Constraints on Inflation Models from Supersymmetry Breakings

Effects of soft SUSY-breaking terms on inflation potentials are discussed. There exist generic constraints that must be satisfied in order not for the inflaton potential to loose its flatness. We examine explicitly the constraints in the case of a hybrid inflation model and find that the coupling constant λ between the inflaton and the “waterfall-direction” field is bounded as 2.0 × 10−6 ∼ λ. This is a highly non-trivial result. Indeed, if we adopt the severest constraint from avoiding the problem of the gravitinos produced non-thermally, λ∼ 7.4 × 10−6 is required, under a reasonable assumption on the reheating process. This means that the hybrid inflation model marginally has a viable parameter space. We also discuss analogous constraints on other inflation models. Supersymmetry (SUSY) is not only motivated from the low-energy particle physics, but also from the primordial inflation. SUSY naturally explains the existence of the inflaton scalar field, and moreover, the cancellation of radiative corrections in supersymmetric theories is preferable for maintaining the flatness of the inflaton potential. There have been constructed various models for inflation in the framework of supergravity such as chaotic [1], hybrid [2], new [3], topological [4], D-term [5] inflation models and so on. Each model, in general, has some free parameters even after the COBE-normalization condition δρ ρ = 1 5 √ 3π V (φNe) M P |V (φNe)| ≃ 1 5 √ 3π × 5.3× 10−4 (1) is imposed. Here, MP denotes the reduced Planck scale 2.4 × 10GeV, V the inflaton potential, V ′ the derivative of the potential with respect to the inflaton φ, and φNe the value of the inflaton field at the time the COBE-scale exited the horizon. Different values of model parameters lead to different reheating processes after those inflations. Successful inflation models must have viable parameter spaces whose resulting reheating processes satisfy all phenomenological constraints such as those from the gravitino problem. Whether a sufficient e-fold number is obtained for a given model parameter must be examined using the full supergravity potential; it must be noted that the inflation sector is not the only one which couples to supergravity. In particular, the hidden sector, which has SUSY-breaking expectation values, is also coupled to supergravity. Then, the inflaton potential receives soft SUSY-breaking contributions, as in the ordinary gravity mediation of the SUSY breaking to the standard-model sector. It is true that the gravity-mediated soft SUSY-breaking terms appear with 1/MP -suppression and are tiny. However, the flatness of the slow-roll-inflaton potential requires a tuning at the level of order of the Hubble parameter H = √ V/3/MP , which is also a Planck-suppressed quantity. Therefore, the perturbations to the inflaton potential by soft SUSY-breaking terms can be crucial for its flatness. This observation gives new constraints on the viable parameter space of successful inflation. In this letter, we show how the existence of the soft SUSY-breaking terms constrains the parameter spaces of inflation models. We apply the above-mentioned new generic constraints to a hybrid inflation model in supergravity, as an example, and derive a lower bound on the Yukawa coupling of the inflaton field φ and the inflaton mass mφ. We show that this new lower bound is highly significant if we take seriously the severest constraint from avoiding the gravitino problems. We also discuss analogous constraints on other inflation models. Ref.[6] pointed out that these SUSY breaking terms can have physical importance.