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
Chain Inflation (Freese and Spolyar 2005)
tunnel through series of vacua:
in string landscape, or with QCD axion
ROLLING MODELS
New inflation, chaotic inflation, hybrid inflation
Natural inflation (Freese, Frieman, Olinto)
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. Old Inflation Universe goes from false vacuum to true vacuum.
Guth (1981)
Universe goes from false vacuum to true vacuum.
Bubbles of true vacuum nucleate in a universe of false vacuum (first order phase transition)

5. Old Inflation Vacuum decay: “swiss cheese problem”
Entire universe is in false vacuum
Nucleate bubbles of true vacuum
Problem: bubbles never percolate & thermalize  NO REHEATING

6. What is needed for tunneling inflation to work?
Two requirements for inflation:
1) Sufficient Inflation: 60 e-foldings
2) The universe must thermalize and reheat; i.e. the entire universe must go through the phase transition at once. Then the phase transition completes.
Can achieve both requirements with
(i) time-dependent nucleation rate in Double-field inflation (Adams and Freese ‘91) with two coupled fields in a single tunneling event
(ii) Chain Inflation (Freese and Spolyar 2005) with multiple tunneling events

7. Rapid phase transition leads to percolation (entire universe goes through phase transition at once)
Vacuum decay: “swiss cheese problem”
Entire universe is in false vacuum
Nucleate bubbles of true vacuum

8. 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

9. Graceful Exit Achieved

10. 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.

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

12. 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)

13. 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

14. 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?

15. Chain Inflation with QCD Axion (Freese,Liu,Spolyar 05)
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

16. Rolling Models of Inflation
Linde (1982)
Albrecht & Steinhardt (1982)
Equation of motion:
Flat region:
V almost constant
rvac dominates energy density
Decay of f:
Particle production
Reheating

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

18. Spectrum of Perturbations
Total number of inflation e-foldings Ntot  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

19. 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)

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

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

22. Different Types of Potentials in the r-n plane
(Dodelson,
Kinney
and Kolb 1997;
Alabidi and
Lyth 2006)

23. Examples of Models

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

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

26. Specific models critically tested
dns/dlnk=0
dns/dlnk=0 r r n n
Models like V(f)~fp
p=4
p=2
For 50 and 60 e-foldings HZ
p fix, Ne varies
(taken from L. Verde)
p varies, Ne fix

27. The full treatment:

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

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

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

31. Fine Tuning due to Radiative Correctionslg +
+   
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!

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

33. 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

34. Shift Symmetries
 “Natural Inflation”
Freese, Frieman & Olinto (1990)
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.

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

36. 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

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

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

39. Sufficient Inflation
f rolls down the hill. The pieces of the universe with f far enough uphill will inflate enough. x
T < L

40. 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 MPl , P = O(1)

41. Density Fluctuations  L ~ 1015 GeV – 1016 GeV (height of potential)
Largest at 60 efolds
before end of inflation
 L ~ GeV – 1016 GeV (height of potential)
 mf = L2/f ~ GeV – 1013 GeV
Density fluctuation spectrum is non-scale invariant with extra power on large length scales
WMAP  f > 0.7 MPL

42. 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)

43. 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.

44. 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,

45. A large number of fields
Assisted Inflation (Liddle and Mazumdar 1998)
N-flation (Dimopoulos, Kachru, McGreevy, Wacker 2005):
Creation of cosmological magnetic fields (Anber and Sorbo 2006)

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

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

48. Tensor Modes in Natural Inflation (original model) (Freese and Kinney 2004)
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.
n.b. not much
running of n
Sensitivity of PLANCK: error bars +/ on r and 0.01 on n.
Next generation expts (3 times more sensitive) must see it.

49. Natural Inflation agrees well with WMAP!

50. r-n plane: Natural inflation after WMAP 3
f > 0.7 MPl allowed

51. Tensor modes

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

53. Running of Spectral Index in Natural Inflation

54. The full treatment:

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

56. Potential
At the end of inflation

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

58. Model Classes

59. Potential
f > few Mpl: V(f) ~ quadratic

60. 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.

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

62. Summary of Natural Inflation confronting data
0) No fine-tuning, naturally flat potential
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.

63. 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
Natural inflation looks good