1. WMAP and smooth hybrid new inflation: Density fluctuations of an effectively single field inflation in a multi-field configuration MASAHIDE YAMAGUCHI (AOYAMA GAKUIN UNIVERSITY) The Joint Meeting of Pacific Region Particle Physics Communities 10/30/06 @ Cosmology Session

2. Contents Introduction WMAP results (particularly, running spectral index) Basics of inflation Inflation models with a running spectral index Smooth hybrid new inflation in supergravity How to calculate density fluctuations of an effectively single inflation in a muti-field configuration Parametric resonance between smooth hybrid and new inflation Discussion and conclusions

3. Introduction

4. Inflation Inflation can naturally solve the problems of the standard big bang cosmology. The horizon problem The flatness problem The origin of density fluctuations The monopole problem …

5. General prediction of inflation Global isotropy and homogeneity Spatially flat universe Almost scale invariant, adiabatic, and Gaussian density fluctuations Inflation causal seed model T=2.725K Cosmic Microwave Background CMB TT correlation TE correlation Causal seed models Inflation models Predictions of inflation are confirmed observationally by WMAP.

6. Inflation model building Next step is To identify which inflation model is realized. Chaotic ? Hybrid ? New ? ….. To realize inflation as a sensible particle physics model. GUT, SUSY, …

7. Inflation models proposed so far Chaotic inflation (Linde). Hybrid inflation (Linde) New inflation (Linde, Albrecht & Steinhardt) σ V[σ] １ σ V Ψ

8. Observational constraints (Dodelson, Kinney, Kolb) Though chaotic inflation with a quartic potential is disfavored, many inflation models are still consistent with observations. 0 0.5 1 0.8 0.9 1 1.1 Hybrid Inflation Chaotic Inflation New Inflation Spectral index n s Tensor-Scalar ratio rTrT

9. Running spectral index WMAP may give us a hint for an inflation model. (Peiris et al.) on larger scaleson smaller scales 1 year result It is a challenge to build such inflation models. at Nearly 2σ hint of running spectral index

10. No running ? Slosar et al. pointed out that the running feature is diminished if correct likelihood functions are used. Bridle et al. pointed out that the running feature has marginal evidence only if the first three WMAP multipoles (l =2, 3, 4), which have large cosmic variance, are included in the analysis. WMAP onlyWMAP-----SDSS n running r ＜ 0.81 at

11. Revival of running in WMAP 3 year results ? Best Fit Inflationary Parameters (WMAP data only) at

12. Robustness of running in WMAP 3 year results Even without tensor fluctuations, negative running is observed at almost or above 2σlevel. Running feature is robust even if low multipoles data are subtracted in the analysis. Though Ly-alpha data tend to diminish the running, systematic uncertainties are discussed. No tensor fluctuations Large-scale scalar perturbation becomes higher smaller running (Feng et al.) (Meiksin & White)

13. Running in WMAP 3 year results II If χ 2 improves by 2 or more, it is worth introducing a new parameter, according to Akaike’s information criteria (AIC). χ 2 improves by 3.6 (WMAP), or 2.4 (WMAP+SDSS) by introducing a running spectral index. Tegmark et al. Vanilla : Inflation-based ΛCDM model with a power law spectrum How much does the likelihood (χ 2 ) improve by introducing a new parameter? : the running spectral index Though further investigations are necessary, it is interesting and a challenge to build inflation models with such running feature.

14. Inflation models with a running spectral index

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

16. Scalar potential in supergravity K : Kähler potentialkinetic terms W : Superpotentialpotential terms V + D-terms

17. Inflation models with a running spectral index on larger scales on smaller scales It is a challenge to build such inflation models. Hybrid inflation in supergravity Smooth hybrid inflation in supergravity 0 0.5 1 0.8 0.9 1 1.1 Hybrid Inflation Chaotic Inflation New Inflation r n (Linde & Riotto, Kawasaki, MY, & Yokoyama) (MY & Yokoyama) no cosmic strings no one-loop correction

18. Smooth hybrid new inflation in supergravity

19. Smooth hybrid inflation in supergravity Superpotential : positive parameters, Kähler Potential S, : superfields Invariant under R-symmetry, gauge symmetry, and discrete symmetry Z_m with (Lazarides and Panagiotakopoulos) Some variants of smooth hybrid inflation is realized in the framework of GUT. (Kyae & Shafi)

20. breaks gauge symmetry during inflation so that no topological defect is formed at the end of inflation. Dynamics of smooth hybrid inflation : Inflaton, For m=2 & D-flat directions : Minimum of ψ : σ ψ

21. Effective potential of smooth hybrid inflation Symmetry breakingSupergravity effect for Setting for σ ψ (MY & Yokoyama) Can we look on this inflation as an effectively single field inflation with the potential to calculate primordial density fluctuations ?

22. Setting During inflation ( :the inflaton), the heavy field χ is stuck to its minimum of the potential within a few hubble time. is light, Is it justfied to calculate the dynamics and the generated density fluctuations of the inflation by use of the single-field potential of, is heavy,

23. rolls slowly. Dynamics of the homogeneous mode Time dependence of is small. χ is quickly stuck to. The dynamics of the inflaton can be described by the single-field potential.

24. Basic equations of primordial density fluctuations The perturbed metric in the longitudinal gauge : : the gravitational potential Equation of motion for the gravitaional potential : Equation of motion for the field fluctuations :

25. Adiabatic density fluctuations 1 Only one light field Only adiabatic mode Adiabatic condition : slow-rolls. is still light.

26. Adiabatic density fluctuations 2 Adiabatic condition : In the longwave limit (k → 0) : The equations (A,B1,C1) determine density fluctuations.

27. Equation of motion for the field fluctuation : Calculation with the single-field potential Equation of motion for the gravitaional potential : In the longwave limit (k → 0) : The equations (B2,C2) completely coincide with (B1,C1). It is justfied to calculate the generated density fluctuations of the inflation by use of the single-field potential.

28. Physical meaning of adiabatic condition During inflation, the heavy field χ stays at its minimum. The adiabatic condition Because Therefore, The perturbed field also stays at the minimum of χ for adiabatic fluctuations.

29. e-fold and density fluctuations Density fluctuations at Number of e-fold ⇔ The desired spectrum can be obtained but another inflation is needed in this case. For example, we try to fit the WMAP 3 year data.

30. Inflation models proposed so far Chaotic inflation (Linde) No fine tuning of initial conditions. Difficult to implement in supergravity.  shift symmetry Hybrid inflation (Linde) Fine tuning of initial conditions. Easy to implement in supergravity and GUT. Too high reheating temperature.  gravitino problem ? New inflation (Linde, Albrecht &Steinhardt) Fine tuning of initial conditions. Low reheating temperature.  No gravitino problem σ V[σ] １ σ V Ψ Each inflation model has its own pros and cons.

31. Gravitino problems (Kawasaki et al. ’04)(Moroi et al. ’93) Stable gravitino (Gauge mediated SUSY breaking) Unstable gravitino (Gravity or Anomaly mediated) Allowed region

32. New inflation We select new inflation as another inflation because New inflation should last about 40 e-folds to make the universe sufficiently large. (Linde, Albrecht & Steinhardt) The reheating temperature is low enough to avoid the gravitino problem. An initial value is set dynamically during smooth hybrid inflation in our model.

33. New inflation model in supergravity Superpotential g, v : positive parameters, Kähler Potential Φ : a superfield Effective potential of the inflaton (Izawa & Yanagida) U(1) R-symmetry is broken into a discrete R-symmetry at a scale v.

34. Initial value for new inflation Generally speaking, new inflation has an initial value problem. But, in our model, an initial value is set dynamically during smooth hybrid inflation. ( Izawa, Kawasaki, Yanagida ) Smooth hybrid new inflation in supergravity After smooth hybrid inflation, inflatons start oscillations, which leads to further suppression by the factor

35. Numerical calculation Smooth Hybrid Inflation Field Oscillation New Inflation Field Oscillation Reheating Numerical analysis is desired σ ψ Smooth hybrid inflationNew inflation

36. Equations for linear perturbations Initial conditions for fluctuations are set during smooth hybrid inflation. We assign the vacuum solution to in the short wavelength regime as usual. is used for accuracy check. comoving curvature perturbation

37. The homogeneous part represents dissipation rate of each scalar field. Reheating with m = 2 & n = 4

38. Numerical Results at Running smooth hybrid inflation new inflation

39. After smooth hybrid inflation, ψ and σ oscillate around their respective minima. to lowest order. These oscillating terms induce parametric amplification of and. Anomalous peak due to parametric resonances ~

40. We find no resonant amplification here. These terms induce forced oscillation. Long period oscillation with a frequency. Forced oscillations As a result, is also enhanced due to the last two terms.. For, σ and oscillate with the frequency while ψ and oscillate with the frequency Due to the slight difference between the two frequencies, also acquires a component with a long period oscillation.

41. Large k cutoff is due to the resonant amplification condition. Longwave modes which were outside the horizon at the onset of the oscillation are irrelevant. Height of the peak is sensitive to the dissipation rate of σ and ψ during oscillation. Understandings of the peak structure Long period oscillation comes from the oscillation of

42. In this particular case, fluctuation peaked at, corresponding to whose abundance is constrained as in order not to overclose the universe. : initial mass fraction of PBHs : peak amplitude of fluctuations PBHs formation In this case, the PBH abundance is sufficiently suppressed. too much PBHs !!

43. Conclusions and Discussion Smooth hybrid inflation in supergravity can reproduce the running spectrum of density fluctuations suggested by WMAP. But, another inflation is necessary to make the universe enough large and hence we take new inflation as a second inflation, which naturally gives low reheating temperature. During an oscillatory phase between double inflation, small-scale fluctuations are amplified due to parametric resonances, which are stretched to the cosmological scales by the second inflation. We have shown that adiabatic density fluctuations of such an inflation can be completely reproduced by looking on it as an effectively single-field model with the potential obtained by inserting the minima of the heavy fields to the original potential.