Higher-Dimensional Gauge Theory/Gravity

Cosmological Inflation/Perturbation from Higher-Dimensional Gauge Theory/Gravity

Takeo Inami, Chuo University Kobe U, March 13 − 14, 2012

Model for inflaton Koyama, Lim, Minakami & Inami, PTP (2009)

Inflaton v.s. curvaton Koyama, C.M. Lin, Minakami & Inami, PTP (2011) Radion inflation Fukazawa, Koyama & Inami, (to appear)

Fuzzy extra dimensions Furuuchi, Okuyama & Inami, JHEP (2011)

1

1. Introduction − Inflation from Gauge Fields in Extra Dim

Scalar fields play as important roles as gauge fields. ₋ Okun, Bonn conf, 81

Higgs determines the symmetry phases Nowadays more scalars are known.

Inflaton makes universe as it is now

Curvaton? is the origin of the fluctuation in the universe?

Question: Where do these scalars come from?

One answer: They may arise from

extra dimensions

of gauge fields.

The talk consists of three parts:

Part I : Inflaton from higher-dim gauge field

»»»»»»»»

»»»»»»»

Part II: Inflaton (inflation) + Curvaton (density/curvature pert ζ) Part III: Radion inflation in higher-dim gravity

2

A. Idea of our work

Fine-tuning problem in gauge theories i) Gauge hierarchy problem

m_{W boson}, m_Higgs ≅ 100 GeV ≪ Λ (M_Planck) More recently, similar one in cosmology

ii) Parameters of

inflaton potential V (φ), δ T/T

We argue

: The two may be solved by one mechanism −

higher-dim

(SUSY) gauge theory.

(4 + n)D gauge field A_M (µ = 0, ..., 3; y = 5, 6, ...) A_M =

(

A_µ A_y

)

→ (

A⁽⁰⁾_µ = A_µ gauge field A⁽⁰⁾_y = h, φ Higgs/inflaton

)

gauge-Higgs/

gauge-inflaton

unification 3

Nice points

^:

⋆ Potential V (h), V (φ)arise at 1-loop; two properties are guaranteed.

Higgs mass m²_h is

small

^.

Inflaton potential V (φ) is

”slow-roll”

^.

Some results

^:

⋆ 1-loop potential is not affected by

other quantum effects, quantum gravity effects ...

gauge symmetry in extra dim

⋆ Our toy model reproduces the inflation parameters without tuning, but ...

g²/4π ≅ 0.003 ←→ g_GUT² /4π ≅ 0.03

⋆ Higher-dim radion model works. (Part III)

4

B. Summary of Higgs/inflaton

¡¡¡⋆¡ Fine-tuning problem in Higgs potential

V_H(h) = m²h^†h + λ(h^†h)²

(δm²)_1−loop ∼ O(Λ²) ≫ m²_h. How is δm² ≅ m²_h realized?

A few alternative solutions

technicolor, supersymmetry, ...

We have proposed higher-dim gauge theory as one solution.

⋆ An analogus fine-tuning problem in inflaton potential is observed.

V (φ) = λφ⁴ * λ ∼ 10⁻¹², extremely small! − unnatural

Question: Construct an inflation model which is free from fine-tuning.

* V = λM_P⁴(φ/M_P)ⁿ chaotic inflation, occurs for φ ≥ M_P

5

A few attempts:

inflation ₋ Sato, Guth (81)

slow-roll (new inflation) ₋ Linde (81)

V = aφⁿ (chaotic), V (φ₁, φ₂) (hybrid)

PNGB (technicolor) ₋ Freese et al (90), SUSY/SUGRA extra-natural ₋ Arkani-Hamed et al (03)

We study a new inflation model: Inflaton φ is a part of gauge fields,

A⁽⁰⁾_M = (A_µ, φ (= h), ...) (1)

6

C. Remark − Other fine tuning parameters

There appear a few fine-tuning problems in other areas of particle physics/cosmology.

i) Gauge hierarchy m_W/M_GUT = 10⁻¹⁴ ii) Cosmological constant

Λ_exp = (2.32meV)⁴ ... Λ > 0 de Sitter space iii) θ angle θ_exp < 10⁻¹⁰

iv) nutrino masses n_ν ∼ 0.1 eV gravinutrino? Ho, Inami (unfinished, 09)

ψ_M⁽⁰⁾ = (ψ_µ⁽⁰⁾, ψ_y⁽⁰⁾ = ν) (2) v) inflaton potential V (φ)

Question: Do any of them (other than i) have their origin in higher-dim theory??

UV

or

IR

problems?

7

2. Basics of inflaton

¡¡¡

¡A. Why inflation?¡

⋆ Historical: Big-bang scenario explains how our universe was made.

Problems remain in BB, horizon problem flatness problem

They are solved if inflation occurred before BB. [Fig]

BB a <¨ 0 (deccel expansion) inflation ¨a > 0 (accel expansion)

⋆ Recent context: Inflation may generate

irregularities (fluct in CMB) ↔ various models of inflation [Fig]

8

9

10

11

12

¡¡¡

¡B. What is inflation?¡

During the inflation epoch, ρ ≅ ρ_vac (const vacuum energy)

a(t) ∼ e^Ht universe expands exponentially. H = ˙a/a ∼ √ρ_vac

A small causally connected region of universe grows for a ”long time”

∆t to solve the horizon/flatness problems.

N ≅ H∆t ≅ 60 e-foldings

Inflation occurs if universe is filled with a scalar field φ. − inflaton

¨

a/a = −(ρ + 3P)/6M_P² > 0 ... P < 0, negative pressure ρ, P = ¹₂φ˙² ± V (φ), ... V (φ) > 0 de Sitter-like

Inflaton potential V (φ) determines how inflation proceeds.

V (φ) should be

nearly flat

to meet the data. [Fig]

sufficiently long inflation period, ∆t ≡ t_f − t_i ≅ 60/H

CMB aniso data, density fluct δT /T (∼ δρ/ρ) ∼ O(10⁻⁵) 13

8Ǟ

Ǟ

8Ǟ

Ǟ

8ǞǞ̉

Ǟ KKPGYKPHNCVKQP KKKEJCQVKE KXJ[DTKF

Ǟ̉

Ǟ㧦KPHNCVQP

8ǞKPHNCVKQP

KPHNCVKQPTGIKQP PGCTN[HNCV

Ǎ8

Ǟ_K Ǟ_H

14

C. Fine-tuning problem

Astrophysics data demand that V (φ) is ”slow-roll”,

n_s = 1 − 6ϵ + 2η, 0.95 ≤ n_s ≤ 0.97 (spectral index) ϵ = [(V ^′/V )M_P]² ≅ 0.01, η = (V ^′′/V )M_P² ≅ 0.01

Past models fail to solve this fine-tuning problem: Naturally ϵ, η ≅ 1.

(learned from Tachikawa, Watari)

We argue

^{: ϵ and η may be} naturally small in model of higher-dim gauge theory.

15

½½½½

¡¡¡

3. Gauge-Higgs unification from higher dim

A. Higher-dim gauge theory Antoniadis (90), ..., Hatanaka, Lim & Inami (98) use of Z₂ orbifold Kawamura

d-dim space-time: X^M = (x^µ, x^m); m = 5, .., d, Gauge field A_M A_M = (A_µ, A_y) → (A⁽⁰⁾_µ = gauge field, A⁽⁰⁾_m = h Higgs) (3)

¡¡¡

¡B. Gauge hierachy problem¡

Higgs mams term gets huge corrections from one-loop effects.

δm² ∼ Λ² ≫ m²_h

Solution in higher-dim gauge theory:

At tree level, δm² = 0 ...

gauge symmetry in extra dim At one-loop δm² = finite ≪ Λ², i.e.,

solved !/?

Remark: A different view was proposed recently. Aoki, Iso, Okada (11).

16

4. Models of inflation

A. Models of inflation Two approaches:

a) Construct a ”good model” of inflation from GUT, SUGRA, string, ...

→ our approach

¡¡¡

¡b) Write a ”good form” of inflaton potential¡¡ V (φ).

ii) new, iii) chaotic, iv) hybrid, ... [Fig]

B. Inflation parameters

From recent astrophysical data:

Temperature of the universe (at it’s age of 390,000 years)

⋆ Same temperature from all directions

→ horizon problem → solved by inflation

⋆ Tiny anisotropy at scales of 10^′ − 5^◦ → constrains inflaton models Parameters: spectral index n_s, ..., curv pert δ_H, (r = P_h/P_ζ tensor/scalar)

17

5. Inflaton model: 5D N=1 SUSY gauge theory on M₄ × S¹

Assumption

^{: A(0)}₅ ^{= h = φ} Higgs, inflaton

Aims and Questions

^:

a) Are two fine-tuning problems solved simultaneously.

Compute Higgs/inflaton potential V (φ) = ¹₂m²_φφ² + λφ⁴ + ...

b) Does SUSY play a role?

B. Model: gauge group SU(2) × global SU(2)_R − toy model Gauge multiplet (+ matter multiplet)

Vector A_M, Real scalar Σ, Majorana spinor λⁱ_L (i = 1, 2) Scalar H_i, Dirac spinor Ψ = (Ψ_L, Ψ_R)^T

C. Action: (in component fields) Pomarol & Quiros (98)

L_5,gauge = Tr [¹₂(F_{M N})² + · · · ]

L_5,mat = |D_MH_i|² − m²|H_i|² + ¯Ψ(iγ^MD_M − m)Ψ + · · ·

18

D. Compactification: M₄ × S¹ x^M = (x^µ, y), y ∼ y + L (L = 2πR)

• Set (twisted) boundary conditions.

λⁱ(x^µ, y + L) = (e^iβaσa)ⁱ _j λ^j(x^µ, y), same for Hⁱ other fields are periodic

• SUSY is broken `a la Sherk-Scharz. susy breaking ∝ β = √

(β^a)²

• Kaluza-Klein expansion:

A_M(x, y) = Σ^∞_n=−∞A⁽ⁿ⁾_M (x) e^iny/R, same for other fields λⁱ(x, y) = Σ^∞_n=−∞λⁱ⁽ⁿ⁾(x) e^{i(n+iβaσa/2π)y/R}, same for Hⁱ Zero modes of A_M are

A⁽⁰⁾_M = (A_µ, φ) (4)

19

E. Inflaton potential

Purpose: Compute effective potential V (φ).

A standard technique to use Wilson line phase.

g₅ ∫ _L

0 dy < A₅ >= g₅L < A⁽⁰⁾₅ >= diag(θ, −θ)

1-loop contributions (gauge multiplet and matter), periodic

V ^gauge(θ) = − · · · L⁻⁴Σ^∞_n=1n⁻⁵[1 + cos(2nθ)](1 − cosβ) + const V ^mat(θ) = · · · m²L⁻² · · ·

We use the n = 1 term.

V ^gauge ∝ β² φ² ∝ (susy breaking)²

= · · · /L⁴(1 − cosβ)(1 − cos(2θ))

θ ∝ vev of A⁽⁰⁾₅ as above, and θ = φ/f. (f is a scale parameter)

20

21

F. Constraints on potential from astrophysics data (and theory) 2) Slow-roll condition:

ϵ ≡ ¹₂((V ^′/V )M_P)² ≅ 0.01 η ≡ (V ^′′/V )M_P² ≅ 0.01

3) Spectral index: n_s = 1 − 6ϵ + 2η 0.95 ≤ n_s ≤ 0.97

4) e-folds: N ≡ ∫ _tf

t_i Hdt ∼ M_P⁻²| ∫ φ_f

φ_i (V /V ^′)dφ| ≅ 50 − 60 5) Curvature pert: δ_H = · · · V ^1/2/M_P²ϵ^1/2 = 1.91 × 10⁻⁵ 6) Quantum gravity effects may be ignored safely.

G₄/L²₅ ≪ 1 → R ≥ 0.8 × 10⁻²⁰ GeV⁻¹ Appelquist & Chodos (82)

22

6. Application

Does our toy model of inflaton has any realistic meaning in the context of GUT?

1) One scale parameter of the model is f ≡ 1/(2πgR), g = g₅/√

L φ = f θ inflaton field

2) The potential differs a bit depending on the value of f.

I) For f ≅ 10M_P, V is cosine function, deviates from chaotic.

¡¡¡

¡II) For¡¡ f ≫ 10M_P, V is chaotic.

3) We recall that m_φ is constrained by the value of δ_H, I), II) m_φ = (1.8 − 1.3) × 10¹³ GeV

23

4) We have applied our gauge-Higgs-inflaton unification model to gauge theories for typical values of f.

I) f = 10 M_P. g ≅ 0.2 − 0.0005, β/R ≅ 10¹⁵ − 10¹⁸ GeV

The 4D gauge coupling constant g² is somewhat smaller than the realistic value. In case II) (non-chaotic),

g²/4π ≅ 0.003 (or smaller) ←→ g_GUT² /4π ≅ 0.03

24

7. Summary and outlook + Part III

♥ Inflation parameters ϵ, η, n, N, δ_H

are reproduced without fine tuning by gauge-Higgs-inflaton unification picture.

This good accord may be by chance, or encouraging?

♥ Inflaton mass is m_φ ∼ 10¹³ GeV.

What does this value mean in GUT is unclear to us.

♥ SUSY helps us obtain a better value of the gauge coupl const.

25

B. More recent works, unfinished works

♠ We may extend the gauge theory on M₄ × S¹ to that on M₄ × T². Then what roles the A₅ and A₆ may play? → Part II

A⁽⁰⁾₅ = φ (inflaton), A⁽⁰⁾₆ = σ (curvaton)

Question: φ and σ, which is responsible for the curvature perturbation?

♠ We may embed higher-dim gauge theory in string theory, and compute string-loop effective action in open string field theory. − unfinished work

♠ The possibility that inflaton is a scalar field from higher-dim gravity.

g_{M N} = (

g_µν + A_µA_ν A_µn A_mν g_mn

)

(mod φⁿ) g_mn⁽⁰⁾ → inflaton (radion)

♠ Radion potential from 5D gravity loop + matter one-loop

26

27

♠ Higgs/inflaton from fuzzy extra dimensions

1-loop term in gauge theory on M₄ × fuzzy T²

©©©©

©©©©

CP ∝ 1/N (fuzzyness) Furuuchi, Okuyama & Inami (11)

28

Part II

₋ inflation versus curvature perturbation 1. Introduction

It is natural to extend the gauge theory on M₄×S¹ to higher dimensions, eg, that on M₄ × Tⁿ.

Question: Then, what roles the scalar fields A₅, A₆, … may play?

Answer: Two scalars may play the roles of inflaton and curvaton.

A₅ = φ (inflaton) responsible for inflation

A₆ = σ (curvaton) generates curv pert, P_ζ ∼< ζ(p₁)ζ(p₂) >

Origin of the large scale structure

Tiny anisotropy in CMB, δT /T ∼ δρ/ρ ∼ O(10⁻⁵) P_ζ ≅ 10⁻⁹ (WMAP)

Remark 1: In our 5D model inflaton φ is assumed to explain both inflation and curv pert.

29

Remark 2: Curvaton σ is introduced in 02 (Lyth), but without referring to inflation.

Aim: Evaluate the potential for both φ and σ, V (φ, σ), at 1-loop in higher-dim gauge theory.

2. Model

We construct 6D model and compute 1-loop potential.

· 6D SU(2) gauge theory

· Compactify on M₄ × T². KK modes A^(m,n)_M

two compactification radii: R₅, R₆

· Massless scalars :

A^(0,0)₅ = φ, A^(0,0)₆ = σ

30

3. Numerical analysis

We are concerned with inflation and curv pert.

Two situations for curv pert : I P_ζ = P_σ

II P_ζ = P_σ + P_φ

Results in case II :

small P_σ/P_φ (small R₅/R₆) ...

3) n_s = 0.960 ± 0.013 (→ 0.98 − 1.0 ?)

31