Axion monodromy and inflation

Transcription

1 Axion monodromy and inflation Albion Lawrence, Brandeis/NYU arxiv: with Nemanja Kaloper (UC Davis), AL, and Lorenzo Sorbo (U Mass Amherst) Silverstein and Westphal, arxiv: McAllister, Silverstein, and Westphal, arxiv: others Related work in progress with Sergei Dubovsky (NYU), AL, and Matthew Roberts (NYU) and with Dubovsky, Raphael Flauger (Yale), Kaloper, and AL 0/31

2 I. Introduction: high scale inflation in UVcomplete theories II. Monodromy inflation in string theory III. 4d models of axion monodromy IV. Quantum corrections V. Conclusions 1

3 I. Introduction Slow roll inflation matches CMB/LSS data well V (ϕ) δϕ ϕ(t) ϕ Slow roll + vacuum dominance: ɛ = m 2 p ( V V ) 2 1 η = m 2 p V V 1 Spacetime approximately de Sitter: ds 2 = dt 2 + e 2 R dth d x 2 3, H 2 = V m 2 p Observed flatness of space: ϕ must roll long enough for space to expand by e 60 Quantum fluctuations generate structure: Quantum fluctuations generate gravitational waves: δρ ρ V 3/2 m 3 p V P g V m 4 p δt CMB T CMB Detectable via CMB polarization experiments 2

4 Scale of inflation Observational upper bound on GW: Close to unification scale V GeV M GUT α i = e 2 i / c α 3 α 2 α α grav log 10 (E/GeV ) See also: p decay ν mass Couplings unify (assuming MSSM above 1 T ev ) at approximately GeV. Graph not to scale. If V near upper bound: detectable by PLANCK or ground-based CMB polarization experiments 3

5 Detectable gravitational radiation requires large fields ( δρ ρ ) = c V 1/2 m 3 pl V V 10 5 from observations Lyth, hep-ph/ upper bound on V upper bound on V /V N e = dth = dφ φ H = 3 m 2 pl dφ V V 60 to match observed flatness upper bound on dφ dn, φ during inflation ϕ m pl 4

6 Effective field theory and large φ Effective field theory: expansion in 1/M for some UV scale M V = n g n φn M n 4 generically g n 1 M m pl unless forbidden by symmetry Expansion breaks down for φ >M New degrees of freedom could become light Relevant d.o.f. very different 5

7 Inflation is a highly nongeneric theory Consider V m 2 φ 2 or V λφ 4 δρ/ρ 10 5,N e 60 m2 m 2 pl λ Corrections δv = n g n φn M n 4 all g n must be small: infinite fine tuning! else e.g. η = m 2 pl V V 1 Slow roll inflation requires approximate shift symmetry φ φ + a 6

8 Perturbative quantum corrections Small couplings m 2 m 2 pl, λ m pl -suppressed couplings to gravity loops of inflaton, graviton gives suppressed couplings V loop = V class F ( V m 4 pl, V m 2 pl ),... Coleman and Weinberg; Smolin; Linde Slow roll inflation safe against inflaton, graviton loops perturbative corrections preserve symmetries 7

9 UV completions make slow roll difficult to maintain Continuous global symmetries like φ φ + a are always (we think) broken Gravity breaks continuous global symmetries (Hawking radiation/virtual black holes, wormholes,...) Holman et al; Kamionkowski and March-Russell; Barr and Seckel; Lusignoli and Roncadelli; Kallosh, Linde, and Susskind String theory: continuous global symmetries tend to be gauged, anomalous Anomalous symmetries broken by nonperturbative effects (e.g. Peccei-Quinn symmetry of axion) δv Λ 4 n c n cos(nφ/f φ ) 8

10 One attempt: pseudonatural inflation Use anomalous symmetry to generate potential V = Λ 4 cos ( φ f φ ) +... δv Λ 4 n>1 c n cos(nφ/f φ ) c n e ns, ( fφ M ) n Λ some dynamical scale; slow roll for f φ Λ large field if f φ m pl Problem: f φ >m pl with c n f φ M 1 small does not seem to be allowed Banks, Dine, Fox, and Gorbatov; Arkani-Hamed, Motl, Nicolis, and Vafa 9

11 II. Monodromy inflation in string theory Consider compact scalar field ϕ ϕ + f ; f m pl Silverstein and Westphal; McAllister, Silverstein, and Westphal Theory invariant under shift ϕ ϕ + f physical state need not be V (φ) n = 2 n =1 n = 1 n =0 f φ φ Let axion wind N times such that Nf φ m pl Compactness of field space seems to control quantum corrections 10

12 Cartoon Most models to date constructed within string theory But see Berg, Pajer, and Sors; Kaloper and Sorbo Illustrative example: type IIA with D4-brane wrapped on 2-torus τ has period =1 φ = m pl τ canonically normalized scalar τ C τ +1 = Shift τ τ + 1 is symmetry of torus, but stretches D-brane. V (φ) m4 s g s 1+(mpl φ) 2 n = # of D4 windings Shift τ n times; D-brane becomes n times as long. Doesn t quite work but illustrates point 11

13 Example: nilmanifold N!!"!#!%!!& u Silverstein and Westphal T 2 fibration of S 1 with monodromy τ τ + M u Red: D4-brane wrapped on single cycle of torus =!!"!$!!"!# Consider M 6 = (Nil) 1 (Nil) 2 Wrap D4 on linear combinations of S1,2 1 T1,2 2 S DBI d 4 xm 4 (A + Bu 2 )(1 C( u) 2 ) Inflation works at large u: S au( u) 2 bu ( φ) 2 µ 10/3 φ 2/3 12

14 Example: axion monodromy McAllister, Silverstein, and Westphal S 2 M 6 ; a = S 2 C (2) RR a a +2π L = 1 2 f 2 a( a) 2 = 1 2 ( φ)2 Now add NS5-brane wrapping S 2 a a +2π induces D3-brane charge V 1 g 2 s C + Da 2 Again, slow roll inflation works only at large a L 1 2 ( φ)2 µ 3 φ 13

15 Known string realizations seem to give flat potentials, with relatively small powers V M 4 p ϕ p<2 Seems to the result of coupling to moduli, KK modes Is a quadratic potential viable? Dong, Horn, Silverstein, and Westphal CMB data: p <=2 viable, smaller p more viable Quantum corrections studied model by model: these are complicated, and physical reason for flat potentials is not completely transparent. 14

16 Effective field theory approach Input basic fields, symmetries, topology of field space Expand action in powers of 1/M (M = UV scale), include all terms consistent with symmetries Pinpoints physics behind suppressing corrections to slow roll Isolates fine tuning required. Provides a framework for building new string models String theory has a complicated landscape Realistic models very hard to construct Quantum corrections difficult to compute 4d effective field theory analysis is always important 15

17 III. 4d models of axion monodromy Axion-four form model Kaloper and Sorbo S class = d 4 x ( ) g m 2 pl R 1 48 F ( ϕ)2 + µ 24 ϕ F F µνλρ = [µ A νλρ] U(1) gauge symmetry: δa µνλ = [µ Λ νλ] ϕ periodic: ϕ ϕ + f ϕ F does not propagate. U(1) quantized F µνλρ = ne 2 ɛ µνλρ ; n Z n can jump across domain walls/membranes 16

18 Dynamics Single massive scalar degree of freedom Dvali; Kaloper and Sorbo Hamiltonian: H tree = 1 2 p2 φ (p A + µφ) 2 + grav. Compact U(1): p A = ne 2 p A conserved by H tree Jumps by membrane nucleation Consistency condition: µf ϕ = e 2 V (φ) n = 2 n =1 Realizes monodromy inflation: theory invariant if ϕ ϕ + f ϕ ; n n 1 n = 1 n =0 f φ φ Good model for inflation: fits data well if + observable GW µ 10 6 m pl 17

19 Large-N gauge dynamics S class = d 4 x g ( m 2 pl R 1 4g 2 YM ) trg ( ϕ)2 + ϕ f ϕ trg G G: field strength for U(N) gauge theory with N large; strong coupling in IR Instanton expansion breaks down Witten; Giusti, Petrarca, and Taglienti ( nλ 2 + µϕ ) 2 H tree = H gauge p2 ϕ Λ strong coupling scale of U(N) theory µ = Λ 2 /f ϕ Can be related to 4-form version: F µνλρ tr G [µν G λρ] Dvali 18

20 IV. Quantum corrections S class = d 4 x g ( m 2 pl R 1 48 F ( ϕ)2 + µ 24 ϕ F µ 10 6 m pl to match constraints on δρ/ρ, N e ) What are the possible corrections? Effective field theory: Allow all terms consistent with symmetries, topology of field space Dimenson-d operators suppressed by M d 4 uv Corrections controlled by: Compactness of scalar, U(1) Small coupling µ/m uv 1 19

21 Direct corrections to V (ϕ) Periodicity of ϕ quantum corrections to S must be Functions of n ϕ periodic functions of ϕ δv Λ 4 n>1 c n cos(nϕ/f ϕ ) V (ϕ) f φ m pl Monodromy potential modulated by periodic effects ϕ V corr 1 2 µ2 ϕ 2 Λ 4 M 4 gut η = m 2 pl V V 1 Λ4 f 2 ϕ V m 2 pl = H 2 Example: feasible if Λ.1 M gut, f >.01 m pl 20

22 Gauge dynamics: Λ =Λ QCD from couplings ϕ f ϕ tr G G instanton corrections take above form (if dilute gas approx good) strong coupling effects (when dilute gas aprox fails) ( ) δv Λ 4 min k F ϕ + k f ϕ multibranched function of ϕ When using this effect to generate monodromy potential: mixing between branches must be weak Witten; Giusti, Petrarca, and Taglienti! "#$#& "#$#!' When this generates corrections: mixing must be strong (else trapped in a fixed branch) "#$#!& "#$#% (! Gravitational dynamics: Λ 4 f n+4 ϕ m n pl gravitational instantons, wormholes, etc. 21

23 Caveat: moduli stabilization In any string theory: couplings in V will depend on moduli ψ V = V 0 (ψ)+ 1 2 µ2 ( ψ m pl ) ϕ 2 + Λ 4 n c n ( ψ m pl ) cos ( ) nϕ f ϕ Periodic corrections change sign many times since f φ m pl Moduli must be stabilized by different effects than instantons coupling to inflaton M 2 ψ V 0 (ψ) Λ4 m 2 pl Large ϕ m pl sources potential for ψ Stability requires M 2 ψ µ2 ϕ 2 /m 2 pl µ2 /ɛ H 2 22

24 Indirect corrections to V (ϕ) Additional corrections must respect periodicity of ϕ corrections to dynamics of four-form F S class = d 4 x g ( m 2 pl R 1 48 F ( ϕ)2 + µ 24 ϕ F ) Consider δl = n d n F 2n M 4n 4 Integrate out F: F µϕ +... ( ) δv eff = V class n=1 d n+1 V n class M 4n Safe if: M 4 V class Mgut 4 ( Corrections of the form δl = n=1 d n+1 F 2n M 4n ) ( ϕ) 2 Gives same effect after redefining ϕ to be canonically normalized 23

25 Small M not always fatal Many string theory scenarios: V (ϕ) =M ϕ2 M 2 2 M 2 m pl Silverstein and Westphal; McAllister, Silverstein, and Westphal For small ϕ V 1 2 µ2 ϕ 2 ; µ = M 4 1 M 2 2 For ϕ m pl V m 3 ϕ; m 3 = M 4 1 M 2 Out of range of 4d effective field theory; requires understanding of UV completion (eg 10d SUGRA) to compute 26

26 Example: backreaction on compactification Consider string modulus ψ determines KK scale: L 0 e ψ/m pl ; V D L D ; m 2 pl = md+2 V D L ψ = 1 2 ( ψ)2 1 2 M 2 ψ ψ2 + c ψ m pl F Integrate out ψ : ψ F m pl = c 2 Mψ 2 m2 pl c V m 2 pl M 2 ψ = c H2 M 2 ψ δm 2 pl m 2 pl H2 M 2 ψ Since η = m 2 pl V V ; ɛ = m2 pl (V ) 2 V 2 We must have δm 2 pl m 2 pl H2 M 2 ψ 1 Moduli coupling to inflaton must be fairly heavy If coupling to F is: (ψ ψ 0) 2 m 2 pl F 2 corrections proportional to ψ 0 m pl Dong, Horn, Silverstein, and Westphal ψ 0 m pl 1 also edge of validity of effective field theory 25

27 Example: Coleman-Weinberg corrections Consider scalar fields ψ n (e.g. moduli, KK states, etc.) δl 1 2 ( ψ n) M 2 nψ 2 n k d n,k F 2n M 4n 2 ψ 2 n Integrate out F: F 2 V class = 1 2 µ2 ϕ 2 Effective mass for ψ : M 2 eff = M 2 ψ + M 2 k d n,k V 2 Integrate out ψ n : δv CW (ϕ) M eff (φ) 4 ln M eff M Must include all such states with M 2 n <M 2 M 4n Corrections safe if n eff M 2 ψ M 2 ; V M 4 26

28 Kaluza-Klein corrections Roughly n eff = m2 pl m 2 ; m =(m s,m pl,10 ) M gut V CW = KK V D d 4 q ln ( q 2 + Mn,eff 2 ) ( d D+4 q ln q 2 + k m D+4 V D (ψ)+m 2 V D δv (ψ)+v tree F k ( Vtree M 4 ) d k d k ) Vtree k M 4k 4 m 2 pl V k tree M 4k 4 m 2 pl Corrections safe if V class M 4 NB if KK mode couples to F as (ψ n ψ 0,n ) 2 m 2 pl F 2 tree level corrections subleading if H 2 <M 2 KK ; ψ n,0 <m pl,10 27

29 Additional stringy light states W p Shift τ n times; D-brane becomes n times as long. Consider square torus with sides of length L; D4 wrapped n times m 2 W = m4 s L2 1+n 2 ; m 2 p = 1 L(1+n 2 ) ; n = ϕ f ϕ = F µf ϕ n >> 1: strings have spectrum of asymmetric torus with sides of length L W = n m 2 s L ; L p n L and volume V eff n2 m 2 s F 2 m 2 s e4 where e 2 = µf ϕ is unit of quantization of F flux 28

30 Leading quantum correction V CW = k,l d 4 q ln ( q 2 + m 2 W,k + m 2 p,k) +... F 2 m 2 se 4 d 6 q lnq m4 s e 4 F Effect is to renormalize e 2 m 2 s M 2 gut 10 4 m 2 pl Dangerous: µ = 10 6 m pl to match observation f ϕ 10 2 m pl Must ensure renormalization of e is suppressed: f ϕ.1 m pl e 2 (.1M gut ) 2 29

31 NB model above is crude (and known not to work for other reasons) so this is a caveat and not a fatal flaw Even if µ 2 pushed above 10 6 m pl we may still get successful large field inflation of the form, e.g. V (ϕ) =M ϕ2 M 2 2 but this requires more than our 4d EFT can do at present 30

32 Stability Sergei Dubovsky, AL, Matthew Roberts; SD, AL, Raphael Flauger, in progress V (φ) n = 2 n =1 n = 1 n =0 f φ φ Success of monodromy inflation requires that transition between branches is slow compared to time scale of inflation (must complete 60 efolds before such transitions) 31

33 Bounds on membrane tension Transitions occur by bubble nucleation. Let: T = tension of bubble wall E = energy difference between branches Decay probability: Γ exp ( 27π2 2 ) T 4 E 3 (thin wall) Coleman Phenomenological bound on T ϕ = Nf ϕ ; ϕ = f ϕ E V V (ϕ)f ϕ V N Γ 1 T 1/3 ( 2 27π 2 N 3 ) 1/4 V 1/4 Let: f φ.1 m pl ; N 100; V M 4 gut T (.2V 3 ) 1/4 (.9M gut ) 3 Borderline; should check against explicit models 32

34 V. Conclusions Check stability in explicit string, field theory models General issue: monodromy inflation does not seem parametrically safe. Should we worry? Dubovsky, AL, Roberts; SD, Flauger, NK, AL, in progress Perhaps this is interesting: Implies number of e-foldings could be close to lower bound Implications for measurements of curvature, pre-inflation transients Other interesting applications of axion monodromy Kerr black holes; axion condensation via Penrose process. Instability can lead to observable axion decays Dubovsky and Gorbenko 33