## Hilltop Supernatural

## Inflation and Gravitino Problem

### Chia-Min Lin 林家民

### Physics Department

### 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

## Plan of this talk

### Part I: Basic (Inflationary) Cosmology

### Part II: Hilltop (Supernatural) Inflation

### Part III: Gravitino problem

## Part I: Basic

## (Inflationary) Cosmology

## Hot Big Bang

### Hubble expansion

### Big Bang Nucleosynthesis

### CMB

## Hubble expansion

qwickstep.com

### H = ˙a a

H: Hubble parameter

Edwin Hubble

The biggest blunder of my life ...

## Big Bang Nucleosynthesis

astro-ph/0601514

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

### (n/p) � e

^{−∆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

## The CMB (WMAP 7- year)

### A baby picture of our universe.

2.73 K

## Cosmic Inflation

### H ≡ ˙a

### a = const.

### a ∝ e

^{H∆t}

### ≡ e

^{N}

## Why Inflation?

### Inflation solves many problems of hot big

### bang (flatness problem, horizon problem, and monopole problem etc.).

### Provide the seeds of structure formation.

## The Basic Equations

### � ≡ M

_{P}

^{2}

### 2

### � V

^{�}

### V

### �

2### η ≡ M

_{P}

^{2}

### V

^{��}

### V P

_{ζ}

### = 1

### 24π

^{2}

### M

_{P}

^{4}

### V

### � � (5 × 10

^{−5}

### )

^{2}

### N =

### �

_{φ}

φ_{e}

### V

### V

^{�}

### dφ n

_{s}

### = 1 + 2η − 6�

### r = 16�

Slow roll parameters

The Spectrum

Number of e-folds

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

## WMAP 7 result

E. Komatsu et al., 1001.4538

## Inflation Models

### V = V

_{0}

### − 1

### 4 λφ

^{4}

### λ ∼ 10

^{−13}

### V = 1

### 4 λφ

^{4}

### λ ∼ 10

^{−13}

old inflation

new inflation

chaotic inflation

φ φ

φ

V V

V

## Hybrid Inflation

A. Linde, astro-ph/9307002

### V (Φ, φ) = M

_{I}

^{4}

### � φ

^{2}

### M

^{2}

### − 1

### �

^{2}

### + m

^{2}

### 2 Φ

^{2}

### + g

^{2}

### 2 φ

^{2}

### Φ

^{2}

### V = M

_{I}

^{4}

### + 1

### 2 m

^{2}

### Φ

^{2}

V

Φ

φ

### Φ

^{2}

_{end}

### = 8M

_{S}

^{4}

### g

^{2}

### M

^{2}

Andrei Linde

## Why Hybrid Inflation?

### The inflaton field value could be smaller.

### Couplings could be not very small.

### The inflation scale could be lowered.

## Part II: Hilltop

## (Supernatural) Inflation

## Hilltop Inflation

L. Boubekeur and D. H. Lyth, 1001.4538

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

V (Φ) = V_{0} ± 1

2m^{2}Φ^{2} − λ Φ^{p}

M_{P}^{p}^{−4} + · · ·

≡ V^{0}

�

1 + 1

2η_{0} Φ^{2}
M_{P}^{2}

�

− λ Φ^{p}

M_{P}^{p}^{−4} + · · ·

V (Φ) = V_{0} − 1

2m^{2}Φ^{2} + · · ·

≡ V^{0}

�

1 − 1

2|η^{0}| Φ^{2}
M_{P}^{2}

�

+ · · ·

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

Hill Top - the home of Beatrix Potter and Peter Rabbit

## Three classes of hilltop inflation

λ = 0

η_{0} = 0

Φ Φ Φ

V(Φ) V(Φ) V(Φ)

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

(Hilltop) D-term inflation

### η

_{0}

### ≤ 0 p > 2

### η

_{0}

### < 0 p < 0

### η

_{0}

### > 0

### p > 2

## Why Hilltop?

### To make the spectral index to be less than one.

### Initial condition is topological eternal inflation.

### It may make the inflation scale even lower.

## How to lower the spectral index?

### ∆N ∼ 1

### √ 2� ∆φ n

_{s}

### ∼ 1 + 2η

### η ≡ V

^{��}

### V

### An intuitive way to understand:

### If the potenital becomes convave downward,

### we can have a red spectrum.

### This is the idea of hilltop inflation.

## Supersymmetry (SUSY)

### Fermion Boson quark squark lepton slepton photino photon gravitino graviton

### Gravitino has spin 3/2

SUSY has to be broken

## Why SUSY?

### Hierarchy problem

### Gauge coupling unification

### Connected to gravitiy (SUGRA) Candidate of Dark matter (LSP)

### Many scalar fields and Flat directions

## SUSY scalar potential

### V = �

i

### |F

^{i}

### |

^{2}

### + 1 2

### �

a

### g

_{a}

^{2}

### D

^{a}

### D

^{a}

### F

_{i}

### ≡ ∂W

### ∂φ

_{i}

### D

^{a}

### ≡ φ

^{†}

### T

^{a}

### φ

broken SUSY

broken gauge symmetry V

V

φ

φ

F-term D-term

### Q |B� ∼ |F � Q |F � ∼ |B�

### Φ = {φ, ψ, F }

### V = {V

µ^{a}

### , λ

^{a}

### , D

^{a}

### }

Chiral supermultiplets

Vector supermultiplets

### {Q

^{α}

### , ¯ Q

_{β}

### } = 2σ

_{αβ}

^{µ}

### P

_{µ}

## SUSY flat directions

### (F-term)=(D-term)=0

### W

_{M SSM}

### = λ

_{u}

### QH

_{u}

### u + λ ¯

_{d}

### QH

_{d}

### d + λ ¯

_{e}

### LH

_{d}

### e + µH ¯

_{u}

### H

_{d}

For example:

### LH

_{u}

### F

_{H}

^{∗}

_{u}

### = λ

_{u}

### Q¯ u + µH

_{d}

### = F

_{L}

^{∗}

### = λ

_{d}

### H

_{d}

### e ¯ ≡ 0

D_{SU (2)}^{a} = H_{u}^{†}τ_{3}H_{u} + L^{†}τ_{3}L = 1

2 |φ|^{2} − 1

2 |φ|^{2} ≡ 0

F-term:

D-term:

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

### H

_{u}

### = 1

### √ 2

### � 0 φ

### �

### L = 1

### √ 2

### � φ 0

### �

## MSSM flat directions

## Lifting the Flat Direction

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

### W = λ

_{p}

### Φ

^{p}

### pM

_{P}

^{p}

^{−3}

### p > 3

### λ

_{p}

### ∼ O(1)

### V (Φ) = 1

### 2 m

^{2}

### Φ

^{2}

### − A λ

_{p}

### Φ

^{p}

### pM

_{P}

^{p}

^{−3}

### + λ

^{2}

_{p}

### Φ

^{2(p}

^{−1)}

### M

_{P}

^{2(p}

^{−3)}

soft

mass A-term F-term

## 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 M_{I} or smaller. If we also take M ≈ M^{I}, we find δρ/ρ ≈ M^{I}/M_{p}
or bigger (rather than (M_{I}/M_{p})^{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) = M_{I}^{4}f (φ/M_{p}, ψ/M_{p}) (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 M_{p},
and the resulting δρ/ρ will be of order (M_{I}/M_{p})^{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 M_{p}, M_{G}, 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 = φ^{2}ψ^{2}

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 = M^{4} cos^{2}

!|Φ|

f

"

+ m^{2}_{ψ}|Ψ|^{2} + |Φ|^{4}|Ψ|^{2} + |Ψ|^{4}|Φ|^{2}

M^{!}^{2} (9)

where again we assume (and verify for consistent inflation) M ≈ M^{I}. 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

5

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 + iψ^{I})/√

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

V = M^{4} cos^{2} ^{!}φ/√

2f ^{"} + m^{2}_{ψ}

2 ψ^{2} + ψ^{4}φ^{2} + φ^{4}ψ^{2}

8M^{!2} (10)

The ψ mass is m_{ψ} and the magnitude of the (imaginary) φ mass term (at the origin) is
m_{φ} ≡ M^{2}/f . During inflation, φ is confined near the origin. The field ψ slowly rolls towards
the origin and inflation ends about when ψ = ψ_{c} = ^{#}2M^{!}m_{φ}. It will turn out that either
m_{ψ}/m_{φ} or M^{!}/M_{p} is small, so that during inflation the term m^{2}_{ψ}ψ^{2} is small relative to M^{4}.
The Hubble parameter during inflation is therefore approximately H = ^{#}8π/3M^{2}/M_{p}.

We expect f is of order M_{P} , or equivalently, m^{2}_{φ} is of order m^{2}_{3/2}. 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}

M˜_{p}^{3}(dV /dψ) = 6 · 10^{−4} (11)

where ˜M_{p} ≡ M^{p}/√

8π. This gives the constraint
M^{5}

m^{2}_{ψ}M_{p}^{3}

$ f

M^{!} e^{rN} = 6.7 · 10^{−6} (12)

where

r = −3 2 +

$9

4 + µ^{2}_{ψ} ≈ µ^{2}_{ψ}

3 (13)

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

Here we have defined µ_{ψ} = m_{ψ}/H and have measured time in e-foldings away from the
time N = 0 when ψ = ψ_{c} (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 α_{s} is readily determined from the scale dependence of the density per-
turbations to be −µ^{2}ψ/3. This can be seen directly from the formula for density perturbations
above. Alternatively, it is extracted from the general formula [14]

6

## SUSY Breaking

### We consider Gravity Mediation SUSY breaking.

### Observable

### sector Hiddem

### sector

Through Gravity

### M

SUSY### =10

^{10}

### -10

^{11}

### GeV

### M

squarks### =M

sleptons### =M

3/2### =(M

SUSY### )

^{2}

### /M

P### =10

^{2}

### -10

^{3}

### GeV

## Waterfall field

### V (φ) = M

_{S}

^{4}

### f (φ/M

_{P}

### )

V (φ) = M_{S}^{4}

� φ^{2}

M_{P}^{2} − 1

�^{2}

V (φ) = M_{S}^{4} cos^{2}(φ/√

2M_{P} )

### W = Φ

^{2}

### φ

^{2}

### 2M

^{�}

φ V

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

## Supernatural Inflation

### V = M

_{S}

^{4}

### + 1

### 2 m

^{2}

_{sof t}

### Φ

^{2}

### However the spectral index is larger than one.

Φ V

(MS)^{4}

## Hilltop Supernatural Inflation

V (Φ) = V_{0} + 1

2m^{2}Φ^{2} − λ_{4}AΦ^{4}
4M_{P}

≡

�

1 + 1

2η_{0} Φ^{2}
M_{P}^{2}

�

− λΦ^{4}

η_{0} ≡ m^{2}M_{P}^{2}

V_{0} λ ≡ λ_{4}A

4M_{P}

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

## Solutions

### � Φ M

_{P}

### �

2### =

### � V

_{0}

### M

_{P}

^{4}

### � η

_{0}

### e

^{2N η}

^{0}

### η

_{0}

### x + 4λ(e

^{2N η}

^{0}

### − 1) x ≡

### � V

_{0}

### M

_{P}

^{4}

### � � M

_{P}

### Φ

_{end}

### �

2### P

_{R}

### = 1

### 12π

^{2}

### e

^{−2Nη}

^{0}

### [4λ(e

^{2N η}

^{0}

### − 1) + η

^{0}

### x]

^{3}

### η

_{0}

^{3}

### (η

_{0}

### x − 4λ)

^{2}

### n

_{s}

### = 1 + 2η

_{0}

### �

### 1 − 12λe

^{2N η}

^{0}

### η

_{0}

### x + 4λ(e

^{2N η}

^{0}

### − 1)

### �

## Result

### η

_{0}

### ≡ m

^{2}

### M

_{P}

^{2}

### V

_{0}

### λ ≡ λ

_{4}

### A 4M

_{P}

### m ∼ A ∼ O(TeV) ∼ 10

^{−15}

### M

_{P}

### λ

_{4}

### ∼ O(1)

### λ ∼ O(10

^{−15}

### ) V

_{0}

### = M

_{S}

^{4}

### M

_{S}

### ∼ 10

^{11}

### GeV ∼ 10

^{−7}

### M

_{P}

### η

_{0}

### ∼ O(10

^{−2}

### )

## Part III: Gravitino

## Problem

## Reheating

### Reheating happens

### when , and the reheating temperature is

### Γ ∼ H

### T

_{R}

^{4}

### ∼ H

^{2}

### M

_{P}

^{2}

### ∼ Γ

^{2}

### M

_{P}

^{2}

## Gravitino Production

### Thermal Production

### Nonthermal Production

### Y

_{3/2}

^{(th)}

### = Y

_{3/2}

^{(th)}

### (T

_{R}

### )

### Y

_{3/2}

^{(nt)}

### = Y

_{3/2}

^{(nt)}

### (T

_{R}

### , m

_{φ}

### , �φ�)

## Thermal Production

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

### dn

_{3/2}

### dt + 3Hn

_{3/2}

### = −�σv�(n

^{2}

_{3/2}

### − n

^{2}

_{rad}

### )

Effect of Hubble expansion

source term sink term

(negligible)

ex:

ex: gψ˜ _{3/2} → gg gg → ˜gψ^{3/2}
Changing

vareibles t → T

n_{3/2} → Y3/2 ≡ n_{3/2}
s

n_{rad} ∝ T^{3}
H ∝ T^{2}
s ∝ T^{3}

�σv� ∼ 1
M_{P}^{2}

Y_{3/2} � 2 × 10^{−16} ×

� T_{R}

10^{6} GeV

�

Provide an upper bound to reheating temperature.

dY_{3/2}

dT = − �σv�

HT sn^{2}_{ST D}

## Gravitino Constraint (thermal)

Constraint from big bang nucleosynthesis is roughly:

### Y

_{3/2}

### ≤ 10

^{−17}

Kawasaki, Kohri, and Moroi astro-ph/0408426

Kawasaki, Kohri, and Moroi astro-ph/0402490

## Gravitino Pair-Production

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

e^{−1}L = −1

8�^{µνρσ}(G_{φ}∂_{ρ}φ + G_{φ}^{†}∂_{ρ}φ^{†}) ¯ψ_{µ}γ_{ν}ψ_{σ}

= −1

8e^{G/2}(G_{φ}φ + G_{φ}^{†}φ^{†}) ¯ψ_{µ}[γ^{µ}, γ^{ν}]ψ_{ν}
G ≡ K + ln |W |^{2}

Γ_{3/2} ≡ Γ(φ → 2ψ^{3/2}) � |G^{φ}|^{2}
288π

m^{5}_{φ}

m^{2}_{3/2}M_{P}^{2} � 1
32π

� �φ�

M_{P}

�2 m^{3}_{φ}
M_{P}^{2}

## Nonthermal Production

### φ → 2ψ

^{3/2}

### n

_{3/2}

### = 2n

_{φ}

### B

_{3/2}

### B

_{3/2}

### ≡ Γ

_{φ}

_{→2ψ}

_{3/2}

### Γ

_{φ}

### Γ

_{φ}

### ∼ H ∼ T

_{R}

^{2}

### M

_{P}

### s ∝ T

^{3}

### ρ

_{φ}

### ∝ T

^{4}

### n

_{φ}

### = ρ

_{φ}

### m

_{φ}

### Y

_{3/2}

### = 2B

_{3/2}

### n

_{φ}

### s � 3 2

### M

_{P}

### m

_{φ}

### Γ

_{φ}

_{→2ψ}

_{3/2}

### T

_{R}

Y_{3/2} � 10^{−17}

� T_{R}

10^{3} GeV

�_{−1} �

�φ�

10^{18} GeV

�2 � m_{φ}
10 TeV

�2

where

Provide a lower bound to reheating temperature.

## Result

## Result

## Result

## Conclusion

### We have shown hilltop supernatural inflation can produce n=0.96 for natural values of the parameters.

### We also investigate the allowed region for

### reheating temperature and found the model

### is safe from gravitino problem.

## Constraints to other models

from Fuminobu Takahashi’s talk

## Candidates of flat directions

Example 1:

### Φ = ¯ u

^{1}

_{1}

### d ¯

^{2}

_{1}

### d ¯

^{3}

_{2}

### φ = Q

^{1}

_{2}

### d ¯

^{1}

_{3}

### L

_{2}

### W = Q¯ uQ ¯ d/M

^{�}

Example 2:

### Φ = LH

_{u}

### φ = H

_{u}

### H

_{d}

color

flavor

Interaction is from D-term Example 3:

### Φ = ¯ u ¯ d ¯ d φ = LH

_{u}

### W = λ

_{u}

### Q

_{u}

### H

_{u}