Particle Cosmology
Nov.6 2011
@第２４回理論懇シンポジウム

Fuminobu TAKAHASHI(Tohoku University and IPMU)SUSY and Inflation
Nov.6 2011
@第２４回理論懇シンポジウム

Fuminobu TAKAHASHI(Tohoku University and IPMU)Inflation

Starobinsky,`80, K.Sato, `81
Guth, `81, Albrecht and Steinhardt `82: a phase of the exponential expansion.
Inflation can solve the horizon andflatness problems.

Inflation: a phase of the exponential expansion.
Inflation can solve the horizon andflatness problems.
One way to realize the inflation is the slow-rollinflation.
9Hδφ = 2πDensity
perturbations

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What is the inflaton?

Inflation
Density
perturbation

Theory beyond SM
BaryonAsymmetry
Dark Mattere.g. SUSY, string

Inflationary Zoo
V0 − m2 φ2m2 φ2Large-field inflation

OthersSmall-field inflation
V0 + m2 φ2 + (φ2 − µ2 )χ2Hybrid inflation

Model 1
New inflation

Let us consider an extension of SM,
SM + RH neutrinos + U(1)B-L

Let us consider an extension of SM,
SM + RH neutrinos + U(1)B-L

The seesaw mechanism
2�h�mν ∼MR

GUT & Charge quantization
T. Yanagida `79, M.Gell-Mann, P.Ramondand R.Slansky `79, Minkowski, `77

yNc
νR + h.c..
L = − φ ν̄R2MR ∼ �φ� ∼ 1015GeV

B-L Higgs inflation
9Hδφ = 2π

However, this simple model
does not work because of theColeman-Weinberg correction.

F15�φ�∼10GeV
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Radiative Correction
The B-L Higgs boson is coupled to the B-L gauge boson(and the right-handed neutrinos), leading to�

3 4 43g ϕVCW =g ϕ ln 2 −264πµ222ϕ≡�

√2|φ|The density perturbation would be too large unlessg is smaller than O(10-3).
Similar contribution (with negative sign) from the RH
neutrino loop.

どうするか1. Gauge singlet inflatonを考える。
2. SUSYにする。
Ellis, Nanopoulos, Olive and Tamvakis `83

ここでは２を考える。ただしSUSYは破れているから、おつりが残るはず。

In SUSY, two Higgs are required for anomaly
cancellation.
Φ(+2), Φ̄(−2)

We identify the inflaton with the D-flat direction

φ ≡ ΦΦ̄
2Bosons

B-L gauge boson
The scalar perpendicular
to the D-flat direction
22 2mB = g ϕFermionsB-L gaugino
B-L Higgsino

mF = gϕ ± m̃soft SUSY breaking mass輻射補正はだいぶ小さくなって、おつりは��
22 2gg ϕ
2 2VCW (ϕ) �
1 − 3 ln 2m̃ ϕ28πµ√ϕ≡2|φ|となる。インフレーションがうまくいくにはm̃ � 0.1Hinf
1108.0070, K. Nakayama and FTが必要。至極もっともな条件。
今回（多分）初めて導かれた。

B-L new inflation model
κ3 4
κ1 42222
模型の詳細はともかく、SUSYにすると
K = |φ| + |S| + |φ| + κ2 |φ| |S| + |S| ,W44� 2�= インフレーションがうまくいく。
S v − kφ2n ,φ2 ≡ ΦΦ̄ Asaka et al `99Senoguz and Shafi, `04

22 H 2 2n∗2n24nV (φ) � v 4 − m|φ|−kv(φ+φ)+k|φ|δφ0 =92πF15�φ�∼
10GeV
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Inflation scale
従ってSUSY粒子はTeV-PeV以下ということが階層性問題と関係なく、インフレーションから要求される。
�ϕ� = 1015 GeVSUSYの破れはHubbleの一桁落ち以下

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SM Particles

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再加熱
B-L Higgsは右巻きニュートリノとの相互作用を通じて崩壊する。Non-thermal leptogenesisが自然に起こり、バリオン数を生成できる。

yNc
L = − φ ν̄R νR + h.c..
2φνRνRinflaton mass

TRn ≥ 3 is required for non-thermal leptogenessSM Higgs boson mass

1108.3762, K. Nakayama and FTModel 2
Hybrid inflation

Hybrid inflation
We can identify the gauge singlet S with theBuchmuller, Covi, Delepine `01
inflaton.
Nakayama, FT, Yanagida, `10K
W

= |S| + |φ| + |φ̄|��2= κS φφ̄ − M + W022

2

where W0 � m3/2 Mp2
is needed to cancel CC.
V = e (Di W gKij ∗Dj ∗ W ∗ − 3|W |2 )φS

The inflaton potentialσ iθS=√ e2σ : 動径方向
θ : 位相方向��22 24σκκ σ
24V =κ M 1++ln
428MP
16π
2Λ2

√
2+2 2κM m3/2 σ cos θ.

The linear term affects both the radial andangular components of S.

Radial component
The linear term generates a local maximumπ
3π<θ<and a local minimum for
22V(S)
44SSlog S

m3/2 M 2 σ log SRe[S]Angular component
The phase of S evolves since the linear term
breaks the U(1)R symmetry.
√3H φ̇ = 2 2κM 2 m3/2 cos θ,√sin θ
23H θ̇ = 2 2κM m3/2φ√
2 2m3/2 MP2 cos θ∆θ � tan θ ln (1 + δN ) ,
δ=.
3
κM

∆θ � 1 is needed for successful inflation

Fine-tuning of the initial conditionθiκ = 10−3
M = 1014 GeV

σiFine-tuning of the initial conditionθiκ = 10−3
M = 1014 GeVσiκ

M [GeV]κ

M [GeV]κ

M [GeV]まとめB-L new inflationの為にはinflation scale以下にSUSYが必要。

m3/2 � TeV − PeV
B-L hybrid inflationの為には
m3/2 � 100 TeV(m3/2 � 100 GeV (stable))が必要。

Inflation?

Low-scaleSUSYback-up slides

Gravitino Pair-Production
Kawasaki, F.T. and Yanagida, hep-ph/0603265, 0605297
Asaka, Nakamura and Yamaguchi, hep-ph/0604132

Relevant interactions:−1eL=

1 µνρσ^− �(Gφ ∂ρ φ + Gz ∂ρ z − h.c.) ψ̄µ γν ψσ
8
1 G/2
^
− e
(Gφ φ + Gz z + h.c.) ψ̄µ [γ µ , γ ν ] ψν ,8φ : inflaton fieldz : SUSY breaking field, w/ Gz Gz � 3G ≡ K + ln |W |2Taking account of the mixings,

m3/2Gφ ∼ �φ�mφ

for mφ < mzGravitino Pair Production Rate:
5m|Gφ |φ
Γ3/2 �
22288π m3/2 MP
2Endo, Hamaguchi and F.T., hep-ph/0602061
Nakamura and Yamaguchi, hep-ph/0602081

1
�32π�

�φ�MP

�2m3φ2MP

for mφ < mz

Gravitino pair production is effective especiallyfor low-scale inflation models.
Gravitino abundance is inversely proportional tothe reheating temperature!

Gravitino Abundance:
Y3/2

Γ3/2 3 TR� 2,Γtotal 4 mφ
�−1
� g �− 12 � T∗
R−14∼ 10200106 GeV��2 ��2�φ�mφ×1015 GeV1010 GeV

TR2
Note: Γtotal ∼MPGravitino Abundance
Y3/2

non-thermalthermalTRConservative
Constraints on the inflation models;
1019
10C

18

: new(single);1TeV: new(single);100TeV: new(multi): hybrid: smooth hyb.
: chaotic (w/o Z 2 )D<φ > [GeV]

1017
101610151014

10131012 8AB10 10 9 1010 1011 1012 1013 1014 1015 1016 1017mφ [GeV]

A: m 3/2 = 1TeV; Bh = 1-3
B: m 3/2 = 1TeV; Bh = 10
C: m 3/2 = 100TeV
D: m 3/2 = 1GeV

Implications for inflation models(i) Low-sclae inflation (small VEV, light mass), andrelatively strong couplings with the SM sector.
e.g.) SUSY new inflation by B-L (or GUT) Higgs inflation
Nakayama and FT, 1108.0070(ii) Large-field inflation, but with a vanishing vev.
e.g.) Chaotic inflation

V = 12 m2 φ2 w/ φ ↔ −φe.g.) Running kinetic chaotic inflation

FT 1006.2801, Nakayama and FT, 1008.2956

Other solutions (not related toinflation models)(1) Heavy or light gravitino
Cosmological constraints are then relaxed.

(2)Maybe thermal history is more complicated.
Gravitinos can be diluted by late-time
entropy production.

Ellis, Nanopoulos, Olive and Tamvakis, PLB 118 335 (1982)

These are valid iff

mB , mF ∝ �φ�However, the soft massneeds not be proportional tothe Higgs vev, which resultsin the underestimate of theSUSY breaking mass.

gmS < 1014 GeV
gmS < 2 × 109 GeVfrom density perturbation.

Split SUSY case

Case with non-minimal Kahler potential∆K=22622|S||φ||S||
φ̄||S|
k1 |S|4 + k2+ k3+ k4 4 + · · · ,22MPMPMP��
422 22σσκκ σ24V = κ M 1 − k1+γ+ln2422MP8MP
16π2Λ2√+2 2κM 2 m3/2 σ cos θ.
γ = 1 − 7k1 /2 − 3k4 + 2k12

For k1 > 0, the spectral index becomes redder,and there appear local maximum and minimum.

k1 = 0.01