Can the Vacuum Energy be Dark Energy?

1. Can the Vacuum Energy be Dark Energy? Sang Pyo Kim Kunsan Nat’l Univ. Seminar at Yonsei Univ. Oct. 29,2010 (Talk at COSMO/CosPA, Sept. 30, 2010, U. Tokyo)

2. Outline • Motivation • Classical and Quantum Aspects of de Sitter Space • Polyakov’s Cosmic Laser • Effective Action for Gravity • Conclusion

3. FLRW Universe • The large scale structure of the universe is homogeneous and isotropic, described by the metric • The theory for gravity is Einstein gravity • Friedmann equations in terms of the redshift

4. Hubble Parameter & Dark Energy • Radiation • Matter • Curvature • Cosmological constant WMAP-5 year data

5. Dark Energy Models[Copeland, Sami, Tsujikawa, hep-th/0603057] • Cosmological constant w/wo quantum gravity. • Modified gravity: how to reconcile the QG scale with ? • f(R) gravities • DGP model • Scalar field models: wheredo these fields come from?(origin) • Quintessence • K-essence • Tachyon field • Phantom (ghost) field • Dilatonic dark energy • Chaplygin gas

6. Vacuum Energy and  • Vacuum energy of fundamental fields due to quantum fluctuations (uncertainty principle): • massive scalar: • Planck scale cut-off: • present value: • order of120 difference for the Planck scale cut-off and order 40 for the QCD scale cut-off • Casimir force from vacuum fluctuations is physical.

7. Vacuum Energy in an Expanding Universe • What is the effect of the expansion of the universe on the vacuum energy? • Unless it decays into light particles, it will fluctuate around the minimum forever! • The vacuum energy from the effective action in an expanding universe?

8. Vacuum Energy and  • The uncertainty principle prevents the vacuum energy from vanishing, unless some mechanism cancels it. • Cosmological constant problem • how to resolve the huge gap? • renormalization, for instance, spinor QED • SUSY, for instance, scalar and spinor QED with the same spin multiplicity (nature breaks SUSY if any)

9. Why de Sitter Space in Cosmology? • The Universe dominated by dark energy is an asymptotically de Sitter space. • CDM model is consistent with CMB data (WMAP+ACT+) • The Universe with  is a pure de Sitter space with the Hubble constant H= (/3). . • The “cosmic laser” mechanism depletes curvature and may help solving the cosmological constant problem [Polyakov, NPB834(2010); NPB797(2008)]. • de Sitter/anti de Sitter spaces are spacetimes where quantum effects, such as IR effects and vacuum structure, may be better understood.

10. Classical de Sitter Spaces • Global coordinates of (D=d+1) dimensional de Sitter embedded into (D+1) dimensional Minkowskispacetime has the O(D,1) symmetry. • The Euclidean space (Wick-rotated) has the O(D+1) symmetry (maximally spacetime symmetry).

11. BD-Vacuum in de Sitter Spaces • The quantum theory in dS spaces is still an issue of controversy and debates since Chernikov and Tagirov (1968): -The Bunch-Davies vacuum (Euclidean vacuum, in-/in-formalism) leads to the real effective action, implying no particle production in any dimensions, but exhibits a thermal state: Euclidean Green function (KMS property of thermal Green function) has the periodicity -The BD vacuum respects the dS symmetry in the same way the Minkowski vacuum respects the Lorentz symmetry.

12. BD-Vacuum in de Sitter Spaces • BUT, in cosmology, an expanding (FRW) spacetime does not have a Euclidean counterpart for general a(t). The dS spaces are an exception: Further, particle production in the expanding FRW spacetime [L. Parker, PR 183 (1969)] is a concept well accepted by GR community.

13. Polyakov’s Cosmic Laser • Cosmic Lasers: particle production a la Schwinger mechanism -The in-/out-formalism (t = ) predicts particle production only in even dimensions [Mottola, PRD 31 (1985); Bousso, PRD 65 (2002)]. -The in-/out-formalism is consistent with the composition principle [Polyakov,NPB(2008),(2008)]: the Feynman prescription for a free particle propagating on a stable manifold

14. Radiation in de Sitter Spaces • QFT in dS space: the time-component equation for a massive scalar in dS

15. Radiation in de Sitter Spaces • The Hamilton-Jacobi equation in complex time

16. Stokes Phenomenon • Four turning points • Hamilton-Jacobi action [figure adopted from Dumlu & Dunne, PRL 104 (2010)]

17. Radiation in de Sitter Spaces • One may use the phase-integral approximation and find the mean number of produced particles [SPK, JHEP09(2010)054]. • The dS analog of Schwinger mechanism in QED: the correspondence between two accelerations (Hawking-Unruh effect)

18. Radiation in de Sitter Spaces • The Stokes phenomenon explains why there is NO particle production in odd dimensional de Sitter spaces - destructive interference between two Stokes’s lines -Polyakovintepreted this as reflectionless scattering of KdV equation [NPB797(2008)]. • In even dimensional de Sitter spaces, two Stokes lines contribute constructively, thus leading to de Sitter radiation.

19. Vacuum Persistence • Consistent with the one-loop effective action from the in-/out-formalism in de Sitter spaces: -the imaginary part is absent/present in odd/even dimensions. • Does dS radiation imply the decay of vacuum energy of the Universe? -A solution for cosmological constant problem[Polyakov]. Can it work?

20. Effective Action for Gravity • Charged scalar field in curved spacetime • Effective action in the Schwinger-DeWitt proper time integral • One-loop corrections to gravity

21. One-Loop Effective Action • The in-/out-state formalism [Schwinger (51), Nikishov (70), DeWitt (75), Ambjorn et al (83)] • The Bogoliubov transformation between the in-state and the out-state:

22. One-Loop Effective Action • The effective action for boson/fermion [SPK, Lee, Yoon, PRD 78, 105013 (`08); PRD 82, 025015, 025016 (`10); ] • Sum of all one-loops with even number of external gravitons

23. Effective Action for de Sitter • de Sitter space with the metric • Bogoliubov coefficients for a massive scalar

24. Effective Action for dS [SPK, arXiv:1008.0577] • The Gamma-function Regularization and the Residue Theorem • The effective action per Hubble volume and per Compton time

25. Effective Action for de Sitter • The vacuum structure of de Sitter in the weak curvature limit (H<<m) • The general relation holds between vacuum persistence and mean number of produced pairs

26. No Quantum Hair for dS Space? [SPK, arXiv:1008.0577] • The effective action per Hubble volume and per Compton time, for instance, in D=4 • Zeta-function regularization [Hawking, CMP 55 (1977)]

27. Effective Action of Spinor [W-Y.Pauchy Hwang, SPK, in preparation] • The Bogoliubov coefficients • The effective action

28. QED vs QG

29. Conformal Anomaly, Black Holes and de Sitter Space

30. Conformal Anomaly • An anomaly in QFT is a classical symmetry which is broken at the quantum level, such as the energy momentum tensor, which is conserved due to the Bianchi identity even in curved spacetimes. • The conformal anomaly is the anomaly under the conformal transformation:

31. FLRW Universe and Conformal Anomaly • The FLRW universe with the metric has the conformal Killing vector: • The FLRW metric in the conformal time • The scale factor of the universe is just a conformal one, which leads to conformal anomaly.

32. FLRW Universe and Conformal Anomaly • At the classical level, the QCD Lagrangian is conformally invariant for m=0: • At the quantum level, the scale factor leads to the conformal anomaly [Crewther, PRL 28 (72)] • The FLRW universe leads to the QCD conformal anomaly [Schutzhold, PRL 89 (02)]

33. Conformal Anomaly • The conformal anomaly from the nonperturbative renormalized effective action is • The first term is too small to explain the dark energy at the present epoch; but it may be important in the very early stage of the universe even up to the Planckian regime. The trace anomaly may drive the inflation [Hawking, Hertog, Reall, PRD (01)].

34. Canonical QFT for Gravity • A free field has the Hamiltonian in Fourier-mode decomposition in FLRW universe • The quantum theory is the Schrodinger equation and the vacuum energy density is [SPK et al, PRD 56(97); 62(00); 64(01); 65(02); 68(03); JHEP0412(04)]

35. Canonical QFT for Gravity • Assume an adiabatic expansion of the universe, which leads to • The vacuum energy density given by is the same as by Schutzhold if but the result is from nonequilibrium quantum field theory in FLRW universe. • Equation of state:

36. Conclusion • The effective action for gravity may provide a clue for the origin of . • Does dS radiation imply the decay of vacuum energy of the Universe? And is it a solver for cosmological constant problem? [Polyakov] • dS may not have a quantum hair at one-loop level and be stable for linear perturbations. • What is the vacuum structure at higher loops and/or with interactions? (challenging question)