Features in the primordial power spectrum: a running spectral index and formation of primordial black holes Jun’ichi Yokoyama (RESCEU, The University of Tokyo) M. Kawasaki, T. Takayama, M. Yamaguchi & JY Physical Review D74 (2006) 043525

Inflation in the early Universe simple slow-roll Inflation causal seed model TE correlation Global Isotropy & Homogeneity Spatial flatness T=2.725K Cosmic Microwave Background CMB Generation mechanism of Density/Curvature/Tensor fluctuations Next issue is to find out which, if any, the correct mechanism of inflation is and identify the inflaton in particle-physics context.

Each inflation model has its own pros and cons. V[φ] φ R2 inflation (conformal tr.) Chaotic inflation can be realized naturally without fine-tuning of the initial condition. Supergravity ←shift symmetry. σ V[σ] Chaotic inflation φ V[φ] １ (Linde) (Starobinsky) １ Hybrid inflation fine-tuning of initial condition may be required. easy to implement in Supergravity. reheat temperature may be too high gravitino problem. σ V Ψ (Linde) New inflation fine-tuning of initial condition may be required. reheat temperature can be low enough gravitino problem. (Linde, Albrecht & Steinhardt)

Inflation models may be distinguished by observations. slow-roll parameters Observable quantities Amplitude of comoving curvature perturbation on scale r=2π/k. spectral index its scale dependence, “running” inflaton potential and its derivatives (< particle physics) tensor-scalar ratio

Inflation models may be distinguished by observations. rT Tensor-Scalar ratio 1 Chaotic Inflation Hybrid Inflation 0.5 New Inflation 0.8 0.9 1 1.1 Spectral index ns (Dodelson, Kinney, Kolb 97) While many models are still consistent with the data,...

The most surprising result of WMAP1 was,... Running spectral index WMAP1 n >1 on larger scales n <1 on smaller scales This is a challenge to the model building of inflation.

“n >1 on larger scales n <1 on smaller scales” can be realized in… 0.5 1 0.8 0.9 1 1.1 Hybrid Inflation Chaotic New Inflation rT n a class of hybrid inflation Linde-Riotto model in supergravity σ: inflaton for Hybrid Inflation smooth hybrid inflation (Lazarides and Panagiotakopoulos)

Large enough running and large enough e-folds are incompatible. A more general discussion can be found in Easther and Peiris (2006). (ex) Linde-Riotto model in supergravity σ: inflaton for Hybrid Inflation Running spectral index and the number of e-folds have the same parameter dependence: Large enough running is incompatible with large enough e-folds NH～50. The same problem applies to the smooth hybrid inflation as well. Another inflation is necessary after hybrid inflation.

Inflation with a running spectral index in supergravity. M. Kawasaki, Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D68:023508,2003 Chaotic hybrid new inflation in supergravity with a running spectral index. Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D68:123520,2003 more natural initial condition Smooth hybrid inflation in supergravity with a running spectral index and early star formation. Masahide Yamaguchi, Jun'ichi Yokoyama Phys.Rev.D70:023513,2004 natural initial condition no cosmic strings no one-loop correction

“Evolution” of the Observational Results… WMAP1 Papers 　　　(Spergel etal) 　　　(Peiris etal) WMAP only WMAP-----Lyα n 0.93±0.07 0.93±0.03 running r ＜0.71 WMAP only WMAP-----Lyα n 1.13±0.08 running r ＜1.28 ＜0.90 “Nearly 2 σ” hint of running spectral index

with a correct likelihood function Reanalysis of WMAP1 with a correct likelihood function (Slosar etal 2004) WMAP only WMAP-----SDSS n 1.02±0.07±0.15 0.97±0.06+0.16-0.11 running r ＜0.81 Running disappeared!?

Revival of running in WMAP3? ３年目　WMAPonly

running (no tensor) (with tensor) WMAP3 only WMAP3+CBI +VSA WMAP3+BOOM+ACBAR Add tensor fluctuations Large-scale scalar perturbation lowered Larger running

dataset running WMAP only + 2dF +SDSS +Boom+Acbar +CBI+VSA Models with no tensor modes also give negative running around or above 2σ with various datasets. dataset running WMAP only + 2dF +SDSS +Boom+Acbar +CBI+VSA +weak lensing +ALL

Vanilla= Inflation-based ΛCDM model with a power law spectrum How much does the likelihood (χ2 ) improve by introducing a new parameter? Vanilla= Inflation-based ΛCDM model with a power law spectrum (Tegmark et al 06) If χ2 improves by 2 or more, it is worth introducing a new parameter, according to Akaike’s information criteria (AIC). χ2 improves by 3.6 (WMAP), or 2.4 (WMAP+SDSS) by introducing a running spectral index.

Φ: New inflaton superfield revisit Smooth Hybrid New Inflation Model We use the Planckian unit . superpotential Kähler potential Smooth hybrid inflation sector S, : superfields energy scale of inflation cutoff scale : integer discrete symmetry Zm New inflation sector energy scale of new inflation (lower than μ) Φ: New inflaton superfield

Scalar potential in supergravity

Potential of smooth hybrid inflation For D-flat directions : : Inflaton, Minimum of is nonvanishing even when σ is large. breaks gauge symmetry so that no topological defect is formed at the end of inflation.

Dynamics of smooth hybrid inflation σ ψ two critical values: from equating two terms in the potential force from , which determines the end of inflation Setting with m =2, Supergravity effect Symmetry breaking effect

Density fluctuations and # of e-folds For example, if we take m = 2 and try to fit the WMAP1 data on a scale , we would find ⇔ Number of e-folds of smooth hybrid inflation The desired spectrum with a large running can be obtained, but another inflation is needed also in this case.

General m case Even if we take a larger m, the number of e-folds cannot be much larger than 10. So, another inflation is necessary. We choose new inflation as the second inflation. (Later we set m = 2 for simplicity)

New inflation model in supergravity energy scale of new inflation (lower than μ) (Izawa & Yanagida) Potential for the new inflaton Its initial condition is set by the interaction with the smooth hybrid inflaton σ.

After smooth hybrid inflation… The superpotential of the smooth hybrid inflation sector vanishes, and the New Inflaton starts oscillation around the minimum with a mass . New inflation starts when the vacuum energy dominates over the field oscillation energy with the initial condition . Smooth Hybrid Inflation Field Oscillation New Reheating

Numerical analysis is desired Smooth hybrid inflation σ ψ New inflation Smooth Hybrid Inflation Field Oscillation New Inflation Field Oscillation Reheating Numerical analysis is desired

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. comoving curvature perturbation is used for accuracy check.

Solved with the homogeneous part. represents dissipation rate of each scalar field. Reheating

Result I @ smooth hybrid inflation new inflation Running

Result II (A larger dissipation rate is required to suppress peak amplitude.)

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

to lowest order. These oscillating terms induce parametric amplification of and . Comparing these equations with the Mathieu equation, we find these fluctuations are amplified when , corresponding to the instability band near in the Mathieu equation .

also acquires a component with a long period oscillation. We find no resonant amplification here. These terms induce forced oscillation. As a result, is also enhanced. For , and oscillate with the frequency while and oscillate with the frequency . Long period oscillation with a frequency . also acquires a component with a long period oscillation.

This explains why oscillatory structure appears around the peak. Longwave modes which were outside the horizon at the oscillation regime are irrelevant. Height of the peak is sensitive to the dissipation rate of σ and ψ during oscillation, which determines the abundance of PBHs with mass in this particular case. Large k cutoff is due to the resonant amplification condition .

Fraction of PBHs β constrains the dissipation rate. In this particular case, fluctuation peaked at . corresponding to whose abundance is constrained as : peak amplitude of fluctuations OK

Conclusion Inflation models with a large running spectral index. Double inflation with an oscillatory period in between. Resonant amplification of small-scale fluctuations that are stretched to the cosmological scales by the second inflation.

A nice feature of these models were... WMAP observation of running spectral index Multiple inflation in supergravity NEW HYBRID k WMAP1 observation of early reionization

(Bridle, Lewis, Weller and Efstathiou 2003) WMAP1: running disappears if we omit . WMAP3: running may be present even when we drop components with . (Bridle, Lewis, Weller and Efstathiou 2003) WMAP3 only full range dropped (Feng, Xia, JY 2006)

General m case Even if we take a larger m, the e-folding numbers are still 10 or so. So, another inflation is necessary. We use new inflation as the second inflation. (Later we set m = 2 for simplicity)

running spectral index WMAP observation of running spectral index Multiple inflation in supergravity NEW HYBRID k WMAP observation of early reionization

Numerical analysis is desired New inflation ends at . In order to have the number of e-folds 40, keeping we must have a non vanishing . At the onset of new inflation Smooth Hybrid Inflation Field Oscillation New Inflation Field Oscillation Reheating Numerical analysis is desired