Axion Inflation and the Lattice Weak Gravity Conjecture. Tom Rudelius Harvard/IAS

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1 Axion Inflation and the Lattice Weak Gravity Conjecture Tom Rudelius Harvard/IAS

2 Based On /hep-th, /hep-th /hep-th, /hep-th with Ben Heidenreich, Matthew Reece To appear with Ben Heidenreich, Cody Long, Liam McAllister, Matthew Reece, John Stout

3 Outline I. The Weak Gravity Conjecture II. Axion Inflation III. The WGC and Inflation 1. Mild WGC and Inflation 2. Lattice WGC and Inflation

4 The Weak Gravity Conjecture

5 The Weak Gravity Conjecture The WGC: In any U(1) gauge theory that admits a UV completion with gravity, there must exist a state of charge q, mass m, such that q m Q M extremal Arkani-Hamed, Motl, Nicolis, Vafa, 06 (see talks by Gary Shiu, Ben Heidenreich)

6 Why should the WGC be true? If not, extremal BH will be unable to decay. If not, near-extremal BH will move towards extremality. M = Q M > Q M < Q

7 Why should the WGC be true? Many examples from string theory, KK theory: M Q = M Q

8 The WGC for Multiple U(1)s 1. Take theory with N 1-form U(1)s. 2. Consider charge-to-mass vectors ~z i = ~q i m i M p. 3. WGC: Convex Hull of {~z i } must contain unit ball in R N. Z 2 BH region Z 1 Cheung, Remmen 14

9 The slwgc Sublattice Weak Gravity Conjecture: There exists some sublattice of the full charge lattice with particles satisfying ~q /m ~ Q /M ext.

10 Axion Inflation

11 Inflation Problem: Why is the universe so flat and so homogeneous? Solution: Inflation postulated period of exponential expansion in the early universe

12 Axion Inflation Axions (scalars with perturbative shift symmetry) acquire periodic potential via instanton effects: V ( )= 4 (1 cos f )+... axion decay constant Need f>m p for inflation V( )

13 The WGC and Inflation

14 The Generalized WGC The generalized WGC: In any p-form theory in d dimensions, there must exist an electrically charged object of dimension p 1 and a magnetically charged object of dimension d p 1 with T el. g 2 G N 1/2, T mag. 1 g 2 G N 1/2

15 WGC and Axion Inflation Consider 0-form with decay constant f. Charged object is an instanton of action S. WGC ) 1/(fS) & 1/M p. S>1+WGC) f. M p Incompatible with inflation Agrees with string theory Banks, Dine, Fox, Gorbatov 03

16 Multi-Axion Inflation Recruit additional axions: Decay Constant Alignment Kim, Nilles, Peloso 04 N-flation Dimopoulos, Kachru, McGreevy, Wacker 05

17 Multi-Axion Inflation Simplest models violate convex hull condition Decay Constant Alignment N-flation

18 Multi-Axion Inflation Field space diameters bounded TR 14, 15 Brown, Cottrell, Shiu, Soler 15 Montero, Uranga, Valenzuela 15 Decay Constant Alignment N-flation

19 (s)lwgc and Inflation Model with P instantons, N axions, WGC bounds assume (s)lwgc says P = N P = 1 Need to revisit WGC constraints in light of new conjecture

20 Two Setups Consider square lattice saturating WGC bound: of instantons 1)S = ~ Q := q ~Q ~Q 2)S = ~ Q := X i Q i t i

21 First Setup Consider square lattice saturating WGC bound: of instantons ~ Q := q ~Q K 1 ~Q L 1 2 K X ab@ b ~Q2 e ~ Q cos( ~ Q ~ + ~Q )

22 A Loophole Set, N =1, ~Q = = const. V ( )= X n2z e n/f 0 cos n f 0 + = V 1 + V 0 cos( /f 0 + ) e 1/f 0 cos cosh 1/f 0 cos /f 0. V ( )

23 A Second Loophole Scrunch lattice along one diagonal: Effect on potential is subleading in scrunched direction, find N 1 in f e p NM p

24 A Third Loophole Set N =2, take two aligned instantons to have S ~Q Q ~ and dominate potential Gives rise to standard KNP alignment Kim, Nilles, Peloso 04

25 Generic Constraints More generally, take random phases Given direction ~e 0 in charge lattice, decompose potential into harmonics ~Q ~e 0

26 Phasor Notation Introduce phasor notation, L 1 2 K X ab@ b Z ~Q = e ~ Q +i ~Q ~Q2 Z ~Q e i ~ Q ~ Total contribution to nth harmonic is then X Z n = ~Q, P i Q i= n Z ~Q

27 Gaussian Approximation For large number of instantons, can use CLT to write: 2 n = Z n N (0, X ~Q, P i Q i= n 2 n) e 2 ~ Q Higher harmonics suppressed, 2 n 2 1

28 Estimating 2 n Want to estimate sum X 2 n = ~Q, P i Q i= n e 2 ~ Q Method 1: use instantons of small charges and ignore the rest (valid for small decay constants) Method 2: use Poisson resummation (valid for large decay constants) c.f. talks by Liam McAllister, John Stout

29 For general Volume Bound K ab vol, define = M N p (2 )N with the volume of the unit cell, the fundamental domain of axion moduli space: Decay Constant Alignment N-flation

30 Volume Bound (cont.) Demanding 2 n 1 and approximating, get log[vol ]. log[vol D N (2M p )] + O(log N) with D N (R) the N-ball of radius R.

31 Volume Bound and N-flation Isotropic N-flation violates the volume bound M p p NMp But, decay constant alignment survives

32 Volume Bound and Alignment Set K ab 2 Wish N ( 2,N) ) f e. O(1M p ) Set K 1 ab 2 Wish N ( 2,N) ) f e. O(NM p )

33 Second Setup Square lattice of instantons saturating WGC bound, now with linear action sum: ~ Q := X i Q i t i L 1 2 K X ab@ b ~Q2 e ~ Q cos( ~ Q ~ + ~Q )

34 Convex Hull Condition Expect CHC satisfied by charge 1 instantons (c.f. mild WGC constraints): Would need higher harmonics suppressed to get N-flation

35 N-flation Higher harmonics actually become worse as N increases: 2 2/ N= N= No N-flation N= t

36 Summary The WGC is incompatible with some models of axion inflation If the (s)lwgc is true: Isotropic N-flation would be ruled out Some decay constant alignment models would be ruled out But, some would be allowed See talks by Liam McAllister, John Stout for further details

37 Open Questions Can one close the aforementioned loopholes? If not, can one exploit one of these opportunities in a stringy model of inflation? What more can the WGC tell us about lowenergy physics? (see talks by Valenzuela, Blumenhagen, Hebecker, Montero, and Ooguri)