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Making Precise the Nothing at the Beginning of the Universe

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lester
25 diapositives · 380 Views · Sep 09, 2014

This work explores the initial singularity at the universe's birth, leveraging string theory and higher-dimensional frameworks. We present theorems about singularities, their implications in cosmology, and the role of winding tachyon condensation as a potential solution to resolve ambiguities. The discourse extends to black hole and orbifold singularities, emphasizing localized tachyon condensation's significance. We also delve into open and closed string tachyon condensation, examining their cosmological applications, resolution of singularities, and the fate of unstable D-branes in the universe's evolution.

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  1. Making Precise the Nothing at the Beginning of the Universe Yu Nakayama, hep-th/0606127 (Collaboration with S.J. Rey, Y. Sugawara)

  2. Introduction • Universe begins from the singularity. • Theorem: under some assumptions, the universe (cosmological solution of Einstein’s GR) has an initial singularity (Penrose, Hawking) • Several ways out in string/higher dimensional theories • String cosmology (T-duality, dilaton) • Brane cosmology (cyclic universe) • Winding tachyon condensation

  3. String Theory at the singularity Can string theory coexist with singularities? • (time-like) orbifold singularity?  YES (with SUSY) • Black hole singularity?  Probably yes (dual D-brane) • (space-like) singularity?  time-dependent system. Based on exact construction (orbifold, coset), the theory is defined, but divergence in amplitudes?

  4. Tachyon censorship Localized Tachyon condensation provides a new way to resolve singularities. • Big-bang / Big-crunch singularities (MS) • Naked Singularities (ASP…) • Singularities inside the blackhole (Horowitz)

  5. Plan of the Talk • Introduction • Winding tachyon at the beginning of the universe (McGreevy-Silverstein scenario) • Time-like sine-Liouville theory and resolution of the singularity • Summary

  6. Winding tachyon at the beginning of the universe

  7. Closed string tachyon condensation • Open string tachyon condensation  Decay of unstable D-brane (Sen’s conjecture) • Checked in many ways • Open string field theory • Rolling Tachyon • Many applications • (Brane) Inflational Cosmology • Classification of D-branes (K-theory, Derived category…) • Closed string tachyon condensation  Decay of unstable space-time? • Many applications? • Resolving singularity? Cosmological applications? • Classification of space-time??

  8. Decay of D-brane Open String Tachyon Condensation Closed String Tachyon Condensation Decay of Space-(time) ?

  9. The tachyon at the end of the universe (MS) • Consider expanding universe (with S1 circle) • If we choose SS-like compactification, winding tachyon appears t  0. • Classical singularity in GR is removed by winding tachyon condensation! ~ Initial singularity of space-time would be resolved by the winding tachyon condensation.

  10. Time-like Sine-Liouville Theory • As a toy model of MS scenario, we consider time-like Sine-Liouville theory (analytic continuation of 3-sine-Liouville: Kim et al) • Obtained by • Fermionize by , so we obtain 2 fermions • MS studied the model with non-conventional Wick rotation in the semiclassical approach. ~

  11. Analytic continuation of Liouville theory Revival of old idea that Liouville direction might be time • Idea: noncritical string needs Liouville direction to compensate Weyl anomaly. • Take Q  0 or b  i so that we have critical string • For Hermiticity of the action, we need to Wick rotate • Worldsheet cosmological constant becomes real time tachyon condensation • The structure of Liouville theory has been well- understood in this ten years • Suitable analytic continuation will be useful to understand the real time tachyon condensation problem.

  12. Time-like Liouville Field Theory Wick rotate the Liouville action • Action • C=1 theory with time-dependent tachyon condensation • Minisuperspace approximation: • Euclidean continuation is given by the Liouville theory with negative cosmological const:

  13. Interpretation of 2pt function • Vertex operator • V (Euclidean mode) is expanded by later (free) mode • R is related to Bogoliubov coefficient • In the minisuperspace approximation (not a phase!) Minisuperspace 2pt function governs vacuum particle production as Bogoliubov coefficient This should also hold in string theory (conjecture: GS)

  14. Beyond minisuperspace Adopting GS conjecture, where does non-phase come from? • Exact 2pt function • Substitute • Bogoliubov coefficient • Carefully regularizing, renormalized cosmological const is negative • Then we reproduce minisuperspace result (ST) • Higher correlation functions are much subtler (ST, Schomerus…)

  15. Time-like sine-Liouville theory and resolution of the singularity

  16. 3. Sine-Liouville Theory • 3-parameter action • Vertex operator: • Conformal condition: • Symmetry: U(1) conserved current

  17. 2-parameter model (BF) • Suppose • Infinitely many symmetry appears • Due to the duality, 2-parameter sine-Liouville is much better-understood. For this value of q, model is rotation of usual sine-Liouville + free boson. So FZZ dual to coset

  18. 2pt function for neutral sector (KLPR) • Can be computed by Teschner’s trick (at least in the neutral sector) • Vertex operator: • Remarks • No dual relation. Answer is not unique. • Agreement with BF in 2-parameter limit. • Renormalized cosmological constant should be correct.

  19. Time-like Sine-Liouville Theory • As a toy model of MS scenario, we consider time-like Sine-Liouville theory (MS) • Obtained by • Fermionize by , so we obtain 2 fermions • MS studied the model with non-conventional Wick rotation in the semiclassical approach. ~

  20. 2pt function for neutral sector • We compute 2pt function (Bogoliubov coefficient) by the analytic continuation from 3-parameter sine-Liouville • Apart from the renormalized cosmological constant part, integral converges and gives a phase (as Q  0).

  21. When not a phase? • Renormalized cosmological const governs the qualitative feature of Bogoliubov coefficient • Depending on the sign, particle production is drastically different.

  22. Is the singularity resolved? • Due to the tachyon condensation, the geometry is effectively cut-off around • We can freely take a weak coupling limit near the singularity. • Bogoliubov particle production is a function of A. • If the transverse dimension is less than 4. The theory shows no diverging particle production (small back reaction). • Torus partition function also shows an imaginary part when

  23. Summary • Closed string tachyon condensation is interesting • Resolution of singularity • New geometrical interpretation • Time-like Liouville approach is promising • Exact in alpha’ corrections • Beyond the minisuperspace approximation • End (beginning) of the universe • Time-like sine-Liouville approach • Exact 2pt function • Evaluation of particle production

  24. Conclusion • String Theory is a candidate for Theory of Everything But…

  25. Conclusion • String Theory is a candidate for Theory of Everything But… • Exact treatment of α’ is very important! also provides a Theory of Nothing