Large-field Inflation with Multiple Axions and the Weak Gravity Conjecture

Daniel Junghans

Ludwig-Maximilians-Universität München

Based on: arXiv:1504.03566 & work in progress

Outline

Introduction

Field excursion in multi-axion models

Quantum gravity constraints

Conclusions

Introduction

Large-field inflation (Δ𝜙 > 𝑀𝑝) is sensitive to Planck-suppressed operators

→ use an axion as the inflaton (natural inflation)

Shift symmetry protects inflaton potential from perturbative corrections

Axions are ubiquitous in string theory

However: Δ𝜙 > 𝑀𝑝 difficult to realize

Two main proposals to overcome this problem:

1. Axion monodromy (not in this talk)

2. Use multiple axions → field space diagonals with enhanced field range

Inflation with Axions

Banks, Dine, Fox, Gorbatov 03

𝑉 𝜙 ∼ cos𝜙

𝑓Non-perturbative scalar potential 𝑓: axion decay constant

Is Δ𝜙 > 𝑀𝑝 consistent in quantum gravity?

Silverstein, Westphal 08McAllister, Silverstein, Westphal 08...see also talks by Wrase and Herschmann

Kim, Nilles, Peloso 04Dimopoulos, Kachru, McGreevy, Wacker 05...

Freese, Frieman, Olinto 90

In any U(1) gauge theory coupled to gravity, ∃ at least one charged particle satisfying

𝑚 ≲ 𝑞𝑀𝑝

Main motivation: forbid infinite number of remnants

Can be generalized to multiple U(1)‘s

Axion version of the conjecture:

∃ at least one instanton satisfying

Intuitively:

instanton tension 𝑆 ↔ mass 𝑚, instanton-axion coupling 1/𝑓 ↔ electric charge 𝑞

Precise map to U(1)‘s via T-dualities

Implications for maximal axion field excursion?

The Weak Gravity Conjecture

Arkani-Hamed, Motl, Nicolis, Vafa 06

Brown, Cottrell, Shiu, Soler 15

Cheung, Remmen 14

𝑆 ≲𝑀𝑝𝑓

(Multi-)instanton corrections to scalar potential:

Instantons are unsuppressed for 𝑛𝑆 ≲ 1 ↔ 𝑛 ≲𝑓

𝑀𝑝

→ Δ𝜙 constrained to be sub-Planckian

Possible loophole: instanton satisfying the WGC could be subdominant

→ bound on Δ𝜙 would be relaxed

Is such a loophole realized in string theory?

Status quo: ongoing debate, outcome unclear

this talk: different perspective on the problem

Bound on field excursion?

(Heidenreich, Reece,) Rudelius 14, 15de la Fuente, Saraswat, Sundrum 14Montero, Uranga, Valenzuela 15Brown, Cottrell, Shiu, Soler 15Bachlechner, Long, McAllister 15Hebecker, Mangat, Rompineve, Witkowski 15Palti 15

Brown, Cottrell, Shiu, Soler 15

WGC

𝑉 𝜙 ∼ e−𝑛𝑆 1 − cos𝑛𝜙

𝑓

𝑉(𝜙)

𝜙

Field excursion in multi-axion models

General scalar potential: 𝑉 𝜙𝑖 = 𝑗=1𝑃 Λ𝑗

4 1 − cos 𝑖=1𝑁 𝑐𝑖𝑗𝜙𝑖

Fundamental domain: 𝑁-polytope, 𝑃 instantons ↔ 2𝑃 facets

Max. field excursion ∼ 𝑓eff is towards a vertex

→ Compute 𝑓eff in terms of 𝑁 distances 𝑓(𝑖)

and 𝑁(𝑁−1)2

angles 𝛼(𝑖𝑗) encoded in 𝑉(𝜙𝑖)

In general: any enhancement possible depending on

• Amount of alignment

• Number of instantons 𝑃 (facets)

Special cases:

𝑁-flation

power-law enhancement

exponential enhancement

„Bottom-up“ perspective

Dimopoulos, Kachru, McGreevy, Wacker 05

Bachlechner, Long, McAllister 14

Choi, Kim, Yun 14

𝑓eff ∼ 𝑁

𝑓eff ∼ 𝑁,𝑁3/2

𝑓eff ∼ 𝑛𝑁

Bachlechner, Long, McAllister 14

Kim, Nilles, Peloso 04

𝛼(12)

𝑓eff

𝑓(2)

𝑓(1)

𝑓(3)

𝛼(13) 𝛼(23)

Quantum gravity constraints

Assuming a strict bound on Δ𝜙, how would string theory forbid a parametric

enhancement 𝑓eff ∼ 𝑁𝑥 at large 𝑁?

Several possibilities:

• No string compactifications exist beyond a certain 𝑁

No evidence, Calabi-Yau‘s can easily yield ∼1000 axions. Moduli stabilization?

• 4D Planck mass renormalization: 𝑀p ∼ 𝑁𝑥

Argued to scale at most like 𝑀p ∼ 𝑁, milder in actual string models (at NLO)

• Individual 𝑓(𝑖)‘s are downscaled: 𝑓(𝑖) ∼ 𝑁−𝑥

Counter-examples seem to exist

• New dominant instanton contributions to 𝑉(𝜙)

• Loophole?

Quantum gravity constraints

Dvali 07; Bachlechner, Long, McAllister 14

large 𝑁

large 𝑁

Bachlechner, Long, McAllister 14Hebecker, Mangat, Rompineve, Witkowski 15

cf. Palti 15

How many instantons are required in order to bound the field range

(= no parametric enhancement) at large 𝑁?

Simple example: 𝑁-cube (𝑃 = 𝑁)

cap off 2𝑁 vertices along which 𝑓eff ∼ 𝑁

→ exponentially large number 𝑃 ∼ 2𝑁 of dominant terms in the scalar potential

cf. typical string compactifications: 𝑃 ≈ 𝑁

Similar conclusion applies to general polytopes:

Use algorithm for 𝑓eff to test polytopes with 𝑃 ∼ 𝑁, 𝑃 ∼ 𝑁2, 𝑃 ∼ 2𝑁

→ parametric enhancement unless 𝑃 grows faster than quadratically with 𝑁

suggests loophole such that bound Δ𝜙 < 𝑀𝑝 can be violated at large 𝑁

Quantum gravity constraints

large 𝑁

Conclusions

• Models of large-field inflation with multiple axions admit a variety of regimesranging from no enhancement to exponentially large enhancement of 𝑓eff

• The weak gravity conjecture may constrain the maximally allowed field excursion in such models

Open question: Is large-field inflation consistent with quantum gravity?

Is there a strict bound on the axion field range or do loopholes exist?

• Explaining such a possible bound at large 𝑁 is difficult

Enormous number of instanton corrections required, alternative explanations also

problematic → suggests existence of a loophole

• Future work: construct explicit string theory models realizing a loophole or else show that any such model must fail

Conclusions

• Models of large-field inflation with multiple axions admit a variety of regimesranging from no enhancement to exponentially large enhancement of 𝑓eff

• The weak gravity conjecture may constrain the maximally allowed field excursion in such models

Open question: Is large-field inflation consistent with quantum gravity?

Is there a strict bound on the axion field range or do loopholes exist?

• Explaining such a possible bound at large 𝑁 is difficult

Enormous number of instanton corrections required, alternative explanations also

problematic → suggests existence of a loophole

• Future work: construct explicit string theory models realizing a loophole or else show that any such model must fail

Conclusions

Thank you!