Hilltop Supernatural Inflation and Gravitino Problem

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

� 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

≡ e

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

2

� V

V

�

η ≡ M

V

V P

= 1

24π

M

V

� � (5 × 10

)

N =

�

φ_e

V

V

dφ n

= 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

− 1

4 λφ

λ ∼ 10

V = 1

4 λφ

λ ∼ 10

old inflation

new inflation

chaotic inflation

φ φ

φ

V V

V

Hybrid Inflation

A. Linde, astro-ph/9307002

V (Φ, φ) = M

� φ

M

− 1

�

+ m

2 Φ

+ g

2 φ

Φ

V = M

+ 1

2 m

Φ

V

Φ

φ

Φ

= 8M

g

M

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) = V₀ ± 1

2m²Φ² − λ Φ^p

M_P^p−4 + · · ·

≡ V⁰

�

1 + 1

2η₀ Φ² M_P²

�

− λ Φ^p

M_P^p−4 + · · ·

V (Φ) = V₀ − 1

2m²Φ² + · · ·

≡ V⁰

�

1 − 1

2|η⁰| Φ² M_P²

�

+ · · ·

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

Hill Top - the home of Beatrix Potter and Peter Rabbit

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

η

≤ 0 p > 2

η

< 0 p < 0

η

> 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

∼ 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

|

+ 1 2

�

a

g

D

D

F

≡ ∂W

∂φ

D

≡ φ

T

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

, λ

, D

}

Chiral supermultiplets

Vector supermultiplets

{Q

, ¯ Q

} = 2σ

P

SUSY flat directions

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

W

= λ

QH

u + λ ¯

QH

d + λ ¯

LH

e + µH ¯

H

For example:

LH

F

= λ

Q¯ u + µH

= F

= λ

H

e ¯ ≡ 0

D_{SU (2)}^a = H_u^†τ₃H_u + L^†τ₃L = 1

2 |φ|² − 1

2 |φ|² ≡ 0

F-term:

D-term:

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

H

= 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

Φ

pM

p > 3

λ

∼ O(1)

V (Φ) = 1

2 m

Φ

− A λ

Φ

pM

+ λ

Φ

M

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_I⁴f (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 = 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

"

+ m²_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 = M⁴ cos^{2 !}φ/√

2f ^" + m²_ψ

2 ψ² + ψ⁴φ² + φ⁴ψ²

8M^!2 (10)

The ψ mass is m_ψ and the magnitude of the (imaginary) φ mass term (at the origin) is m_φ ≡ M²/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 m²_ψψ² is small relative to M⁴. The Hubble parameter during inflation is therefore approximately H = ^#8π/3M²/M_p.

We expect f is of order M_P , or equivalently, m²_φ is of order m²_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˜_p³(dV /dψ) = 6 · 10⁻⁴ (11)

where ˜M_p ≡ M^p/√

8π. This gives the constraint M⁵

m²_ψM_p³

$ f

M^! e^rN = 6.7 · 10⁻⁶ (12)

where

r = −3 2 +

$9

4 + µ²_ψ ≈ µ²_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 −µ²ψ/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

=10

-10

GeV

M

=M

=M

=(M

)

/M

=10

-10

GeV

Waterfall field

V (φ) = M

f (φ/M

)

V (φ) = M_S⁴

� φ²

M_P² − 1

�²

V (φ) = M_S⁴ cos²(φ/√

2M_P )

W = Φ

φ

2M

φ V

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

Supernatural Inflation

V = M

+ 1

2 m

Φ

However the spectral index is larger than one.

Φ V

(MS)⁴

Hilltop Supernatural Inflation

V (Φ) = V₀ + 1

2m²Φ² − λ₄AΦ⁴ 4M_P

≡

�

1 + 1

2η₀ Φ² M_P²

�

− λΦ⁴

η₀ ≡ m²M_P²

V₀ λ ≡ λ₄A

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

�

=

� V

M

� η

e

η

x + 4λ(e

− 1) x ≡

� V

M

� � M

Φ

�

P

= 1

12π

e

[4λ(e

− 1) + η

x]

η

(η

x − 4λ)

n

= 1 + 2η

�

1 − 12λe

η

x + 4λ(e

− 1)

�

Result

η

≡ m

M

V

λ ≡ λ

A 4M

m ∼ A ∼ O(TeV) ∼ 10

M

λ

∼ O(1)

λ ∼ O(10

) V

= M

M

∼ 10

GeV ∼ 10

M

η

∼ O(10

)

Part III: Gravitino

Problem

Reheating

Reheating happens

when , and the reheating temperature is

Γ ∼ H

T

∼ H

M

∼ Γ

M

Gravitino Production

Thermal Production

Nonthermal Production

Y

= Y

(T

)

Y

= Y

(T

, m

, �φ�)

Thermal Production

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

dn

dt + 3Hn

= −�σv�(n

− n

)

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 ∝ T³ H ∝ T² s ∝ T³

�σv� ∼ 1 M_P²

Y_3/2 � 2 × 10⁻¹⁶ ×

� T_R

10⁶ GeV

�

Provide an upper bound to reheating temperature.

dY_3/2

dT = − �σv�

HT sn²_{ST D}

Gravitino Constraint (thermal)

Constraint from big bang nucleosynthesis is roughly:

Y

≤ 10

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⁻¹L = −1

8�^µνρσ(G_φ∂_ρφ + G_φ^†∂_ρφ^†) ¯ψ_µγ_νψ_σ

= −1

8e^G/2(G_φφ + 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

m⁵_φ

m²_3/2M_P² � 1 32π

� �φ�

M_P

�2 m³_φ M_P²

Nonthermal Production

φ → 2ψ

n

= 2n

B

B

≡ Γ

Γ

Γ

∼ H ∼ T

M

s ∝ T

ρ

∝ T

n

= ρ

m

Y

= 2B

n

s � 3 2

M

m

Γ

T

Y_3/2 � 10⁻¹⁷

� T_R

10³ GeV

�₋₁ �

�φ�

10¹⁸ 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

d ¯

d ¯

φ = Q

d ¯

L

W = Q¯ uQ ¯ d/M

Example 2:

Φ = LH

φ = H

H

color

flavor

Interaction is from D-term Example 3:

Φ = ¯ u ¯ d ¯ d φ = LH

W = λ

Q

H

u ¯