Brane-World Cosmology Misao Sasaki Osaka University

JGRG12, 2002/11/25-28

Brane-World Cosmology

Misao Sasaki Osaka University 2 §1. Historical Notes Progress in Particle Physics Supergravity ⇒ String theory ⇒ ‘M’ theory • Hoˇrava & Witten (1996) 1 11D Supergravity with 1D compactiﬁed as S /Z2 gives a desirable string theory on the 10D boundary spacetime. (Standard matter ﬁelds live on the boundary.)

1 1 S S Z2 fold

⇓ Compactify extra 6D ∂(11D) = 10D ⇒ 6D ⊗ 4D Putting aside the extra 6D, we have ∂(5D) = 4D ··· brane world! 3 • Self-gravitating brane ≈ 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 6= cosmological constant on the brane ? causality on the brane 6= causality in the bulk H. Ishihara, PRL86 (2001)

brane's lightcone

A new picture of the universe! bulk's lightcone 4 §2. Randall-Sundrum (RS) Brane World PRL 83, 3370 (1999) [RS1]; 83, 4690 (1999) [RS2] 4 1 • 5D-AdS bounded by 2 branes: R × (S /Z2) 2 2 −2|y|/` µ ν ds = dy + e ηµνdx dx (−rc ≤ y ≤ rc) 2 6 3 ` = − , brane tension: σ± = ± Λ5 4πG5` (b(y) = e−|y|/` is called the warp factor) b( y )

negative tension positive tension negative tension

y y 0 y - 0 0 5 • Possible solution to the Mass hierarchy problem in particle physics if we live on the negative tension brane [RS1: 2-brane model].

? rc is arbitrary ⇒ Existence of Radion mode Brans-Dicke type gravity on the branes unless ∃stabilization mechanism. ωBD < 0 on the negative tension brane (Garriga & Tanaka, ’99) • If we live on the positive tension brane, the negative tension brane may be absent (rc → ∞ in RS1) [RS2: 1-brane model]. ··· 5th dimension can be non-compact −1/2 ? Gravity conﬁned within ` ∼ |Λ5| from the brane ? No “radion” modes (no relative motion) Einstein gravity is recovered on scales À `

G M G M G M Φ = − 5 ⇒ − 5 = − 4 Newton r2 rÀ` ` r r The positive tension brane seems cosmologically favored. 6

§3. Brane Cosmology in AdS5-Schwarzschild Bulk Kraus (’99); Ida (’00); ··· • 5D AdS-Schwarzschild in Static Chart: dR2 ds2 = −A(R)dT 2 + + R2dΩ2 A(R) K R2 α2 A(R) = K + − (K = ±1, 0, α2 = 2G M) `2 R2 5 ? For α2 = 0 (K = 1), ds2 = dr2 + (H`)2 sinh2(r/`)[−dt2 + cosh2(Ht)dΩ2 ] µ ¶ (3) R = ` sinh(r/`) cosh(Ht) H is arbitrary. T = ` arctan (tanh(r/`) sinh(Ht)) Any r =const. timelike hypersurface is 4D de Sitter space.

· Pure de Sitter brane at r = r0 (with Z2 symmetry): (Tµν = −σ gµν) 2 3 3 2 2 1 ` σ = coth(r0/`) ≡ ,H ` = 2 = 2 − 1 4πG5` 4πG5`σ sinh (r0/`) `σ 7

? Deviates from de Sitter if Tµν is non-trivial:

T n τ

brane trajectory :

"bulk" ½ R = R(τ) T = T (τ)

R Ã ! R˙ 2 ds2| = −A(R)T˙ 2 + dτ 2 + R2(τ)dΩ2 brane A(R) K · Choose τ to be Proper time on the Brane: R˙ 2 −A(R)T˙ 2 + = −1 A(R) 8

⇒ A2(R)T˙ 2 = R˙ 2 + A(R)

· Junction condition under Z2-symmetry: + [Kµν]− = 2Kµν(+0) = −8πG5[Tµν − (1/3)T gµν] 1 K = –L q ; n = (R,˙ −T,˙ 0, 0, 0) µν 2 n µν a qµν ··· induced metric on the brane µ µ 3 T ν = diag(−ρ, p, p, p) − σc δ ν ; σc = 4πG5`

A(R) 4πG5 4πG5 1 ⇒ T˙ = (ρ + σc) = ρ + R 3 3 `

G4 = G5/` ··· 4D Newton const. α2 ⇓ A(R) = K + R2`2 − R2

Ã !2 µ ¶ R˙ K 8πG 4πG 2 α2 + = 4ρ + `2 4ρ + R R2 3 3 R4 9 • Friedmann equation on the brane: Ã !2 µ ¶ R˙ K 8πG 4πG 2 α2 + = 4ρ + `2 4ρ + R R2 3 3 R4

? presence of ∝ ρ2 and α2/R4 terms. · ρ2-term dominates in the early universe: H ∝ ρ. µ ¶ 2t2 1/4 For ρ ∝ R−4, K = α2 = 0; R ∝ t + ` 2 · reduces to standard Friedmann equation for ` G4ρ ¿ 1. 2 (` G4ρ ¿ 1 ⇔ ρ ¿ σc) 4 6 BBN constraint: σc & (100 MeV) (⇔ ` . 10 cm) 2 4 2 · α /R -term: “dark radiation” α = 2G5M ∼ BH mass 2 α Ett (5) a b = − ; Eµν = C n n (5D Weyl contribution) R4 3 µaνb Hawking radiation from BH? ··· AdS/CFT (T. Tanaka, gr-qc/0203082) 10 §4. Quantum Brane Cosmology

• Euclidean AdS: H5 (O(5, 1)-symmetric) 2 2 2 2 2 2 2 ds = dr + ` sinh (r/`)(dχ + sin χ dΩ(3))

? Brane at r = r0 (with Z2-symmetry) = de Sitter-brane instanton

r = 0 r = 0

χ π = 2 r = r0

topology ∼ S5 χ = π/2 : totally geodesic 4-surface 11 • Creation of inﬂating brane-world Garriga & MS (’00); Koyama & Soda (’00) Analytic continuation: χ → iHt + π/2 2 2 2 2 2 −2 2 2 ds = dr + (H`) sinh (r/`)(−dt + H cosh Ht dΩ(3)) (r ≤ r0)

r = 0 r = 0 r = r 0 (Identify) Spatially Compact 5D Universe

(RS ﬂat brane is recovered in the limit r0 → ∞) 12 §5. 4D Graviton and Kaluza-Klein Excitations

• Gravitational perturbation of de Sitter brane universe: ds2 = dr2 + (H`)2 sinh2(r/`)ds2 + h dxadxb ¡ ¢ dS4 ab = b2(η) dη2 + H2ds2 + h dxadxb dS4 ab · dr = b(r)dη (conformal radial coordinate): ` b(r) = ` sinh(r/`) ⇒ b(η) = µ sinh(|η| + η0) ¶ 1 sinh η0 = ; −∞ < η < ∞ sinh(r0/`) · (generalized) Randall-Sundrum gauge: µ µν h55 = h5µ = h µ = Dµh = 0 ;

Dµ : 4D covariant derivative 13 • Perturbation equations:   (b3/2)00 m2  −ϕ00 + ϕ = ϕ 1/2 3/2 2 hµν = b ϕ(η)Hµν(x) ⇒ b H (4)  2 2 (− ¤ +2H + m )Hµν = 0 ? “Volcano” potential for ϕ: (b3/2)00 15 9 V (η) ≡ = + − 3 coth η δ(η) 3/2 2 0 b 4 sinh (|η| + η0) 4

V( η )

V( η ) 9 4 for η + 9 4 η

3/2 2 ? 4D graviton (zero mode m = 0): ϕ ∝ b ⇒ hµν ∝ b ? KK excitations (m > 0): V → (9/4) ⇒ m > (3/2)H 14 · Zero mode is conﬁned just like the case of the RS ﬂat brane. When quantized, however, the normalization (amplitude) is non-trivial: µ ¶ P (k)k3 H 2 gw ∼ F (H`) (Langlois, Maartens & Wands (’00)) G4 2π

F(x)

7 6 Ã " √ #!−1 5 √ 1 + 1 + x2 4 F (x) = 1 + x2 − x2 ln 3 x 2 1 x 1 2 3 4 5 3 – large enhancement at H` À 1: F ∼ H`. 2

· Mass gap in KK spectrum: ∆m = (3/2)H – The same is true for any bulk scalar ﬁeld with m . H. – No ‘zero-mode’ (bound state mode) for m À H. 15 §6. Brane-world Inﬂation Driven by a Bulk Scalar Field Himemoto & MS (’00), Himemoto, Sago & MS (’01), ··· ? Randall-Sundrum’s “default” parameters: ¯ ¯ ¯ ¯1/2 3 ¯ 6 ¯ brane tension: σc = ; ` = ¯ ¯ . 4πG5` Λ5

If |σ| > σc, then inﬂation occurs on the brane:

2 1 1 1 |Λ5| 3 H = 2 − 2 = 2 − ; σ ≡ `σ ` `σ 6 4πG5`σ

If |σ| = σc but |Λ5,eff | < |Λ5|, inflation also occurs on the brane: |Λ − Λ | H2 = 5 5,eff 6 ⇓ Brane-world inflation can be driven solely by bulk (gravitational) fields.

cf. Hawking, Hertog & Reall (’01) , Nojiri, Odintsov & Osetrin (’01), ··· 16 • Toy Model 1 1 ab L5 = R − g ∂aφ∂bφ − U(φ) 16πG5 2 ∼ a conformally transformed scalar-tensor gravity

If φ varies very slowly,

|Λ5,eff | = |Λ5 + 8πG5U(φ)| < |Λ5|,

4πG U(φ) 8πG ` H2 = 5 = 4U ; G = G `, U = U(φ). 3 3 4 5 4 4 2 (` is arbitrary here.)

Λ 5 + 8πG 5 U( φ ) φ

Λ 5 17 • Friedmann equation in the presence of a bulk scalar: µ ¶ a˙ 2 K K 8πG 1 + ≡ H2 + = 4 ρ˜ + Et ; a a2 a2 3 φ 3 t µ ¶ (5) 1 ˙2 1 t t ρ˜φ = ` φ + U(φ) ,E t = C rtr . 4 2 From Bianchi Ids. on the brane and φ equation in the bulk, Z 4πG t a˙ Et = − 5 a4φ˙(∂2φ + φ˙) dt t a4 r a C 1 = 2πG φ˙2 + , if φ¨ + 3Hφ˙ + ∂ U = 0 on the brane. 5 a4 2 φ t µ ¶ E t 1 ˙2 1 C ⇒ ρeﬀ =ρ ˜φ + = ` φ + U(φ) + 4 . 8πG4 2 2 a √ ? 5D scalar φ behaves like a 4D scalar Φ = ` φ with ` √ U (Φ) = U(Φ/ `) eﬀ 2 When and in what situations will this be true? 18 µ ¶ 1 • “Zero mode” and KK modes U = M 2φ2 2 ? For de Sitter brane at r = r0, Z ∞ µ µ µ Φ(r, x ) = u0(r)φ0(x ) + dλ uλ(r)φλ(x ) 3/2

φ0 : “zero mode” (bound state mode) 2 2 2 φλ : Kaluza-Klein modes Mλ = λ H (λ > 3/2) Effective 4d mass of zero mode when M 2 . H2: ½ M 2/2 for H2`2 ¿ 1 M 2 = 0 3M 2/5 for H2`2 À 1 2 2 ? No bound state when M > H . √ (But there is a quasi-normal mode with M0 = M/ 2 − iΓ) · Zero-mode dominance ⇔ consistent with the effective potential picture. · KK modes are important when H` À 1 (⇔ gravity zero-mode is non-trivial when H` À 1). Quantum fluctuations of KK modes need be evaluated. 19 • Case of a scalar coupled to brane tension Langlois and MS, in prep. Z µ ¶ Z 5 √ 1 1 2 4 √ S = d x −g R − (∇φ) − V5(φ) − d x −q σ(φ) 16πG5 2 2 8πG5V5(0) = Λ5 = −6 ` ? Friedmann equation on the brane: Ã µ ¶ ! 8πG 1 1 8πG 1 ∂σ 2 E0 H2 = 5 φ˙2 + V + 5σ2 − + 0 3 4 2 5 12 16 ∂φ 3 √ ∃ ¨ ˙ 0 If Veff s.t. Φ + 3HΦ + Veff = −J, Φ ≡ ` φ  · ¸  2 8πG4 1 ˙ 2  H = Φ + Veff(Φ) + ρE , 3 2 ⇒ µ ¶  1 8πG 1 ∂σ 2  ρ˙ + 4Hρ = JΦ;˙ V = V + 5σ2 − , E E eff 2 5 12 16 ∂φ µ ¶ 1 where G = G /` and E0 = 8πG ρ + Φ˙ 2 . 4 5 0 4 E 4

Does this hold whenever H2`2 ¿ 1? AdS/CFT? 20 §7. Large-scale Cosmological Perturbations on the Brane · General formalism ⇒ Kodama et al., Langlois, Mukohyama, ··· . · Essentially a 5-dimensional, PDE problem. · However, some simpliﬁcations on super-horizon scales. Langlois, Maartens, MS & Wands (’01)

• Basic equations (in AdS5 bulk background; no bulk scalar) 2 Gµν + Λ4 qµν = 8πG4 Tµν + (8πG5) Πµν − Eµν tot ≡ 8πG4 Tµν (Πµν ∼ quadratic in Tµν)

“−Eµν” : Weyl ﬂuid (or “dark radiation”)

For Tµν = ρ uµuν + P hµν + πµν (πµν: anisotropic stress = O(²)) µ ν 2 u D Eµν = O(² ) Weyl ﬂuid is decoupled on superhorizon scales.

0 standard 4-d theory is applicable (E 0 = 8πG4 ρE , etc.) 21 • Large angle CMB anisotropy µ ¶ Z δT η0 (~γ, η0) = (ζr + Θ)(ηdec, ~x(ηdec)) + dη ∂ηΘ(η, ~x(η)). T sw ηdec (Sachs-Wolfe) (Integrated Sachs-Wolfe)

ζr ∼ curvature perturbation on ρphoton = const. surfaces Θ = Ψ − Φ Ψ ∼ Newton potential Φ ∼ curvature perturbation in Newton gauge

η x ( dec )

Last Scattering Surface 22 For a dust-dominated universe at decoupling,

1 2 8 ρr SW: ζr + Θ = − ζ∗ − Sdr − SE 5 5 3ρd Z 16πG a −8πG a2 δπ + 4 δπ a7/2da 4 tot 5/2 tot · a 0 Z ¸ 8 ρ 16πG a ISW: ∂ Θ = −∂ r S + 8πG a2 δπ − 4 δπ a7/2da η η E 4 tot 5/2 tot 3ρd a 0 where ζ∗ is the adiabatic curvature perturbation,

δρE ρE δρ SE := − ∼ Weyl entropy perturbation 4ρr 3ρr(ρ + P )

δρd 3δρr Sdr := − ∼ standard entropy perturbation µρd 4 ρr ¶ ρ + 3P δπ = 1 − δπ + δπ ∼ anisotropic stress tot 2σ E 1 δπ : traceless part of Weyl ﬂuid (Ei − δiEk ) E j 3 j k

δπE cannot be determined within this 4-d approach 23 §8. Summary Brane-world gives a new picture of the universe

Can we ﬁnd cosmological evidence?

• Quantum brane cosmology ? 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 H` À 1 · Effects of KK modes need to be investigated. ? Inflation without inflaton on the brane · Inflation as a result of 5D gravitational dynamics ? Mass gap (∆m = (3/2)H) in the KK spectrum · Isolation of the zero mode ⇔ Stability of the brane world 24 • Evolution of a brane universe ? Presence of ρ2 term in Friedmann equation 2 · Modified evolution when ` G4ρ & 1 ? Weyl fluid term (dark radiation) in Friedmann equation · Effect of 5D bulk gravity ? Large scale perturbation can be solved without 5D equations. · 5D effect is encoded in CMB through Weyl anisotropy. • Some issues on brane-world cosmology ? Search for a natural brane-world inflation model related to cosmological constant problem ? ? Quantitative analysis of cosmological perturbations 5D dynamics of “Eµν” (poster by Minamitsuji) ? Analysis of the two-brane cosmological model (radion dynamics) “born-again braneworld” (talk by Kanno)

Need a good approximation method, e.g., Kanno-Soda’s (Shiromizu-Koyama’s) approach