1. CMB beyond linear order arXiv : 1409.2464 Atsushi NARUKO (TiTech, Tokyo)

2. CMB beyond linear order → Cosmology in Modified Background arXiv : 1409.2464 Atsushi NARUKO (TiTech, Tokyo)

3. in collaboration with Eiichiro KOMATSU (MPA, Munich) Masahide YAMAGUCHI (TiTech) Anisotropic inflation reexamined : upper bound on broken rotational invariance during inflation arXiv : 1411.5489

4. our main result Anisotropic inflation (BG : Bianchi FLRW) : Watanabe, Kanno, Soda. ( 2009 ) v : preferred direction prediction : The original prediction for g * : @ attractor sol. Watanabe, Kanno Soda. ( 2010 ) N k : e-folds Our result : sorry m(__)m we couldn’t calculate the final value of g *..

5. our findings attractor mechanism doesn’t work (or induce too big anisotropy) (precise) in-in computation doesn’t yield N in g * need a careful study to predict the final g * with in-in comp. → needs δN (if it works)

6. 내용 1, Introduction (cosmic no-hair theorem, statistical anisotropy, …) 2, Anisotropic inflation --- background 3, Anisotropic inflation --- perturbations 4, Conclusion

7. Introduction

8. cosmic no-hair conjecture cosmological constant & ordinary matter fields → the universe will be isotropized and evolve toward de Sitter ＝ initial anisotropy disappear Gibbons & Hawking (1977), Wald (1983) inflaton potential (in a simple model) behaves like a c.c. Naively no-hair conjecture can be applied to inflationary universe. → anisotropy & anisotropic matter (cf. vector field) disappear ??

9. symmetry during inflation de Sitter spacetime ( ≒ the universe during inflation) has time-translation invariance How about rotation symmetry ? is invariant under 3d-rotation ? if it is broken, how much extent can it be broken ? O (ε) ?? invariant under and → scale-invariant spectrum But this symmetry is actually broken → spectrum is not exactly scale- invariant !! @ 5σ by PLANCK

10. observationally testable ansiotropy of P(k) = anisotropy in Fourier space (≠ real space) amplitude of ---- ≠ Kim & Komatsu (2013) Making the precise theoretical prediction of g * is essential !!

11. 이방성 인플레이션 --- background ---

12. Anisotropic inflation Consider a homogeneous vector field, Watanabe et al. (2009) The BG spacetime is described by a Bianchi type I universe ; Equation for a homogeneous vector field :

13. sketch of BG equations Hamiltonian constraint : equation for H : equation for φ : equation for σ : Interestingly, 4 quantities (ρ, P, π and V eff ) can be expressed in terms of a single quantity : V (a function of φ and b) ← sol. Shear will rapidly converge to a terminal value :

14. attractor solution ?? equation of scalar field : → I (= ρ V & shear) @attractor is determined only by “c” !! ＋ slow- roll : (c ≠ 1) Watanabe et al, (2009) However, this solution exists only if 1 >> e - 4(c - 1) α, c – 1 = O (1)

15. Anisotropic inflation --- perturbations ---

16. number of degrees of freedom In total, 15 = 10 (metric) + 1 (scalar field) + 4 (vector field) metric :vector field : scalar field : (3d-tensor, GW) (3d-scalar) 5 constraints = 1 (Hamiltonian) + 3 (momentum) + 1 (U(1)) 5 gauge d.o.f = 1 (temporal) + 3 (spatial) + 1 (U (1)) In summary, 5 (= 15 - 5 - 5) degrees of freedom

17. eom of δφ After “some” computation, In isotropic limit, δφ ⇔ standard scalar perturbation with → “I” must be small, O(ε) (← δφ should be scale-invariant) ε H : slow-roll parameter → nature of δφ ≒ that of CMB !!

18. (anisotropic) Power spectrum power spectrum sol of δφ : → homogeneous = free (tree) inhomogeneous = interaction

19. evaluation of integral interaction part : (longwave limit) H -1 L t η = η ε ΔN ε g * : cf.

20. g* → EXCLUDED Anisotropy during inflation is incredibly small << O(ε) !!! current observational upperbound of g* : Kim & Komatsu (2013) <<< breaking of rotational sym. breaking of t-translation sym. σ = O (10 -8 ε) n s – 1 = O (ε) attractor case (exists only if c – 1 = O (1) )

21. 결론 We have reexamined a model of anisotropic inflation. We have derived the new formula for g * finding that the attractor branch is ruled out by the constraint of g *. g * depends on 1/ε 2 in stead of N 2. To predict the final value of g *, it needs the precise estimation of fluctuations on large scales or δN formalism (gradient expansion in anisotropic universe)

22. 감사합니다

24. crucial points attractor mechanism doesn’t work (or too big anisotropy) (precise) in-in computation doesn’t yield N in g * needs numerics to predict the final value of g* with in-in comp. At an intermediate time, g * is H -1 L t t = t ε ΔN ε → needs δN (if it works)

25. (anisotropic) Power spectrum power spectrum sol of δφ : → homogeneous = free (tree) inhomogeneous = interaction after some computation N ε : ε できまる。 N ε < 60 cf.

26. evaluation of integral Φ のふるまいは、 1, ε N 〜 O (1) なので、 e の肩を展開することはできない。 (longwave limit) g * の評価は、 ( 先に ε = 0 をとる ) 2, 自由場の解 ( ハンケル ) の長波長での取り扱いは注意が必 要！ ν は定数 ν が変化すると、解にならない 3, ⇔ N -> 1/ε !! でも g * のオーダーはかわらない。。

27. Inflation … + α ?? Standard theory of the early universe : Big bang cosmology + Inflation cosmology ☝ solve several problems (horizon, flatness …) ☝ give the seed of density fluctuations (CMB, large-scale structures) Theory of a scalar field (s?) in light of PLANCK (BICEP2 ??) → standard scenario (canonical, m 2 φ 2, BD vacuum…) How about other fields (vector or gauge fields) ?

28. cosmic no-hair conjecture In the presence of cosmological constant & matter fields respect - Dominant energy condition : ρ ≥ 0, ρ ≥ |P| - Strong energy condition : ρ + 3 P ≥ 0, ρ + P ≥ 0 Gibbons & Hawking (1977), Wald (1983) → the universe will be isotropized and evolve toward de Sitter σ : anisotropy (shear) (vector field)

29. Inflation with anisotropic hair Very recently, a counter example was found !! Watanabe, Kanno Soda. (2009) The model with non-trivial kinetic term : ☝ f (φ) breaks the conformal invariance of vector field ☝ no ghost/tachyon instabilities non-trivial kinetic function v : preferred direction This model predicts (statistically) anisotropic power spectrum

30. ① Conformal invariance Vector field : conformal invariance of vector field → vector field is decoupled from cosmic expansion and hence no interesting effect can be expected. In Friedmann universe : cancel out ＝ → S V FLRW = S V Min

31. ② Ghost (tachyon) instability several models breaking conformal invariance All these models (with standard kinetic term) suffer from ghost (tachyon) instabilities in perturbations … Himmetoglu et al. (2009) -- Lorentz violation : -- non-minimal coupling : Golovnev et al. (2008) -- with vector potential : Ford (1989) Ackerman et al. (2007)

32. vector field on FLRW The model : scalar + vector fields with a special coupling vector field on FLRW universe (on large scales) : → vector field can survive on large scales if 4c – 1 > 0. large scale fluctuations 〜 (locally) homogeneous vector field !! → realization of Bianchi universe (homo. but anisotropic) !

33. attractor solution ? scalar field equation under slow-roll approximation Watanabe et al. (2009) I (= ρ V and σ) @ attractor is determined only by c !! where exact solution ( ⊃ attractor solution) : (c ≠ 1) However, this solution exists only if 1 >> e - 4(c - 1) α, c – 1 = O (1)

34. linear perturbation analysis Now let us study perturbations !!  (statistically) anisotropy in PS for scalar and tensor  cross correlation between scalar and tensor -- g * could be order of ε ? cf. n S – 1 = O (ε) Bianchi universe has less symmetry than FLRW (3D -> 2D ). → only 2 types of perturbations = 2d-scalar + 2d-vector 4 2d-scalar 2 2d-vector 2 3d-scalar 2 3d-vector 2 3d-tensor coupling

35. number of degrees of freedom In GR, 1S & 1V (2 3d-tensor) modes can propagate in vacuum. adding scalar field φ → 1S (3d-scalar) propagating mode. adding vector field A μ → 1S and 1V modes (1 constraint + 1 gauge d.o.f reduce 2 modes) To summarize, 1 + 1 + 1 = 3 scalar and 1 + 1 = 2 vector metric :vector field : scalar field : scalar type (3d-tensor, GW) (3d-scalar)

36. expansion of the action ① After expanding the action and solving constraints, the action is auto-correlationcross-correlation Since φ corresponds to the standard scalar type perturbation, its property must be consistent with that of CMB. cf. nearly scale-invariance of φ → “I” must be ≦ O(ε) ε H : slow-roll parameter

37. expansion of the action ② L AA depends on the branches ( I = const. or e 4(c - 1)α )!! Hierarchy among couplings : → the strongest one is the coupling b/w Φ and A μ ＜＜ → vector on FLRW ?? Bartolo, Matarrese, Peloso, Ricciardone (2012) ☝ attractor (I = const.) : = scale invariant ! ☝ new branch (I ∝ e 4(c - 1)α ) : 〜 2 (c → 1)

38. anisotropic power spectrum The spectrum is anisotropic cf. O (I 2 ) g* (after a bit annoying but straightforward computations…) : O (I) >>

39. g* → EXCLUDED Anisotropy during inflation is incredibly small << O(ε) !!! current observational constraint of g* : Kim & Komatsu (2013) current observational constraint of g* : <<< breaking of rotational sym. breaking of t-translation sym. σ = O (10 -8 ε) n s – 1 = O (ε) attractor case (exists only if c – 1 = O (1) )

40. comment on N -> 1/ε Why we have obtained 1/ε instead of N ?? How much different ?? very m---ild correction → monomial potential : U (φ) ∝ φ n ☝ no divergence (≠ N 2 ), merely suppressed by 1/ε 2. → (qualitatively) quite different !! mode func. of φ or A μ ↑

41. references papers by original authors ( Watanabe, Kanno, Soda) -- CQG 29 (2012) arXiv:1201.6434 -- Rhys. Rept. 528 (2013) arXiv:1212.2921 -- PRL 102 (2009) arXiv:0902.2833 [anisotropic inf. (BG)] -- PTP 123 (2010) arXiv:1003.0056 [perturbation] reviews on anisotropic inflation by Soda et al. paper by us -- arXiv:1n??.???? [superhorizon dynamics, δN-like] -- arXiv:1411.5489 [perturbation, in-in computation ] in principle, n can be 4 but …