8/11/2012 A Stringy Mechanism for A Small Cosmological Constant - PowerPoint PPT Presentation

8/11/2012 A Stringy Mechanism for A Small Cosmological Constant Yoske Sumitomo X. Chen, Shiu, Sumitomo, Tye , • IAS, The Hong Kong University of arxiv:1112.3338, JHEP 1204 (2012) 026 Science and Technology Sumitomo, Tye, • arXiv:1204.5177 Sumitomo, Tye, in preparation 1 •

2 8/11/2012 Contents  Motivation  Moduli stabilization ~random approach~  Moduli stabilization ~concrete models~  Statistical approach  More on product distribution  Multi-moduli analyses  Summary & Discussion

3 8/11/2012 Motivation

4 Dark Energy Late time expansion Awarded Nobel Prize in 2011! What can be a source for this?

5 Acceleration The universe is accelerating if or pressure-density ratio: Cosmological scale EOM (Friedmann eq.) for flat background Observationally DE domination

6 Two possibilities For cosmological constant • WMAP+BAO+SN suggests for a flat universe Cosmological constant For time-varying DE • Time varying DE WMAP+BAO+H0+D Δt+SN suggests e.g. Stringy Quintessence models [Kiwoon, 99], [Svrcek, 06], [Kaloper, Sorbo, 08], [Panda, YS, Trivedi, 10], [Cicoli, Pedro, Tasinato , 12]…

7 8/11/2012 Landscape Metastable vacua in moduli space dS dS Inflation • rolling down Low energy (& tunneling) dS vacua • AdS tunneling AdS vacua? • We may stay here for a while. But how likely with tiny CC?

8 8/11/2012 Stringy Landscape There are many types of vacua in string theory, as a result of a variety of (Calabi-Yau) compactification. 2 = 𝑒𝑡 4 2 + 𝑒𝑡 6 2 𝑒𝑡 10 A class of Calabi-Yau gives Swiss-cheese type of volume. 1 − 𝛿 𝑗 𝑈 𝑗 + 𝑈 𝑗 , 𝒲 6 = 𝛿 1 𝑈 1 + 𝑈 𝑗=2 E.g. workable models: [Denef, Douglas, Florea, 04] : ℎ 1,1 = 2, ℎ 2,1 = 272 4 • ℙ 1,1,1,6,9 All can be stabilized ℱ 11 : ℎ 1,1 = 3, ℎ 2,1 = 111 • (a la KKLT) , ℱ 18 : ℎ 1,1 = 5, ℎ 2,1 = 89 but in various way. • More recently, for 2 ≤ ℎ 1,1 ≤ 4 , 418 manifolds ! [Gray, He, Jejjala, Jurke, Nelson, Simon, 12] Any implication of multiple vacua?

9 8/11/2012 Keys in this talk Product distribution Assuming products of random variables: 𝑨 = 𝑧 1 𝑧 2 𝑧 3 ⋯ Many terms? Correlation through stabilization 𝑨 = 𝑧 1 𝑧 2 𝑧 3 ⋯ 𝑔(𝑧 1 , 𝑧 2 , 𝑧 3 , ⋯ ) still peaked We apply this mechanism for cosmological constant (CC)

10 8/11/2012 Before proceeding… I have to say we don’t solve cosmological constant problem completely. But here, we introduce a tool to make cosmological constant smaller, maybe up to a certain value. “ A Stringy Mechanism for A Small Cosmological C onstant”

11 8/11/2012 Moduli stabilization ~random approach~

12 8/11/2012 Gaussian suppression on stability Various vacua in string landscape Mass matrix given randomly at extrema How likely stable minima exist? min 𝜖 𝜚 𝑗 𝜖 𝜚 𝑘 𝑊 Positivity of Hessian Positivity of mass matrix Real/complex symmetric matrix Gaussian Orthogonal Emsemble • [Aazami, Easther, 05], [Dean, Majumdar, 08], [Borot, Eynard, Majumdar, Nadal, 10] 2tr 𝑁 2 , 𝑁 = 𝑁 𝑈 𝑎 = 𝑒𝑁 𝑗𝑘 𝑓 −1 Gaussian term dominates even at lower 𝑂 . ln 3 ln 2 3−3 4 ∼ 0.275 , ∼ −0.384 2

13 8/11/2012 Hierarchical setup Assuming hierarchy between diag. and off-diag. comp. • Actual models are likely to have minima at AdS. + uplifting term toward dS vacua. Hessian = 𝐵 + 𝐶 where 𝐵 : diagonal positive definite with 𝜏 𝐵 𝐶 : GOE with 𝜏 𝐶 Still Gaussianly suppressed, but a chance for dS 𝒬 = 𝑏 𝑓 −𝑐𝑂 2 −𝑑𝑂 hierarchy Larger [X. Chen, Shiu, YS, Tye, 11] When applying a model in type IIA, quite tiny chance remains. Assuming more randomness in SUGRA at SUSY AdS • 𝒬 = 𝑓 −𝑐𝑂 2 [Bachlechner, Marsh, McAllister, Wrase, 12]

14 8/11/2012 Moduli stabilization ~concrete models~

15 8/11/2012 Type IIB 5 , dilaton, localized sources Sources: 𝐼 3 , 𝐺 1 , 𝐺 3 , 𝐺 2 = 𝑓 2𝐵 𝑒𝑡 4 2 + 𝑓 −2𝐵 𝑒𝑡 6 2 Metric: 𝑒𝑡 10 Calabi-Yau Then EOM becomes [Giddings, Kachru, Polchinski, 02] e 2A 2 e 4𝐵 − 𝛽 = 6 Im 𝜐 𝑗𝐻 3 −∗ 6 𝐻 3 2 + 𝑓 −6𝐵 𝜖 𝑓 4𝐵 − 𝛽 2 + (local sources) 𝛼 LHS=0 when integrating out positive contributions 𝑓 4𝐵 = 𝛽, 𝑗𝐻 3 =∗ 6 𝐻 3 : imaginary self-dual condition 5 , 𝐻 3 = 𝐺 3 − 𝜐 𝐼 3 , 𝜐 = 𝐷 0 + 𝑗 𝑓 −𝜚 where 𝛽 is a function in 𝐺

16 8/11/2012 No-scale structure Take a scaling: 𝑕 𝑛𝑜 𝑛𝑜 → 𝜇 𝑕 𝑓 4𝐵 = 𝛽, 𝑗𝐻 3 =∗ 6 𝐻 3 : invariant The other equations are also unchanged. No-scale structure superpotential 𝑋 0 = ∫ 𝐻 3 ∧ Ω is independent of Kahler , 𝑋 4D effective potential with 𝐿 = −3 ln 𝑈 + 𝑈 0 = const 0 − 3 2 𝑊 = 𝑓 𝐿 𝑁 𝑄 𝐿 𝐽𝐾 𝐸 𝐽 𝑋 2 𝑋 2 = 0 0 𝐸 𝐾 𝑋 𝑁 𝑄 Kahler directions remain flat.

17 8/11/2012 A bonus in type IIB Hierarchical structure of mass matrix/potential helps to stabilize moduli at positive cosmological constant. [X. Chen, Shiu, YS, Tye, 12] Moduli stabilization with positive cosmological constant Fluxes Complex structure & dilaton • Non-perturbative effect, 𝛽 ′ -correction, localized branes • [KKLT, 03], [Balasubramanian, Berglund, Conlon, Quevedo, 05], Kahler [Balasubramanian , Berglund, 04]… 𝛽′ + ⋯ 𝑊 = 𝑊 Flux + 𝑊 NP + 𝑊 Complex Kahler Hierarchy No scale structure between Kahler and Complex

18 8/11/2012 KKLT Non-trivial potential for Kahler is generated by NP-corrections. E.g. Gluino condensation on D7-branes 8𝜌 2 𝑕 𝐸7 𝑂𝑄 = 𝐵 𝑓 −𝑏 = 𝐵 𝑓 −𝑏 𝑈 D7-branes wrapping the four cycle: 𝑋 Together with the superpotential from fluxes: 𝑋 = 𝑋 𝑂𝑄 0 + 𝑋 Supersymmetric vacuum 𝐸 𝑈 𝑋 = 0 existes. But exponentially small 𝑋 0 is 0 | ∼ 10 −4 |𝑋 required. 0 | ∼ 𝐵 𝑓 −𝑏 𝑈 , naturally realized? |𝑋

19 8/11/2012 Large Volume Scenario [Balasubramanian, Beglund, Conlon, Quevedo, 05] 𝛽 ′ -corrections can break no-scale structure too. 𝒫 𝛽 ′3 -correction in type II action [Becker, Becker, Haack, Louis, 02] 𝐿 = −2 ln 𝒲 + 𝜊 3 2 − ln(−𝑗 𝜐 + 𝜐 ) + ⋯ 2 −𝑗 𝜐 + 𝜐 scales differently 4 E.g. ℙ 1,1,1,6,9 model (assuming complex sector is stabilized) 1 − 𝑢 2 3 2 3 2 1 + 𝐵 2 𝑓 −𝑏 2 𝑈 0 + 𝐵 1 𝑓 −𝑏 1 𝑈 2 𝒲 = 𝑢 1 , 𝑋 = 𝑋 9 2 0 ∼ −20, 𝐵 1 ∼ 1, 𝑢 1 ∼ 10 6 , 𝑢 2 ∼ 3 Solution: 𝑋 𝑊min ∼ −10 −25 : AdS vacua 𝑂𝑄 |, 𝒲 ≫ 𝜊 : naturally realized |𝑋 0 | ≫ |𝑋

20 8/11/2012 Kahler uplifting [Balasubramanian, Berglund, 04], [Westphal, 06], [Rummel, Westphal, 11], [de Alwis, Givens, 11] Same setup as that of LVS 𝐿 = −2 ln 𝒲 + 𝜊 1 − 𝛿 𝑗 𝑈 𝑗 + 𝑈 𝑗 , 2 + ⋯ , 𝒲 = 𝛿 1 𝑈 1 + 𝑈 𝑗=2 1 + 𝐵 𝑗 𝑓 −𝑏 𝑗 𝑈 𝑗 0 + 𝐵 1 𝑓 −𝑏 1 𝑈 𝑋 = 𝑋 𝑗=2 Interested in a region where this term plays a roll. less large volume than LVS, but still |𝑋 𝑂𝑄 |, 𝒲 ≫ 𝜊 0 | ≫ |𝑋 E.g. single modulus [Rummel, Westphal, 11] 3 2 3 𝐵 1 − 𝑓 −𝑦 1 𝑊 ∼ − 𝑋 0 𝑏 1 2𝐷 𝐷 = −27 𝑋 0 𝜊 𝑏 1 , 𝑦 1 = 𝑏 1 𝑢 1 , 2 2 9 2 2 𝛿 𝑦 1 64 2𝛿 1 𝐵 1 9𝑦 1 1 When 𝑋 0 𝐵 1 < 0 , the 𝐷 ∝ 𝜊 term contributes the uplifting.

21 8/11/2012 KKLT vs Kahler uplifting KKLT • Add an uplifting potential by hand 𝐸3−𝐸3 𝑊 = 𝑊 𝑇𝑉𝐻𝑆𝐵 + 𝑊 𝐸3−𝐸3 = 2𝑈 3 𝑒 4 𝑦 −𝑕 4 𝑊 Backreaction of 𝐸3 ? A singularity exists, but finite action [DeWolfe, Kachru, Mulligan, 08], [McGuirk, Shiu, YS, 09], Safe or not? [Bena, Giecold, Grana, Halmagyi, Massai, 09- 12], [Dymarsky, 11],… Kahler uplifting • 𝑇𝑉𝐻𝑆𝐵 SUGRA + 𝛽′ -correction 𝑊 = 𝑊 Owing to |𝑋 𝑂𝑄 | 0 | ≫ |𝑋 No fine-tuning for 𝑋 0

22 8/11/2012 Statistical approach

23 8/11/2012 Further approximation 3 3 𝐵 1 − 𝑓 −𝑦 1 𝑊 4 = − 𝑋 0 𝑏 1 𝐷 C = −27𝑋 0 𝜊𝑏 1 2 𝑦 1 = 𝑏 1 𝑢 1 , , 2 2 𝛿 1 9𝑦 19 2 2 𝐵 1 𝑁 𝑄 𝑦 1 64 2𝛿 1 [Rummel, Westphal, 11] The stability constraint with positive CC at stationery points: 2 𝑊 > 0 𝑊 ≥ 0 3.65 ≤ 𝐷 < 3.89 𝜖 𝑦 Further focusing on smaller CC region: 𝐷 ∼ 3.65 9 3 𝐵 1 𝑊 4 ∼ 1 2 −𝑋 0 𝑏 1 2 𝐷 − 3.65 2 𝑁 𝑄 9 5 𝛿 1 Neglecting the parameters 𝑏 1 , 𝛿 1 , 𝜊 , the model is simplified to be 𝑑 0 ≤ 𝑑 = 𝑥 1 0 , 𝑥 2 = 𝐵 1 , 𝑑 ∝ 𝐷) < 𝑑 1 (𝑥 1 = −𝑋 Λ = 𝑥 1 𝑥 2 𝑑 − 𝑑 0 , 𝑥 2

24 8/11/2012 Stringy Random Landscape [YS, Tye, 12] Starting with the simplified potential: 𝑑 0 ≤ 𝑑 = 𝑥 1 < 𝑑 1 Λ = 𝑥 1 𝑥 2 𝑑 − 𝑑 0 , 𝑥 2 Since 𝑋 0 , 𝐵 1 are given model by model (various ways of stabilizing complex moduli), here we impose reasonable randomness on parameters. 𝑥 1 , 𝑥 2 ∈ [0, 1] , uniform distribution (for simplicity) Probability distribution function 𝑄 Λ = 𝑂 0 𝑒𝑑 𝑒𝑥 1 𝑒𝑥 2 𝜀 𝑥 1 𝑥 2 𝑑 − 𝑑 0 − Λ 𝜀 𝑥 1 − 𝑑 𝑥 2 𝑂 0 : normalization constant