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

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

Graceful Exit Achieved

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)

Examples of ModelsExamples of Models

Effect of more data

WMAP IWMAP I ExtWMAP II

Reducing the noise by 3 degeneracies broken

LCDM model

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

Specific models critically tested

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.



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)

The full treatment:



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


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

T < x


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:


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.

Natural Natural Inflation Inflation

agrees wellagrees wellwith WMAP!with WMAP!

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

Spectral Index Runningsmall f :

(exponentially suppressed)

large f :

22

ln Nkdnd s −→

kd

nd s

ln

The full treatment:



Potential

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

Potential

At the end of inflation

Model Classes

Kinney & collaborators

Large-field

Small-field

Hybrid

0)( >′′ φV

0)( <′′ φVεη −<

εηε ≤<−

ηε <<0

Model Classes

0)( >′′ φV0)( <′′ φV

Potential

f > few Mpl: V() ~ quadratic

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

QuickTime™ and aTIFF (Uncompressed) decompressor




Kamionkowski, Kosowsky, Stebbings 1997, Zaldarriga & Seljak 1997

TT

TE

EE

BB

WMAP3 data

Future prospects: gravity waves



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)



11



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

STOP HERESTOP HERE

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)

Polarization for density perturbation

• Radial (tangential) pattern around hot (cold) spots.

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

Clean BB after FG removal.

3-sigma detection of EE.

The “Gold” multipoles: l=3,4,5,6.

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





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



Tev3.2x10

1.7x10

9.7x10

5.5x10

3x10

13

13

12

12

12

Verde Peiris Jimenez 05



end

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::


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

The The Microwave Microwave

SkySky

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)

PolarizationPolarization

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

natural inflationnatural inflation(Freese andKinney 2004)

n.b. This is asmall-fieldmodel

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

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