Naturalness in Inflation Katherine Freese Michigan Center for Theoretical Physics University of Michigan Ann Arbor, MI

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

Old Inflation (Guth 1981) Enough inflation requires the scale factor to grow at least 60 e-foldings.

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

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

What’s 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

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!

Need 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)

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

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

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

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

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

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

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

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 et al 2004 (Racetrack inflation)

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.

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,

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 f≈mpl

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/ ; Freese, Liu, and Spolyar hep-ph/

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.

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)

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

Topics Why Old Inflation Fails What’s Needed: Time Dependent This model: Multiple tunneling events each with less than one e-fold provide graceful exit

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).

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: 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

Graceful Exit Achieved

For large

Two Requirements for Inflation Lifetime of field in metastable state: Number of e-folds from single tunneling event: Sufficient Inflation: Reheating:

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

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)

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!

Asymmetric Well is energy d difference between vacua (thin wall) (Callan and Coleman; Voloshin, Okun, and Obzarev)) Nucleation rate of true vacuum:

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: N=1000 for N=0.01 for To go from enough inflation to percolation, need this ratio to change by less that 2%

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

Inflating with the QCD axion

Invisible Axion (DFSZ) Axion is identified as phase of a complex SU(2) U(1) singlet scalar s below PQ symmetry breaking scale s=v/√2 Soft breaking: Phase shift

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

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.

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 CHAIN INFLATING WITH THE QCD AXION: Conclusion

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)

Toy Model in Landscape

Toy model

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

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.

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.

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

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 Adder’s fork and blind-worm’s sting Lizards’s leg and howlet’s wing For a charm of powerful trouble Like a hell-broth boil and bubble Shakespeare (Macbeth)