IPMU-10-0220 1 Strong dynamics at the Planck scale as a solution to the 1 0 cosmological moduli problem 2 n a J 0 Fuminobu Takahashia and Tsutomu T. Yanagidaa,b 3 ] aInstitute for the Physics and Mathematics of the Universe, University of Tokyo, h p Kashiwa 277-8568, Japan - p e bDepartment of Physics, University of Tokyo, Tokyo 113-0033, Japan h [ 3 v 7 Abstract 2 2 We argue that strong dynamics at the Planck scale can solve the cosmological 3 moduli problem. We discuss its implications for inflation models, and find that a . 2 certain type of multi-field inflation model is required for this mechanism to work, 1 since otherwise it would lead to the serious η-problem. Combined with the inflaton- 0 1 induced gravitino problem, we show that a chaotic inflation with a discrete symme- : v try naturally avoids both problems. Interestingly, the focus point supersymmetry i is predicted when this mechanism is applied to the Polonyi model. X r a 1 Introduction In supergravity/superstring theories, there exist moduli fields, collectively denoted by Z, which have flat potentials and obtain masses from supersymmetry (SUSY) breaking. In gravitymediation,forinstance, thereistypicallyapseudomoduluswithamassoforderthe gravitinomassm asinthePolonyi model. Duringinflationtheminimumoftheeffective 3/2 potential for such a modulus is generically deviated from the low-energy minimum. After inflation, the modulus starts to oscillate about the minimum with an amplitude of order the Planck scale M , and soon dominates the energy density of the Universe. Since p interactions are suppressed by the Planck scale, the modulus lifetime is very long, leading to an onset of a radiation-dominated Universe with a low temperature, typically below MeV. Such a low temperature would dramatically alter the big bang nucleosynthesis (BBN) predictions of light element abundances in contradiction with observations. This is the cosmological moduli problem or the Polonyi problem [1, 2]. Several solutions to the problem have been proposed so far. The modulus abundance can be diluted if there is a huge entropy production at late times by e.g. thermal infla- tion [3, 4, 5], which however also dilutes the pre-existing baryon asymmetry [6]. Another possible solution is to suppose that the modulus mass is as heavy as 100TeV, or heavier, so that the modulus decays before BBN [7]. However, the cosmological moduli problem still persists because decays of moduli to gravitinos andgauginosare not suppressed [8]. If themodulusmass isheavier thanthegravitino, apair ofgravitinosisgenerically produced by the modulus decay [8, 9]. Those gravitinos typically spoil the success of BBN or the lightest supersymmetric particles (LSPs) produced by the gravitino decay exceed the dark matter abundance, unless the gravitino is much heavier than O(100)TeV. On the other hand, the gravitino production from the modulus decay can be kinematically forbidden if the modulus mass is lighter than twice the gravitino mass. However, the branching fraction of the modulus decay into SUSY particles is not suppressed [8], leading to an overproduction of the LSPs [10].1 1 In Refs. [11, 12], it was pointed out that the Wino-like LSP of mass about 100GeV can account for the present dark matter abundance if the modulus decay rate is enhanced by O(10). If we adopt the cosmologicalconstraintsontheWinomass[13,14],themodulus decayratemustbe enhancedbyO(100) or greater. We wouldlike to thank Bobby Acharya,GordonKane, PiyushKumar,and Scott Watson for 2 In fact, there is a simple and elegant solution to the cosmological moduli problem proposed long ago by Linde [15]. If the inflaton and the modulus have a large quartic coupling, = κ d4θχ†χZ†Z (1) L − Z with κ = O(100), the amplitude of the modulus oscillations becomes exponentially sup- pressed, where we adopt the Planck units, M = 1. Here χ denotes a chiral superfield p which dominates the energy density of the Universe when Z starts to oscillate. In prin- ciple one can introduce such an enhanced operator by hand, however the origin of such large coupling remains a puzzle. In this paper we argue that the operator (1) may result from a strong dynamics at the Planck scale. We will show that, using the naive dimensional analysis (NDA) [16], the operator (1) with a desired magnitude arises in the effective theory below the dynamical scale, if both χ and Z are involved in the strong dynamics. Here the strong dynamics should not be confused with any other interactions such as the usual QCD interactions or those in the dynamical SUSY breaking, which become strong at scales much lower than the Planck scale. Throughout this paper we do not specify the strong dynamics which is supposed to take place at a scale close to the Planck scale. Such dynamics may originate from the UV theory such as string theory [17]. We will discuss the implications of such strong dynamics for the inflation models and find that a certain class of multi-field inflation model is required for the mechanism to work, sinceotherwiseitwouldmaketheη problemmuchworsethanusual. Inthefollowing sectionswemainlyfocusonthePolonyiproblemasanexplicit example, however, ourmain result can be straightforwardly applied to the generic moduli problem. The rest of the paper is organized as follows. In Sec. 2, we describe a modified version of the Polonyi model with a strong dynamics at the Planck scale and show that the Polonyi problem can be solved in the model. We shall see in Sec. 3 that the operator (1) as well as many other ones are in general present, which restrict possible inflation models to a certain class of multi-field inflation models in order not to make the η problem worse. The last section is devoted to discussion and conclusions. their comment on this issue. 3 2 A Polonyi model with strong dynamics Inthis section we will show that strong dynamics at thePlanck scale cansolve the Polonyi problem. First we review the original Polonyi model and its cosmological problem, and explain the Linde’s proposal for a solution to the Polonyi problem. We will then see that an enhancement of the operator (1) is naturally realized if both χ and Z are strongly coupled at the Planck scale. Here and in what follows χ denotes a field that dominates the energy density of the Universe when the Z starts to oscillate after inflation. We will discuss whether the χ can represent the slow-rolling inflaton in Sec. 3. 2.1 The Polonyi problem and the Linde’s solution We introduce a pseudomodulus Z which is a singlet under any symmetries and therefore has no special point in the field space. This property is required to give a soft mass to the SSM gauginos from the following operator, d2θZW W , (2) α α Z where W is a chiral superfield for the SM gauge multiplets. The only scale associated α with Z is considered to be the Planck scale. These properties lead to the moduli (or Polonyi) problem as we shall see below. The Polonyi model is given by K = a Z +a Z† + Z 2 + (3) 10 01 | | ··· W = Ξ(1+c Z + ), (4) 1 ··· whereΞisaspectatorfieldwithanon-vanishingvacuumexpectationvalue(VEV) Ξ = µ h i and carries an R charge 2, a (= a∗ ) and c are numerical coefficients of order unity, and 10 01 1 the kinetic term of Z is set to be canonically normalized. Since Z has an R-charge 0, the Ka¨hler potential is considered to be a generic function of Z and Z†. The dots represent higher order terms of Z suppressed by the Planck scale, which are not relevant as long as we consider Z < 1. Here and in what follows we set the origin of Z to be the potential | | minimum for simplicity. 4 The requirement of the vanishing cosmological constant relates µ and c to the grav- 1 itino mass, µ = m , c µ = √3m , (5) 3/2 1 3/2 up to a phase factor. The F-term of Z is given by F = √3m . For a generic Ka¨hler Z 3/2 − potential squarks, sleptons and Higgs bosons acquire the SUSY-breaking soft masses of O(m ). The SSM gauginos acquire a mass of the same order from the interaction (2). 3/2 Examining the Ka¨hler and super-potentials (3) and(4), one can see that the mass of Z is of order the gravitino mass. In the early Universe, however, the effective potential of Z is affected by the Planck suppressed interactions with the inflaton sector. Since there is no special point in the field space of Z, the potential minimum during inflation is generically deviated from the origin. When H m , the Z starts to oscillate about the origin with 3/2 ∼ an amplitude of order the Planck scale, and soon dominates the energy density of the Universe after reheating.2 The couplings of Z to the visible sector are suppressed by the Planck scale, and the decay rate is roughly estimated by c Γ m3, (6) Z ≃ 4π Z where c is of order unity. The decay temperature is given by π2g −41 ∗ T = Γ Z Z 90 (cid:18) (cid:19) p m 3/2 0.002 MeVc1/2 3/2 , (7) ≃ 100GeV (cid:16) (cid:17) where g counts the relativistic degrees of freedom, and m = m was used. The ∗ Z 3/2 successful BBN requires T & 5MeV [18]. Thus, the onset of a radiation-dominated Z Universe is too late to be consistent with observations, for the gravitino mass of order the weak scale as in the gravity mediation. This is the notorious Polonyi problem. It was pointed out by Linde [15] that if the Z has an enhanced coupling to χ as in (1), the Z follows a time-dependent minimum and the effective amplitude of oscillations is exponentially suppressed. The enhanced quartic coupling (1) generates a mass of Z as m = CH, (8) Z 2 Around Z 1, there will be SUSY vacua in general. Here we assume that Z settles down to the | | ∼ SUSY breaking minimum at the origin. 5 where C √κ. The amplitude is suppressed by the following factor [15] ∼ √3π 3 πC C2 exp , (9) S ≃ 2 − 3 (cid:18) (cid:19) where we have assumed an inflaton-matter domination at the onset of oscillations of Z.3 The abundance of Z is estimated as 1 T Y R ( Z )2, (10) Z 0 ∼ 8m S Z where T denotes the reheating temperature and Z 1 is the oscillation amplitude R 0 ∼ in the absence of the enhanced coupling. It depends on the mass of Z how much its oscillation amplitude should be suppressed to be consistent with BBN. For instance, the BBN constraint reads, Y . O(10−16), for m = 100GeV [19]. Thus the suppression Z Z factor needs to satisfy 1 m 1 T −12 Y(BBN) 2 . 3 10−10Z−1 3/2 2 R Z , (11) S × 0 100GeV 106GeV 10−16 ! (cid:18) (cid:19) (cid:16) (cid:17) (BBN) where Y represents the BBN constraint on the abundance of Z. In order to achieve Z such suppression with the use of the above mechanism, we need C 25, or equivalently, ∼ κ 600. For a heavier gravitino mass, the BBN bound is relaxed, and the required value ∼ of C becomes slightly smaller accordingly. We comment on the validity of this mechanism. The essence of the suppression is the adiabatic invariance. Namely, in the above Polonyi model, the number density of Z becomes a good adiabatic invariant in the limit of C 1, and that is why the coherent ≫ oscillations of Z is suppressed. The exponential factor in (9) reflects a well-known fact that the variation of the adiabatic invariant is exponentially suppressed. Because of this, there is a limitation to the potential of Z where the mechanism applies. In the event that the effective potential of Z is extremely flat, the minimum of the effective potential may change rapidly, and as a result the coherent oscillations of Z are induced even in the presence of the enhanced coupling [20]. This is expected to be the case in a certain class of dynamical SUSY breaking models such as [21], and the Polonyi problem still 3 This condition is necessary since too many gravitinos are produced if the reheating is already completed when H m3/2. ∼ 6 persists [22]. Thus, the suppression mechanism applies only to the case in which the low-energy potential of Z is not much flatter than the quadratic potential in the entire region where Z moves, as in the Polonyi model. The requisite for the Linde’s solution is a quartic coupling with a large coefficient. As long as we work in the low-energy effective theory with a Planck-scale cut-off, such a large coefficient may look a puzzle. However, if there is a strong dynamics at the Planck scale, the large coefficient may arise from the strong interaction. In fact, using the NDA, we can roughly estimate the size of the coupling. In the next subsection we will see the coefficient obtained by NDA can meet the requirement. 2.2 Strong dynamics solves the Polonyi problem Now we show that the Polonyi problem can be solved if both Z andχ are strongly coupled at a scale Λ close to the Planck scale. We will see that, using NDA, a coupling (1) with a desired magnitude arises from the strong dynamics. Let us first describe a modified version of the Polonyi model. We assume that the Polonyi field Z carries a vanishing R charge as usual, but it is assumed strongly coupled at a scale Λ O(1). In addition to the Z we introduce spectator fields carrying an R ≃ charge 2, Ξ and Ξ′, which have a non-vanishing VEV, Ξ = µ and Ξ′ = µ′. Using NDA, h i h i we have a Ka¨hler potential and a superpotential Λ2 4πZ i 4πZ† j K a ≈ 16π2 ij Λ Λ i,j=0 (cid:18) (cid:19) (cid:18) (cid:19) X Λ = a Z +a Z† + a Z2 +a Z†2 + Z 2 + (12) 10 01 20 02 4π | | ··· (cid:0) 4π(cid:1)Z (cid:0)i (cid:1) W Ξ′ +Ξ c i ≈ · Λ i=0 (cid:18) (cid:19) X 2 3 4πZ 4πZ 4πZ = (Ξ′ +Ξ)+Ξ c +c +c + (13) 1 2 3 Λ Λ Λ ··· ! (cid:18) (cid:19) (cid:18) (cid:19) where Ξ′ is assumed to be decoupled from Z, a (= a∗ ) and c are numerical coefficients ij ji i of order unity, and we have normalized a = 1 and c = 1. We have included a factor of 11 0 1/16π2, which usually appears in NDA, in the definition of Ξ and Ξ′. For the above NDA 7 to be valid, the value of the Polonyi field is constrained as Λ Z . . (14) | | 4π There is in general a SUSY breaking meta-stable vacuum at Z < Λ/4π, while there are | | SUSY preserving vacua at Z Λ/4π. In supergravity, the scalar potential is considered | | ∼ to increase exponentially for Z > Λ/4π because of an exponential pre-factor, eK, in | | the scalar potential. As we shall see later, this constraint on the variation of Z will be important when we discuss the implications for inflation models. We will set the SUSY breaking minimum to be at the origin, which places a certain relation among the coefficients a and c , but the following argument is not affected. The requirement of the ij i vanishing cosmological constant is satisfied if 4πc µ µ+µ′ = m , 1 = √3m . (15) 3/2 3/2 Λ While the F-term of Z is given by F = √3m as before, the soft masses of SUSY Z 3/2 − particles are modified. Assuming that the SSM particles are not involved in the strong dynamics at the Planck scale, the scalars acquire a soft SUSY breaking mass of order the gravitino mass for a generic Ka¨hler potential. On the other hand, the gaugino mass arises from the following operator instead of (2), Z d2θ W W . (16) α α 4πΛ Z The gaugino mass is of order O(m /4πΛ), an order of magnitude lighter than the scalar 3/2 mass. The gauginomasses aretypically of O(100)GeV,while the scalar masses areseveral TeV. Thus, the SUSY mass spectrum is that in the focus point region [23], which has phenomenological virtues. It is interesting that the focus point SUSY naturally appears from the strong dynamics at the Planck scale. The mass of Z about the origin mainly arises from the quartic coupling in the Ka¨hler potential, 2 4π K a Z 4, (17) 22 ⊃ Λ | | (cid:18) (cid:19) leading to 4π m m . (18) Z 3/2 ∼ Λ 8 Here we have assumed that the sign of a is negative for the stability of the SUSY 22 breaking vacuum. If the χ is not involved in the strong dynamics, the coefficient κ of (1) is expected to be of order unity. Then, the Polonyi field Z is generically away from the origin during inflation, and starts to oscillate after inflation when the Hubble parameter becomes com- parable to its mass. Since it has a large initial amplitude Z Λ/4π and it decays only 0 ∼ through interactions suppressed by Λ, the Polonyi problem still persists. The decay rate of Z to the SSM particles is modified from (6) c m3 cm3 Γ(Z SSM particles) Z 3/2, (19) → ∼ 4π(4πΛ)2 ∼ Λ5 where (18) is used in the last equality. One may expect that the decay temperature becomes higher than 5MeV, since the mass of Z is larger than before. Actually, however, the main decay mode of Z is to a pair of gravitinos through (17), and the decay rate is given by [8] 1 m5 Γ(Z 2ψ ) Z (20) → 3/2 ≃ 96πm2 3/2 which is several orders of magnitudes larger than (19). Thus, the Universe will be dom- inated by the gravitinos produced by decay of Z. The gravitino of mass O(1)TeV ≃ decays during BBN and it becomes inconsistent with observations. In order to avoid the BBN constraint, the suppression factor for the oscillation amplitude Z must satisfy 0 S 1 m 1 T −12 Λ −32 Y(BBN) 2 . 3 10−8 3/2 2 R Z . (21) S × 1 TeV 106GeV M 10−16 (cid:18) (cid:19) (cid:18) p(cid:19) ! (cid:16) (cid:17) Thus the Polonyi problem is not solved if only Z is strongly coupled at the Planck scale. Now let us assume that χ is also involved in the strong dynamics. Using NDA, we expect that there is a quartic coupling (1) with 16π2 κ . (22) ∼ Λ2 Using (9), the suppression of O(10−8) is achieved for C 22 or κ 500. If we take ∼ ∼ Λ 0.6, this condition is satisfied. Thus, the strong dynamics at the Planck scale indeed ∼ solvesthePolonyiproblem. Itisstraightforwardtocheckthatonearrivesattheessentially same conclusion for the generic moduli problem, except for the prediction of the SUSY mass spectrum. 9 3 Implications for inflation models We have seen that the strong dynamics can solve the moduli problem. Here we discuss its implications for the inflation models. The crucial assumption was that both χ and Z are strongly coupled at the Planck scale. For the moment we assume that χ and Z are the only particles in the low energy effective theory, which are involved in the strong dynamics. The NDA provides us with a prescription to estimate the size of interactions allowed by the symmetry. While Z must be a singlet under any symmetries to generate gaugino masses through (16), we assume a non-trivial charge on χ to forbid unwanted couplings like W = χZ. This is because such an operator induces a large SUSY mass for Z and Z should no longer be treated as a modulus field. In general, however, we expect that there are unsuppressed interactions involving χ 2, Z and Z† in the Ka¨hler potential. | | Since χ gives a main contribution to the energy density of the Universe at the onset of the modulus oscillations, it is natural to expect that χ is a part of the inflation sector. Let us first consider a single-field inflation model in which χ is to be identified with the inflaton. One can easily see, however, that the χ cannot be responsible for the slow-rolling inflaton, because it would have a too large mass from the following operator 4, 16π2 d4θ χ 4. (23) Λ2 | | Z The mass of χ is then m O(10)H , (24) χ inf ≃ where H represents the Hubble parameter during inflation. Thus, the η-problem be- inf comes worse than usual, and the slow-roll inflation does not occur unless the above cou- pling (23) is suppressed somehow or there is an accidental cancellation between (23) and other contributions. So we conclude that our mechanism does not fit with the single-field inflation model in which χ is identified with the inflaton. In principle, there could be another weakly-coupled scalar field responsible for the slow-rolling inflaton. In this case, the η-problem does not necessarily become worse. So we are led to consider an inflation model in which there are multiple fields, and one (or 4 One exception is the case in which χ has a shift symmetry [24]. 10