the two parts14ecomes dominant.If the value of c is close to the critical value √
3, the potential is dominated by the deriva-tive term. To obtain an asymptotically flat potential, the superpotential has to approacha linear function asymptotically. Besides, the value and the slope of the superpotential atthe origin should vanish. Those superpotentials are shown in Fig. 5a. - - ( ) (a) The asymptotically linear superpotentials. ϕ ( ϕ ) (b) The potentials of the log cosh -type. Figure 5: The examples of asymptotically linear superpotentials and the correspondingpotentials. (5a) The yellow solid (blue dashed) line corresponds to W = log(cosh √ W = √ − c, a ) = (1 . , , (1 . , , and (1 . , m = 1 /a .A good example is given by Type 2a: the log cosh model W = m (cid:16) log (cid:16) cosh √ a Φ (cid:17)(cid:17) , (30)where a is a real parameter, m sets the scale of the inflationary potential. The leading-orderscalar potential (3) takes the form V /m = a tanh aφ + (cid:0) c − (cid:1) (log (cosh aφ )) . (31)This potential is shown in Fig. 5b. When the second term is negligible, the potential isthat of the T-model [59, 65]. When the second term dominates, the potential becomes a15uadratic function. It is also possible to interpolate between the plateau potential and thequadratic one, as the limiting cases. Taking the small a limit, we get the potential that isclose to a quartic one near the origin. Those potentials have a rich structure dependingon the values of the parameters.Yet another case is Type 2b: the square-root-model with W = m (cid:16) √ a Φ − (cid:17) , (32)which are similar to the previous type-2a (log cosh) model, see Fig. 5a. The potential (3)in this case reads V /m = 2 a φ a φ + ( c − (cid:16) a φ − (cid:112) a φ + 2 (cid:17) . (33)The last two types have a singularity and/or a branch cut off the inflationary trajectory.There exist infinitely many superpotentials with similar predictions in the field regionrelevant to observations.Next, we consider the case when c is sufficiently large so that the non-derivative termdominates and asymptotes to a constant in the large-field region. For example, it is wellrepresented by the following model: Type 3a: the tanh model with W = m tanh √ a Φ , (34)where m sets the scale of inflation, and a is a real parameter. The potential (3) becomes V /m = tanh aφ (cid:20) a cosh aφ + (cid:0) c − (cid:1) tanh aφ (cid:21) . (35)It yields a flat potential for inflation, V ∼ tanh aφ , but in the c → √ a limit leads to a quartic potential. This is becausethe first term in the expansion of the SUSY-restoring superpotential (8) is quadratic, whilethe main part of the scalar potential is proportional to its square.16ne may also expand an asymptotically constant superpotential as a constant plus aseries of decaying functions. Type 3b: the models with exponentials , W = a + a e − b Φ + a e − b . (36)This case was studied in Section 3 in detail.Finally, we consider the following type of models. Type 3c: the models with a rational function , W = a Φ + a Φ + · · · + a n Φ n b Φ + b Φ + · · · + b m Φ m . (37)This can be viewed as the Pad´e approximation of order [ n / m ] of some holomorphic function.The numerator begins with the quadratic term to satisfy (4). Here, we take n = m toobtain an asymptotically constant potential, and set n = 2 as the simplest choice. Then,the leading-order potential is V /a = 4(2 √ b φ ) φ (2 + √ b φ + b φ ) + ( c − φ (2 + √ b + b φ ) . (38)To further simplify the model, consider the case of b = 0. The asymptotic form ofthe potential is a constant plus a fall-off like φ − . This inverse-hilltop potential yields n s (cid:39) − N and r (cid:39) (cid:113) b N / .Needless to say, our classification here is incomplete, being the first step in that direc-tion. References [1] A. A. Starobinsky, “A New Type of Isotropic Cosmological Models WithoutSingularity,”
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