Multi-field -attractor in fundamental theory Yusuke Yamada - - PowerPoint PPT Presentation

Multi-field α-attractor in fundamental theory

Yusuke Yamada (Stanford Univ.)

collaborators: R. Kallosh, A. Linde, D. Roest, A. Westphal, T. Wrase

Outline

1. 2.

ds2 = 3α(dr2 + r2dθ2) 1 − r2

ds2 = 3α(dτ 2 + dχ2) 4τ 2

r = tanh ✓ φ √ 6α ◆

τ = e−√

2 3α ϕ

inflation dark matter

V (r) ∼ V (0)(1 − ae−√

2 3α φ + · · · )

V (τ) ∼ V (0)(1 − ˜ ae−√

2 3α ϕ + · · · )

Outline

1. 2.

inflation dark matter

α-attractor from fundamental theory

Hyperbolic geometry M-theory/11D supergravity ➡ 4D N=8 supergravity

T2 × T2 × T2

superstring ➡10D supergravity on

From

4D N=1 supergravity reduction

L ⊃

7

X

i=1

∂τi∂τi + ∂χi∂χi 4τ 2

i

7-disk moduli with αi = 1

3

We find α=1/3 (r~0.001) in fundamental theory

Is it possible to have r > 0.001?

r = 12α N 2

Merger of α-attractors

We find α=1/3 (r~0.001) in fundamental theory

Is it possible to have r > 0.001?

r = 12α N 2

Merger of α-attractors

We find α=1/3 (r~0.001) in fundamental theory

Is it possible to have r > 0.001?

L = −3α1 4τ 2

1

∂τ1∂τ1 − 3α2 4τ 2

2

∂τ2∂τ2 − V τ1 = τ2 = τ

if

L = −3(α1 + α2) 4τ 2 ∂τ∂τ − V

Effective value increases!

• S. Ferrara, R. Kallosh (2016)
• R. Kallosh, A. Linde, T. Wrase, YY (2017)

r = 12α N 2

Merger of α-attractors

V = m2|1 − T1|2 + m2|1 − T2|2 + M 2|T1 − T2|2

Merger of α-attractors

e.g.

Ti = τi + iχi

α1 = α2 = 1 3

τi = e−

√ 2φi

χi = 0

For

T1 = T2

Inflation takes place along e.g.

Ti = τi + iχi

τ1 = τ2 → αeff = α1 + α2

α1 = α2 = 1 3

τi = e−

√ 2φi

χi = 0

Two directions merge

V = m2|1 − T1|2 + m2|1 − T2|2 + M 2|T1 − T2|2

m ⌧ M

Merger of α-attractors

T1 = T2

Inflation takes place along e.g.

Ti = τi + iχi

τ1 = τ2 → αeff = α1 + α2

α1 = α2 = 1 3

τi = e−

√ 2φi

χi = 0

Two directions merge

V = m2|1 − T1|2 + m2|1 − T2|2 + M 2|T1 − T2|2

m ⌧ M

Merger of α-attractors

More general merger takes place in string theory: Fibre inflation

τ1τ 2

2 ∼ V2

V : Calabi-Yau volume

Moduli = (6D) CY volume: Coupling to other sector stabilizes the volume:

V = const

L ∼ −∂τ1∂τ1 4τ 2

1

− 2∂τ2∂τ2 4τ 2

2

− V

τ1 ∼ τ −2

2

Generalized merger:

α1 = 1 3, α2 = 2 3 → αeff = 2

Merger of α-attractors may have important meanings in fundamental theories

−∂τ1∂τ1 4τ 2

1

− 2∂τ2∂τ2 4τ 2

2

∼ −6∂τ2∂τ2 4τ 2

2

C.P. Burgess, M. Cicoli, F. Quevedo (2008)

• R. Kallosh, A. Linde, D. Roest, A. Westphal, YY (2017)

τ1 = τ p

2

Merger of α-attractors

Cascade inflation

V = Vinf + Vmerger

Vmerger = Vmerger ✓ e

− q

2 3α1 φ1, e

− q

2 3α2 φ2

◆

Two (or more) inflationary region with different heights

• R. Kallosh, A. Linde, D. Roest, YY (2017)

Application to Initial condition problem, Dark energy, low-l power suppression etc.

• R. Kallosh, A. Linde, D. Roest, YY (2017) Y. Akrami et al (2017)
• T. Fujita, S. Mizuno, YY, work in progress

Outline

1. 2.

inflation dark matter

Axions in α-attractor

• LHC has not yet discovered
• Cosmological problems (e.g. gravitino/moduli)
• Decompactification problem in string models (KKLT/LVS)

What can be dark matter?

some issues of low scale supersymmetry (breaking)

• R. Kallosh, A. Linde (2004)
• J. Conlon, R. Kallosh, A. Linde, F. Quevedo (2008)

a simple solution to these issues: m3/2 ≥ H

Axions in hyperbolic geometry

?

Axions in α-attractor

ds2 = 3αdTd ¯ T (T + ¯ T)2

ds2 = 3αdZd ¯ Z (1 − Z ¯ Z)2

(nonlinear) U(1) symmetry ➡ (light) axion field

Light axion & α-attractor inflation

T = τ + iχ Z = reiθ

L = − 3α∂T∂ ¯ T (T + ¯ T)2 − V (T + ¯ T) L = − 3α∂Z∂ ¯ Z (1 − Z ¯ Z)2 − V (Z ¯ Z)

small correction (~ axion potential) gives an axion mass

χ → χ + c

θ → θ + η

Axions in α-attractor

ds2 = 3αdTd ¯ T (T + ¯ T)2

ds2 = 3αdZd ¯ Z (1 − Z ¯ Z)2

T → T + ic Z → Zeiη T = τ + iχ Z = reiθ

θ

χ

CDM = axion oscillation

PS = ✓δΩa Ωa ◆2 / f 2

ahδθ2 ∗i

PS < 0.03Pζ

Constraint on isocurvature perturbation

Suppression of isocurvature perturbation

Isocurvature perturbation

e.g. for QCD axion

Hinf < 0.86 × 107GeV ✓ fa 1011GeV ◆0.408

h(faδθ∗)2i = ✓ H 2π ◆2

• P. A. R. Ade et al. [Planck collaboration] (2015)
• P. A. R. Ade et al. [Planck collaboration] (2015)

usual case:

Suppression of isocurvature perturbation

axion has large kinetic coefficient

L = − 1 2∂φ∂φ − 3α 4 sinh2 r 2 3αφ ! ∂θ∂θ − V (φ)

a = faθ

* cano. axion (today)

f 2

∗ = 3α

2 M 2

pl sinh2

r 2 3αφ∗ !

Compared with the usual case, quantum fluctuation is extremely suppressed

c.f. usual case

h(faδθ∗)2i = ✓fa f∗ ◆2 h(f∗δθ∗)2i = ✓fa f∗ ◆2 ✓ H 2π ◆2

h(faδθ∗)2i = ✓ H 2π ◆2

where

∼ 8N 2M 2

pl

3α

• Y. Ema, K.Hamaguchi, T. Moroi, K. Nakayama (2016)
• A. Linde, YY, work in progress

Summary

Multiple hyperbolic moduli from fundamental theory Merger of α-attractors

• increases PGW amplitude
• is realized as e.g. moduli stabilization of extra dim.
• leads to cascade inflation

Axions in hyperbolic geometry is a good dark matter candidate:

• It can be naturally light even for high SUSY breaking
• Isocurvature perturbation is suppressed by geometric effect
• If U(1) = PQ sym., strong CP is also solved

Various mysteries (DM, DE, strong CP…etc) might be explained by (multiple) hyperbolic moduli field!

Appendix

Coupling α-attractor to SUSY breaking field No superpotential for inflaton-axion multiplet U(1) symmetric Kahler potential

U(1) symmetric α-attractor

Mass splitting between inflaton and axion due to SUSY

K = K(T + ¯ T, S, ¯ S)

W = W0(1 + S)

• r

K = K(Z ¯ Z, S, ¯ S)

V |S-fixed = V (T + ¯ T) or V (Z ¯ Z)

inflaton potential purely from SUSY axion potential is introduced as small correction

YY (2018)