CORE An Inflaton Candidate in Gauge Mediated SupersymmetryMetadata, Breakingcitation and similar papers at core.ac.uk 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.