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PHYSICAL REVIEW D 100, 066005 (2019)

Deriving the in compactified M-theory with a de Sitter vacuum

Gordon Kane1 and Martin Wolfgang Winkler2,* 1Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 2The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Alba Nova, 10691 Stockholm, Sweden

(Received 13 February 2019; published 4 September 2019)

Compactifying M-theory on a manifold of G2 holonomy gives a UV complete 4D theory. It is supersymmetric, with soft breaking via condensation that simultaneously stabilizes all moduli and generates a hierarchy between the Planck and the Fermi scale. It generically has gauge , chiral , and several other important features of our world. Here we show that the theory also contains a successful inflaton, which is a linear combination of moduli closely aligned with the overall volume modulus of the compactified G2 manifold. The scheme does not rely on ad hoc assumptions, but derives from an effective quantum theory of . arises near an inflection point in the potential which can be deformed into a local minimum. This implies that a de Sitter vacuum can occur in the moduli potential even without uplifting. Generically present charged hidden sector matter generates a de Sitter vacuum as well.

DOI: 10.1103/PhysRevD.100.066005

I. INTRODUCTION interacting via Yang-Mills forces. It generically has radiative electroweak symmetry breaking and cor- Countless papers have suggested or fields that can lead to an inflating universe. Most have used ad hoc rectly anticipates the ratio of the Higgs mass to the Z CP mechanisms without identifying a physical origin—what is mass [2]. It also solves the strong problem [3]. the inflaton? Such bottom-up descriptions, furthermore, In this theory, a particular linear combination of moduli, rely on strong hidden assumptions on the theory of that which describes the volume of the compactified region, quantum gravity. More thorough proposals have identified generates inflation. By means of Kähler geometry, we will the inflaton as part of a construction in which prove that a tachyonic instability develops if the inflaton is the ultraviolet (UV) physics can be addressed. In this case, not “volume-modulus-like.” In contrast to related proposals the inflaton arises in a theory that itself satisfies major in type II string theory [4–7], volume modulus inflation on consistency conditions and tests. The theory should also G2 does not rely on uplifting or higher order corrections to connect with the Standard Models of physics and the Kähler potential. This follows from the smaller curva- cosmology. Ideally, its properties would uniquely deter- ture on the associated Kähler submanifold. mine the nature of the inflaton. Besides being intuitively a likely inflaton, the volume In this work, we focus on M-theory compactified modulus also resolves a notorious problem of string infla- spontaneously on a manifold of G2 holonomy. The result- tion: the energy density injected by inflation can destabilize ing quantum theory is UV complete and describes gravity moduli fields and decompactify the extra dimensions. plus the plus Higgs physics. When its Prominent moduli stabilization schemes including KKLT hidden sector matter is included it has a de Sitter vacuum [8], the large volume scenario [9], and Kähler uplifting [1]. It stabilizes all the moduli and is supersymmetric with [10,11] share the property that the volume modulus partic- supersymmetry softly broken via condensation [1]. ipates in supersymmetry breaking. Its stability is threatened It produces a hierarchy of scales and has and once the Hubble scale of inflation H exceeds m3=2 [12–14]. In contrast, the volume modulus of the compactified G2 *[email protected] manifold drives inflation in the models that we will discuss. Thereby, the inflationary energy density stabilizes the system Published by the American Physical Society under the terms of and H ≫ m3=2 is realized. The supersymmetry breaking the Creative Commons Attribution 4.0 International license. — Further distribution of this work must maintain attribution to fields light moduli and of a strong hidden sector the author(s) and the published article’s title, journal citation, gauge theory—receive stabilizing Hubble mass terms on the and DOI. Funded by SCOAP3. inflationary trajectory.

2470-0010=2019=100(6)=066005(17) 066005-1 Published by the American Physical Society GORDON KANE and MARTIN WOLFGANG WINKLER PHYS. REV. D 100, 066005 (2019)

Inflation takes place close to an inflection point in the sector induces a nonperturbative superpotential due to potential and lasts for 100–200 e-foldings. If we impose the gaugino condensation [17,18], observational constraints on the spectral index, we can −6 − 1 2πf r ∼ 10 N−N predict the tensor-to-scalar ratio . It is unlikely that W ¼ A det ðQQ¯ Þ f exp − ; ð3Þ other observables will directly probe the nature of the N − Nf inflaton. However, inflation emerges as a piece of a theory which also implies low energy supersymmetry with a where Nf denotes the number of flavors. The A mass m3=2 ≲ 100 TeV and a specific pattern of coefficient is calculable, but depends on the renormal- masses. are at the TeV scale and ization scheme as well as threshold corrections to the gauge observable at LHC. Furthermore, a matter dominated cos- coupling. The gauge kinetic function f is a linear combi- mological history is predicted. In a sense, all aspects and nation of the moduli [19], tests of the theory are also tests of the nature of its inflaton, f ¼ c T ; ð Þ although technically they may not be closely related. i i 4 Less is known about G2 manifolds than about Calabi- c Yau manifolds. This is being at least partially remedied via with integer coefficients i. We now turn to the construction a 4-year, $9 million study sponsored by the Simons of de Sitter vacua with broken supersymmetry. Foundation started in 2017, focusing on G2 manifolds. Remarkably, the above successes were achieved without B. Constraints on de Sitter vacua detailed knowledge of the properties of the manifolds. In this section we introduce some tools of Kähler geometry which can be used to derive generic constraints on de Sitter vacua in supergravity [20]. The same frame- II. DE SITTER VACUA IN G2 COMPACTIFICATIONS work also applies to inflationary solutions (see, e.g., [5]) and will later be employed to identify the inflaton field. In A. The moduli sector order to fix our notation, we introduce the (F-term part) of We study M-theory compactifications on a flux-free the scalar potential in supergravity, G2-manifold. The size and the shape of the manifold is G i controlled by moduli Ti. In our convention, the imaginary V ¼ e ðG Gi − 3Þ; ð5Þ 1 parts of the Ti are fields. A consistent set of Kähler 2 potentials is of the form [15,16] with the function G ¼ K þ log jWj . The subscript i indicates differentiation with respect to the complex scalar 1=3 K ¼ −3 log ð4π VÞ; ð1Þ field ϕi. Indices can be raised and lowered by the Kähler ij¯ metric Kij¯ and its inverse K . Extrema of the potential V where denotes the volume of the manifold in units of the satisfy the stationary conditions Vi ¼ 0 which can be 11-dimensional Planck length. Since the volume must be a expressed as homogeneous function of the ReTi of degree 7=3, the G j following simple ansatz has been suggested [16]: e ðGi þ G ∇iGjÞþGiV ¼ 0; ð6Þ π Y X ∇ K ¼ − ðT¯ þ T Þai ; a ¼ 7; ð Þ where we introduced the Kähler covariant derivatives i. log 2 i i i 2 i i The mass matrix at stationary points derives from the Q second derivatives of the potential [21], a =3 which corresponds to V ¼ ðReTiÞ i . We will drop the i G k m n¯ factor π=2 in the following since it merely leads to an Vij¯ ¼ e ðGij¯ þ ∇iGk∇j¯G − Rijm¯ n¯ G G Þ Oð1Þ overall factor in the potential not relevant for this þðGij¯ − GiGj¯ÞV; ð7Þ discussion. A realistic vacuum structure with stabilized G k moduli is realized through hidden sector strong dynamics Vij ¼ e ð2∇iGj þ G ∇i∇jGkÞþð∇iGj − GiGjÞV; ð8Þ such as gaugino condensation. The resulting theory generi- cally has massless quarks and leptons, and Yang-Mills where Rijm¯ n¯ denotes the Riemann tensor of the Kähler forces [1], and it has generic electroweak symmetry break- manifold. (Meta)stable vacua are obtained if the mass matrix CP ing, and no strong problem [3]. is positive semidefinite. A weaker necessary condition We consider one or several hidden sector SUðNÞ gauge requires the submatrix Vij¯ to be positive semidefinite. All Q Q¯ theories. These may include massless quark states , complex scalars orthogonal to the sgoldstino may acquire a N N¯ transforming in the and representations. Each hidden large mass from the superpotential. In addition, the above mass matrix contains the standard soft terms relevant, e.g., 1 Ti in this work corresponds to izi defined in [1]. for the superfields of the visible sector.

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Stability constraints apply in particular to the sgoldstino previously mentioned string theory examples, condition direction which does not receive a supersymmetric mass. (11) can hence be satisfied even in the absence of Via appropriate field redefinitions, we can set all derivatives corrections to the Kähler potential. The modular inflation of G to zero, except for one which we choose to be Gn. The models we discuss in Sec. IV are of this type. We will show curvature scalar of the one-dimensional submanifold asso- that, by a small parameter deformation, the inflationary ciated with the sgoldstino is defined as plateau can be turned into a metastable de Sitter minimum. Let us also briefly allude to the controversy on the K K K R ¼ nnn¯ n¯ − nnn¯ nn¯ n¯ : ð Þ existence of de Sitter vacua in string/M-theory [26].Itis n 2 3 9 Knn¯ Knn¯ known that de Sitter vacua do not arise in the classical limit of string/M-theory [27]. This, however, leaves the pos- From the necessary condition, it follows that Vnn¯ ≥ 0 and, sibility to realize de Sitter vacua at the quantum level. hence, Indeed, in the G2 compactification we describe, the scalar

G potential is generated by quantum effects. The quantum e ð2 − 3RnÞ − VRn ≥ 0: ð10Þ nature is at the heart of the proposal and tied to the origin of physical scales. For a tiny positive as in the observed Universe, the constraint essentially becomes [20] C. Minimal example of modulus stabilization 2 R < : ð Þ We describe the basic mechanism of modulus stabili- n 11 2 3 zation in G2 compactifications leaning on [1]. Some key features are illustrated within a simple one-modulus This condition restricts the Kähler potential of the field example. Since the single-modulus case faces cosmo- responsible for supersymmetry breakdown. Indeed, it logicalproblemswhichcanberesolvedinasetupwith invalidates some early attempts to incorporate supersym- S two or more moduli, we will later introduce a two-moduli metry breaking in string theory. For the in example and comment on the generalization to many heterotic string theory, one can, e.g., derive the curvature ¯ moduli. scalar RS ¼ 2 from its Kähler potential K ¼ − logðS þ SÞ. 3 The minimal example of modulus stabilization in G2 The scenario of dilaton-dominated supersymmetry break- compactifications invokes two hidden sector gauge groups ing [22] is, hence, inconsistent with the presence of a SUðN1 þ 1Þ,SUðN2Þ with gauge kinetic functions de Sitter minimum [20,23]. Kähler potentials of the no- scale type K ¼ −3 logðT¯ þ TÞ, with T denoting an overall f1 ¼ f2 ¼ T: ð12Þ Kähler modulus, feature RT ¼ 2=3. In this case (11) is marginally violated. Corrections to the Kähler potential ðN þ 1Þ and/or subdominant F or D terms from other fields may The SU 1 gauge theory shall contain one pair of Q Q¯ then reconcile T-dominated supersymmetry breaking with massless quarks , transforming in the fundamental and ðN þ 1Þ the bound. Examples of this type include the large volume antifundamental representation of SU 1 . When the ðN þ 1Þ scenario [9] as well as Kähler uplifting [10,11]. SU 1 pcondenses,ffiffiffiffiffiffiffiffiffiffi the quarks form an effective ¯ A less constrained possibility to realize de Sitter vacua field ϕ ¼ 2QQ. Taking SUðN2Þ to be matter free, the consists in the supersymmetry breaking by a hidden sector superpotential and Kähler potential read matter field. Hidden sector matter is present in compactified 2 2πT 2πT −N − N − N M-theory. When it is included using the approach of W ¼ A1ϕ 1 e 1 þ A2e 2 ; Seiberg [18], it generically leads to a de Sitter vacuum. K ¼ −7 ðT¯ þ TÞþϕϕ¯ : ð Þ The identification of the Goldstino with the meson of a log 13 hidden sector strong gauge group allows for a natural explanation of the smallness of the supersymmetry break- We neglected the volume dependence of the matter Kähler ing scale (and correspondingly the weak scale) through potential which does qualitatively not affect the modulus dimensional transmutation. The simple canonical Kähler stabilization [28]. The scalar potential including the modu- potential, for instance, yields a vanishing curvature scalar lus and meson field is consistent with (11). Matter supersymmetry breaking is G T ϕ also employed in KKLT modulus stabilization [8] with V ¼ e ðG GT þ G Gϕ − 3Þ: ð14Þ F-term uplifting [24] and in heterotic string models [25]. G We note, however, that in 2 compactifications of 2 M-theory, de Sitter vacua can arise even if the hidden Some differences occur since [1] mostly focused on the case G of two hidden sector gauge groups with equal gauge kinetic sector matter decouples. As we show in Sec. IV, the 2 functions, while we will consider more general cases. Kähler potential (2) features linear combinations of moduli 3Due to the absence of a constant term in the superpotential, a with curvature scalar as small as 2=7. In contrast to the single gaugino condensate would give rise to a runaway potential.

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CP G G ¯ The scalar mass spectrum contains two even and two δT ¼ ϕT ϕ þ OðT−4Þ: ð Þ CP TT¯ 0 18 odd (axion) states which are linear combinations of GTTK GT¯ T¯ ReT, jϕj and ImT, arg ϕ, respectively. We will denote the CP s φ −2 even and odd mass eigenstates by 1;2 and 1;2, The leading contribution to the shift is δT ¼ OðT0 Þ. This respectively. The scalar potential is invariant under the shift justifies our expansion in δT. In the next step, we want to determine the location of the minimum. As an additional N2 constraint, we require a vanishing vacuum energy. In order T → T þ i Δ; ϕ → eiπΔϕ: ð15Þ N1 − N2 to provide simple analytic results, we will perform a volume expansion which is equivalent to an expansion T−1 OðT−1Þ This can easily be seen from the fact that the superpotential in 0 . We include terms up to 0 . Notice that, at this T ¼ T merely picks up an overall phase under this transformation. order, the modulus minimum satisfies 0 susy.We, The light axion nevertheless, have to keep track of the shift carefully since it may appear in a product with the inverse Kähler φ ∝ N T þ πðN − N Þ ϕ ð Þ metric which compensates its suppression. The conditions 1 2Im 1 2 arg 16 −1 VT ¼ Vϕ ¼ V ¼ 0 lead to the set of equations at order T0 is, hence, massless which makes it a natural candidate G2 pffiffiffi for the QCD axion [3]. The remaining axionic degree of G ¼ 0;Gþ1− ϕT ¼ 0;G¼ 3: ð Þ T ϕϕ G ϕ 19 freedom (d.o.f.) receives a periodic potential which has an TT extremum at the origin of field space. Without loss of The solutions for the modulus and meson minimum read generality, we require signðA1=A2Þ¼−1 such that the extremum is a minimum.4 This allows us to set ImT ¼ pffiffiffi ϕ ¼ 0 CP 3 14 N2 arg when discussing the stabilization of the even ϕ0 ¼ ;T0 ¼ : ð20Þ scalars. 2 π 3ðN2 − N1Þ − 8 We now want to prove that this setup allows for the presence of a (local) de Sitter minimum consistent with The meson naturally settles at a finite field value slightly 1M observation. For practical purposes, we can neglect the below P due to the interplay between terms involving W tiny and require the presence of a ϕ (which drive the meson out in field space) and the K Minkowski minimum with broken supersymmetry. There is supergravity prefactor e (which rapidly grows towards generically no supersymmetric minimum at finite field large meson field values). Despite the large meson vacuum values. Since the negative sign of A1=A2 is required for expectation value, higher order terms in the meson Kähler axion stabilization, a solution to GT ¼ 0 only exists if potential, which we neglected in this work, leave the picture N2 >N1. With this constraint imposed, there is no simul- qualitatively unchanged as shown in [28]. Notice, however, taneous solution to Gϕ ¼ 0 with positive jϕj.However,a that a minimum of the full system only exists for N ≥ N þ 3 N − N ≲ 10 minimum ðT0; ϕ0Þ with broken supersymmetry may occur 2 1 . On the other hand 2 1 since the T G nonperturbative terms in the superpotential would other- close to the field value susy at which T vanishes. This is T wise exceed unity. Equations (19) fix one additional because the modulus mass term at susy dominates over the parameter which can be taken to be the ratio A1=A2. linear term which drives it away from this point. Given a δT ¼ T − T We find minimum with a small shift susy 0, we can expand 1 A1 N1 3 N1 28 N2 − N1 ¼ − exp : ð21Þ A2 N2 4 N1 3ðN2 − N1Þ − 8 GT ¼ GT¯ ¼ −ðGTT þ GTT¯ ÞδT: ð17Þ

A suppressed vacuum energy can be realized on those G2 Here and in the following, all terms are evaluated at the manifolds which fulfill the above constraint5 with accept- minimum if not stated otherwise. Since T0, ϕ0, δT are real, able precision. We now turn to the details of supersym- there is no need to distinguish between GT and GT¯ . In order metry breaking. The gravitino mass is defined as to determine the shift, we insert (17) into the minimization V ¼ 0 δT m ¼jeG=2j : ð Þ condition T and keep terms up to linear order in . 3=2 T0;ϕ0 22 Notice that all derivatives of G with respect to purely holomorphic or purely antiholomorphic variables are of Throughout this work, m3=2 refers to the gravitino mass in −1 zeroth order in T0 . We find the vacuum of the theory. We will later also introduce the

4 5 If this condition is not satisfied, the relative sign of A1 and A2 More accurately, the exact version of the above approximate can be inverted through field redefinition. constraint.

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m ≃ m gravitino mass during inflation, but will clearly indicate the s2 φ2 latter by an additional superscript I. Within the analytic sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G KTT¯ G approximation, the gravitino mass determined from (19) ≃ eG=2 TT T¯ T¯ and (21) is KTT¯ 56 3N2 − 3N2 − 8N e3=8π7=2 3N − 3N − 8 7=2 ≃ 2 1 1 m : ð Þ m ≃ jA j 2 1 N ð3N − 3N − 8Þ2 3=2 27 3=2 1 48N 7N 1 2 1 1 2 N2 28 The mesonlike axion φ1 is massless due to the shift × exp − : ð23Þ N1 3ðN2 − N1Þ − 8 symmetry. Since the meson is the dominant source of supersymmetry breaking, the supertrace of masses in the Up to the overall prefactor, the gravitino mass is fixed by meson multiplet must approximately cancel. This implies the rank of the hidden sector gauge groups. A hierarchy m ≃ 2m : ð Þ between the Planck scale and the supersymmetry breaking s1 3=2 28 scale naturally arises from the dimensional transmutation. If we require a gravitino mass close to the electroweak The scalar potential vanishes towards large modulus field scale, this singles out the choice N2 ¼ N1 þ 4. While this values. Hence, the minimum (T0, ϕ0) is only protected by a particular result only holds for the single-modulus case, finite barrier. We first keep the meson fixed and estimate similar relations between the gravitino mass and the hidden its height in a leading order volume expansion.7 Then, we sector gauge theories can be established in realistic systems allow the meson to float, in order to account for a decrease with many moduli [1].6 In order to determine the pattern of of the barrier in the mixed modulus-meson direction. supersymmetry breaking we evaluate the F terms which are Numerically, we find that the shifting meson generically −1 defined in the usual way, reduces the barrier height by another factor ∼T0 . Our final estimate thus reads i G=2 ij¯ F ¼ e K Gj¯: ð24Þ 16π2T V ≃ 0 m2 : ð Þ barrier 2 2 3=2 29 From (17) and (18), we derive 7e N1 pffiffiffi T 2N2 ϕ The prefactor in front of the gravitino mass is of order jF j ≃ m3=2; jF j ≃ 3m3=2 ð25Þ πðN2 − N1Þ unity. Notice that the above expression is multiplied by two powers of the Planck mass which is set to unity in our at leading order. The meson provides the dominant source convention. of supersymmetry breaking as can be seen by comparing For illustration, we now turn to an explicit numerical the canonically normalized F terms example. We choose the following parameter set: pffiffiffiffiffiffiffiffiffi jFT K j 3N − 3N − 8 N1 ¼ 8;N2 ¼ 12;A1 ¼ 0.0001: ð30Þ TT¯ ≃ pffiffiffiffiffi2 1 : ð Þ ϕ 26 jF j 2 21ðN2 − N1Þ The prefactor A2 is fixed by requiring a vanishing vacuum energy. Numerically, we find This has important implications for the mediation of supersymmetry breaking to the visible sector. Since A1=A2 ¼ −20.9; ð31Þ gravity-mediated gaugino masses only arise from moduli F terms, they are suppressed against the gravitino and in good agreement with the analytic approximation (21). masses. We refer to [29] for details. We list the resulting minimum, particle masses, supersym- As stated earlier, the modulus and the meson are subject metry breaking pattern, and barrier height in Table I. The to mixing. However, the mixing angle is suppressed T CP numerical results are compared with the analytic expres- by 0, and the heavy even and odd mass eigenstates sions provided in this section. The approximations are valid s2 and φ2 are moduluslike. Since their mass is dominated to within a few percent precision. Only for m3=2 is the error by the supersymmetric contribution m ¯ , they are nearly TT larger due to its exponential dependence on the modulus degenerate with minimum. The scalar potential in the modulus-meson plane is 6 In realistic G2 compactifications, the gauge kinetic function is depicted in Fig. 1. Also shown is the potential along the set by a linear combination of many moduli. We can effectively account for this by modifying the gauge kinetic function to f ¼ 7 Oð10 − 100ÞT in the one-modulus example. In this case, the We also assumed N1;2 ≫ N2 − N1 when estimating the preferred value of N2 − N1 changes to 3 in agreement with [1]. barrier height.

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TABLE I. Location of the minimum, mass spectrum, F terms, and height of the potential barrier for the parameter choice (30). The upper and lower rows correspond to the exact numerical result and analytic approximation, respectively. T ϕ m m m m m FT Fϕ V 0 0 3=2 φ1 φ2 s1 s2 barrier 2 12.9 0.85 57 TeV 0 77.1m3=2 1.98m3=2 75.4m3=2 1.98m3=2 1.72m3=2 0.5m3=2 2 13.4 0.87 33 TeV 0 77m3=2 2m3=2 77m3=2 1.91m3=2 1.73m3=2 0.6m3=2

“most shallow” mixed modulus-meson direction. The latter The superpotential is assumed to be of the form was determined by minimizing the potential in the meson direction for each value of T. W ¼ WðT ÞþwðT ;T Þ: ð Þ H H L 33

D. Generalization to several moduli W T The part only depends on H and provides the large Realistic G2 manifolds must contain the full minimal supersymmetric mass for the heavy linear combination. supersymmetric standard model spectrum with its Oð100Þ The part w is responsible for supersymmetry breaking couplings. They will generically feature a large number of and its magnitude is controlled by the (much smaller) T moduli and nonperturbative terms in the superpotential. gravitino mass. We require that H be stabilized super- The low energy phenomenology, however, mostly depends symmetrically at a high mass scale. For this we impose that on the lightest modulus. In this sense, the mass spectrum the high energy theory defined by W has a supersymmetric derived in the previous section is realistic, once T is Minkowski minimum, i.e., identified with the lightest modulus. However, in the early Universe, high energy scales are accessed. This implies W ¼ W ¼ 0; ð Þ H 34 that, for cosmology, the heavier moduli do actually matter. We will later see that inflation in M-theory relies on large where the subscript H indicates differentiation with respect mass hierarchies in the moduli sector. In order to motivate T to H. The above condition has to be fulfilled at the their existence, we now introduce an example with two T T minimum which we denote by H;0. It ensures that H can moduli T1;2. be integrated out at the superfield level. The mass of the One linear combination of moduli T plays the role of L heavy modulus is given as the light modulus as in the previous section. It participates (subdominantly) in supersymmetry breaking and its mass is 1 1 tied to the gravitino mass. The orthogonal linear combi- m ≃ eK=2W þ ; ð Þ T HH 35 T H 4K ¯ 4K ¯ nation H can, however, be decoupled through a large 11 22 supersymmetric mass term from the superpotential. In order to be explicit, we will identify with Kii¯ denoting the entries of the Kähler metric in the m original field basis. Since TH is unrelated to the gravitino T1 þ T2 T1 − T2 T ¼ ;T¼ : ð32Þ mass, it can be parametrically enhanced against the light H 2 L 2 modulus mass. The construction of a Minkowski minimum

2 FIG. 1. Left: The scalar potential (in Planck units) in the modulus and meson direction rescaled by m3=2. A local minimum with broken supersymmetry is located at T0 ¼ 12.9, ϕ0 ¼ 0.85. The field direction with the shallowest potential barrier is indicated by the red line. Right: The potential along this direction is shown.

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T T T for L with softly broken supersymmetry proceeds analo- derives from the set of equations (19) with replaced by L. gously to the one-modulus case. We find As an example we consider five hidden sector gauge pffiffiffi groups SUðN1 þ 1Þ and SUðNiÞ (i ¼ 2; … 5) with gauge 3 4K T N ϕ ¼ ;T¼ − L L;0 2 ; ð Þ 0 L;0 42 kinetic functions 2 π 3ðN2 − N1Þ − 8 f ¼ 2T þ T ;f¼ T þ T : ð Þ T 1;2 1 2 3;4;5 1 2 36 where we wrote the equation for L;0 in implicit form. In contrast to the single-modulus example, values N2

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TABLE II. Minimum and mass spectrum for the parameter set (45). In the original basis, the minimum is located at T1;0 ¼ 5.6, T2;0 ¼ 13.4. All masses are given in TeV. T T ϕ m m m m m m m H;0 L;0 0 3=2 φ1 φ2 φ3 s1 s2 s3 9.5 −3.9 0.78 82 2.4 1.4 × 103 3.3 × 106 148 1.2 × 103 3.3 × 106 wonder whether the two-moduli example introduces addi- light moduli can be fixed through the sum of two gaugino tional tuning compared to the one-modulus case, since two condensates in complete analogy to the examples discussed of the Ai’s are now fixed in order to realize a vanishing in this work. cosmological constant. However, deviations from (39) and (43) can compensate without spoiling the moduli stabiliza- III. MODULUS (DE)STABILIZATION tion.8 Effectively, there is still only a single condition which DURING INFLATION? must be fulfilled to the precision to which the vacuum energy As was shown in the previous section, the lightest cancels. In Table II we provide the location of the minimum modulus is only protected by a barrier whose size is and the resulting mass spectrum for the choice (45).An — controlled by the gravitino mass. There is danger that, important observation is that large mass hierarchies in during inflation, the large potential energy lifts the modulus Oð103Þ— this example a factor of can indeed be realized over the barrier and destabilizes the extra dimensions. in the moduli sector. The origin of such hierarchies lies in We will show that in the single-modulus case, indeed, the the dimensional transmutation of the hidden sector gauge bound H

8In the low energy theory, such deviations would manifest as a 9Another option may consist in fine-tuning several gaugino constant in the superpotential which is acceptable as long as the condensates in order to increase the potential barrier as in models latter is suppressed against the other superpotential terms. with strong moduli stabilization [13,31].

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FIG. 3. Scalar potential in the modulus direction for different choices of the Hubble scale. For each value of T, the potential in the meson direction was minimized. The remaining parameters are chosen as in Fig. 1.

FIG. 2. Scalar potential along the TL-direction. The remaining fields were set to their TL-dependent minima (see text). The Minkowski minimum with softly broken supersymmetry is where we employed V ¼ 3H2. The constraint remains located at T ;0 ¼ −3.9. L qualitatively unchanged if we couple a different inflation sector to the modulus.11 respectively. The modulus superpotential w and Kähler potential k are defined as in (13). As an example inflaton B. Two or more moduli sector, we consider the class of models with a stabilizer field defined in [32]. These feature In the previous example, the single modulus T is apparently the field which sets the overall volume of the manifold. Destabilization of T, which occurs at H ∼ m3=2, W ¼ K ¼ Kα ¼ 0 ð47Þ triggers unacceptable decompactification of the extra along the inflationary trajectory.10 For now, we focus on dimensions. However, once we extend our consideration modulus destabilization during inflation. Whether this par- to multiple fields, the modulus participating in supersym- ticular inflation model can be realized in M-theory does not metry breaking and the modulus controlling the overall matter at this point. In fact, we merely impose the conditions volume can generically be distinct. Consider a simple two- (47) for convenience since they lead to particularly simple modulus example for which the volume is determined as analytic expressions. The important element, which appears K ¯ −7 a1=3 a2=3 universally, is the e ∝ ðT þ TÞ factor which multiplies V ¼ðReT1Þ ðReT2Þ : ð50Þ all terms in the scalar potential. The latter reads The scalar potential (before including the inflaton sector) jϕj2 e α shall have a minimum at ðT1;0;T2;0Þ. At the minimum, we V ¼ V þ W Wα; ð48Þ mod ðT¯ þ TÞ7 may then define the overall volume modulus where V coincides with the scalar potential without the T T mod T ¼ a 1 þ a 2 ; ð Þ inflaton as defined in (14). The second term on the right- V 1 2 51 T1;0 T2;0 hand side sets the energy scale of inflation. It displaces the modulus and the meson. Once the inflationary energy reaches the height of the potential barrier defined in (29), such that for an infinitesimal change of the volume dV ∝ dT T the minimum in the modulus direction gets washed out and V. Let us assume V receives a large supersym- the system is destabilized. This is illustrated in Fig. 3.The metric mass and decouples from the low energy theory. The constraint can also be expressed in the form orthogonal linear combination shall be identified with the light modulus which is stabilized by supersymmetry break- ing. It becomes clear immediately that in this setup the H ≲ m3=2; ð49Þ bound H

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T FIG. 4. Scalar potential during inflation in the light modulus (left panel) and heavy modulus direction (right panel). For each H;L and H, the remaining fields have been set to their corresponding minima.

breaking, large hierarchies between H and m3=2 can in light modulus direction for different choices of H.For 12 T H principle be realized. each value of L and , we have minimized the potential In reality, the heavy modulus which protects the extra in the meson and heavy modulus direction. It can be dimensions does not need to coincide with the volume seen that the light modulus remains stabilized even for modulus. One can easily show that V in (50) remains finite H>m3=2.WithgrowingH it becomes heavier due to the given that an arbitrary linear combination T1 þ αT2 with Hubble mass term induced by inflation. This holds as α > 0 is fixed. If the heavy linear combination is mis- long as the heavy modulus is not pushed over its aligned with the volume modulus, the light modulus still potential barrier. For our numerical example, destabili- remains protected, but receives a shift during inflation. zation of the heavy modulus occurs at H ≃ 470m3=2 as In order to be more explicit, let us consider the two- can be seen in the right panel of the same figure. The T T ϕ modulus example of Sec. II D. We add the inflation sector minima of H, L, as a function of the Hubble scale again imposing (47). The scalar potential along the infla- are depicted in Fig. 5 up to the destabilization point. It T T tionary trajectory is can be seen that L slowly shifts from L;0 to the field value maximizing the volume as given in (53).Our ejϕj2 V ¼ V þ WαW : ð Þ findings can easily be generalized to systems with many mod ¯ ¯ 6 α 52 ðT1 þ T1ÞðT2 þ T2Þ moduli. In this case, an arbitrary number of light moduli remains stabilized during inflation, given at least one m ≫ H Inflation tends to destabilize moduli since the potential heavy modulus ( TH ) which bounds the overall energy is minimized at T1;2 → ∞. However, the direction volume. T ¼ T þ T H 1 2 is protected by the heavy modulus mass m H ≪ m TH . As long as TH , the heavy modulus remains T close to its vacuum expectation value. For fixed H, the inflaton potential energy term [second term on the right- hand side of (52)] is minimized at 5 T ¼ − T : ð Þ L 7 H 53

T T Hence, L remains protected as long as H is stabilized. T It, nevertheless, receives a shift during inflation since H is not exactly aligned with the volume modulus. In the left panel of Fig. 4, we depict the scalar potential in the T T ϕ FIG. 5. Minima of H, L, as a function of the Hubble scale. 12The idea of trapping a light modulus through a heavy Moduli destabilization occurs at H ≃ 470m3=2 as indicated by modulus during inflation has also been applied in [34]. the stars.

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A particularly appealing possibility is that the modulus The number of e-folds N corresponding to a certain field which protects the extra dimensions is itself the inflaton. value can be approximated analytically, In particular, it would seem very natural to identify the 1 1 N ðρ − ρ Þ inflaton with the overall volume modulus. We will prove in N ≃ N þ max 0 ; max arctan 3 the next section that this simple picture is also favored by 2 π 2πρ0 G pffiffiffi the Kähler geometry of the 2 manifold. Indeed, we will 2πρ2 N ¼ pffiffiffi 0 ; ð Þ show that inflationary solutions only arise in moduli max 57 directions closely aligned with the overall volume modulus. δ N e where max denotes the maximal -fold number. Since we IV. MODULAR INFLATION IN M-THEORY assume ρ0 to be sub-Planckian, the slope parameter δ must So far we have discussed modulus stabilization during be strongly suppressed for inflation to last 60 e-folds or inflation without specifying the inflaton sector. In this longer. The CMB observables, namely, the normalization section, we will select a modulus as the inflaton. The of the scalar power spectrum As, the spectral index of scalar resulting scheme falls into the class of “inflection point perturbations ns, and the tensor-to-scalar ratio r, are inflation” which we will briefly review. We will then determined by the standard expressions identify the overall volume modulus (or a closely aligned V direction) as the inflaton by means of Kähler geometry, A ≃ ;n≃1−6ϵ þ2η ;r≃16ϵ : ð Þ s 2 s V V V 58 before finally introducing explicit realizations of inflation 24π ϵV and moduli stabilization. For comparison with observation, these quantities must be evaluated at the field value for which the scales A. Inflection point inflation relevant to the CMB cross the horizon, i.e., at N ¼ Observations of the cosmic microwave background 50–60 according to (57). We can use the Planck measured −9 (CMB) suggest an epoch of slow roll inflation in the very values ns ¼ 0.96–0.97, As ≃ 2.1 × 10 [35] to fix two early Universe. The nearly scale-invariant spectrum of parameters of the inflaton potential. This allows us to density perturbations sets constraints on the first and predict the tensor-to-scalar ratio second derivative of the inflaton potential, 6 ρ0 −11 0 00 r ∼ × 10 : ð59Þ jV j; jV j ≪ V: ð54Þ 0.1

Unless the inflaton undergoes trans-Planckian excursions, Inflation models rather generically require some degree of the above conditions imply a nearly vanishing slope and fine-tuning. This is also the case for inflection point inflation curvature of the potential at the relevant field value. An and manifests in the (accidental) strong suppression of the obvious possibility to realize successful inflation invokes slope at the inflection point. In addition, the slow roll an inflection point with small slope, i.e., an approximate analysis only holds for the range of initial conditions which saddle point. Most features of this so-called inflection point enable the inflaton to dissipate (most of) its kinetic energy inflation can be illustrated by choosing a simple poly- before the last 60 e-folds of inflation. While initial con- nomial potential, ditions cannot meaningfully be addressed in the effective description (55), we note that the problem gets ameliorated δ 1 V ¼ V 1 þ ðρ − ρ Þþ ðρ − ρ Þ3 þ Oððρ − ρ Þ4Þ; if the inflationary plateau spans a sizable distance in field 0 0 3 0 0 ρ0 6ρ0 space. This favors large ρ0 as is, indeed, expected for a ð55Þ modulus field. In this case, the typical distance between the minimum of the potential and an inflection point relates to ρ ≲ 1 where ρ is the inflaton which is assumed to be canonically the Planck scale (although 0 to avoid uncontrollable ρ corrections to the setup). Setting ρ0 to a few tens of MP,we normalized and 0 is the location of the inflection point. −8 −6 4 expect r ∼ 10 –10 according to (59). The maximal The coefficient in front of ðρ − ρ0Þ can be chosen such that e N ¼ 100–200 the potential has a minimum with vanishing vacuum energy number of -folds is max . While the modulus at the origin. Since the quartic term does not play a role potential differs somewhat from (55) (e.g., due to nonca- r during inflation, it has not been specified explicitly. The nonical kinetic terms), we will still find similar values of in the M-theory examples of the next sections. height of the inflationary plateau is set by V0. The potential slow roll parameters follow as B. Identifying the inflaton 1 V0 2 V00 ϵV ¼ ; ηV ¼ : ð56Þ We now want to realize inflation with a modulus field as 2 V V inflaton. Viable inflaton candidates shall be identified by

066005-11 GORDON KANE and MARTIN WOLFGANG WINKLER PHYS. REV. D 100, 066005 (2019) means of Kähler geometry. This will allow us to derive X 6O4 X 4O3O3 R ¼ i − i j : ð Þ some powerful constraints on the nature of the inflaton ρ pffiffiffiffiffiffiffiffiffi 62 ai aiaj without restricting to any particular superpotential. i i;j Inflationary solutions feature nearly vanishing slope and curvature of the inflaton potential in some direction of field Since successful inflation singles out field directions with space. To very good approximation we can neglect the tiny small curvature scalar, it is instructive to identify the linear R slope and apply the supergravity formalism for stationary combination of moduli with minimal ρ. The latter is R O points (see Sec. II B). All field directions orthogonal to the obtained byp minimizingffiffiffiffiffiffiffiffiffi ρ with respect to the i,which inflaton must be stabilized. Hence, the modulus mass yields Oi ¼ ai=7 and matrix during inflation should at most have one negative X a eigenvalue corresponding to the inflaton mass. The latter ρ ∝ i T : ð Þ TI i 63 must, however, be strongly suppressed against V due to the i i nearly scale-invariant spectrum of scalar perturbations caused by inflation. We can hence neglect it against the By comparison with (51), we can identify this combination last term in (7) and require the mass matrix to be positive as the overall volume modulus (defined at the field location semidefinite. This leads to the same necessary condition as of inflation). The corresponding minimal value of Rρ ¼ 2=7. for the realization of de Sitter vacua, namely, that Vij¯ must Hence, inflation must take place in the direction of be positive semidefinite. During inflation, we expect the the overall volume modulus or a closely aligned field ρ potential energy to be dominated by F . The curvature direction—as was independently suggested by modulus scalar of the one-dimensional submanifold associated with stabilization during inflation (see Sec. III B). In order to be the inflaton ρ [cf. (9)] should, hence, fulfill condition (10). more explicit, we define θ as the angle14 between ρ and the The latter can be rewritten as volume modulus TV, 3 3 H 2 rffiffiffiffi R−1 > þ : ð Þ ai ρ I 60 cos θ ¼ Oi : ð64Þ 2 2 m3=2 7

Here we introducedpffiffiffiffiffiffiffiffiffi the inflationary Hubble scale through In other words, cos2 θ is the fraction of volume modulus the relation H ¼ V=3 and the “gravitino mass during contained in the inflaton. The constraint on the angle I G=2 I inflation” m3=2 ¼ e . Note that m3=2 is evaluated close to depends on the properties of the manifold. However, one the inflection point. It is generically different from the can derive the lower bound m gravitino mass in the vacuum which we denoted by 3=2. 7 We notice that field directions with a small Kähler curvature R−1 < ð1 þ 2 2 θÞ; ð Þ ρ 6 cos 65 scalar are most promising for realizing inflation. For a simple logarithmic Kähler potential K ¼ −a logðρ¯ þ ρÞ, one finds R ¼ 2=a a>3 which holds for an arbitrary number of moduli and inde- ρ . Condition (60) then imposes at least . a a ρ pendent of the coefficients i (only requiring that the i sum However, more generically, we expect to be a linear up to 7). If we combine this constraint with (60),wefind combination of the moduli Ti appearing in the G2 Kähler potential (2). We perform the following field redefinition: 1 9 H 2 2 θ > þ : ð Þ pffiffiffiffiffi cos I 66 X a 7 14 m3=2 ρ ¼ O j T : ð Þ i ij 2TI j 61 j j From this condition, it may seem sufficient to have a moderate volume modulus admixture in the inflaton. TI T Here j denotes the field value of j during inflation (more However, in the absence of fine-tuning, the second term precisely, at the quasistationary point). Without loss of on the right-hand side is not expected to be much smaller TI 13 O generality, we assume that j is real. The matrix is than unity. Furthermore, for any concrete set of ai, a stronger an element of SOðMÞ, where M denotes the number of bound than (66) may arise. Therefore, values of cos θ close moduli. The coefficients ai mustagainsumto7forG2.The to unity—corresponding to near alignment between the above field redefinition leads to canonically normalized ρi inflaton and volume modulus—are preferred. at the stationary point. We now choose ρ1 ≡ ρ to be the inflaton and abbreviate O1i by Oi. The curvature scalar can 14The angle θ is defined in the M-dimensional space spanned then be expressed as by the canonically normalized Ti. For two linear combinations of ˆ ˆ moduli ρ1 ¼ αiTi and ρ2 ¼ βiTi, it is obtained from the scalar 13 I αβ ¼jαjjβj θ Tˆ Imaginary parts of T can be absorbed by shifting Tj along product cos . Here, i denotes the canonically j ˆ pffiffiffiffi I the imaginary axis, which leaves the Kähler potential invariant. normalized moduli Ti ¼ð ai=Ti ÞTi=2.

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TABLE III. Input parameter sets PS1 and PS2 which give rise to the potential shown in Fig. 6. Two input parameters are specified with higher precision. This is required to (nearly) cancel the cosmological constant and to ensure that the spectral index matches precisely with observation.

PS a1 a2 A1 A2 A3 A4 N1 N2 N3 N4 TH;0 1161 −1.18 0.719766 −0.178645 11 15 19 23 7.8 225−1.35 2.16245 −0.918729 … 15 17 19 … 8.2

Let us, finally, point out that the lower limit on the As an example, we take the superpotential and Kähler curvature scalar also implies the following bound on the potential to be of the form, Hubble scale: X −2πT =N W ¼ WðT1 þ T2Þþ Aie 1 i ; I 2m3=2 i H< pffiffiffi ; ð67Þ ¯ ¯ 3 K ¼ −a1 log ðT1 þ T1Þ − a2 log ðT2 þ T2Þ; ð69Þ

T ¼ which must hold for an arbitrary superpotential. One may respectively. The heavy modulus can be defined as H ðT þ T Þ=2 T now worry that this constraint imposes either low scale 1 2 in this case. In the limit where H becomes T inflation or high scale supersymmetry breaking. This is, infinitely heavy, integrating out H at the superfield level is mI T T however, not the case since 3=2 can be much larger than the equivalent to replacing H by H;0 in the superpotential and T → T þ ρ T → T − ρ gravitino mass in the true vacuum. Indeed, if the inflaton is Kähler potential, i.e., 1 H;0 and 2 H;0 . not identified with the lightest, but with a heavier modulus, it We consider the case where inflation proceeds along the I appears natural to have m3=2 ≫ m3=2. Nevertheless, (67) real axis. The scalar potential features terms which decrease exponentially towards large ρ which originate from W imposes serious restrictions on the superpotential. In order K for the potential energy during inflation to be positive, while and its derivatives. At the same time, the prefactor e has satisfying (60), one must require15 positive slope if we choose a2 >a1. For appropriate parameters, the interplay between the superpotential and ρ 3

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FIG. 6. Left: The inflaton potential for the two parameter sets of Table III. The inflection point at ρ − ρ0 ¼ 4 is indicated by the thick gray dot. Right: The squared mass of the axion direction in units of H2. angle [analogously, the curvature scalar is small enough D. Inflation and supersymmetry breaking to satisfy (60)]. Successful inflation can, therefore, be In the final step, we wish to construct a more realistic realized. For PS2, the situation is different since the same model which incorporates inflation and supersymmetry condition is violated. The tachyonic instability which breaking simultaneously. The plan is to augment the prevents inflation for PS2 is, hence, due to the misalign- inflation sector of the previous section by the supersym- ment between the (would-be) inflaton and the volume metry breaking sector comprised of the light modulus and modulus. the meson field (cf. Sec. II). For the parameter choice PS1, the inflationary observ- The minimal example contains three moduli fields T1;2;3 ables can be determined from the slow roll expressions T ρ T which form the linear combinations H, , and L. The (58), where the normalization of the kinetic term has to be inflaton ρ must be approximately aligned with the volume taken into account [the slow roll parameters are defined as T modulus. An orthogonal light modulus L participates in derivatives with respect to the canonically normalized T supersymmetry breaking. The third modulus direction H inflaton in (56)]. The observables are consistent with is stabilized supersymmetrically at a mass scale above the present CMB bounds; specifically we find inflationary Hubble scale. While it does not play a dynamical role, its vacuum expectation value manifests −7 −9 ns ¼ 0.96;r¼ 3 × 10 ;As ¼ 2 × 10 : ð71Þ in the Kähler potential of the lighter d.o.f. It assists in generating the plateau in the inflaton potential. The super- m ≫ potential is chosen such that a mass hierarchy TH The tensor-to-scalar ratio falls in the expected range for m ≫ m ρ TL arises in the vacuum. This can be achieved via inflection point inflation with a modulus (see Sec. IVA). the form From a theoretical point of view, it is interesting that the inflationary plateau can be turned into a de Sitter W ¼ WðT ÞþWðT ; ρÞþwðT ;T Þ: ð72Þ minimum through a small parameter deformation. If we, H H H L e.g., increase the value of T ;0 (or change one of the Ai) H All three superpotential parts originate from gaugino for PS1 slightly, the potential develops a minimum condensation. The desired mass pattern is realized through close to the inflection point. The consistency of de an appropriate hierarchy in the condensation scales in W, Sitter vacua in the moduli potential follows from the W, and w, respectively. For concreteness, we will make the G2 Kähler potential which has a curvature scalar as small following identification: as 2=7 on the submanifold associated with the volume modulus—in contrast to other prominent string theory T ρ T T ρ T T ρ T ¼ H þ þ L ;T¼ H þ − L ;T¼ H − ; constructions (see Sec. II B). 1 3 6 2 2 3 6 2 3 3 3 ð73Þ TABLE IV. Derived parameters for the inputs PS1 and PS2 from Table III. which is just one of many possibilities. Without specifying 2 I W W ¼ W ¼ 0 T ;0 PS O1 O2 cos θ Rρ Hm3=2 explicitly, we assume H at H . As shown previously, this can, e.g., be achieved via three gaugino 1 0.12 −0.99 0.76 0.34 5.9 × 10−8 1.7H −0 96 5 9 10−8 1 1H condensation terms (see Sec. II D). In the limit of very large 2 0.29 . 0.42 0.47 . × . m mass TH , integrating out the heavy modulus then simply

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TABLE V. Parameter choice giving rise to the inflaton potential shown in Fig. 8. The parameter A5 has been specified with higher precision in order to ensure that inflation with the correct spectral index arises. Cancellation of the cosmological constant fixes the remaining input parameter, A6 ¼ 2.4213062895.

a1 a2 a3 A1 A2 A3 A4 A5 N1 N2 N3 N4 N5 N6 TA;0 1 9 2 2 2 −7 0.117 −4.9 22.52 −20.52678 8 10 24 30 32 38 21.7

T T amounts to replacing H by H;0 in the superpotential and outside the validity of the supergravity approximation Kähler potential. In addition, we choose (T2 < 1). In the Minkowski minimum, where we can trust our calculation, the mass spectrum shown in Fig. 7 arises. 6 2 2πf1 2πf2 X 2πfi The light modulus and meson are responsible for super- −N − N − N − N w ¼ A1ϕ 1 e 1 þA2e 2 ; W ¼ Aie i : ð74Þ symmetry breaking. Their masses cluster around the i¼3 gravitino mass m3=2 ∼ 200 TeV. A slight suppression of the mesonlike axion mass arises due to an approximate shift The gauge kinetic functions are defined as symmetry (see Sec. II D). The inflaton is substantially heavier compared to the other fields since it decouples from f ¼ 2T þ T ¼ T þ T ; 1;2 1 3 H L supersymmetry breaking. 2 1 Inflation occurs along the real axis of ρ. The poten- f ¼ T þ T ¼ T þ ρ; ð Þ 3;4;5;6 1 2 3 H 3 75 tial along this direction is shown in Fig. 8, where the remaining fields have been set to their ρ-dependent such that W only depends on ρ, while w only depends on minima. A (quasistationary) inflection point occurs at T ϕ T G ρ − ρ ¼ 15 5 L and (once H has been integrated out). The 2 Kähler 0 . , where we can still trust the supergravity potential, approximation. Corrections to the moduli Kähler potential, which are expected at small compactification volume, X3 are suppressed in this regime. Even if they slightly ¯ K ¼ − ai log ðTi þ TiÞ; ð76Þ perturbed the inflaton potential, this could easily be i¼1 compensated for by adjusting the superpotential parame- ters. Inflation, hence, appears to be robust with respect to ρ T can be expressed in terms of , L via (73). For an exact any higher order effects. numerical evaluation, we choose the parameter set of For applying the constraints from Kähler geometry, we Table V. express the inflaton in terms of the (canonically normal- The latter gives rise to a Minkowski minimum with ized) original field basis ϕ ¼ 0 78 ρ ¼ −3 5 T ¼ broken supersymmetry at 0 . , 0 . , L;0 6 7 T ¼ 10 T ¼ 3 3 T ¼ 8 4 ˆ ˆ ˆ . (corresponding to 1 , 2 . , 3 . in the ρ ∝ 0.09T1 þ 0.07T2 − 0.99T3; ð77Þ original field basis). An additional AdS minimum appears ˆ where the Ti have been defined in (70). As can be seen, the inflaton is dominantly T3. The curvature scalar along the inflaton direction is Rρ ¼ 0.45. The Hubble scale and

FIG. 7. Spectrum of scalar (þ) and pseudoscalar (−) masses in the vacuum and during inflation. The dominant field components of the mass eigenstates are given in the plot legend (the orange lines, e.g., refer to the mesonlike mass eigenstates). Also shown FIG. 8. Scalar potential in the inflaton direction with the other are the gravitino mass and the Hubble parameter during inflation. fields eliminated through their minimization condition.

066005-15 GORDON KANE and MARTIN WOLFGANG WINKLER PHYS. REV. D 100, 066005 (2019) the gravitino mass during inflation are depicted in Fig. 7. One easily verifies that the curvature constraint (60) is satisfied and viable inflation without can thus be achieved. This can be related to the fact that the inflaton is sufficiently aligned with the volume modulus. The fraction of volume modulus contained in the inflaton is given by cos2 θ ¼ 0.54, in agreement with (66). In Fig. 7, we also provide the scalar mass spectrum during inflation. The inflaton mass is not shown since its squared mass is negative as required by the constraints 2 2 on the spectral index, specifically mρ ¼ −0.05H during inflation (corresponding to ηV ¼ −0.015). The other sca- FIG. 9. Solution to the coupled system of equations of motion lars receive positive Hubble scale masses during inflation for the fields ρ, T , ϕ. (as described in Sec. III B). Only the mesonlike axion is L about an order of magnitude lighter than H due to the approximate shift symmetry. The resulting isocurvature minimum, the postinflationary field evolution should 16 perturbations in the light axion are not expected to be ensure that the Universe ends up in the desired vacuum. dangerous since they are transferred into adiabatic pertur- This might impose additional constraints on the moduli bations once the axion has decayed into radiation. For the couplings including those to the visible sector. A compre- parameter example, this decay occurs before primordial hensive discussion of the reheating process is, however, nucleosynthesis. beyond the scope of this work. Let us just note that the In order to describe the dynamics of the multifield energy density stored in the light d.o.f. redshifts slower than system, the coupled equations of motion need to be solved. the thermal energy of the radiation bath and may dominate For noncanonical fields, the most general set of equations the energy content of the Universe before they decay. We, reads [36] therefore, expect a nonstandard cosmology with late time entropy production to occur (see [37]). Notice that this ∂V scenario is consistent with the observed element abundan- ψ̈α þ Γα ψ_ βψ_ γ þ 3Hψ_ α þ Gαβ ¼ 0: ð Þ βγ ∂ψ β 78 ces since all particles are sufficiently heavy to decay before primordial nucleosynthesis. Here the fields ψ α label the real and imaginary parts of ρ, T ϕ G L, . The field space metric αβ can be determined from V. CONCLUSION the Kähler metric and Γα is the Christoffel symbol with βγ M-theory compactified on a manifold of G2 holonomy αβ respect to the field metric Gαβ and its inverse G . The successfully describes many microphysical features of our solution to the field equations is depicted in Fig. 9. For a world. It has chiral fermions interacting via gauge forces range of initial conditions, the fields approach the infla- and explains the hierarchy of scales. We have now T ϕ tionary attractor solution. This means that L, settle at identified the inflaton within this theory. The latter is finite field values which do not depend on the initial essentially the overall volume modulus of the compactified condition after a few oscillations. Their minima during region (or a closely aligned field direction). This statement inflation, however, differ from their vacuum expectation is model independent and derives from the Kähler geom- values. The inflaton ρ slowly rolls down its potential close etry of the G2 manifold. to the inflection point. Inflation ends when it reaches the We provided concrete realizations of volume modulus steeper part of the potential. Then, ρ oscillates around its inflation which satisfy all consistency conditions. Inflation vacuum expectation value with the amplitude decreasing occurs close to an inflection point in the scalar potential. In due to the Hubble friction. The inflationary observables the relevant parameter regime, string theory corrections to can again be determined from a slow roll analysis. The the supergravity approximation are under full control. We parametric example was chosen to be consistent with solved the system of coupled field equations and proved observation. It has that all moduli are stabilized during inflation. The scalar fields orthogonal to the inflaton receive Hubble mass terms −7 −9 ns ¼ 0.97;r¼ 5 × 10 ;As ¼ 2 × 10 : ð79Þ such that inflation is effectively described as a single field slow roll model. However, several scalar fields are dis- The field evolution shown in Fig. 9 spans 5 orders of placed from their vacuum expectation values during magnitude in energy. All scalar fields remain stabilized over the full energy range. After inflation, the volume of 16In the parameter example, an additional AdS minimum the compactified manifold remains protected by the large occurs. It may, however, get lifted since it appears outside the inflaton mass. If the scalar potential features more than one range, where we can trust the supergravity calculation.

066005-16 DERIVING THE INFLATON IN COMPACTIFIED M-THEORY … PHYS. REV. D 100, 066005 (2019) inflation. They are expected to undergo coherent oscilla- While experiments will not directly probe the inflaton of tions when the Hubble scale drops below their mass. compactified M-theory, indirect evidence may be collected. The energy stored in these d.o.f. generically induces late This is because inflation sets the initial conditions for a time entropy production at their decay (which happens nonthermal cosmology which affects many other phenom- before primordial nucleosynthesis). ena including baryogenesis and dark matter. Further pre- The scale of inflation emerges from hidden sector strong dictions of the compactified M-theory will soon be tested dynamics. The Planck scale is the only dimensionful input to by laboratory experiments. the theory. We predict V1=4 ∼ 1015 GeV and the correspond- r ∼ 10−6 ing tensor-to-scalar ratio . Despite the large energy ACKNOWLEDGMENTS density of inflation, the theory is consistent with, and generically has low energy supersymmetry. It has a de We would like to thank Scott Watson for helpful com- Sitter vacuum in which the (s)goldstino dominantly descends ments on the manuscript. M. W. acknowledges support from a hidden sector meson field. Supersymmetry breaking by the Vetenskapsrådet (Swedish Research Council) is transmitted to the visible sector via gravity mediation. through Contract No. 638-2013-8993 and the Oskar It generates a hierarchy with heavy and lighter Klein Centre for Cosmoparticle Physics and the LCTP at gauginos. The gauginos are expected to reside at the TeV the University of Michigan. Both authors acknowledge scale, close to the present LHC sensitivity. support from DoE Grant No. DE-SC0007859.

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