Thoughts on realistic inflation models 2016 - PowerPoint PPT Presentation

現実的なインフレーション模型は何か Thoughts on realistic inflation models 2016 寺田 隆広 Korean Institute of Advanced Study 素粒子物理学の進展 2016, 基礎物理学研究所 , 2016/9/6

DISCLAIMER • 主にインフレーションのレビューです。 • 限られた経験 / 知識に基づき、偏見に満ちています。 • 個々の模型では、議論に色々な抜け道があります。 • 皆様の模型が出なくても怒らないでください。 
 （コメントは歓迎。） • 寄り道して関連した自分の仕事を紹介します。

3 Outline 1. Introduction : inflation in a nutshell 2. Universality classes of inflation 
 Realization by “pole inflation” 3. Initial conditions : Small-field or Large-field? 4. Shift symmetry and its origin 
 U(1): pNGB or Wilson line 
 Weak Gravity Conjecture 
 R : scale invariant models 5. Summary & Conclusion

Introduction: inflation in a nutshell

5 宇宙の加速膨張 動機・利点 一様等方宇宙 • 指数関数的膨張によって、一様性問題、平坦性問題、モノポール問題を解決する。 • インフラトンの量子揺らぎにより、宇宙の大規模構造の「種」をつくる。 Einstein eq. Friedmann eq. ✓ ˙ R µ ν − 1 ◆ 2 a = 8 π G ρ FLRW universe 2 g µ ν R = 8 π GT µ ν a 3 a = − 4 π G ¨ T µ ν = diag( − ρ , P, P, P ) a d s 2 = − d t 2 + a ( t ) 2 (d x 2 + d y 2 + d z 2 ) ( ρ + 3 P ) 3 Slow-roll P ' � ρ ' const. ρ = 1 φ 2 + V ˙ slow-roll 近似 2 a ( t ) ' e Ht ◆ 2 P = 1 ✓ V 0 ✏ = 1 ˙ φ 2 − V ⌧ 1 2 2 V Z φ Z t end � � V 00 d � � � | η | = � ⌧ 1 φ + V 0 = 0 φ + 3 H ˙ ¨ � � N = H d t = V � √ 2 ✏ φ end t

観測者 加速する観測者 解析接続された世界 ホワイトホール ブラックホール 6 観測者 [Hawking, Commun.Math.Phys. 43 (1975) 199, [Unruh, PRD14 (1976) 870] [Gibbons, Hawking, PRD15 (1977) 2738] Erratum: ibid. 46 (1976) 206] Minkowski 時空 Black Hole de Sitter 宇宙 r = 0 t = ∞ r = ∞ r = 0 Cosmological horizon Event horizon Rindler horizon T = a T = H T = κ Hawking radiation Gibbons-Hawking radiation Unruh radiation 2 π 2 π 2 π δφ = H inflaton fluctuation 2 π ζ = δ N = H δ t = H curvature perturbation δφ ˙ φ ✓ k ◆ n s − 1 Power spectra n s − 1 = − 6 ✏ + 2 ⌘ P s ( k ) = A s k ∗ ✓ k r ≡ A t ◆ n t = 16 ✏ P t ( k ) = A t k ∗ A s

7 Excellent fit by Λ CDM Scale invariant ( n s ∼ 1) (Planck TT+ low P) n s = 0 . 9655 ± 0 . 0062 Adiabatic ( β iso ∼ 0) β iso (0 . 002 Mpc − 1 ) < 4 . 1 × 10 − 2 (for CDM) Gaussian ( f NL ∼ 0) f local (Planck TT+ low P) = 0 . 8 ± 5 . 0 NL r n s [Planck collaboration, 1502.02114, 1502.01592]

Universality classes of inflation

9 Analogy to Renormalization Group The Hamilton-Jacobi formalism φ ( t ) ↔ t ( φ ) “superpotential” W ( φ ) ≡ − H W φ = ˙ → This implies: V = 3 W 2 − 2 W 2 φ / 2 φ d g d φ d ln µ = β ( g ) d ln a = β ( φ ) cf .) s ˙ 3( P + ⇢ ) � ( � ) = − 2 W φ � √ where W = H = ± = ± 2 ✏ ⇢ → classified by the behavior near the fixed point (de Sitter).

10 Underlying connections? [McFadden, Skenderis, 0907.5542, 1001.2007] See also, dS/CFT and FRW/CFT. [Strominger, hep-th/0106113] [Freivogel et al., hep-th/0606204] [Witten hep-th/0106109] [Sekino et al., 0908.3844] [Larsen et al., hep-th/0202127] [Halyo, hep-th/0203235]

11 Universality classes of inflation [Mukhanov, 1303.3925] [Roest, 1309.1285] [Garcia-Bellido et al., 1402.2059] [Binetruy et al., 1407.0820]

12 Universality classes of inflation Figures from [Garcia-Bellido, Roest, 1402.2059]

13 観測的ステータス ◎ ◯ △ × △ △ Universality classes of inflation [Mukhanov, 1303.3925] [Roest, 1309.1285] [Garcia-Bellido et al., 1402.2059] [Binetruy et al., 1407.0820] Any underlying mechanism for the universality?

14 Inflationary Attractor Models Model space observables r predictions n s a limit of a parameter “attraction” r n s

15 Unity of cosmological attractors ( √− g ) − 1 L = − 1 2 Ω ( φ ) R − 1 2 K J ( φ )( ∂ µ φ ) 2 − V J ( φ ) [Galante, Kallosh, Linde, Roest, 1412.3797]

16 Unity of cosmological attractors ( √− g ) − 1 L = − 1 2 R − 1 2 K E ( ϕ )( ∂ µ ϕ ) 2 − V E ( ϕ ) [Galante, Kallosh, Linde, Roest, 1412.3797] 3 α / 2 K E ( φ ) ' ( φ � φ 0 ) 2 2nd order pole(s) in ! K E Inflation occurs near the pole. Canonical normalization makes the potential flat.

17 Pole inflation � − 1 L = − a p � √− g 1 − ϕ + O ( ϕ 2 ) � � 2 ϕ p ∂ µ ϕ∂ µ ϕ − V 0 2 ✓ ◆ p − 2 + · · · 8 ⇣ ⌘ − p − 2 1 � ( p 6 = 2) , V 0 2 √ a p φ < V = 1 � e − φ / √ a p + · · · � � ( p = 2) , V 0 : p p ✓ ◆ r = 8 a p p − 1 n s = 1 − ( p − 1) N ( p − 1) N a p [Galante, Kallosh, Linde, Roest, 1412.3797] [Broy, Galante, Roest, Westphal, 1507.02277]

18 Change of potential shape The original potential 3.5 3.0 2.5 2.0 1.5 1.0 0.5 - 1.0 - 0.5 0.5 1.0 0 < p < 2 p ≥ 2 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 - 0.8 - 0.6 - 0.4 - 0.2 2 3 4 5 6 “hilltop” “inverse-hilltop”

19 Inflation with a singular potential The original diverging potential 40 30 20 10 - 1.0 - 0.5 0.0 0.5 1.0 p = 2 p > 2 140 50 120 40 100 80 30 60 20 40 10 20 2 3 4 5 6 0.5 1.0 1.5 “power-law” “chaotic”

20 Inflation with a singular potential [Rinaldi, L. Vanzo, S. Zerbini, and G. Venturi, 1505.03386] [TT, 1602.07867] � − 1 L = − a p 2 ϕ p ∂ µ ϕ∂ µ ϕ − C � √− g ϕ s (1 + O ( ϕ )) 8 2 s p − 2 + · · · ⇣ ⌘ p − 2 ( p 6 = 2) , C 2 √ a p φ < V = Ce s φ / √ a p + · · · ( p = 2) . : Potentials for monomial chaotic and power-law inflation 8 s n s =1 − p + s − 2 r = ( p − 2) N ( p − 2) N

21 Summary of general pole inflation For more details, see [TT, arXiv:1602.07867]. p=1 1<p<2 p=2 2<p non-singular potential alpha-attractor 2nd order hilltop xi-attractor hilltop inverse-hilltop generalization of Starobinsky model natural inflation Higgs inflation singular potential power-law inflation monomial run-away run-away (exponential potential) chaotic

22 Correspondence to universality classes of inflation

23 General Pole Inflation As a realization of Universality Classes Figures from [Garcia-Bellido, Roest, 1402.2059] p > 2 w/ sing. pot. w/ sing. pot. p = 2 p < 2 p = 1 p > 2 w/ log. corr. p = 2 p = 2

ここまでのまとめ インフラトン作用の極と次数による分類は、 
 さて、どのクラスが現実的でしょう？ 24 インフレーション模型は Universality class に分類できる。 • • それを実現する具体例となっている。 •

Initial condition problems: Small-field or Large-field?

26 How likely the slow-roll is? Small field Large field Figures from a review [Brandenberger, 1601.01918]

３＋１次元数値計算で 非一様性が大きくてもインフレーションが起こる事を示した。 27 Inhomogeneous initial conditions [East, Kleban, Linde, and Senatore, 1511.05143] スカラー場の揺らぎがスローロールの領域を越えない限り hr φ · r φ i = 10 3 Λ 結論 small field: チューニングが必要 large field: robust

28 △ ◎ ◎ ◎ ◎ 初期条件 △ × △ ◯ ◎ 観測的ステータス × Universality classes of inflation [Mukhanov, 1303.3925] [Roest, 1309.1285] [Garcia-Bellido et al., 1402.2059] [Binetruy et al., 1407.0820] Any underlying mechanism for the universality?

Shift symmetry and its origin

30 Planck-suppressed terms are NOT suppressed enough! ✓ φ ◆ n V ∼ m 2 φ 2 + λ 3 φ 3 + λ 4 φ 4 + X λ 4+ n φ 4 M P n> 4 In particular, V φ 2 η = O (1) M 2 P Also a naturalness question: m ⌧ M P (or Λ ) Why ? See e.g. a good review [Westphal, 1409.5350].

31 Shift symmetry φ → φ + c with an explicit, soft breaking V ( φ ) ⌧ 1 Perturbative quantum gravity corrections: [Smolin, PLB 93, 95 (1980)] ✓ a V 00 ◆ + b V δ V ∼ V [Linde, PLB 202 (1988) 194] M 2 M 4 [Kaloper, Lawrence, Sorbo, 1101.0026] P P Technically natural. [’t Hooft, NATO Sci.Ser.B 59 (1980) 135]

32 Shift symmetry in SUGRA SUSY breaking e ff ects m soft ∼ O ( H ) Why ( ) ? m ⌧ H η ⌧ 1 SUGRA scalar potential + 1 V = e K ⇣ j W − 3 | W | 2 ⌘ ¯ ji D i W ¯ 2 f AB D A D B K D ¯ where . D i W = W i + K i W shift symmetry [Kawasaki, Yamaguchi, Yanagida, hep-ph/0004243] K ( Φ , ¯ Φ ) = K ( i ( Φ − ¯ Φ )) Φ → Φ + c

33 U(1) R Origin of the shift symmetry? shift symmetry … non-linearly realized symmetry … any linear realization? [Freese, Frieman, Olinto, PRL 65 (1990) 3233] Axion Dilaton pNG boson of U(1) or pNG boson of scale invariance Wilson line of extra dimensions Inverse-hilltop class と chaotic class は、このどちらかに帰着するべき？