Naturalness in Inflation

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Naturalness in Inflation

• Katherine Freese
• Michigan Center for Theoretical
  Physics
• University of Michigan
• Ann Arbor, MI

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Outline

• Brief review of inflation
• Naturalness in rolling models
• flat potential required, i.e., two
  disparate mass scales, natural inflation uses
  shift symmetries, new twists in new contexts
• New paradigm for tunneling models
• Chain Inflation
• Nice features no fine-tuning, single mass
  scale for potential can be 10 MeV-GUT scale,
  graceful exit is successful

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Old Inflation (Guth 1981)

Enough inflation requires the scale factor to
grow at least 60 e-foldings.
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Inflation Resolves Cosmological Problems

• Horizon Problem (homogeneity and isotropy) small
  causally connected region inflates to large
  region containing our universe
• Flatness Problem
• Monopole Problem tightest bounds on GUT
  monopoles from neutron stars (Freese, Schramm,
  and Turner 1983) monopoles inflated away
  (outside our horizon)
• BONUS Density Perturbations that give rise to
  large scale structure are generated by inflation

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Shortcomings of Inflationary Models

• Tunneling Fields Inflation Fails no graceful
  exit
• except through a time-dependent nucleation
  rate (double-field)
  .
• F. Adams and K.
  Freese 1991
• A. Linde 1991
• Rolling Field Inflation
• Linde 1981
  Albrecht and Steinhardt 1981
• Fine-Tuned
• Except natural inflation (shift
  symmetry) .
  Freese, Frieman, and Olinto 1991

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Whats new in inflation?

• Observational Tests
• spectral index, tensor
  modes
• New physical setup
• extra dimensions, braneworlds
• New solutions to old problems, new ideas
• naturalness in rolling models
• graceful exit in tunneling models
• new paradigm chain inflation

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I. Fine Tuning in Rolling Models

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

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Need 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)

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

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Two Mass Scales Provide required heirarchy

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

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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

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Density Fluctuations in Natural Inflation

• Power Spectrum
• WMAP data
• implies

(Freese and Kinney 2004)
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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.
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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 et al 2004 (Racetrack inflation)

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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.

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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,

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Natural Inflation (again) Shift Symmetries

• Inflationary Potentials in rolling models must be
  flat I.e. have two disparate mass scales
• Shift (axionic) symmetries protect flatness of
  inflaton potential (inflaton is Goldstone boson)
• Original model of natural inflation is testable
  in CMB in next decade
• New implementation in extra dimensions and with
  multiple fields allows fmpl

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II) New Framework for Inflation Chain Inflation

• No fine-tuning even with only one mass scale in
  the potential
• Large Range of Energy Scales for Potential
• Saves Old Inflation
  Graceful
  Exit each stage of phase transition occurs very
  quickly
• E.g. can inflate with QCD axion or in stringy
  landscape

Freese and Spolyar hep-ph/0412145 Freese, Liu,
and Spolyar hep-ph/0502177
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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.

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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 ?)

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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

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Topics

• Why Old Inflation Fails
• Whats Needed Time Dependent
• This model
• Multiple tunneling events each with less
  than one e-fold provide graceful exit

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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).
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Swiss Cheese Problem of Old Inflation no
graceful exit
Bubbles of true vacuum nucleate in a sea of
false vacuum.

• PROBLEM Bubbles never percolate and thermalize
• REHEATING FAILS we dont
  live in a vacuum

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What is needed for tunneling inflation to work?

• Probability of a point remaining in false vacuum
  phase
• is the nucleation rate of T bubbles and
  H is the expansion rate of the universe
• Theories with constant fail (e.g. old
  inflation)
• Small slow phase transition, inflation
  but no reheating
• Large fast phase transition, not enough
  inflation, yes there is reheating
• Need time-dependent ,first small then large
  

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Graceful Exit Achieved
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For large
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Two Requirements for Inflation

• Lifetime of field
• in metastable state
• Number of e-folds from
• single tunneling event
• Sufficient Inflation
• Reheating

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How to achieve both criteria

• Sufficient inflation
• Reheating
• With single tunneling event
• Double Field Inflation (Adams Freese 91
  Linde 91) time-dependent nucleation rate,
  couple two scalar fields
• With multiple tunneling events
• CHAIN INFLATION
• get a fraction of an e-fold at each stage, adds
  to more than 60 in the end

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Double Field Inflation (Adams and Freese 1991)

• Time dependent nucleation rate
• Couple 2 scalar fields
• Once the roller reaches its min,
• grows, tunneling rate
  increases. The tunneling rate is zero for at
  top of potential, large as approaches min
  (then, nucleation)

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Required time dependence

• Need small initially to inflate.
• Then, suddenly, gets larger so that all of
  universe goes from false to true vacuum at once.
  All bubbles of same size, get percolation and
  thermalization.

No Swiss Cheese!
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Asymmetric Well

• is
  energy
  d
  difference between vacua
  

Nucleation rate of true vacuum
(thin wall)
(Callan and Coleman Voloshin, Okun, and Obzarev))
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Sensitivity of nucleation rate to parameters in
the potential

• Sufficient inflation
• number of e-folds
• Followed by rapid nucleation
• Both achieved by small change in
• e.g. consider TeV, 100 fields
• N1000 for
• N0.01 for
• To go from enough inflation to percolation, need
  this ratio to change by less that 2

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How to achieve both criteria

• Sufficient inflation
• Reheating
• With single tunneling event
• Double Field Inflation (Adams Freese 91
  Linde 91) time-dependent nucleation rate,
  couple two scalar fields
• With multiple tunneling events
• CHAIN INFLATION
• get a fraction of an e-fold at each stage, adds
  to more than 60 in the end

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Inflating with the QCD axion
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Invisible Axion (DFSZ)

• Axion is identified as phase of a complex SU(2)
  U(1) singlet scalar s below PQ symmetry breaking
  scale sv/v2
• Soft breaking
• Phase shift

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REHEATING radiation is produced in last few
stages PROBLEM Get stuck in last minimum before
the bottom (tunneling becomes too slow), How stop
inflating?
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How to get out of last minimum before the bottom?
Possibilities

• 1. Set so that minima of two cosines
  line up artificial.
• 2. Energy of last minimum is very small, e.g.
  (dark energy)
• 3. Couple several axions
• 4. Different soft PQ breaking term
• 5. Go in new direction in potential near the
  bottom (axion couples)
• Etc.

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CHAIN INFLATING WITH THE QCD AXION
Conclusion

• Can inflate with the QCD axion, a particle
  proposed for independent reasons (strong CP
  problem) (Wilczek,Weinberg)
• Scale of inflation is low testable
• Need tilted cosine (soft breaking of PQ symmetry)
  with many minima, tunnel from one minimum to the
  next
• Graceful exit is resolved, reheating is successful

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Chain Inflation in the Stringy Landscape

• Our universe (a causal patch) starts in a
  high-energy (local) mininum, tunnels from bowl to
  bowl to ever lower energies
• Takes single path through the various vacuum
  states in the landscape
• Can model as large number of coupled fields whose
  interactions provide graceful exit (rapid enough
  tunneling) model as coupled scalar fields in
  asymmetric double wells
• (Freese and Spolyar
  2004)

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Toy Model in Landscape
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Toy model
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Enhanced tunneling
In the language of the landscape, the field
chooses the path of least resistance, I.e. the
fastest tunneling rate, i.e. a direction in which
interaction with other fields enhances the
tunneling
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NO fine-tuning

• The height and the width of the potential can be
  the same. In fact, it is when these two
  quantities are roughly comparable that the field
  is on the border of tunneling rapidly or never
  tunneling at all, so that any interaction is
  likely to cause the phase transition to proceed
  rapidly.

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Key ingredients provided by the landscape

• Many minima required for chain inflation to work
• Many interacting fields tend to drive the
  tunneling rate to speed up
• Will the field get stuck in a minimum and
  overinflate there? Unlikely because it will
  choose the path of least resistance, I.e. go to a
  minimum out of which it can tunnel quickly.
• Will the field skip ahead and leap over many
  minima? For equal parameters for all mimina, NO
  the fastest path is to move sequentially. For
  unequal parameters, to move through largest
  potentials first and then smaller ones.

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Naturalness in Inflation (Conclusion)

• Natural Inflation Rolling Models with Shift
  Symmetries
• testable in CMB, many variants in
  braneworld contexts, multi-field models
• Chain Inflation
• any scale above MeV, no fine-tuning even
  with only one scale in potential, reheating
  successful, can work with QCD axion or in stringy
  landscape

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Bubble Bubble Toil and Trouble

• Bubble bubble toil and trouble
• Fire burn and cauldron bubble
• Fillet of a fenny snake
• In the cauldron boil and bake
• Eye of newt and toe of frog
• Wool of bat and tongue of dog
• Adders fork and blind-worms sting
• Lizardss leg and howlets wing
• For a charm of powerful trouble
• Like a hell-broth boil and bubble
• 
  Shakespeare (Macbeth)