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# Hilltop Supernatural Inﬂation and Gravitino Problem

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## Inflation and Gravitino Problem

### National Tsing Hua University Seminar at NTU on 10/29

Based on arXiv: 1008.3200 [hep-ph] with Kazunori Kohri arXiv: 0901.3280 [hep-ph] with Kingman Cheung

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## Hubble expansion

qwickstep.com

### H = ˙a a

H: Hubble parameter

Edwin Hubble

The biggest blunder of my life ...

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## Big Bang Nucleosynthesis

astro-ph/0601514

Alpher-Bethe-Gamow paper (known as αβγ) 1948

−∆m/T

### � 1/7

Every 16 nucleons (2 neutrons and 14 protons), 4 of these (25%) combined into one helium-4 nucleus.

At the temperature of helium-4 formation:

We cannot explain this by stellar nucleosynthesis.

This happened when the universe is 3 to 20 minutes old.

∆m = 1.293 MeV

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2.73 K

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H∆t

N

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## The Basic Equations

P2

2

P2

��

ζ

2

P4

−5

2

φ

φe

s

### r = 16�

Slow roll parameters

The Spectrum

Number of e-folds

The spectral index Tensor to scalar ratio We call this CMB normalization.

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## WMAP 7 result

E. Komatsu et al., 1001.4538

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## Inflation Models

0

4

−13

4

### λ ∼ 10

−13

old inflation

new inflation

chaotic inflation

φ φ

φ

V V

V

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## Hybrid Inflation

A. Linde, astro-ph/9307002

I4

2

2

2

2

2

2

2

2

I4

2

2

V

Φ

φ

2end

S4

2

2

Andrei Linde

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## Hilltop Inflation

L. Boubekeur and D. H. Lyth, 1001.4538

K. Kohri, C. M. Lin, and D. H. Lyth, 0707.3826

V (Φ) = V0 ± 1

2m2Φ2 − λ Φp

MPp−4 + · · ·

≡ V0

1 + 1

2η0 Φ2 MP2

− λ Φp

MPp−4 + · · ·

V (Φ) = V0 1

2m2Φ2 + · · ·

≡ V0

1 1

20| Φ2 MP2

+ · · ·

CML, D. H. Lyth, and K. Kohri

Hill Top - the home of Beatrix Potter and Peter Rabbit

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## Three classes of hilltop inflation

λ = 0

η0 = 0

Φ Φ Φ

V(Φ) V(Φ) V(Φ)

(Hilltop) Supernatural Inflation New Inflation (Hilltop) F-term inflation

(Hilltop) D-term inflation

0

0

0

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s

��

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## Supersymmetry (SUSY)

### Gravitino has spin 3/2

SUSY has to be broken

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## SUSY scalar potential

i

i

2

a

a2

a

a

i

i

a

a

### φ

broken SUSY

broken gauge symmetry V

V

φ

φ

F-term D-term

µa

a

a

### }

Chiral supermultiplets

Vector supermultiplets

α

β

αβµ

µ

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## SUSY flat directions

M SSM

u

u

d

d

e

d

u

d

For example:

u

H u

u

d

L

d

d

### e ¯ ≡ 0

DSU (2)a = Huτ3Hu + Lτ3L = 1

2 |φ|2 1

2 |φ|2 ≡ 0

F-term:

D-term:

K. Enqvist and A. Mazumdar hep-ph/0209244

u

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## Lifting the Flat Direction

Flat directions can be lifted by SUSY breaking terms and nonrenormalizable terms

p

p

Pp−3

p

2

2

p

p

Pp−3

2p

2(p−1)

### M

P2(p−3)

soft

mass A-term F-term

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## Supernatural Inflation

Lisa Randall, Marin Soljacic, and Alan H. Guth hep-ph/9512439

Lisa Randall Alan Guth A hybrid inflation.

Both inflaton and waterfall fields are flat directions (or moduli).

with input parameters suggested from SUSY breaking with no (very) small parameters.

inflation is of the order MI or smaller. If we also take M ≈ MI, we find δρ/ρ ≈ MI/Mp or bigger (rather than (MI/Mp)2 as was the case in single field models). Although the coupling between the fields can have a coefficient which varies by many orders of magnitude, as does ψ in eqn. (6), the strong M dependence of Eqn. (6) allows for agreement with the COBE constraint with only a relatively small M variation. This is very promising from the perspective of relating inflation models to real scales of particle physics. To answer the questions of how well these ideas really work, and how constrained the parameters of the models really are, requires a detailed investigation of particular examples of these ideas.

III. SUPERNATURAL INFLATION

We define Flat Direction Hybrid Inflation (FDHI) models as those motivated by the properties of moduli fields or flat directions of the standard model. For moduli fields with no gauge charge or superpotential, the whole potential arises from the Kahler potential once supersymmetry is broken. This potential for φ and ψ will take the form

V (φ, ψ) = MI4f (φ/Mp, ψ/Mp) (7) where the dimensionless coefficients in f should be of order unity.

However, it is clear that a model of this sort will not give rise to inflation with sufficiently large density fluctuations, since during the relevant period ψ will typically be of order Mp, and the resulting δρ/ρ will be of order (MI/Mp)2. We conclude that it is essential to have an additional interaction between ψ and φ. In this model, we assume the existence of a superpotential which couples ψ and φ but which is suppressed by a large mass scale M!. For standard model flat directions, such higher dimension operators are to be expected, with M! equal to Mp, MG, or some dynamical scale. In the case of moduli fields, it might be that this scale is of dynamical origin; one can readily determine how the answer changes with the form of the superpotential and the size of the mass scale.

We therefore assume the presence of a superpotential. The example we take is W = φ2ψ2

2M! (8)

We now need to specify the form of the supersymmetry breaking potential. We assume both ψ and φ have mass of order the soft SUSY breaking scale of order 1 TeV (where we will need to test the consistency of this assumption). We assume that the potential for the ψ field gives a positive mass squared at the origin, while the φ field has negative mass squared at the origin. Furthermore, we assume that the cosmological constant is zero at the minimum of both ψ and φ. The specific form of the potential we choose is

V = M4 cos2

!|Φ|

f

"

+ m2ψ|Ψ|2 + |Φ|4|Ψ|2 + |Ψ|4|Φ|2

M!2 (9)

where again we assume (and verify for consistent inflation) M ≈ MI. When the parameters are motivated by supersymmetry breaking, we refer to our models by the name supernatural inflation. We will see that one very naturally obtains the correct magnitude of density

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perturbations, and sufficiently many e-foldings of inflation, using parameters and a potential which are well motivated in supersymmetric models.

For the purpose of the inflationary model, the scalar field can be assumed to be real.

Defining Ψ ≡ (ψ + iψI)/√

2 and a similar equation for Φ, the potential for the real fields becomes

V = M4 cos2 !φ/√

2f " + m2ψ

2 ψ2 + ψ4φ2 + φ4ψ2

8M!2 (10)

The ψ mass is mψ and the magnitude of the (imaginary) φ mass term (at the origin) is mφ ≡ M2/f . During inflation, φ is confined near the origin. The field ψ slowly rolls towards the origin and inflation ends about when ψ = ψc = #2M!mφ. It will turn out that either mψ/mφ or M!/Mp is small, so that during inflation the term m2ψψ2 is small relative to M4. The Hubble parameter during inflation is therefore approximately H = #8π/3M2/Mp.

We expect f is of order MP , or equivalently, m2φ is of order m23/2. Although it looks like we took a very special form for the φ-potential in Eq. (9), the use of the cosine is not essential. As can be seen from a Taylor expansion, only at the very late stages of inflation are terms other than the constant and mass term relevant. We could equally well have specified a potential which is truncated at fifth order in the fields, or which has different higher order terms. Although both ψ and φ might be moduli or standard model flat direction fields, we assume their potentials are of very different form; the particular case we assume is illustrative of how a model could work.

The constraint from density perturbations in the slow-roll regime is [12,13]

V 3/2

p3(dV /dψ) = 6 · 10−4 (11)

where ˜Mp ≡ Mp/√

8π. This gives the constraint M5

m2ψMp3

\$ f

M! erN = 6.7 · 10−6 (12)

where

r = −3 2 +

\$9

4 + µ2ψ ≈ µ2ψ

3 (13)

where the approximation in Eq. (13) is required if the slow roll conditions are satisfied.

Here we have defined µψ = mψ/H and have measured time in e-foldings away from the time N = 0 when ψ = ψc (where inflation ends at positive N). It is clear that a lower M! makes the value of ψ at the end of inflation lower, which in turn increases the density perturbations. The exponential in Eq. (12) determines the scale dependence of the density perturbations, characterized by the scalar index.

The scalar index αs is readily determined from the scale dependence of the density per- turbations to be −µ2ψ/3. This can be seen directly from the formula for density perturbations above. Alternatively, it is extracted from the general formula [14]

6

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Through Gravity

SUSY

10

11

squarks

sleptons

3/2

SUSY

2

P

2

3

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## Waterfall field

S4

P

### )

V (φ) = MS4

φ2

MP2 − 1

2

V (φ) = MS4 cos2(φ/

2MP )

2

2

### 2M

φ V

In order to have hybrid inflation, waterfall field should couple to inflaton.

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S4

2sof t

2

Φ V

(MS)4

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## Hilltop Supernatural Inflation

V (Φ) = V0 + 1

2m2Φ2 λ44 4MP

1 + 1

2η0 Φ2 MP2

− λΦ4

η0 m2MP2

V0 λ λ4A

4MP

We choose p=4 and neglect the last term (due to the smallness of Φ) Φ

V

(MS)4

CML and Kingman Cheung 0901.3280

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P

2

0

P4

0

2N η0

0

2N η0

0

P4

P

end

2

R

2

−2Nη0

2N η0

0

3

03

0

2

s

0

2N η0

0

2N η0

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0

2

P2

0

4

P

−15

P

4

−15

0

S4

S

11

−7

P

0

−2

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R4

2

P2

2

P2

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3/2(th)

3/2(th)

R

3/2(nt)

3/2(nt)

R

φ

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## Thermal Production

M. Kawasaki and T. Moroi hep-ph/9403364 Boltzmann Equation:

3/2

3/2

23/2

2rad

### )

Effect of Hubble expansion

source term sink term

(negligible)

ex:

ex: ˜ 3/2 → gg gg → ˜gψ3/2 Changing

vareibles t → T

n3/2 → Y3/2 n3/2 s

nrad ∝ T3 H ∝ T2 s ∝ T3

�σv� ∼ 1 MP2

Y3/2 � 2 × 10−16 ×

TR

106 GeV

Provide an upper bound to reheating temperature.

dY3/2

dT = �σv�

HT sn2ST D

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## Gravitino Constraint (thermal)

Constraint from big bang nucleosynthesis is roughly:

3/2

### ≤ 10

−17

Kawasaki, Kohri, and Moroi astro-ph/0408426

Kawasaki, Kohri, and Moroi astro-ph/0402490

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## Gravitino Pair-Production

M. Endo, K. Hamaguchi, and F. Takahashi hep-ph/0602061 S. Nakamura and M. Yamaguchi hep-ph/0602081

e−1L = −1

8µνρσ(Gφρφ + Gφρφ) ¯ψµγνψσ

= 1

8eG/2(Gφφ + Gφφ) ¯ψµµ, γνν G ≡ K + ln |W |2

Γ3/2 ≡ Γ(φ → 2ψ3/2) |Gφ|2 288π

m5φ

m23/2MP2 1 32π

�φ�

MP

2 m3φ MP2

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## Nonthermal Production

3/2

3/2

φ

3/2

3/2

φ→2ψ3/2

φ

φ

R2

P

3

φ

4

φ

φ

φ

3/2

3/2

φ

P

φ

φ→2ψ3/2

### T

R

Y3/2 � 10−17

TR

103 GeV

−1

�φ�

1018 GeV

2 � mφ 10 TeV

2

where

Provide a lower bound to reheating temperature.

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## Constraints to other models

from Fuminobu Takahashi’s talk

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## Candidates of flat directions

Example 1:

11

21

32

12

13

2

Example 2:

u

u

### H

d

color

flavor

Interaction is from D-term Example 3:

u

u

u

u

### u ¯

Al atoms are larger than N atoms because as you trace the path between N and Al on the periodic table, you move down a column (atomic size increases) and then to the left across

Accordingly, we reformulate the image deblur- ring problem as a smoothing convex optimization problem, and then apply semi-proximal alternating direction method of multipliers

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Particularly, combining the numerical results of the two papers, we may obtain such a conclusion that the merit function method based on ϕ p has a better a global convergence and

Then, it is easy to see that there are 9 problems for which the iterative numbers of the algorithm using ψ α,θ,p in the case of θ = 1 and p = 3 are less than the one of the

To compare different models using PPMC, the frequency of extreme PPP values (i.e., values \0.05 or .0.95 as discussed earlier) for the selected measures was computed for each

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