Brane Inflation: Observational Signatures and Non-Gaussianities. Gary Shiu. University of Wisconsin

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1 Brane Inflation: Observational Signatures and Non-Gaussianities Gary Shiu University of Wisconsin

2 Collaborators Reheating in D-brane inflation: D.Chialva, GS, B. Underwood Non-Gaussianities in CMB: X.Chen, M. Huang, S. Kachru, GS DBI Inflation in Warped Throats: S.Kecskemeti, J.Maiden, GS, B.Underwood

3 Two popular themes in String Phenomenology: Construct realistic particle physics models: Not enough (realistic) vacua Landscape (statistics, wave function, swampland,...): Too many vacua. String theory: great scenario generator! SUSY, brane world,...

4

5 ... in the year 1BC

6 ... in the year 1BC

7 ... in the year 1BC LHC

8 WMAP3 Strong and growing evidence for inflation

9 Goals and Motivation Construct & study well motivated inflationary scenarios (incorporate SM, reheating,...) Look for distinctive observational signatures Building realistic models Many interesting possibilities with branes and fluxes

10 Brane Inflation Dvali and Tye Animation by A. Miller DD Inflation [Burgess, Majumdar, Nolte, Quevedo, Rajesh, Zhang];[Dvali, Shafi, Solganik], [Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi] and many others.

11 Brane Inflation Is this scenario viable/robust? e.g., number of e-folds, reheating,... Observational signatures/constraints? e.g., cosmic strings (Tye s talk), non- Gaussianities,... Model building? constraints on compactification geometry?

12 Warped Throats Hierarchies from fluxes Giddings, Kachru, Polchinski... S 3 size e K Mgs Strong dynamics scale IR AdS5 UV e.g., Klebanov, Strassler warped deformed conifold

13 Warped Reheating Reheating by DD annihilation Shiu, Tye, Wasserman Barnaby, Burgess, Cline Kofman and Yi Chialva, Shiu, Underwood Frey, Mazumdar, Myers Chen and Tye Langfelder... Accommodate different hierarchies. Cosmic strings spatially separated from SM branes: not susceptible to breakage. Reheating via tunneling is efficient, can avoid overproduction of gravitational waves.

14 A Cartoon of Reheating Gravitons Decay KK modes Annihilation Massive Closed Strings Sen; Lambert, Liu, Maldacena;.. Tunneling Excite SM

15 Warped Reheating %%&'#()&*+,-./0, #!"!$ #!$ /,* # /,* #!" 4 1)20&"03)&'#()&*+,-./0, Production rate, interaction cross sections among KK modes enhanced relative to gravitons. For moderate warping of inflationary throat, KK preferably tunnel rather than decay to gravitons. $ c.f. Dimopoulos, Kachru, Kaloper, Lawrence, Silverstein

16 Is brane inflation robust? Helps flatten the potential Casual speed limit DBI Slow-roll e.g., KKLMMT,... Silverstein, Tong; Alishahiha, Silverstein, Tong

17 Derivative terms sum to a DBI action: S = d 4 x a 3 (t) [ T (φ) 1 φ 2 /T (φ) + V (φ) T (φ) T (φ) = T 3 h 4 (φ) ] Casual speed limit: φ 2 T (φ) warp factor γ = 1 1 φ 2 /T (φ) D3 Slow-roll + DBI : inflation is robust D3 Relativistic even when φ is small. Shandera & Tye

18 Non-Gaussianities

19 Non-Gaussianities Power spectrum: ζ k1 ζ k2 δ 3 (k 1 + k 2 ) P ζ k k 3 1 Bi-spectrum contain much richer info: ζ k1 ζ k2 ζ k3 = (2π) 3 δ 3 (k 1 + k 2 + k 3 )F (k 1, k 2, k 3 ) size ~ f NL and shape. Slow-roll: full functional form derived in f NL O(ɛ) Maldacena 02 Acquaviva et al 02 DBI inflation for γ >> 1 : f NL 0.32γ 2 Alishahiha, Silverstein, Tong Chen Chen, Huang, Kachru, GS

20 Non-Gaussianities For a general single field Lagrangian: L(φ, X) where X = 1 2 g µν µ φ ν φ Bi-spectrum depends on 5 parameters: [Chen, Huang, Kachru, GS] c 2 s = L,X L,X + 2XL,XX 1 for DBI γ2 λ/σ = X2 L,XX X3 L,XXX XL,X + 2X 2 L,XX and slow variation parameters: ɛ = Ḣ H 2 η = ɛ ɛh, s = ċs c s H.

21 Shape of Non-Gaussianities F (k 1, k 2, k 3 ) = (2π) 4 (P ζ k )2 1 i k 3 i (A λ + A c + A ɛ + A η + A s ) where A λ = ( 1 c 2 s 1 2λ Σ ) 1 1 K ) 3k 2 1 k 2 2 k2 3 2K 3, ( 1 A c = k c 2 i 2 kj k s 2K 2 i 2 kj i>j i j i A ɛ = ɛ 1 k c 2 i k i kj ki 2 k 2 j, s 8 8 K i i j i>j ( ) A η = η 1 k c 2 i 3, s 8 i A s = s 1 k c 2 i 3 1 ki 2 kj k 2 s 4 K 2K 2 i kj 3. i i>j i j k 3 i, and K = k 1 + k 2 + k 3, Σ = XP,X + 2X 2 P,XX, λ = X 2 P,XX X3 P,XXX.

22 Correction Terms Solution to the quadratic part of the action: π u k (y) 2 2 where y c sk ah H 1 ɛcs k 3/2(1 + ɛ 2 + s 2 ) ei π 2 (ɛ+ η 2 ) y 3/2 H (1) 3 2 +ɛ+ η 2 + s 2 ((1 + ɛ + s)y) Slowly-varying parameters H, c s, λ and ɛ f(τ) f(τ K ) f(τ K ) f 1 ln τ + O(ɛ 2 f) t H K τ K The scale factor a 1 H K τ 1 H K τ ɛ H K τ + ɛ H K τ ln(τ/τ K) + O(ɛ 2 )

23 Final Results F (k 1, k 2, k 3 ) = (2π) 4 ( P ζ K )2 1 i k 3 i (A λ + A c + A o + A ɛ + A η + A s ) ( 1 A λ = 1 λ ) c 2 s Σ [2 (3 2c 3k1 2 1)l] k2 2 k2 3, K 2K 3 ( ) 1 A c = 1 1 k c 2 i 2 s K K k2 j + 1 k 2K 2 i 2 k3 j i>j i j i ( 1 A o = 1 2λ ) ( ) ɛfλɛ + ηf c 2 λη + sf λs s Σ ( ) K (ɛf c 2 cɛ + ηf cη + sf cs ), s K A ɛ = ɛ 1 ki k i kj ki 2 K k2 j, i i j i>j ( ) 1 A η = η ki 3, 8 A s = sf s. i k 3 i,

24 Experimental Bound WMAP ansatz for the primordial non-gaussianities ζ(x) = ζ g (x) 3 5 f NL(ζ g (x) 2 ζ 2 g (x) here ζ g (x) is purely Gaussian with vanishing three point functions. The size of non-gaussianities is measured by the parameter f NL in the above ansatz. Current experimental bound (from WMAP3) is 54 < f NL < 114 at 95% C.L. Future experiments can eventually reach the sensitivity of f NL < 20 (WMAP) and f NL < 5 (PLANCK). However, the experimental bound depends on the shape of F (k 1, k 2, k 3 ). Creminelli, Nicolis, Senatore, Tegmark, and Zaldarriaga

25 Due to the symmetry and scaling property of F (k 1, k 2, k 3 ), all info about the shape can be viewed by plotting [Babich, Creminelli, Zaldarriaga] F (1, k 2, k 3 )k 2 2 k2 3 For the WMAP ansatz: ( ) ζ 2 k 3 F (k 1, k 2, k 3 ) f NL P 1 + k2 3 + k3 3 k k1 3k3 2 k k k3

26 Slow Roll Shapes The relevant shapes are F (k 1, k 2, k 3 ) 1 A ɛ = ɛ c 2 s A η = η c 2 s A s = s c 2 s i k3 i 1 ki k i kj K ( 1 8 i 1 4 i k 3 i ) i j A(k 1, k 2, k 3 ) where i>j k 2 i k2 j, ki 3 1 ki 2 kj k 2 K 2K 2 i kj 3. i i>j i j

27 Consistency Condition In the squeeze triangle limit : one momentum mode is much smaller than the other two: k 3 k 1, k 2 k 1 k 2 Maldacena During inflation, the comoving Hubble scale decreases with time. The long wavelength mode k 3 crosses the horizon much earlier than the other two modes k 1, k 2. After horizon crossing, the long wavelength mode k 3 acts as background whose effect is to introduce a time variation at which k 1,2 cross the horizon. d ζ k1 ζ k2 ζ k3 ζ k3 ζ -k3 ζ k1 ζ k2 (n s 1) 1 1 d ln k 1 k1 3 k3 3

28 DBI Shape Non-Gaussianities are generically quite large f NL 1 c 2 s γ 2 The shape of non-gaussianities vanishes in the squeeze triangle limit k 3 k 1, k 2, as required by Maldacena s consistency relation: F (k 1, k 2, k 3 )k 3 1 k3 3 n s 1 This contradicts that the non-gaussianities are large, unless the shape vanishes in the squeeze limit.

29 The shape of non-gaussianities for DBI inflation f NL Peak at the equilateral triangle limit and vanishes in the squeeze limit. If non-gaussianities of this shape is measured, gives interesting constraint on m 2 φ 2 term and in turn 4-cycles of CY. [Baumann, Dymarsky, Klebanov, Maldacena, McAllister, and Murugan] Also: [Berg, Haack, Kors]

30 More Shapes Not realized in D-brane inflation. Similar to the DBI inflation but with an opposite sign. ( 1 A λ = 1 2λ ) 3k1 2k2 2 k2 3 Σ 2(k 1 + k 2 + k 3 ) 3 c 2 s

31 Confronting Data ɛ D 2M 2 p γ η D 2M 2 p γ κ D 2M 2 p γ ä a = H2 (1 ɛ D ) ( H (φ) H(φ) ( ) H (φ) H(φ) ( ) H H n s 1 = γ γ ) 2 ( ) r = 16ɛ D γ f NL 0.3γ 2 (1 + ɛ D + κ D )( 4ɛ D + 2η D 2κ D ) + If r saturates the observational bound, non-gaussianity is small.

32 Warped Deformed Conifold r=0 r=! r=constant 4 i=1 z 2 i = ε 2 ds 2 10 = h 1/2 (τ)dx n dx n + h 1/2 (τ)ds 2 6 where ds 2 6 = 1 [ 1 ( τ ) 2 ε4/3 K(τ) 3K 3 (τ) (dτ 2 + (g 5 ) 2 ) + cosh 2 [(g 3 ) 2 + (g 4 ) 2 ] 2 ( τ ) ] + sinh 2 [(g 1 ) 2 + (g 2 ) 2 ], 2 K(τ) = (sinh(2τ) 2τ)1/3 2 1/3 sinh τ. h(τ) = α 22/3 4 I(τ) = (g smα ) 2 2 2/3 ε 8/3 I(τ), I(τ) dx x coth x 1 τ sinh 2 (sinh(2x) 2x) 1/3. x ceeded in evaluating this integral in terms of element

33 DBI ultra-relativistic region f NL [ ( m M p m s > 10 2 M p ity, m 10 6 ) M ( p ) ) 2 ( Mp m s h A To fit a KS-like throat inside the bulk : ) (Gµ s ) 2 - other warped throats? - Z p -orbifold the KS-like throat? N A h A m s M p M. Alishahiha, E. Silverstein and D. Tong, hep-th/ S. Kecskemeti, J. Maiden, G. Shiu, B. Underwood, hep-th/ Need a long narrow throat :

34 Red or blue tilt in DBI? h 4 (φ) (φ2 + b) 2 λ h 4 (φ) φ4 λ cut off at φ A Red tilt A small blue tilt KS throat?

35 Red or blue tilt in DBI? red (small) blue

36 Tip from the Sky? Bret Under wood

37 Red or blue tilt in DBI-KS? red blue For example, if h tip 10 2 and M s 10 2 M P red tilt dominates for KS throat

38 Summary Brane inflation is robust: number of e-foldings, reheating,... Interesting signatures: can lead to large tensorscalar ratio r, or large non-gaussianities, cosmic strings... Data probe warped geometry. [c.f. talks of Giddings, Hebecker] Large influx of data from Cosmology + LHC!