Scale hierarchies and string cosmology I. Antoniadis Albert - PowerPoint PPT Presentation

Scale hierarchies and string cosmology I. Antoniadis Albert Einstein Center, University of Bern and LPTHE, UPMC/CNRS, Sorbonne Universit´ es, Paris 9th Mathematical Physics Meeting Belgrade, Serbia, 18-23 September 2017 I. Antoniadis (mphy9 2017) 1 / 33

String theory: Quantum Mechanics + General Relativity Main predictions → inspirations for BSM physics Spacetime supersymmetry but arbitrary breaking scale Extra dimensions of space six or seven in M-theory Brane-world description of our Universe matter and gauge interactions may be localised in less dimensions Landscape of vacua · · · I. Antoniadis (mphy9 2017) 2 / 33

Connect string theory to the real world Is it a tool for strong coupling dynamics or a theory of fundamental forces? If theory of Nature can it describe both particle physics and cosmology? I. Antoniadis (mphy9 2017) 3 / 33

Problem of scales describe high energy (SUSY?) extension of the Standard Model unification of all fundamental interactions incorporate Dark Energy simplest case: infinitesimal (tuneable) +ve cosmological constant describe possible accelerated expanding phase of our universe models of inflation (approximate de Sitter) = > 3 very different scales besides M Planck : DarkEnergy QuantumGravity ElectroWeak Inflation ✲ meV TeV M I M Planck I. Antoniadis (mphy9 2017) 4 / 33

Relativistic dark energy 70-75% of the observable universe negative pressure: p = − ρ = > cosmological constant > ρ Λ = c 4 Λ R ab − 1 2 Rg ab + Λ g ab = 8 π G c 4 T ab = 8 π G = − p Λ Two length scales: [Λ] = L − 2 ← size of the observable Universe 0 / c 2 ≃ 1 . 4 × (10 26 m ) − 2 Λ obs ≃ 0 . 74 × 3 H 2 տ Hubble parameter ≃ 73 km s − 1 Mpc − 1 G × c 3 � ] = L − 4 ← dark energy length ≃ 85 µ m [ Λ I. Antoniadis (mphy9 2017) 5 / 33

Problem of scales DarkEnergy QuantumGravity ElectroWeak Inflation ✲ meV TeV M I M Planck they are independent 1 possible connections 2 M I could be near the EW scale, such as in Higgs inflation but large non minimal coupling to explain M Planck could be emergent from the EW scale in models of low-scale gravity and TeV strings What about M I ? can it be at the TeV scale? Can we infer M I from cosmological data? I.A.-Patil ’14 and ’15 connect inflation and SUSY breaking scales I. Antoniadis (mphy9 2017) 6 / 33

Inflation in supergravity: main problems slow-roll conditions: the eta problem = > fine-tuning of the potential V F = e K ( | DW | 2 − 3 | W | 2 ) , DW = W ′ + K ′ W η = V ′′ / V , K : K¨ ahler potential, W : superpotential canonically normalised field: K = X ¯ X = > η = 1 + . . . trans-Planckian initial conditions = > break validity of EFT no-scale type models that avoid the η -problem stabilisation of the (pseudo) scalar companion of the inflaton chiral multiplets = > complex scalars moduli stabilisation, de Sitter vacuum, . . . I. Antoniadis (mphy9 2017) 7 / 33

Starobinsky model of inflation L = 1 2 R + α R 2 > L = 1 4 α φ 2 1 Lagrange multiplier φ = 2 (1 + 2 φ ) R − Weyl rescaling = > equivalent to a scalar field with exponential potential: � 2 2( ∂φ ) 2 − M 2 L = 1 2 R − 1 � � M 2 = 3 2 1 − e − 3 φ 12 4 α Note that the two metrics are not the same supersymmetric extension: R because F-term R 2 does not contain R 2 add D-term R ¯ = > brings two chiral multiplets I. Antoniadis (mphy9 2017) 8 / 33

SUSY extension of Starobinsky model W = MC ( T − 1 K = − 3 ln( T + ¯ T − C ¯ C ) ; 2) � 2 3 φ T contains the inflaton: Re T = e C ∼ R is unstable during inflation = > add higher order terms to stabilize it e.g. C ¯ C → h ( C , ¯ C ) = C ¯ C − ζ ( C ¯ C ) 2 Kallosh-Linde ’13 SUSY is broken during inflation with C the goldstino superfield → model independent treatment in the decoupling sgoldstino limit = > minimal SUSY extension that evades stability problem [12] I. Antoniadis (mphy9 2017) 9 / 33

Non-linear supersymmetry = > goldstino mode χ Volkov-Akulov ’73 Effective field theory of SUSY breaking at low energies Analog of non-linear σ -model = > constraint superfields Rocek-Tseytlin ’78, Lindstrom-Rocek ’79, Komargodski-Seiberg ’09 Goldstino: chiral superfield X NL satisfying X 2 NL = 0 = > X NL ( y ) = χ 2 √ y µ = x µ + i θσ µ ¯ 2 θχ + θ 2 F 2 F + θ χ = F Θ 2 √ Θ = θ + 2 F � 1 �� � d 4 θ X NL ¯ d 2 θ X NL + h . c . L NL = X NL − √ = L Volkov − Akulov 2 κ 1 R-symmetry with [ θ ] R = [ χ ] R = 1 and [ X ] R = 2 F = 2 κ + . . . √ I. Antoniadis (mphy9 2017) 10 / 33

Non-linear SUSY in supergravity I.A.-Dudas-Ferrara-Sagnotti ’14 K = − 3 log(1 − X ¯ X ) ≡ 3 X ¯ X ; W = f X + W 0 X ≡ X NL V = 1 3 | f | 2 − 3 | W 0 | 2 m 2 3 / 2 = | W 0 | 2 = > ; V can have any sign contrary to global NL SUSY NL SUSY in flat space = > f = 3 m 3 / 2 M p R-symmetry is broken by W 0 Dual gravitational formulation: ( R − 6 W 0 ) 2 = 0 I.A.-Markou ’15 տ chiral curvature superfield Minimal SUSY extension of R 2 gravity I. Antoniadis (mphy9 2017) 11 / 33

Non-linear Starobinsky supergravity [9] K = − 3 ln( T + ¯ T − X ¯ X ) ; W = M XT + f X + W 0 = > � 2 2( ∂φ ) 2 − M 2 3 φ ( ∂ a ) 2 − M 2 L = 1 2 R − 1 � � − 1 � � 2 2 2 1 − e − 3 φ 2 e − 2 18 e − 2 3 φ a 2 12 axion a much heavier than φ during inflation, decouples: � 2 3 φ 0 << m a = M 3 e − m φ = M 3 inflation scale M independent from NL-SUSY breaking scale f = > compatible with low energy SUSY however inflaton different from goldstino superpartner also initial conditions require trans-planckian values for φ ( φ > 1) [18] I. Antoniadis (mphy9 2017) 12 / 33

Inflation from supersymmetry breaking I.A.-Chatrabhuti-Isono-Knoops ’16, ’17 Inflaton : goldstino superpartner in the presence of a gauged R-symmetry linear superpotential W = f X = > no η -problem | DW | 2 − 3 | W | 2 � e K � V F = | 1 + K X X | 2 − 3 | X | 2 � | f | 2 K = X ¯ e K � = X 1 − | X | 2 + O ( | X | 4 � | f | 2 = O ( | X | 4 ) e | X | 2 � = = > η = 0 + . . . inflation around a maximum of scalar potential (hill-top) = > small field no large field initial conditions gauge R-symmetry: (pseudo) scalar absorbed by the U (1) R vacuum energy at the minimum: tuning between V F and V D I. Antoniadis (mphy9 2017) 13 / 33

Two classes of models Case 1: R-symmetry is restored during inflation (at the maximum) Case 2: R-symmetry is (spontaneously) broken everywhere (and restored at infinity) example: toy model of SUSY breaking [18] [27] I. Antoniadis (mphy9 2017) 14 / 33

Case 1: R-symmetry restored during inflation K ( X , ¯ X ) = κ − 2 X ¯ X + κ − 4 A ( X ¯ X ) 2 A > 0 W ( X ) = κ − 3 f X = > f ( X ) = 1 (+ β ln X to cancel anomalies but β very small ) V = V F + V D � 2 � 1 + X ¯ X (1 + 2 AX ¯ � � X ) V F = κ − 4 f 2 e X ¯ X ( 1+ AX ¯ X ) − 3 X ¯ X + 1 + 4 AX ¯ X V D = κ − 4 q 2 � 2 1 + X ¯ X (1 + 2 AX ¯ � X ) 2 Assume inflation happens around the maximum | X | ≡ ρ ≃ 0 = > I. Antoniadis (mphy9 2017) 15 / 33

Case 1: predictions slow-roll parameters � V ′′ � − 4 A + x 2 η = 1 � � + O ( ρ 2 ) = 2 x = q / f κ 2 2 + x 2 V � 2 � 2 � V ′ � − 4 A + x 2 1 ρ 2 + O ( ρ 4 ) ≃ η 2 ρ 2 ǫ = = 4 2 κ 2 2 + x 2 V η small: for instance x ≪ 1 and A ∼ O (10 − 1 ) inflation starts with an initial condition for φ = φ ∗ near the maximum and ends when | η | = 1 � start 1 1 � ρ end � V � √ = > number of e-folds N = V ′ = κ ≃ | η ∗ | ln ρ ∗ 2 ǫ end I. Antoniadis (mphy9 2017) 16 / 33

Case 1: predictions A s = κ 4 V ∗ = κ 2 H 2 ∗ amplitude of density perturbations 24 π 2 ǫ ∗ 8 π 2 ǫ ∗ n s = 1 + 2 η ∗ − 6 ǫ ∗ ≃ 1 + 2 η ∗ spectral index tensor − to − scalar ratio r = 16 ǫ ∗ Planck ’15 data : η ≃ − 0 . 02, A s ≃ 2 . 2 × 10 − 9 , N > ∼ 50 ∼ 10 12 GeV > r < ∼ 10 − 4 , H ∗ < = Question: can a ‘nearby’ minimum exist with a tiny +ve vacuum energy? Answer: Yes in a ‘weaker’ sense: perturbative expansion [14] [19] valid for the K¨ ahler potential but not for the slow-roll parameters > 10 − 9 < ∼ 10 − 4 , 10 10 < ∼ 10 12 GeV [33] ∼ r < ∼ H ∗ < generic V (not fine-tuned) = I. Antoniadis (mphy9 2017) 17 / 33

impose independent scales: proceed in 2 steps SUSY breaking at m SUSY ∼ TeV 1 with an infinitesimal (tuneable) positive cosmological constant Villadoro-Zwirner ’05 I.A.-Knoops, I.A.-Ghilencea-Knoops ’14, I.A.-Knoops ’15 Inflation connected or independent? 2 [7] [10] [27] I. Antoniadis (mphy9 2017) 18 / 33

Toy model for SUSY breaking Content (besides N = 1 SUGRA): one vector V and one chiral multiplet S with a shift symmetry S → S − ic ω ← transformation parameter String theory: compactification modulus or universal dilaton s = 1 / g 2 + ia ← dual to antisymmetric tensor ahler potential K : function of S + ¯ K¨ S string theory: K = − p ln( S + ¯ S ) Superpotential: constant or single exponential if R-symmetry W = ae bS d 2 θ W invariant � b < 0 = > non perturbative can also be described by a generalized linear multiplet [24] I. Antoniadis (mphy9 2017) 19 / 33