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 1 Standard single field inflation = (@')2 V (')+... L 2

V () t ds2 dt2 + e2 dt0H(t0)d~x2 ⇠ R 2 V ⇥ H = 2 3mpl (t)

Basic observational input:

⇥ H2 V 3/2 5 3 10 • ⇥ ⇤˙ mpV 0 2 N = dtH = d H = d H 60 • e ˙ H ˙ & Small density perturbationsR R and vacuumR dominance: 2 = m2 V 1 • p V ⇥ ⇥ ⇥ = m2 V 1 • p V ⇥

2 4 V/mpl CMB “B-mode” polarization measures r = 16✏ / (⇢/⇢)2

Current upper bound from direct searches r 0.09 V (1.7 1016 GeV )4 . ) . ⇥ (PLANCK+BICEP2+Keck)

Close to GUT scale, string scale, 10d Planck scale in many string models

3 Future experiments could sweep out range 10

3 Why we should care:

⇥ H2 V 3/2 5 3 10 • ⇥ ⇤˙ mpV 0 2 N = dtH = d H = d H 60 • e ˙ H ˙ & R R R Determined by V

Ties range in field space to V Lyth; Efstathiou and Mack

3 r>10 ' m “High scale inflation” ) & pl

Constraints on inflaton potential 'n EFT parametrization V = cn n 4 n mpl X Exquisitely small for all n

Need control of quantum gravity corrections

4 Perturbative quantum corrections

1 Slow roll inflation models such as V = m 2 ' 2 “technically natural” 2

5 m 10 m softly broken shift symmetry ' ' + a ⇠ pl ) !

Loops of inflaton, graviton,… (if other couplings shift symmetric)

Vtree Vtree00 Coleman and Weinberg; Smolin; Linde V = Vtree 1+a 4 + b 2 + ... mpl mpl !

2 m 10 power series in 2 10 mpl ⇠

5 Natural inflation: inflaton = periodic pseudoscalar (aka “axion”) Freese, Frieman, and Olinto 4 ' 1 4 (k 1)S k' V (')=⇤ cos + c ⇤ e cos f k f ✓ ◆ kX=2 ✓ ◆ ' ' +2⇡f ⌘ Can be kept small if S << 1

⇤ M ,f m ⇠ GUT & pl

n Nonrenormalization theorem: prevents ' V = cn n 4 mpl

Potential can arise from gauge instantons ' = trG G L f ^

f: strong coupling scale for axion

6 Current lore: nonperturbative quantum gravity breaks global symmetries

Giddings and Strominger; Abbott and Wise; Coleman and Lee • Black hole formation and evaporation

• Gravitational instantons: break axion shift symmetries esp when f>mpl

2 m Banks, Dine, Fox, Gorbatov; S pl inst ⇠ f 2 Arkani-Hamed, Motl, Nicolis, Vafa

• String theory: many global symmetries = gauge symmetries

Banks; Dixon

Some mechanism is required to suppress QG effects

7 Candidate models

4 1 Arkani-Hamed, Cheng, • Extranatural inflation: M5 = R S Creminelli, Randall ⇥ R 5 ' = dx A5 I 5d gauge symmetry protects shift symmetry

Dimopoulos, Kachru, • N-flation McGreevy, Wacker

4 'n N axions ' with f

'2 pNf f ⇠ i ⇠ ⌘ eff i sX

feff >mpl even if f

8 Silverstein and Westphal; McAllister, Silverstein, • Axion monodromy and Westphal; Kaloper, Lawrence, and Sorbo

V(

n=1 n=2

n=0 n=3

f 2f 3f

Symmetry: ' ' +2⇡kf ; n n k ; k Z KLS ! ! 2 Upshot: this protects theory from large corrections

9 Possible constraints

1. Entropic bounds Conlon

Claim: in de Sitter patch: f 2 S N inflaton,dS ⇠ H2 m2 Covariant entropy bound: S pl dS ⇠ H2 2 2 Nf >mpl violates this

2. Weak Gravity Conjecture Arkani-Hamed, Motl, Nicolis, Vafa Upper bound on mass/tension/action of charged objects (avoid stable remnants, etc)

Eg for (unbroken) U(1) gauge theory, must exist charged particle (D 2)/2 m . gmpl,D 4 1 Extranatural inflation: 5d charged particle on R S 2 ⇥ mpl 4d instanton with Sinst ) ⇠ f 2

10 II. Entropy bounds Kaloper, Kleban, Lawrence, and Sloth

Basic point: A S = is computed in low energy theory, via classical gravity 4GN

Should use physical, renormalized value m 2.4 1018 GeV pl ⇠ ⇥ as measured by low-energy observer

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

The cutoff should be used in computing QFT contributions to entropy M 2 S N UV Nfields ⇠ H2

2 2 We claim, roughly: NMUV

11 A. Perturbative graviton loops

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

Seff = Sbare(gµ⌫ )+1 loop(g)+...

4 ⇤bare 2 1 2 2 = d xpg + aMUV + + cMUV R +(↵bare + ln MUV )R + ... GN,bare 16⇡GN,bare Z ✓ ◆ ✓ 2 ◆ = mpl,phys

Physical couplings

minimally coupled scalars + Weyl fermions:

c N + N = N ⇠ 0 1/2 m2 m2 < pl,phys UV N

12 B. Example: compactification

D+4-dim theory: S mD+2 dD+4pgR grav ⇠ pl,D+4 Z

fundamental scale/ 4 UV cutoff S4d = mpl,D+4VD d xpgR Z

2 D+2 mpl,4 = mpl,D+4VD

1 Number of KK modes with

N (L m )D = V mD ⇠ KK pl,D+4 D pl,D+4

m2 2 pl,4 2 Lesson: cutoff of EFT is something MUV = mpl,D+4 ⌘ N physical and obvious in UV completion

13 C. de Sitter entropy: N scalar fields

1 r2 r2 ds2 = 1 dt2 + 1 dr2 + r2d⌦2 r2 r2 2 ✓ h ◆ ✓ h ◆

• Impose Dirichlet (“brick wall”) b.c. at r = r ✏ h • Count number of field modes at fixed E ’t Hooft (for BHs) Blueshifting near horizon: can use WKB

• Repeat for Pauli-Villars regulator fields; then take ✏ 0 !

Quadratic area law divergence

• Compute corrections to G N using same (PV) scheme Kabat; Demers, LaFrance, and A A Myers (for BH) + S1 loop = 4GN,bare 4GN,ren

14 III. Axion monodromy

Silverstein and Westphal; McAllister, Original idea arose from string models Silverstein and Westphal 4d effective field theory version Kaloper and Sorbo; Kaloper, Lawrence and Sorbo

4 2 1 2 1 2 µ Sclass = d x⇥g m R F (⇥) + F pl 48 2 24 ⇤ ⇥ Fµ⇤⌅ = [µ A ⇤⌅]

Aµ⇤ = ⇥[µ ⇤] U(1) gauge symmetry ' ' +2⇡f ⌘ '

V( 1 2 1 2 Htree = 2 p + 2 (pA + µ⇥) + grav. 2 pA = ne Compactness of U(1) n=1 n=2

2 µf = ke n=0 n=3 Dim. red. to 0+1: ' ' +2⇡f' ; n n k charged particle ! ! in B-field on torus. k = magnetic flux quantum discrete gauge symmetry in phase space = LLL degeneracy 15 6 4 f .1m ; µ = 10 m ; e (few) 10 m ' ⇠ pl pl ⇠ ⇥ pl At fixed n, good model of chaotic inflation ) Discrete gauge symmetry: no dangerously large Planck-suppressed operators 'n V (1) n 4 ⇠ O mpl

Allowed corrections to Lagrangian: F 2n = c 4n 4 • L MUV 2 2n n 1 (µ' + ne ) Vtree H = 4n 4 Vtree 4 ) M ⇠ M UV ✓ UV ◆

Slow roll safe if M 4 >V M 4 UV ⇠ GUT ' V () = c⇤4 cos • L f Small sinusoidal ✓ ◆ modulation ... • 16 Weak gravity conjecture?

V(

n=1 n=2

n=0 n=3

Membranes charged under F: n n 1 ! ± Coleman; Brown and Teitelboim; via nucleation of bubbles of lower branch Coleman and de Luccia

Naive expectation: WGC upper bound on membrane tension ) Brown, Cottrell, Shiu, and Soler; Ibanez, Montero, Uranga, Valenzuela

17 1 (4) 2 1 2 = (F ) (@') µ'⇤F L 48 2

d' = H(3) = dB(2)

1 1 = (F (4))2 µ2A2 B is longitudinal mode of A L 48 2 “London equation” for axion monodromy Marchesano, Shiu, and Uranga; Kaloper and Lawrence, in progress

Standard WGC argument does not apply to massive gauge fields Cheung and Remmen

Stable charged black objects do not exist Bekenstein

Are there other constraints? See also Hebecker, Moritz, Westphal, and Witkowski

18 Julia-Toulouse mechanism Julia and Toulouse; Quevedo and Trugenberger

Membranes electrically charged under A 1 1 = (F (4))2 µ2A2 L 48 2

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 ?

19 2d analog: Schwinger model Lawrence; Seiberg

Charged fermions = 2d domain walls 1 = F F µ⌫ + ¯µ (@ iA ) L 4 µ⌫ µ µ Bosonization: 1 1 = F F µ⌫ (@')2 '✏µ⌫ F L 4 µ⌫ 2 µ⌫

1 1 = F F µ⌫ A2 L 4 µ⌫ 2