Signatures of Axion Monodromy Inflation

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1 Signatures of Axion Monodromy Inflation Gang Xu Cornell University based on arxiv: with Flauger, McAllister, Pajer and Westphal McGill University December 2009 Gang Xu Signatures of Axion Monodromy Inflation 1/44

2 Outline Motivations 1 Motivations Gang Xu Signatures of Axion Monodromy Inflation 2/44

3 Inflation Motivations Gang Xu Signatures of Axion Monodromy Inflation 3/44

4 Brief Introduction to Inflation A period of accelerated expansion ds 2 = dt 2 + e 2Ht d x 2 H const Generic Predictions: Scalar perturbations are approximately scale-invariant: n s 1 Tensor to scalar ratio r = 16ɛ, where ɛ Ḣ H 2. Scalar perturbations are approximately Gaussian. Gang Xu Signatures of Axion Monodromy Inflation 4/44

5 Motivations Cosmological Data Planck was launched on May 14th 2009 and started collecting data since August Gang Xu Signatures of Axion Monodromy Inflation 5/44

6 Motivations Cosmological Data Planck was launched on May 14th 2009 and started collecting data since August loc < 6 Planck is able to detect r > 0.05 and constrain fnl Gang Xu Signatures of Axion Monodromy Inflation 5/44

7 r and the Field Excursion L = 1/2( φ) 2 V (φ) 1 Lyth, 1996 Gang Xu Signatures of Axion Monodromy Inflation 6/44

8 r and the Field Excursion L = 1/2( φ) 2 V (φ) r = 16ɛ = 8 φ M p ( φ = N end N CMB dn H ) 2 with dn = Hdt r(n) 8 1 Lyth, 1996 Gang Xu Signatures of Axion Monodromy Inflation 6/44

9 r and the Field Excursion L = 1/2( φ) 2 V (φ) r = 16ɛ = 8 φ M p φ M p ( φ = N end N CMB dn O(1) ( r 0.01 H ) 2 with dn = Hdt r(n) 8 ) 1/2 1 Lyth, 1996 Gang Xu Signatures of Axion Monodromy Inflation 6/44

10 r and the Field Excursion L = 1/2( φ) 2 V (φ) r = 16ɛ = 8 φ M p φ M p ( φ = N end N CMB dn O(1) ( r 0.01 H ) 2 with dn = Hdt r(n) 8 ) 1/2 Detectable gravitational waves suggests superplanckian field variation: φ > M p 1 1 Lyth, 1996 Gang Xu Signatures of Axion Monodromy Inflation 6/44

11 UV Sensitivity An inflaton potential in EFT: V = 1/2m 2 φ 2 + 1/4λφ 4 + φ 4 Σ p=1 λ p ( φ Λ ) 2p Gang Xu Signatures of Axion Monodromy Inflation 7/44

12 UV Sensitivity An inflaton potential in EFT: ) 2p ( V = 1/2m 2 φ 2 + 1/4λφ 4 + φ 4 Σ p=1 λ φ p Λ One good idea: shift symmetry φ φ + const e.g. axions Gang Xu Signatures of Axion Monodromy Inflation 7/44

13 UV Sensitivity An inflaton potential in EFT: ) 2p ( V = 1/2m 2 φ 2 + 1/4λφ 4 + φ 4 Σ p=1 λ φ p Λ One good idea: shift symmetry φ φ + const e.g. axions Still not satisfactory b/c EFT is ignorant about Planck scale. Gang Xu Signatures of Axion Monodromy Inflation 7/44

14 UV Sensitivity An inflaton potential in EFT: ) 2p ( V = 1/2m 2 φ 2 + 1/4λφ 4 + φ 4 Σ p=1 λ φ p Λ One good idea: shift symmetry φ φ + const e.g. axions Still not satisfactory b/c EFT is ignorant about Planck scale. We need a candidate UV completed theory, string theory. Gang Xu Signatures of Axion Monodromy Inflation 7/44

15 Shift symmetry of axions Axions are pseudoscalar fields with only derivative couplings. Continuous shift symmetry: S(a) = S(a + const). 2 Freese,Frieman and Olinto, 1990 Gang Xu Signatures of Axion Monodromy Inflation 8/44

16 Shift symmetry of axions Axions are pseudoscalar fields with only derivative couplings. Continuous shift symmetry: S(a) = S(a + const). Non-perturbative effect breaks it to a discrete shift symmetry, a a + 2π. 2 Freese,Frieman and Olinto, 1990 Gang Xu Signatures of Axion Monodromy Inflation 8/44

17 Shift symmetry of axions Axions are pseudoscalar fields with only derivative couplings. Continuous shift symmetry: S(a) = S(a + const). Non-perturbative effect breaks it to a discrete shift symmetry, a a + 2π. ( )) Natural Inflation 2 with potential V (φ) = Λ (1 4 cos φ f 0 2πf 2 Freese,Frieman and Olinto, 1990 Gang Xu Signatures of Axion Monodromy Inflation 8/44

18 Axion inflation Problem with Natural Inflation: Axions with super-planckian decay constant 3 are elusive in controllable string theory setup. 3 Banks, Dine, Fox and Gorbatov, 2003; Svrcek and Witten, Dimopoulos, Kachru, McGreevy and Wacker, 2005 Gang Xu Signatures of Axion Monodromy Inflation 9/44

19 Axion inflation Problem with Natural Inflation: Axions with super-planckian decay constant 3 are elusive in controllable string theory setup. A good idea: N-flation 4 use N 10 3 axions collectively to inflate 3 Banks, Dine, Fox and Gorbatov, 2003; Svrcek and Witten, Dimopoulos, Kachru, McGreevy and Wacker, 2005 Gang Xu Signatures of Axion Monodromy Inflation 9/44

20 Axion inflation Problem with Natural Inflation: Axions with super-planckian decay constant 3 are elusive in controllable string theory setup. A good idea: N-flation 4 use N 10 3 axions collectively to inflate Our proposal: recycle an axion N times via monodromy 3 Banks, Dine, Fox and Gorbatov, 2003; Svrcek and Witten, Dimopoulos, Kachru, McGreevy and Wacker, 2005 Gang Xu Signatures of Axion Monodromy Inflation 9/44

21 Real-life Monodromy: Waterride A waterride I have taken when I was young Gang Xu Signatures of Axion Monodromy Inflation 10/44

22 Real-life Monodromy: Spiral Stairs or the spiral stairs once walked on Gang Xu Signatures of Axion Monodromy Inflation 11/44

23 Summary of Motivations Large field inflation is both interesting for observations and theoretical concerns. Gang Xu Signatures of Axion Monodromy Inflation 12/44

24 Summary of Motivations Large field inflation is both interesting for observations and theoretical concerns. Our model= An axion from string theory with shift symmetry + Its non-periodic potential from monodromy + Modulations computed with string theory setup Gang Xu Signatures of Axion Monodromy Inflation 12/44

25 Axions in String Theory String theory provides many axions: b(x) = B 2, c(x) = Σ 2 Σ p C p 5 Wen and Witten, 1986 Gang Xu Signatures of Axion Monodromy Inflation 13/44

26 Axions in String Theory String theory provides many axions: b(x) = B 2, c(x) = Σ 2 Σ p C p Worldsheet vertex operator 5 : V (k) = d 2 ξexp(ik X (ξ))ɛ αβ α X µ β X ν B µν (X ) WS V (k = 0) = Σ 2 B 2 = 0 indicates axion b can only have derivative couplings. 5 Wen and Witten, 1986 Gang Xu Signatures of Axion Monodromy Inflation 13/44

27 Shift Symmetry Breaking We need to break shift symmetry to obtain an inflaton potential. Gang Xu Signatures of Axion Monodromy Inflation 14/44

28 Shift Symmetry Breaking We need to break shift symmetry to obtain an inflaton potential. From D-branes The presence of worldsheet boundaries such as D-branes. We use wrapped branes to create a monodromy. From instantons Worldsheet instantons, or D-brane instantons introduce sinusoidal modulations. Gang Xu Signatures of Axion Monodromy Inflation 14/44

29 Monodromy from a Five Brane Consider a D5-brane wrapping on a two cycle 6, S DBI = T 5 d 4 x g 4 d 2 ξ det(g ind + B ind ) V (b) = T 5 l 4 + b 2 Take large b limit: V (φ) = µ 3 φ Gang Xu Signatures of Axion Monodromy Inflation 15/44

30 Monodromy from a Five Brane Consider a D5-brane wrapping on a two cycle 6, S DBI = T 5 d 4 x g 4 d 2 ξ det(g ind + B ind ) V (b) = T 5 l 4 + b 2 Take large b limit: V (φ) = µ 3 φ NS5 NS5 Gang Xu Signatures of Axion Monodromy Inflation 15/44

31 Fixing Parameters A linear potential V = µ 3 φ with ɛ = 1 2φ 2, η = 0 N(φ) = φ φ end dφ 2ɛ 60 efolds indicates φ = 11Mp COBE normalization: P s = 1 V 24π 2 ɛ = ( ) 2 fixes µ = Mp Gang Xu Signatures of Axion Monodromy Inflation 16/44

32 Predictions of the Linear Potential V = µ 3 φ Tensor mode: r = We will soon know! Tilt:n s = Gang Xu Signatures of Axion Monodromy Inflation 17/44

33 Predictions of the Linear Potential V = µ 3 φ Tensor mode: r = We will soon know! Tilt:n s = Chaotic Inflation Linear Axion Inflation µ 3! N = 50 N = 60 IIA Nil manifolds µ 10/3! 2/3 N = 50 N = Gang Xu ns Signatures of Axion Monodromy Inflation 17/44

34 Modulations V (φ) = µ 3 φ + µ 3 bf cos( φ f ) Monotonicity: b < 1 Number of oscillations in CMB: 1 10f Gang Xu Signatures of Axion Monodromy Inflation 18/44

35 Background Evolution Solve φ + 3H φ + V (φ) = 0 perturbatively in b φ = φ 0 + φ 1 Zeroth order solution: φ 0 (t) = (φ 3/2 First order solution: φ 1 3bf 2 φ 0 sin ( φ0 3 2 µ3/2 ) t) 2/3 f Gang Xu Signatures of Axion Monodromy Inflation 19/44

36 Background Evolution Slow roll parameters ɛ = ɛ 0 + ɛ osci 1 2φ 2 3bf φ η = η 0 + η osci 0 6b sin cos ( φ0 ( φ0 f f ) ) Gang Xu Signatures of Axion Monodromy Inflation 20/44

37 Resonant Mechanism Background oscillates at frequency ω = φ f Perturbation mode of comoving momentum k oscillates at frequency k a until freezes at k = ah Gang Xu Signatures of Axion Monodromy Inflation 21/44

38 Resonant Mechanism Background oscillates at frequency ω = φ f Perturbation mode of comoving momentum k oscillates at frequency k a until freezes at k = ah So when H < ω < M p, every mode will resonate with the background at some time. Gang Xu Signatures of Axion Monodromy Inflation 21/44

39 Postulated Power Spectrum Oscillations in the power spectrum ( ) ns 1 ( )) P s = A k s k (1 + δn s cos φk ln k/k f where φ k φ φ Gang Xu Signatures of Axion Monodromy Inflation 22/44

40 Analytical Power Spectrum Solution: δn s = 12b (1+(3f φ ) 2 ) ( ) π 8 coth π 2f φ f φ 24b f ns f Gang Xu Signatures of Axion Monodromy Inflation 23/44

41 Predicted Angular Power Spectrum Gang Xu Signatures of Axion Monodromy Inflation 24/44

42 Compare with Unbinned WMAP Gang Xu Signatures of Axion Monodromy Inflation 25/44

43 Improvement from Planck 7 7 Planck Collaboration, 2006 Gang Xu Signatures of Axion Monodromy Inflation 26/44

44 Power Spectrum: Likelihood Analysis ns 0.1 bf f log 10 f Figure of merit: bf < 10 4 for f < 0.01 Gang Xu Signatures of Axion Monodromy Inflation 27/44

45 Lightning review of non-gaussianity Simple case: φ(x) = φ G (x) + f local NL φ 2 G (x) 8 f NL, amplitude of bispectrum, is usually a function of three momenta k 1 k 2 k 1 k 3 k 2 k 3 Equilateral Local 8 Komatsu and Spergel, Komatsu et. al. WMAP Gang Xu Signatures of Axion Monodromy Inflation 28/44

46 Lightning review of non-gaussianity Simple case: φ(x) = φ G (x) + f local NL φ 2 G (x) 8 f NL, amplitude of bispectrum, is usually a function of three momenta k 1 k 2 k 1 k 3 k 2 k 3 Equilateral Local WMAP5 data gives: 9 < f local 9 NL Planck will constraint f local NL < 6 8 Komatsu and Spergel, Komatsu et. al. WMAP < 111, 151 < f equil NL < 253 Gang Xu Signatures of Axion Monodromy Inflation 28/44

47 f NL from Inflation Models Canonical single-field slow-roll 10 : f NL O(ɛ), ɛ 10 2 Curvaton, non-canonical kinetic terms(k-inflation, DBI) or features in the potential: 10 Maldacena, 2003 Gang Xu Signatures of Axion Monodromy Inflation 29/44

48 Non-Gaussianity: Oscillatory Resonant non-gaussianity condition 11 : H < ω < M p for linear potential: < f < Chen, Easther, and Lim, Flauger and Pajer,to appear Gang Xu Signatures of Axion Monodromy Inflation 30/44

49 Non-Gaussianity: Oscillatory Resonant non-gaussianity condition 11 : H < ω < M p for linear potential: < f < R(τ, k 1 )R(τ, k( 2 )R(τ, k 3 ) = ) Ps 2 (k 1 k 2 k 3 f ) 2 res sin 1 φf ln K + phase δ 3 (K) (2π) 7, 12 f res 3 η osci 8H φf, 11 Chen, Easther, and Lim, Flauger and Pajer,to appear Gang Xu Signatures of Axion Monodromy Inflation 30/44

50 Non-Gaussianity: Oscillatory Resonant non-gaussianity condition 11 : H < ω < M p for linear potential: < f < R(τ, k 1 )R(τ, k( 2 )R(τ, k 3 ) = ) Ps 2 (k 1 k 2 k 3 f ) 2 res sin 1 φf ln K + phase δ 3 (K) (2π) 7, 12 f res 3 η osci 8H φf, Our model: f res 9b 4φ 3/2 0 f = 9 ( ω ) 3/2 3/2 4 b H 11 Chen, Easther, and Lim, Flauger and Pajer,to appear Gang Xu Signatures of Axion Monodromy Inflation 30/44

51 Summary of the Phenomenology r=0.07: Gravitational waves detectable by Planck. Potentially detectable oscillations in scalar power spectrum. Potentially detectable resonantly enhanced non-gaussianity. Gang Xu Signatures of Axion Monodromy Inflation 31/44

52 Parameter Space 0-1 TTT TT log 10 b log 10 f Gang Xu Signatures of Axion Monodromy Inflation 32/44

53 Cartoon of Embedding Gang Xu Signatures of Axion Monodromy Inflation 33/44

54 The Plan Motivations Calculate the decay constant f and amplitude of modulation bf and constrain them Gang Xu Signatures of Axion Monodromy Inflation 34/44

55 The Plan Motivations Calculate the decay constant f and amplitude of modulation bf and constrain them Decay constant f Calculate the decay constant from the kinetic term: 1 2 f 2 ( c) 2 Recall in SUSY we have L = K aā Φ a Φā for superfields Gang Xu Signatures of Axion Monodromy Inflation 34/44

56 The Plan Motivations Calculate the decay constant f and amplitude of modulation bf and constrain them Decay constant f Calculate the decay constant from the kinetic term: 1 2 f 2 ( c) 2 Recall in SUSY we have L = K aā Φ a Φā for superfields Size of the modulation Stabilize the moduli following KKLT Scalar potential: V = e K ( DW 2 3 W 2 ) Need to know D=4 N = 1 data. Gang Xu Signatures of Axion Monodromy Inflation 34/44

57 D = 4, N = 1 data from Type IIB O3/O7 orientifolds Under orientifold action, cohomology classes split,h r,s = H r,s + + Hr,s Corresponding two cycle bases ω A also split into ω a, a = 1,, h 1,1,and ω α, α = 1,, h 1, Grimm and Louis,2004 Gang Xu Signatures of Axion Monodromy Inflation 35/44

58 D = 4, N = 1 data from Type IIB O3/O7 orientifolds Under orientifold action, cohomology classes split,h r,s = H r,s + + Hr,s Corresponding two cycle bases ω A also split into ω a, a = 1,, h 1,1,and ω α, α = 1,, h 1, Chiral multiplet G a 1 2π (ca i ba g s ) T α iρ α c αβγv β v γ + g s 4 c αbcg b (G Ḡ) c 13 Grimm and Louis,2004 Gang Xu Signatures of Axion Monodromy Inflation 35/44

59 D = 4, N = 1 data from Type IIB O3/O7 orientifolds Under orientifold action, cohomology classes split,h r,s = H r,s + + Hr,s Corresponding two cycle bases ω A also split into ω a, a = 1,, h 1,1,and ω α, α = 1,, h 1, Chiral multiplet G a 1 2π (ca i ba g s ) T α iρ α c αβγv β v γ + g s 4 c αbcg b (G Ḡ) c K = log ( g s ) 2 2logVE V E = 1 6 c αβγv α (T, G)v β (T, G)v γ (T, G), note specifically v = v(ret, ImG) 13 Grimm and Louis,2004 Gang Xu Signatures of Axion Monodromy Inflation 35/44

60 Calculation of Axion Decay Constant 1 2 f 2 ( c) 2 K GḠ G Ḡ f 2 Mp 2 = g s 8π 2 c α v α V E Gang Xu Signatures of Axion Monodromy Inflation 36/44

61 Calculation of Axion Decay Constant 1 2 f 2 ( c) 2 K GḠ G Ḡ f 2 Mp 2 = g s 8π 2 c α v α V E Validity of α expansion: use worldsheet instanton to estimate: e S WS < e 2 2 < 1 gstring 2πα = g s v α 2π Gang Xu Signatures of Axion Monodromy Inflation 36/44

62 Calculation of Axion Decay Constant 1 2 f 2 ( c) 2 K GḠ G Ḡ f 2 Mp 2 = g s 8π 2 c α v α V E Validity of α expansion: use worldsheet instanton to estimate: e S WS < e 2 2 < 1 gstring 2πα = g s v α 2π Lower bound on f: f 2 Mp 2 > gs (2π) 3 V E Gang Xu Signatures of Axion Monodromy Inflation 36/44

63 Warmup: Review of KKLT Non-perturbative superpotential: W = W h 1,1 α=1 A α e aαtα, a α = 2π N α Gang Xu Signatures of Axion Monodromy Inflation 37/44

64 Warmup: Review of KKLT Non-perturbative superpotential: W = W h 1,1 α=1 A α e aαtα, a α = 2π N α D α W Tα W + W Tα K = A α a α e aαtα W v α 2V E = 0 stabilizes the volume of all 4-cycles Gang Xu Signatures of Axion Monodromy Inflation 37/44

65 Warmup: Review of KKLT Non-perturbative superpotential: W = W h 1,1 α=1 A α e aαtα, a α = 2π N α D α W Tα W + W Tα K = A α a α e aαtα W v α 2V E = 0 stabilizes the volume of all 4-cycles D a W = iw cαacv α b c 4πV E = 0 stablizes b a = 0 Gang Xu Signatures of Axion Monodromy Inflation 37/44

66 Find bf : instanton correction Educated guess: K = 2log(V E + e S ED1cos(c)) Gang Xu Signatures of Axion Monodromy Inflation 38/44

67 Find bf : instanton correction Educated guess: K = 2log(V E + e S ED1cos(c)) Perturbative moduli stabilization like τ = τ 0 + e S τ 1 Gang Xu Signatures of Axion Monodromy Inflation 38/44

68 Find bf : instanton correction Educated guess: K = 2log(V E + e S ED1cos(c)) Perturbative moduli stabilization like τ = τ 0 + e S τ 1 ( ) bf = U mod φ µ 3 φ e S ED1 K (1) + 2Re W (1) W (0) Gang Xu Signatures of Axion Monodromy Inflation 38/44

69 Find bf : instanton correction Educated guess: K = 2log(V E + e S ED1cos(c)) Perturbative moduli stabilization like τ = τ 0 + e S τ 1 ( ) bf = U mod φ µ 3 φ e S ED1 K (1) + 2Re W (1) W (0) ( ) U mod = gs 3 W 2 2 from AdS minimum+uplifting V 2 E (0) bf < 2c gs e ( VE 2 2/gs W 0.1 ) 2 Gang Xu Signatures of Axion Monodromy Inflation 38/44

70 Consistency Conditions g s 1 Euclidean instanton:e aαtα < e 2 τ α > Nα π Back reaction on the throat N w = φ 2πf R4 perpx 4πg s f M p > 0.09 V inf U mod τ α 73 8 log N α < 50 X 1/3 V 2/3 E ( v α π g s Back reaction of the 4-cycle volume. Higher derivative terms 2g s ) where we used Gang Xu Signatures of Axion Monodromy Inflation 39/44

71 Backreaction of 4-cycle volume Effect of N w D3 branes 14 NS5 NS5 14 Alternative: Berg, Pajer and Sjörs, to appear Gang Xu Signatures of Axion Monodromy Inflation 40/44

72 Higher derivative terms S 10D,E α 3 R 4 PQ R MN [M H PQ N] and H C[P [M H H = db S 4D = [ d 4 x 1 φ ] φ 8 + φ 4, MI 12 MII 8 at most a few percent. ω M I, ω M II Q] N]C +..., 15 where 15 Gross and Sloan, 1987 Gang Xu Signatures of Axion Monodromy Inflation 41/44

73 A Toy Example 0-1 TTT TT V =113 e g =0.134 s log 10 b log 10 f Gang Xu Signatures of Axion Monodromy Inflation 42/44

74 Conclusions Axion monodromy inflation is falsifiable by its prediction of r = In addition: possible detectable oscillations in the power spectrum and resonantly enhanced non-gaussianity. We study the microscopic constraints on specific class of models in string theory. Gang Xu Signatures of Axion Monodromy Inflation 43/44

75 Future directions Study of oscillatory type non-gaussianity with data. Study of reheating. Realization of chain inflation. Other compactification models. Gang Xu Signatures of Axion Monodromy Inflation 44/44

Phenomenology of Axion Inflation

Phenomenology of Axion Inflation based on Flauger & E.P. 1002.0833 Flauger, McAllister, E.P., Westphal & Xu 0907.2916 Barnaby, EP & Peloso to appear Enrico Pajer Princeton University Minneapolis Oct 2011