String Moduli Stabilization and Large Field Inflation

Transcription

1 Kyoto, p.1/32 String Moduli Stabilization and Large Field Inflation Ralph Blumenhagen Max-Planck-Institut für Physik, München based on joint work with A.Font, M.Fuchs, D. Herschmann, E. Plauschinn, Y. Sekiguchi, R. Sun, F. Wolf

2 Introduction Kyoto, p.2/32

3 Kyoto, p.2/32 Introduction Inflationary scenario in the early universe: not only solves problem of homogeneous universe for causally disconnected regions flatness problem problem of structure formation but is experimentally testable via the fluctuations in the CMB

4 Introduction Kyoto, p.3/32

5 Kyoto, p.3/32 Introduction PLANCK 2015, BICEP2 results: upper bound: r < 0.07 spectral index: n s = ±0.004 and its running α s = 0.002± amplitude of the scalar power spectrum P = (2.142±0.049) 10 9 Single field slow role inflation V ւ Φ Φ

6 Introduction Kyoto, p.4/32

7 Introduction If r is detected large field inflation: Lyth bound implies Φ > M pl and φ r > O(1) M pl 0.01 ( M inf = (V inf ) 1 r ) Inflationary mass scales: GeV Hubble constant during inflation: H GeV. mass scale of inflation: V inf = M 4 inf = 3M2 Pl H2 inf M inf GeV mass of inflaton during inflation: M 2 Θ = 3ηH2 M Θ GeV Kyoto, p.4/32

8 UV sensitivity Kyoto, p.5/32

9 Kyoto, p.5/32 UV sensitivity Quantum gravity generates Planck suppressed operators of the form (Φ/M pl ) n V Φ Impossible to control flatness over a large region in field space. Makes it important to control Planck suppressed operators (eta-problem) Invoking a symmetry like the shift symmetry of axions helps

10 Axion inflation Kyoto, p.6/32

11 Kyoto, p.6/32 Axion inflation Axions are ubiquitous in string theory so that many scenarios have been proposed Natural inflation with a potential V(θ) = Ae S E (1 cos(θ/f)). Hard to realize in string theory, as f > 1 lies outside perturbative control. (Freese,Frieman,Olinto) Aligned inflation with two axions, f eff > 1. (Kim,Nilles,Peloso) N-flation with many axions and f eff > 1. (Dimopoulos,Kachru,McGreevy,Wacker) Comment: These models have come under pressure by the weak gravity conjecture, which for instantons was proposed to be f S E < 1. (Rudelius),(Montero,Uranga,Valenzuela),(Brown,Cottrell,Shiu,Soler)

12 Natural inflation Kyoto, p.7/32

13 Kyoto, p.7/32 Natural inflation Shift symmetry θ θ +c is broken via a non-perturbative effect V(θ) = Ae S inst (1 cos(θ/f)). Weak gravity conjecture: gravity is the weakest force, i.e. for a U(1) gauge theory m q (Arkani-Hamad,Motl,Nicolis,Vafa) strong version: the lightest particle must satisfy this weak version: this holds for some particle Claim: Any consistent theory of quantum gravity must satisfy the WGC. Via T-duality is has been argued that there should exist such a relation for any p-form gauge field.

14 Natural inflation Kyoto, p.8/32

15 Kyoto, p.8/32 Natural inflation For a 0-form m S inst q 1/f so that fs inst 1. Large field inflation requires θ > 1 f > 1 S inst < 1. However, this spoils the instanton expansion, as higher order terms cannot be neglected, i.e. large field regime θ > 1 is not controlled. More refined but similar arguments have been applied to aligned inflation.

16 Axion monodromy Kyoto, p.9/32

17 Kyoto, p.9/32 Axion monodromy A second mechanism to generate a potential for axions: axion monodromy Field theory: Axion φ and a four-form field strength F 4 = dc 3 and a Lagrangian (Kaloper, Sorbo) L = f 2 dφ dφ F 4 F 4 +2F 4 (mφ+f 0 ) Equation of motion for C 3 d F 4 = d(mφ+f 0 ) F 4 = f 0 +mφ where f 0 can be considered as background value of the flux. Scalar potential V = (f 0 +mφ) 2

18 Axion monodromy Kyoto, p.10/32

19 Kyoto, p.10/32 Axion monodromy The scalar potential and F 4 is invariant under the extended shift symmetry φ φ c/m f 0 f 0 +c The system still preserves the shift symmetry, that is broken spontaneously by a choice of branch f 0 This shift symmetry and the gauge symmetry of C 3 highly constrains higher order corrections: they must be functions of F 4, i.e. δv (F 4 ) 2n (V 0 ) n Even for δφ 1, as long as δv 1 one controls the expansion.

20 Axion monodromy Kyoto, p.11/32

21 Kyoto, p.11/32 Axion monodromy Monodromy inflation: Shift symmetry is broken by branes or fluxes unwrapping the compact axion polynomial potential for θ. (Kaloper, Sorbo), (Silverstein,Westphal) non-pert. fluxes Polynomial inflation see (T.Kobayashi,O.Seto)

22 Axion monodromy Kyoto, p.12/32

23 Kyoto, p.12/32 Axion monodromy Discrete shift symmetry acts also on the fluxes, i.e. one gets different branches tunneling à la Coleman-de Lucia n = 0 n = 0

24 Axion monodromy inflation Kyoto, p.13/32

25 Kyoto, p.13/32 Axion monodromy inflation Recent proposal: Realize axion monodromy inflation via the F-term scalar potential induced by background fluxes. (Marchesano.Shiu,Uranga),(Hebecker, Kraus, Wittkowski),(Bhg, Plauschinn) Example: in Type IIB one has the axio-dilaton modulus S = e Φ +ic 0 with Kähler potential K = log(s +S) and three-form flux induced GVW superpotential W = Ω (F 3 +ish) = f 0 +ihs. (after freezing the complex structure moduli).

26 Axion monodromy inflation Kyoto, p.14/32

27 Axion monodromy inflation The resulting axion part of the Lagrangian reads L = 1 Re(S) 2 µ c µ c+ α Re(S) (f 0 +hc) 2 The parameters in the Kaloper-Sorba model are given by flux quanta and VEVs of saxionic fields like exp( Φ) The four-forms are not visible, yet. They follow from the 10D coupling (Bielleman,Ibanez,Valenzuela) G 7 (F 3 +C 0 H 3 ) Here G 7 = G 3 with G 3 = F 3 +C 0 H 3. Thus, the Kaloper-Sorba four-form is F 4 = Σ 3 G 7 Kyoto, p.14/32

28 Axion monodromy inflation Kyoto, p.15/32

29 Axion monodromy inflation Advantages for application to axion-monodromy inflation Axions are generic in string theory: all R-R forms C p in Type IIA/B are axions As the field strengths G p+1 = dc p +H C p 2 involves the gauge potentials C p 2, kinetic terms G p+1 G p+1 induce scalar potential for C p 2 Generating the inflaton potential, supersymmetry is broken spontaneously by the very same effect by which usually moduli are stabilized Alternative scenario: Fiber inflation based on the large volume scenario (Cicoli, Conlon, Quevedo,...) Kyoto, p.15/32

30 Inflation Kyoto, p.16/32

31 Kyoto, p.16/32 Inflation For a controllable single field inflationary scenario, all moduli need to be stabilized such that M Pl > M s > M KK > M inf > M mod > H inf > M Θ Aim: Systematic study of realizing single-field fluxed F-term axion monodromy inflation, taking into account the interplay with moduli stabilization. series of papers by (Bhg,Font,Fuchs,Herschmann,Plauschinn,Sekiguchi,Sun,Wolf) Paradigm: Since the inflaton receives its potential/mass from a tree-level flux, generically all other moduli need to be stabilized at tree level, as well.

32 Framework Kyoto, p.17/32

33 Kyoto, p.17/32 Framework Framework: Type IIB orientifolds on CY threefolds with geometric and non-geometric fluxes. (Shelton,Taylor,Wecht), (Aldazabal,Camara,Ibanez,Font), (Grana, Louis, Waldram), (Benmachiche, Grimm), (Micu, Palti, Tasinato) Scalar potential splits V N=2 = V F +V D +V NS tad Fits into 4d N = 1 SUGRA, i.e. V F can be computed via a Kähler and superpotential ( K = log i ) Ω Ω log ( S +S ) 2logV, and the flux-induced schematic superpotential W = Ω (F 3 SH +T Q)

34 Relation to DFT Kyoto, p.18/32

35 Kyoto, p.18/32 Relation to DFT Compactifying the double field theory action on a fluxed CY three-fold, the result can be expressed as (Bhg, Font, Plauschinn, arxiv: ) S NSNS 1 4 e 2φ [ 1 2 χ χ Ψ Ψ ( Ω χ ) ( Ω χ ) 1 4 with χ = De ij and Ψ = DΩ, with Scalar potential: D = d H F Q R ( Ω χ ) ( Ω χ ) ]. related to gauged supergravity: V = V N=2 GSUGRA Orientifold projection: V proj = V F +V D +V NS tad

36 Moduli stabilization and inflation Kyoto, p.19/32

37 Kyoto, p.19/32 Moduli stabilization and inflation Scheme of moduli stabilization such that the following aspects are realized: There exist non-supersymmetric minima stabilizing the saxions in their perturbative regime. All mass eigenvalues are positive semi-definite, where the massless states are only axions. For both the values of the moduli in the minima and the mass of the heavy moduli one has parametric control in terms of ratios of fluxes. One has parametric control over the mass of the lightest (massive) axion, i.e. the inflaton candidate. The mass scales satisfy M Pl > M s > M KK > M inf > M mod > H inf > M Θ

38 Kyoto, p.20/32

39 Second part: Simple toy model Kyoto, p.20/32

40 A representative model Kyoto, p.21/32

41 Kyoto, p.21/32 A representative model Kähler potential is given by K = 3log(T +T) log(s +S). Fluxes generate superpotential W = i f+ihs +iqt, with f,h,q Z. Resulting scalar potential V = (hs+ f) 2 16sτ 3 6hqs 2q f 16sτ 2 5q2 48sτ + θ2 16sτ 3

42 A representative model Kyoto, p.22/32

43 Kyoto, p.22/32 A representative model Non-supersymmetric, tachyon-free minimum with τ 0 = 6 f 5q, s 0 = f h, θ 0 = 0. Mass eigenvalues M 2 mod,i = µ i hq 3 MPl 2 16 f 2 4π, with µ i 0. Gravitino-mass scale: M 3 2 p M mod

44 Mass scales Kyoto, p.23/32

45 Kyoto, p.23/32 Mass scales Cosmological constant in AdS minimum: V 0 = µ C hq 3 MPl 4 16 f 2 4π Uplift: It is possible to find (new) scaling type minima with V 0 0 by including an uplifting D3-brane (D-term) V = V F + A V 4 3 +(V D ) (Bhg, Damian, Font, Fuchs, Herschmann, Sun, arxiv: ) Relation of mass scales: M s p M KK p M mod

46 Axion inflaton Kyoto, p.24/32

47 Kyoto, p.24/32 Axion inflaton Generate a non-trivial scalar potential for the massless axion Θ by turning on additional fluxes f ax and deform This quite generically leads to W inf = λw +f ax W. M mod p M Θ = M mod p M KK Thus, there exist a tension between a light inflaton and keeping control over the effective action.

48 Toy model Kyoto, p.25/32

49 Kyoto, p.25/32 Toy model Toy model with θ as the inflaton and an uplifted scalar potential ( (hs+ f) V = λ sτ 3 6hqs 2q f 16sτ 2 5q2 48sτ )+ θ2 16sτ 3 +V up. Here λ M mod M θ Backreaction of θ on other moduli flattens the potential (Dong,Horn,Silverstein,Westphal) τ 0 (θ) = 3 20q s 0 (θ) = 1 4h ( 4 f+ 10 ( θλ ) f 2 ) 10 ( θ λ ) f 2.,

50 Toy model Kyoto, p.26/32

51 Kyoto, p.26/32 Toy model For large-field regime θ/λ f the backreacted potential simplifies V back (θ) hq 3 λ 2 f hq 3 λ 4 θ 2. 3 With τ 0 (θ) 2 10qλ θ and s 0(θ) θ is 10 4hλθ, the kinetic term for L kin λ 2 ( θ θ Axion decay constant f depends on θ! ) 2

52 Toy model Kyoto, p.27/32

53 Kyoto, p.27/32 Toy model Canonically normalizing the inflaton via θ fλ exp(θ/λ) the potential becomes Starobinsky-like V back (Θ) M2 mod f 2 M 2 θ ( 1 e 2Θ/λ). V back θ

54 Toy model Kyoto, p.28/32

55 Toy model Note: Proper field distance scales only logarithmically Θ λ log(θ) (Baume, Palti),(Kläwer, Palti) For having control over one needs V back (Θ) M2 mod f 2 M 2 θ ( 1 e 2Θ/λ). M 2 mod f 2 M 2 θ 1 For Θ Θ crit λ: 60 e-foldings from the quadratic potential Intermediate regime: linear inflation For Θ Θ crit : Starobinsky inflation Kyoto, p.28/32

56 Tensor-to-scalar ratio Kyoto, p.29/32

57 Kyoto, p.29/32 Tensor-to-scalar ratio r With decreasing λ the model changes from chaotic to Starobinsky-like inflation. λ

58 Parametric control Kyoto, p.30/32

59 Kyoto, p.30/32 Parametric control From UV-complete theory point of view, large-field inflation models require a hierarchy of the form M Pl > M s > M KK > M inf > M mod > H inf > M Θ, where neighboring scales can differ by (only) a factor of O(10). Main observation the larger λ, the more difficult it becomes to separate the high scales on the left for small λ, the smaller (Hubble-related) scales on the right become difficult to separate.

60 Conclusions Kyoto, p.31/32

61 Kyoto, p.31/32 Conclusions Large single field inflation cannot be straightforwardly be constructed in the UV complete string theory. Can the situation be improved considering other moduli like open string moduli? However, in F-theory they are on the same footing. Does there exist a clean Quantum Gravity argument why also axion monodromy inflation cannot be realized in a controlled way? Of course, the most important question though is: What is the experimental value of r?

62 Kyoto, p.32/32

63 Thank You! Kyoto, p.32/32