Brane-World Cosmology and Inflation

Transcription

1 ICGC04, 2004/1/5-10 Brane-World Cosmology and Inflation Extra dimension G µν = κ T µν? Misao Sasaki YITP, Kyoto University

2 1. Introduction Braneworld domain wall (n 1)-brane = singular (time-like) hypersurface embedded in (n + 1)-dim spacetime brane tension (σ) = vacuum energy (σ > 0? or σ < 0?) vacuum energy cosmological constant on the brane causality on the brane causality in the bulk H. Ishihara, PRL86 (2001) 2 brane's lightcone A new picture of the universe! bulk's lightcone

3 Randall-Sundrum (RS) Brane-World PRL 83, 3370 (1999); 83, 4690 (1999) 5D-AdS bounded by 2 branes: R 4 (S 1 /Z 2 ) ds 2 = dy 2 + e 2 y /l η µν dx µ dx ν ( r c y r c ) l 2 = 6 Λ 5, brane tension: σ ± = ± 3 4πG 5 l b(y) = e y /l is called the warp factor b( y ) 3 negative tension positive tension negative tension y 0-0 y 0 y

4 vacuum energy (brane tension) cosmological constant. Λ 4 = 1 2 Λ 5 + (8πG 5 σ) 2 /12 r c is arbitrary Existence of Radion mode. 4 Brans-Dicke type gravity on the branes unless stabilization mechanism. (Garriga & Tanaka, 99) In the limit r c, the negative tension brane disappears. single-brane model 5th dimension is non-compact Gravity confined within l Λ 5 1/2 from the brane No radion modes (no relative motion) Einstein gravity is recovered on scales l Φ Newton = G 5M G 5M r 2 r l l r = G 4M r The single-brane model is cosmologically more attractive.

5 2. Brane Cosmology in AdS 5 -Schwarzschild Bulk Kraus ( 99); Ida ( 00); 5D AdS-Schwarzschild in Static Chart: ds 2 = A(R)dT 2 + dr2 A(R) + R2 dω 2 K A(R) = K + R2 l α2 (K = ±1, 0, α 2 = 2G 2 R 2 5 M) For α 2 = 0 and K = 1, (T, R) (t, r); R(t, r) = l sinh (r/l) cosh (Ht) T (t, r) = l arctan ( tanh (r/l) sinh (Ht)) 5 ds 2 = dr 2 + l 2 sinh 2 (r/l)( H 2 dt 2 + cosh 2 (Ht)dΩ 2 (3) ) Any r =const. timelike hypersurface is 4D de Sitter space. de Sitter brane at r = r 0 (with Z 2 symmetry): (T µν = σ g µν ) σ = 3 4πG 5 l coth (r 0/l), H 2 l 2 = 1 sinh 2 (r 0 /l)

6 6 Deviates from de Sitter if T µν is non-trivial: T n τ brane trajectory : "bulk" { R = R(τ ) T = T (τ ) ds 2 brane = ( A(R) T 2 + R Ṙ2 A(R) ) dτ 2 + R 2 (τ )dω 2 K Choose τ to be Proper time on the Brane: A(R) T 2 + Ṙ2 A(R) = 1

7 7 A 2 (R) T 2 = Ṙ 2 + A(R) Junction condition under Z 2 -symmetry: [K µν ] + = 2K µν(+0) = 8πG 5 [T µν (1/3)T g µν ] K µν = 1 2 L nq µν ; n a = (Ṙ, T, 0, 0, 0) q µν induced metric on the brane T µ ν = diag( ρ, p, p, p) σ c δ µ ν ; σ c = 3 (Ṙ R A(R) R T = 4πG 5 3 (ρ + σ c) = 4πG 5 3 ρ + 1 l G 4 = G 5 /l 4D Newton const. A(R) = K + R 2 l 2 α2 R 2 ) 2 + K R 2 = 8πG 4 3 ρ + l2 ( 4πG4 3 ρ ) 2 + α2 R 4 4πG 5 l

8 Friedmann equation on the brane: Binetruy et al. ( 99) (Ṙ ) 2 + K R R = 8πG ( ) ρ + 4πG4 l2 3 ρ + α2 R 4 8 presence of ρ 2 and α 2 /R 4 terms. ρ 2 -term dominates in the early universe: H ρ. ) 1/4 For ρ R 4, K = α 2 = 0; R (t + 2t2 l reduces to standard Friedmann equation for l 2 G 4 ρ 1. (l 2 G 4 ρ 1 ρ σ c ) α 2 /R 4 -term: dark radiation (or Weyl fluid) α 2 = 2G 5 M 5D (bulk) BH mass α 2 R = E tt 4 3 ; E µν = (5) C µaνb n a n b (5D Weyl contribution) Shiromizu, Maeda & MS ( 99), Maartens ( 00),...

9 3. Quantum Creation of Inflationary Brane-World 9 Euclidean AdS: H 5 (O(5, 1)-symmetric) ds 2 = dr 2 + l 2 sinh 2 (r/l)(dχ 2 + sin 2 χ dω 2 (3) ) Brane at r = r 0 (with Z 2 -symmetry) = des brane instanton r = 0 r = 0 r = r 0 χ = π 2 topology S 5 χ = π/2 : totally geodesic 4-surface

10 Creation of inflationary brane-world Garriga & MS ( 99); Koyama & Soda ( 00) Analytic continuation: χ iht + π/2 ds 2 = dr 2 + (Hl) 2 sinh 2 (r/l)( dt 2 + H 2 cosh 2 Ht dω 2 (3) ) (r r 0) 10 r = 0 r = r 0 r = 0 Spatially Compact 5D Universe (Identify) Universe is naturally born with inflation. Initial value (Cauchy) problem is well-posed. (RS flat brane is recovered in the limit r 0 )

11 D Graviton and Kaluza-Klein Excitations 5D gravitational perturbation of de Sitter brane universe: ds 2 = dr 2 + (Hl) 2 sinh 2 (r/l)ds 2 ds 4 + h ab dx a dx b = b 2 (z) ( ) dz 2 + H 2 ds 2 ds 4 + hab dx a dx b dr dz = (conformal radial coordinate): l sinh(r/l) l b(z) = (sinh z 0 = Hl ; < z < ) sinh( z + z 0 ) (generalized) Randall-Sundrum gauge: h 55 = h 5µ = h µ µ = D µ h µν = 0 ; D µ : 4D covariant derivative h µν 5 degrees of freedom (= 1 scalar + 2 vector + 2 tenser)

12 Perturbation equations: h µν = b 1/2 ϕ(z)h µν (x) Volcano potential for ϕ: V (z) (b3/2 ) b 3/2 = ϕ + (b3/2 ) b 3/2 ϕ = m2 H 2ϕ ( (4) + 2H 2 + m 2 )H µν = sinh 2 ( z + z 0 ) coth η 0δ(z) 12 V( z ) 9 4 z V( z ) 9 4 for z + 4D graviton (zero mode m = 0): ϕ b 3/2 h µν b 2 KK excitations (m > 0): V (9/4) m > (3/2)H

13 Zero mode is confined just like the case of the RS flat brane. When quantized, however, the normalization (amplitude) is non-trivial: Langlois, Maartens & Wands ( 00) P gw (k)k 3 ( ) H 2 F (Hl) 2π F(x) G x F (x) = ( 1 + x 2 x 2 ln [ x 2 x ]) 1 large enhancement at Hl 1: F 3 2 Hl. Mass gap in KK spectrum: m KK (3/2)H The same is true for any bulk scalar field with M H. No zero-mode (bound-state mode) for M H.

14 5. Bulk Inflaton Model (Brane Inflation without Inflaton on the Brane) Randall-Sundrum s default parameters: brane tension: σ c = 3 4πG 5 l ; l = If σ > σ c, then inflation occurs on the brane: H 2 = 1 1 l 2 σ l = 1 Λ 5 2 l 2 σ 6 ; σ 3 4πG 5 l σ If σ = σ c but Λ 5,eff < Λ 5, inflation also occurs on the brane: H 2 = Λ 5 Λ 5,eff 6 Brane-world inflation can be driven solely by bulk (gravitational) dynamics. 6 Λ 5 1/2. 14 Kobayashi & Soda ( 00), Himemoto & MS ( 00) Himemoto, Sago & MS ( 01), Himemoto, MS & Tanaka ( 02)

15 5D Einstein-scalar system with a brane 15 G ab + Λ 5 g ab = κ 2 5 T ab ; κ 2 5 = 8πG 5 (4 + 1)-decomposition (Gaussian Normal Coordinates) ( ) 1 0 g ab = ; q 0 q µν 4D metric µν ds 2 = dr 2 + q µν dx µ dx ν ; r 5th dimension Energy-momentum tensor ( ) 1 T ab = φ,a φ,b g ab 2 gcd φ,c φ,d + V (φ) +S ab δ(r r 0 ), S ab = σq ab. (Standard matter is assumed to be in the vacuum on the brane.)

16 Z 2 -symmetry and RS brane tension q ab (r 0 + y) = q ab (r 0 y), φ,r (r 0 ) = 0, 16 σ = σ 0 = 6 κ 2 5 l 0, l 2 0 = 6 Λ 5. 4D Einstein-scalar equations on the brane G µν = κ 2 5 T µν (bulk) E µν, T (bulk) µν = 1 6 ( 4φ,µ φ,ν E µν = (5) C rbrd q b µ q d ν. ( ) ) 5 2 qαβ φ,α φ,β + 3V (φ) q µν, Unconventional form of T (bulk) µν E µν is determined by the bulk scalar dynamics

17 6. Quadratic Potential Model L 5 = 1 16πG 5 R 1 2 gab a φ b φ U(φ) a conformally transformed scalar-tensor gravity 17 If φ varies very slowly (near and on the brane), Λ 5,eff = Λ 5 + 8πG 5 U(φ) < Λ 5, H 2 = 4πG 5U(φ) 3 = 8πG 4 3 U 4 ; G 4 G 5 /l, U 4 = l 2 U(φ). Quadratic potential: (l here is arbitrary; G 4 U 4 = 1 2 G 5U.) U = U M 2 φ 2

18 18 Λ πG U( φ ) φ Λ 5 When M 2 φ 2 U 0, one can solve the field equations iteratively. 0-th order: U = U 0, φ = 0, ds 2 = dr 2 + l 2 sinh 2 (r/l)( H 2 dt 2 + cosh 2 Ht dω 2 (3) ) (r r 0 ) l 2 6 = Λ 5 + κ 2 5 U 0, H2 = κ2 5 6 U 0 This is just an AdS 5 -ds brane system with a modified AdS curvature: l > l 0

19 1st order: O(φ) For de Sitter brane at r = r 0, φ(r, t) = u 0 (r)φ 0 (t) + 3/2 dλ u λ (r)φ λ (t) 19 φ 0 : zero mode (bound-state mode) φ λ : Kaluza-Klein modes M 2 λ = λ 2 H 2 (λ > 3/2) Effective 4d mass of bound-state mode when M 2 H 2 : { M M0 2 = 2 /2 for H 2 l 2 1 3M 2 /5 for H 2 l 2 1 No bound state when M 2 > H 2. (But there is a quasi-normal mode with M 0 = M/ 2 iγ) Γ/M = O(M 2 l 2 ) for H 2 l 2 1 For M 2 H 2, slow-roll inflation is realized on the brane, irrespective of the value of Hl. The bound-state mode dominates at late times if H 2 l 2 1.

20 2nd order: O(φ 2 ) (for H 2 l 2 1) [ (ȧ ) K ] ( ) = κ 2 a a 2 4 ρ eff = κ2 5 φ U(φ) + E t t. From Bianchi Ids. on the brane, t E t t = κ2 5 a 4 φ( rφ 2 + ȧ φ) dt = κ2 5 2a 4 a 4 φ 2 + X, ( ) κ 2 4 = κ2 5 l 0 where φ + 3H φ M 2 φ = Γ φ (Γ 0 only when M H) Ẋ + 4HX = Γ φ (X dark radiation) ( 1 ρ eff = ρ φ + X = l 0 φ ) 2 2 U(φ) + X. Consistent with 1st order (bound-state) solution when H 2 l ρ φ = ϕ2 2 + V (ϕ) ; ϕ l 0 φ, V (ϕ) l 0 2 U(ϕ/ l 0 ) = l 0 2 U m2 eff ϕ2, m 2 eff = M d scalar behaves like a 4d scalar on the brane!

21 7. Cosmological Perturbations in the Bulk Inflaton Model Essentially a 5-dimensional, PDE problem. However, some simplifications on super-horizon scales. Langlois, Maartens, MS & Wands ( 01) Basic equations: G µν = κ 2 4 T eff µν (ϕ) + X µν ; T eff Kanno & Soda, hep-th/ µν = µ ϕ ν ϕ 1 2 ( α ϕ α ϕ + 2V (ϕ)) effective 4d field ( X µν = E µν κ2 4 µ ϕ ν ϕ 1 ) 3 4 g µν α ϕ α ϕ dark radiation where X µ µ = 0, and T eff µν and X µν are conserved separately at O(H 2 l 2 ) : µ T eff µν = O ( H 4 l 4), µ X µν = O ( H 4 l 4). (during inflation when Γ = 0 M 2 H 2 ) standard 4d theory is applicable 21

22 Where is the effect of 5D bulk? 22 X µν Isotropic part of X µν (X 0 0 or X i i) can be determined from µ X µν = 0 (on superhorizon scales). (Only energy conservation law is essential on superhorizon) Anisotropic stress part of X µν cannot be determined from 4d equations. X aniso ij X ij 1 3 g ijx k k ( X aniso µν = Eµν aniso eff, aniso Tµν (ϕ) 0 at linear order ) Need to solve 5D equations for E µν

23 Evaluation of E µν in the bulk inflaton model Minamitsuji, Himemoto & MS, PRD68, (2003) Full background spacetime: ( ) ds 2 = dr 2 + b 2 (r, t) dt 2 + a 2 (r, t)dω 2 (3), φ = φ(r, t). 23 where b(r, t) = b(r) + O(φ 2 ), a(r, t) = a(t) + O(φ 2 ) ; b(r) = Hl sinh (r/l), a(t) = H 1 cosh Ht perturbations to O(φ 2 ) as well as to O(δφ). Lowest order background approximated by AdS 5 : ds 2 = dr 2 + b 2 (r)( dt 2 + a 2 (t)dω 2 (3) ) Then we have E µν = O(φ 2 ), δe µν = O(φδφ) ( δg µν = O(φδφ))

24 To O(φ 2 ), δφ satisfies the linearized field equation on AdS 5 : ( AdS5 + M 2) δφ = δe µν also satisfies a similar equation on AdS 5 : L [δe µν ] = S µν ; L Lichnerowicz ( AdS5 -like) operator S µν source term of O(φδφ) with the boundary condition at the brane, r (b 2 δe µν ) = δσ µν ; σ µν (energy momentum of φ ) Strategy: 1. Solve δφ in the AdS bulk. 2. Solve L [δe µν ] = S µν [δφ] by the Green function method: δe(x) d 5 x G(x, x )δs(x ) + d 4 x r (b 2 δe(x ))G(x, x ) bulk brane 3. Analyze the late time behavior of δe µν on the brane on superhorizon scales.

25 After a long and tedious calculation, we find δx µν δe µν 8πG ( 4 3 δ µ ϕ ν ϕ 1 ) 4 g µν α ϕ α ϕ = O(H 4 l 4 )! 25 δx 0 0 = δx i i part decays like radiation ( a 4 ). No anisotropic stress (δx i j 1 3 δi j δxk k) remains at late times. Up through O(H 2 l 2 ), Bulk Inflaton Model is completely equivalent to standard 4d inflation Difference appears only at O(H 4 l 4 ) or when H 2 l 2 1.

26 8. Summary Brane-world gives a new picture of the universe 26 Quantum brane cosmology Can we find cosmological evidence? Spatially compact 5D Universe created from nothing Well-posed initial value problem 4D Universe created in de Sitter (inflationary) phase Non-trivial quantum fluctuations if Hl 1 Effects of KK modes need to be investigated. Inflation without inflaton on the brane (bulk inflaton model) Inflation as a result of 5D gravitational dynamics Mass gap ( m = (3/2)H) in the KK spectrum Isolation of the zero mode Decoupling of 5D effects at low energy scales

27 Evolution of a brane universe Presence of ρ 2 term in Friedmann equation Modified evolution when l 2 G 4 ρ 1 Dark radiation (Weyl fluid) term in Friedmann equation Effect of 5D bulk gravity Bulk inflaton model is equivalent to standard 4d model up through O(H 2 l 2 ) 5D effect is encoded in Weyl anisotropy, but absent at O(H 2 l 2 ). 27 No trace of braneworld if H 2 l 2 1 However, for time-varying H, 5d mode mixing will occur. generation of KK modes from zero (bound-state) mode. Hiramatsu, Koyama & Taruya, hep-th/ , Easther, Langlois, Maartens & Wands, hep-th/ (for tensor KK mode generation) Generation of scalar KK modes needs to be studied.