High Scale Inflation with Low Scale Susy Breaking

Transcription

1 High Scale Inflation with Low Scale Susy Breaking Joseph P. Conlon (DAMTP, Cambridge) Nottingham University, September 2007 High Scale Inflation with Low Scale Susy Breaking p. 1/3

2 Two paradigms: inflation... High Scale Inflation with Low Scale Susy Breaking p. 2/3

3 Two paradigms: inflation... slow roll inflation generates density perturbations oscillation and reheating Inflationary potential V inf High Scale Inflation with Low Scale Susy Breaking p. 3/3

4 Two paradigms: inflation... Slow-roll inflation homogenises the universe and generates density perturbations. η = ( 1 M 2 P V V ) 1, ǫ = 1 2M 2 P n s = 1 + 2η 6ǫ. The inflationary energy scale is V inf ǫ 1/4 ( GeV). ( V ) 2 1. Unless ǫ is extremely small, the inflationary energy scale is high, V inf GeV V High Scale Inflation with Low Scale Susy Breaking p. 4/3

5 ...and TeV supersymmetry t H t H Λ 2 t H t H Λ 2 High Scale Inflation with Low Scale Susy Breaking p. 5/3

6 ...and TeV supersymmetry t t ln Λ H H + H 2 H t t TeV supersymmetry cancels the quadratic divergences in the Higgs potential. High Scale Inflation with Low Scale Susy Breaking p. 6/3

7 ...and TeV supersymmetry High Scale Inflation with Low Scale Susy Breaking p. 7/3

8 ...and TeV supersymmetry TeV supersymmetry stabilises the Higgs mass against radiative corrections gives a good dark matter candidate explains gauge symmetry breaking through radiative electroweak symmetry breaking is compatible with LEP I precision electroweak data is the prime candidate for new physics discovered at the LHC High Scale Inflation with Low Scale Susy Breaking p. 8/3

9 Tension The supergravity scalar potential can be written as V susy = i F i 2 3m 2 3/2 M2 P. Gravity-mediation : F i GeV,m 3/2 1TeV, Gauge mediation : F i GeV,m 3/2 1TeV. V susy has natural scale F 2 m 2 3/2 M2 P (1016 GeV) 4. Sets scale of barrier height to overshooting Sets scale of structure in the potential High Scale Inflation with Low Scale Susy Breaking p. 9/3

10 Tension The supergravity scalar potential can be written as V susy = i F i 2 3m 2 3/2 M2 P. Gravity-mediation : F i GeV,m 3/2 1TeV, Gauge mediation : F i GeV,m 3/2 1TeV. V susy has natural scale F 2 m 2 3/2 M2 P (1016 GeV) 4. Sets scale of barrier height to overshooting Sets scale of structure in the potential Most models of high-scale inflation are incompatible with TeV supersymmetry High Scale Inflation with Low Scale Susy Breaking p. 9/3

11 Proposed Solution 1. Inflation at E GeV with m 3/2 1TeV 2. Inflation ends with runaway and decompactification. 3. The global minimum has E GeV and m 3/2 1TeV 4. The global minimum lies many Planckian distances from where inflation occurred. 5. A tracker solution guides the moduli into the global minimum and avoids overshooting. High Scale Inflation with Low Scale Susy Breaking p. 10/3

12 Proposed Evolution V 12 4 P ( 10 M ) Inflation 42 4 P ( 10 M ) Low scale vacuum 7M P 26 M P Φ~ Log(Volume) High Scale Inflation with Low Scale Susy Breaking p. 11/3

13 Inflation with Runaway Inflation near an inflection point leads to runaway High Scale Inflation with Low Scale Susy Breaking p. 12/3

14 Inflation with Runaway To get an inflection point: K = 3 ln(t + T) + A (T + T) 3/2 + B (T + T) 2 + C +... (T + T) 5/2 W = W 0 A, B and C can emerge from higher α corrections, higher loop corrections... Ugly but doable. High Scale Inflation with Low Scale Susy Breaking p. 13/3

15 Inflation with Runaway Potential is V = A W 0 2 (T + T) 9/2 + B W 0 2 (T + T) 5 + C W (T + T) 11/2 Canonically normalise Φ = 6 (T+ T) : V = A e 27/2Φ + B e 10Φ/ 6 + C e 11Φ/ 6. Tuning A, B and C gives the simplest model of inflation with runaway. A, B and C can be tuned to get n s = 0.95 with the correct scale and number of efoldings. High Scale Inflation with Low Scale Susy Breaking p. 14/3

16 Inflation with Runaway This model does not work. Problems: There is no vacuum and no low-energy supersymmetry. The fields decompactify to infinity. Solution: embed into the large-volume models. High Scale Inflation with Low Scale Susy Breaking p. 15/3

17 Large Volume Models These arise in IIB flux compactifications and are characterised by exponentially large volume and broken supersymmetry. K = 2 ln (V + ξ), W = W 0 + A s e a st s V = e K ( K i j D i WD j W 3 W 2). Simplest model (P 4 [1,1,1,6,9] ): V = ( ) τ 3/2 b τs 3/2 High Scale Inflation with Low Scale Susy Breaking p. 16/3

18 Large Volume Models The supergravity potential is: V = 8 τ s a 2 s A s 2 e 2a sτ s 3V 4a s A s W 0 τ s e a sτ s V 2 } {{ } integrate out τ s + ξ W 0 2 g 3/2 s V 3. V = W 0 2 (ln V) 3/2 A minimum exists at V 3 + ξ W 0 2 V W 0 e c/g s, τ s ln V. This minimum is non-supersymmetric AdS and at exponentially large volume. g 3/2 s V 3. High Scale Inflation with Low Scale Susy Breaking p. 17/3

19 Large Volume Models High Scale Inflation with Low Scale Susy Breaking p. 18/3

20 Large Volume Models BULK BLOW UP U(2) Q L e L U(3) U(1) Q R U(1) e R High Scale Inflation with Low Scale Susy Breaking p. 19/3

21 Large Volume Models V = W 0 2 (ln V) 3/2 V 3 + ξ W 0 2 g 3/2 s V 3. Canonically normalise Φ = 3 2 ln(t + T): V = ( ǫφ 3/2 + 1)e 27 2 Φ. To solve gauge hierarchy problem, need V 10 15, with Φ 26M P. Note: single-exponential potential! High Scale Inflation with Low Scale Susy Breaking p. 20/3

22 The Model 1 ( ) V = 9 τ 3/2 b τs 3/2, 2 ( K = 2 ln V + ξ + C V 1/3 + D ) V 2/3 W = W 0 + Ae a st s, V = e K ( K i j D i WD j W 3 W 2)., Parameters: ξ = 9,C = ,D = 14,W 0 = 0.1,A = 1,a s = 2π 4. Initial conditions: τ b,init = , τ s,init = 14. High Scale Inflation with Low Scale Susy Breaking p. 21/3

23 Inflationary Potential 5 10 V Α Σ High Scale Inflation with Low Scale Susy Breaking p. 22/3

24 Moduli Evolution: Initial Behaviour Τb High Scale Inflation with Low Scale Susy Breaking p. 23/3

25 Moduli Evolution: Initial Behaviour Scale factor log a t Τb τ b Τs τ s t t High Scale Inflation with Low Scale Susy Breaking p. 24/3

26 Moduli Evolution: Initial Behaviour Moduli evolution starts with a phase of slow-roll inflation. Parameters and initial conditions can be tuned to get sixty e-folds. Inflation occurs on an inflection point and ends with runaway in the volume direction and oscillations in the τ s direction. The τ s oscillations generate a post-inflationary radiation background. High Scale Inflation with Low Scale Susy Breaking p. 25/3

27 Moduli Evolution: Initial Behaviour Moduli evolution starts with a phase of slow-roll inflation. Parameters and initial conditions can be tuned to get sixty e-folds. Inflation occurs on an inflection point and ends with runaway in the volume direction and oscillations in the τ s direction. The τ s oscillations generate a post-inflationary radiation background. Inflation Runaway + radiation High Scale Inflation with Low Scale Susy Breaking p. 25/3

28 Moduli Evolution: Runaway During runaway, m τb m τs. We integrate out τ s to get V = W 0 2 (ln V) 3/2 V 3 + ξ W 0 2 g 3/2 s V 3. Canonically normalise Φ = 3 2 ln(t + T): V = ( ǫφ 3/2 + 1)V 0 e 27 2 Φ. During most of the evolution, the effective potential is V = V 0 e 27 2 Φ! High Scale Inflation with Low Scale Susy Breaking p. 26/3

29 Moduli Evolution: Runaway Evolution of field in exponential potential and background radiation V = V 0 e 27 Φ 2 An attractor scaling solution exists (Copeland, Liddle, Wands): Ω γ = 19 27, Ω kin,φ = 16 81, Ω V (φ) = This attractor solution is valid for 7M P Φ 26M P. High Scale Inflation with Low Scale Susy Breaking p. 27/3

30 Moduli Evolution: Runaway The full 2-modulus potential V = τs a 2 s A s 2 e 2a sτ s V a s A s W 0 τ s e a sτ s V 2 + ξ W 0 2 g 3/2 s V 3. can be shown to have a tracker solution in the presence of radiation. Numerically this is found to be an attractor with Ω γ = 19 27, Ω kin,φ = 16 81, Ω V (φ) = High Scale Inflation with Low Scale Susy Breaking p. 28/3

31 Moduli Evolution: Runaway The full 2-modulus potential V = τs a 2 s A s 2 e 2a sτ s V a s A s W 0 τ s e a sτ s V 2 + ξ W 0 2 g 3/2 s V 3. can be shown to have a tracker solution in the presence of radiation. Numerically this is found to be an attractor with Ω γ = 19 27, Ω kin,φ = 16 81, Ω V (φ) = As expected - we can integrate out τ s as m τs Vm τb m τb. High Scale Inflation with Low Scale Susy Breaking p. 28/3

32 Moduli Evolution: Final Full potential is V = ( ǫφ 3/2 + 1)V 0 e 27 Φ 2 + ǫ e 6Φ }{{}. uplif t Φ 25M P : potential deviates from pure exponential. Φ 25.7M P : minimum exists Φ 26.5M P : barrier to decompactification V barrier m 3 3/2 M P m 2 τ b M 2 P. The tracker solution is radiation-dominated and never overshoots the barrier. High Scale Inflation with Low Scale Susy Breaking p. 29/3

33 Moduli Evolution: Final Φ N High Scale Inflation with Low Scale Susy Breaking p. 30/3

34 Moduli Evolution: Final To avoid overshooting must locate tracker. This requires sufficient initial radiation. We take Φ(t = 0) = 5, Φ(t = 0) = 0 and vary Ω γ,0 Note: moduli potential is a single exponential e Φ rather than double exponential e exp Φ. Avoiding overshooting is much easier. High Scale Inflation with Low Scale Susy Breaking p. 31/3

35 Moduli Evolution: Final Φ Modulus evolution with Ω γ,0 = 10 7, 10 6, 10 4, 10 2, High Scale Inflation with Low Scale Susy Breaking p. 32/3

36 Comparison with KKLT ln Γ Φ init Large-volume KKLT (from hep-th/ , Barreiro, de Carlos, Copeland, Nunes) Overshooting avoided for Ω γ,init 1! High Scale Inflation with Low Scale Susy Breaking p. 33/3

37 Overshooting Problem Across the whole parameter space only trace initial amounts of radiation (Ω γ 1) are required to avoid overshooting. The single-exponential potential is much shallower than double exponential potentials (KKLT, gaugino condensation). The large volume models solve the cosmological overshoot problem. High Scale Inflation with Low Scale Susy Breaking p. 34/3

38 Conclusions Want high-scale inflation with low-scale supersymmetry breaking. To achieve this inflation should end with runaway towards decompactifications. End of inflation generates trace amounts of radiation. Radiation gives a tracker solution. Moduli evolve in the tracker solution to a susy-breaking minimum at m 3/2 1TeV. Tracker solution does not overshoot minimum. High Scale Inflation with Low Scale Susy Breaking p. 35/3

39 Conclusion Runaway first, reheat later! High Scale Inflation with Low Scale Susy Breaking p. 36/3