Effective field theory for axion monodromy inflation Albion Lawrence Brandeis University Based on work in progress with Nemanja Kaloper and L.orenzo Sorbo
Outline I. Introduction and motivation II. Scalar + 4-form dynamics III. Corrections from UV completions IV. Conclusions
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
Large field inflation Observational upper bound on primordial gravity waves: V 10 16 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 3 16 18 log 10 (E/GeV ) Couplings unify (assuming MSSM above 1 T ev ) at approximately 10 16 GeV. Graph not to scale. Could be detectable by PLANCK, ground-based experiments
Detectable gravitational waves require large fields Lyth, hep-ph/9606387 ( 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
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.
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 10 14 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
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
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
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
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.
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
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.
II. Inflaton-4 form dynamics Kaloper and Sorbo, 0810.5346 and 0811.1989 4d 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 2 1 2 ( 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
Hamiltonian dynamics H = 1 2 (p + µφ)2 + 1 2 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.
III. Corrections from UV completions S = d 4 x g ( m 2 pr 1 48 F 2 1 2 ( 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.
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 10 15 GeV
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)
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 2 1 2 ( 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
Small Λ not always bad String scenario with φ a D-brane modulus: V (φ) = M 8 1 + M 6 2 φ2 V µ 2 φ 2 for small φ m pl Large φ: V M 3 φ Silverstein and Westphal, 0803.3085 McAllister, Silverstein, and Westphal, 0808.0706
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 2 +... 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.
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?