1. D-term chaotic inflation in supergravity Masahide Yamaguchi (Aoyama Gakuin University) arXiv:0706.2676 Collaboration with Kenji Kadota 21st Aug ’07 @COSMO07

2. Outline Introduction Inflation models How difficult to realize chaotic inflation in supergravity D-term chaotic inflation model in supergravity Model Dynamics and density perturbations Summary and Discussion

3. General prediction of inflation Global isotropy and homogeneity Spatially flat universe Almost scale invariant, adiabatic, and Gaussian density fluctuations TT correlation WMAP results strongly support inflation. TE correlation Causal seed models Inflation models

4. Old Inflation New Inflation Hybrid inflation Chaotic Inflation … Guth, Sato ’81 Linde, Albrecht & Steinhardt ’82 Linde ’91 Linde ’83 Inflation does not end. Fine tuning of initial conditions. No fine tuning of initial conditions. Inflation models proposed so far

5. Inflation model building Slow-roll Nearly flat potential Keep flat against radiative corrections Supersymmetry (Supergravity)

6. Scalar potential in supergravity K : Kähler potentialW : Superpotential Due to the factor e^K, it has been considered almost impossible to realize chaotic inflation in supergravity. g a : gauge coupling

7. F-term chaotic inflation in supergravity Nambu-Goldstone-like shift symmetry : (C: a dimensionless real parameter) Chaotic inflation can take place naturally. (Kawasaki, MY, Yanagida) e.x. (m represents the breaking of the shift symmetry) However, it may be difficult to associate it with the low energy effective theory of particle physics such as GUT.

8. D-term chaotic inflation in supergravity Q QRQR Φ1Φ1 Φ2Φ2 Φ3Φ3 Φ4Φ4 +1+2- 2- 1 00+2+1 We introduce four superfields charged under U(1) gauge symmetry and U(1) R symmetry, (a, b, c are real and positive constants, for simplicity) We take the canonical Kahler potential, the minimal gauge kinetic function, and the vanishing FI term for simplicity. Model : The general (renormalizable) superpotential :

9. Scalar potential The global minimum of the potential is given by

10. Instead, the almost F-flat condition is first realized due to the exponential factor e^K of the F-term, if, and, for example. Realization of chaotic inflation The potential is mostly dominated by the D-term and chaotic inflation can take place. plays the role of an inflaton. However, when the universe starts around the Planck scale, the global minimum of the potential is not necessarily realized. (Note that the dynamics is essentially the same even if we interchange by.) Next, we investigate the dynamics in detail.

11. Dynamics of the homogeneous mode I The actual inflation trajectory is slightly deviated from the exact F-flat direction due to the presence of the D-term and given by (Note that.) In fact, the mass matrix of along this trajectory : Both of these masses are much larger than

12. Dynamics of the homogeneous mode II The dynamics is described by the following effective potential, Here, we redefine the fields The eigenvalues of the mass matrix of The inflaton in this chaotic inflation corresponds to this effectively massless mode, which is well parametrized by the field phi_1. (We name this trajectory M)

13. Dynamics of the homogeneous mode III The dynamics of the homogeneous mode is completely described by the following reduced potential, The potential is dominated by the D-term during inflation, In fact, the equation of motion for the homogeneous mode of the inflaton phi_1 along the rolling direction (M) is approximated as

14. Primordial density fluctuations I Equation of motion for the field fluctuations in the longitudinal gauge : Only one light field Only adiabatic mode Adiabatic condition :

15. Primordial density fluctuations II By taking into account the adiabatic condition, Note that this value is exactly the effective mass squared along the rolling direction M. The equation of motion for the perturbation becomes Thus, the density fluctuations are completely determined by the reduced potential. On the other hand, by use of the adiabatic condition, the gravitational potential in the long wave limit is given by

16. Summary and Discussion In this talk, we have presented a D-term chaotic inflation model in supergravity. The gauge coupling g should be g = O(10^{-6}) in order to explain the primordial density fluctuations. Even though we presented a toy model of the quartic potential chaotic inflation, the leading order polynomial can be different by choosing the non-minimal gauge kinetic function. As for the reheating, it may require the breaking of the gauge symmetry after inflation because the inflaton cannot directly decay into the standard particles due to the charge conservation. Such a breaking can occur, for instance, by the introduction of the FI term or a Higgs-like field.