2. 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)

3. 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.

4. 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)

5. 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

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

7. 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.

8. “n >1 on larger scales n <1 on smaller scales”
can be realized in…
0.5 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)

9. 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.

10. 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

11. “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

12. 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±
running r
＜0.81
Running disappeared!?

13. Revival of running in WMAP3?
３年目　WMAPonly

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

15. 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

16. 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.

17. 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

18. Scalar potential in supergravity

19. 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.

20. Dynamics of smooth hybrid inflationσ 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

21. 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.

22. 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)

23. 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.

24. 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

25. 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

26. 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.

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

28. Result I @
smooth hybrid
inflation
new inflation
Running

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

30. 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

31. 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

32. 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.

33. 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 .

34. 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

35. 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.

37. 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

38. (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)

39. 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)

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

41. Numerical analysis is desired
New inflation ends at
In order to have the number of e-folds , 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