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Natural Inflation after Natural Inflation after WMAPWMAP
Katherine FreeseMichigan Center for Theoretical PhysicsUniversity of Michigan
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TWO TYPES OF INFLATION TWO TYPES OF INFLATION MODELSMODELS
TUNNELING MODELSTUNNELING MODELS
Old Inflation (Guth 1981Old 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 axionin string landscape, or with QCD axion
ROLLING MODELSROLLING MODELS
Predictions being tested with CMBPredictions being tested with CMB
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).
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.
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
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.
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 Ntot > 60
Freese & Spolyar (2005)Freese, Liu, & Spolyar (2005)
Relevant to:
• stringy landscape
• QCD (or other) axionMultiple tunneling
events
Basic Scenario: Inflation with the QCD axion or in the Stringy
LandscapeChain 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
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
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?
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
Rolling Models of Inflation
Equation of motion:
Flat region: V almost constant vac dominates
energy density
Decay of : Particle production Reheating
0)(3 =′+Γ++ φφφφ VH &&&&
Linde (1982)Albrecht & Steinhardt (1982)
Htiaa e≈→
On: the role of observationsOn: the role of observations
“ “Faith is a fine inventionFaith is a fine invention
When Gentlemen can see ---When Gentlemen can see ---
But Microscopes are prudentBut Microscopes are prudent
In an EmergencyIn an Emergency
Emily Dickinson, 1860Emily Dickinson, 1860
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
Tensor (gravitational wave) Tensor (gravitational wave) modesmodes
In addition to density fluctuations, inflation In addition to density fluctuations, inflation also predicts the generation of tensor also predicts the generation of tensor fluctuations with amplitude fluctuations with amplitude
For comparison with observation, the tensor For comparison with observation, the tensor amplitude is conventionally expressed as:amplitude is conventionally expressed as:
(denominator: scalar modes)(denominator: scalar modes)
Gravity Modes are (at least) two Gravity Modes are (at least) two orders of magnitude smaller than orders of magnitude smaller than density fluctuations: hard to find!density fluctuations: hard to find!
Four parameters from Four parameters from inflationary perturbations:inflationary perturbations:
I. Scalar perturbations: I. Scalar perturbations: amplitude spectral indexamplitude spectral indexII. Tensor (gravitational wave) modes: II. Tensor (gravitational wave) modes: amplitude spectral indexamplitude spectral indexExpressed asExpressed as
Inflationary consistency condition:Inflationary consistency condition:Plot in r-n planePlot in r-n plane
Different Types of Potentials in Different Types of Potentials in the r-n planethe r-n plane
(KINNEY 2002)
Tensor-to-scalar ratio r vs. Tensor-to-scalar ratio r vs. scalar spectral index nscalar spectral index n
Specific models critically tested
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n n
r r
dns/dlnk=0
Models like V( )~ p
dns/dlnk=0
HZ
p=4 p=2 For 50 and 60 e-foldings
p fix, Ne variesp varies, Ne fix (taken from L. Verde)
Natural Inflation after WMAPNatural Inflation after WMAP
Chris Savage, K. Freese, W. Kinney,Chris Savage, K. Freese, W. Kinney,
hep-ph/ 0609144hep-ph/ 0609144
Theoretical motivation: no fine-tuningRecent interest in light of theoretical developmentsUnique predictions:Looks good compared to data
Fine Tuning in Rolling ModelsFine Tuning in Rolling Models
The potential must be very flat:The potential must be very flat:
(Adams, Freese, and Guth 1990)(Adams, Freese, and Guth 1990)But particle physics typically gives this ratio = 1!But particle physics typically gives this ratio = 1!
Inflationary Model Constraints
Success of inflationary models with rolling fields constraints on V()
Enough inflationScale factor a must grow enough
Amplitude of density fluctuations not too large
5
horizonexit
2
horizonenter
10~~ −≤TTH δ
φδ
&
60)(
)(8ln
end
beginbegin
end ≥′
−==⎟⎟⎠
⎞⎜⎜⎝
⎛∫∫ φ
φφπ d
VV
GdtHaa t
t
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
Inflation needs small ratio of Inflation needs small ratio of mass scalesmass scales
Two attitudes:Two attitudes: 1) We know there is a heirarchy problem, wait 1) We know there is a heirarchy problem, wait
until it’s explaineduntil it’s explained 2) Two ways to get small masses in particles 2) Two ways to get small masses in particles
physics:physics: (i) supersymmetry(i) supersymmetry (ii) Goldstone bosons (shift symmetries)(ii) Goldstone bosons (shift symmetries)
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
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)
64
PQ
QCD 10~~
4−
⎟⎟⎠
⎞⎜⎜⎝
⎛Λfa
e.g., mimic the physics of the e.g., mimic the physics of the axion axion (Weinberg; Wilczek)(Weinberg; Wilczek)
Natural InflationNatural Inflation(Freese, Frieman, and Olinto 1990; (Freese, Frieman, and Olinto 1990;
Adams, Bond, Freese, Frieman and Olinto 1993)Adams, Bond, Freese, Frieman and Olinto 1993)
Two different mass scales:Two different mass scales: Width f is the scale of SSB of some global Width f is the scale of SSB of some global
symmetrysymmetry Height is the scale at which some gauge Height is the scale at which some gauge
group becomes stronggroup becomes strong
Two Mass Scales Provide Two Mass Scales Provide required heirarchyrequired heirarchy
For QCD axion,For QCD axion,
For inflation, needFor inflation, need
Enough inflation requires width = f ≈ mpl, Enough inflation requires width = f ≈ mpl, Amplitude of density fluctuations requires Amplitude of density fluctuations requires height = height =
Sufficient Inflation
initially randomly distributed between 0 and fat 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
Sufficient Inflation
rolls down the hill.The pieces of the universe with far enough uphill will inflate enough.
T < x
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
( ) [ ] 60lnln )2/sin()2/sin(
1
22Pl
2
2Pl1
2 168 ≥===≡ ∫∫ ′−
ff
VV
aa
M
f
MddtHN φ
φππ φ
[ ]
[ ]∫∫
−= f
H
f
Nd
NdP π
π
π
φ
φφ
φφ
2/ 11
11
)(3exp
)(3exp1
max1
For f 0.06 MPl ,P = O(1)
Density Fluctuations
~ 1015 GeV – 1016 GeV (height of potential)
m = 2/ ~ 1011 GeV – 1013 GeV
Density fluctuation spectrum is non-scale invariant with extra power on large length scales
[ ] 5max
1
2/3max1
3Pl
22
10~)/sin(
)/cos(13 −+Λ≈≈
f
f
M
fH
φ
φ
φρ
δρ&
snkk kP ~
2δ= )(for 1with Pl2
2Pl
8Mfn
fM
s <−≈ π
WMAP f > 0.7 MPL
Largest at 60 efolds before end of inflation
Implementations of natural Implementations of natural inflation’s shift symmetryinflation’s shift symmetry
Natural chaotic inflation in SUGRA using shift symmetry in Natural chaotic inflation in SUGRA using shift symmetry in Kahler potentialKahler potential (Gaillard, Murayama, Olive 1995; Kawasaki, (Gaillard, Murayama, Olive 1995; Kawasaki, Yamaguchi, Yanagida 2000)Yamaguchi, Yanagida 2000)
In context of extra dimensions: Wilson line withIn context of extra dimensions: Wilson line with (Arkani-Hamed et al 2003)(Arkani-Hamed et al 2003) but Banks et al (2003) showed it but Banks et al (2003) showed it fails in string theory.fails in string theory.
““Little” field modelsLittle” field models (Kaplan and Weiner 2004)(Kaplan and Weiner 2004) In brane Inflation ideasIn brane Inflation ideas (Firouzjahi and Tye 2004)(Firouzjahi and Tye 2004) Gaugino condensation in SU(N) SU(M):Gaugino condensation in SU(N) SU(M):Adams, Bond, Freese, Frieman, Olinto 1993;Adams, Bond, Freese, Frieman, Olinto 1993;Blanco-Pillado, Linde et al 2004Blanco-Pillado, Linde et al 2004 ((Racetrack inflationRacetrack inflation))
Legitimacy of large axion scale?
Natural Inflation needsIs 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.
A large effective axion scaleA large effective axion scale(Kim, Nilles, Peloso(Kim, Nilles, Peloso 2004) 2004)
Two or more axions with low PQ scale can Two or more axions with low PQ scale can provide large provide large
Two axions Two axions
Mass eigenstates are linear combinations ofMass eigenstates are linear combinations of Effective axion scale can be large, Effective axion scale can be large,
A large number of fieldsA large number of fields
Assisted Inflation (Liddle and Mazumdar Assisted Inflation (Liddle and Mazumdar 1998)1998)
N-flation (Dimopoulos, Kachru, McGreevy, N-flation (Dimopoulos, Kachru, McGreevy, Wacker 2005): Shamit’s talk this morningWacker 2005): Shamit’s talk this morning
Creation of cosmological magnetic fields Creation of cosmological magnetic fields (Anber and Sorbo 2006)(Anber and Sorbo 2006)
Density Fluctuations and Tensor Density Fluctuations and Tensor ModesModes
Density Fluctuations and Density Fluctuations and Tensor Modes can determine Tensor Modes can determine
which model is rightwhich model is right Density FluctuationsDensity Fluctuations::
WMAP data:WMAP data:
Slight indication of running of spectral indexSlight indication of running of spectral index
Tensor ModesTensor Modes gravitational wave modes, gravitational wave modes,
detectable in upcoming experimentsdetectable in upcoming experiments
Density Fluctuations in Natural Density Fluctuations in Natural InflationInflation
Power Spectrum:Power Spectrum:
WMAP data:WMAP data:
impliesimplies
(Freese and Kinney2004)
Tensor Modes in Natural InflationTensor Modes in Natural Inflation(original model)(original model) (Freese and Kinney 2004)(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 n.b. not much running of nrunning 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.
r-nr-n plane: Natural inflation after WMAP 3 plane: Natural inflation after WMAP 3f > 0.7 MPl allowed
Model Classes
Kinney & collaborators
Large-field
Small-field
Hybrid
0)( >′′ φV
0)( <′′ φVεη −<
εηε ≤<−
ηε <<0
Natural Inflation Summary
No fine tuning,naturally flat potential
WMAP 3-year data:
f < 0.7 MPl excluded
f > 0.7 MPl consistent Tensor/scalar ratio r Spectral index ns
Spectral index running dns/d lnk
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.
E and B modes polarization
E polarization from scalar, vector and tensor modes
B polarization only from (vector) tensor modes
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Kamionkowski, Kosowsky, Stebbings 1997, Zaldarriga & Seljak 1997
Future prospects: gravity waves
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Tev3.2x10
1.7x10
9.7x10
5.5x10
3x10
13
13
12
12
12
Verde Peiris Jimenez 05
Summary of Natural Inflation Summary of Natural Inflation confronting dataconfronting data
1) Matches data in r-n plane for f>0.7mpl1) Matches data in r-n plane for f>0.7mpl 2) Tensor modes may be as small as 0.0012) Tensor modes may be as small as 0.001 3) Small running, an order of magnitude 3) Small running, an order of magnitude
below sensitivity of WMAP3, not detectable below sensitivity of WMAP3, not detectable any time soon. Big running in the data would any time soon. Big running in the data would kill the model.kill the model.
ConclusionConclusion
Tunneling Models: Chain Inflation in Tunneling Models: Chain Inflation in Landscape and with QCD Axion. TO DO: Landscape and with QCD Axion. TO DO: perturbations (with S. Watson)perturbations (with S. Watson)
Rolling Models: Rolling Models:
Generic predictions of inflation match the Generic predictions of inflation match the datadata
Natural inflation looks goodNatural inflation looks good
ConclusionConclusion
An early period of inflation resolves An early period of inflation resolves cosmological puzzles: homogeneity, isotropy, cosmological puzzles: homogeneity, isotropy, oldness, and monopoles. It also generates oldness, and monopoles. It also generates density perturbations for galaxy formation.density perturbations for galaxy formation.
Details of density and gravitational wave modes Details of density and gravitational wave modes can be used to test inflation as well as individual can be used to test inflation as well as individual models.models.
Predictions of inflation are confirmed!Predictions of inflation are confirmed! Natural inflation, which was theoretically well-Natural inflation, which was theoretically well-
motivated, fits the data very well.motivated, fits the data very well.
DARK ENERGY (w=p/rho)DARK ENERGY (w=p/rho)
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SUMMARY:SUMMARY:
I. The predictions of inflation are right: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 iv) detection of polarization (from gravitational wave modes) in
upcoming data may provide smoking gun for inflationupcoming data may provide smoking gun for inflation
II. Polarization measurements will tell us which II. Polarization measurements will tell us which model is right.model is right.
WMAP already selects between models.WMAP already selects between models. Natural inflation (Freese, Frieman, Olinto) looks greatNatural inflation (Freese, Frieman, Olinto) looks great
Generation of CMB polarization
• Temperature quadrupole at the surface of last scatter generates polarization.
Potential wellPotential hill
From Wayne Hu
At the last scattering surface
At the end of the dark ages (reionization)
E and B modes polarization
E polarization from scalar, vector and tensor modes
B polarization only from (vector) tensor modes
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Kamionkowski, Kosowsky, Stebbings 1997, Zaldarriga & Seljak 1997
Comparison with WMAP I
Best fit LCDM WMAP IBest fit LCDM WMAP I ExtBest fit LCDM WMAP II
WMAP II
WMAP I
Specific models critically tested
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n n
r r
dns/dlnk=0
Models like V( )~ p
dns/dlnk=0
HZ
p=4 p=2 For 50 and 60 e-foldings
p fix, Ne variesp varies, Ne fix
Future prospects: gravity waves
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Tev3.2x10
1.7x10
9.7x10
5.5x10
3x10
13
13
12
12
12
Verde Peiris Jimenez 05
Density Fluctuations and Tensor Density Fluctuations and Tensor ModesModes
Density Fluctuations and Density Fluctuations and Tensor Modes can determine Tensor Modes can determine
which model is rightwhich model is right Density FluctuationsDensity Fluctuations::
WMAP data:WMAP data:
Slight indication of running of spectral indexSlight indication of running of spectral index
Tensor ModesTensor Modes gravitational wave modes, gravitational wave modes,
detectable in upcoming experimentsdetectable in upcoming experiments
1 sigma reconstruction of 1 sigma reconstruction of potential from 1-year WMAP datapotential from 1-year WMAP data
(KINNEY,KOLB,MELCHIORRI,AND RIOTTO2003)
IV. From Theory to Observation: IV. From Theory to Observation: Predictions of InflationPredictions of Inflation
1) flat universe:1) flat universe:
2) Specrum of density perturbations:2) Specrum of density perturbations:
3) gravitational wave modes3) gravitational wave modes Individual models make specific predictions.Individual models make specific predictions. Can test inflation as a concept and can Can test inflation as a concept and can
differentiate between models. differentiate between models.
Prediction 1 of Inflation:Prediction 1 of Inflation:
The geometry of the universe is flat;The geometry of the universe is flat;
i.e. the density is critical and i.e. the density is critical and
WMAP SatelliteWMAP Satellite
Launched June 2002Launched June 2002 Data released Feb. 2003Data released Feb. 2003
Prediction 1 is confirmedPrediction 1 is confirmed
WMAP confirms the inflationary WMAP confirms the inflationary
prediction thatprediction that
Prediction 2 of InflationPrediction 2 of Inflation
Spectral index of density perturbations Spectral index of density perturbations (scalar modes) is near n=1(scalar modes) is near n=1
n.b. individual models make specific n.b. individual models make specific predictions for n which can be used to predictions for n which can be used to differentiate between modelsdifferentiate between models
Prediction 2 of inflation is Prediction 2 of inflation is confirmedconfirmed
Multiple data sets (WMAP, large scale Multiple data sets (WMAP, large scale structure, etc) confirm n near 1.structure, etc) confirm n near 1.
More detail shown in a minute to More detail shown in a minute to differentiate between modelsdifferentiate between models
Prediction 3 of Inflation:Prediction 3 of Inflation:
Existence of gravitational wave Existence of gravitational wave perturbations (tensor modes)perturbations (tensor modes)
Status of Rolling Status of Rolling Models:Models:
I. The predictions of inflation are right: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 iv) detection of polarization (from gravitational wave modes) in
upcoming data may provide smoking gun for inflationupcoming data may provide smoking gun for inflation
II. Polarization measurements will tell us which II. Polarization measurements will tell us which model is right.model is right.
WMAP already selects between models.WMAP already selects between models. Natural inflation (Freese, Frieman, Olinto) looks greatNatural inflation (Freese, Frieman, Olinto) looks great
Predictions: Density and Gravity Predictions: Density and Gravity Fluctuations in Natural InflationFluctuations in Natural Inflation Power Spectrum:Power Spectrum:
(not quite scale invariant, n<1)(not quite scale invariant, n<1) Gravitational wave modes extremely Gravitational wave modes extremely
suppressed (smaller than in most models)suppressed (smaller than in most models)