Duality Cascade in the Sky

Transcription

1 Duality Cascade in the Sky (R.Bean, X.Chen, G.Hailu, S.H.Tye and JX, to appear) Jiajun Xu Cornell University 02/01/2008

2 Our Current Understanding of the Early Universe - Homogeneous and isotropic δh 2 = Pretty flat Ω K < O(0.02) - Primordial perturbations are: nearly Gaussian f NL < O(100) nearly adiabatic S/R < O(0.2) nearly scale invariant n s 1 < O(0.05), dn s /d ln k < O(0.05) - Tensor perturbation r < O(0.5)

3 to explain the local feature in WMAP data, we need local feature in the slow-roll inflaton potential. (Adams, Ross, Sarkar 1997, Leach, Liddle, 2001 Hunt, Sakar, 2004, 2007 Adams, Creswell, Easther, 2001 Peiris et al, 2003 Covi et al, 2006, 2007) local features in the potential also generates large non-gaussianity. (Chen, Easther, Lim, 2006) f NL ɛ, η V (φ)(1 + δ(φ)) P R = H2 2π φ In brane inflation, local features arise both in slowroll scenario and DBI scenario, due to gauge/gravity duality.

4 Brane Inflation (Dvali & Tye) throat r = r 0 anti-d3 φ D3 r ~ R Calabi! Yau

5 Brane Inflation (KKLMMT) Consider D3 Branes moving in the AdS 5 X 5 background ds 2 = h 2 (r)( dt 2 + a(t) 2 dx 2 ) + h 2 (r)(dr 2 + r 2 ds 2 X 5 ), S = in the UV region, d 4 x g ds 2 X5 = ds 2 T 1,1 e Φ T (φ) 1 φ 2 T (φ) + T (φ) V (φ) T 3 m4 s g s = m4 s T (φ) = T 3 h 4 (φ) g s e Φ(r) V (φ) = β 2 H2 φ 2 + V D D(φ) ( V D D(φ) = V 0 1 V ) 0 1 4π 2 v φ 4 V 0 = 2T 3 h 4 A

6 Klebanov-Strassler Throat S3 IR r UV S2 1 2πα A F 3 = 2πM, 1 2πα B H 3 = 2πK size of S3 at the bottom = e 2πK/(Mg s)

7 Gauge Gravity Duality gauge theory SU(N+M)XSU(N), N=KM, with bifundamental chiral fields A1, A2, B1, B2 T = 8π2 g 2 SU(N M) T 1 (2) SU(N) T2(2) SU(N) T2(1) SU(N + M) T 1 (1) ˆb = 2 β 1 = µ dt 1(1) dµ β 2 = µ dt 2(1) dµ ˆb = 0 = 3(N + M) 2N(1 γ 1 (1)), = 3N 2(N + M)(1 γ 2 (1)), ln(r 2 /r 0 ) T 1 + T 2 = 2π g s e Φ, ln(r 1 /r 0 ) 0 ln(r/r 0 ) T 1 T 2 = 2π g s e Φ (ˆb 1) = 2π g s e Φ ( b 2 (mod 2)) R = 1 2 ( Φ) g2 se 2Φ ( C 0 ) e Φ H g2 se Φ F 2 3.

8 The Slow-Roll Scenario with Running Dilaton expand the DBI action in non-relativistic limit e Φ T (φ) 1 φ 2 T (φ) + T (φ) V (φ) = 1 2 e Φ φ2 [ T (φ)(e Φ 1) + V (φ) ] T (φ) = T 3 h 4 (φ) sharp features in the warp factor translates into the effective potential T (φ) φ 4 φ 4 A V (φ), too strong for slow-roll unless φ M pl. However, φ 1 1 M pl KM Need e Φ 1, so that T (φ)(e Φ 1) V (φ)

9 h 4 (r) r4 R 4 B the warp factor A series of steps in the warp factor, spaced according to ln(r p+1 ) ln(r p ) 2π 3g s M K p l (1 + ), = R B = 27 4 πg skmα 2 K p i 3g s M 16π 1 p 3 [ ( )] r rp 1 + tanh d p h we expect sharp step features in the slow roll potential too c V V = T (φ)(e Φ 1) V r

10 DBI Inflation S = d 4 x g e Φ T (φ) 1 φ 2 T (φ) + T (φ) V (φ) the exact equation of motion from the DBI action is V (φ) + T (φ)(c 1 s 1) = 3H 2, φ 3 T (φ) 2 T (φ) φ 2 + 3Hc 2 φ s + T (φ) + c 3 s[v (φ) T (φ)] = 0 c s = γ 1 = 1 φ 2 /T T (φ) sets the speed limit, φ 2 < T (φ) the brane moves relativistically, c s 1, γ 1 a sharp step in T (φ) non-gaussian power spectrum sharp change in c s f NL c 2 s 10 2

11 Observable effects (I): the power spectrum the power spectrum v k + ( k 2 c 2 s z z v k zu k, ( ) ) v k = 0 z a 2ɛ/c s Define three parameters P R (k) v k e ikτ, c 2 sk 2 z /z v k z, c 2 sk 2 z /z k3 2π 2 u k 2 the time dependent mass z /z, encodes all the information of the background space-time z z = 2a2 H 2 ζ(τ, k) = u(τ, k)a(k) + u (τ, k)a ( k) = H 2 8π 2 M 2 pl ɛc s ɛ Ḣ ɛ, η H2 Hɛ, s ċs. Hc s ( 1 ɛ η 4 3s 2 ɛ η 4 + ɛs 2 + η2 8 ηs 2 + s2 2 + η 4H, ṡ 2H ) dominant in slow-roll dominant in DBI inflation

12 potential step in slow-roll V (φ) c 1 2 V V = z z /(a2 H 2 ) z ( z 2a2 H 2 1 V ) (φ) 2H N e ( ) φ warp factor step in IR-DBI h(φ) b 1 2 h h = z z /(a2 H 2 ) z z 2a2 H (1 2 T ) 2H 2 T T 3 h(φ) N e φ

13 Slow-roll power spectrum moving across the potential step generates efolds z z N e = Hdt = H ḋ φ ( = 2a 2 H 2 1 V ) H 2 2a 2 H 2 (1 c d 2 1 H 2 ) 2a 2 H 2 (1 c ɛ 1 N 2 e ) d ɛ c ɛ 1, N e 1 observable effects in brane inflation models, is tiny conservatively, take In KKLMMT, ɛ ɛ Ḣ H 2 = 1 2c s ɛ φ2 H 2 ( dφ dn e ) 2 ( φ) 2 = 1 KM = 10 4, N e = 10 2, ɛ 10 8 we are able to detect a potential step with c 10 11

14 Numerical power spectrum in slow-roll ɛ = ( φ 2 ) 9 2H 2 3 V V 5c R P 4 11! 10 7 relaxation V (φ) φ = V ɛ/ k(/mpc) relaxation deviate from attractor φ = V (c + ɛ/3) φ !1 t/h φ = V ɛ/3 P R = H2 2π φ c + ɛ/3 ɛ/3 = 1 + 3c/ɛ

15 IR-DBI power spectrum moving across the step in warp factor N e H t H ḋ φ = d 2cs ɛ the sound speed changes sharply upon crossing the step c s b = 0.1, N e = c s = γ 1 = 1 φ 2 /T z N e ρ = V (φ) + T (φ)(c 1 s 1) z 2a2 H 2 ( 1 ṡ 2H ( ) ( ) ( ) ) ( ṡ T H 2c sɛ T 1 T 2 2 T 2 ) s = c s c s ċs = 3(1 c 2 Hc s) + c s V s ( T ) 2c s ɛ T = T T b c sɛ d = 2b O T H + ( b 2 N e T T H (1 c s) )

16 9 8 7 b=0 b=0.1,!n =0. e 1 b=0.05,!n =0. e 1 b="0.05,!n =0. e 1 9!10 6 R P k(/mpc) P R = H2 2π φ φ = T (φ)

17 9 8 b=0 b=0.1,!n =0. e 1 b=0.1,!n =10 e "4 7 9!10 6 P R c s c s k max k min = k(/mpc) z z b N e = T T = 2b always saturated oscillation amplitude weakly depends on step width

18 need b=-0.3, too large for steps in duality cascade

19 Multiple Steps duality cascade gives a series of K steps, spaced according to ln(r p+1 ) ln(r p ) k 2π 3g s M feature on scale in the power spectrum, shows up on angular scale l on WMAP π k 1 l H 1 0 dn e d ln k d ln l Hdt Ḣ φ dφ, d ln l dφ φ p φ p+1 φ p+1 φ p+2 φ p+2 φ p+3 φ p φ p+1 φ p+1 φ p+2 exp( 2π 3g s M ) 1 + δ 1 + δ take l=2, l=20 as two steps for example, ln(2) ln(20) = ln(20) ln(l 3 ) l 3 = 200, l 4 = 2000

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21 Observable effects (II): non-gaussianity the three-point correlation function in slow-roll, ζ(τ end, k 1 )ζ(τ end, k 2 )ζ(τ end, k 3 ) ( ) τend = i u ki (τ end ) dτa 2 ɛ η ( u k 1 (τ)u k 2 (τ) d ) dτ u k 3 (τ) + perm (2π) 3 δ 3 ( i i ζ(τ, k) = u(τ, k)a(k) + u (τ, k)a ( k) k i ) + c.c., in DBI case, leading term is (a 3 ɛ/c 4 s )ζ ζ 2 f NL 1/c 2 s. d dt ( η c 2 s ) f feature NL small by construction in usual slow-roll, can be large locally at sharp features a 3 ɛ 2c 2 s d dt ( η c 2 s ) ζ 2 ζ

22 slow roll case data fitting give f NL = O( η) ɛ V/H 2 5c t accel φ/ φ d/ cv, c/ɛ = 0.2 ( ) ( ) η η = c/d = O(1) ɛ Hɛ 7c3/2 dɛ f NL = O(1), IR-DBI case f feature NL d dt f feature NL ( η c 2 s η c s s ) ( η t s c s 1 c s b N e c 2 s ) b = 0.01, c s = 0.1, N e = 0.01, f feature NL = O(10)

23 Conclusions Duality cascade predicts a series of steps in the warp geometry, the steps are equally spaced in ln(r). Steps are generic, with KS a calculable example. Generically, dilaton runs, and features in the warp geometry becomes features in slow-roll potential. In the slow-roll power spectrum, the sensitivity to the steps is controlled by c/ɛ. Brane inflation is highly sensitive to small features with ɛ In DBI inflation, sharp features in warp factor may not be observed in the power spectrum, but gives detectable level non-guassianity. The steps features in the power spectrum are always accompanied by large non-guassianity on the same scale => chances to tell the feature from statistical fluctuation / cosmic variance