Axion-driven inflation and quantum gravity Albion Lawrence, - - PowerPoint PPT Presentation

Axion-driven inflation and quantum gravity

Albion Lawrence, Brandeis University

Kaloper, Lawrence, and Sorbo 1101.0026 Kaloper and Lawrence 1404.2912 Kaloper, Kleban, Lawrence, and Sloth 1511.05119 Kaloper and Lawrence, in progress

1

• I. Inflation and sensitivity to QG

V (ϕ)

ϕ

ϕ(t)

δϕ

• = m2

p

• V

V

⇥2 ⇥ 1

• ⇥ = m2

p V V

⇥ 1

L = 1 2(∂ϕ)2 − V (ϕ) + . . .

Standard single field inflation Small density perturbations and vacuum dominance: Basic observational input:

• δρ

ρ ∼ H2 ˙ ϕ ∼ V 3/2 m3

pV 0 ∼ 10−5

• Ne =

R dtH = R dϕ

˙ ϕ H =

R dϕ

H H2 ˙ ϕ & 60

2

H2 = V 3m2

pl

ds2 ∼ −dt2 + e2

R t dt0H(t0)d~

x2

3

CMB “B-mode” polarization measures

r = 16✏ ∝ V/m4

pl

(⇢/⇢)2

Current upper bound from direct searches (PLANCK+BICEP2+Keck)

r . 0.09 ⇒ V . (1.7 × 1016 GeV )4

Close to GUT scale, string scale, 10d Planck scale in many string models Future experiments could sweep out range

10−3 < r < .1

Why we should care:

• δρ

ρ ∼ H2 ˙ ϕ ∼ V 3/2 m3

pV 0 ∼ 10−5

• Ne =

R dtH = R dϕ

˙ ϕ H =

R dϕ

H H2 ˙ ϕ & 60

Determined by V

Ties range in field space to V

Lyth; Efstathiou and Mack

r > 10−3 ⇒ ∆ϕ & mpl

Constraints on inflaton potential

V = X

n

cn ϕn mn−4

pl

EFT parametrization Exquisitely small for all n

4

Need control of quantum gravity corrections “High scale inflation”

Perturbative quantum corrections Slow roll inflation models such as “technically natural” V = 1 2m2ϕ2

m ∼ 10−5mpl ⇒

softly broken shift symmetry ϕ → ϕ + a Loops of inflaton, graviton,… (if other couplings shift symmetric)

V = Vtree 1 + aVtree m4

pl

+ bV 00

tree

m2

pl

+ ... !

5

Coleman and Weinberg; Smolin; Linde

power series in

m2 m2

pl

∼ 10−10

Natural inflation: inflaton = periodic pseudoscalar (aka “axion”) Nonrenormalization theorem: prevents

δV = cn ϕn mn−4

pl

Can be kept small if S << 1

ϕ ≡ ϕ + 2πf

Freese, Frieman, and Olinto

f: strong coupling scale for axion Potential can arise from gauge instantons

δL = ϕ f trG ∧ G

Λ ∼ MGUT , f & mpl

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V (ϕ) = Λ4 cos ✓ϕ f ◆ +

∞

X

k=2

ckΛ4e−(k−1)S cos ✓kϕ f ◆

Current lore: nonperturbative quantum gravity breaks global symmetries

• Black hole formation and evaporation
• Gravitational instantons: break axion shift symmetries esp when

Giddings and Strominger; Abbott and Wise; Coleman and Lee Banks; Dixon

f > mpl

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Some mechanism is required to suppress QG effects

• String theory: many global symmetries = gauge symmetries

Sinst ∼ m2

pl

f 2

Banks, Dine, Fox, Gorbatov; Arkani-Hamed, Motl, Nicolis, Vafa

Candidate models

• Extranatural inflation: M5 = R4 × S1

R

ϕ = I dx5A5

5d gauge symmetry protects shift symmetry

Arkani-Hamed, Cheng, Creminelli, Randall

• N-flation

Dimopoulos, Kachru, McGreevy, Wacker

N axions with : f < mpl

ϕi

Total possible distance in field space:

∆Φ ∼ sX

i

δϕ2

i ∼

√ Nf ≡ feff

even if

feff > mpl f < mpl

V = X

n

Λ4

n cos

✓ϕn fn ◆

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• Axion monodromy

Silverstein and Westphal; McAllister, Silverstein, and Westphal; Kaloper, Lawrence, and Sorbo

n=0 n=1 n=2 n=3

• V(

f 2f 3f

Symmetry: ϕ → ϕ + 2πkf ; n → n − k ; k ∈ Z

KLS

Upshot: this protects theory from large corrections

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Possible constraints

• 1. Entropic bounds

Conlon

Claim: in de Sitter patch:

• 2. Weak Gravity Conjecture

Arkani-Hamed, Motl, Nicolis, Vafa

Upper bound on mass/tension/action of charged objects

Eg for (unbroken) U(1) gauge theory, must exist charged particle Extranatural inflation: 5d charged particle on

m . gm(D−2)/2

pl,D

R4 × S1

⇒

4d instanton with Sinst ∼

m2

pl

f 2 10

Nf 2 > m2

pl

Sinflaton,dS ∼ N f 2 H2 SdS ∼ m2

pl

H2 Covariant entropy bound: violates this (avoid stable remnants, etc)

• II. Entropy bounds

Basic point:

S = A 4GN

is computed in low energy theory, via classical gravity Should use physical, renormalized value as measured by low-energy observer

mpl ∼ 2.4 × 1018 GeV

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This may be different from bare value and much higher than cutoff marking validity of 4d semiclassical gravity

Kaloper, Kleban, Lawrence, and Sloth

The cutoff should be used in computing QFT contributions to entropy

SNfields ∼ N M 2

UV

H2

We claim, roughly:

NM 2

UV < m2 pl,phys

• A. Perturbative graviton loops

Integrate out matter fields to get effective action for gravitational sector:

Seff = Sbare(gµν) + Γ1−loop(g) + . . .

= Z d4x√g ✓ − Λbare GN,bare + aM 2

UV

◆ + ✓ 1 16πGN,bare + cM 2

UV

◆ R + (αbare + β ln MUV )R2 + . . .

• Physical couplings

minimally coupled scalars + Weyl fermions:

c ∼ N0 + N1/2 = N = m2

pl,phys

m2

UV <

m2

pl,phys

N

12

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• B. Example: compactification

D+4-dim theory: Sgrav ∼ mD+2

pl,D+4

Z dD+4√gR

S4d = mpl,D+4VD Z d4x√gR

m2

pl,4 = mD+2 pl,D+4VD

Number of KK modes with

1 LKK < m < mpl,D+4

fundamental scale/ UV cutoff N ∼ (LKKmpl,D+4)D = VDmD

pl,D+4

M 2

UV ≡

m2

pl,4

N = m2

pl,D+4

Lesson: cutoff of EFT is something physical and obvious in UV completion

• C. de Sitter entropy: N scalar fields

ds2 = − ✓ 1 − r2 r2

h

◆ dt2 + ✓ 1 − r2 r2

h

◆−1 dr2 + r2dΩ2

2

• Impose Dirichlet (“brick wall”) b.c. at r = rh − ✏
• Count number of field modes at fixed E

Blueshifting near horizon: can use WKB

’t Hooft (for BHs)

• Repeat for Pauli-Villars regulator fields; then take ✏ → 0
• Compute corrections to using same (PV) scheme

GN

Kabat; Demers, LaFrance, and Myers (for BH)

A 4GN,bare + δS1−loop = A 4GN,ren

Quadratic area law divergence

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• III. Axion monodromy

Original idea arose from string models

Silverstein and Westphal; McAllister, Silverstein and Westphal

4d effective field theory version

Kaloper and Sorbo; Kaloper, Lawrence and Sorbo

Sclass = ⇤ d4x√g

• m2

plR − 1 48F 2 − 1 2(∂ϕ)2 + µ 24ϕ∗F

⇥

Fµνλρ = ∂[µ A νλρ] δAµνλ = ∂[µ Λ νλ]

U(1) gauge symmetry ϕ ≡ ϕ + 2πfϕ

Htree = 1

2p2 φ + 1 2 (pA + µφ)2 + grav.

pA = ne2 Compactness of U(1)

discrete gauge symmetry in phase space

n=0 n=1 n=2 n=3

• V(

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µfφ = ke2

ϕ → ϕ + 2πfϕ ; n → n − k

• Dim. red. to 0+1:

charged particle in B-field on torus. k = magnetic flux quantum = LLL degeneracy

Discrete gauge symmetry: no dangerously large Planck-suppressed operators

fϕ ∼ .1mpl ; µ = 10−6mpl ; e ∼ (few) × 10−4mpl

⇒ At fixed n, good model of chaotic inflation

δV ∼ O(1) ϕn mn−4

pl

Allowed corrections to Lagrangian:

• δL = c F 2n

M 4n−4

UV

⇒ δH = (µϕ + ne2)2n M 4n−4

UV

∼ Vtree ✓ Vtree M 4

UV

◆n−1

Slow roll safe if

M 4

UV > V ∼ M 4 GUT

• δL = cΛ4 cos

✓ϕ f ◆

V (ϕ)

ϕ

Small sinusoidal modulation

16

• . . .

Weak gravity conjecture?

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n=0 n=1 n=2 n=3

• V(

Membranes charged under F: n → n ± 1

Coleman; Brown and Teitelboim; Coleman and de Luccia

via nucleation of bubbles of lower branch Naive expectation: WGC upper bound on membrane tension

⇒

Brown, Cottrell, Shiu, and Soler; Ibanez, Montero, Uranga, Valenzuela

dϕ = H(3) = dB(2)

L = − 1 48(F (4))2 − 1 2(∂ϕ)2 − µϕ∗F

L = − 1 48(F (4))2 − 1 2µ2A2

B is longitudinal mode of A “London equation” for axion monodromy

Marchesano, Shiu, and Uranga; Kaloper and Lawrence, in progress

Standard WGC argument does not apply to massive gauge fields Stable charged black objects do not exist

Cheung and Remmen Bekenstein

Are there other constraints?

See also Hebecker, Moritz, Westphal, and Witkowski

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Julia-Toulouse mechanism

Julia and Toulouse; Quevedo and Trugenberger

Membranes electrically charged under A

L = − 1 48(F (4))2 − 1 2µ2A2

4-form coupled to membrane condensate?

• UV complete model (eg via string theory)?

D-brane condensates often dual to fundamental fields

Strominger; Witten; ...

• Mechanism for small ?

µ

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2d analog: Schwinger model

Charged fermions = 2d domain walls

L = −1 4FµνF µν + ¯ ψγµ (∂µ − iAµ) ψ

Bosonization:

L = −1 4FµνF µν − 1 2(@')2 − '✏µνFµν

Lawrence; Seiberg

L = −1 4FµνF µν − 1 2A2