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

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

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

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 *..

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)

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

Introduction

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 ??

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

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 !!

이방성 인플레이션 --- background ---

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 :

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 :

attractor solution ?? equation of scalar field : → I (= ρ V & 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)

Anisotropic inflation --- perturbations ---

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 (= ) degrees of freedom

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 !!

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

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

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) )

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)

감사합니다

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)

(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.

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。。

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) ?

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)

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

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

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)

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) !

attractor solution ? scalar field equation under slow-roll approximation Watanabe et al. (2009) I (= ρ V and 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)

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

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, = 3 scalar and = 2 vector metric :vector field : scalar field : scalar type (3d-tensor, GW) (3d-scalar)

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

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)

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

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) )

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 ↑

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