Large-field Inflation with Multiple Axions and the Weak ... · Status quo: ongoing debate, outcome...
Transcript of Large-field Inflation with Multiple Axions and the Weak ... · Status quo: ongoing debate, outcome...
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!