Inflation from supersymmetry breaking

Transcription

1 Inflation from supersymmetry breaking I. Antoniadis Albert Einstein Center, University of Bern and LPTHE, Sorbonne Université, CNRS Paris I. Antoniadis (Athens Mar 018) 1 / 0

2 In memory of Ioannis Bakas 8th Regional meeting, Nafplion, July 015 I. Antoniadis (Athens Mar 018) / 0

3 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 ElectroWeak Inflation QuantumGravity mev TeV M I M Planck I. Antoniadis (Athens Mar 018) 3 / 0

4 Problem of scales DarkEnergy ElectroWeak Inflation QuantumGravity mev TeV M I M Planck 1 they are independent possible connections 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 connect inflation and SUSY breaking scales while accommodating observed vacuum energy I. Antoniadis (Athens Mar 018) 4 / 0

5 Inflation in supergravity: main problems slow-roll conditions: the eta problem => fine-tuning of the potential η = V /V, V F = e K ( DW 3 W ), DW = W + K W K: Kähler potential, W : superpotential canonically normalised field: K = X X => η = trans-planckian initial conditions => break validity of EFT no-scale type models that avoid the η-problem K = 3 ln(t + T ) stabilisation of the (pseudo) scalar companion of the inflaton chiral multiplets => complex scalars moduli stabilisation, de Sitter vacuum,... I. Antoniadis (Athens Mar 018) 5 / 0

6 Starobinsky model of inflation L = 1 R + αr Lagrange multiplier φ => L = 1 1 (1 + φ)r 4α φ Weyl rescaling => equivalent to a scalar field with exponential potential: L = 1 R 1 ( φ) M 1 Note that the two metrics are not the same supersymmetric extension: (1 e 3 φ ) M = 3 4α add D-term R R because F-term R does not contain R => brings two chiral multiplets I. Antoniadis (Athens Mar 018) 6 / 0

7 SUSY extension of Starobinsky model K = 3 ln(t + T C C) ; W = MC(T 1 ) T contains the inflaton: Re T = e C R is unstable during inflation 3 φ => add higher order terms to stabilize it e.g. C C h(c, C) = C C ζ(c C) Kallosh-Linde 13 SUSY is broken during inflation with C the goldstino superfield model independent treatment in the decoupling sgoldstino limit replace C by a constrained superfield X satisfying X = 0 => sgoldstino = (goldstino) /F => minimal SUSY extension that evades stability problem I. Antoniadis (Athens Mar 018) 7 / 0

8 Non-linear Starobinsky supergravity I.A.-Dudas-Ferrara-Sagnotti 14 K = 3 ln(t + T X X ) ; W = M XT + f X + f /3 => L = 1 R 1 ( φ) M 1 ) (1 e 3 φ 1 e 3 φ ( a) M axion a much heavier than φ during inflation, decouples: m φ = M 3 e 3 φ 0 << m a = M 3 18 e 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) 3 φ a I. Antoniadis (Athens Mar 018) 8 / 0

9 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 V F = e K ( DW 3 W ) = e K ( 1 + K X X 3 X ) f K = X X = e X ( 1 X + O( X 4) f = O( X 4 ) => η = 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 (Athens Mar 018) 9 / 0

10 Two classes of models Case 1: R-symmetry is restored during inflation (at the maximum) Case : R-symmetry is (spontaneously) broken everywhere (and restored at infinity) example: toy model of SUSY breaking I. Antoniadis (Athens Mar 018) 10 / 0

11 Case 1: R-symmetry restored during inflation [13] K(X, X ) = κ X X + κ 4 A (X X ) A > 0 [17] W (X ) = κ 3 f X => f (X ) = 1 (+β ln X to cancel anomalies but β very small) V = V F + V D V F = κ 4 f X X(1+AX e X) [ 3X X + ( 1 + X X (1 + AX X ) ) ] 1 + 4AX X 4 q [ V D = κ 1 + X X (1 + AX X ) ] [14] Assume inflation happens around the maximum X ρ 0 => I. Antoniadis (Athens Mar 018) 11 / 0

12 Case 1: predictions slow-roll parameters η = 1 ( ) ( ) V 4A + x κ = V + x + O(ρ ) x = q/f [14] ɛ = 1 ( ) V ( ) 4A + x κ = 4 V + x ρ + O(ρ 4 ) η ρ 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 => number of e-folds N = start end V V = κ 1 1 ɛ η ln ( ρend ρ ) [19] I. Antoniadis (Athens Mar 018) 1 / 0

13 Case 1: predictions amplitude of density perturbations A s = κ4 V 4π ɛ = κ H 8π ɛ spectral index tensor to scalar ratio n s = 1 + η 6ɛ 1 + η r = 16ɛ Planck 15 data : η 0.0, A s. 10 9, N > 50 => r < 10 4, H < 10 1 GeV assuming ρ end < 1/ Question: can a nearby minimum exist with a tiny +ve vacuum energy? Answer: Yes in a weaker sense: perturbative expansion [11] valid for the Kähler potential but not for the slow-roll parameters generic V (not fine-tuned) => 10 9 < r < 10 4, < H < 10 1 GeV [0] I. Antoniadis (Athens Mar 018) 13 / 0

14 Fayet-Iliopoulos (FI) D-terms in supergravity D-term contribution: positive contribution to η => should stay small [1] its role: not important for inflation U(1) absorbs the pseudoscalar partner of inflaton allows tuning the EW vacuum energy at a tiny positive value in case Question: is it possible to have inflation by SUSY breaking via D-term? the inflaton should belong to a massive vector multiplet as before FI-term in supergravity very restrictive: constant FI term exists only by gauging the R-symmetry [11] A new FI term was written recently Cribiori-Farakos-Tournoy-Van Proeyen 18 gauge invariant at the Lagrangian level but non-local becomes local and very simple in the unitary gauge I. Antoniadis (Athens Mar 018) 14 / 0

15 A new FI term Global supersymmetry: gauge field-srength superfield L new FI = ξ 1 d 4 W W θ D W D W DW = ξ 1D + fermions It makes sense only when < D > 0 => SUSY broken by a D-term Supergravity generalisation: straightforward unitarity gauge: goldstino = U(1) gaugino = 0 => standard sugra ξ 1 D Pure sugra + one vector multiplet => L = R + ψ µ σ µνρ D ρ ψ ν + m 3/ ψ µ σ µν ψ ν 1 4 F µν ξ 1 = 0 => AdS supergravity ξ 1 0 uplifts the vacuum energy and breaks SUSY ( ) 3m3/ +1 ξ 1 e.g. ξ 1 = 6m 3/ => massive gravitino in flat space I. Antoniadis (Athens Mar 018) 15 / 0

16 New FI term with matter net result: ξ 1 ξ 1 e K/3 Not invariant under Kähler transformations K(X, X ) K + J(X ) + J( X ) W e J W U(1) cannot be an R-symmetry however R-symmetry becomes ordinary U(1) by a Kähler transformation: J = ln(w /W 0 ) => W W 0 constant and K K + ln W /W 0 The new and standard FI terms can co-exist in this basis I.A.-Chatrabhuti-Isono-Knoops 18 Case 1 model for A = 0 and W = f X b (W 0 = f, κ = 1) => [11] I. Antoniadis (Athens Mar 018) 16 / 0

17 Model of inflation on D-terms K = X X + b ln X X ; W = f (b: standard FI constant) => V F = f e ρ [ ρ (b 1) (b + ρ ) 3ρ b] V D = q (ρ + b + ξρ 4b 3 e 3 ρ) ξ = ξ 1 /q Case f = 0 (pure D-term potential): maximum at ρ = 0 => b = 3/ and ξ 1 (or b = 0 and /3 ξ 0) V D = q [b + ρ ( 1 + ξe 3 ρ)] ξ = 1: effective charge of X vanishes supersymmetric minimum at D=0 I. Antoniadis (Athens Mar 018) 17 / 0

18 Pure D-term potential Case f 0: maximum is shifted at ρ = 3f 4(1+ξ)q minimum is lifted up and SUSY is broken by both D and F of O(f ) I. Antoniadis (Athens Mar 018) 18 / 0

19 Predictions for inflation slow-roll parameters η = 4(1 + ξ) 3 + O(ρ ) ɛ = 16 9 (1 + ξ) ρ + O(ρ 4 ) η ρ N 1 ( ) η ln ρend => same main results as before (F-term dominated inflation)!! [1] However allowing higher order correction to the Kähler potential one can obtain r as large as (near the experimental bound) ρ I. Antoniadis (Athens Mar 018) 19 / 0

20 Conclusions Challenge of scales: at least three very different (besides M Planck ) electroweak, dark energy, inflation, SUSY? their origins may be connected or independent General class of models with inflation from SUSY breaking: identify inflaton with goldstino superpartner (gauged) R-symmetry restored (case 1) or broken (case ) during inflation small field, avoids the η-problem, no (pseudo) scalar companion D-term inflation is also possible using a new FI term I. Antoniadis (Athens Mar 018) 0 / 0