Non-Gaussianities in String Inflation. Gary Shiu

Transcription

1 Non-Gaussianities in String Inflation Gary Shiu University of Wisconsin, Madison Frontiers in String Theory Workshop Banff, February 13, 2006 Collaborators: X.G. Chen, M.X. Huang, S. Kachru

2 Introduction from WMAP An almost scale invariant, adiabatic, Gaussian primordial fluctuation predicted by inflation is in good agreement with CMB data. A tantalizing upper bound on the energy density during inflation: V M 4 GUT (1016 GeV ) 4 i.e., H GeV. The relevant energy scale is close to the scale where stringy physics becomes important.

3 Inflation as a Probe of Stringy Physics Quantum Fluctuations "Freeze In" Structure -1 H ~ constant -1 H ~ constant λ < Η -1 λ Η -1-1 H increases λ < Η -1

4 Inflation as a Probe of Stringy Physics Quantum Fluctuations "Freeze In" Structure -1 H ~ constant -1 H ~ constant λ < Η -1 λ Η -1-1 H increases λ < Η -1 Imprints of short distance physics. [More later in this workshop]

5 Inflation as a Probe of Stringy Physics Quantum Fluctuations "Freeze In" Structure -1 H ~ constant -1 H ~ constant λ < Η -1 λ Η -1-1 H increases λ < Η -1 Imprints of short distance physics. [More later in this workshop] Studies focused mainly on the power spectrum: primordial non-gaussian fluctuations predicted by inflation are typically too small.

6 String Inflation A plethora of string inflationary models. Two broad classes:

7 String Inflation A plethora of string inflationary models. Two broad classes: Slow roll inflation (e.g., KKLMMT):

8 String Inflation A plethora of string inflationary models. Two broad classes: Slow roll inflation (e.g., KKLMMT): Our Brane Extra Brane Extra Anti Brane Brane inflation Dvali and Tye

9 String Inflation A plethora of string inflationary models. Two broad classes: Slow roll inflation (e.g., KKLMMT): Our Brane Extra Brane Extra Anti Brane Brane inflation Dvali and Tye

10 String Inflation A plethora of string inflationary models. Two broad classes: Slow roll inflation (e.g., KKLMMT): Our Brane radiation + D strings + F strings Brane inflation Dvali and Tye

11 String Inflation A plethora of string inflationary models. Two broad classes: Slow roll inflation (e.g., KKLMMT): Our Brane radiation + D strings + F strings Brane inflation Dvali and Tye Flux compactification: stabilizes moduli and generates warped throats.

12 Warped throats: D3 D3 Help flatten the potential, though some degree of fine-tuning is still needed : usual η problem. Reheating and suppression of gravitational wave production. Barneby, Burgess, and Cline Kofman and Yi Chialva, GS, and Underwood Frey, Mazumdar, and Myers

13 DBI Inflation [Silverstein and Tong] Different regime: higher derivative terms enforce a casual speed limit. S DBI = d 4 x g [ ] T 1 φ 2 /T + V (φ) T where T = T 3 h(φ) 4 with T 3 = D3 tension, h(φ) is the warping factor.

14 DBI Inflation [Silverstein and Tong] Different regime: higher derivative terms enforce a casual speed limit. S DBI = d 4 x g [ ] T 1 φ 2 /T + V (φ) T where T = T 3 h(φ) 4 with T 3 = D3 tension, h(φ) is the warping factor.

15 Distinctive signatures: Alishahiha, Silverstein, and Tong See also: Chen Large Non-Gaussianties with a characteristic shape of ζ k1 ζ k2 ζ k3 : f NL γ 2 γ = 1 1 φ 2 /T The numerical coefficient of γ 2 in f NL is [Chen, Huang, GS] Modified consistency relation : P h k P ζ k = 8 γ n T

16 Distinctive signatures: Alishahiha, Silverstein, and Tong See also: Chen Large Non-Gaussianties with a characteristic shape of ζ k1 ζ k2 ζ k3 : f NL γ 2 γ = 1 1 φ 2 /T The numerical coefficient of γ 2 in f NL is [Chen, Huang, GS] Modified consistency relation : P h k P ζ k = 8 γ n T More generally: by varying γ, one can interpolate between slow roll (γ 1) to DBI inflation (γ >> 1) including intermediate regime (γ > 1). Inflation in this general setup is robust. Shandera and Tye

17 Our Results General analysis for an arbitrary action of the form: S = 1 2 d 4 x g[mpl 2 R + 2P(X, φ)] X = 1 2 gµν µ φ ν φ Our results are applicable to the intermediate regime, as well as in extracting subleading (but potentially observable) non-gaussianities. We obtain all known shapes of non-gaussianities plus more. Laboratory for testing the ds/cft proposal Strominger [Larsen, van der Schaar, Leigh]; [Maldacena]; [Larsen, McNees]; [van der Schaar] f k1 f k2 f k3 = 2Re O k 1 O k2 O k3 i( 2Re O ki O ki ) [Bousso, Maloney, Strominger]... Maldacena, astro-ph/

18 The General Setup Consider an action of the form: S = 1 d 4 x g[mpl 2 R + 2P(X, φ)] 2 whose Gaussian perturbations have been considered by Garriga and Mukhanov.

19 The General Setup Consider an action of the form: S = 1 d 4 x g[mpl 2 R + 2P(X, φ)] 2 whose Gaussian perturbations have been considered by Garriga and Mukhanov. Define the energy E and the sound speed c s as E = 2XP,X P c 2 s = P,X P,X + 2XP,XX

20 The General Setup Consider an action of the form: S = 1 d 4 x g[mpl 2 R + 2P(X, φ)] 2 whose Gaussian perturbations have been considered by Garriga and Mukhanov. Define the energy E and the sound speed c s as E = 2XP,X P c 2 s = P,X P,X + 2XP,XX Friedman equation and the continuity equation 3M 2 pl H2 = E Ė = 3H(E + P)

21 The General Setup Consider an action of the form: S = 1 d 4 x g[mpl 2 R + 2P(X, φ)] 2 whose Gaussian perturbations have been considered by Garriga and Mukhanov. Define the energy E and the sound speed c s as E = 2XP,X P c 2 s = P,X P,X + 2XP,XX Friedman equation and the continuity equation 3M 2 pl H2 = E Ė = 3H(E + P) For a Lagrangian with standard kinetic term: P(X, φ) = X V (φ) and hence c s = 1. For DBI action, c s = 1/γ.

22 Generalized slow roll parameters: ɛ = Ḣ H 2 = XP,X M pl H 2, η = ɛ ɛh, s = ċs c s H.

23 Generalized slow roll parameters: ɛ = Ḣ H 2 = XP,X M pl H 2, η = ɛ ɛh, s = ċs c s H. The non-gaussianities that we found depend on 5 parameters: c 2 s, ɛ, η, s, and λ/σ (to be defined). Potentially observable when these parameters are sufficiently large.

24 Non-Gaussianities Primordial power spectrum: ζ k1 ζ k2 δ 3 (k 1 + k 2 ) P ζ k k 3 1 Non-Gaussianity contains potentially more info because of its shape: ζ k1 ζ k2 ζ k3 = (2π) 3 δ 3 (k 1 + k 2 + k 3 )F(k 1,k 2,k 3 ) Scaling and symmetries imply that F(k 1,k 2,k 3 ) is a symmetric, homogeneous function of degree 6. Primordial non-gaussianities come from cubic terms in the Lagrangian.

25 It is useful to work in the ADM metric formalism: ds 2 = N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt) where inflaton φ and metric h ij are dynamical variables, N and N i are Lagrange multipliers.

26 It is useful to work in the ADM metric formalism: ds 2 = N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt) where inflaton φ and metric h ij are dynamical variables, N and N i are Lagrange multipliers. Focus on scalar perturbations. In the comoving gauge: δφ = 0 h ij = a 2 e 2ζ δ ij with ζ being the gauge invariant scalar perturbation which remains constant outside horizon. Calculations greatly simplified.

27 It is useful to work in the ADM metric formalism: ds 2 = N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt) where inflaton φ and metric h ij are dynamical variables, N and N i are Lagrange multipliers. Focus on scalar perturbations. In the comoving gauge: δφ = 0 h ij = a 2 e 2ζ δ ij with ζ being the gauge invariant scalar perturbation which remains constant outside horizon. Calculations greatly simplified. To obtain the cubic terms: we plug the metric ansatz into the Lagragian, and substitute the Lagrangian multipliers N and N i with the solutions to their equations of motion.

28 It is useful to work in the ADM metric formalism: ds 2 = N 2 dt 2 + h ij (dx i + N i dt)(dx j + N j dt) where inflaton φ and metric h ij are dynamical variables, N and N i are Lagrange multipliers. Focus on scalar perturbations. In the comoving gauge: δφ = 0 h ij = a 2 e 2ζ δ ij with ζ being the gauge invariant scalar perturbation which remains constant outside horizon. Calculations greatly simplified. To obtain the cubic terms: we plug the metric ansatz into the Lagragian, and substitute the Lagrangian multipliers N and N i with the solutions to their equations of motion. To compute the effective action to order O(ζ 3 ), we need only the solutions of N and N i to order O(ζ). Maldacena

29 The solution to the equation of motion for scalar perturbation ζ(t, k) at quadratic order gives the power spectrum. The primordial non-gaussianities are: ζ(t,k 1 )ζ(t,k 2 )ζ(t,k 3 ) = i t t 0 dt [ζ(t,k 1 )ζ(t,k 2 )ζ(t,k 3 ), H int (t )], where H int (t) is the cubic Hamiltonian for the scalar perturbation ζ.

30 The solution to the equation of motion for scalar perturbation ζ(t, k) at quadratic order gives the power spectrum. The primordial non-gaussianities are: ζ(t,k 1 )ζ(t,k 2 )ζ(t,k 3 ) = i t t 0 dt [ζ(t,k 1 )ζ(t,k 2 )ζ(t,k 3 ), H int (t )], where H int (t) is the cubic Hamiltonian for the scalar perturbation ζ. A lot of pain and sweat...

31 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.

32 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 is 58 < f NL < 134 at 95% C.L. Future experiments can eventually reach the sensitivity of f NL < 20 (WMAP) and f NL < 5 (PLANCK).

33 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 is 58 < f NL < 134 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

34 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

35 Slow Roll Limit Maldacena Seery and Lidsey... Slow roll inflation predicts non-gaussianties of order f NL ɛ 1, which is too small to be observed. The relevant shapes are F(k 1, k 2, k 3 ) 1 A ɛ A η = ɛ c 2 s = η c 2 s k 3 i i ki 3. i i j i k3 i k i k 2 j + 1 K A(k 1, k 2, k 3 ) where i>j ki 2 k2 j,

36 The shape of A ɛ is Similar shape for A η. The shapes of slow roll inflation look similar to that of the WMAP ansatz: peak at the squeeze limit.

37 Consisteny Relation for Non-Gaussianities Maldacena 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

38 Consisteny Relation for Non-Gaussianities Maldacena 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 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.

39 Consisteny Relation for Non-Gaussianities Maldacena 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 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 dln k 1 k1 3 k3 3

40 DBI Limit Alishahiha, Silverstein, and Tong Chen Non-Gaussianities are generically quite large f NL 1 c 2 s

41 DBI Limit Alishahiha, Silverstein, and Tong Chen Non-Gaussianities are generically quite large f NL 1 c 2 s The shape of non-gaussianities vanishes in the squeeze triangle limit k 3 k 1, k 2. This is required by Maldacena s consistency relation: F(k 1, k 2, k 3 )k 3 1 k3 3 n s 1 ɛ This contradicts that the non-gaussianities are large, unless the shape vanishes in the squeeze limit.

42 The shape of non-gaussianities for DBI inflation Peak at the equilateral triangle limit. F(k 1,k 2,k 3 ) vanishes in the squeeze limit: higher derivative interactions favor correlations between modes with comparable wavelengths.

43 More Shapes Another shape of potentially large non-gaussianities: A λ = ( 1 c 2 s 1 2λ Σ ) 3k 2 1 k2 2 k2 3 2(k 1 + k 2 + k 3 ) 3 Gruzinov

44 More Shapes Another shape of potentially large non-gaussianities: A λ = ( 1 c 2 s 1 2λ Σ ) 3k 2 1 k2 2 k2 3 2(k 1 + k 2 + k 3 ) 3 Gruzinov The Gaussianities are not multiplied by any slow roll parameter, so can be potentially large. The shape also vanishes in the squeeze limit due to the consistency relation of Maldacena.

45 More Shapes Another shape of potentially large non-gaussianities: A λ = ( 1 c 2 s 1 2λ Σ ) 3k 2 1 k2 2 k2 3 2(k 1 + k 2 + k 3 ) 3 Gruzinov The Gaussianities are not multiplied by any slow roll parameter, so can be potentially large. The shape also vanishes in the squeeze limit due to the consistency relation of Maldacena. For DBI inflation, A λ vanishes but it should be possible to construct realistic models where this shape is large.

46 The shape looks similar to the DBI inflation

47 Relative Sizes For general shapes, f NL is defined in the equilateral triangle limit: fnl λ = 5 81 fnl c = fnl ɛ = f η NL = 5 12 fnl s = 85 s 54 c 2 s ( 1 c 2 ( s 1 ɛ c 2 s η c 2 s c 2 s.,, 1 2λ Σ 1 ), ),

48 Relative Sizes For general shapes, f NL is defined in the equilateral triangle limit: fnl λ = 5 81 fnl c = fnl ɛ = f η NL = 5 12 fnl s = 85 s 54 c 2 s ( 1 c 2 ( s 1 ɛ c 2 s η c 2 s c 2 s.,, 1 2λ Σ 1 ), ), If the sound speed is sufficiently small c 2 s ɛ, η it is possible to observe the slow roll shapes A ɛ, A η. This can be realized in DBI inflation, consistent with current experimental bound on c s.

49 Choice of Vacuum There have been some friendly debates on whether the initial state of inflation can deviate from the Bunch-Davies vacuum.

50 Choice of Vacuum There have been some friendly debates on whether the initial state of inflation can deviate from the Bunch-Davies vacuum. A different perspective: We investigate whether there are pronounced effects of non-standard vacua to be observed in non-gaussianities.

51 Choice of Vacuum There have been some friendly debates on whether the initial state of inflation can deviate from the Bunch-Davies vacuum. A different perspective: We investigate whether there are pronounced effects of non-standard vacua to be observed in non-gaussianities. The quantum state of inflaton is u k = u(τ,k) = ih 4ɛc s k 3(C +(1 + ikc s τ)e ikc sτ + C (1 ikc s τ)e ikc sτ ) where C + and C satisfy the normalization condition C + 2 C 2 = 1. Bunch-Davies vacuum corresponds to C + = 1, C = 0.

52 Two potentially observable contributions à λ and à c due to deviation from Bunch-Davies vacuum. The size of the non-gaussianities are: f λ NL = 5Re(C )( 1 c 2 s 1 2λ Σ ) f c NL = 25 4 Re(C )( 1 c 2 s 1)

53 Two potentially observable contributions à λ and à c due to deviation from Bunch-Davies vacuum. The size of the non-gaussianities are: f NL λ = 5Re(C )( 1 c 2 1 2λ s Σ ) f c NL = 25 4 Re(C )( 1 c 2 s 1) More importantly, the shapes are very distinctive:

54 These shapes are peaked at the folded triangle limit: for arbitrary k 2 and k 3. k 1 = k 2 + k 3 A feature not shared by other sources of non-gaussianities, potentially more pronounced than modulation in power spectrum.

55 ds/cft Strominger Unlike AdS/CFT, the ds/cft proposal suffers from many objections: absence of supersymmetry, no concrete string example, and other conceptual issues. Nevertheless, it is useful to recast problems in (approximate) ds space in terms of a dual 3D CFT as the results are sometimes a consequence of the underlying symmetries. The two point functions and three point functions of the inflaton f are related to the correlators of the CFT operators by the following f k f k 1 = 2Re O k O k f k1 f k2 f k3 = 2Re O k 1 O k2 O k3 i( 2Re O ki O ki ) Maldacena

56 In conformal field theory, the two point and three point correlation functions are constrained by conformal symmetry. O(x)O(y) O(x)O(y)O(z) 1 x y 2 1 x y x z y z For 3, the two point function on the CFT gives the correct scaling for the powers spectrum. [Larsen, McNees];[van der Schaar] We are testing whether conformal symmetries can determine the universal shapes of non-gaussianities.

57 Summary Observational signatures and non-gaussianities of general single field inflation. Size and Shape of non-gaussianities depend on 5 parameters: c 2 s, ɛ, η, s, λ/σ By varying these parameters, can recover all known shapes and more. Deviation from Bunch-Davies vacuum can lead to pronounced signatures in non-gaussianities. Interesting to see whether the ds/cft proposal can shed light on the universality of the shape of non-gausianities. This will be very useful especially for multi-field inflation.

58 Thank You