Natural Inflation after WMAP

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 MODELSOld 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
Guth (1981)

• Universe goesfrom false vacuumto 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

Problem bubbles never percolate thermalize ?
NO REHEATING
6
Old Inflation

• Bubbles inflate away faster than they form
  grow ? no end to inflation no reheating

7
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

8
Rapid phase transition leads to
percolation (entire universe goes through
phase transition at once)

• Vacuum decay swiss cheese problem

9
Rapid Phase Transition

• Need bubbles to form and grow faster than
  inflation ? inflation comes to an end and
  reheating occurs

10
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

11
Graceful Exit Achieved
12
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.

13
Chain Inflation
Freese Spolyar (2005) Freese, Liu, Spolyar
(2005)

• Graceful exitrequires that the number of
  e-foldings per stage is N lt 1/3
• Sufficient inflationtotal number of e-foldings
  is Ntot gt 60

Relevant to stringy landscape QCD (or
other) axion
14
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) V01- cos (Na /v) - ? cos(a/v ?)

15
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 lt
  1/3
• Sufficient Inflation requires a total number of
  e-folds gt 60, hence there are many tunneling
  events

16
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 10200 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?

17
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

18
Rolling Models of Inflation
Linde (1982) Albrecht Steinhardt (1982)

• Equation of motion
• Flat region
• V almost constant
• rvac dominatesenergy density
• Decay of f
• Particle production
• Reheating

19
On the role of observations

• Faith is a fine invention
• When Gentlemen can see ---
• But Microscopes are prudent
• In an Emergency
• Emily Dickinson, 1860

20
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)

21
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)

22
Gravity Modes are (at least) two orders of
magnitude smaller than density fluctuations hard
to find!
23
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

24
Different Types of Potentials in the r-n plane
(Dodelson, Kinney and Kolb 1997 Alabidi
and Lyth 2006)
25
Examples of Models
26
Effect of more data
LCDM model
Reducing the noise by 3 degeneracies
broken
27
Tensor-to-scalar ratio r vs. scalar spectral
index n
28
Specific models critically tested
dns/dlnk0
dns/dlnk0
r
r
n
n
Models like V(f)fp
p4
p2
For 50 and 60 e-foldings
p fix, Ne varies
HZ
(taken from L. Verde)
p varies, Ne fix
29
The full treatment
30
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/ 0609144

31
Fine Tuning in Rolling Models

• The potential must be very flat
• (Adams, Freese, and Guth 1990)
• But particle physics typically gives this ratio
  1!

32
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

33
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!

34
Inflation needs small ratio of mass scales

• Two attitudes
• 1) We know there is a heirarchy problem, wait
  until its explained
• 2) Two ways to get small masses in particles
  physics
• (i) supersymmetry
• (ii) Goldstone bosons (shift symmetries)

35
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 1990Adams, Bond,
Freese, Frieman and Olinto 1993
36
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.

37
e.g., mimic the physics of the axion (Weinberg
Wilczek)
38
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

39
Two Mass Scales Provide required heirarchy

• For QCD axion,
• For inflation, need
• Enough inflation requires width f mpl,
• Amplitude of density fluctuations requires
• height

40
Sufficient Inflation

• f initially randomly distributed between 0 and
  pfat different places in the universe.
• T lt ? f rolls down the hill. The pieces of the
  universe with f far enough uphill will inflate
  enough.

T gt L
T lt L
41
Sufficient Inflation

• f rolls down the hill.The pieces of the universe
  with f far enough uphill will inflate enough.

T lt L
42
Sufficient Inflation

• A posteriori probabilityThose 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)
43
Density Fluctuations
Largest at 60 efolds before end of inflation

• ? L 1015 GeV 1016 GeV (height of
  potential)
• ? mf L2/f 1011 GeV 1013 GeV
• Density fluctuation spectrum is non-scale
  invariant with extra power on large length scales

WMAP ? f gt 0.7 MPL
44
Implementations of natural inflations 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)

45
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.

46
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,

47
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)

48
Density Fluctuations and Tensor Modes
Density Fluctuations and Tensor Modes can
determine which model is right

• Density Fluctuations
• WMAP
  data
• Slight indication of running of spectral index
• Tensor Modes
• 
  gravitational wave modes, detectable in upcoming
  experiments

49
Density Fluctuations in Natural Inflation

• Power Spectrum
• WMAP data
• implies

(Freese and Kinney 2004)
50
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 /- 0.05 on r
and 0.01 on n. Next generation expts (3 times
more sensitive) must see it.
51
Natural Inflation agrees wellwith WMAP!
52
r-n plane Natural inflation after WMAP 3
53
Tensor modes
54
Spectral Index Running
55
Runningof SpectralIndexin NaturalInflation
56
The full treatment
57
Potential

• 60 e-foldings before the end of inflation
  present day horizon

58
Potential

• At the end of inflation

59
Model Classes

• Kinney collaborators
• Large-field
• Small-field
• Hybrid

60
Model Classes
61
Potential

• f gt few Mpl V(f) quadratic

62
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.

63
Future prospects gravity waves
Tev
13
3.2x10
13
1.7x10
12
9.7x10
12
5.5x10
12
3x10
Verde Peiris Jimenez 05
64
Summary of Natural Inflation confronting data

• 0) No fine-tuning, naturally flat potential
• 1) Matches data in r-n plane for fgt0.7mpl
• 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.

65
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