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 L = 1 2( ∂ϕ ) 2 − V ( ϕ ) + . . . Standard single field inflation R t dt 0 H ( t 0 ) d ~ V ( ϕ ) ds 2 ∼ − dt 2 + e 2 x 2 V H 2 = δϕ 3 m 2 pl ϕ ( t ) ϕ Basic observational input: ϕ ∼ V 3 / 2 ρ ∼ H 2 • δρ p V 0 ∼ 10 − 5 m 3 ˙ R d ϕ R d ϕ H 2 R ϕ & 60 • N e = dtH = ϕ H = ˙ H ˙ Small density perturbations and vacuum dominance: ⇥ 2 � V � • � = m 2 ⇥ 1 p V V �� • ⇥ = m 2 ⇥ 1 p V 2

V/m 4 pl r = 16 ✏ ∝ CMB “B-mode” polarization measures ( �⇢ / ⇢ ) 2 r . 0 . 09 ⇒ V . (1 . 7 × 10 16 GeV ) 4 Current upper bound from direct searches (PLANCK+BICEP2+Keck) Close to GUT scale, string scale, 10d Planck scale in many string models 10 − 3 < r < . 1 Future experiments could sweep out range 3

Why we should care: ϕ ∼ V 3 / 2 ρ ∼ H 2 • δρ p V 0 ∼ 10 − 5 m 3 ˙ R d ϕ R d ϕ H 2 R ϕ & 60 • N e = dtH = ϕ H = ˙ H ˙ Determined by V Ties range in field space to V Lyth; Efstathiou and Mack r > 10 − 3 ⇒ ∆ ϕ & m pl “High scale inflation” Constraints on inflaton potential ϕ n X EFT parametrization V = c n m n − 4 pl n Exquisitely small for all n Need control of quantum gravity corrections 4

Perturbative quantum corrections V = 1 2 m 2 ϕ 2 Slow roll inflation models such as “technically natural” m ∼ 10 − 5 m pl ⇒ softly broken shift symmetry ϕ → ϕ + a Loops of inflaton, graviton,… (if other couplings shift symmetric) ! + bV 00 1 + aV tree tree V = V tree + ... Coleman and Weinberg; Smolin; Linde m 4 m 2 pl pl m 2 power series in ∼ 10 − 10 m 2 pl 5

Natural inflation: inflaton = periodic pseudoscalar (aka “axion”) ∞ Freese, Frieman, and Olinto ✓ ϕ ◆ ✓ k ϕ ◆ V ( ϕ ) = Λ 4 cos c k Λ 4 e − ( k − 1) S cos X + f f k =2 ϕ ≡ ϕ + 2 π f Can be kept small if S << 1 Λ ∼ M GUT , f & m pl ϕ n Nonrenormalization theorem: prevents δ V = c n m n − 4 pl Potential can arise from gauge instantons δ L = ϕ f tr G ∧ G 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 > m pl m 2 Banks, Dine, Fox, Gorbatov; pl S inst ∼ Arkani-Hamed, Motl, Nicolis, Vafa f 2 • String theory: many global symmetries = gauge symmetries Banks; Dixon Some mechanism is required to suppress QG effects 7

Candidate models • Extranatural inflation: M 5 = R 4 × S 1 Arkani-Hamed, Cheng, R Creminelli, Randall I dx 5 A 5 ϕ = 5d gauge symmetry protects shift symmetry Dimopoulos, Kachru, • N-flation McGreevy, Wacker ✓ ϕ n ◆ X Λ 4 f < m pl N axions with : V = ϕ i n cos f n n Total possible distance in field space: sX √ δϕ 2 ∆Φ ∼ Nf ≡ f eff i ∼ i f eff > m pl even if f < m pl 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: ϕ → ϕ + 2 π kf ; n → n − k ; k ∈ Z KLS Upshot: this protects theory from large corrections 9

Possible constraints 1. Entropic bounds Conlon Claim: in de Sitter patch: S inflaton,dS ∼ N f 2 H 2 m 2 pl Covariant entropy bound: S dS ∼ H 2 Nf 2 > m 2 violates this pl 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 m . gm ( D − 2) / 2 pl,D R 4 × S 1 Extranatural inflation: 5d charged particle on m 2 pl ⇒ 4d instanton with S inst ∼ f 2 10

II. Entropy bounds Kaloper, Kleban, Lawrence, and Sloth Basic point: A S = is computed in low energy theory, via classical gravity 4 G N m pl ∼ 2 . 4 × 10 18 GeV Should use physical, renormalized value 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 S Nfields ∼ N M 2 UV H 2 NM 2 UV < m 2 We claim, roughly: pl,phys 11

A. Perturbative graviton loops Integrate out matter fields to get effective action for gravitational sector: S eff = S bare ( g µ ν ) + Γ 1 − loop ( g ) + . . . ✓ ◆ ✓ 1 ◆ � Z − Λ bare R + ( α bare + β ln M UV ) R 2 + . . . d 4 x √ g + aM 2 + cM 2 = + UV UV G N,bare 16 π G N,bare = m 2 pl,phys Physical couplings minimally coupled scalars + Weyl fermions: c ∼ N 0 + N 1 / 2 = N m 2 pl,phys m 2 UV < N 12

B. Example: compactification Z d D +4 √ gR D+4-dim theory: S grav ∼ m D +2 pl,D +4 fundamental scale/ Z d 4 x √ gR UV cutoff S 4 d = m pl,D +4 V D m 2 pl, 4 = m D +2 pl,D +4 V D 1 Number of KK modes with < m < m pl,D +4 L KK N ∼ ( L KK m pl,D +4 ) D = V D m D pl,D +4 m 2 Lesson: cutoff of EFT is something pl, 4 M 2 = m 2 UV ≡ pl,D +4 N physical and obvious in UV completion 13

C. de Sitter entropy: N scalar fields ◆ − 1 1 − r 2 1 − r 2 ✓ ◆ ✓ ds 2 = − dt 2 + dr 2 + r 2 d Ω 2 2 r 2 r 2 h h • Impose Dirichlet (“brick wall”) b.c. at r = r h − ✏ ’t Hooft (for BHs) • Count number of field modes at fixed E Blueshifting near horizon: can use WKB • Repeat for Pauli-Villars regulator fields; then take ✏ → 0 Quadratic area law divergence • Compute corrections to using same (PV) scheme G N Kabat; Demers, LaFrance, and Myers (for BH) A A + δ S 1 − loop = 4 G N,bare 4 G N,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 � ⇥ 2 ( ∂ϕ ) 2 + µ d 4 x √ g ⇤ pl R − 1 48 F 2 − 1 m 2 S class = 24 ϕ ∗ F F µ νλρ = ∂ [ µ A νλρ ] δ A µ νλ = ∂ [ µ Λ νλ ] U(1) gauge symmetry ϕ ≡ ϕ + 2 π f ϕ 2 ( p A + µ φ ) 2 + grav. V( �� H tree = 1 φ + 1 2 p 2 p A = ne 2 Compactness of U(1) n=1 n=2 µf φ = ke 2 n=0 n=3 Dim. red. to 0+1: ϕ → ϕ + 2 π f ϕ ; n → n − k charged particle � in B-field on torus. discrete gauge symmetry in phase space k = magnetic flux quantum = LLL degeneracy 15

f ϕ ∼ . 1 m pl ; µ = 10 − 6 m pl ; e ∼ ( few ) × 10 − 4 m pl ⇒ At fixed n, good model of chaotic inflation Discrete gauge symmetry: no dangerously large Planck-suppressed operators δ V ∼ O (1) ϕ n m n − 4 pl Allowed corrections to Lagrangian: • δ L = c F 2 n M 4 n − 4 UV ✓ V tree ◆ n − 1 ⇒ δ H = ( µ ϕ + ne 2 ) 2 n ∼ V tree M 4 n − 4 M 4 UV UV M 4 UV > V ∼ M 4 Slow roll safe if GUT V ( ϕ ) ✓ ϕ ◆ • δ L = c Λ 4 cos Small sinusoidal f modulation ϕ • . . . 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

L = − 1 48( F (4) ) 2 − 1 2( ∂ϕ ) 2 − µ ϕ ∗ F d ϕ = H (3) = dB (2) L = − 1 48( F (4) ) 2 − 1 2 µ 2 A 2 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 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 L = − 1 48( F (4) ) 2 − 1 2 µ 2 A 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 L = − 1 4 F µ ν F µ ν + ¯ ψγ µ ( ∂ µ − iA µ ) ψ Bosonization: L = − 1 4 F µ ν F µ ν − 1 2( @' ) 2 − '✏ µ ν F µ ν L = − 1 4 F µ ν F µ ν − 1 2 A 2