1. Natural Inflation after WMAP Katherine Freese Michigan Center for Theoretical Physics University of Michigan

2. TWO TYPES OF INFLATION MODELS  TUNNELING MODELS Old Inflation (Guth 1981 Old Inflation (Guth 1981 Chain Inflation (Freese and Spolyar 2005) Chain Inflation (Freese and Spolyar 2005) tunnel through series of vacua: tunnel through series of vacua: in string landscape, or with QCD axion in string landscape, or with QCD axion ROLLING MODELS ROLLING MODELS Predictions being tested with CMB Predictions being tested with CMB

3. I. TUNNELING MODELS Old Inflation (Guth 1981) Universe goes from false vacuum to true vacuum. Bubbles of true vacuum nucleate in a sea of false vacuum (first order phase transition).

4. Swiss Cheese Problem of Old Inflation: no graceful exit PROBLEM: Bubbles never percolate and thermalize: REHEATING FAILS; we don’t live in a vacuum Bubbles of true vacuum nucleate in a sea of false vacuum.

5. What is needed for tunneling inflation to work? Probability of a point remaining in false vacuum phase: where is the nucleation rate of bubbles and H is the expansion rate of the universe The number of e-foldings per tunneling event is Graceful exit: Critical value of is required to get percolation and reheating. In terms of number of efolds, this is Sufficient Inflation requires

6. Graceful Exit Achieved

7. Inflation Requires Two Basic Ingredients 1. Sufficient e-foldings of inflation 2. The universe must thermalize and reheat Old inflation, wih a single tunneling event, failed to do both. Here, MULTIPLE TUNNELING events, each responsible for a fraction of an e-fold (adds to enough). Graceful exit is obtained: phase transition completes at each tunneling event.

8. Chain Inflation Graceful exit: requires that the number of e-foldings per stage is N < 1/3 Sufficient inflation: total number of e-foldings is N tot > 60 Freese & Spolyar (2005) Freese, Liu, & Spolyar (2005) Relevant to: stringy landscape QCD (or other) axion Multiple tunneling events

9. Basic Scenario: Inflation with the QCD axion or in the Stringy Landscape Chain Inflate: Tunnel from higher to lower minimum in stages, with a fraction of an efold at each stage Freese, Liu, and Spolyar (2005) V (a) = V0[1− cos (Na /v)] − η cos(a/v +γ)

10. Chain Inflation: Basic Setup The universe transitions from an initially high vacuum down towards zero, through a series of tunneling events. The picture to consider: tilted cosine Solves old inflation problem: Graceful Exit requires that the number of e-folds per stage < 1/3 Sufficient Inflation requires a total number of e-folds > 60, hence there are many tunneling events

11. Chain Inflation in String Landscape Chain inflation is generic in the string landscape, as the universe tunnels through a series of metastable vacua, each with different fluxes. There appear to be at least 10^200 vacua. Vacua of different fluxes are disconnected in the multidimensional potential, with barriers in between them. Chain inflation is the result of tunneling between these vacua. N.b. Quantized drops in four-form field strength. Tunneling can be fast early on; can it stop without going through intermediate slow stage?

12. Chain Inflation with QCD Axion Low scale inflation at 200 MeV: axion can simultaneously solve strong CP problem and provide inflation In addition to standard QCD axion, need (i) new heavy fermions to get many bumps in the theta field and (ii) tilt from soft breaking of underlying PQ symmetry

13. Rolling Models of Inflation Equation of motion: Flat region: V almost constant  vac dominates energy density Decay of  : Particle production Reheating Linde (1982) Albrecht & Steinhardt (1982)

14. On: the role of observations “Faith is a fine invention “Faith is a fine invention When Gentlemen can see --- When Gentlemen can see --- But Microscopes are prudent But Microscopes are prudent In an Emergency In an Emergency Emily Dickinson, 1860 Emily Dickinson, 1860

15. Spectrum of Perturbations Total number of inflation e-foldings N tot  60 Spectrum of observable scales is produced ~ 50 – 60 e-foldings before the end of inflation 50: later during inflation  smaller scales (~1 Mpc) 60: earlier during inflation  larger scales (~3000 Mpc) 50-60 e-foldings

16. Tensor (gravitational wave) modes  In addition to density fluctuations, inflation also predicts the generation of tensor fluctuations with amplitude  For comparison with observation, the tensor amplitude is conventionally expressed as:  (denominator: scalar modes)

17. Gravity Modes are (at least) two orders of magnitude smaller than density fluctuations: hard to find!

18. Four parameters from inflationary perturbations: I. Scalar perturbations: amplitude spectral index amplitude spectral index II. Tensor (gravitational wave) modes: amplitude spectral index amplitude spectral index Expressed as Inflationary consistency condition: Plot in r-n plane

19. Different Types of Potentials in the r-n plane (KINNEY 2002)

20. Examples of Models

21. Effect of more data WMAP I WMAP I Ext WMAP II Reducing the noise by 3 degeneracies broken LCDM model

22. Tensor-to-scalar ratio r vs. scalar spectral index n scalar spectral index n

23. Specific models critically tested nn rr dn s /dlnk=0 Models like V(  )~  p dn s /dlnk=0 HZ p=4p=2 For 50 and 60 e-foldings p fix, Ne varies p varies, Ne fix (taken from L. Verde)

24. The full treatment:

25. Natural Inflation after WMAP Chris Savage, K. Freese, W. Kinney, hep-ph/ 0609144 hep-ph/ 0609144 Theoretical motivation: no fine-tuning Recent interest in light of theoretical developments Unique predictions: Looks good compared to data

26. Fine Tuning in Rolling Models Fine Tuning in Rolling Models  The potential must be very flat: (Adams, Freese, and Guth 1990) But particle physics typically gives this ratio = 1!

27. Inflationary Model Constraints Success of inflationary models with rolling fields  constraints on V(  ) Enough inflation Scale factor a must grow enough Amplitude of density fluctuations not too large

28. Fine Tuning due to Radiative Corrections Perturbation theory: 1-loop, 2-loop, 3-loop, etc. To keep must balance tree level term against corrections to each order in perturbation theory. Ugly! g 

29. Inflation needs small ratio of mass scales  Two attitudes: 1) We know there is a heirarchy problem, wait until it’s explained 1) We know there is a heirarchy problem, wait until it’s explained 2) Two ways to get small masses in particles physics: 2) Two ways to get small masses in particles physics: (i) supersymmetry (i) supersymmetry (ii) Goldstone bosons (shift symmetries) (ii) Goldstone bosons (shift symmetries)

30. Natural Inflation: Shift Symmetries Shift (axionic) symmetries protect flatness of inflaton potential (inflaton is Goldstone boson) Additional explicit breaking allows field to roll. This mechanism, known as natural inflation, was first proposed in Freese, Frieman, and Olinto 1990; Adams, Bond, Freese, Frieman and Olinto 1993

31. Shift Symmetries We know of a particle with a small ratio of scales: the axion IDEA: use a potential similar to that for axions in inflation  natural inflation (no fine-tuning) Here, we do not use the QCD axion. We use a heavier particle with similar behavior.  “Natural Inflation” Freese, Frieman & Olinto (1990)

32. e.g., mimic the physics of the axion (Weinberg; Wilczek)

33. Natural Inflation (Freese, Frieman, and Olinto 1990; Adams, Bond, Freese, Frieman and Olinto 1993)  Two different mass scales:  Width f is the scale of SSB of some global symmetry  Height is the scale at which some gauge group becomes strong

34. Two Mass Scales Provide required heirarchy  For QCD axion,  For inflation, need Enough inflation requires width = f ≈ mpl, Amplitude of density fluctuations requires height =

35. Sufficient Inflation  initially randomly distributed between 0 and  f at different places in the universe. T <  :  rolls down the hill. The pieces of the universe with  far enough uphill will inflate enough. T >  T <  x

36. Sufficient Inflation  rolls down the hill. The pieces of the universe with  far enough uphill will inflate enough. T <  x

37. Sufficient Inflation A posteriori probability: Those pieces of the universe that do inflate end up very large. Slice the universe after inflation and see what was probability of sufficient inflation. Numerically evolved scalar field For f  0.06 M Pl, P = O(1)

38. Density Fluctuations   ~ 10 15 GeV – 10 16 GeV (height of potential)  m  =  2 /  ~ 10 11 GeV – 10 13 GeV Density fluctuation spectrum is non-scale invariant with extra power on large length scales WMAP  f > 0.7 M PL Largest at 60 efolds before end of inflation

39. Implementations of natural inflation’s shift symmetry  Natural chaotic inflation in SUGRA using shift symmetry in Kahler potential (Gaillard, Murayama, Olive 1995; Kawasaki, Yamaguchi, Yanagida 2000)  In context of extra dimensions: Wilson line with (Arkani-Hamed et al 2003) but Banks et al (2003) showed it fails in string theory.  “Little” field models (Kaplan and Weiner 2004)  In brane Inflation ideas (Firouzjahi and Tye 2004)  Gaugino condensation in SU(N) SU(M): Adams, Bond, Freese, Frieman, Olinto 1993; Blanco-Pillado, Linde et al 2004 (Racetrack inflation)

40. Legitimacy of large axion scale? Natural Inflation needs Is such a high value compatible with an effective field theory description? Do quantum gravity effects break the global axion symmetry? Kinney and Mahantappa 1995: symmetries suppress the mass term and is OK. Arkani-Hamed et al (2003):axion direction from Wilson line of U(1) field along compactified extra dimension provides However, Banks et al (2003) showed it does not work in string theory.

41. A large effective axion scale (Kim, Nilles, Peloso 2004)  Two or more axions with low PQ scale can provide large  Two axions  Mass eigenstates are linear combinations of  Effective axion scale can be large,

42. A large number of fields  Assisted Inflation (Liddle and Mazumdar 1998)  N-flation (Dimopoulos, Kachru, McGreevy, Wacker 2005): Shamit’s talk this morning  Creation of cosmological magnetic fields (Anber and Sorbo 2006)

43. Density Fluctuations and Tensor Modes Density Fluctuations and Tensor Modes can determine which model is right  Density Fluctuations : WMAP data: WMAP data: Slight indication of running of spectral index  Tensor Modes gravitational wave modes, detectable in upcoming experiments gravitational wave modes, detectable in upcoming experiments

44. Density Fluctuations in Natural Inflation  Power Spectrum:  WMAP data: implies implies (Freese and Kinney 2004)

45. Tensor Modes in Natural Inflation (original model) (Freese and Kinney 2004) Sensitivity of PLANCK: error bars +/- 0.05 on r and 0.01 on n. Next generation expts (3 times more sensitive) must see it. n.b. not much running of n running of n Two predictions, testable in next decade: 1) Tensor modes, while smaller than in other models, must be found. 2) There is very little running of n in natural inflation.

46. Natural Inflation agrees well with WMAP!

47. r-n plane: Natural inflation after WMAP 3 f > 0.7 M Pl allowed

48. Spectral Index Running small f : (exponentially suppressed) large f :

49. The full treatment:

50. Potential 60 e-foldings before the end of inflation ~ present day horizon

51. Potential At the end of inflation

52. Model Classes Kinney & collaborators Large-field Small-field Hybrid

53. Model Classes

54. Potential f > few M pl : V(  ) ~ quadratic

55. Natural Inflation Summary No fine tuning, naturally flat potential WMAP 3-year data: f < 0.7 M Pl excluded f > 0.7 M Pl consistent Tensor/scalar ratio r Spectral index n s Spectral index running dn s /d lnk

56. To really test inflation need B modes, which can only be produced by gravity waves. Will confirm key prediction of inflation. Will differentiate between models. Need next generation experiments.

59. E and B modes polarization E polarization from scalar, vector and tensor modes B polarization only from (vector) tensor modes Kamionkowski, Kosowsky, Stebbings 1997, Zaldarriga & Seljak 1997

60. TT TE EE BB WMAP3 data

62. Future prospects: gravity waves Tev 3.2x10 1.7x10 9.7x10 5.5x10 3x10 13 12 Verde Peiris Jimenez 05

63. Summary of Natural Inflation confronting data  1) Matches data in r-n plane for f>0.7mpl  2) Tensor modes may be as small as 0.001  3) Small running, an order of magnitude below sensitivity of WMAP3, not detectable any time soon. Big running in the data would kill the model.

64. Conclusion  Tunneling Models: Chain Inflation in Landscape and with QCD Axion. TO DO: perturbations (with S. Watson)  Rolling Models: Generic predictions of inflation match the data Generic predictions of inflation match the data Natural inflation looks good Natural inflation looks good

65. Conclusion  An early period of inflation resolves cosmological puzzles: homogeneity, isotropy, oldness, and monopoles. It also generates density perturbations for galaxy formation.  Details of density and gravitational wave modes can be used to test inflation as well as individual models.  Predictions of inflation are confirmed!  Natural inflation, which was theoretically well- motivated, fits the data very well.

66. DARK ENERGY (w=p/rho)

68. 1

69. SUMMARY:  I. The predictions of inflation are right: (i) the universe has a critical density (i) the universe has a critical density (ii) Gaussian perturbations (ii) Gaussian perturbations (iii) density perturbation spectrum nearly scale invariant (iii) density perturbation spectrum nearly scale invariant iv) detection of polarization (from gravitational wave modes) in upcoming data may provide smoking gun for inflation iv) detection of polarization (from gravitational wave modes) in upcoming data may provide smoking gun for inflation  II. Polarization measurements will tell us which model is right. WMAP already selects between models. WMAP already selects between models. Natural inflation (Freese, Frieman, Olinto) looks great Natural inflation (Freese, Frieman, Olinto) looks great

70. STOP HERE

71. Generation of CMB polarization Temperature quadrupole at the surface of last scatter generates polarization. Potential well Potential hill From Wayne Hu At the last scattering surface At the end of the dark ages (reionization)

72. Polarization for density perturbation Radial (tangential) pattern around hot (cold) spots.

73. E and B modes polarization E polarization from scalar, vector and tensor modes B polarization only from (vector) tensor modes Kamionkowski, Kosowsky, Stebbings 1997, Zaldarriga & Seljak 1997

74. Clean BB after FG removal. 3-sigma detection of EE. The “Gold” multipoles: l=3,4,5,6.

75. Comparison with WMAP I Best fit LCDM WMAP I Best fit LCDM WMAP I Ext Best fit LCDM WMAP II WMAP II WMAP I

76. Specific models critically tested nn rr dn s /dlnk=0 Models like V(  )~  p dn s /dlnk=0 HZ p=4p=2 For 50 and 60 e-foldings p fix, Ne varies p varies, Ne fix

77. Future prospects: gravity waves Tev 3.2x10 1.7x10 9.7x10 5.5x10 3x10 13 12 Verde Peiris Jimenez 05

78. end

79. Density Fluctuations and Tensor Modes Density Fluctuations and Tensor Modes can determine which model is right  Density Fluctuations : WMAP data: WMAP data: Slight indication of running of spectral index  Tensor Modes gravitational wave modes, detectable in upcoming experiments gravitational wave modes, detectable in upcoming experiments

80. 1 sigma reconstruction of potential from 1-year WMAP data (KINNEY, KOLB, MELCHIORRI, AND RIOTTO 2003)

81. IV. From Theory to Observation: Predictions of Inflation  1) flat universe:  2) Specrum of density perturbations:  3) gravitational wave modes  Individual models make specific predictions.  Can test inflation as a concept and can differentiate between models.

82. Prediction 1 of Inflation: The geometry of the universe is flat; i.e. the density is critical and i.e. the density is critical and

83. WMAP Satellite  Launched June 2002  Data released Feb. 2003

84. The Microwave Sky

85. Prediction 1 is confirmed  WMAP confirms the inflationary prediction that prediction that

86. Prediction 2 of Inflation  Spectral index of density perturbations (scalar modes) is near n=1  n.b. individual models make specific predictions for n which can be used to differentiate between models

89. Prediction 2 of inflation is confirmed  Multiple data sets (WMAP, large scale structure, etc) confirm n near 1.  More detail shown in a minute to differentiate between models

90. Prediction 3 of Inflation: Existence of gravitational wave perturbations (tensor modes) Existence of gravitational wave perturbations (tensor modes)

91. Status of Rolling Models:  I. The predictions of inflation are right: (i) the universe has a critical density (i) the universe has a critical density (ii) Gaussian perturbations (so far) (ii) Gaussian perturbations (so far) (iii) density perturbation spectrum nearly scale invariant (iii) density perturbation spectrum nearly scale invariant iv) detection of polarization (from gravitational wave modes) in upcoming data may provide smoking gun for inflation iv) detection of polarization (from gravitational wave modes) in upcoming data may provide smoking gun for inflation  II. Polarization measurements will tell us which model is right. WMAP already selects between models. WMAP already selects between models. Natural inflation (Freese, Frieman, Olinto) looks great Natural inflation (Freese, Frieman, Olinto) looks great

92. Predictions: Density and Gravity Fluctuations in Natural Inflation  Power Spectrum: (not quite scale invariant, n<1) (not quite scale invariant, n<1)  Gravitational wave modes extremely suppressed (smaller than in most models)

93. Polarization

94. Tensor-to-scalar ratio r vs. scalar spectral index n for natural inflation (Freese and Kinney 2004) n.b. This is a small-field model