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Provided by CERN Document Server Micheal Dine(1) and Antonio Riotto(2) (1)Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064 (2)NASA/Fermilab Astrophysics Center, Fermilab National Accelerator Laboratory, Batavia, Illinois 60510-0500 (May 15, 1997) Inflation, as currently understood, requires the presence of fields with very flat potentials. Su- persymmetric models in which supersymmetry breaking is communicated by supergravity naturally yield such fields, but the scales are typically not suitable for obtaining both sufficient inflation and a suitable fluctuation spectrum. In the context of recent ideas about gauge mediation, there are new candidates for the inflaton. We present a simple model for slow-rollover inflation where the vacuum energy driving inflation is related to the same F -term responsible for the spectrum of super- symmetric particles in gauge mediated supersymmetry breaking models. The inflaton is identified with field responsible for the generation of the µ-term. This opens the possibility of getting some knowledge about the low-energy supersymmetric theory from measurements of the cosmic microwave background radiation. Gravitinos do not pose a cosmological problem, while the moduli problem is ameliorated.

PACS: 98.80.Cq FERMILAB–Pub–97/138-A SCIPP 97/14 hep-ph/yymmnn

The existence of an inflationary stage during the evolu- in this latter class of models, those with gauge-mediated tion of the early Universe is usually invoked to solve the supersymmetry breaking (GMSB) [8]. flatness and the horizon problems of the standard big In this paper we would like to follow the minimal prin- bang cosmology [1]. During inflation the energy density ciple that the inflaton field should not be an extra de- is dominated by the vacuum energy and comoving scales gree of freedom inserted ad hoc in some (supersymmet- suffer an rapid growth. As a result, any harmful topo- ric) theory of particle physics just to drive inflation. We logical defects left as remnants after some Grand Unified will argue that this is indeed possible in the framework of phase transition, such as monopoles, are diluted. Quan- GMSB models and show that a successful inflation model tum fluctuations of the inflaton field imprint a nearly may be constructed. The energy density driving infla- scale invariant spectrum of fluctuations on the back- tion may correspond to the F -term responsible for the ground space time metric. These fluctuations may be re- spectrum of the superpartners in the low-energy effective sponsible for the generation of structure formation. How- theory. Moreover, we will be able to identify the inflaton ever, the level of density and temperature fluctuations field with the same scalar responsible for the generation observed in the present Universe, δρ/ρ ∼ 10−5,require of the µ-term present in the effective superpotential of the inflaton potential to be extremely flat. For instance, the minimal supersymmetric standard model (MSSM), in the chaotic inflationary scenario [2] where the inflaton δW ∼ µHU HD. This raises the possibility of someday potential is V = λφ4 and the condensate sits initially connecting a theory which could be tested at accelera- at scales of order of the Planck scale, the dimensionless tors with measurements of the temperature anisotropy self-coupling λ must be of order of 10−13 to be consistent in the cosmic microwave background and related mea- with observations. The inflaton field must be coupled surements of the two-point correlation function! We will to other fields in order to ensure the conversion of the see that in such a picture, one can predict the fundamen- vacuum energy into radiation at the end of inflation, but tal scale of supersymmetry breaking (which will turn out these couplings must be very small otherwise loop cor- to be about 108 GeV). As it stands, our model is not rections to the inflaton potential spoil its flatness. Such completely satisfactory. As in virtually all models of in- flatness is difficult to understand in generic field theories, flation based on supersymmetry, a fine tuning is required but is quite common in supersymmetric theories [3,4]. to obtain a sufficiently small curvature for the potential Supersymmetry, then, might be expected to play a fun- near the origin. This tuning, at the level of a part in 102, damental role during inflation [5], but what this role must ultimately be given some deeper explanation. might be will almost certainly depend on the way in In general, the effective low-energy theory that emerges which supersymmetry is broken. It is usually assumed from GMSB models contains soft SUSY-breaking mass that supersymmetry is broken in a hidden sector and terms for the scalar superpartners which carry informa- then transmitted to observable fields either through grav- tion about the scale and nature of the hidden-sector itational [6] or gauge interactions [7]. The cosmology of theory. Typical soft breaking terms for sfermions re- this latter class of models has only been partially ex- sulting from GMSB models where SUSY is broken at plored. In this paper we will look for inflaton candidates the scale Λ. The magnitude of these terms is given

1 α 2 √ by m ∼ 4π Λ. This has the interesting consequence hSi /Mp ∼ βFX. Notice that we have added the con- 2 that flavour-changing-neutral-current (FCNC) processes stant ∼ FX in such a way that the the cosmological con- are naturallye suppressed in agreement with experimental stant in the true vacuum is zero, V (hSi) = 0. As we will bounds. In the minimal GMSB models, a singlet field X see, it is necessary that α be small, of order 10−2.On acquires a vacuum expactation value (VEV) for both its the other hand, in the framework of supergravity, it is scalar and auxiliary components, hXi and hFX i respec- well-known that there are contributions to the mass of tively and the spectrum for the messangers is rendered particles during inflation of order H from the terms S†S non-supersymmetric. Integrating out the messenger sec- which must be present in the Kahler potential. These tor gives rise to gaugino masses at one loop and scalar can be cancelled by terms of the form S†SX†X.Thisis masses at two loops, with Λ = FX /X. As we shall see, the fine tuning we have alluded to earlier. It is a generic the energy density V0 driving inflation may be identified feature of models based on supersymmetry [10]. One with the F -term responsible for the spectrum of super- positive feature of these models is that the curvature at 2 symmetric particles, V0 ∼ FX . This is one of the key the minimum of the potential is automatically far larger results of our paper. than the curvature near the origin. This is important for Before launching ourselves into the inflationary aspects reheating. of this paper, let us discuss the issue of the generation Around S = 0 we may simplify the potential as of the µ-term in the context of GMSB models. Leurer et al., although in a different context, have suggested a µ- m2 λ V (S) ' V − S2 − S4, (3) term generation which is relevant for us [9]. In addition 0 2 4 to the usual MSSM fields, there is another singlet, S. 2 2 2 2 FX FX As a consequence of discrete symmetries, the coupling where V0 ∼ β F , m ∼ α 2 and λ ∼ 4β 2 .Ifthe X Mp Mp SHUHD is forbidden in the superpotential. There are, S-field starts sufficiently close to the origin the system however, various higher dimension couplings which can may inflate. S rolls very slowly toward S = hSi during drive hSi. In particular, consider terms in the effective inflation. Its potential energy dominates the energy den- Lagrangian of the form sity of the Universe driving a nearly constant expansion rate 1 1 1 d2θ XS2+p + Sm+3 + Sn+1H H p m n U D 8πV(S) 8πV Mp Mp Mp 2 0 Z   H (S)= 2 ' 2. (4) 1 3Mp 3Mp + d4θX†XS†S, (1) M 2 p Z In the slow-roll approximation, we may neglect S¨ in the ˙ 0 where M 1.2 1019 GeV is the Planck mass. This equation of motion so that S '−V /3H. This approx- p ' × 0 00 structure can be enforced by discrete symmetries. The imation is consistent as long as V ,V  V for S ∼ 0. first and the fourth terms can contribute to the effective Notice that the quadratic and the quartic terms√ in the negative curvature terms to the S potential. For exam- expression (4) become comparable for S∗ ∼ FX .Since ple, if p = m =2,andn= 1, then the µ-term turns this value is much smaller than hSi and all the dynam- √ ics giving rise to density perturbations takes place in the out to be of the order of FX (times powers of coupling constants). vicinity of the origin, we prefer not to neglect the quartic The field S, although very weakly coupled to ordinary term in the rest of our analysis. matter, may play a significant role in cosmology. We will During the slow-roll phase, when S is near to the ori- devote the rest of the paper to explore the cosmological gin, the cosmic scale factor may grow by N(S) e-foldings: implications of such a field and to show that a succes- Se 8π V0 2πV0 full inflationary scenario may be constructed out of the N ' ' M 2 −V 0(S) m2M2 potential for the field S. p ZSN p Let us suppose that the field X has acquired a vacuum SN × 1 − 4ln , (5) expactation value (VEV) for both its scalar and auxiliary S components and restrict ourselves to the case p = m =2,   ∗  and n = 1. The potential along the real component of where Se denotes the value of the field when infla- the field S reads tion ends. Successfull inflation requires N ' 60. 2 We can express now SN in terms of S∗: SN ' 2 2 FX 2 FX 4 2 2 V (S) ∼ β F − α S − β S S∗exp 1/4 − N(m Mp /8πV0) . This expression is valid X M 2 M 2 p p as long as SN

2 2 00 η =(Mp/8π)(V /V ) and the slow-roll approximation is that, in order to achieve the 60 or so e-foldings of in- valid as long as , η  1. In the model under considera- flation required, the initial value of the scalar field must 5 √ 8 tion, however, the slow-roll limit breaks down well before be less than about 2 × 10 α/β FX /10 GeV GeV. the end of inflation. Using the slow-roll solution for  As a result, only regions where the initial value of the p
results in understimating the value Se at which inflation field is small enough will undergo inflation. These re- ends [11], the correct value being Se of the order of hSi. gions have grown exponentially in size and they should Even though the correct value of η becomes large before occupy most of the physical volume of the Universe. We the end of inflation, the slow-roll approximation is valid notice that the small value of the field is not spoiled by for S hSi, which is the region where observable pa- quantum fluctuations which are of the order of H/2π ∼ −4 √ 8 2 rameters must be computed [11]. In this region one may 4 × 10 β FX /10 GeV GeV. Thermal fluctuations compute the power spectrum, which is the Fourier trans- √ might spoil such a localization since S2 1/2 T F . form of the two-point density autocorrelation function. It
h iT ∼ ∼ X However, the inflaton field is so weakly coupled (its cou- has the primordial form P (k) ∝ kn.wherekis the ampli- plings are all suppressed by powers of M that it is not tude of the Fourier wavevector and n denotes the spectral p in thermal contact with the rest of the Universe [13,14]. index. Fluctuations arise due to quantum fluctuations in After inflation ends, the S-field starts oscillating the scalar field S. The measurement of the quadrupole around the minimum of its potential and the vacuum anisotropy in the cosmic microwave background radia- energy that drives inflation is converted into coherent tion detected by COBE allows us to fix the parameters scalar field oscillations. The Universe undergoes a pe- of the model: riod of matter domination. Reheating takes place when √ 3/2 7/2 S decays into light fields, which will eventually thermal- ∆T 32π V0 β FX − 1 +N|η| = e 4 . (6) 0 3 ' 3/2 ize and give rise to a thermal bath of radiation. The T 45 V (SN )Mp α Mp   r reheating temperature TR is determined by the decay ∆T −6 −2 width of the scalar oscillations ΓS, TR ∼ 0.1 ΓSMp Imposing T ' 6 × 10 and fixing β ∼ 1, α ∼ 10 √ 8 −1 [15]. At the minimum of its potential, the scalar field and FX ∼ 10 GeV gives |η|∼10 .Itiseasyto 3/2 p
2 00 FX verify that for such a value of |η| the condition SN Higgsinos S → ψHu ψHd the framework of GMSB models. However, the spectrum µ hSi 2 √ with a rate ΓS = g mS/4π with g ∼ M and we of the superparticles depends on the ratio Λ = FX /X. √ FX p The latter is fixed to be relatively small and in the range have taken into account that µ  FX . The resulting 3 −2 1/8 −1/4 (10 − 10 ) TeV. This may be easily accomplished if X reheating temperature is then TR ∼ 10 µF Mp ' √ X acquires a large vacuum expectation value via nonrenor- 2 3 8 1/4 10 µ/10 GeV FX/10 GeV MeV, which is √ 8 malizable operators. Notice that FX ∼ 10 GeV cor- large enough to preserve the classical cosmology begin- responds to a very low value of the Hubble parameter ning with the era
of nucleosynthesis.
It seems difficult, during inflation, H ∼ 1 MeV. Note also that it implies however, to push the reheating temperature above the that there is a small dimensionless number in the cou- electroweak scale. Thus, it appears that electroweak pling of S to the Higgs. This number is comparable to baryogenesis is not a viable option for the generation of the electron Yukawa coupling. the baryon asymmetry. On the other hand, the decays The COBE satellite measured fluctuations in the cos- of the inflaton themselves might be responsible for the mic microwave background with a spectral index n = baryon asymmetry [16]. The couplings by which the in- 1.2 ± 0.6(at2σlevel). In our model the spectral in- flaton decays may contain CP-violation and baryon num- dex n ' 1 − 2|η|∼0.8 is noticeably smaller than 1 and ber violation. In order to produce a baryon asymmetry, gives rise to a red spectrum. The amplitude of the grav- we must have baryon number violating operators in the itational waves produced by quantum fluctuations is far Lagrangian, such as δW ∼ S U¯D¯D¯. The presence of too small to be detected since the variation of the field Mp such operator is compatible with the stability of the pro- during inflation is much smaller than Mp [12].   Let us briefly address the issue of the initial condi- ton and the experimental absence of neutron-antineutron tion for the field S. We have assumed that underlying oscillations. We can estimate the baryon asymmetry pro- the model are discrete symmetries under which S trans- duced by the inflaton decay simply. We assume that the forms non-trivially. As a result, S = 0 is a special point, amount of baryon number produced per decay is ε. ε and it is natural that S may “sit” at this point initially. is the product of CP-violating phases δ times the ra- This despite the fact that it is very weakly coupled to tio of the baryon number violating decay rate over the −2 √ 2 2 ordinary matter, and might not be in thermal equilib- total decay rate ΓB/Γtot ∼ 10 µ/ FX (mS/hSi) . rium. Many models of slow-rollover inflation require a The number of massless particles produced per decay is
fine-tuning in the initial value for the field to be suc- ∼ mS/TR. Plugging in the expected values of the infla- √ 8 cessfull and the smaller is the scale of inflation the more ton mass and the reheating temperature for FX ∼ 10 severe is the fine-tuning [13]. From Eq. (7) we may infer GeV,wefindabaryontoentropyratioB∼10−8δ,which

3 is compatible with the observed value for δ ∼ 10−2. and NASA under grant NAG5–2788. MD thanks the Another remarkable effect of a late period inflation is Institute for Theoretical Physics at Stanford University that the gravitino problem is solved. As is well known, in for its hospitality, and Scott Thomas for discussions. GMSB scenarios the gravitino is the lightest supersym- metric particles with mass m3/2 ∼ FX /Mp (of order of 1 MeV in our case). If a stable gravitino is thermalized in the early Universe and not diluted by any mechanism, < its mass density may exceed the closure limit Ω3/2 ∼ 1. Since the number density of gravitinos is fixed once they are thermalized, the above argument sets a stringent up- [1] A. H. Guth, Phys. Rev. D 23, 347 (1981). For a recent < review of inflation and list of references, see A. D. Linde, per bound on the gravitino mass, m3/2 ∼ 2keV(with- out dilution) [17]. However, gravitinos are efficiently di- Particle Physics and Inflationary Cosmology (Harwood luted during the inflationary stage driven by the field S Academic, New York, 1990). and they are not produced in the subsequent stage of re- [2] A.D. Linde, Phys. Lett. B129 (1993) 177. heating. Indeed, light gravitinos (or, better to say, the [3] J. Ellis, D. Nanopoulos, K. Olive and K. Tamvakis, Phys. longitudinal components of them) may be regenerated Lett. B118 (1982) 335. [4] M. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B159 during reheating either by the decays of sparticles (or (1979) 429. particles in the messanger sector) or by scatterings pro- [5] For a review with extensive references, see D. H. Lyth, cesses. However, the first mechanism requires the reheat Report No. hep-ph/9609431. temperature to be at least of order of the typical sparticle [6] For a review, see H. P. Nilles, Phys. Rep. 110 (1984) 1; mass, m ∼ 100 GeV, and scattering processes regenerate H.E. Haber and G. L. Kane, Phys. Rep. 117 (1985) 75. > 2 the light gravitinos only if TR ∼ 10 m3/2/1MeV TeV [7] M. Dine, W. Fischler, and M. Srednicki, Nucl. Phys. [18]. Since in our scenario the reheat temperature turns B189 (1981) 575; S. Dimopoulos and S. Raby, Nucl. e 2
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