Effective field theory for axion monodromy inflation

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1 Effective field theory for axion monodromy inflation Albion Lawrence Brandeis University Based on work in progress with Nemanja Kaloper and L.orenzo Sorbo

2 Outline I. Introduction and motivation II. Scalar + 4-form dynamics III. Corrections from UV completions IV. Conclusions

3 I. Introduction and motivation Single field inflation V (ϕ) δϕ ϕ(t) Slow roll and vacuum dominance: ( ) 2 ɛ = m 2 pl 1 V V ϕ η = m 2 pl V V 1 Spacetime approximately de Sitter: ds 2 dt 2 + e 2 R Hdt d x 2, H 2 = V. m 2 pl Quantum fluctuations of φ generate observed density fluctuations: δρ ρ V 3/2 m 3 pl V Quantum fluctuations of metric produce gravity waves detectable via CMB polarization P g V m 4 pl

4 Large field inflation Observational upper bound on primordial gravity waves: V GeV M gut Close to unification scale α i = e 2 i / c α 3 α 2 α 1 α grav Consistent with Proton decay Neutrino masses log 10 (E/GeV ) Couplings unify (assuming MSSM above 1 T ev ) at approximately GeV. Graph not to scale. Could be detectable by PLANCK, ground-based experiments

5 Detectable gravitational waves require large fields Lyth, hep-ph/ ( δρ ρ ) 2 V 3 (V ) 2 measured by CMB temperature fluctuations Primordial gravitational waves P g V m 4 pl Upper bound on V upper bound on V V N e = dth = dφ φ H = 3 m 2 pl dφ V V 60 Upper bound on dφ dn, φ during inflation φ m pl

6 Effective field theory and large φ Effective field theory: Allow all terms in action consistent with symmetries V = n g n φn M n 4 M m pl dynamical scale of UV physics Generic theory: g n 1 Expansion breaks down for φ >M New degrees of freedom become light. Relevant degrees of freedom could be very different.

7 Inflation is a highly nongeneric theory Consider V m 2 φ 2 or V λφ 4 Give observable GW N e 60, δρ ρ 10 5 m 10 6 m pl, λ Very finely tuned! But in fact the situation is worse: All coefficients g n in V = n g n φn M n 4 must be exquisitely small For example such corrections give η 1

8 NB: quantum loops of inflatons and gravitons are not dangerous m, λ give small breaking of shift symmetry: Guarantees loops of φ, gravitons do not make g n too large V loop = V class (φ)f ( V m 4 pl, V m 3 pl ),... Coleman and Weinberg; Smolin; Linde This approximate shift symmetry is the key to building chaotic inflation models

9 Fine tuning hard to justify Coupling to other degrees of freedom is the problem! Tends to give unacceptable breaking of shift symmetry Gravity breaks global symmetries: wormholes, virtual black holes,... Holman et al; Kamionkowski and March- Russell; Barr and Seckel; Lusignoli and Roncadelli; Kallosh,Linde,Linde,Susskind String theory: global symmetries tend to be gauged or anomalous. Anomalous shift symmetry broken by instantons (eg axion): V Λ 4 n c n cos(nφ/f φ ) Λ 4 cos(φ/f φ )+... Arkani-Hamed, Cheng, Creminelli, Randall f φ m pl is hard to realize (eg c n tends to be large). Banks, Dine, Fox, Gorbatov; Arkani-Hamed, Motl, Nicolis, Vafa

10 Solution: monodromy inflation Silverstein and Westphal; McAllister, Silverstein and Westphal; Kaloper and Sorbo; Berg, Pajer, and Sjors; KLS Consider axion with period (decay constant) f φ In this scenario, physics invariant under φ φ + f φ, but states are not periodic under continuous shift V (φ) n = 2 n =1 Spectral flow V (φ,n)=µ 2 (φ nf φ ) 2 n Z discrete variable n = 1 n =0 τ n τ inflation φ f φ Inflation: φ ranges over many periods Compact field space (nb: must include n) may keep EFT under control

11 String theory example Type II on torus; unit volume and complex structure τ in string units Silverstein and Westphal; McAllister, Silverstein and Westphal τ C τ +1 τ has period 1. = Canonically normalized scalar φ = m pl τ Shift τ τ + 1 is symmetry of torus, but stretches D-brane. V m4 s g s 1+τ 2 Shift τ n times; D-brane becomes n times as long.

12 Goal: These scenarios receive quantum corrections from integrating out UV degrees of freedom: moduli Kaluza-Klein modes light string modes... Additional effects: instantons semiclassical gravity These were analyzed model by model in the string constructions. These models are of necessity complicated. Silverstein and Westphal; McAllister, Silverstein and Westphal

13 We study 4d effective field theory to : Better understand the physics behind suppressing corrections Better understand the degree of fine tuning still needed Provide a framework for building and comparing models.

14 II. Inflaton-4 form dynamics Kaloper and Sorbo, and d mechanism for generating a potential via spectral flow. F µνλρ totally antisymmetric 4-form field strength F µνλρ = [µ A νλρ] S = d 4 x g ( m 2 pr 1 48 F ( φ)2 + µ 24 φ F ) + boundary terms U(1) gauge invariance A µνρ [µ Λ νρ] (Will assume U(1) compact as in string theory) F sourced by membranes: S membrane = e 6 Σ 3 d 3 σɛ ijk i x µ j x ν k x ρ A µνρ Compact U(1) quantized membrane charge Theory has 1 scalar degree of freedom with mass µ Bousso and Polchinski Dvali; Kaloper and Sorbo

15 Hamiltonian dynamics H = 1 2 (p + µφ) π2 φ + grav. π φ : conjugate momentum for φ p: conjugate momentum for A 123 Compact U(1) p is quantized in units of e 2. p is conserved by H: jumps via membrane nucleation. φ periodicity: f φ = e 2 /µ. Realizes monodromy inflation: φ f φ, p p + e 2 leaves H invariant.

16 III. Corrections from UV completions S = d 4 x g ( m 2 pr 1 48 F ( φ)2 + µ 24 φ F ) +bndry terms+uv corrections µ 10 6 m pl for slow roll inflation: Effective field theory: Allow all terms in action consistent with symmetries, topology of field space Corrections controlled by: Compactness of φ (and of U(1)). Small coupling µ m pl,m GUT.

17 Direct corrections Direct corrections to V must be periodic in φ n c n φn M n 4 = Λ 4 cos(φ/f φ )+... Generally f < m pl ; µ 2 φ 2 potential modulated by oscillations Gauge instantons: Λ Λ QCD V (φ) Gravitational effects: Λ 4 f n+4 /m n pl φ V corr V class Λ 4 M 4 gut η 1 Λ4 f 2 V m 2 pl = H 2 Example: feasible if f M gut, Λ GeV

18 Must be careful with moduli stabilization Coefficients of V (φ) typically depend on moduli ψ V = V 0 (ψ)+c 1 (ψ)λ 4 cos(φ/f)+... We must have V 0 (ψ) c 1 Λ 4 Otherwise modulus destabilized whenever cos(φ/f) 1 (which will happen many times during inflation)

19 Indirect corrections Additional corrections must respect periodicity of φ. Instead we can correct dynamics of 4-form sector. δl = n c n F 2n Λ 4n 4 (Λ some UV scale) S classical = d 4 x g ( m 2 pr 1 48 F ( φ)2 + µ 24 φ F ) +... We can guess effects of δl by integrating out F classically: F µφ δl V (φ) n c n V n 1 Λ 4n 4 Multiplicative correction to V : safe if Λ 4 >V Corrections of the form δl n d n F 2n Λ 4n ( φ) 2 give identical story after 1. Integrating out F 2. Canonically normalizing φ

20 Small Λ not always bad String scenario with φ a D-brane modulus: V (φ) = M M 6 2 φ2 V µ 2 φ 2 for small φ m pl Large φ: V M 3 φ Silverstein and Westphal, McAllister, Silverstein, and Westphal,

21 Source of corrections ψ a field (modulus, Kaluza-Klein state, etc.) with mass M class δl n d n F 2n Λ 4n ( ψ) 2 + n d n Canonically normalize ψ: effective mass M 2 = M 2 classical + Λ2 n d n V n F 2n Λ 4n 2 ψ Integrate out ψ: induce Coleman-Weinberg potential V CW M(φ) 4 ln M 2 Λ 2 Λ 4n Must sum over all such fields with M class < Λ UV Safe if n species M 2 class Λ2 ; V Λ 4. Eg: n species m 2 pl /M 2 gut; Λ M gut M 2 class H2.

22 III. Conclusions 1. Large scale inflation feasible without infinite numbers of fine tunes. 2. Compactness of field space, small µ controls corrections Additional questions: 1. Lifetime of p due to membrane nucleation should be longer than inflation. 2. Compare to explicit string models with and without 4-forms. 3. Is µ 2 φ 2 inflation possible or do we always get V φ p<2? 4. Can we find a way to make µ naturally small?